% -*- coding: iso-latin-1; time-stamp-format: "%02d-%02m-%:y at %02H:%02M:%02S %Z" -*-
%<*dtx>
\def\xintdtxtimestamp  {Time-stamp: <31-08-2015 at 11:53:55 CEST>}
%</dtx>
%<*drv>
%% ---------------------------------------------------------------
\def\xintdocdate {2015/08/31}
\def\xintbndldate{2015/08/31}
\def\xintbndlversion {1.1b}
%</drv>
%<*dtx>
\iffalse
%</dtx>
%<readme>% README
%<changes>% CHANGE LOG
%<readme|changes>% xint v1.1b
%<readme|changes>% 2015/08/31 (doc 2015/08/31)
%<*readme|changes>

    Source:  xint.dtx (v1.1b 2015/08/31, doc of 2015/08/31)
    Author:  Jean-Francois Burnol
    Info:    Expandable operations on big integers, decimals, fractions
    License: LPPL 1.3c or later

%</readme|changes>
%<*!readme&!changes&!dohtmlsh&!dopdfsh&!makefile>
%% ---------------------------------------------------------------
%% The xint bundle v1.1b (2015/08/31; doc: 2015/08/31)
%% Copyright (C) 2013-2015 by Jean-Francois Burnol
%<xintkernel>%% xintkernel: Paraphernalia for the xint packages
%<xinttools>%% xinttools: Expandable and non-expandable utilities
%<xintcore>%% xintcore: Expandable arithmetic on big integers
%<xint>%% xint: Expandable operations on big integers
%<xintfrac>%% xintfrac: Expandable operations on fractions
%<xintexpr>%% xintexpr: Expandable expression parser
%<xintbinhex>%% xintbinhex: Expandable binary and hexadecimal conversions
%<xintgcd>%% xintgcd: Euclidean algorithm with xint package
%<xintseries>%% xintseries: Expandable partial sums with xint package
%<xintcfrac>%% xintcfrac: Expandable continued fractions with xint package
%% ---------------------------------------------------------------
%</!readme&!changes&!dohtmlsh&!dopdfsh&!makefile>
%<*readme>
This README is also available as README.pdf and README.html. 

Change log is in CHANGES.pdf and CHANGES.html.

Usage
=====

## With LaTeX

    \usepackage{xint}       % expandable arithmetic with big integers
    \usepackage{xintfrac}   % decimal numbers, fractions, floats
    \usepackage{xintexpr}   % expressions with infix operators

Further packages: `xintbinhex`, `xintgcd`, `xintseries` and
`xintcfrac`. All dependencies are handled automatically. For example
`xintexpr` automatically loads `xintfrac` which itself loads `xint`.
Package `xintcore` is the subset of `xint` providing only the five
operations on big integers: `\xintiiAdd`, `\xintiiMul`,\ ...
There is also `xinttools` which is a separate package providing,
among others, expandable and non-expandable loops such as `\xintFor`.

## With TeX

One does for example:

    \input xintexpr.sty\relax

Again, all dependencies are handled automatically. The packages may
be loaded in any catcode context such that letters, digits, `\` and
`%` have their standard catcodes.

`xintcore.sty` and `xinttools.sty` both import `xintkernel.sty`
which has the catcode handler and package identifier and defines a
few utilities such as `\oodef` or `\xint_dothis/\xint_orthat`.

Installation
============

## Method A: using the package manager of your TeX distribution

`xint` is included in [TeXLive](http://tug.org/texlive/) (hence also
[MacTeX](http://tug.org/mactex/)) and [MikTeX](http://www.miktex.org/).

There can be a few days of delay between apparition of a new version on
[CTAN](http://www.ctan.org/pkg/xint) and availability via the distribution
package manager.

## Method B: manual installation using `xint.tds.zip` and `unzip`

Assumes a GNU/Linux-like system (or Mac OS X).

1. obtain `xint.tds.zip` from CTAN:
  <http://mirror.ctan.org/install/macros/generic/xint.tds.zip>

2. cd to the download repertory and issue:

        unzip xint.tds.zip -d <TEXMF>

    where `<TEXMF>` is a suitable TDS-compliant destination repertory.
    For example, with TeXLive:

    - Linux, standard access rights, hence sudo is needed, installation
      into the "local" tree:

            sudo unzip xint.tds.zip -d /usr/local/texlive/texmf-local
            sudo texhash /usr/local/texlive/texmf-local

    - Mac OS X, installation into user home folder (no sudo needed,
      and it is recommended to not have a ls-R file there, hence no texhash):

            unzip xint.tds.zip -d  ~/Library/texmf

## Method C: manual installation using `Makefile` and `xint.dtx`

The Makefile automatizes rebuilding from `xint.dtx` all documentation
files as well as `xint.tds.zip`. It is for GNU/Linux-like (inc. Mac OS X)
systems, with a teTeX like installation such as TeXLive. Furthermore the
[Pandoc](http://johnmacfarlane.net/pandoc/) software is required.

1. obtain `xint.dtx` and `Makefile` from <http://www.ctan.org/pkg/xint>.

2. put them in an otherwise empty working repertory, run `make` or
   equivalently `make help` for further instructions.

## Method D: installation starting with only `xint.dtx`

Run `"tex xint.dtx"` or `"etex xint.dtx"` to extract from `xint.dtx`
all packages as well as these files:

`README.md`
  : the current README with Markdown formatting.

`CHANGES.md`
  : the changes across successive releases.

`xint.tex`
  : used to generate `xint.pdf` via `"latex xint.tex"` (thrice) then
    `"dvipdfmx xint.dvi"`. For successful compilation, packages
    `newtxtt`, `newtxmath`, `etoc`, `mathastext` are needed. Inclusion
    of the source code is off by default, but the toggle can be set in
    `xint.tex`.

    It is also possible to compile `xint.tex` with `pdflatex`.

    A third option is to generate `xint.pdf` via `pdflatex xint.dtx`.
    Source code is then included by default.

`Makefile.mk`
  : this is for UNIX-like systems. Note: this file is only produced
  with `"etex xint.dtx"`, not with `"tex xint.dtx"`. Rename it to
  `Makefile` and run `make` on the command line for further help.

`doHTMLs.sh` and `doPDFs.sh`
  : these are scripts (for UNIX-like systems) which can be used to
  convert the `README.md` and `CHANGES.md` to HTML and PDF formats.
  They require [Pandoc](http://johnmacfarlane.net/pandoc/).

`pandoctpl.latex`
  : a Pandoc template used by `doPDFs.sh`.

Finishing the installation in a TDS hierarchy:

- move the style files to `TDS:tex/generic/xint/`

- `xint.dtx` goes to `TDS:source/generic/xint/`

- the documentation (xint.pdf, README.md,...) goes to `TDS:doc/generic/xint/`

Depending on the destination, it may then be necessary to refresh a
filename database.

License
=======

<div class="mono">
Copyright (C) 2013-2015 by Jean-Francois Burnol

This Work may be distributed and/or modified under the
conditions of the LaTeX Project Public License, either
version 1.3c of this license or (at your option) any later
version. This version of this license is in

>   <http://www.latex-project.org/lppl/lppl-1-3c.txt>

and the latest version of this license is in

>   <http://www.latex-project.org/lppl.txt>

and version 1.3 or later is part of all distributions of
LaTeX version 2005/12/01 or later.

This Work has the LPPL maintenance status `author-maintained`.

The Author of this Work is Jean-Francois Burnol.

This Work consists of the source file xint.dtx and of its derived
files: xintkernel.sty, xintcore.sty, xint.sty, xintfrac.sty,
xintexpr.sty, xintbinhex.sty, xintgcd.sty, xintseries.sty,
xintcfrac.sty, xinttools.sty, xint.ins, xint.tex, README, README.md,
README.html, README.pdf, CHANGES.md, CHANGES.html, CHANGES.pdf,
pandoctpl.latex, doHTMLs.sh, doPDFs.sh, xint.dvi, xint.pdf,
Makefile.mk.</div>
%</readme>--------------------------------------------------------
%<*changes>-------------------------------------------------------
`1.1b (2015/08/31)`
----

 - bugfix: some macros needed by the integer division routine from 
   **xintcore** had been left in **xint.sty** since release `1.1`. This
   for example broke the `\xintGCD` from **xintgcd** if package **xint**
   was not loaded.

 - Slight enhancements to the documentation, particularly in the 
   `Read this first` section.


`1.1a (2014/11/07)`
----

 - fixed a bug which prevented `\xintNewExpr` from producing correctly working
   macros from a comma separated replacement text.

 - new `\xintiiSqrtR` for rounded integer square root; former `\xintiiSqrt`
   already produced truncated integer square root; corresponding function
   `sqrtr` added to `\xintiiexpr..\relax` syntax.

 - use of straight quotes in the documentation for better legibility.

 - added `\xintiiIsOne`, `\xintiiifOne`, `\xintiiifCmp`, `\xintiiifEq`,
    `\xintiiifGt`, `\xintiiifLt`, `\xintiiifOdd`, `\xintiiCmp`, `\xintiiEq`,
    `\xintiiGt`, `\xintiiLt`, `\xintiiLtorEq`, `\xintiiGtorEq`, `\xintiiNeq`,
    mainly for efficiency of `\xintiiexpr`.

 - for the same reason, added `\xintiiGCD` and `\xintiiLCM`.

 - added the previously mentioned `ii` macros, and some others from `v1.1`, to
   the user manual. But their main usage is internal to `\xintiiexpr`, to skip
   unnecessary overheads.

 - various typographical fixes throughout the documentation, and a bit
   of clean up of the code comments. Improved `\Factors` example of nested
   `subs`, `rseq`, `iter` in `\xintiiexpr`. 

`1.1 (2014/10/28)`
----

bug fixes

:   - `\xintZapFirstSpaces` hence also `\xintZapSpaces` from package **xinttools**
        were buggy when used with  an argument either empty or containing only
        space tokens. 

    - `\xintiiexpr` did not strip leading zeroes, hence
      `\xinttheiiexpr 001+1\relax` did not obtain the expected result ...

    - `\xinttheexpr \xintiexpr 1.23\relax\relax` should have produced `1`,
    but it produced `1.23`

    - the catcode of `;` was not set at package launching time.

    - the `\XINTinFloatPrd:csv` macro name had a typo, hence `prd` was
      non-functional in `\xintfloatexpr`. 

breaking changes

:   - in `\xintiiexpr`, `/` does _rounded_ division, rather than the
    Euclidean division (for positive arguments, this is truncated division).
    The new `//` operator does truncated division,

    - the `:` operator for three-way branching is gone, replaced with `??`,

    - `1e(3+5)` is now illegal. The number parser identifies `e` and `E`
    in the same way it does for the decimal mark, earlier versions treated
    `e` as `E` rather as infix operators of highest precedence,

    - the `add` and `mul` have a new syntax, old syntax is with `` `+` `` and
    `` `*` `` (left quotes mandatory), `sum` and `prd` are gone,

    - no more special treatment for encountered brace pairs `{..}` by the
    number scanner, `a/b[N]` notation can be used without use of braces (the
    `N` will end up as is in a `\numexpr`, it is not parsed by the
    `\xintexpr`-ession scanner),
    
    -  although `&` and `|` are still available as Boolean operators the
    use of `&&` and `||` is strongly recommended. The single
    letter operators might be assigned some other meaning in later releases
    (bitwise operations, perhaps). Do not use them.

    - in earlier releases, place holders for `\xintNewExpr` could either
    be denoted `#1`, `#2`, ... or also `$1`, `$2`, ... 
    Only the usual `#` form is now accepted and the special cases previously
    treated via the second form are now managed via a `protect(...)` function.

**novelties :**

  * new package **xintcore** has been split off **xint**. It contains the
      core arithmetic macros. It is loaded by package **bnumexpr**,

  * neither **xint** nor **xintfrac** load **xinttools**. Only
      **xintexpr** does,
	  
  * whenever some portion of code has been revised, often use has been made of
      the `\xint_dothis` and `\xint_orthat` pair of macros for expandably
      branching,

  * these tiny helpful macros, and a few others are in package
      **xintkernel** which contains also the catcode and loading order
      management code, initially inspired by code found in Heiko Oberdiek's
      packages,

  * the source code, which was suppressed from `xint.pdf` in release
      `1.09n`, is now compiled into a separate file `sourcexint.pdf`,

  * faster handling by `\xintAdd`, `\xintSub`, `\xintMul`, ... of the case
      where one of the arguments is zero,

  * the `\xintAdd` and `\xintSub` macros from package **xintfrac** check if
      one of the denominators is a multiple of the other, and only if this is
      not the case do they multiply the denominators. But systematic reduction
      would be too costly,

  * this naturally will be also the case for the `+` and `-` operations
      in `\xintexpr`,

  * new macros `\xintiiDivRound`, `\xintiiDivTrunc` and `\xintiiMod` for
      rounded and truncated division of big integers (now in **xintcore**),
      alongside the earlier `\xintiiQuo` and `\xintiiRem`,

  * with **xintfrac** loaded, the `\xintNum` macro does `\xintTTrunc`
      (which is truncation to an integer, same as `\xintiTrunc {0}`),

  * new macro `\xintMod` in **xintfrac** for modulo operation with
      fractional numbers,

  * `\xintiexpr`, `\xinttheiexpr` admit an optional argument within brackets
      `[d]`, they round the computation result (or results, if comma separated)
      to `d` digits after decimal mark, (the whole computation is done exactly,
      as in `xintexpr`),

  * `\xintfloatexpr`, `\xintthefloatexpr` similarly admit an optional
      argument which serves to keep only `d` digits of precision, getting rid
      of cumulated uncertainties in the last digits (the whole computation is
      done according to the precision set via `\xintDigits`),

  * `\xinttheexpr` and `\xintthefloatexpr` _pretty-print_ if possible, the
      former removing unit denominator or `[0]` brackets, the latter avoiding
      scientific notation if decimal notation is practical,

  * the `//` does truncated division and `/:` is the associated modulo,

  * multi-character operators `&&`, `||`, `==`, `<=`, `>=`, `!=`,
      `**`,

  * multi-letter infix binary words `'and'`, `'or'`, `'xor'`, `'mod'`
      (straight quotes mandatory),

  * functions `even`, `odd`, 

  * `\xintdefvar A3:=3.1415;` for variable definitions (non expandable,
    naturally), usable in subsequent expressions; variable names may contain
    letters, digits, underscores. They should not start with a digit, the `@`
    is reserved, and single lowercase and uppercase Latin letters are
    predefined to work as dummy variables (see next),

  * generation of comma separated lists `a..b`, `a..[d]..b`,

  * Python syntax-like list extractors `[list][n:]`, `[list][:n]`,
    `[list][a:b]` allowing negative indices, but no optional step argument,
    and `[list][n]` (`n=0` for the number of items in the list),

  * functions `first`, `last`, `reversed`,

  * itemwise operations on comma separated lists `a*[list]`, etc.., possible
    on both sides `a*[list]^b`, and obeying the same precedence rules as with
    numbers,

  * `add` and `mul` must use a dummy variable: `add(x(x+1)(x-1), x=-10..10)`,

  * variable substitutions with `subs`:
    `subs(subs(add(x^2+y^2,x=1..y),y=t),t=20)`,

  * sequence generation using `seq` with a dummy variable: `seq(x^3,
    x=-10..10)`,

  * simple recursive lists with `rseq`, with `@` given the last value,
     `rseq(1;2@+1,i=1..10)`,

  * higher recursion with `rrseq`, `@1`, `@2`, `@3`, `@4`, and `@@(n)`
     for earlier values, up to `n=K` where `K` is the number of terms of the
     initial stretch `rrseq(0,1;@1+@2,i=2..100)`,

  * iteration with `iter` which is like `rrseq` but outputs only the
     last `K` terms, where `K` was the number of initial terms,

  * inside `seq`, `rseq`, `rrseq`, `iter`, possibility to use `omit`,
     `abort` and `break` to control termination,

  * `n++` potentially infinite index generation for `seq`, `rseq`,
     `rrseq`, and `iter`, it is advised to use `abort` or `break(..)` at
     some point,

  * the `add`, `mul`, `seq`, ... are nestable,

  * `\xintthecoords` converts a comma separated list of an even number
     of items to the format expected by the `TikZ` `coordinates` syntax,

  * completely new version `\xintNewExpr`, `protect` function to handle
     external macros. The dollar sign
     `$` for place holders is not accepted anymore, only the standard macro
     parameter `#`. Not all constructs are compatible with `\xintNewExpr`.

`1.09n (2014/04/01)`
----

  * the user manual does not include by default the source code
    anymore: the `\NoSourceCode` toggle in file `xint.tex` has to
    be set to 0 before compilation to get source code inclusion
    (later release `1.1` made source code available as `sourcexint.pdf`).

  * bug fix (**xinttools**) in `\XINT_nthelt_finish` (this bug was
    introduced in `1.09i` of `2013/12/18` and showed up when the index
    `N` was larger than the number of elements of the list).

`1.09m (2014/02/26)`
----

  * new in **xinttools**: `\xintKeep` keeps the first `N` or last
    `N` elements of a list (sequence of braced items); `\xintTrim`
    cuts out either the first `N` or the last `N` elements from a
    list.

  * new in **xintcfrac**: `\xintFGtoC` finds the initial partial
    quotients common to two numbers or fractions `f` and `g`;
    `\xintGGCFrac` is a clone of `\xintGCFrac` which however does not
    assume that the coefficients of the generalized continued
    fraction are numeric quantities. Some other minor changes.

`1.09kb (2014/02/13)`
----

  * bug fix (**xintexpr**): an aloof modification done by `1.09i` to
    `\xintNewExpr` had resulted in a spurious trailing space present
    in the outputs of all macros created by `\xintNewExpr`, making
    nesting of such macros impossible.

  * bug fix (**xinttools**): `\xintBreakFor` and `\xintBreakForAndDo`
    were buggy when used in the last iteration of an `\xintFor` loop.

  * bug fix (**xinttools**): `\xintSeq` from `1.09k` needed a `\chardef`
    which was missing from `xinttools.sty`, it was in `xint.sty`.

`1.09k (2014/01/21)`
----

  * inside `\xintexpr..\relax` (and its variants) tacit multiplication is
    implied when a number or operand is followed directly with an
    opening parenthesis,

  * the `"` for denoting (arbitrarily big) hexadecimal numbers is
    recognized by `\xintexpr` and its variants (package
    **xintbinhex** is required); a fractional hexadecimal part
    introduced by a dot `.` is allowed.

  * re-organization of the first sections of the user manual.

  * bug fix (**xinttools**, **xint**, ...): forgotten catcode check of
    `"` at loading time has been added.

`1.09j (2014/01/09)`
----

  * (**xint**) the core division routines have been re-written for some
    (limited) efficiency gain, more pronounced for small divisors. As a
    result the *computation of one thousand digits of $\pi$* is close
    to three times faster than with earlier releases.

  * some various other small improvements, particularly in the power
    routines.

  * (**xintfrac**) a new macro `\xintXTrunc` is designed to produce
    thousands or even tens of thousands of digits of the decimal
    expansion of a fraction. Although completely expandable it has its
    use limited to inside an `\edef`, `\write`, `\message`, \dots. It
    can thus not be nested as argument to another package macro.

  * (**xintexpr**) the tacit multiplication done in `\xintexpr..\relax`
    on encountering a count register or variable, or a `\numexpr`,
    while scanning a (decimal) number, is extended to the case of a sub
    `\xintexpr`-ession.

  * `\xintexpr` can now be used in an `\edef` with no `\xintthe` prefix;
    it will execute completely the computation, and the error message
    about a missing `\xintthe` will be inhibited. Previously, in the
    absence of `\xintthe`, expansion could only be a full one (with
    ``\romannumeral-`0``), not a complete one (with `\edef`). Note
    that this differs from the behavior of the non-expandable
    `\numexpr`: `\the` or `\number` are needed not only to print but
    also to trigger the computation, whereas `\xintthe` is mandatory
    only for the printing step.

  * the default behavior of `\xintAssign` is changed, it now does not
    do any further expansion beyond the initial full-expansion which
    provided the list of items to be assigned to macros.

  * bug fix (**xintfrac**): `1.09i` did an unexplainable change to
    `\XINT_infloat_zero` which broke the floating point routines for
    vanishing operands =:(((

  * bug fix: the `1.09i` `xint.ins` file produced a buggy `xint.tex` file.

`1.09i (2013/12/18)`
----

  * (**xintexpr**) `\xintiiexpr` is a variant of `\xintexpr` which is
    optimized to deal only with (long) integers, `/` does a euclidean
    quotient.

  * `\xintnumexpr`, `\xintthenumexpr`, `\xintNewNumExpr` are renamed,
    respectively, `\xintiexpr`, `\xinttheiexpr`, `\xintNewIExpr`. The
    earlier denominations are kept but to be removed at some point.

  * it is now possible within `\xintexpr...\relax` and its variants to
    use count, dimen, and skip registers or variables without
    explicit `\the/\number`: the parser inserts automatically
    `\number` and a tacit multiplication is implied when a register
    or variable immediately follows a number or fraction. Regarding
    dimensions and `\number`, see the further discussion in
    *Dimensions*.

  * (**xintfrac**) new conditional `\xintifOne`; `\xintifTrueFalse`
    renamed to `\xintifTrueAelseB`; new macros `\xintTFrac`
    (`fractional part`, mapped to function `frac` in
    `\xintexpr`-essions), `\xintFloatE`.

  * (**xinttools**) `\xintAssign` admits an optional argument to
    specify the expansion type to be used: `[]` (none, default), `[o]`
    (once), `[oo]` (twice), `[f]` (full), `[e]` (`\edef`),... to define
    the macros

  * **xinttools** defines `\odef`, `\oodef`, `\fdef` (if the names have
    already been assigned, it uses `\xintoodef` etc...). These tools
    are provided for the case one uses the package macros in a
    non-expandable context, particularly `\oodef` which expands twice
    the macro replacement text and is thus a faster alternative to
    `\edef` taking into account that the **xint** bundle macros expand
    already completely in only two steps. This can be significant when
    repeatedly making `\def`-initions expanding to hundreds of digits.

  * some across the board slight efficiency improvement as a result of
    modifications of various types to *fork macros* and *branching
    conditionals* which are used internally.

  * bug fix (**xint**): `\xintAND` and `\xintOR` inserted a space token
    in some cases and did not expand as promised in two steps `:-((`
    (bug dating back to `1.09a` I think; this bug was without
    consequences when using `&` and `|` in `\xintexpr-essions`, it
    affected only the macro form).

  * bug fix (**xintcfrac**): `\xintFtoCCv` still ended fractions with
    the `[0]`'s which were supposed to have been removed since release
    `1.09b`.

`1.09h (2013/11/28)`
----

  * parts of the documentation have been re-written or re-organized,
    particularly the discussion of expansion issues and of input and
    output formats.

  * the expansion types of macro arguments are documented in the margin
    of the macro descriptions, with conventions mainly taken over
    from those in the `LaTeX3` documentation.

  * a dependency of **xinttools** on **xint** (inside `\xintSeq`) has
    been removed.

  * (**xintgcd**) `\xintTypesetEuclideAlgorithm` and
    `\xintTypesetBezoutAlgorithm` have been slightly modified
    (regarding indentation).

  * (**xint**) macros `xintiSum` and `xintiPrd` are renamed to
    `\xintiiSum` and `\xintiiPrd`.

  * (**xinttools**) a count register used in `1.09g` in the `\xintFor`
    loops for parsing purposes has been removed and replaced by use of
    a `\numexpr`.

  * the few uses of `\loop` have been replaced by `\xintloop/\xintiloop`.

  * all macros of **xinttools** for which it makes sense are now declared
    `\long`.

`1.09g (2013/11/22)`
----

  * a package **xinttools** is detached from **xint**, to make tools such
    as `\xintFor`, `\xintApplyUnbraced`, and `\xintiloop` available
    without the **xint** overhead.

  * new expandable nestable loops `\xintloop` and `\xintiloop`.

  * bugfix: `\xintFor` and `\xintFor*` do not modify anymore the value of
    `\count 255`.

`1.09f (2013/11/04)`
----

  * (**xint**) new `\xintZapFirstSpaces`, `\xintZapLastSpaces`,
    `\xintZapSpaces`, `\xintZapSpacesB`, for expandably stripping away
    leading and/or ending spaces.

  * `\xintCSVtoList` by default uses `\xintZapSpacesB` to strip away
    spaces around commas (or at the start and end of the comma
    separated list).

  * also the `\xintFor` loop will strip out all spaces around commas and
    at the start and the end of its list argument; and similarly for
    `\xintForpair`, `\xintForthree`, `\xintForfour`.

  * `\xintFor` *et al.* accept all macro parameters from `#1` to
    `#9`.

  * for reasons of inner coherence some macros previously with one extra
    `i` in their names (e.g. `\xintiMON`) now have a doubled
    `ii` (`\xintiiMON`) to indicate that they skip the overhead of
    parsing their inputs via `\xintNum`. Macros with a *single*
    `i` such as `\xintiAdd` are those which maintain the
    non-**xintfrac** output format for big integers, but do parse
    their inputs via `\xintNum` (since release `1.09a`). They too may
    have doubled-`i` variants for matters of programming optimization
    when working only with (big) integers and not fractions or
    decimal numbers.

`1.09e (2013/10/29)`
----

  * (**xint**) new `\xintintegers`, `\xintdimensions`, `\xintrationals`
    for infinite `\xintFor` loops, interrupted with `\xintBreakFor` and
    `\xintBreakForAndDo`.

  * new `\xintifForFirst`, `\xintifForLast` for the `\xintFor` and
    `\xintFor*` loops,

  * the `\xintFor` and `xintFor*` loops are now `\long`, the
    replacement text and the items may contain explicit `\par`'s.

  * new conditionals `\xintifCmp`, `\xintifInt`, `\xintifOdd`.

  * bug fix (**xint**): the `\xintFor` loop (not `\xintFor*`) did
    not correctly detect an empty list.

  * bug fix (**xint**): `\xintiSqrt {0}` crashed. `:-((`

  * the documentation has been enriched with various additional examples,
    such as the *the quick sort algorithm
    illustrated* or the various ways of *computing prime numbers*.

  * the documentation explains with more details various expansion
    related issues, particularly in relation to conditionals.

`1.09d (2013/10/22)`
----

  * bug fix (**xint**): `\xintFor*` is modified to gracefully
    handle a space token (or more than one) located at the very end of
    its list argument (as the space before `\do` in `\xintFor* #1 in
    {{a}{b}{c}<space>} \do {stuff}`; spaces at other locations were
    already harmless). Furthermore this new version _f-expands_ the
    un-braced list items. After `\def\x{{1}{2}}` and `\def\y{{a}\x
    {b}{c}\x }`, `\y` will appear to `\xintFor*` exactly as if it had
    been defined as `\def\y{{a}{1}{2}{b}{c}{1}{2}}`.

  * same bug fix for `\xintApplyInline`.

`1.09c (2013/10/09)`
----

  * (**xintexpr**) added `bool` and `togl` to the `\xintexpr` syntax;
    also added `\xintboolexpr` and `\xintifboolexpr`.

  * added `\xintNewNumExpr` (now `\xintNewIExpr` and `\xintNewBoolExpr`),

  * the factorial `!` and branching `?`, `:`, operators (in
    `\xintexpr...\relax`) have now less precedence than a function
    name located just before,

  * (**xint**) `\xintFor` is a new type of loop, whose replacement text
    inserts the comma separated values or list items via macro
    parameters, rather than encapsulated in macros; the loops are
    nestable up to four levels (nine levels since `1.09f`) and their
    replacement texts are allowed to close groups as happens with the
    tabulation in alignments,

  * `\xintForpair`, `\xintForthree`, `\xintForfour` are experimental
    variants of `\xintFor`,

  * `\xintApplyInline` has been enhanced in order to be usable for
    generating rows (partially or completely) in an alignment,

  * new command `\xintSeq` to generate (expandably) arithmetic sequences
    of (short) integers,

  * again various improvements and changes in the documentation.

`1.09b (2013/10/03)`
----

  * various improvements in the documentation,

  * more economical catcode management and re-loading handling,

  * removal of all those `[0]`'s previously forcefully added at the end
    of fractions by various macros of **xintcfrac**,

  * `\xintNthElt` with a negative index returns from the tail of the
    list,

  * new macro `\xintPRaw` to have something like what `\xintFrac` does in
    math mode; i.e. a `\xintRaw` which does not print the denominator
    if it is one.

`1.09a (2013/09/24)`
----

  * (**xintexpr**) `\xintexpr..\relax` and `\xintfloatexpr..\relax`
    admit functions in their syntax, with comma separated values as
    arguments, among them `reduce, sqr, sqrt, abs, sgn, floor, ceil,
    quo, rem, round, trunc, float, gcd, lcm, max, min, sum, prd, add,
    mul, not, all, any, xor`.

  * comparison (`<`, `>`, `=`) and logical (`|`, `&`) operators.

  * the command `\xintthe` which converts `\xintexpr`essions into
    printable format (like `\the` with `\numexpr`) is more efficient,
    for example one can do `\xintthe\x` if `\x` was defined to be an
    `\xintexpr..\relax`:

        \def\x{\xintexpr 3^57\relax}
        \def\y{\xintexpr \x^(-2)\relax}
        \def\z{\xintexpr \y-3^-114\relax}
        \xintthe\z

  * `\xintnumexpr .. \relax` (now renamed `\xintiexpr`) is `\xintexpr
    round( .. ) \relax`.

  * `\xintNewExpr` now works with the standard macro parameter character
    `#`.

  * both regular `\xintexpr`-essions and commands defined by
    `\xintNewExpr` will work with comma separated lists of
    expressions,

  * new commands `\xintFloor`, `\xintCeil`, `\xintMaxof`, `\xintMinof`
    (package **xintfrac**), `\xintGCDof`, `\xintLCM`, `\xintLCMof`
    (package **xintgcd**), `\xintifLt`, `\xintifGt`, `\xintifSgn`,
    `\xintANDof`, ...

  * The arithmetic macros from package **xint** now filter their operands
    via `\xintNum` which means that they may use directly count
    registers and `\numexpr`-essions without having to prefix them by
    `\the`. This is thus similar to the situation holding previously
    already when **xintfrac** was loaded.

  * a bug (**xintfrac**) introduced in `1.08b` made `\xintCmp` crash
    when one of its arguments was zero. `:-((`

`1.08b (2013/06/14)`
----

  * (**xintexpr**) Correction of a problem with spaces inside
    `\xintexpr`-essions.

  * (**xintfrac**) Additional improvements to the handling of floating
    point numbers.

  * new section *Use of count registers* documenting how count
    registers may be directly used in arguments to the macros of
    **xintfrac**.

`1.08a (2013/06/11)`
----

  * (**xintfrac**) Improved efficiency of the basic conversion from
    exact fractions to floating point numbers, with ensuing speed gains
    especially for the power function macros `\xintFloatPow` and
    `\xintFloatPower`,

  * Better management by `\xintCmp`, `\xintMax`, `\xintMin` and
    `\xintGeq` of inputs having big powers of ten in them.

  * Macros for floating point numbers added to the **xintseries**
    package.

`1.08 (2013/06/07)`
----

  * (**xint** and **xintfrac**) Macros for extraction of square roots,
    for floating point numbers (`\xintFloatSqrt`), and integers
    (`\xintiSqrt`).

  * New package **xintbinhex** providing *conversion routines* to and from
    binary and hexadecimal bases.

`1.07 (2013/05/25)`
----

  * The **xintexpr** package is a new core constituent (which loads
    automatically **xintfrac** and **xint**) and implements the
    expandable expanding parser

        \xintexpr . . . \relax,

    and its variant

        \xintfloatexpr . . . \relax

    allowing on input formulas using the infix operators `+`, `-`, `*`,
    `/`, and `^`, and arbitrary levels of parenthesizing. Within a
    float expression the operations are executed according to the
    current value of `\xintDigits`. Within an `\xintexpr`-ession the
    binary operators are computed exactly.

    To write the `\xintexpr` parser I benefited from the commented
    source of the `l3fp` parser; the `\xintexpr` parser has its own
    features and peculiarities. *See its documentation*.

  * The floating point precision `D` is set (this is a local assignment
    to a `\mathchar` variable) with `\xintDigits := D;` and queried
    with `\xinttheDigits`. It may be set to anything up to
    `32767`.[^1] The macro incarnations of the binary operations
    admit an optional argument which will replace pointwise `D`; this
    argument may exceed the `32767` bound.

  * The **xintfrac** macros now accept numbers written in scientific
    notation, the `\xintFloat` command serves to output its argument
    with a given number `D` of significant figures. The value of `D`
    is either given as optional argument to `\xintFloat` or set with
    `\xintDigits := D;`. The default value is `16`.

[^1]: but values higher than 100 or 200 will presumably give too slow
evaluations.

`1.06b (2013/05/14)`
----

  * Minor code and documentation improvements. Everywhere in the source
   code, a more modern underscore has replaced the @ sign.

`1.06 (2013/05/07)`
----

  * Some code improvements, particularly for macros of **xint** doing loops.

  * New utilities in **xint** for expandable manipulations of lists:

        \xintNthElt, \xintCSVtoList, \xintRevWithBraces

  * The macros did only a double expansion of their arguments. They now
    fully expand them (using ``\romannumeral-`0``). Furthermore, in the
    case of arguments constrained to obey the TeX bounds they will be
    inserted inside a `\numexpr..\relax`, hence completely expanded, one
    may use count registers, even infix arithmetic operations, etc...

`1.05 (2013/05/01)`
----

Minor changes and additions to **xintfrac** and **xintcfrac**.

`1.04 (2013/04/25)`
----

  * New component **xintcfrac** devoted to continued fractions.

  * bug fix (**xintfrac**): `\xintIrr {0}` crashed.

  * faster division routine in **xint**, new macros to deal expandably with
    token lists.

  * `\xintRound` added.

  * **xintseries** has a new implementation of `\xintPowerSeries` based
on a Horner scheme, and new macro `\xintRationalSeries`. Both to help
deal with the *denominator buildup* plague.

  * `tex xint.dtx` extracts style files (no need for a `xint.ins`).

`1.03 (2013/04/14)`
----

  * new modules **xintfrac** (expandable operations on fractions) and
    **xintseries** (expandable partial sums with xint package).

  * slightly improved division and faster multiplication (the best
ordering of the arguments is chosen automatically).

  * added illustration of Machin algorithm to the documentation.

`1.0 (2013/03/28)`
----

Initial announcement:

>   The **xint** package implements with expandable TeX macros the basic
    arithmetic operations of addition, subtraction, multiplication
    and division, as applied to arbitrarily long numbers represented
    as chains of digits with an optional minus sign.

>  The **xintgcd** package provides implementations of the Euclidean
    algorithm and of its typesetting.

>  The packages may be used with Plain and with LaTeX.

%</changes>-------------------------------------------------------
%<*dtx>
\fi
\catcode`+ 0 \catcode`\\ 12 +iffalse
%</dtx>
%<*makefile>------------------------------------------------------
# This file: Makefile.mk (generated from xint.dtx)
# Tested on Mac OS X Mavericks with GNU Make 3.81,
# TeXLive 2014 and Pandoc 1.13.1.
# Either download the master Makefile from
#    http://www.ctan.org/pkg/xint
# or rename the present one as "Makefile".
# Then run "make" or equivalently "make help" to
# get instructions. Compilation of the complete
# documentation requires Pandoc.

# Note to myself: I wanted to use .RECIPEPREFIX = > but it is supported
# only with GNU Make 3.82 and later.

# this crazyness is to circumvent a problem with docstrip generation
# of the Makefile; we do not want two empty lines becoming only one
nullstring :=
define newline
$(nullstring)

endef
# will speed-up a little, I think.
newline := $(newline)

define helptext
==== INSTRUCTIONS

The Makefile is to automatize the extraction and compilation from
xint.dtx of package files and documentation files, and for producing
xint.tds.zip. It is for GNU/Linux like systems, with a teTeX like
installation such as TeXLive. Tested on Mac OS X Mavericks with TL2014.

For compiling the PDF files, packages newtx, newtxtt, etoc,... are used
and should be up-to-date (as of 2014/10). Conversion to plain, html and
pdf format of README.md and CHANGES.md (make PanPDF, make PanHTML)
require Pandoc software. (tested with Pandoc 1.13.1).

It is recommended to work with xint.dtx and Makefile in an otherwise
initially empty temporary repertory.

make help
    prints this help (using more). It will also have already extracted
    all files from xint.dtx.

make helpless
    prints this help (using less).

make xint.pdf
    extracts files and produces only xint.pdf, via latex+dvipdfmx.
    No Pandoc needed.

make sourcexint.pdf
    extracts files and produces only sourcexint.pdf, via latex+dvipdfmx.
    No Pandoc needed.

make PanPDF
    produces README.pdf and CHANGES.pdf, requires Pandoc.

make PanHTML
    produces README.html and CHANGES.html, requires Pandoc.

make doc
    produces all documentation.

make all
    produces all documentation, and creates xint.tds.zip.

make xint.tds.zip
    same as "make all"

make cleanall
    removes all generated files

==== INSTALLING

The following has been tested on a TeXLive installation:

make installhome
    creates xint.tds.zip, and unzips it in <TEXMFHOME>
        (it assumes there is no ls-R file there)

make installlocal
    creates xint.tds.zip, and unzips it in <TEXMFLOCAL>
        (and then does texhash <TEXMFLOCAL>)
    IT MIGHT BE NEEDED TO RUN IT AS "sudo make installlocal"
    This depends on how the access rights are configured.
    In case of doubt run first "make doc" and then "make
    installlocal". If the latter fails, "sudo make installlocal".

make uninstallhome
    removes all xint files and repertories from <TEXMFHOME>

make uninstalllocal
    removes all xint files and repertories from <TEXMFLOCAL>
        (and then does texhash <TEXMFLOCAL>)
    IT MIGHT BE NEEDED TO RUN IT AS "sudo make uninstalllocal"

endef

.PHONY: help helpless all extract doc PDF HTML clean cleanall\
        installhome uninstallhome installlocal uninstalllocal

# for printf with subst and \n, got it from
#     http://stackoverflow.com/a/5887751

# I could do the trick with := here, for \n substitution, but this would add
# tiny overhead to all other operations of make

help:
	@printf '$(subst $(newline),\n,$(helptext))' | more

helpless:
	@printf '$(subst $(newline),\n,$(helptext))' | less

# RM         = rm -f
JF_tmpdir  := $(shell mktemp -d jfbu_XXX)
TEXMF_local = $(shell kpsewhich -var-value TEXMFLOCAL)
TEXMF_home  = $(shell kpsewhich -var-value TEXMFHOME)
packages = xintkernel.sty xintcore.sty xint.sty xintfrac.sty xintexpr.sty\
             xintgcd.sty xintbinhex.sty xintseries.sty xintcfrac.sty\
             xinttools.sty
# Makefile.mk is not included in $(extracted). Its extraction rule is in
# master Makefile file. We can not extract Makefile from xint.dtx via
# docstrip, as .tex is always appended if a filename with no extension is
# specified. If "make -f Makefile.mk" is run, Makefile.mk will not be
# overwritten because tex xint.dtx does not extract it (etex xint.dtx does).
extracted  = $(packages) xint.tex xint.ins README.md CHANGES.md\
             doHTMLs.sh doPDFs.sh pandoctpl.latex
doc_pdf  = README.pdf CHANGES.pdf
doc_html = README.html CHANGES.html
filesfortex    = $(packages)
filesforsource = xint.dtx xint.ins Makefile
filesfordoc    = xint.pdf sourcexint.pdf README $(doc_pdf) $(doc_html)
auxiliaryfiles = xint.tex xint.dvi xint.aux xint.toc xint.log\
     sourcexint.dvi sourcexint.aux sourcexint.toc sourcexint.log\
     README.tex README.dvi README.aux README.toc README.out README.log\
     CHANGES.tex CHANGES.dvi CHANGES.aux CHANGES.toc CHANGES.out CHANGES.log
xint_cmd       = latex -interaction=nonstopmode xint
sourcexint_cmd = latex -interaction=nonstopmode -jobname=sourcexint\
                "\chardef\dosourcexint=1 \input{xint}"

all: $(extracted) doc xint.tds.zip
	@echo 'make all done.'

extract: $(extracted)

$(extracted): xint.dtx
	tex xint.dtx

doc: xint.pdf sourcexint.pdf README PanPDF PanHTML
	@echo 'make doc done.'

xint.pdf: xint.dtx xint.tex
	rm -f xint.aux xint.toc
	$(xint_cmd)
	$(xint_cmd)
	$(xint_cmd)
	dvipdfmx xint.dvi
	@echo 'Warnings of dvipdfmx may presumably be safely ignored,'
	@echo 'but if the pdf has (font) problems do "pdflatex xint.dtx"'

sourcexint.pdf: xint.dtx xint.tex
	rm -f sourcexint.aux sourcexint.toc
	$(sourcexint_cmd)
	$(sourcexint_cmd)
	$(sourcexint_cmd)
	dvipdfmx sourcexint.dvi
	@echo 'Warnings of dvipdfmx may presumably be safely ignored,'
	@echo 'but if the pdf has (font) problems do "pdflatex xint.dtx"'

README: README.md
	pandoc -t plain -o README README.md

PanPDF: $(doc_pdf)

$(doc_pdf): doPDFs.sh
	chmod u+x doPDFs.sh && ./doPDFs.sh

PanHTML: $(doc_html)

$(doc_html): doHTMLs.sh
	chmod u+x doHTMLs.sh && ./doHTMLs.sh

xint.tds.zip: $(filesfordoc) $(filesforsource) $(filesfortex)
	rm -fr $(JF_tmpdir)
	mkdir -p $(JF_tmpdir)/doc/generic/xint
	mkdir -p $(JF_tmpdir)/source/generic/xint
	mkdir -p $(JF_tmpdir)/tex/generic/xint
	chmod -R ugo+rwx $(JF_tmpdir)
	cp -a $(filesfordoc)    $(JF_tmpdir)/doc/generic/xint
	cp -a $(filesforsource) $(JF_tmpdir)/source/generic/xint
	cp -a $(filesfortex)    $(JF_tmpdir)/tex/generic/xint
	cd $(JF_tmpdir); chmod -R ugo+r doc source tex
	umask 0022 && cd $(JF_tmpdir) &&\
                zip -r xint.tds.zip doc source tex &&\
                mv -f xint.tds.zip ../
	rm -fr $(JF_tmpdir)
	@echo 'make xint.tds.zip done.'

xint.zip: $(filesfordoc) $(filesforsource) $(filesfortex) xint.tds.zip
	mkdir -p              $(JF_tmpdir)/xint
	chmod ugo+rwx         $(JF_tmpdir)/xint
	cp -a $(filesfordoc)    $(JF_tmpdir)/xint
	cp -a $(filesforsource) $(JF_tmpdir)/xint
	chmod -R ugo+r        $(JF_tmpdir)/xint
	mv xint.tds.zip       $(JF_tmpdir)/
	umask 0022 && cd $(JF_tmpdir) && zip -r xint.zip xint.tds.zip xint
	mv     $(JF_tmpdir)/xint.tds.zip ./
	mv -f  $(JF_tmpdir)/xint.zip ./
	rm -fr $(JF_tmpdir)
	@echo 'make xint.zip done.'

installhome: xint.tds.zip
	unzip xint.tds.zip -d $(TEXMF_home)

uninstallhome:
	cd $(TEXMF_home) && rm -fr  doc/generic/xint \
	                            source/generic/xint \
	                            tex/generic/xint

# cf http://stackoverflow.com/a/1909390
# as kpsewhich is very slow (.5s) I want to evaluate once only.
installlocal: xint.tds.zip
	$(eval $@_tmp := $(TEXMF_local))
	unzip xint.tds.zip -d $($@_tmp) && texhash $($@_tmp)

uninstalllocal:
	cd $(TEXMF_local) && rm -fr doc/generic/xint \
	                            source/generic/xint \
	                            tex/generic/xint && texhash .
clean:
	rm -f $(auxiliaryfiles)

cleanall: clean
	rm -f $(extracted) $(doc_pdf) $(doc_html)\
          README xint.pdf sourcexint.pdf xint.tds.zip
%</makefile>$------------------------------------------------------
%<*pandoctpl>-----------------------------------------------------
$if(dvipdfmx)$
{\csname @for\endcsname\x:=hyperref,graphicx,color,xcolor\do
    {\PassOptionsToPackage{dvipdfmx}\x}}
   \PassOptionsToPackage{dvipdfmx-outline-open}{hyperref}
   \PassOptionsToPackage{dvipdfm}{geometry}
$endif$
\documentclass[$papersize$,fontsize=$fontsize$]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}

\usepackage{newtxtext}
\usepackage{newtxtt}
\usepackage{newtxmath}

\usepackage{upquote}

% pour les \texttt venant de la conversion par pandoc des `...`:
\begingroup\makeatletter
  \catcode`\'\active
  \catcode`\*\active
  \catcode`\`\active
\@firstofone {\endgroup
  \def\dostraightquotesandstar{% textcomp package is loaded by newtxtext
     \let`\textasciigrave 
     \let'\textquotesingle
     \edef*{\noexpand\raisebox{-.25\noexpand\height}{\string*}}%
     \catcode39\active % '
     \catcode96\active % `
     \catcode42\active }% *
}% for \texttt, let's just forget about math and italic correction things
\DeclareRobustCommand\texttt {\bgroup 
   \dostraightquotesandstar\afterassignment\ttfamily\let\next=}

$if(geometry)$
\usepackage[$for(geometry)$$geometry$$sep$,$endfor$]{geometry}
$endif$
$if(tables)$
\usepackage{longtable,booktabs}
$endif$
\usepackage[unicode=true,bookmarks]{hyperref}
\hypersetup{breaklinks=true,%
            pdfauthor={Jean-Fran\c cois Burnol},%
            pdftitle={$title$ $author$ $date$},%
            colorlinks=true,%
            citecolor=$if(citecolor)$$citecolor$$else$blue$endif$,%
            urlcolor=$if(urlcolor)$$urlcolor$$else$blue$endif$,%
            linkcolor=$if(linkcolor)$$linkcolor$$else$magenta$endif$,%
            pdfborder={0 0 0},%
            pdfstartview=FitH,%
            pdfpagemode=UseOutlines}
%%\urlstyle{same}  % don't use monospace font for urls

\setlength{\parindent}{0pt}
\setlength{\emergencystretch}{3em}  % prevent overfull lines
\usepackage{enumitem}
%% reduce LaTeX's insane vertical spacing around verbatim blocks
\setlength{\parskip}{\medskipamount}
\setlist[trivlist]{topsep=0pt,partopsep=0pt,itemsep=0pt,parsep=0pt}

$if(numbersections)$
\setcounter{secnumdepth}{5}
$else$
\setcounter{secnumdepth}{0}
$endif$

$if(etoc)$\usepackage{etoc}$endif$

\title{$title$}
\author{$author$}
\date{$date$}

$for(header-includes)$
$header-includes$
$endfor$

\begin{document}
$if(title)$
\maketitle
$endif$

$for(include-before)$
$include-before$

$endfor$

$if(toc)$
\setcounter{tocdepth}{$toc-depth$}
$if(etoc)$
\etocdefaultlines
\etocmulticolstyle[$etoc$]{}
$endif$
\tableofcontents
$endif$

$body$

$for(include-after)$
$include-after$

$endfor$
\end{document}
%</pandoctpl>-----------------------------------------------------
%<*dohtmlsh>------------------------------------------------------
#! /bin/sh
# produces README.html and CHANGES.html from README.md and CHANGES.md
# tested with pandoc 1.13.1

pandoc -o README.html -s --toc -V highlighting-css='    body{margin-left : 10%; margin-right : 15%; margin-top: 4ex; font-size: 14pt;}
    pre   {white-space: pre-wrap; }
    code  {white-space: pre-wrap; } 
    .mono {font-family: monospace;}' README.md

pandoc -o CHANGES.html -s --toc -V highlighting-css='    body{margin-left : 10%; margin-right : 15%; margin-top: 4ex; font-size: 14pt;}
    pre  {white-space: pre-wrap;}
    code {white-space: pre-wrap;}
    #TOC {float: right; position: relative; top: 100px; margin-bottom: 100px;}' CHANGES.md

%</dohtmlsh>------------------------------------------------------
%<*dopdfsh>-------------------------------------------------------
#! /bin/sh
# produces README.pdf and CHANGES.pdf from README.md and CHANGES.md
# via latex+dvipdfmx and custom pandoc latex template

pandoc -o README.tex --template=pandoctpl --toc -V papersize=a4paper -V fontsize=11pt -V dvipdfmx --variable=geometry:footskip=1cm,left=2.5cm,right=2.5cm,top=2cm,bottom=3cm -V etoc=1 README.md
rm -f README.aux README.toc README.out
latex -interaction=nonstopmode README
latex -interaction=nonstopmode README
latex -interaction=nonstopmode README
dvipdfmx README.dvi

pandoc -o CHANGES.tex --template=pandoctpl --toc -V papersize=a4paper -V fontsize=11pt -V dvipdfmx --variable=geometry:footskip=1cm,left=2.5cm,right=2.5cm,top=2cm,bottom=3cm -V etoc=2 CHANGES.md
rm -f CHANGES.aux CHANGES.toc CHANGES.out
latex -interaction=nonstopmode CHANGES
latex -interaction=nonstopmode CHANGES
latex -interaction=nonstopmode CHANGES
dvipdfmx CHANGES.dvi
%</dopdfsh>-------------------------------------------------------
%<*drv>-----------------------------------------------------------
%%
%% Run latex thrice on this file xint.tex then run dvipdfmx on 
%% xint.dvi to produce the documentation xint.pdf.
%%
%% Or if preferred, run pdflatex (setting of \Withdvipdfmx will be
%% automatically corrected).
%% It does not extract Makefile.mk. For this, run rather etex xint.dtx.
%%
\NeedsTeXFormat{LaTeX2e}
\ProvidesFile{xint.tex}%
[\xintbndldate\space v\xintbndlversion\space driver file for xint documentation (jfB)]%
\PassOptionsToClass{a4paper,fontsize=10pt}{scrdoc}
\chardef\Withdvipdfmx 1 % 0 for pdflatex or latex+dvips
\chardef\NoSourceCode 1 % 0 for source code inclusion
\input xint.dtx
%%% Local Variables:
%%% mode: latex
%%% End:
%</drv>----------------------------------------------------------------------
%<*ins>-------------------------------------------------------------------------
%%
%% tex xint.ins extracts all package files from xint.dtx, as well as
%% xint.tex, README.md, CHANGES.md, doPDFs.sh, doHTMLs.sh. 
%%
%% tex xint.ins does not extract Makefile.mk, but etex xint.ins does
%%
\input docstrip.tex
\askforoverwritefalse
\generate{\nopreamble\nopostamble
\file{README.md}{\from{xint.dtx}{readme}}
\file{CHANGES.md}{\from{xint.dtx}{changes}}
\file{doHTMLs.sh}{\from{xint.dtx}{dohtmlsh}}
\file{doPDFs.sh}{\from{xint.dtx}{dopdfsh}}
\ifx\numexpr\undefined\else\catcode9 11
            \file{Makefile.mk}{\from{xint.dtx}{makefile}}\fi
\usepreamble\defaultpreamble
\usepostamble\defaultpostamble
\file{pandoctpl.latex}{\from{xint.dtx}{pandoctpl}}
\file{xint.tex}{\from{xint.dtx}{drv}}
\file{xintkernel.sty}{\from{xint.dtx}{xintkernel}}
\file{xinttools.sty}{\from{xint.dtx}{xinttools}}
\file{xintcore.sty}{\from{xint.dtx}{xintcore}}
\file{xint.sty}{\from{xint.dtx}{xint}}
\file{xintbinhex.sty}{\from{xint.dtx}{xintbinhex}}
\file{xintgcd.sty}{\from{xint.dtx}{xintgcd}}
\file{xintfrac.sty}{\from{xint.dtx}{xintfrac}}
\file{xintseries.sty}{\from{xint.dtx}{xintseries}}
\file{xintcfrac.sty}{\from{xint.dtx}{xintcfrac}}
\file{xintexpr.sty}{\from{xint.dtx}{xintexpr}}}
\catcode32=13\relax% active space
\let =\space%
\Msg{************************************************************************}
\Msg{*}
\Msg{* To finish the installation you have to move the following}
\Msg{* files into a directory searched by TeX:}
\Msg{*}
\Msg{*     xintkernel.sty}
\Msg{*     xintcore.sty}
\Msg{*     xint.sty}
\Msg{*     xintbinhex.sty}
\Msg{*     xintgcd.sty}
\Msg{*     xintfrac.sty}
\Msg{*     xintseries.sty}
\Msg{*     xintcfrac.sty}
\Msg{*     xintexpr.sty}
\Msg{*     xinttools.sty}
\Msg{*}
\Msg{* To produce the documentation run latex thrice on xint.tex}
\Msg{* then dvipdfmx on xint.dvi. Edit xint.tex to get the code}
\Msg{* source included.}
\Msg{* dvipdfmx warnings may be ignored, but if the produced pdf}
\Msg{* has font problems, run rather pdflatex on xint.tex}
\Msg{*}
\Msg{* Happy TeXing!}
\Msg{*}
\Msg{************************************************************************}
\endbatchfile
%</ins>-------------------------------------------------------------------------
%<*dtx>
+fi % end of \iffalse block
+catcode`\ 0 \catcode`\+ 12
\chardef\noetex 0
\ifx\numexpr\undefined\chardef\noetex 1 \fi
\ifnum\noetex=1 \chardef\extractfiles 0 % extract files, then stop
\else
  \ifx\ProvidesFile\undefined
      \chardef\extractfiles 0 % no LaTeX2e. etex, xetex, ... on xint.dtx
  \else % latex/pdflatex on xint.tex or on xint.dtx
      \ifx\Withdvipdfmx\undefined
        % latex run is on xint.dtx, we will extract all files
        \chardef\extractfiles 1 % 1 = extract and typeset, 2 = only typeset
        \chardef\Withdvipdfmx 0 % 0 = pdflatex or latex+dvips, 1 = dvipdfmx
        \chardef\NoSourceCode 0 % 0 = include source code, 1 = do not
        \NeedsTeXFormat{LaTeX2e}%
        \PassOptionsToClass{a4paper,fontsize=10pt}{scrdoc}%
      \else % latex run is on xint.tex,
        \chardef\extractfiles 2 % no extractions, but typeset
        %  \Withdvipdfmx and \NoSourceCode are set-up there
      \fi
    \ProvidesFile{xint.dtx}[bundle source (\xintbndlversion, \xintbndldate) %
                             and documentation (\xintdocdate)]%
  \fi
\fi
\ifnum\extractfiles<2 % extract files
\def\MessageDeFin{\newlinechar10 \let\Msg\message
\Msg{^^J}%
\Msg{********************************************************************^^J}%
\Msg{*^^J}%
\Msg{* To finish the installation you have to move the following^^J}%
\Msg{* files into a directory searched by TeX:^^J}%
\Msg{*^^J}%
\Msg{*\space\space\space\space xintkernel.sty^^J}%
\Msg{*\space\space\space\space xintcore.sty^^J}%
\Msg{*\space\space\space\space xint.sty^^J}%
\Msg{*\space\space\space\space xintbinhex.sty^^J}%
\Msg{*\space\space\space\space xintgcd.sty^^J}%
\Msg{*\space\space\space\space xintfrac.sty^^J}%
\Msg{*\space\space\space\space xintseries.sty^^J}%
\Msg{*\space\space\space\space xintcfrac.sty^^J}%
\Msg{*\space\space\space\space xintexpr.sty^^J}%
\Msg{*\space\space\space\space xinttools.sty^^J}%
\Msg{*^^J}%
\Msg{* To produce the documentation run latex thrice on xint.tex^^J}%
\Msg{* then dvipdfmx on xint.dvi. Edit xint.tex to get the code^^J}%
\Msg{* source included.^^J}%
\Msg{* dvipdfmx warnings may be ignored, but if the produced pdf^^J}%
\Msg{* has font problems, run rather pdflatex on xint.tex^^J}%
\Msg{*^^J}%
\Msg{* Happy TeXing!^^J}%
\Msg{*^^J}%
\Msg{********************************************************************^^J}%
}%
\begingroup
    \input docstrip.tex
    \askforoverwritefalse
    \catcode9 11 % do not kill TAB in producing Makefile.mk
    \generate{\nopreamble\nopostamble
    \file{README.md}{\from{xint.dtx}{readme}}
    \file{CHANGES.md}{\from{xint.dtx}{changes}}
    % pure tex will use ^^I notation for TAB character, don't want that.
    \ifnum\noetex=0 \file{Makefile.mk}{\from{xint.dtx}{makefile}}\fi
    \file{doHTMLs.sh}{\from{xint.dtx}{dohtmlsh}}
    \file{doPDFs.sh}{\from{xint.dtx}{dopdfsh}}
    \usepreamble\defaultpreamble
    \usepostamble\defaultpostamble
    \file{pandoctpl.latex}{\from{xint.dtx}{pandoctpl}}
    \file{xint.ins}{\from{xint.dtx}{ins}}
    \file{xint.tex}{\from{xint.dtx}{drv}}
    \file{xintkernel.sty}{\from{xint.dtx}{xintkernel}}
    \file{xinttools.sty}{\from{xint.dtx}{xinttools}}
    \file{xintcore.sty}{\from{xint.dtx}{xintcore}}
    \file{xint.sty}{\from{xint.dtx}{xint}}
    \file{xintbinhex.sty}{\from{xint.dtx}{xintbinhex}}
    \file{xintgcd.sty}{\from{xint.dtx}{xintgcd}}
    \file{xintfrac.sty}{\from{xint.dtx}{xintfrac}}
    \file{xintseries.sty}{\from{xint.dtx}{xintseries}}
    \file{xintcfrac.sty}{\from{xint.dtx}{xintcfrac}}
    \file{xintexpr.sty}{\from{xint.dtx}{xintexpr}}}
\endgroup
\fi % end of file extraction (from etex/latex/pdflatex run on xint.dtx)
\ifnum\extractfiles=0 % no LaTeX, files now extracted. Stop.
  \MessageDeFin\expandafter\end
\fi
% From this point on, run is necessarily with e-TeX.
% Check if \MessageDeFin got defined, if yes put it at end of run.
\ifdefined\MessageDeFin\AtEndDocument{\MessageDeFin}\fi
%-------------------------------------------------------------------------------
\documentclass {scrdoc}
\ifdefined\dosourcexint % this toggle is set by corresponding make rule
    \chardef\NoSourceCode 0
\else
    \chardef\dosourcexint 0
\fi
\ifnum\NoSourceCode=1 \OnlyDescription\fi
\usepackage{ifpdf}% also loaded later by other packages (such as xcolor)
\ifpdf\chardef\Withdvipdfmx 0 \fi
\makeatletter
\ifnum\Withdvipdfmx=1
   \@for\@tempa:=hyperref,bookmark,graphicx,xcolor,pict2e\do
            {\PassOptionsToPackage{dvipdfmx}\@tempa}
   %
   \PassOptionsToPackage{dvipdfm}{geometry}
   \PassOptionsToPackage{bookmarks=true}{hyperref}
   \PassOptionsToPackage{dvipdfmx-outline-open}{hyperref}
   \PassOptionsToPackage{dvipdfmx-outline-open}{bookmark}
   %
   \def\pgfsysdriver{pgfsys-dvipdfm.def}
\else
   \PassOptionsToPackage{bookmarks=true}{hyperref}
\fi
\makeatother

\pagestyle{headings}
\makeatletter
\def\buggysectionmark #1{% KOMA 3.12 as released to CTAN December 2013
    \if@twoside\expandafter\markboth\else\expandafter\markright\fi
    {\MakeMarkcase{\ifnumbered{section}{\sectionmarkformat\fi}{}#1}}{}}
\ifx\buggysectionmark\sectionmark
\def\sectionmark #1{%
    \if@twoside\expandafter\markboth\else\expandafter\markright\fi
    {\MakeMarkcase{\ifnumbered{section}{\sectionmarkformat}{}#1}}{}}
\fi
\makeatother

\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}

%\usepackage{array}
\usepackage{multicol}

\usepackage{geometry}
% 11 octobre 2014
\AtBeginDocument {\ttzfamily
                  \newgeometry{textwidth=\dimexpr92\fontcharwd\font`X\relax,
                               vscale=0.75}}

\unless\ifnum\dosourcexint=1
\usepackage{xintexpr}
\usepackage{xintbinhex}
\usepackage{xintgcd}
\usepackage{xintseries}
\usepackage{xintcfrac}
\usepackage{amsmath} % for use of \cfrac in the documentation
\usepackage{pifont}  % pour la hollow star \ding{73}
\fi

\usepackage{xinttools}

\usepackage{enumitem}% à partir d'octobre 2014

\usepackage{varioref}
%\vrefwarning

\usepackage{etoolbox}
% Est-ce que je l'utilise vraiment? oui, \ifnumequal dans un \IsPrime

\usepackage{tocloft}% pour la TOC de la section locale du code pour
% xintexpr, il vaut mieux un look standard, mais je dois customiser la
% "numwidth", trop petite.


\catcode`| \active
\usepackage{etoc}[2013/10/16] % I need \etocdepthtag.toc
\catcode`| 12

%---- USE OF ETOC FOR THE TABLES OF CONTENTS

\def\gobbletodot #1.{}

\newif\ifindescription % 1 avril 2014
\ifnum\dosourcexint=0
    \indescriptiontrue
\fi
\def\sectioncouleur{{cyan}}

\def\MARGEPAGENO {1.5em}% changera pour la partie implémentation

% 1er avril 2014, je fais un vrai style, un peu grossier cependant,
% alors qu'avant j'utilisais savedsectionline, par paresse.

% 12 octobre 2014, emploi \llap, \leftmargini et aussi de \MARGEPAGENO ici aussi
\etocsetstyle{section}{}
     {\normalfont}
     {\kern\bigskipamount
         \rightskip    \MARGEPAGENO\relax
         \parfillskip  -\MARGEPAGENO\relax
      \bfseries
         \leftskip \leftmargini
      \noindent\llap            %  \llap
      {\makebox[\leftmargini][l]%  et \leftmargini le 12/10/2014
              {\expandafter\textcolor\sectioncouleur {\etocnumber}}}%
      \strut\etocname
      \mdseries\nobreak\leaders\etoctoclineleaders\hfill\nobreak\strut
                             \makebox[\MARGEPAGENO][r]{\etocpage}\par
% pour les sous-sections (1 avril 2014)
      \let\ETOCsectionnumber\etocthenumber
      }%
     {}%

% Octobre 2014: emploi de \leftmargini et ajout de \parskip\z@skip.
\makeatletter
\etocsetstyle{subsection}
    {\begingroup\normalfont
     \setlength{\premulticols}{0pt}%
     \setlength{\multicolsep}{0pt}%
     \setlength{\columnsep}{1em}%
     \setlength{\columnseprule}{.4pt}%
     % Octobre 2014 mes problèmes d'alors étaient liés à la glue dans \parskip
     \parskip\z@skip
     % j'avais seulement ceci avant, je laisse les deux
     \raggedcolumns
     \addvspace{\smallskipamount}%
     \begin{multicols}{2}
     \leftskip  \leftmargini % 12 octobre 2014
     \rightskip \MARGEPAGENO plus 2em minus 1em % 18 octobre 2013
     \parfillskip -\MARGEPAGENO\relax
    }
    {}
    {\noindent
     \llap{\makebox[\leftmargini][l]{\ttzfamily\bfseries\etoclink
             {\ifindescription\expandafter\textcolor\sectioncouleur
               {\normalfont\bfseries\ETOCsectionnumber}\fi
              .\expandafter\gobbletodot\etocthenumber}}}%
     \strut\etocname\nobreak
     \unless\ifindescription\leaders\etoctoclineleaders\fi
     \hfill\nobreak
     \strut\makebox[\MARGEPAGENO][r]{\small\etocpage}\endgraf }
    {\end{multicols}\endgroup\addvspace{\smallskipamount}}%

\makeatother

\addtocontents{toc}{\protect\hypersetup{hidelinks}}

\usepackage[zerostyle=a,straightquotes,scaled=0.95]{newtxtt}
% j'ai essayé zerostyle=e, finalement je reviens à =a
\usepackage{newtxmath}

\makeatletter

% I need also the font with a slashed zero, for verbatim code.

% Mardi 18 novembre 2014 à 09:06:44
% Test de newtxtt 1.05, 'q' pour uprightquotes
% (maintenant: straightquotes)

\DeclareFontFamily{T1}{newtxttb}{\hyphenchar\font\m@ne}

\DeclareFontShape{T1}{newtxttb}{m}{n}{
      <-> s*[\newtxtt@scale]newtxttbq
}{}
\DeclareFontShape{T1}{newtxttb}{b}{n}{
      <-> s*[\newtxtt@scale]newtxbttbq
}{}
\DeclareFontShape{T1}{newtxttb}{bx}{n}{
      <-> ssub * newtxttb/b/n
}{}
\DeclareFontShape{T1}{newtxttb}{m}{sl}{
     <-> s*[\newtxtt@scale]newtxttslbq
}{}
\DeclareFontShape{T1}{newtxttb}{m}{it}{
     <-> ssub * newtxttb/m/sl
}{}

\makeatother

% this is with a slashed 0 like the original txtt.
\newcommand\ttbfamily {\fontfamily{newtxttb}\selectfont }

% I will leave this old markup here for the time being, in case there
% is later some use to it. 
% 11 octobre, j'essaie couleur, YellowOrange, CadetBlue
% \def\digitstt {\bgroup \color[named]{OrangeRed}\let\next=}

% Mardi 18 novembre 2014 à 09:07:30
% test des old style figures par \textsc
% ATTENTION à cause emploi d'argument pouvant contenir des tokens comme \if 
% (cf lignes environ 5319)
% Finalement pour release doc du 7 mars 2015, je n'utilise pas old style
%\def\digitstt #1{\begingroup\color[named]{OrangeRed}%
%    \unless\ifmmode\scshape\fi #1\endgroup}
\def\digitstt #1{\begingroup\color[named]{OrangeRed}#1\endgroup}

\let\dtt\digitstt

\ifnum\dosourcexint=1
\else
% Septembre 2014 emploi de mathastext
\renewcommand\familydefault\ttdefault
\usepackage[noendash]{mathastext}% pas de endahs dans newtxtt
\fi
\renewcommand\familydefault\sfdefault % <-- sans-serif in footnotes, TOC,
                                % headers etc...

\usepackage{xspace}
\usepackage[dvipsnames]{xcolor}
\usepackage{framed}

% 14 octobre 2014
% copié de snugshade de framed.sty
\makeatletter
\newenvironment{snugframed}{%
% transféré ici 17 octobre 
  \fboxsep \dimexpr2\fontcharwd\font`X\relax
  \advance\linewidth-2\fboxsep
  \advance\csname @totalleftmargin\endcsname \fboxsep
  \def\FrameCommand##1{\hskip\@totalleftmargin
                       \hskip-\fboxsep
                       \fbox{##1}\hskip-\fboxsep
      % There is no \@totalrightmargin, so:
      \hskip-\linewidth \hskip-\@totalleftmargin \hskip\columnwidth}%
    \MakeFramed {\advance\hsize-\width \@totalleftmargin\z@ \linewidth\hsize
    \@setminipage}%
 }{\par\unskip\@minipagefalse\endMakeFramed}
\makeatother

\definecolor{joli}{RGB}{225,95,0}
\definecolor{JOLI}{RGB}{225,95,0}
\definecolor{BLUE}{RGB}{0,0,255}
\definecolor{niceone}{RGB}{38,128,192}

\usepackage[para]{footmisc}

\usepackage[english]{babel}
\usepackage[autolanguage,np]{numprint}
\AtBeginDocument{\npthousandsep{,\hskip .5pt plus .1pt minus .1pt}}

\usepackage[pdfencoding=pdfdoc]{hyperref}

\hypersetup{%
linktoc=all,%
breaklinks=true,%
colorlinks=true,%
urlcolor=niceone,%
linkcolor=blue,%
pdfauthor={Jean-Fran\c cois Burnol},%
pdftitle={The xint bundle},%
pdfsubject={Arithmetic with TeX},%
pdfkeywords={Expansion, arithmetic, TeX},%
pdfstartview=FitH,%
pdfpagemode=UseOutlines}

\ifnum\dosourcexint=1 
\hypersetup{pdftitle={The xint bundle source code}}
\fi

\usepackage{bookmark}

\usepackage{picture} % permet d'utiliser des unités dans les dimensions de la
                     % picture et dans \put
\usepackage{graphicx}
\usepackage{eso-pic}

%---- \MyMarginNote: a simple macro for some margin notes with no fuss
% je m'aperçois que je peux l'utiliser dans les footnotes...
\makeatletter
\def\MyMarginNote {\@ifnextchar[\@MyMarginNote{\@MyMarginNote[]}}%
% 18 janvier 2014, j'ai besoin d'un raccourci.
\let\inmarg\MyMarginNote
\def\@MyMarginNote [#1]#2{%
    \vadjust{\vskip-\dp\strutbox
             \smash{\hbox to 0pt
                       {\color[named]{PineGreen}\normalfont\small
                        \hsize 1.6cm\rightskip.5cm minus.5cm
                        \hss\vtop{\offinterlineskip #2}\ $\to$#1\ }}%
             \vskip\dp\strutbox }\strut{}}
\def\MyMarginNoteWithBrace #1{%
    \vadjust{\vskip-\dp\strutbox
             \smash{\hbox to 0pt
                       {\color[named]{PineGreen}%\normalfont\small
                        \hss #1\ $\bigg\{$\ }}%
             \vskip\dp\strutbox }\strut{}}
\def\IMPORTANT {\MyMarginNoteWithBrace 
   {\raisebox{-.5\height}{\resizebox{2\width}{!}{\ding{43}}}}}
% 26 novembre 2013:
\def\etype #1{%
    \vadjust{\vskip-\dp\strutbox
             \smash{\hbox to 0pt {\hss\color[named]{PineGreen}%
                    \itshape \xintListWithSep{\,}{#1}\ $\star$\quad }}%
             \vskip\dp\strutbox }\strut{}}
\def\retype #1{%
    \vadjust{\vskip-\dp\strutbox
             \smash{\hbox to 0pt {\hss\color[named]{PineGreen}%
                    \itshape \xintListWithSep{\,}{#1}\ \ding{73}\quad }}%
             \vskip\dp\strutbox }\strut{}}
\def\ntype #1{%
    \vadjust{\vskip-\dp\strutbox
             \smash{\hbox to 0pt {\hss\color[named]{PineGreen}%
                    \itshape \xintListWithSep{\,}{#1}\quad }}%
             \vskip\dp\strutbox }\strut{}}
%-------------------------------------------------------------------------------
\def\Numf {{\vbox{\halign{\hfil##\hfil\cr \footnotesize
    \upshape Num\cr
    \noalign{\hrule height 0pt \vskip1pt\relax}
    \itshape f\cr}}}}
\def\Ff {{\vbox{\halign{\hfil##\hfil\cr \footnotesize
    \upshape Frac\cr
    \noalign{\hrule height 0pt \vskip1pt\relax}
    \itshape f\cr}}}}
\def\numx {{\vbox{\halign{\hfil##\hfil\cr \footnotesize
    \upshape num\cr
    \noalign{\hrule height 0pt \vskip1pt\relax}
    \itshape x\cr}}}}
%-------------------------------------------------------------------------------
% 24 février 2014. J'ai besoin de me débarasser du \to
\def\NewWith #1{%
    \vadjust{\vskip-\dp\strutbox
             \smash{\hbox to 0pt {\hss\color[named]{PineGreen}%
                        \normalfont\small
                        \hsize 1.5cm\rightskip.5cm minus.5cm
                        \vtop{\noindent New with #1}\ }}%
             \vskip\dp\strutbox }\strut{}}

\makeatother

% \centeredline: OUR OWN LITTLE MACRO FOR CENTERING LINES
% =======================================================

% 7 mars 2013
% -----------
%
% This macro allows to conveniently center a line inside a paragraph and still
% allow use therein of \verb or other commands changing catcodes.
% A proposito, the \LaTeX \centerline uses \hsize and not \linewidth !
% (which in my humble opinion is bad)

% Actually my \centeredline works nicely in list environments.

% \ignorespaces ajouté le 9 juin 2013.

% Note: \centeredline creates a group

\makeatletter
\newcommand*\centeredline {%
      \ifhmode \\\relax
        \def\centeredline@{\hss\egroup\hskip\z@skip\ignorespaces }%
      \else
        \def\centeredline@{\hss\egroup }%
      \fi
      \afterassignment\@centeredline
      \let\next=}
\def\@centeredline
    {\hbox to \linewidth \bgroup \hss \bgroup \aftergroup\centeredline@ }

% 12 octobre 2014
% ---------------
% 
% \centeredline-->\leftedline. 
% And I add colored background. I have more sophisticated approaches, but the
% mark-up was essentially already there, thus I just wanted to exploit the
% manual from 1.09n for the transition to 1.1.
%

\newif\ifinlefted

\newcommand*\leftedline {%
      \ifhmode \\\relax
        \def\leftedline@{\hss\egroup\hskip\z@skip\ignorespaces }%
      \else
        \def\leftedline@{\hss\egroup }%
      \fi
      \afterassignment\@leftedline
      \let\next=}
\def\@leftedline
    {\hbox to \linewidth \bgroup \inleftedtrue
                         \everbatimeverypar
                         \bgroup 
                         \aftergroup\leftedline@ }

\makeatother

% verbatim environments
% =====================
% 
% June 2013, then October 2014.
% -----------------------------
%
\makeatletter
\catcode`_ 11

% some of my verbatim environments do not make the space active (\lverb e.g.). Then
% \do@noligs must be modified, \char`#1 must be followed by a space token, else,
% the `#1 expansion will swallow one space.
\def\do@noligs #1{%
  \catcode`#1\active
  \begingroup
     \lccode`~`#1\relax
     \lowercase{%
  \endgroup\def~{\leavevmode\kern\z@\char`#1 }}%
}

%--- \lowast
\def\lowast{\raisebox{-.25\height}{*}}
\catcode`* 13
\def\makestarlowast {\let*\lowast\catcode`\*\active}%
\catcode`* 12

%--- straight quotes, added (finally...) Nov 2, 2014

% \begingroup\makeatletter
%   \catcode`\'\active
%   \catcode`\`\active
% \@firstofone {\endgroup
%   \def\makequotesstraight{% assumes textcomp package
%      \let`\textasciigrave 
%      \let'\textquotesingle
%      \catcode39\active
%      \catcode96\active }%
% }
    
%--- for soft-wrapping. I will use discretionaries.

\DeclareFontFamily{U}{MdSymbolC}{}
\DeclareFontShape {U}{MdSymbolC}{m}{n}{<-> MdSymbolC-Regular}{}

\newbox\SoftWrapIcon
\colorlet {softwrapicon}{blue}

\def\SetSoftWrapIcon{%
    \setbox\SoftWrapIcon\hb@xt@\z@ 
    {\hb@xt@\fontdimen2\font
        {\hss{\color{softwrapicon}\usefont{U}{MdSymbolC}{m}{n}\char"97}\hss}%
     \hss}%
   }

\AtBeginDocument {\SetSoftWrapIcon }% ttzfamily déjà fait

%--- \MacroFont, and a \MicroFont
% 
% Ne PAS mettre de changement de taille de police dans \MacroFont.

% 17/10/2014, essai avec CadetBlue après Purple. Puis Blue

\def\restoreMicroFont {\def\MicroFont {\ttbfamily\makestarlowast
    \ifinlefted\else\ifineverb\else\color[named]{Blue}\fi\fi}}
\restoreMicroFont

% Notice that \macrocode uses \macro@font which freezes \MacroFont
% at \begin{document}. But doc.sty verbatim uses \MacroFont which is not
% frozen. Comprenne qui pourra...

\def\restoreMacroFont {\def\MacroFont {\ttbfamily
    \ifinlefted\else\ifineverb\else\color[named]{Blue}\fi\fi}}
\restoreMacroFont

%--- straight quotes, also in macrocode (2014/11/04)
%
% There is no hook in \macrocode after \dospecials, which is done *after*
% \macro@font. Thus I will need to take the risk that some future evolution of
% doc.sty (or perhaps scrdoc) invalidates the following.
%
% Note: in contrast, \MacroFont in \verbatim is done *after* \dospecials (but
% before the space becomes active), thus I could use \MacroFont there. But as
% there is no verbatim environment anymore in xint.dtx, I don't need to take
% care of it.
%
% Actually, I should not at all rely on the doc class, I should do it all by
% myself. As I don't use at all \DocInput (which caused me loads of problems
% back then when I was trying to get a workflow satisfying my views on how
% .dtx files should be structured), there is not much rationale for using the
% doc class.
%

\def\macrocode{\macro@code
               \frenchspacing \@vobeyspaces
               \makestarlowast %\makequotesstraight
               \xmacro@code }


% ---- a new \verb

% Initially, June 2013, then Sep 9, 2014, and Oct 9-12 2014
%
% Initial motivation was simply that doc.sty and related classes \verb
% command is with a hard-coded \ttfamily. There were further issues.
%
% 1. with |stuff with space|, paragraph reformatting in the Emacs/AUCTeX
% buffer caused havoc. Thus I wanted to be able to have the input across
% lines.
%
% 2. Hence I did not want to have spaces obeyed, as often the
% reformatting added spaces at the beginning of a line.
%
% 3. Also I wanted to allow hyphenation on output, at least at some
% locations. I did a first version which treated spaces, \, {, and }
% specially.
%
% 4. at some point I wanted to add some colored background (I have
% dropped that since due to pdf file size increase).
%
% 5. and also I got fed up from the non-compatibility with footnotes due
% to catcode freeze.
%
% Because of 5. I opted for a \scantokens approach, hence for a macro
% with delimited argument. Here is what I do now, this is compatible
% with short verbs.

\def\verb
{%
  \relax \ifmmode\else\leavevmode\null\fi
  \bgroup
  \let\do\@makeother \dospecials
  %\makequotesstraight  % belatedly added for 1.1a release
  \MicroFont % change font, color, catcode hooks, ...
  \catcode 32 10
  \endlinechar 32
  \@@jfverb 
}% 
% Note (Oct 12, 2014): in the improbable situation a newlinechar is
% found in the ##1, \scantokens will convert this to an end of line in
% its "write" phase, which will be then ignored in its "read" phase due
% to \endlinechar-1. This also avoids possible creation of \par which
% would defeat \@@jfverb@@. Thus it is good.
\def\@@jfverb #1{%
   \ifcat\noexpand#1\noexpand~\catcode`#1\active\fi
% No problem with the EOL for the line where the short verb delimiter stands.
   \def\next ##1#1{%
            \@vobeyspaces\everyeof{\relax}\endlinechar\m@ne
            \expandafter\@@jfverb_a\scantokens\expandafter{##1}}%
% hack with \@empty to prevent brace stripping if catcodes have been
% frozen earlier, like in footnotes.
   \next \@empty
}

% We don't want a \discretionary at the very start.
% But then an empty argument is forbidden!
\def\@@jfverb_a #1{#1\@@jfverb_b }

\def\@@jfverb_b #1{\ifx\relax #1%
        \egroup
      \else
% \penalty\z@, or rather (Oct 11, 2014) but I then adjust the textwidth
% precisely: 
      \discretionary{\copy\SoftWrapIcon}{}{}%
        #1\expandafter\@@jfverb_b\fi 
}

\catcode`_ 8
\makeatother

% --- \lverb
% Définition de \lverb
\makeatletter
\long\def\lverb {%
  \relax\par\smallskip\noindent\null
  \begingroup
  \bgroup
    \aftergroup\@@par \aftergroup\endgroup \aftergroup\medskip
    \let\do\do@noligs  \verbatim@nolig@list
    \let\do\@makeother \dospecials
    %\makequotesstraight  % belatedly added for 1.1a release
    \catcode32 10 \catcode`\% 9 \catcode`\& 14 \catcode`\$ 0
  \MicroFont % sera donc en couleur.
    \@lverb
}

\def\@lverb #1{\catcode`#1\active
               \lccode`\~`#1\lowercase{\let~\egroup}}%
\makeatother
%--- everbatim environment
% October 13-14, 2014
% Verbatim with an \everypar hook, mainly to have background color, followed by
% execution of the code 

\makeatletter
\catcode`_ 11

\def\everbatimtop {\MacroFont\small }
\let\everbatimbottom\relax
\let\everbatimhook\relax

\newif\ifineverb

\def\everbatim {\s@everbatim\@everbatim }
\@namedef{everbatim*}{\s@everbatim\expandafter\@everbatimx\expandafter
                      {\the\newlinechar}}

\def\everbatimeverypar{\strut
                   {\color{yellow!5}\vrule\@width\linewidth }%
                   \kern-\linewidth
                   \kern\everbatimindent }
\def\everbatimindent {\z@}
% voir plus loin atbegindocument

\def\endeverbatim    {\if@newlist \leavevmode\fi\endtrivlist }
\expandafter\let\csname endeverbatim*\endcsname \endeverbatim

\def\s@everbatim {%
     \ineverbtrue
     \everbatimtop % put there size changes
       \topsep    \z@skip
       \partopsep \z@skip
       \itemsep   \z@skip
       \parsep    \z@skip
       \parskip   \z@skip
       \lineskip  \z@skip
     \let\do\@makeother \dospecials
     \let\do\do@noligs  \verbatim@nolig@list 
     \makestarlowast
     \everbatimhook
     \trivlist\item\relax
       \leftskip   \@totalleftmargin
       \rightskip  \z@skip
       \parindent  \z@
       \parfillskip\@flushglue
       \parskip    \z@skip
       \@@par
       \def\par{\leavevmode\null\@@par\pagebreak[1]}% 
       \everypar\expandafter{\the\everypar \unpenalty
                \everbatimeverypar 
                \everypar \expandafter{\the\everypar\everbatimeverypar}%
       }%
       \obeylines \@vobeyspaces 
}

\begingroup
\lccode`X 13
\catcode`X \active
\lccode`Y `* % this is because of \makestarlowast.
% I have to think whether this is useful: obviously if I were to provide
% everbatim and everbatim*  in a package I wouldn't do that.
\catcode`Y  \active
\catcode`| 0 \catcode`[ 1 \catcode`] 2 \catcode`* 12
\catcode`{ 12 \catcode`} 12 |catcode`\\ 12
|lowercase[|endgroup% both freezes catcodes and converts X to active ^^M
|def|@everbatim #1X#2\end{everbatim}%
  [#2|end[everbatim]|everbatimbottom ]
|def|@everbatimx #1#2X#3\end{everbatimY}]%
  {#3\end{everbatim*}%
     \everbatimbottom
     % No group here: this allows executed code to make macro
     % definitions which may reused in later uses of everbatim.
     % But the problem is with colors... j'ai visiblement un problème
     % avec le color stack pour dvipdfmx avec les \colorlet/\color
     \newlinechar 13
     % Indentation of next paragraph produced from execution of #3 is
     % suppressed, if #3 by itself or \everbatimbottom does no \par, 
     % from \@endparenv done by \endtrivlist
     \everbatimxprehook
     \scantokens {#3}% there will typically be an EOL space after this if one
                     % continues after \end{everbatim} on same line, which
                     % is allowed.
     \newlinechar #1\relax % restores \newlinechar to previous value.
     \everbatimxposthook
}%

\def\everbatimxprehook  {\colorlet{everbsavedcolor}{.}\color[named]{OrangeRed}}
\def\everbatimxposthook {\color{everbsavedcolor}}


% L'espace venant du endofline final mis par \scantokens sera inhibé si #3 se
% termine par un % ou un \x, etc... 

% --- \everb
% Original was called \dverb and I did it in June 2013.
% Then after doing everbatim, I transformed \dverb, now called \everb
% for itself being as compatible as standard verbatim with list making
% surrounding environments.
% Supposed to be used as 
% \everb|@ this will be ignored
% stuff
% escape character: "
% | not necessarily starting a line.
% I chose @ as comment character, mainly for pretty-formatting of the
% source, this can be changed by \everbhook.

% " comme caractère d'échappement. Par exemple pour colorier des parties.
\def\restoreeverbhook{\def\everbhook{%
     \def\"{\begingroup\catcode123 1 \catcode 125 2 \everbescape }%
     \catcode`\" 0 \catcode`\@ 14
}}\restoreeverbhook

\def\everbescape #1;!{#1\endgroup }

\def\everb {%
    \bgroup
    \let\everbatimhook\everbhook
    \s@everbatim
    \@everb
}

\def\@everb #1{\catcode`#1\active
               \lccode`\~`#1%
               \lowercase{\def~{\if@newlist \leavevmode\fi
                                \endtrivlist 
                                \egroup
                                \@doendpe
                                \everbatimbottom }}%
              }%

\catcode`_8
\makeatother

%--- \csa, \csbxint, \csh, \csbh
% dates back to earliest versions of this manual, but I changed things a bit
% (back then for example @ was active throughout the document...)
% The mark-up being in place, I only have to use it here.

\DeclareRobustCommand\csa [1]
% attention que le \expandafter est nécessaire ici, sinon \scantokens n'agit
% pas sur ce que produit \detokenize. Par ailleurs \MicroFont fait
% \makestarlowast (et c'est seulement à cause de * que je fais \scantokens)
% Attention cependant aux _ maintenant dans les noms de macros
    {{\MicroFont\char92\endlinechar-1 \catcode`_ 11
                       \scantokens\expandafter{\detokenize{#1}}}}

\DeclareRobustCommand\csbnolk [1]
    {{\MicroFont\color{blue}\char92\endlinechar-1 \scantokens\expandafter{\detokenize{#1}}}}

\DeclareRobustCommand\csbxint [1]
        {\hyperref[\detokenize{xint#1}]%
          {{\ttzfamily\char92\mbox{xint}\-\endlinechar-1 \makestarlowast
                 \scantokens\expandafter{\detokenize{#1}}}}}

\newcommand\csh[1]
    {\texorpdfstring{\csa{#1}}{\textbackslash\detokenize{#1}}}
\newcommand\csbh[1]
    {\texorpdfstring{\csbnolk{#1}}{\textbackslash\detokenize{#1}}}

% --- \xintname, \xintnameimp etc... 

% 7 mars 2015, je résous (non!) un problème de color stack overflow avec
% dvipdfmx qui venait au final des page headers, et à cause d'un brace
% stripping qui enlevait la protection de mes \color ci-dessous. Il a suffi de
% rajouter un \empty pour me débarrasser finalement du problème.
%
% Bon c'est bizarre, en fait le problème n'est pas résolu. Après avoir
% supprimé fichiers auxiliaires et recompilé, il revient.

% J'ai dû remettre
%\def\everbatimxprehook  {}
%\def\everbatimxposthook {}
% avant la section Command of xintseries

% et aussi j'ai dû supprimer la couleur dans \digitstt là-bas.
% C'est très étrange que j'ai cru le problème réglé un temps.

\xintForpair #1#2 in
{(xintkernel,kernel),
 (xinttools,tools),
 (xintcore,core),(xint,xint),(xintbinhex,binhex),(xintgcd,gcd),%
 (xintfrac,frac),(xintseries,series),(xintcfrac,cfrac),(xintexpr,expr)}
\do
{%
 \expandafter\def\csname #1name\endcsname
   {\texorpdfstring
                  {\hyperref[sec:#2]%
                     {\relax{\color{joli}\ttzfamily #1}}}
                  {#1}%
    \xspace }%
 \expandafter\def\csname #1nameimp\endcsname
   {\texorpdfstring
                  {\hyperref[sec:#2imp]%
                    {\relax{\color[named]{RoyalPurple}\ttzfamily #1}}}
                  {#1}%
    \xspace }%
}%

%--- \printnumber
% new version, october 11, 2014
\catcode`_ 11
\makeatletter
\hyphenpenalty \z@

\def\allowsplits_a #1{\ifx \relax #1\xint_dothis\xint_gobble_i\fi
                    \if ,#1\xint_dothis {\discretionary{\rlap,}{}{,} }\fi
                    \xint_orthat{\discretionary
                           {\copy\SoftWrapIcon}%
                           {}%
                           {}#1}\allowsplits_a }%

\def\allowsplits #1{#1\allowsplits_a}% évite un discretionary au premier.

\def\printnumber #1{\expandafter\allowsplits \romannumeral-`0#1\relax }%

\makeatother
\catcode`_ 8

%--- counts used in particular in the samples from the documentation of the
%    xintseries.sty package
\newcount\cnta
\newcount\cntb
\newcount\cntc

%--- \fexpan 22 octobre 2013
\newcommand\fexpan {\textit{f}-expan}

\ifnum\dosourcexint=1
\else
% Dependency graph done using TikZ (manually)
  \usepackage{tikz}
  \usetikzlibrary{shapes,arrows}
\fi

\colorlet {codeboxbg}{yellow!10}
\colorlet {codeboxframe}{black!30}
\colorlet {execboxfringe}{black!10}

% 12 octobre 2014
\AtBeginDocument{%
    \leftmargini \dimexpr4\fontcharwd\font`X\relax
    \leftmarginii\dimexpr3\fontcharwd\font`X\relax
    \leftmarginiii \leftmarginii
    \leftmarginiv  \leftmarginii
    \parindent\dimexpr2\fontcharwd\font`X\relax
    \leftmargin\leftmargini % pourquoi pas 0?
% attention à un deuxième relax!
%\advance\linewidth\dimexpr-\everbatimindent-\everbatimindent\relax
% était alors bogué! 
    \edef\everbatimindent{\the\dimexpr\leftmargini\relax\space }%
    \cftsubsecnumwidth    2\leftmarginii
    \cftsubsubsecnumwidth 2\leftmargini
    \cftsubsecindent 0pt
    \cftsubsubsecindent \cftsubsecnumwidth
}%

%--- CodeBox

\frenchspacing
\renewcommand\familydefault\sfdefault

\begin{document}\thispagestyle{empty}% \ttzfamily already done
\pdfbookmark[1]{Title page}{TOP}
% \makeatletter % @ n'est plus actif dans dtx 1.1, ouf!

{%
\normalfont\Large\parindent0pt \parfillskip 0pt\relax
 \leftskip 2cm plus 1fil \rightskip 2cm plus 1fil
\ifnum\dosourcexint=1
 The \xintnameimp source code\par
\else
 The \xintname bundle\par
\fi
}

{\centering
  \textsc{Jean-François Burnol}\par
  \footnotesize
  jfbu (at) free (dot) fr\par
  Package version: \xintbndlversion\ (\xintbndldate);
            documentation date: \xintdocdate.\par
  {From source file \texttt{xint.dtx}. \xintdtxtimestamp.}\par
}

\bigskip

% Mercredi 08 octobre 2014 à 22:03:19
% Skips safely.
\ifnum\dosourcexint=1
\catcode`+ 0
\catcode`\\ 12
+expandafter+iffalse+fi
\fi

% ---- 
% Fibonacci code
% December 7, 2013. Expandably computing a big Fibonacci number
% with the help of TeX+\numexpr+\xintexpr, (c) Jean-François Burnol
\catcode`_ 11
%
% ajouté 7 janvier 2014 au xint.dtx pour 1.07j.
%
% Le 17 janvier je me décide de simplifier l'algorithme car l'original ne tenait
% pas compte de la relation toujours vraie A=B+C dans les matrices symétriques
% utilisées en sous-main [[A,B],[B,C]].
%
% la version ici est celle avec les * omis: car multiplication tacite devant les
% sous-expressions depuis 1.09j, et aussi devant les parenthèses depuis 1.09k.
% (pour tester)
\def\Fibonacci #1{%
    \expandafter\Fibonacci_a\expandafter
        {\the\numexpr #1\expandafter}\expandafter
        {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter
        {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter
        {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter
        {\romannumeral0\xintiieval 0\relax}}
%
\def\Fibonacci_a #1{%
    \ifcase #1
          \expandafter\Fibonacci_end_i
    \or
          \expandafter\Fibonacci_end_ii
    \else
          \ifodd #1
              \expandafter\expandafter\expandafter\Fibonacci_b_ii
          \else
              \expandafter\expandafter\expandafter\Fibonacci_b_i
          \fi
    \fi {#1}%
}%
\def\Fibonacci_b_i #1#2#3{\expandafter\Fibonacci_a\expandafter
  {\the\numexpr #1/2\expandafter}\expandafter
  {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter
  {\romannumeral0\xintiieval (2#2-#3)#3\relax}%
}% end of Fibonacci_b_i
\def\Fibonacci_b_ii #1#2#3#4#5{\expandafter\Fibonacci_a\expandafter
  {\the\numexpr (#1-1)/2\expandafter}\expandafter
  {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter
  {\romannumeral0\xintiieval (2#2-#3)#3\expandafter\relax\expandafter}\expandafter
  {\romannumeral0\xintiieval #2#4+#3#5\expandafter\relax\expandafter}\expandafter
  {\romannumeral0\xintiieval #2#5+#3(#4-#5)\relax}%
}% end of Fibonacci_b_ii
\def\Fibonacci_end_i  #1#2#3#4#5{\xintthe#5}
\def\Fibonacci_end_ii #1#2#3#4#5{\xinttheiiexpr #2#5+#3(#4-#5)\relax}
\catcode`_ 8

\def\Fibo #1.{\Fibonacci {#1}}

% nice background added for 1.09j release, January 7, 2014.
% superbe, non? moi très content!
% bon je peaufine ce background le 17 janvier, c'est hard-coded mais je ne veux
% pas y passer plus de temps (ce qui est amusant c'est que j'ai constaté a
% posteriori qu'il y a 17 chiffres par lignes donc 1 chiffre avec son padding =
% 1cm...
% *\message{\xinttheexpr round(\dimexpr 8cm\relax/17,3)\relax}
% 877496.353
\def\specialprintone #1%
{%
    \ifx #1\relax \else \makebox[877496sp]{#1}\hskip 0pt plus 2sp\relax
    \expandafter\specialprintone\fi
}%
\def\specialprintnumber #1% first ``fully'' expands its argument.
{\expandafter\specialprintone \romannumeral-`0#1\relax }%

\AddToShipoutPicture*{%
    \put(10.5cm,14.85cm)
    {\makebox(0,0)
      {\resizebox{17cm}{!}{\vbox
       {\hsize 8cm\Huge\baselineskip.8\baselineskip\color{black!10}%
        \specialprintnumber{F(1250)=}%
                  \specialprintnumber{\Fibonacci{1250}}}\par}%
       }%
    }%
}

% Samedi 27 septembre 2014 à 16:04:52
\pdfbookmark[1]{Dependency graph}{DependencyGraph}

\tikzstyle{block} = [rectangle, draw,
    fill=codeboxbg,
    fill opacity=0.5,% fill opacity Octobre 2014
    draw=codeboxframe,
    line width=2pt,
    text width=6em, text centered, rounded corners, minimum height=4em]
\tikzstyle{line} = [draw, line width=1pt, color=codeboxframe]

\vspace{2\baselineskip}

\begin{figure}[ht!]
  \phantomsection\label{dependencygraph}
\centeredline{%
\begin{tikzpicture}[node distance = 2.5cm]
    % Place nodes
    \node [block] (kernel) {xintkernel};
    \node [left of=kernel] (A) {};
    \node [right of=kernel] (B) {};
    \node [block, below right of=B] (core) {\xintcorename};
    \node [block, below left of=A]  (tools) {\xinttoolsname};
    \node [block, right of=core, xshift=1cm]  (bnumexpr) {\href{http://www.ctan.org/pkg/bnumexpr}{bnumexpr}};
    \node [block, below of=core] (xint) {\xintname};
    \node [block, left of=xint, xshift=-.5cm] (binhex) {\xintbinhexname};
    \node [block, left of=binhex] (gcd) {\xintgcdname};
    \node [block, below of=xint] (frac) {\xintfracname};
    \node [block, below of=frac, yshift=-.5cm] (expr) {\xintexprname};
    \node [block, below right of=frac, xshift=1cm] (cfrac) {\xintcfracname};
    \node [block, right of=cfrac] (series) {\xintseriesname};
    % Draw edges
    \path [line] (kernel) -- (core);
    \path [line] (kernel) -- (tools);
    \path [line] (core) -- (bnumexpr);
    \path [line] (core) -- (gcd.north);
    \path [line] (core) -- (binhex.north);
    \path [line] (core) -- (xint);
    \path [line] (xint) -- (frac);
    \path [line] (frac) -- (expr);
    \path [line] (frac) -- (series.north);
    \path [line] (frac) -- (cfrac.north);
    \path [line,dashed] (binhex.south) -- (expr);
    \path [line,dashed] (gcd.south) -- (expr);
    \path [line,dashed] (tools) -- (gcd);
    \path [line] (tools) to [out=270,in=180] (expr);
  \end{tikzpicture}}\bigskip
\end{figure}

\vspace{2\baselineskip}

\begin{addmargin}{2cm}
\normalfont\footnotesize Dependency graph for the
    \xintname bundle components: modules at the bottom import modules at the
    top when connected by a continuous line. No module will be loaded twice,
    this is managed internally under Plain as well as \LaTeX. Dashed lines
    indicate a partial dependency, and to enable the corresponding
    functionalities of the lower module it is necessary for the user to issue
    the suitable |\usepackage{top_module}| in the
    preamble (or |\input top_module.sty\relax| in Plain
    \TeX). The \href{http://ctan.org/pkg/bnumexpr}{bnumexpr} package is a
    separate package (\LaTeX{} only) by the author.\par
\end{addmargin}

\vfill

\clearpage

\etocsetlevel{toctobookmark}{6} % 9 octobre 2013, je fais des petits tricks.

% 18 novembre 2013, je n'inclus plus la TOC détaillée de xintexpr. Je
% reconfigure la TOC.

\etocsettocdepth {subsection}

\renewcommand*{\etocbelowtocskip}{0pt}
\renewcommand*{\etocinnertopsep}{0pt}
\renewcommand*{\etoctoclineleaders}
              {\hbox{\normalfont\normalsize\hbox to 1ex {\hss.\hss}}}
\etocmulticolstyle [1]{%
    \phantomsection\section* {Contents}
    \etoctoccontentsline*{toctobookmark}{Contents}{1}%
}
   \etocsettagdepth {description}{subsection}
   \etocsettagdepth   {commands} {none}
   \etocsettagdepth {implementation}{none}
\tableofcontents
\renewcommand*\etocabovetocskip{\bigskipamount}
\makeatletter
\etocmulticolstyle [2]{\parskip\z@skip\raggedcolumns }%
\makeatother
   \etocsettagdepth {description}{none}
   \etocsettagdepth {commands}   {section}
\ifnum\NoSourceCode=1
   \etocsettagdepth {implementation}{none}
\else
   \etocsettagdepth {implementation}{section}
\fi
\tableofcontents

% pour la suite: [voir aussi juste avant la section Commandes de xint]
% 12 octobre 2014, je supprime tous les "Contents", maintenant que les
% TOC sont déplacées vers immédiatement après le titre de la section.
\etocignoredepthtags
\etocmulticolstyle [1]{%
    \phantomsection% \section* {Contents}
    \etoctoccontentsline*{toctobookmark}{Contents}{2}%
}

\etocdepthtag.toc {description}

\section{Read this first}\label{sec:quickintro}

This section provides recommended reading on first discovering the package.

% local TOC supprimée 12 octobre 2014
% finalement non, car LaTeX laisse un énorme vide au bas de la page à la
% place!!! Bon, je ne le mets que s'il y a la place.

\ifnum\NoSourceCode=1
{\etocdefaultlines\etocsettocstyle{}{}\localtableofcontents}
\fi

\subsection{The packages of the \xintname bundle}

The components of the \xintname bundle provide macros dedicated to
\emph{expandable} computations on numbers exceeding the \TeX{} (and \eTeX{})
limit of \dtt{\number"7FFFFFFF}.

The \eTeX{} extensions (dating back to 1999) must be enabled; this is the case
by default in modern distributions, except for the |tex| executable itself
which has to be the pure \textsc{D.~Knuth} software with no additions. The
name for the extended binary is |etex|. In |TL2014| for example |etex| is a
symbolic link to the |pdftex| executable which will then run in |DVI| output
mode, the \eTeX{} extensions being automatically active.

All components may be loaded with \LaTeX{} |\usepackage| or
|\RequirePackage| or, for any other format based on \TeX{}, directly via
\string\input{}, e.g. |\input xint.sty\relax|. There are no package
options.
%
% {\makeatother\footnote{\csa{empty}, \csa{space}, \csa{z@},
%     \csa{@ne}, and \csa{m@ne} should have the same meaning as in Plain and
%     \LaTeX.}}
%
Each package automatically loads those not already loaded it depends on (but
in a few rare cases there are some extra dependencies, for example the |gcd|
function in \xintexprname expressions requires explicit loading of package
\xintgcdname for its activation).\smallskip

%% \pdfbookmark[1]{Abstract}{ABSTRACT}

\begin{addmargin}{1cm}\small
  % \begin{center} \bfseries\large Description of the packages\par\smallskip
  % \end{center}\medskip
\makeatletter
\renewenvironment{description}
    {\list{}{\topsep\z@\partopsep\z@
              \parsep\z@ \labelwidth\z@ \itemindent-\leftmargin
             \let\makelabel\descriptionlabel}}
    {\endlist}
\makeatother
\begin{description}
\item[\xinttoolsname] provides utilities of independent interest such as
  expandable and non-expandable loops. It is \fbox{not}
  loaded\MyMarginNote{Changed} automatically (nor needed) by the other bundle
  packages, apart from \xintexprname.

\item[\xintcorename] provides\MyMarginNote{Formerly, part of \xintname.} the
  expandable \TeX{} macros doing additions, subtractions, multiplications,
  divisions and powers on arbitrarily long numbers (loaded automatically by
  \xintname, and also by package \href{http://ctan.org/pkg/bnumexpr}{bnumexpr}
  in its default configuration).

\item[\xintname] extends \xintcorename with additional operations on big
  integers.

\item[\xintfracname] extends the scope of \xintname to decimal numbers, to
  numbers in scientific notation and also to fractions with arbitrarily
  long such numerators and denominators separated by a forward slash.

\item[\xintexprname] extends \xintfracname with expandable parsers doing
  algebra (exact or float, or limited to integers) on comma separated
  expressions using standard infix notations with parentheses, numbers in
  decimal notation, and scientific notation, comparison operators, Boolean
  logic, twofold and threefold way conditionals, sub-expressions, some
  functions with one or many arguments, user-definable variables, evaluation
  of sub-expressions over a dummy variable range, with possible recursion,
  omit, abort, break instructions, nesting.
\end{description}

Further modules:

\begin{description}
\item[\xintbinhexname] is for conversions to and from binary and
  hexadecimal bases.

\item[\xintseriesname] provides some basic functionality for computing in an
  expandable manner partial sums of series and power series with fractional
  coefficients.

\item[\xintgcdname] implements the Euclidean algorithm and its typesetting.

\item[\xintcfracname] deals with the computation of continued fractions.
\end{description}
\end{addmargin}

\subsection{About this manual and a quick first overview}

This documentation was build via successive layers; a complete re-write
would be needed for a more user-friendly access. I will try to add here a
quick expos\'e of the package abilities, starting from the arithmetic
expressions handled by package \xintexprname.

Here is an example of a computation:
\begin{everbatim*}
\xinttheexpr (31.567^2 - 21.56*52)^3/13.52^5\relax 
\end{everbatim*}

The result is a bit frightening but illustrates that |\xinttheexpr..\relax|
does its computations \emph{exactly}. There is a variant which emulates
floating point computations:
\begin{everbatim*}
\xintthefloatexpr (31.567^2 - 21.56*52)^3/13.52^5\relax 
\end{everbatim*}

Such ``floating-point'' computations are done by default with 16 digits of
precision (this can be increased via a prior assignment such as
|\xintDigits:=24;|). \xintname has very few typesetting macros. \LaTeX{} users
can do:
\begin{everbatim*}
\[\xintFrac{\xintthefloatexpr (31.567^2 - 21.56*52)^3/13.52^5\relax }\]
\end{everbatim*}
but it probably is better to use packages dedicated to the typesetting of
numbers in scientific format (but the output of |\xinttheexpr| will
typically be in |A/B[N]| format which is unlikely to be understood by your
favorite number formatting package).
 
The computations are done expandably: you can put them in an |\edef| or a
|\write| or even force complete expansion via |\romannumeral-`0| (if you don't
understand the latter sentence, this doesn't matter; this manual should
contain a description of expandability in \TeX, but this is yet to arise.)
Let's just say that such expandable macros are maximally usable in almost all
locations of \TeX{} code (but \TeX{} will choke on fractions |A/B| or integers
with more than ten digits). However, the implementation is also maximally
complicated as many functionalities of the \TeX{} macro language can not be
used in the source code of \xintname by the poor author of the package.
Besides, there is a speed impact, and although algebraic computations with
numbers of up to one hundred digits are fine, my current implementation of
multiplication needs of the order of the second for inputs with thousand
digits. Macro \csbxint{XTrunc} is however able to compute and ouput tens of
thousands of digits of a decimal expansion in reasonable time, still
expandably, but it does not expand completely via |\romannumeral-`0|. As the
|\xinttheexpr..\relax| mechanism only needs this weaker expandability of the
underlying macros, perhaps some speed gains could be achieved via a complete
rewrite of all basic macros of the package (but nesting of the macros would
not be possible; only the |\xintexpr...\relax| and variants interface would be
functional).

This brings me to my second point here: the underlying macros are provided by
packages \xintcorename, \xintname for integers and \xintfracname for
fractions. For example to do |21+32*43|, the syntax would be (only
\xintcorename needed):
\begin{everbatim*}
\xintiiAdd{21}{\xintiiMul{32}{43}}
\end{everbatim*}

Needless to say this quickly becomes a bit painful. One more example (needs \xintfracname):
\begin{everbatim*}
\xintIrr {\xintiiPrd{{128}{81}{125}}/\xintiiPrd{{32}{729}{100}}}
\end{everbatim*}

This shows that package \xintfracname knows natively how to handle fractions
|A/B|, and has a command to reduce to smallest terms (in an
|\xintexpr..\relax| this would be the |reduce| function).

Again, all computations done by |\xintexpr..\relax| are completely exact.
Thus, very quickly very big numbers are created (and computation times
explode). To compute something like |1.23456789^10000| you will have to opt
for the floating point version:
\begin{everbatim*}
{\xintDigits:=24; \xintthefloatexpr 1.23456789123456789^123456789\relax }
\end{everbatim*}
(notice the size of the power of ten: this surely largely exceeds your pocket
calculator abilities).

Braces limit the scope of the assignment to |\xintDigits|. It is also possible
to do some computer algebra like evaluations (only numerically though):
\begin{everbatim*}
\xinttheiiexpr add(i^5, i=100..150)\relax
\end{everbatim*}

Further examples are to be found in \autoref{sec:expr11} and \autoref{sec:expr}.
Make sure to read \autoref{ssec:userinterface}.


\subsection {Changes}

On |TeXLive| (and presumably |MikTeX|), type |texdoc --list xint| to get
access to the
various included documentation files, among them |CHANGES.html| provides
the cumulative change log since the initial release.

It is also available on internet via
\href{http://mirrors.ctan.org/macros/generic/xint/CHANGES.html}{this link}.

Here is a brief review of recent changes:

\subsubsection{\unexpanded{\unexpanded{Changes from |1.1a 2014/11/07| to 
|1.1b 2015/08/31|}}}

This is a bug-fix release to move from |xint.sty| to |xintcore.sty| some
macros which are used by the division routines included in \xintcorename (as
package \xintgcdname loads only \xintcorename, this dependency on macros which
had been left in |xint.sty| broke \csbxint{GCD} for example).

Slight enhancements to the |Read this first| section of the documentation. 

\subsubsection{\unexpanded{\unexpanded{Changes from |1.1 2014/10/28| to 
|1.1a 2014/11/07|}}}

Numerous small fixes to the documentation.

Bug fix in \csbxint{NewExpr} to get it to correctly accept comma separated
inputs. A new macro \csbxint{iiSqrtR} computes rounded integer square roots,
the earlier \csbxint{iiSqrt} computing truncated integer square roots.

Later release of |2015/03/07| only improved a few things in the documentation
and the package version number was kept unchanged at |1.1a|.

\subsubsection{\unexpanded{\unexpanded{Changes from |1.09n 2014/04/01| to 
|1.1 2014/10/28|}}}

|1.1| brought many additions, most of them to package \xintexprname, for example
the evaluation of expressions with dummy variables, possibly iteratively,
and with allowed nesting. See \autoref{sec:expr11} for a description of these
changes.

Apart from that, the two main other changes were:
\begin{enumerate}
\item |\xintiiexpr...\relax| now associates |/| with the \emph{rounded}
  division (the |//| operator being provided for the \emph{truncated}
  division) to be in synchrony with the habits of |\numexpr|, 
\item the \xintfracname macro \csbxint{Add} (corresponding to |+| in
  expressions) does not anymore blindly multiply denominators but first checks
  if one is a multiple of the other. However doing systematic reduction to
  smallest terms, or systematically computing the |LCM| of the denominators
  would be too costly (I think).
\end{enumerate}
Also worth mentioning is that \xintname does not load \xinttoolsname
anymore (only \xintexprname does) and that the core arithmetic macros have
been moved to a new package \xintcorename which is loaded automatically by
\xintname.

Package \href{http://www.ctan.org/pkg/bnumexpr}{bnumexpr} (which is \LaTeX{}
only) now also loads only \xintcorename.

\subsection{Installation instructions}
\label{ssec:install}

\xintname is made available under the
\href{http://www.latex-project.org/lppl/lppl-1-3c.txt}{LaTeX Project Public
  License 1.3c} (or any later version). It is included in the major \TeX\
distributions, thus there is probably no need for a custom install: just use
the package manager to update if necessary \xintname to the latest version
available.

After installation, issuing in terminal |texdoc --list xint|, on installations
with a |"texdoc"| or similar utility, will offer the choice to display one of
the documentation files: |xint.pdf| (this file), |sourcexint.pdf| (source
code), |README|, |README.pdf|, |README.html|, |CHANGES.pdf|, and
|CHANGES.html|.

For manual installation, follow the instructions from the |README| file which
is to be found on \href{http://www.ctan.org/pkg/xint}{CTAN}; it is also
available there in PDF and HTML formats. The simplest method proposed is to
use the archive file \href{http://www.ctan.org/pkg/xint}{xint.tds.zip},
downloadable from the same location.

The next simplest one is to make use of the |Makefile|, which is also
downloadable from \href{http://www.ctan.org/pkg/xint}{CTAN}. This is
for GNU/Linux systems and Mac OS X, and necessitates use of the command
line. If for some reason you have |xint.dtx| but no internet access,
you can recreate |Makefile| as a file with this name and the following
contents:

{\def\everbatimindent {0pt }%
\begin{everbatim}
include Makefile.mk
Makefile.mk: xint.dtx ; etex xint.dtx
\end{everbatim}}

Then run |make| in a working repertory where there is |xint.dtx| and the file
named |Makefile| and having only the two lines above. The |make| will extract
the package files from |xint.dtx| and display some further instructions.

If you have |xint.dtx|, no internet access and can not use the Makefile method
: |etex xint.dtx| extracts all files and among them the |README| as a file
with name |README.md|. Further help and options will be found therein.

\subsection{FAQ}

We are honoured to present here this interview with the author, dating back to
late March 2014.
\begin{description}
\item[Will \xintexprname implement \texttt{exp}, \texttt{log},
  \texttt{cos}, \texttt{sin} \dots at some point?]
I guess so.

\item[\xintseriesname already provides generic tools.] Right, although the
  casual user of the \xintname bundle will not quite know how to do variable
  reduction expandably in order to use some series or Pad\'e approximants.
  Besides I wrote the code at the beginning of the project and perhaps I could
  do it better now (I have not looked at it for a while). Anyhow, generic things
  do not help much if one wants to optimize.

\item[Optimizing? isn't \TeX's macro expansion mechanism intrinsically slow?]
  Intensive use of \csa{numexpr} and some token manipulation algorithms
  exploiting to the best I could \TeX{} macros with parameters grant \xintname a
  significant speed up in expandable arithmetic on big integers compared to
  previously available implementations. You can do some comparisons with
  multiplication on numbers with $100$ digits or division of one of $100$ digits
  by another of $50$ digits, for example. However expandability is antagonist of
  speed, and I agree it is not very exciting to optimize slow things. And I was
  disappointed last year to realize the slowness of \TeX's mouth when it has to
  keep hundreds of tokens in cheek to mix them later with new aliments. Believe
  me, I try not to think too much about the fact that the whole enterprise is
  made irrelevant by Lua\LaTeX's ability to access external libraries.

\item[Well, why isn't this \texttt{log} etc\dots thing done yet?]
I have to decide on the maximal precision to achieve: $24$, $32$, $48$,
$64$,\dots ; to settle that I would need to implement some initial versions and
benchmark them.

\item[Fair enough. That's the common lot. So why not yet?] I am a bit
  overworked. It is also an opportunity to think over
  the basic underlying mathematics, and will need devoted thinking for some not
  insignificant amount of time. So far I didn't find the time, or rather I found
  out good means to waste it sillily. I also anticipate that originality could
  very
  well not pay off at all, so small is the window for the precision.

\item[Any chance this could be done in time for TL2014?] No,
  sorry.\newline Release |1.09m| of |[2014/02/26]| was the end of a cycle, and
  this |1.09n| of |[2014/04/01]| is only for a bug fix and inclusion of this
  |FAQ| in the documentation.

\footnotesize

\item[and in time for TL2015?] ... (indistinct mumbles, something like
  \emph{too tired}, \emph{I need a life}, \emph{get yourself a calculator},
  \emph{we'll see}\dots)
\end{description}

\section{Introduction via examples}

The main goal is to allow expandable computations with integers and
fractions of arbitrary sizes.

\subsection{Printing big numbers on the page}\label{ssec:printnumber}

When producing very long numbers there is the question of printing them on
  the page, without going beyond the page limits. In this document, I have most
  of the time made use of these macros (not provided by the package:)

%
\everb|@
\def\allowsplits #1{\ifx #1\relax \else #1\hskip 0pt plus 1pt\relax
                    \expandafter\allowsplits\fi}%
\def\printnumber #1{\expandafter\allowsplits \romannumeral-`0#1\relax }%
% \printnumber thus first ``fully'' expands its argument.
|

It may be used like this:
%
\leftedline{|\printnumber {\xintiiQuo{\xintiiPow {2}{1000}}{\xintiFac{100}}}|}
%
or as |\printnumber\mybiginteger| or |\printnumber{\mybiginteger}| if
|\mybiginteger| was previously defined via a |\newcommand|, a |\def| or
an |\edef|.

An alternative is to suitably configure the thousand
separator with the \href{http://ctan.org/pkg/numprint}{numprint} package
(see \autoref{fn:np}. This will not allow linebreaks when used in math
mode; I also tried \href{http://ctan.org/pkg/siunitx}{siunitx} but even
in text mode could not get it to break numbers accross lines). Recently
I became aware of the \href{http://ctan.org/pkg/seqsplit}{seqsplit}
package%
%
\footnote{\url{http://ctan.org/pkg/seqsplit}} 
%
which can be used to achieve this splitting accross lines, and does work
in inline math mode (however it doesn't allow to separate digits by
groups of three, for example).\par

\subsection{User interface}
\label{ssec:userinterface}

The user interface for executing operations on numbers is via macros such as
\csbxint{Add} or \csbxint{Mul} which have two arguments, or via expressions
\csbxint{expr}|..\relax| which use infix notations such as |+|, |-|, |*|, |/|
and |^| (or |**|) for the basic operations, and recognize functions of one or
more comma separated arguments (such as |max|, or |round|, or |sqrt|),
parentheses, logic operators of conjunction |&&|, disjunction \verb+||+, as
well as two-way |?| and three-way |??| conditionals and more. A few examples:
%
\begin{everbatim*}
\begin{enumerate}[nosep]
  \item \xintiiAdd {2719873981798137981381789317981279}{13819093809180120910390190}
  \item \xintiiMul {2719873981798137981381789317981279}{13819093809180120910390190}
  \item \xintthefloatexpr (19317/21913+2198/9291)^3\relax
  \item \xintDigits:=64;\xintthefloatexpr (19317/21913+2198/9291)^3\relax
% Let's compute the inner sum exactly, not as a float, before raising to third power:
  \item \xintDigits:=16;\xintthefloatexpr \xintexpr 19317/21913+2198/9291\relax^3\relax
\end{enumerate}
\end{everbatim*}
In \csbxint{expr}|..\relax| the contents are expanded completely from left to
right until the ending |\relax| is found and swallowed, and spaces and even
(to some extent) catcodes do not matter. Algebraic operations are done
\emph{exactly}. 

The \csbxint{floatexpr} variant is for operations which are done using the
precision set via the |\xintDigits:=N;| assignment (default is with
significands of \dtt{16} digits).

For all |\xintexpr| variants, prefixing with |\xintthe| allows to print
the result or use it in other contexts. Shortcuts \csbxint{theexpr},
\csbxint{thefloatexpr}, \csbxint{theiiexpr}, \dots\ are available.

The \csbxint{iiexpr} variant is only for big integers, it does not know
fractions.

There are some important differences of syntax between |\numexpr| and
|\xintiiexpr| and variants:
\begin{itemize}
\item Contrarily to |\numexpr|, the |\xintiiexpr| parser will stop expanding
  only after having encountered (and swallowed) a \emph{mandatory} |\relax|
  token.
\item In particular, spaces between digits (and not only around infix
  operators or parentheses) do not stop |\xintiiexpr|, contrarily to the
  situation with |numexpr|: |\the\numexpr 7 + 3 5\relax| expands (in one step)
  to \dtt{\detokenize\expandafter{\the\numexpr 7 + 3 5\relax}\unskip}, whereas
  |\xintthe\xintiiexpr 7 + 3 5\relax| expands (in two steps) to
  \dtt{\detokenize\expandafter\expandafter\expandafter {\xintthe\xintiiexpr 7
      + 3 5\relax}}.
  \item Also worth mentioning is the fact that |\numexpr -(1)\relax| is
    illegal. But this is perfectly legal and with the expected result in
    |\xintiiexpr...\relax|.
  \item Inside an |\edef|, expressions |\xintiiexpr...\relax| get fully
    evaluated, but need the prefix |\xintthe| to get printed or used as
    arguments to some macros, whereas expansion of |\numexpr| in an |\edef|
    occurs only if prefixed with |\the| or |\number| (except if in a bigger
    |\numexpr| which will be the one to have to be prefixed\dots .)
\end{itemize}

For macros such as \csbxint{Add} or
\csbxint{Mul} the arguments are each subjected to the process of \fexpan sion:
repeated expansion of the first token until finding something unexpandable (or
being stopped by a space token).

Conversely this process of \fexpan sion always provokes the complete expansion
of the package macros and |\xintexpr..\relax| also will expand completely
under \fexpan sion, but to a private format; the \csbxint{the} prefix allows the
computation result either to be passed as argument to one of the package
macros,\footnote{the \csa{xintthe} prefix \fexpan ds the \csa{xintexpr}-ession
  then unlocks it from its private format; it should not be used for
  sub-expressions inside a bigger one as it is more efficient for the
  expression parser to keep the result in the private format.} or also end up on
the printed page (or in an auxiliary file).
To recapitulate, all macros dealing with computations
\begin{enumerate}
\item \emph{expand completely under the sole process of repeated expansion of
    the first token, (and two expansions suffice)},\footnote{see in
    \autoref{sec:expansions} for more details.}

\item \emph{apply this \fexpan sion to each one of their arguments.}
\end{enumerate}
Hence they can be nested one within the other up to arbitrary
depths. Conditional evaluations either within the macro arguments themselves, or
with branches defined in terms of these macros are made possible via macros such
as as \csbxint{ifSgn} or \csbxint{ifCmp}.

\begin{framed}
  There is no notion of \emph{declaration of a variable} to \xintname,
  \xintfracname, or \xintexprname.
   The user employs the |\def|, |\edef|, or
  |\newcommand| (in \LaTeX) as usual, for example: \IMPORTANT
%
\begin{everbatim*}
\def\x{1729728} \def\y{352827927} \edef\z{\xintMul {\x}{\y}}
\meaning\z
\end{everbatim*}\ (see below for the |A/B[N]| output format; with |\xintiiMul|
in place of |\xintMul| there would not be the strange looking |/1[0]|.)
 
As a faster alternative to |\edef| (when hundreds of digits are involved), the
package provides |\oodef| which only expands twice its argument. This provokes
full expansion of the \xintname \fexpan dable macros (nested to possibly many
levels), inclusive of |\xintexpr| and variants.
\end{framed}

\begingroup % pour \z, \zz
The \xintexprname package has a private internal
representation for the evaluated computation result. With
%
\begin{everbatim*}
\oodef\z {\xintexpr 3.141^18\relax}
\end{everbatim*}
%
the macro |\z| is already fully evaluated (two expansions were applied, and this
is enough), and can be reused in other |\xintexpr|-essions, such as for example
%
\begin{everbatim*}
\edef\zz {\xintexpr \z+1/\z\relax}
  % (using short macro names such as \z and \zz is not too recommended in real
  % life, some may have already definitions; I did it all in a group).
\end{everbatim*}
%
But to print it, or to use it as argument to one of the package macros,
it must be prefixed by |\xintthe| (a synonym for |\xintthe\xintexpr| is
\csbxint{theexpr}). Application of this |\xintthe| prefix outputs the
value in the \xintfracname semi-private internal format
|A/B[N]|,\footnote{there is also the notion of \csbxint{floatexpr}, for
  which the output format after the action of \csa{xintthe} is a number in
  floating point scientific notation.} representing the fraction
$(A/B)\times 10^N$. The |\zz| above produces a somewhat large output:
\begin{everbatim*}
\printnumber{\xintthe\zz }${}\approx{}$\xintFloat{\xintthe\zz}
\end{everbatim*}
\endgroup % pour \z, \zz

  \begin{framed}
    By default, computations done by the macros of \xintfracname or within
    |\xintexpr..\relax| are exact. Inputs containing decimal points or
    scientific parts do not make the package switch to a `floating-point' mode.
    The inputs, however long, are converted into exact internal representations.
%
    % Floating point evaluations are done via special macros containing
    % `Float' in their names, or inside |\xintfloatexpr|-essions.

    Manipulating exactly big fractions quickly leads to \dots bigger fractions.
    There is a command \csbxint{Irr} (or the function |reduce| in an expression)
    to reduce to smallest terms, but it has to be explicitely requested. Prior
    to release |1.1| addition and subtraction blindly multiplied denominators;
    they now check if one is a multiple of the other.\IMPORTANT\ But systematic
    reduction of the result to its smallest terms would be too
    costly.\def\everbatimindent{0pt }
\begin{everbatim*}
\xinttheexpr 27/25+46/50\relax\ is a bit simpler than \xinttheexpr (27*50+25*46)/(25*50)\relax, 
but less so than \xinttheexpr reduce(27/25+46/50)\relax. And \xinttheexpr 3/75+4/50+2/100\relax\
looks weird, but systematically reducing fractions would be too costly. 
\end{everbatim*}
  \end{framed}

%
The |A/B[N]| shape is the output format of most \xintfracname macros, it
benefits from accelerated parsing when used on input, compared to the normal
user syntax which has no |[N]| part. An example of valid user input for a
fraction is
%
\leftedline{|-123.45602e78/+765.987e-123|}
%
where both the decimal parts, the scientific exponent parts, and the whole
denominator are optional components. The corresponding semi-private form in this
case would be
%
\leftedline{\xintRaw{-123.45602e78/+765.987e-123}}
%
The forward slash |/| is simply a delimiter to separate numerator and
denominator, in order to allow inputs having such denominators.

Reduction to the irreducible form of the output must be asked for explicitely
via the \csbxint{Irr} macro or the |reduce| function within
|\xintexpr..\relax|. Elementary operations on fractions do very little of the
simplifications which could be obvious to (some) human beings.


\subsection{Randomly chosen examples}

Here are some examples of use of the package macros. The first one uses only
the base module \xintname, the next two require the \xintfracname package,
which deals with fractions. Then two examples with the \xintgcdname package,
one with the \xintseriesname package, and finally a computation with a float.
Some inputs are simplified by the use of the \xintexprname package.

\begin{itemize}
\item {$123456^{99}$: }\\
|\xintiiPow {123456}{99}|: 
\dtt{\printnumber{\xintiiPow {123456}{99}}}

\item {1234/56789 with 1500 digits after the decimal point: }\\
|\xintTrunc {1500}{1234/56789}\dots|:
\dtt{\printnumber {\xintTrunc {1500}{1234/56789}}\dots }

\item {$0.99^{-100}$ with 200 digits after the decimal point:}\\
|\xinttheexpr trunc(.99^-100,200)\relax\dots|:
\dtt{\printnumber{\xinttheexpr trunc(.99^-100,200)\relax}\dots }


\item Just to show off (again), let's print 300 digits (after the decimal
  point) of the decimal expansion of $0.7^{-25}$:%
%
\footnote{the |\np| typesetting macro is from the |numprint| package.}
%
\begin{everbatim*}
% % in the preamble:
% \usepackage[english]{babel}
% \usepackage[autolanguage,np]{numprint}
% \npthousandsep{,\hskip 1pt plus .5pt minus .5pt}
% \usepackage{xintexpr}
% in the body:
\np {\xinttheexpr trunc(.7^-25,300)\relax}\dots
\end{everbatim*}

This computation is with \csbxint{theexpr} from package \xintexprname, which
allows to use standard infix notations and function names to access the package
macros, such as here |trunc| which corresponds to the \xintfracname macro
\csbxint{Trunc}. 

\begin{snugframed}
  The fraction |0.7^-25| is first evaluated \emph{exactly}; for some more
  complex inputs, such as |0.7123045678952^-243|, the exact evaluation before
  truncation would be rather costly, and one would rather use floating point
  numbers: 
%
\leftedline{|\xintDigits:=20;
               \np{\xintthefloatexpr .7123045678952^-243\relax}|}%
%
\leftedline{\xintDigits:=20;\dtt{\np{\xintthefloatexpr .7123045678952^-243\relax }}} 
%
Side note: the exponent |-243| didn't have to be put inside parentheses,
contrarily to what happens with some professional computational
software. |;-)|
% 6.342,022,117,488,416,127,3  10^35
% maple n'aime pas ^-243 il veut les parenthèses, bon et il donne, en Digits
% = 24: 0.634202211748841612732270 10^36
\end{snugframed}

\xintDigits:=16;

\item Computation of a Bezout identity with  |7^200-3^200| and |2^200-1|:
(with \xintgcdname)\par
\everb|@
\xintAssign \xintBezout {\xinttheiiexpr 7^200-3^200\relax}
                       {\xinttheiiexpr 2^200-1\relax}\to\A\B\U\V\D
$\U\times(7^{200}-3^{200})+\xintiOpp\V\times(2^{200}-1)=\D$
|

\xintAssign \xintBezout {\xinttheiiexpr 7^200-3^200\relax}%
                            {\xinttheiiexpr 2^200-1\relax}\to\A\B\U\V\D
\dtt
{\printnumber\U$\times(7^{200}-3^{200})+{}$%
 \printnumber{\xintiOpp\V}$\times(2^{200}-1)={}$\printnumber\D}

% 11 octobre 2014, je modifie juste d'une unité le deuxième... plus joli.
\item The Euclide algorithm applied to \np{22206980239027589097} and
\np{8169486210102119257}: (with \xintgcdname)%
%
\footnote {this example is computed tremendously faster than the other
  ones, but we had to limit the space taken by the output hence picked
  up rather small big integers as input.}\par
\noindent\begingroup\parskip0pt\relax
|\xintTypesetEuclideAlgorithm {22206980239027589097}{8169486210102119257}|\par
\dtt
{\xintTypesetEuclideAlgorithm {22206980239027589097}{8169486210102119257}} 
\endgroup
\smallskip

\item $\sum_{n=1}^{500} (4n^2 - 9)^{-2}$ with each term rounded to twelve digits,
and the sum to nine digits:
\begin{everbatim*}
\def\coeff #1{\xintiRound {12}{1/\xintiiSqr{\the\numexpr 4*#1*#1-9\relax }[0]}}
\xintRound {9}{\xintiSeries {1}{500}{\coeff}[-12]}
\end{everbatim*}

The complete series, extended to
infinity, has value
$\frac{\pi^2}{144}-\frac1{162}={}$%
\dtt{\np{0.06236607994583659534684445}\dots}\,%
%
\footnote{\label{fn:np}This number is typeset using the
  \href{http://www.ctan.org/pkg/numprint}{numprint} package, with
  |\npthousandsep{,\hskip 1pt plus .5pt minus .5pt}|. But the breaking
  across lines works only in text mode. The number itself was (of
  course...) computed initially with \xintname, with 30 digits of $\pi$
  as input. See \hyperref[ssec:Machin]{{how {\xintname} may compute
      $\pi$ from scratch}}.}
%
I also used (this is a lengthier computation
than the one above) \xintseriesname to evaluate the sum with \np{100000} terms,
obtaining 16
correct decimal digits for the complete sum. The
coefficient macro must be redefined to avoid a |\numexpr| overflow, as
|\numexpr| inputs must not exceed $2^{31}-1$; my choice
was:
\everb|@
\def\coeff #1%
{\xintiRound {22}{1/\xintiiSqr{\xintiiMul{\the\numexpr 2*#1-3\relax}
                                         {\the\numexpr 2*#1+3\relax}}[0]}}
|

\restoreMacroFont
\edef\Temp {\xintFloatPow [24]{2}{999999999}}

\item {Computation of $2^{\np{999999999}}$ with |24| significant
  figures:}
%
\leftedline{|\numprint{\xintFloatPow [24]{2}{999999999}}|}
\leftedline{\dtt{\numprint{\Temp}}}
%
where the \href{http://www.ctan.org/pkg/numprint}{numprint} package was used
(\autoref{fn:np}), directly in text mode (it can also naturally be used from
inside math mode). \xintname provides a simple-minded \csbxint{Frac}
typesetting macro,%
%
\footnote{Plain \TeX{} users of \xintname have \csbxint{FwOver}.}
%
which is math-mode only:
%
\leftedline{|$\xintFrac{\xintFloatPow [24]{2}{999999999}}$|}
\leftedline{\dtt{$\xintFrac{\Temp}$}}
%
The exponent differs, but this is because
|\xintFrac| does not use a decimal mark in the significand of the output.
Admittedly most users will have the need of more powerful (and customizable)
number formatting macros than |\xintFrac|.
%
\footnote{There should be a |\xintFloatFrac|, but it is lacking.}
%
We have already mentioned
|\numprint| which is used above, there is also |\num| from package
\href{http://www.ctan.org/pkg/siunitx}{siunitx}. The raw output from
%
\leftedline{\detokenize{\xintFloatPow[24]{2}{999999999}}}
%
is $\Temp$.

\edef\x{\xintiiQuo{\xintiiPow {2}{1000}}{\xintiFac{100}}}
\edef\y{\xintLen{\x}}

\item As an example of nesting package macros, let us consider the following
code snippet within a file with filename |myfile.tex|:
\everb|@
\newwrite\outstream
\immediate\openout\outstream \jobname-out\relax
\immediate\write\outstream {\xintiiQuo{\xintiiPow{2}{1000}}{\xintiFac{100}}}
% \immediate\closeout\outstream
|
\noindent
The tex run creates a file |myfile-out.tex|, and then writes to it the
quotient from the euclidean division of $2^{1000}$ by $100!$. The number of
digits is |\xintLen{\xintiiQuo{\xintiiPow{2}{1000}}{\xintiFac{100}}}| which
expands (in two steps) and tells us that $[2^{1000}/100!]$ has \dtt{\y}
digits. This is not so many, let us print them here:
\dtt{\printnumber\x}.%
%
% \footnote{See \autoref{ssec:printnumber} and \hyperref[fn:np]{a previous
%     footnote}.} 

\end{itemize}

\subsection {More examples, some quite elaborate, within this document}
\label{sec:awesome}

\begin{itemize}
\item The utilities provided by \xinttoolsname (\autoref{sec:tools}), some
  completely expandable, others not, are of independent interest. Their use
  is illustrated through various examples: among those, it is shown in
  \autoref{ssec:quicksort} how to implement in a completely expandable way
  the \hyperref[quicksort]{Quick Sort algorithm} and also how to illustrate
  it graphically. Other examples include some dynamically constructed
  alignments with automatically computed prime number cells: one using a
  completely expandable prime test and \csbxint{ApplyUnbraced}
  (\autoref{ssec:primesI}), another one with \csbxint{For*} (\autoref{ssec:primesIII}).

\item  One has also a \hyperref[edefprimes]{computation of primes within an
    \csa{edef}} (\autoref{xintiloop}), with the help of \csbxint{iloop}.
  Also with \csbxint{iloop} an
  \hyperref[ssec:factorizationtable]{automatically generated table of
    factorizations} (\autoref{ssec:factorizationtable}).

\item  The code for the title page fun with Fibonacci numbers is given in
  \autoref{ssec:fibonacci} with \csbxint{For*} joining the game.

\item  The computations of \hyperref[ssec:Machin]{ $\pi$ and $\log 2$}
  (\autoref{ssec:Machin}) using \xintname and the computation of the
  \hyperref[ssec:e-convergents]{convergents of $e$} with the further help of
  the \xintcfracname package are among further examples. 

\item There is also an
  example of an \hyperref[xintXTrunc]{interactive session}, where results
  are output to the log or to a file.

\item The new functionalities of \xintexprname are illustrated with various
  examples in \autoref{sec:expr11}.
\end{itemize}
Almost all of the computational results interspersed throughout the
documentation are not hard-coded in the source of the document. They are
the result of evaluation of the package macros, and were selected to not
impact too much the compilation time of this documentation.
Nevertheless, there are so many computations done that compilation time
is significantly increased compared to a \LaTeX\ run on a typical
document of about the same size.

\section{The \xintname bundle}

\subsection{General overview}

The main characteristics are:
\begin{enumerate}
\item exact algebra on arbitrarily big numbers, integers as well as fractions,
\item floating point variants with user-chosen precision,
\item implemented via macros compatible with expansion-only
  context.
\end{enumerate}

`Arbitrarily big' means with less than
    |2^31-1|\dtt{=\number"7FFFFFFF} digits, as most of the macros will
    have to compute the length of the inputs and these lengths must be treatable
    as \TeX{} integers, which are at most \dtt{\number "7FFFFFFF}
    in absolute value.
    This is a distant irrelevant upper bound, as no such thing can fit
    in \TeX's memory! And besides,
the true limitation is from the \emph{time} taken by the
expansion-compatible algorithms, as will be commented upon soon.

As just recalled, ten-digits numbers starting with a $3$ already exceed the
\TeX{} bound on integers; and \TeX{} does not have a native processing of
floating point numbers (multiplication by a decimal number of a dimension
register is allowed --- this is used for example by the
\href{http://mirror.ctan.org/graphics/pgf/base}{pgf} basic math
engine.)

\TeX{} elementary operations on numbers are done via the non-expandable
\emph{advance, multiply, \emph{and} divide} assignments. This was changed with
\eTeX{}'s |\numexpr| which does expandable computations using standard infix
notations with \TeX{} integers. But \eTeX{} did not modify the \TeX{} bound on
acceptable integers, and did not add floating point support.

The \href{http://www.ctan.org/pkg/bigintcalc}{bigintcalc} package by
\textsc{Heiko Oberdiek} provided expandable operations (using some of |\numexpr|
possibilities, when available) on arbitrarily big integers, beyond the \TeX{}
bound. The present package does this again, using more of |\numexpr| (\xintname
requires the \eTeX{} extensions) for higher speed, and also on fractions, not
only integers. Arbitrary precision floating points operations are a derivative,
and not the initial design goal.%
%
\footnote{currently (|v1.08|), the only non-elementary operation
  implemented for floating point numbers is the square-root extraction;
  no signed infinities, signed zeroes, |NaN|'s, error trapes\dots, have
  been implemented, only the notion of `scientific notation with a given
  number of significant figures'.}%
%
${}^{\text{,\,}}$%
%
\footnote{multiplication of two floats with |P=\xinttheDigits| digits is
  first done exactly then rounded to |P| digits, rather than using a
  specially tailored multiplication for floating point numbers which
  would be more efficient (it is a waste to evaluate fully the
  multiplication result with |2P| or |2P-1| digits.)}

The \LaTeX3 project has implemented expandably floating-point computations with
16 significant figures
(\href{http://www.ctan.org/pkg/l3kernel}{l3fp}), including
special functions such as exp, log, sine and cosine.%
%
\footnote{at the time of writing the
  \href{http://www.ctan.org/pkg/l3kernel}{l3fp} (exactly represented)
  floating point numbers have their exponents limited to
  $\pm$\dtt{9999}.} There is also
  \href{http://latex-project.org/svnroot/experimental/trunk/l3trial/l3bigint}{l3bigint},
  which (a.t.t.o.w.) is part of the experimental trunk of the
  \href{http://latex-project.org}{\LaTeX3 Project}. Like
  \href{http://www.ctan.org/pkg/bigintcalc}{bigintcalc} and \xintname it
  provides macros for big integer arithmetics. All three implementations of the
  basic arithmetic macros can be mapped to easier infix notations via the
  services of the \href{http://www.ctan.org/pkg/bnumexpr}{bnumexpr} package.

The \xintname package can be used for $24$, $40$, etc\dots{} significant figures
but one rather quickly (not much beyond $100$ figures) hits against a
`wall' created by the constraint of expandability: currently, multiplying out
two one-hundred digits numbers takes circa $80$ or $90$ times longer than for
two ten-digits numbers, which is reasonable, but multiplying out two
one-thousand digits numbers takes more than $500$ times longer than for two one
hundred-digits numbers. This shows that the algorithm is drifting from quadratic
to cubic in that range. On my laptop multiplication of two $1000$-digits numbers
takes some seconds, so it can not be done routinely in a
document.%
%
\footnote{without entering into too much technical details, the source
  of this `wall' is that when dealing with two long operands, when one wants to
  pick some digits from the second one, one has to jump above all digits
  constituting the first one, which can not be stored away: expandability
  forbids assignments to memory storage. One may envision some sophisticated
  schemes, dealing with this problem in less naive ways, trying to move big
  chunks of data higher up in the input stream and come back to it later,
  etc...; but each `better' algorithm adds overhead for the smaller inputs. For
  example, I have another version of addition which is twice faster on inputs
  with 500 digits or more, but it is slightly less efficient for 50 digits or
  less. This `wall' dissuaded me to look into implementing `intelligent'
  multiplication which would be sub-quadratic in a model where storing and
  retrieving from memory would not cost much.}

The conclusion perhaps could be that it is in the end lucky that the speed gains
brought by \xintname for expandable operations on big numbers do open some
non-empty range of applicability in terms of the number of kept digits for
routine floating point operations.

The second conclusion, somewhat depressing after all the hard work, is
that if one really wants to do computations with \emph{hundreds} of digits, one
should drop the expandability requirement. And indeed, as clearly
demonstrated long ago by the \href{http://www.ctan.org/pkg/pi}{pi
  computing file} by \textsc{D. Roegel} one can program \TeX{} to
compute with many digits at a much higher speed than what \xintname
achieves: but, direct access to memory storage in one form or another
seems a necessity for this kind of speed and one has to renounce at the
complete expandability.%
%
\footnote{I could, naturally, be proven wrong!}\,%
%
\footnote{The Lua\TeX{} project possibly makes endeavours such as
  \xintname appear even more insane that they are, in truth.}

% \section{Missing things}

% `Arbitrary-precision' floating-point
% operations are currently limited to the basic four operations, the power
% function with integer exponent, and the extraction of square-roots.

\subsection{Origins of the package}

Package |bigintcalc| by \textsc{Heiko Oberdiek} already
provides expandable arithmetic operations on ``big integers'',
exceeding the \TeX{} limits (of $2^{31}-1$), so why another%
%
\footnote{this section was written before the \xintfracname package; the
  author is not aware of another package allowing expandable
  computations with arbitrarily big fractions.}
%
one?

I got started on this in early March 2013, via a thread on the
|c.t.tex| usenet group, where \textsc{Ulrich D\,i\,e\,z} used the
previously cited package together with a macro (|\ReverseOrder|)
which I had contributed to another thread.%
%
\footnote{the \csa{ReverseOrder} could be avoided in that circumstance,
  but it does play a crucial r\^ole here.}
%
What I had learned in this
other thread thanks to interaction with \textsc{Ulrich D\,i\,e\,z} and
\textsc{GL} on expandable manipulations of tokens motivated me to
try my hands at addition and multiplication.

I wrote macros \csa{bigMul} and \csa{bigAdd} which I posted to the
newsgroup; they appeared to work comparatively fast. These first
versions did not use the \eTeX{} \csa{numexpr} primitive, they worked
one digit at a time, having previously stored carry-arithmetic in
1200 macros.

I noticed that the |bigintcalc| package used \csa{numexpr}
if available, but (as far as I could tell) not
to do computations many digits at a time. Using \csa{numexpr} for
one digit at a time for \csa{bigAdd} and \csa{bigMul} slowed them
a tiny bit but avoided cluttering \TeX{} memory with the 1200
macros storing pre-computed digit arithmetic. I wondered if some speed
could be gained by using \csa{numexpr} to do four digits at a time
for elementary multiplications (as the maximal admissible number
for \csa{numexpr} has ten digits).

The present package is the result of this initial questioning.

% \begin{framed}\centering
%   \xintname requires the \eTeX{} extensions.
% \end{framed}

\subsection{Expansion matters}
\label{sec:expansions}

By convention in this manual \fexpan sion (``full expansion'' or ``full first
expansion'') is the process of expanding repeatedly the first token seen until
hitting against something not further expandable like an unexpandable
\TeX-primitive or an opening brace |{| or a character (inactive). For
  those familiar with \LaTeX3 (which is not used by \xintname) this is what is
  called in its documentation full expansion. Technically, macro arguments in
  \xintname which are submitted to such a \fexpan sion are so via prefixing them
  with |\romannumeral-`0|. An explicit or implicit space token stops such an
  expansion and is gobbled.
%
Most of the package macros, and all those dealing with computations, are
expandable in the strong sense that they expand to their final result via this
\fexpan sion. Again copied from \LaTeX3 documentation conventions, this will be
signaled in the description of the macro by a \etype{}star in the margin.
All%
%
\footnote{except \csbxint{loop} and \csbxint{iloop}.}
%
expandable macros of the \xintname packages completely expand in two steps.

Furthermore the macros dealing with computations, as well as many utilities from
\xinttoolsname, apply this process of \fexpan sion to their arguments. Again
from \LaTeX3's conventions this will be signaled by a%
%
\ntype{{\setbox0 \hbox{\Ff}\hbox to \wd0 {\hss f\hss}}}
%
margin annotation. Some additional parsing
which is done by most macros of \xintname is indicated with a
variant\ntype{\Numf{\kern.5cm}}; and the extended fraction parsing done by most
macros of \xintfracname has its own symbol\ntype{\Ff}. When the argument has a
priori to obey the \TeX{} bound of \dtt{\number"7FFFFFFF} it is
systematically fed to a |\numexpr..\relax| hence the expansion is then a
\emph{complete} one, signaled with an \ntype{\numx}\emph{x} in the margin. This
means not only complete expansion, but also that spaces are ignored, infix
algebra is possible, count registers are allowed, etc\dots

The \csbxint{ApplyInline} and \csbxint{For*} macros from \xinttoolsname
apply a special iterated \fexpan sion, which gobbles spaces, to all
those items which are found \emph{unbraced} from left to right in the
list argument; this is denoted specially as here\ntype{{\lowast f}} in
the margin. Some other macros such as \csbxint{Sum} from \xintfracname
first do an \fexpan sion, then treat each found (braced or not) item
(skipping spaces between such items) via the general fraction input
parsing, this is signaled as here\ntype{f{$\to$}{\lowast\Ff}} in the
margin where the signification of the \lowast{} is thus a bit different
from the previous case.

A few macros from \xinttoolsname do not expand, or expand only once their
argument\ntype{n{{\color{black}\upshape, resp.}} o}. This is also
signaled in the margin with notations \`a la \LaTeX3.

As the computations are done by \fexpan dable macros which \fexpan d their
argument they may be chained up to arbitrary depths and still produce expandable
macros.

Conversely, wherever the package expects on input a ``big'' integers, or a
``fraction'', \fexpan sion of the argument \emph{must result in a complete
  expansion} for this argument to be acceptable.%
%
\footnote{this is not quite as
  stringent as claimed here, see \autoref{sec:useofcount} for more details.}
The
main exception is inside \csbxint{expr}|...\relax| where everything will be
expanded from left to right, completely.

Summary of important expansion aspects:
\begin{enumerate}
\item the macros \fexpan d their arguments, this means that they expand the
  first token seen (for each argument), then expand, etc..., until something
  un-expandable such as a\strut{} digit or a brace is hit against. This
  example
%
  \leftedline{|\def\x{98765}\def\y{43210}| |\xintAdd {\x}{\x\y}|} 
%
  is \emph{not} a legal construct, as the |\y| will remain untouched by
  expansion and not get converted into the digits which are expected by the
  sub-routines of |\xintAdd|. It is a |\numexpr| which will expand it and an
  arithmetic overflow will arise as |9876543210| exceeds the \TeX{} bounds.

  \begingroup\slshape
  With \csbxint{theexpr} one could write |\xinttheexpr \x+\x\y\relax|, or
  |\xintAdd\x{\xinttheexpr\x\y\relax}|.\hfill
  \endgroup

\item\label{fn:expansions} using |\if...\fi| constructs \emph{inside} the
  package macro arguments requires suitably mastering \TeX niques
  (|\expandafter|'s and/or swapping techniques) to ensure that the \fexpan sion
  will indeed absorb the \csa{else} or closing \csa{fi}, else some error will
  arise in further processing. Therefore it is highly recommended to use the
  package provided conditionals such as \csbxint{ifEq}, \csbxint{ifGt},
  \csbxint{ifSgn}, \csbxint{ifOdd}\dots, or, for \LaTeX{} users and when dealing
  with short integers the
  \href{http://www.ctan.org/pkg/etoolbox}{etoolbox}%
%
\footnote{\url{http://www.ctan.org/pkg/etoolbox}}
  expandable conditionals (for small integers only) such as \texttt{\char92
    ifnumequal}, \texttt{\char92 ifnumgreater}, \dots . Use of
  \emph{non-expandable} things such as \csa{ifthenelse} is impossible inside the
  arguments of \xintname macros.

  \begingroup\slshape
  One can use naive |\if..\fi| things inside an \csbxint{theexpr}-ession
  and cousins,  as long as the test is
  expandable, for example\upshape
%
\leftedline{|\xinttheiexpr\ifnum3>2 143\else 33\fi
  0^2\relax|$\to$\dtt{\xinttheiexpr \ifnum3>2 143\else 33\fi 0^2\relax
    =1430\char`\^2}}
%
  \endgroup

\item after the definition |\def\x {12}|, one can not use
  {\color{blue}|-\x|} as input to one of the package macros: the \fexpan sion
  will act only on the minus sign, hence do nothing. The only way is to use the
  \csbxint{Opp} macro, or perhaps here rather \csbxint{iOpp} which does
    maintains integer format on output, as they  replace a number with
    its opposite.

  \begingroup\slshape
  Again, this is otherwise inside an \csbxint{theexpr}-ession or
  \csbxint{thefloatexpr}-ession. There, the
  minus sign may prefix macros which will expand to numbers (or parentheses
  etc...)
  \endgroup

\def\x {12}%
\def\AplusBC #1#2#3{\xintAdd {#1}{\xintMul {#2}{#3}}}%

\item \label{item:xpxp} With the definition 
%
\leftedline{|\def\AplusBC #1#2#3{\xintAdd {#1}{\xintMul {#2}{#3}}}|}
%
one obtains an
  expandable macro producing the expected result, not in two, but rather in
  three steps: a first expansion is consumed by the macro expanding to its
  definition. As the package macros expand their arguments until no more is
  possible (regarding what comes first), this |\AplusBC| may be used inside
  them: {|\xintAdd {\AplusBC {1}{2}{3}}{4}|} does work and returns
  \dtt{\xintAdd {\AplusBC {1}{2}{3}}{4}}.

  If, for some reason, it is important to create a macro expanding in two steps
  to its final value, one may either do:
%
\smallskip
%
\leftedline {|\def\AplusBC #1#2#3{\romannumeral-`0\xintAdd {#1}{\xintMul
      {#2}{#3}}}|}
%
or use the \emph{lowercase} form of \csa{xintAdd}: 
%
\smallskip
%
\leftedline {|\def\AplusBC #1#2#3{\romannumeral0\xintadd {#1}{\xintMul
      {#2}{#3}}}|}

  and then \csa{AplusBC} will share the same properties as do the
  other \xintname `primitive' macros.

\end{enumerate}

The |\romannumeral0| and |\romannumeral-`0| things above look like an invitation
to hacker's territory; if it is not important that the macro expands in two
steps only, there is no reason to follow these guidelines. Just chain
arbitrarily the package macros, and the new ones will be completely expandable
and usable one within the other.

Since release |1.07| the \csbxint{NewExpr} command automatizes the creation of
such expandable macros: 
%
\leftedline{|\xintNewExpr\AplusBC[3]{#1+#2*#3}|}
%
creates the |\AplusBC| macro doing the above and expanding in two expansion
steps.

\subsection{Efficiency; floating point macros}

The size of the manipulated numbers is limited by two
factors:\footnote{there is an intrinsic limit of
  \dtt{\number"7FFFFFFF} on the number of digits, but it is
  irrelevant, in view of the other limiting factors.} (1.)~\emph{the
available memory as configured in the |tex| executable},
(2.)~\emph{the \emph{time} necessary to fully expand the computations
  themselves}. 
The most limiting factor is the second one, the time
needed (for multiplication and division, and even more for powers)
explodes with increasing input sizes long before the computations could
get limited by constraints on \TeX's available memory:
computations with $100$ digits are still reasonably fast, but the
situation then deteriorates swiftly, as it takes of the order of seconds (on my
laptop) for the package to multiply exactly two numbers each of $1000$ digits
and it would take hours for numbers each of $20000$ digits.\footnote{Perhaps
  some faster routines could emerge from an approach which, while maintaining
  expandability would renounce at \fexpan dability (without impacting the input
  save stack). There is one such routine \csbxint{XTrunc} which is able to write
  to a file (or inside an \csa{edef}) tens of thousands of digits of a
  (reasonably-sized) fraction.}

To address this issue, floating
point macros are provided to work with a given arbitrary precision. The default
size for significands is $16$ digits. Working with significands of $24$, $32$,
$48$, $64$, or even $80$ digits is well within the reach of the package. But
routine multiplications and divisions will become too slow if the precision goes
into the hundreds, although the syntax to set it (|\xintDigits:=P;|) allows
values up to $32767$.\footnote{for a one-shot conversion of a fraction to float
  format, or one addition, a precision exceeding \dtt{32767} may be passed
  as optional argument to the used macro.} The exponents may be as big as
\dtt{$\pm$\number"7FFFFFFF}.\footnote{almost\dots{} as inner manipulations
  may either add or subtract the precision value to the exponent, arithmetic
  overflow may occur if the exponents are a bit too close to the \TeX{} bound
  \dtt{$\pm$\number"7FFFFFFF}.}

Here is such a floating point computation:
%
\leftedline{|\xintFloatPower [48] {1.1547}{\xintiiPow {2}{35}}|}
%
which thus computes $(1.1547)^{2^{35}}=(1.1547)^{\xintiiPow {2}{35}}$ to be
approximately 
%
\leftedline{\dtt{\np{\xintFloatPower [48] {1.1547}{\xintiiPow {2}{35}}}}}
%
Notice that $2^{35}$ exceeds \TeX's bound, but \csa{xintFloatPower} allows it,
what counts is the exponent of the result which, while dangerously close to
$2^{31}$ is not quite there yet. The printing of the result was done via the
|\numprint| command from the \href{http://ctan.org/pkg/numprint}{numprint}
package\footnote{\url{http://ctan.org/pkg/numprint}}.

The same computation can be done via the non-expandable assignment
|\xintDigits:=48;| and then
%
\leftedline{|\xintthefloatexpr 1.1547^(2^35)\relax|}
%
Notice though that |2^35| will be evaluated as a
floating point number, and if the floating point precision had been too
low, this computation would have given an inexact value. It is safer,
and also more efficient to code this as:
%
\leftedline{|\xintthefloatexpr 1.1547^\xintiiexpr 2^35\relax\relax|}
%
The \csbxint{iiexpr} is a cousin of \csbxint{expr} which is big integer-only
and skips the overhead of fraction management. Notice on this example that
being embedded inside the |floatexpr|-ession has nil influence on the
|iiexpr|-ession: expansion proceeds in exactly the same way as if it had been
at the `top' level.

\xintexprname provides \emph{no} implementation of the |IEEE| standard:
no |NaN|s, signed infinities, signed zeroes, error traps, \dots; what is
achieved though is exact rounding for the basic operations. The only
non-algebraic operation currently implemented is square root extraction.
The power functions (there are three of them: \csbxint{Pow} to which |^|
is mapped in |\xintexpr..\relax|, \csbxint{FloatPower} for |^| in
|\xintfloatexpr..\relax|, and \csbxint{FloatPow} which is slighty faster
but limits the exponent to the \TeX{} bound) allow only integral
exponents.


\section{User interface}

Maintaining complete expandability is not for the faint of heart as it excludes
doing macro definitions in the midst of the computation; in many cases, one does
not need complete expandability, and definitions are allowed. In such contexts,
there is no declaration for the user to be made to the package of a ``typed
variable'' such as a long integer, or a (long) fraction, or possibly an
|\xintexpr|-ession. Rather, the user has at its disposal the general tools of
the \TeX{} language: |\def| and |\edef|.  In \LaTeX\ there is |\newcommand| as
wrapper to |\def|,
but \LaTeX\ chose not to provide an analogous wrapper for |\edef|. It can still
be used directly of course.\footnote{I don't know if \LaTeX3 will still allow
  direct use of |\def| and |\edef|\dots} 

The \xinttoolsname package provides |\oodef| which expands twice the replacement
text\footnote{only for parameter less undelimited macros.}, hence forces
complete expansion when the top level of this replacement 
text is a call to one of the \xintname bundle macros, its arguments being
themselves chains of such macros. There is also |\fdef| which will apply \fexpan
sion to the replacement text. Both are in such uses faster alternatives to
|\edef|.

This section will explain the various inputs which are recognized by the package
macros and the format for their outputs. Inputs have mainly five possible
shapes:
\begin{enumerate}
\item expressions which will end up inside a |\numexpr..\relax|,

\item long integers in the strict format (no |+|, no leading zeroes, a count
  register or variable must be prefixed by |\the| or |\number|)

\item long integers in the general format allowing both |-| and |+| signs, then
  leading zeroes, and a count register or variable without prefix is allowed,

\item fractions with numerators and denominators as in the
  previous item, or also decimal numbers, possibly in scientific notation (with
  a lowercase |e|), and
  also optionally the semi-private |A/B[N]| format,

\item and finally expandable material understood by the |\xintexpr| parser.
\end{enumerate}
Outputs are mostly of the following types:
\begin{enumerate}
\item long integers in the strict format,

\item fractions in the |A/B[N]| format where |A| and |B| are both strict long
  integers, and |B| is positive,

\item numbers in scientific format (with a lowercase |e|),

\item the private |\xintexpr| format which needs the |\xintthe| prefix in order
  to end up on the printed page (or get expanded in the log)
  or be used as argument to the package macros.
\end{enumerate}

{\etocdefaultlines\etocsettocstyle{}{}\localtableofcontents}

\subsection {Input formats}\label{sec:inputs}

% \edef\z {\xintAdd
%     {+--0367.8920280/-++278.289287}{-109.2882/+270.12898}}

Some macro arguments are by nature `short' integers,\ntype{\numx} \emph{i.e.}
less than (or equal to) in absolute value \np{\number "7FFFFFFF}. This is
generally the case for arguments which serve to count or index something. They
will be embedded in a |\numexpr..\relax| hence on input one may even use count
registers or variables and expressions with infix operators. Notice though that
|-(..stuff..)| is surprisingly not legal in the |\numexpr| syntax!

But \xintname is mainly devoted to big numbers;
the allowed input formats for `long numbers' and `fractions' are:
\begin{enumerate}
\item the strict format\ntype{f} is for some macros of \xintname which only
  \fexpan d their arguments. After this \fexpan sion the input should be a
  string of digits, optionally preceded by a unique minus sign. The first digit
  can be zero only if the number is zero. A plus sign is not accepted. |-0| is
  not legal in the strict format. A count register can serve as argument of such
  a macro only if prefixed by |\the| or |\number|. Most macros of \xintname are
  like \csbxint{Add} and accept the extended format described in the next item;
  they may have a `strict' variant such as \csbxint{iiAdd} which remains
  available even with \xintfracname loaded, for optimization purposes.

\item the macro \csbxint{Num} normalizes into strict format an input having
  arbitrarily many minus and plus signs, followed by a string of zeroes, then
  digits:%
  %
  \leftedline{|\xintNum
    {+-+-+----++-++----00000000009876543210}|\dtt{=\xintNum
      {+-+-+----++-++----0000000009876543210}}}
  %
  The extended integer format\ntype{\Numf} is thus for the arithmetic macros
  of \xintname which automatically parse their arguments via this
  \csbxint{Num}.%
%
\footnote{A
    \LaTeX{} \texttt{\char 92value\{countername\}} is accepted as macro
    argument.}

\item the fraction format\ntype{\Ff} is what is expected by the macros of
  \xintfracname: a fraction is constituted of a numerator |A| and optionally a
  denominator |B|, separated by a forward slash |/| and |A| and |B| may be
  macros which will be automatically given to \csbxint{Num}. Each of |A| and |B|
  may be decimal numbers (the decimal mark must be a |.|). Here is an
  example:%
  %
  \footnote{the square brackets one sees in various outputs are
    explained
    near the end of this section.} %
  %
  \leftedline{|\xintAdd
    {+--0367.8920280/-++278.289287}{-109.2882/+270.12898}|}
  %
  Scientific notation is accepted for both numerator and denominator of a
  fraction, and is produced on output by \csbxint{Float}:
  %
  \begin{quote}
    |\xintAdd{10.1e1}{101.010e3}|\dtt{=\xintAdd{10.1e1}{101.010e3}}\\
  %
    |\xintFloatAdd{10.1e1}{101.010e3}|\dtt{=\xintFloatAdd{10.1e1}{101.010e3}}\\
  %
    |\xintPow {2}{100}|\dtt{=\xintPow {2}{100}}\\
  %
    |\xintFloat{\xintPow {2}{100}}|\dtt{=\xintFloat{\xintPow {2}{100}}}\\
  %
    |\xintFloatPow {2}{100}|\dtt{=\xintFloatPow {2}{100}}
  \end{quote}
  %
  Produced fractions having a denominator equal to one are, as a general
  rule, nevertheless printed as fractions. In math mode \csbxint{Frac}
  will remove such dummy denominators, and in inline text mode one has
  \csbxint{PRaw} with the similar effect.
  %
  \begin{quote}
     |\xintPRaw{\xintAdd{10.1e1}{101.010e3}}|\dtt{=\xintPRaw{\xintAdd{10.1e1}{101.010e3}}}\\
  %
     |\xintRaw{1.234e5/6.789e3}|\dtt{=\xintRaw{1.234e5/6.789e3}}%
  \end{quote}

\item the \hyperref[xintexpr]{expression format} is for inclusion in an
  \csbxint{expr}|...\relax|, it uses infix notations, function names, complete
  expansion, and is described in \autoref{sec:expr11} and \autoref{sec:expr}.
\end{enumerate}

Generally speaking, there should be no spaces among the digits in the inputs
(in arguments to the package macros).
Although most would be harmless in most macros, there are some cases
where spaces could break havoc. So the best is to avoid them entirely.

This is entirely otherwise inside an |\xintexpr|-ession, where spaces are
ignored (except when they occur inside arguments to some macros, thus
escaping the |\xintexpr| parser). See the \hyperref[sec:expr]{documentation}.

Even with \xintfracname loaded, some macros by their nature can not accept
fractions on input. Those parsing their inputs through \csbxint{Num} will now
accept fractions, truncating them first to integers.

With \xintfracname loaded, a number may be empty or start directly with a
decimal point: 
\begin{quote}
    |\xintRaw{}=\xintRaw{.}|\dtt{=\xintRaw{}}\\
    |\xintPow{-.3/.7}{11}|\dtt{=\xintPow{-.3/+.7}{11}}\\
    |\xinttheexpr (-.3/.7)^11\relax|\dtt{=\xinttheexpr (-.3/.7)^11\relax}
\end{quote}
It is also licit to use |\A/\B| as
input if each of |\A| and |\B| expands (in the sense previously described) to a
``decimal number'' as examplified above by the numerators and denominators
(thus, possibly with a `scientific' exponent part, with a lowercase `e'). Or one
may have just one macro |\C| which expands to such a ``fraction with optional
decimal points'', or mixed things such as |\A 245/7.77|, where the numerator
will be the concatenation of the expansion of |\A| and |245|. But, as explained
already |123\A| is a no-go, \emph{except inside an |\xintexpr|-ession}!

The scientific notation is necessarily (except in |\xintexpr..\relax|) with a
lowercase |e|. It may appear both at the numerator and at the denominator of a
fraction. 
%
\leftedline{|\xintRaw {+--+1253.2782e++--3/---0087.123e---5}|\dtt{=\xintRaw
    {+--+1253.2782e++--3/---0087.123e---5}}}

Arithmetic macros of \xintname which parse their arguments automatically through
\csbxint{Num} are signaled by a special
symbol%\ntype{\Numf{\unskip\kern\dimexpr\FrameSep+\FrameRule\relax}}
\ntype{\Numf} in the margin. This symbol also means that these arguments may
contain to some extent infix algebra with count registers, see the section
\hyperref[sec:useofcount]{Use of count registers}.

  With \xintfracname loaded the symbol \smash{\Numf} means that a fraction is
  accepted if it is a whole number in disguise; and for macros accepting the
  full fraction format with no restriction there is the corresponding symbol
  in the margin\ntype{\Ff}.

The \xintfracname macros generally output
their result in |A/B[n]| format, representing the fraction |A/B| times |10^n|.

This format with a trailing |[n]| (possibly, |n=0|) is accepted on input
but it presupposes that the numerator and denominator |A| and |B| are in
the strict integer format described above. So |16000/289072[17]| or
|3[-4]| are authorized and it is even possible to use |\A/\B[17]| if
|\A| expands to |16000| and |\B| to |289072|, or |\A| if |\A| expands to
|3[-4]|. However, NEITHER the numerator NOR the denominator may then
have a decimal point\IMPORTANT{}. And, for this format, ONLY the
numerator may carry a UNIQUE minus sign (and no superfluous leading
zeroes; and NO plus sign).

It is allowed for user input but the parsing is minimal and it is mandatory to
follow the above rules. This reduced flexibility, compared to the format without
the square brackets, allows nesting package macros without too much speed
impact.

\subsection{Output formats}

With package \xintfracname loaded, the routines \csbxint{Add}, \csbxint{Sub},
\csbxint{Mul}, \csbxint{Pow}, initiallly synonyms in \xintname of
\csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul}, \csbxint{iPow}, are modified
to become the fraction handling routines.%
%
\footnote{the power function does not accept a fractional
  exponent. Or rather, does not expect, and errors will result if one is
  provided.}
%
\footnote{as commented upon more later, for that very reason use of
  \csbxint{Add} etc\dots when only \xintname is loaded is strongly
  discouraged.}\,%
%
% \footnote{macros \csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul},
%   \csbxint{iPow}, are the original ones dealing only with integers. They are
%   available as synonyms, also when \xintfracname is not loaded. With
%   \xintfracname loaded they accept on input also fractions, which they first
%   truncate to integers, and then the output format is the integer one. The macros \csbxint{iiAdd}, \csbxint{iiSub}, \csbxint{iiMul},
%   \csbxint{iiPow}, \csbxint{iiSum}, \csbxint{iiPrd} are strictly integer-only:
%   they skip the overhead of parsing their arguments via
%   \csbxint{Num}.}\,%
%
% \footnote{also \csbxint{Cmp}, \csbxint{Sgn}, \csbxint{Geq},
%   \csbxint{Opp}, \csbxint{Abs}, \csbxint{Max}, \csbxint{Min} are extended to
%   fractions; and the last four have the integer-only variants \csbxint{iOpp},
%   \csbxint{iAbs}, \csbxint{iMax}, \csbxint{iMin}.}\,%
% %
% \footnote{and \csbxint{Fac},
%   \csbxint{Quo}, \csbxint{Rem}, \csbxint{Division}, \csbxint{FDg},
%   \csbxint{LDg}, \csbxint{Odd}, \csbxint{MON}, \csbxint{MMON} all accept a
%   fractional input as long as it reduces to an integer.} 
%
They produce on output a
fractional number |f=A/B[n]| where |A| and |B| are integers, with |B| positive,
and |n| is a ``short'' integer.
%
% (\emph{i.e} less in absolute value than |2^{31}-9|).
%
This represents |(A/B)| times |10^n|.
%
\footnote{at each stage of the
  computations, the sum of |n| and the length of |A|, or of the absolute value
  of |n| and the length of |B|, must be kept less than
  |2^{31}-9|.}

\begin{framed}
  The fraction output format for most \xintfracname macros is {|A/B[n]|} which
  stands for |(A/B)|$\times$|10^n|. The |A| and |B| may end in zeroes
  (\emph{i.e}, |n| does not represent all powers of ten), and will generally
  have a common factor. The denominator |B| is always strictly positive.
  Conversely, this format is accepted on input and is parsed more quickly than
  fractions containing decimal points or in scientific notation; the input
  denominator is optional.
\end{framed}

\begin{itemize}
\item A macro \csbxint{Frac} is provided for the typesetting (math-mode
  only) of such a `raw' output. The command \csbxint{Frac} is not accepted as
  input to the package macros, it is for typesetting only (in math mode).

\item \csbxint{Raw} prints the fraction directly as its internal
  representation |A/B[n]|.
\begin{everbatim*}
$\xintRaw{273.3734e5/3395.7200e-2}=\xintFrac {273.3734e5/3395.7200e-2}$
\end{everbatim*}

\item \csbxint{PRaw} does the same but without printing the |[n]| if |n=0| and
  without printing |/1| if |B=1|.

\item \csbxint{Irr} reduces the fraction to its irreducible form |C/D|
  (without a trailing |[0]|), and it prints the |D| even if |D=1|.
\begin{everbatim*}
$\xintIrr{273.3734e5/3395.7200e-2}$
\end{everbatim*}

\item \csbxint{Num} from package \xintname becomes when \xintfracname is
  loaded a synonym to its macro \csbxint{TTrunc} (same as
  \csbxint{iTrunc}|{0}|) which truncates to the nearest integer.

\item See also the documentations of \csbxint{Trunc}, \csbxint{iTrunc},
\csbxint{XTrunc}, \csbxint{Round}, \csbxint{iRound} and \csbxint{Float}.

\item The \csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul}, \csbxint{iPow}
  macros and some others accept fractions on input which they truncate via
  \csbxint{TTrunc}. On output they still produce an integer with no fraction
  slash nor trailing |[n]|.

\item The \csbxint{iiAdd}, \csbxint{iiSub}, \csbxint{iiMul}, \csbxint{iiPow},
  and others with `\textcolor{blue}{ii}' in their names accept on input only
  integers in the strict format (they skip the overhead of the \csbxint{Num}
  parsing) and naturally they output an integer, with no fraction slash nor
  trailing |[n]|.

\end{itemize}

%\subsection{Multiple outputs}\label{sec:multout}

Some macros return a token list of two or more numbers or fractions; they are
then each enclosed in braces. Examples are \csbxint{iiDivision} which gives
first the quotient and then the remainder of euclidean division,
\csbxint{Bezout} from the \xintgcdname package which outputs five numbers,
\csbxint{FtoCv} from the \xintcfracname package which returns the list of the
convergents of a fraction, ... \autoref{sec:assign} and \autoref{sec:utils}
mention utilities, expandable or not, to cope with such outputs.

Another type of multiple number output is when using commas inside
\csbxint{expr}|..\relax|:
%
\leftedline{|\xinttheiexpr 10!,2^20,lcm(1000,725)\relax|%
     $\to$\dtt{\xinttheiexpr 10!,2^20,lcm(1000,725)\relax}}

This returns a comma separated list, with a space after each comma.

% \section{Use of \TeX{} registers and variables}

% {\etocdefaultlines\etocsettocstyle{}{}\localtableofcontents}

\subsection{Use of count registers}\label{sec:useofcount}

Inside |\xintexpr..\relax| and its variants, a count register or count control
sequence is automatically unpacked using |\number|, with tacit multiplication:
|1.23\counta| is like |1.23*\number\counta|. There
is a subtle difference between count \emph{registers} and count
\emph{variables}. In |1.23*\counta| the unpacked |\counta| variable defines a
complete operand thus |1.23*\counta 7| is a syntax error. But |1.23*\count0|
just replaces |\count0| by |\number\count0| hence |1.23*\count0 7| is like
|1.23*57| if |\count0| contains the integer value |5|.

Regarding now the package macros, there is first the case of arguments having to
be short integers: this means that they are fed to a |\numexpr...\relax|, hence
submitted to a \emph{complete expansion} which must deliver an integer, and
count registers and even algebraic expressions with them like
|\mycountA+\mycountB*17-\mycountC/12+\mycountD| are admissible arguments (the
slash stands here for the rounded integer division done by |\numexpr|). This
applies in particular to the number of digits to truncate or round with, to the
indices of a series partial sum, \dots

The macros allowing the extended format for long numbers or dealing with
fractions will \emph{to some extent} allow the direct use of count
registers and even infix algebra inside their arguments: a count
register |\mycountA| or |\count 255| is admissible as numerator or also as
denominator, with no need to be prefixed by |\the| or |\number|. It is possible
to have as argument an algebraic expression as would be acceptable by a
|\numexpr...\relax|, under this condition: \emph{each of the numerator and
  denominator is expressed with at most \emph{eight}
  tokens}.%
%
\footnote{Attention! there is no problem with a \LaTeX{}
  \csa{value}\texttt{\{countername\}} if if comes first, but if it comes later
  in the input it will not get expanded, and braces around the name will be
  removed and chaos\IMPORTANT{} will ensue inside a \csa{numexpr}. One should
  enclose the whole input in \csa{the}\csa{numexpr}|...|\csa{relax} in such
  cases.} 
%
The slash for rounded division in a |\numexpr| should be written with
braces |{/}| to not be confused with the \xintfracname delimiter between
numerator and denominator (braces will be removed internally). Example:
|\mycountA+\mycountB{/}17/1+\mycountA*\mycountB|, or |\count 0+\count
2{/}17/1+\count 0*\count 2|, but in the latter case the numerator has the
maximal allowed number of tokens (the braced slash counts for only one).
%
\leftedline{|\cnta 10 \cntb 35 \xintRaw
  {\cnta+\cntb{/}17/1+\cnta*\cntb}|\dtt{->\cnta 10 \cntb 35 \xintRaw
    {\cnta+\cntb{/}17/1+\cnta*\cntb}}} 
%
For longer algebraic expressions using
count registers, there are two possibilities:
\begin{enumerate}
\item encompass each of the numerator and denominator in |\the\numexpr...\relax|,
\item encompass each of the numerator and denominator in |\numexpr {...}\relax|.
\end{enumerate}
\everb|@
\cnta 100 \cntb 10 \cntc 1
\xintPRaw {\numexpr {\cnta*\cnta+\cntb*\cntb+\cntc*\cntc+
                    2*\cnta*\cntb+2*\cnta*\cntc+2*\cntb*\cntc}\relax/%
          \numexpr {\cnta*\cnta+\cntb*\cntb+\cntc*\cntc}\relax }
|
\cnta 100 \cntb 10 \cntc 1
%
\leftedline{\dtt{\xintPRaw {\numexpr
      {\cnta*\cnta+\cntb*\cntb+\cntc*\cntc+
                    2*\cnta*\cntb+2*\cnta*\cntc+2*\cntb*\cntc}\relax/%
          \numexpr {\cnta*\cnta+\cntb*\cntb+\cntc*\cntc}\relax }}}
%
The braces would not be accepted
      as regular
|\numexpr|-syntax: and indeed, they
        are removed at some point in the processing.

\subsection{Dimensions}
\label{sec:Dimensions}

\meta{dimen} variables can be converted into (short) integers suitable for the
\xintname macros by prefixing them with |\number|. This transforms a dimension
into an explicit short integer which is its value in terms of the |sp| unit
($1/65536$\,|pt|).
When |\number| is applied to a \meta{glue} variable, the stretch and shrink
components are lost.

For \LaTeX{} users: a length is a \meta{glue} variable, prefixing a
length command defined by \csa{newlength} with \csa{number} will thus discard
the |plus| and |minus| glue components and return the dimension component as
described above, and usable in the \xintname bundle macros.

This conversion is done automatically inside an
|\xintexpr|-essions, with tacit multiplication implied if prefixed by some
(integral or decimal) number.

One may thus compute areas or volumes with no limitations, in units of |sp^2|
respectively |sp^3|, do arithmetic with them, compare them, etc..., and possibly
express some final result back in another unit, with the suitable conversion
factor and a rounding to a given number of decimal places.

A \hyperref[tableofdimensions]{table of dimensions} illustrates that the
internal values used by \TeX{} do not correspond always to the closest rounding.
For example a millimeter exact value in terms of |sp| units is
\dtt{72.27/10/2.54*65536=\xinttheexpr trunc(72.27/10/2.54*65536,3)\relax
  ...} and \TeX{} uses internally \dtt{\number\dimexpr 1mm\relax}|sp| (it
thus appears that \TeX{} truncates to get an integral multiple of the |sp|
unit).

% impossible avec le \ignorespaces mis par LaTeX de faire \number\dimexpr
% idem à la fin avec \unskip, si je veux xinttheexpr
\begin{figure*}[ht!]
\phantomsection\label{tableofdimensions}
\begingroup\let\ignorespaces\empty
           \let\unskip\empty
           \def\T{\expandafter\TT\number\dimexpr}
           \def\TT#1!{\gdef\tempT{#1}}
           \def\E{\expandafter\expandafter\expandafter
                  \EE\xintexpr reduce(}
           \def\EE#1!{\gdef\tempE{#1}}
\centeredline{\begin{tabular}{%
                >{\bfseries\strut}c%
                c%
                >{\E}c<{)\relax!}@{}%
                >{\xintthe\tempE}r@{${}={}$}%
                >{\xinttheexpr trunc(\tempE,3)\relax...}l%
                >{\T}c<{!}@{}%
                >{\tempT}r%
                >{\xinttheexpr round(100*(\tempT-\tempE)/\tempE,4)\relax\%}c}
   \hline
   Unit&%
   definition&%
   \omit &%
   \multicolumn{2}{c}{Exact value in \texttt{sp} units\strut}&%
   \omit &%
   \omit\parbox{2cm}{\centering\strut\TeX's value in \texttt{sp} units\strut}&%
   \omit\parbox{2cm}{\centering\strut Relative error\strut}\\\hline
  cm&0.01 m&72.27/2.54*65536&&&1cm&&\\
  mm&0.001 m&72.27/10/2.54*65536&&&1mm&&\\
  in&2.54 cm&72.27*65536&&&1in&&\\
  pc&12 pt&12*65536&&&1pc&&\\
  pt&1/72.27 in&65536&&&1pt&&\\
  bp&1/72 in&72.27*65536/72&&&1bp&&\\
  \omit\hfil\llap{3}bp\hfil&1/24 in&72.27*65536/24&&&3bp&&\\
  \omit\hfil\llap{12}bp\hfil&1/6 in&72.27*65536/6&&&12bp&&\\
  \omit\hfil\llap{72}bp\hfil&1 in&72.27*65536&&&72bp&&\\
  dd&1238/1157 pt&1238/1157*65536&&&1dd&&\\
  \omit\hfil\llap{11}dd\hfil&11*1238/1157 pt&11*1238/1157*65536&&&11dd&&\\
  \omit\hfil\llap{12}dd\hfil&12*1238/1157 pt&12*1238/1157*65536&&&12dd&&\\
  sp&1/65536 pt&1&&&1sp&&\\\hline
  \multicolumn{8}{c}{\bfseries\large\TeX{} \strut dimensions}\\\hline
\end{tabular}}
\endgroup
\end{figure*}

There is something quite amusing with the Didot point. According to the \TeX
Book, $1157$\,|dd|=$1238$\,|pt|. The actual internal value of $1$\,|dd| in \TeX{} is $70124$\,|sp|. We can use \xintcfracname to display the list of
centered convergents of the fraction $70124/65536$:
%
\leftedline{|\xintListWithSep{, }{\xintFtoCCv{70124/65536}}|}
%
\xintFor* #1 in {\xintFtoCCv{70124/65536}}\do {$\printnumber{#1}$, }%
and we don't find
$1238/1157$ therein, but another approximant $1452/1357$!

And indeed multiplying $70124/65536$ by $1157$, and respectively $1357$, we find
the approximations (wait for more, later):
%
\leftedline{``$1157$\,|dd|''\dtt{=\xinttheexpr trunc(1157\dimexpr
    1dd\relax/\dimexpr 1pt\relax,12)\relax}\dots|pt|}
%
\leftedline{``$1357$\,|dd|''\dtt{=\xinttheexpr trunc(1357\dimexpr
    1dd\relax/\dimexpr 1pt\relax,12)\relax}\dots|pt|}
%
and we seemingly discover that $1357$\,|dd|=$1452$\,|pt| is \emph{far more
  accurate} than
the \TeX Book formula $1157$\,|dd|=$1238$\,|pt|~!
The formula to compute $N$\,|dd| was
%
\leftedline{|\xinttheexpr trunc(N\dimexpr 1dd\relax/\dimexpr
  1pt\relax,12)\relax}|}
%

What's the catch? The catch is that \TeX{} \emph{does not} compute $1157$\,|dd|
like we just did:%
%
\leftedline{$1157$\,|dd|=|\number\dimexpr 1157dd\relax/65536|%
      \dtt{=\xintTrunc{12}{\number\dimexpr 1157dd\relax/65536}}\dots|pt|}
%
\leftedline{$1357$\,|dd|=|\number\dimexpr 1357dd\relax/65536|%
      \dtt{=\xintTrunc{12}{\number\dimexpr 1357dd\relax/65536}}\dots|pt|}
%
We thus discover that \TeX{} (or rather here, e-\TeX{}, but one can check that
this works the same in \TeX82), uses indeed $1238/1157$ as a conversion
factor, and necessarily intermediate computations are done with more precision
than is possible with only integers less than $2^{31}$ (or $2^{30}$ for
dimensions). Hence the $1452/1357$ ratio is irrelevant, a misleading artefact
of the necessary rounding (or, as we see, truncating) for one |dd| as an
integral number of |sp|'s.

Let us now
use |\xintexpr| to compute the value of the Didot point in millimeters, if
the above rule is exactly verified: 
%
\leftedline{|\xinttheexpr
 trunc(1238/1157*25.4/72.27,12)\relax|%
  \dtt{=\xinttheexpr trunc(1238/1157*25.4/72.27,12)\relax}|...mm|} 
%
This fits very well with the possible values of the Didot point as listed in
the
\href{http://en.wikipedia.org/wiki/Point_%28typography%29#Didot}{Wikipedia Article}.
%
The value $0.376065$\,|mm| is said to be \emph{the traditional value in
  European printers' offices}. So the $1157$\,|dd|=$1238$\,|pt| rule refers to
this Didot point, or more precisely to the \emph{conversion factor} to be used
between this Didot and \TeX{} points.

The actual value in millimeters of exactly one Didot point as implemented in
\TeX{} is
%
\leftedline {|\xinttheexpr trunc(\dimexpr
  1dd\relax/65536/72.27*25.4,12)\relax|} 
%
\leftedline{\dtt{=\xinttheexpr trunc(\dimexpr
    1dd\relax/65536/72.27*25.4,12)\relax}|...mm|}
%
The difference of circa $5$\AA\ is arguably tiny!

% 543564351/508000000

By the way the \emph{European printers' offices \emph{(dixit Wikipedia)}
  Didot} is thus exactly
%
\leftedline{|\xinttheexpr reduce(.376065/(25.4/72.27))\relax|%
   \dtt{=\xinttheexpr reduce(.376065/(25.4/72.27))\relax}\,|pt|}
%
and the centered convergents of this fraction are \xintFor* #1 in
{\xintFtoCCv{543564351/508000000}}\do {\dtt{\printnumber{#1}}\xintifForLast{.}{, }} We do
recover the $1238/1157$ therein!

% As a final comment on the \hyperref[tableofdimensions]{table of dimensions}, we
% conclude that the ``Relative Error'' column is misleading as these relative
% errors by necessity decrease for integer multiples of the given dimension units.
% This was already indicated by the \textbf{72bp} row.

% To conclude our comments on the
% \hyperref[tableofdimensions]{table of dimensions}, the big point, now known as
% \emph{Desktop Publishing Point} is less accurately implemented in \TeX{} than
% other units. Let us test for example the relation $1$\,|in|$=72$\,|bp|, the difference is
% %
% \centeredline{|\number\numexpr\dimexpr1in\relax-72*\dimexpr1bp\relax\relax|%
% \dtt{=\number\numexpr\dimexpr1in\relax-72*\dimexpr1bp\relax\relax}\,|sp|}
% \centeredline{|\number\dimexpr1in-72bp\relax|%
% \dtt{=\number\dimexpr1in-72bp\relax}\,|sp|}
% on the other hand
% \centeredline{|\xinttheexpr reduce(\dimexpr1in\relax-72.27*\dimexpr1pt\relax)\relax|}
% \centeredline
% \dtt{=\xinttheexpr reduce(\dimexpr1in\relax-72.27*\dimexpr1pt\relax)\relax}\,|sp|=$-0.72$\,|sp|}
% \centeredline
% {\dtt{=\number\dimexpr1in-72.27pt\relax}\,|sp|=$-0.72$\,|sp|}

\subsection{\csh{ifcase}, \csh{ifnum}, ... constructs}\label{sec:ifcase}

When using things such as |\ifcase \xintSgn{\A}| one has to make sure to leave
a space after the closing brace for \TeX{} to
stop its scanning for a number: once \TeX{} has finished expanding
|\xintSgn{\A}| and has so far obtained either |1|, |0|, or |-1|, a
space (or something `unexpandable') must stop it looking for more
digits. Using |\ifcase\xintSgn\A| without the braces is very dangerous,
because the blanks (including the end of line) following |\A| will be
skipped and not serve to stop the number which |\ifcase| is looking for.
%
\begin{everbatim*}
\begin{enumerate}[nosep]\def\A{1}
\item \ifcase \xintSgn\A 0\or OK\else ERROR\fi
\item \ifcase \xintSgn\A\space 0\or OK\else ERROR\fi
\item \ifcase \xintSgn{\A} 0\or OK\else ERROR\fi
\end{enumerate}
\end{everbatim*}

In order to use successfully |\if...\fi| constructions either as arguments to
the \xintname bundle expandable macros, or when building up a completely
expandable macro of one's own, one needs some \TeX nical expertise (see also
\autoref{fn:expansions} on page~\pageref{fn:expansions}).

It is thus much to be recommended to opt rather for already existing expandable
branching macros, such as the ones which are provided by
\xintname/\xintfracname: among them
\csbxint{SgnFork}, \csbxint{ifSgn}, \csbxint{ifZero}, \csbxint{ifOne},
\csbxint{ifNotZero}, \csbxint{ifTrueAelseB}, \csbxint{ifCmp}, \csbxint{ifGt},
\csbxint{ifLt}, \csbxint{ifEq}, \csbxint{ifOdd}, and \csbxint{ifInt}. See their
respective documentations. All these conditionals always have either two or
three branches, and empty brace pairs |{}| for unused branches should not be
forgotten.

If these tests are to be applied to standard \TeX{} short integers, it is more
efficient to use (under \LaTeX{}) the equivalent conditional tests from the
\href{http://www.ctan.org/pkg/etoolbox}{etoolbox}%
%
\footnote{\url{http://www.ctan.org/pkg/etoolbox}}
package.

\subsection{Expandable implementations of mathematical algorithms}

It is possible to chain |\xintexpr|-essions with |\expandafter|'s, like experts
do with |\numexpr| to compute multiple things at once. See
\autoref{ssec:fibonacci} for an example devoted to Fibonacci numbers (this
section provides the code which was used on the title page for the
\dtt{$F(1250)$} evaluation.) Notice that the $47$th Fibonacci number is
\dtt{\Fibonacci {47}} thus already too big for \TeX{} and \eTeX{}. 

The |\Fibonacci| macro found in \autoref{ssec:fibonacci} is completely
expandable, (it is even \fexpan dable in the sense previously explained) hence
can be used for example within |\message| to write to the log and terminal.

\begingroup
  \def\A {1859}\def\B {1573}
  \edef\C {\xintiiGCD\A\B}
  \edef\X {\Fibonacci\A}
  \edef\Y {\Fibonacci\B}

%
Also, one can thus use it as argument to the \xintname macros: for example if
we are interested in knowing how many digits $F(1250)$ has, it suffices to
issue |\xintLen {\Fibonacci {1250}}| (which expands to \dtt{\xintLen
  {\Fibonacci {1250}}}). Or if we want to check the formula
$gcd(F(1859),F(1573))=F(gcd(1859,1573))=F(143)$, we only need%
%
\footnote{The
  \csa{xintiiGCD} macro is provided by the \xintgcdname package.}
%
\leftedline{|$\xintiiGCD{\Fibonacci{1859}}{\Fibonacci{1573}}=\Fibonacci{\xintiiGCD{1859}{1573}}$|}
%
which outputs:
%
\leftedline{$\dtt{\xintiiGCD{\X}{\Y}}=\dtt{\Fibonacci{\C}}$}

The |\Fibonacci| macro expanded its |\xintiiGCD{1859}{1573}| argument via the
services of |\numexpr|: this step allows only things obeying the \TeX{} bound,
naturally! (but \dtt{F(\xintiiPow2{31}}) would be rather big anyhow...). 

In practice, whenever one typesets things, one has left the expansion only
contexts; hence there is no objection to, on the contrary it is recommended,
assign the result of earlier computations to macros via an |\edef| (or an
|\oodef|, see \ref{oodef}), for later use. The above could thus be coded
\begin{everbatim}
\begingroup
  \def\A {1859}  \def\B {1573}  \edef\C {\xintiiGCD\A\B}
  \edef\X {\Fibonacci\A}  \edef\Y {\Fibonacci\B}
The identity $\gcd(F(\A),F(\B))=F(\gcd(\A,\B))$ can be checked via evaluation
of both sides: $\gcd(F(\A),F(\B))=\gcd(\printnumber\X,\printnumber\Y)=
\printnumber{\xintiiGCD\X\Y} = F(\gcd(\A,\B))$.\par
          % some further computations involving \A, \B, \C, \X, \Y
\endgroup % closing the group removes assignments to \A, \B, ...
% or choose longer names less susceptible to overwriting something. Note that there
% is no LaTeX \newecommand which would be to \edef like \newcommand is to \def
\end{everbatim}
The identity $\gcd(F(\A),F(\B))=F(\gcd(\A,\B))$ can be checked via evaluation
of both sides:
$\gcd(F(\A),F(\B))=\gcd(\dtt{\printnumber\X{\normalcolor,}\printnumber\Y})=
\dtt{\printnumber{\xintiiGCD\X\Y}} = F(\C) = F(\gcd(\A,\B))$.\par


\endgroup
One may thus legitimately ask the author: why expandability to such extremes,
for things such as big fractions or floating point numbers (even
continued fractions...) which anyhow can not be used directly
within \TeX's primitives such as |\ifnum|? the answer is that the author
chose, seemingly, at some point back in his past to waste from then on his time
on such useless things!

\subsection{Possible syntax errors to avoid}

\edef\x{\xintMul {3}{5}/\xintMul{7}{9}}

Here is a list of imaginable input errors. Some will cause compilation errors,
others are more annoying as they may pass through unsignaled.
\begin{itemize}
\item using |-| to prefix some macro: |-\xintiSqr{35}/271|.%
%
\footnote{to the
    contrary, this \emph{is}
    allowed inside an |\xintexpr|-ession.}
\item using one pair of braces too many |\xintIrr{{\xintiPow {3}{13}}/243}| (the
  computation goes through with no error signaled, but the result is completely
  wrong).
\item using |[]| and decimal points at the same time |1.5/3.5[2]|, or with a
  sign in the denominator |3/-5[7]|. The scientific notation has no such
  restriction, the two inputs |1.5/-3.5e-2| and |-1.5e2/3.5| are equivalent:
  |\xintRaw{1.5/-3.5e-2}|\dtt{=\xintRaw{1.5/-3.5e-2}},
  |\xintRaw{-1.5e2/3.5}|\dtt{=\xintRaw{-1.5e2/3.5}}.
\item specifying numerators and
  denominators with macros producing fractions when \xintfracname is loaded:
  |\edef\x{|\allowbreak|\xintMul {3}{5}/\xintMul{7}{9}}|. This expands to
  \texttt{\x} which is
  invalid on input. Using this |\x| in a fraction macro will most certainly
  cause a compilation error, with its usual arcane and undecipherable
  accompanying message. The fix here would be to use |\xintiMul|. The simpler
  alternative with package \xintexprname:
  |\xinttheexpr 3*5/(7*9)\relax|.
\item generally speaking, using in a context expecting an integer (possibly
  restricted to the \TeX{} bound) a macro or expression which returns a
  fraction: |\xinttheexpr 4/2\relax| outputs \dtt{\xinttheexpr 4/2\relax},
  not $2$. Use |\xintNum {\xinttheexpr 4/2\relax}| or |\xinttheiexpr 4/2\relax|
  (which rounds the result to the nearest integer, here, the result is already
  an integer) or |\xinttheiiexpr 4/2\relax|.
\item use of square brackets |[|, |]| in |\xintexpr...\name| has some traps, see
  \autoref{sec:expr}.
\end{itemize}

\subsection{Error messages}

In situations such as division by zero, the package will insert in the
\TeX{} processing an undefined control sequence (we copy this method
from the |bigintcalc| package). This will trigger the writing to the log
of a message signaling an undefined control sequence. The name of the
control sequence is the message. The error is raised \emph{before} the
end of the expansion so as to not disturb further processing of the
token stream, after completion of the operation. Generally the problematic
operation will output a zero. Possible such error message control
sequences:

% \the\parskip\par % attention 0pt plus 1pt

\begin{multicols}{2}\parskip0pt\relax
\begin{everbatim}
\xintError:ArrayIndexIsNegative
\xintError:ArrayIndexBeyondLimit
\xintError:FactorialOfNegativeNumber
\xintError:FactorialOfTooBigNumber
\xintError:DivisionByZero
\xintError:NaN
\xintError:FractionRoundedToZero
\xintError:NotAnInteger
\xintError:ExponentTooBig
\xintError:TooBigDecimalShift
\xintError:TooBigDecimalSplit
\xintError:RootOfNegative
\xintError:NoBezoutForZeros
\xintError:ignored
\xintError:removed
\xintError:inserted
\xintError:unknownfunction
\xintError:we_are_doomed
\xintError:missing_xintthe!
\end{everbatim}
\end{multicols}
% NOTES 12 octobre 2014
% J'ai voulu faire avec verbatim, mais bizarrement il met dans la
% colonne de gauche 10 et 8 dans celle de droite. Je n'ai pas réussi à
% reproduire le problème sur un MWE, en tout cas un exemple naïf ne
% reproduit pas le problème. Ensuite avec \everb c'est beaucoup mieux
% mais j'ai dû mettre \raggedcolumns. J'ai essayé avec \strut. Ah, mais
% le problème c'est \parskip. 

% par ailleurs il y a trop d'espace vertical avant le multicols, mais
% bon.

Don't forget to set |\errorcontextlines| to at least |2| to get from \LaTeX\
more meaningful error messages. Errors occuring during the parsing of
|\xintexpr-essions| try to provide helpful information about the offending
token. 

Release |1.1| employs in some situations delimited macros and there is
the possibility in case of an ill-formed expression to end up beyond the
|\relax| end-marker. The errors inevitably arising could then lead to very
cryptic messages; but nothing unusual or especially traumatizing for the
daring experienced \TeX/\LaTeX\ user.


\subsection{Package namespace, catcodes}

The \xintname bundle packages presuppose that the \csa{space}, \csa{empty} and
|\m@ne| control sequences are pre-defined with meanings as in Plain
\TeX{} or \LaTeX2e.

Private macros of \xintkernelname, \xintcorename, \xinttoolsname,
\xintname, \xintfracname, \xintexprname, \xintbinhexname, \xintgcdname,
\xintseriesname, and \xintcfracname{} use one or more underscores |_| as
private letter, to reduce the risk of getting overwritten. They almost
all begin either with |\XINT_| or with |\xint_|, a handful of these
private macros such as \csa{XINTsetupcatcodes}, \csa{XINTdigits} and
those with names such as |\XINTinFloat...| or |\XINTinfloat...| do not
have any underscore in their names (for obscure legacy reasons).

\xinttoolsname provides \hyperref[odef]{|\odef|}, \hyperref[oodef]{|\oodef|},
\hyperref[fdef]{|\fdef|} (if macros with these names already exist
\xinttoolsname will not overwrite them but provide |\xintodef| etc... ) but
all other public macros from the \xintname bundle packages start with |\xint|.

For the good functioning of the macros, standard catcodes are assumed for the
minus sign, the forward slash, the square brackets, the letter `e'. These
requirements are dropped inside an |\xintexpr|-ession: spaces are gobbled,
catcodes mostly do not matter, the |e| of scientific notation may be |E| (on
input) \dots{} 

If a character used in the |\xintexpr| syntax is made active,
this will surely cause problems; prefixing it with |\string| is one option.
There is \csbxint{exprSafeCatcodes} and \csbxint{exprRestoreCatcodes} to
temporarily turn off potentially active characters (but setting catcodes is an
un-expandable action).

\begin{framed}
  For advanced \TeX\ users. At loading time of the packages the
  catcode configuration may be arbitrary as long as it satisfies the following
  requirements: the percent is of category code comment character, the
  backslash is of category code escape character, digits have category code
  other and letters have category code letter. Nothing else is assumed.
\end{framed}

\section{Some utilities from the \xinttoolsname package}

This is a first overview. Many examples combining these utilities with the
arithmetic macros of \xintname are to be found in \autoref{sec:tools}.

\subsection{Assignments}\label{sec:assign}

\xintAssign \xintBezout{357}{323}\to\tmpA\tmpB\tmpU\tmpV\tmpD

It might not be necessary to maintain at all times complete expandability. A
devoted syntax is provided to make these things more efficient, for example when
using the \csbxint{iDivision} macro which computes both quotient and remainder
at 
the same time:
%
\leftedline{\csbxint{Assign}
  |\xintiiDivision{\xintiiPow {2}{1000}}{\xintFac{100}}|\csbnolk{to}|\A\B|} 
%
give:
\xintAssign\xintiiDivision{\xintiPow {2}{1000}}{\xintFac{100}}\to\A\B
|\meaning\A|\dtt{: \printnumber{\meaning\A}\relax} and
|\meaning\B|\dtt{: \printnumber{\meaning\B}\relax}.
%
Another example (which uses \csbxint{Bezout} from the \xintgcdname package):
%
\leftedline{\csbxint{Assign}
%
    |\xintBezout{357}{323}|\csbnolk{to}|\A\B\U\V\D|} 
%
is equivalent to setting |\A| to \dtt{\tmpA}, |\B| to \dtt{\tmpB}, |\U| to
\dtt{\tmpU}, |\V| to \dtt{\tmpV}, and |\D| to \dtt{\tmpD}. And indeed
\dtt{(\tmpU)$\times$\tmpA-(\tmpV)$\times$\tmpB$=$%
  \xintiSub{\xintiMul\tmpU\tmpA}{\xintiMul\tmpV\tmpB}} is a Bezout Identity.

Thus, what |\xintAssign| does is to first apply an
\hyperref[sec:expansions]{\fexpan sion} to what comes next; it then defines one
after the other (using |\def|; an optional argument allows to modify the
expansion type, see \autoref{xintAssign} for details), the macros found after
|\to| to correspond to the successive braced contents (or single tokens) located
prior to |\to|.

\xintAssign
\xintBezout{3570902836026}{200467139463}\to\tmpA\tmpB\tmpU\tmpV\tmpD

\leftedline
{\csbxint{Assign}|\xintBezout{3570902836026}{200467139463}|%
    \csbnolk{to}|\A\B\U\V\D|}
\noindent
gives then |\U|\dtt{:
    \printnumber\tmpU},
  |\V|\dtt{:
    \printnumber\tmpV} and |\D|\dtt{=\tmpD}.

%
In situations when one does not know in advance the number of items, one has
\csbxint{AssignArray} or its synonym \csbxint{DigitsOf}:
%
\leftedline{\csbxint{DigitsOf}|\xintiPow{2}{100}|\csbnolk{to}\csa{DIGITS}}
%
This defines \csa{DIGITS} to be macro with one parameter, \csa{DIGITS}|{0}|
gives the size |N| of the array and \csa{DIGITS}|{n}|, for |n| from |1| to |N|
then gives the |n|th element of the array, here the |n|th digit of $2^{100}$,
from the most significant to the least significant. As usual, the generated
macro \csa{DIGITS} is completely expandable (in two steps). As it wouldn't make
much sense to allow indices exceeding the \TeX{} bounds, the macros created by
\csbxint{AssignArray} put their argument inside a \csa{numexpr}, so it is
completely expanded and may be a count register, not necessarily prefixed by
|\the| or |\number|. Consider the following code snippet:
%
\begin{everbatim*}
% \newcount\cnta
% \newcount\cntb
\begingroup
\xintDigitsOf\xintiPow{2}{100}\to\DIGITS
\cnta = 1
\cntb = 0
\loop
\advance \cntb \xintiSqr{\DIGITS{\cnta}}
\ifnum \cnta < \DIGITS{0}
\advance\cnta 1
\repeat

|2^{100}| (=\xintiPow {2}{100}) has \DIGITS{0} digits and the sum of their squares is \the\cntb.
These digits are, from the least to the most significant: \cnta = \DIGITS{0} \loop
\DIGITS{\cnta}\ifnum \cnta > 1 \advance\cnta -1 , \repeat.\endgroup
\end{everbatim*}

Warning: \csbxint{Assign}, \csbxint{AssignArray} and \csbxint{DigitsOf}
\emph{do not do any check} on whether the macros they define are already
defined.


\subsection{Utilities for expandable manipulations}\label{sec:utils}

The package now has more utilities to deal expandably with `lists of things',
which were treated un-expandably in the previous section with \csa{xintAssign}
and \csa{xintAssignArray}: \csbxint{ReverseOrder} and \csbxint{Length} since the
first release, \csbxint{Apply} and \csbxint{ListWithSep} since |1.04|,
\csbxint{RevWithBraces}, \csbxint{CSVtoList}, \csbxint{NthElt} since |1.06|,
\csbxint{ApplyUnbraced}, since |1.06b|, \csbxint{loop} and \csbxint{iloop} since
|1.09g|.%
%
\footnote{All these utilities, as well as \csbxint{Assign},
  \csbxint{AssignArray} and the \csbxint{For} loops are now available from the
  \xinttoolsname package, independently of the big integers facilities of
  \xintname.}

As an example the following code uses only expandable operations:
\begin{everbatim*}
$2^{100}$ (=\xintiPow {2}{100}) has \xintLen{\xintiPow {2}{100}} digits and the sum of their
  squares is \xintiiSum{\xintApply {\xintiSqr}{\xintiPow {2}{100}}}. These digits are, from the
  least to the most significant: \xintListWithSep {, }{\xintRev{\xintiPow {2}{100}}}. The thirteenth
  most significant digit is \xintNthElt{13}{\xintiPow {2}{100}}. The seventh least significant one
  is \xintNthElt{7}{\xintRev{\xintiPow {2}{100}}}.
\end{everbatim*}

It would be more efficient to do once and for all
|\oodef\z{\xintiPow {2}{100}}|, and then use |\z| in place of
  |\xintiPow {2}{100}| everywhere as this would  spare the CPU some repetitions.

Expandably computing primes is done in \autoref{xintSeq}.

\subsection{A new kind of for loop}

As part of the \hyperref[sec:tools]{utilities} coming with the \xinttoolsname
package, there is a new kind of for loop, \csbxint{For}. Check it out
(\autoref{xintFor}).

\subsection{A new kind of expandable loop}

Also included in \xinttoolsname, \csbxint{iloop} is an expandable loop giving
access to an iteration index, without using count registers which would break
expandability. Check it out (\autoref{xintiloop}).


% Lundi 06 octobre 2014 à 22:02:44
% je décide de ne plus inclure le README verbatim
% \begingroup
% \makeatletter\def\x{\baselineskip10pt
%                     \ttfamily
%                     %\settowidth\dimen@{X}%
%                     %\parindent \dimexpr.5\linewidth-33\dimen@\relax
%                     \parindent\z@
%                     \let\do\do@noligs\verbatim@nolig@list
%                     \let\do\@makeother\dospecials
%                     \def\par{\leavevmode \null\@@par\penalty\interlinepenalty}%
%                     \makestarlowast
%                     \@vobeyspaces\obeylines
%                     \noindent\kern\parindent\input README.md
% \endgroup }\x

\section{New features of the \xintexprname package}
\label{sec:expr11}

Release |1.1| brought many changes to \xintexprname. This chapter is
for people already familiar with earlier versions. A more systematic
item per item syntax description is provided in \autoref{sec:expr}. But be
sure to come back here as the latter chapter has not been compeletely updated.

First, there were some breaking changes:
\begin{itemize}
  \item in |\xintiiexpr|, |/| does \emph{rounded} division, rather than as
    in earlier releases the
    Euclidean division (for positive arguments, this is truncated division).
    The new |//| operator does truncated division,
  \item the |:| operator for three-way branching is gone, replaced with |??|,
  \item |1e(3+5)| is now illegal. The number parser identifies |e| and |E|
    in the same way it does for the decimal mark, earlier versions treated
    |e| as |E| rather as postfix operators,
  \item the |add| and |mul| have a new syntax, old syntax is with |`+`| and
    |`*`| (quotes mandatory), |sum| and |prd| are gone,
  \item no more special treatment for encountered brace pairs |{..}| by the
    number scanner, |a/b[N]| notation can be used without use of braces (the
    |N| will end up space-stripped in a |\numexpr|, it is not parsed by the
    |\xintexpr|-ession scanner).
  \item although |&| and \verb+|+ are still available as Boolean operators the
    use of |&&| and \verb+||+ is strongly recommended. The single
    letter operators might be assigned some other meaning in later releases
    (bitwise operations, perhaps). Do not use them.
  \item place holders for |\xintNewExpr|
    could be denoted |#1|, |#2|, ... or also, for special purposes |$1|, |$2|,
    ... Only the first form is now accepted and the special cases previously
    treated via the second form are now managed via a |protect(...)| function.
\end{itemize}

Let's now describle some of the numerous additional functionalities.

\begin{itemize}[parsep=0pt]
  \item |\xintiexpr|, |\xinttheiexpr| admit an optional argument within
    brackets |[d]|, it then presents the computation result (or results, if
    comma separated) after rounding to |d| digits after decimal mark, (the
    whole computation is done exactly, as in |xintexpr|),
\begin{everbatim*}
\xinttheiexpr [32] 1.23^50, 1.231^50\relax
\end{everbatim*}
  \item |\xintfloatexpr|, |\xintthefloatexpr| similarly admit an optional
    argument which serves to keep only |d| digits of precision, getting rid
    of cumulated uncertainties in the last digits (the whole computation is
    done according to the precision set via |\xintDigits|),
\begin{everbatim*}
\xintDigits:=32;\xintthefloatexpr 1.010101^10-1.0101^10\relax

\xintDigits:=16;\xintthefloatexpr 1.010101^10-1.0101^10\relax

\xintthefloatexpr [12] 1.010101^10-1.0101^10\relax
\end{everbatim*}

  \item |\xinttheexpr| and |\xintthefloatexpr| ``pretty-print'' if possible,
    the former removing unit denominator or |[0]| brackets, the latter
    avoiding scientific notation if decimal notation is practical,
  \item the |//| does truncated division and |/:| is the associated modulo,
  \item multi-character operators |&&|, \verb+||+, |==|, |<=|, |>=|, |!=|,
    |**|,
  \item multi-letter infix binary words |'and'|, |'or'|, |'xor'|, |'mod'|
    (quotes mandatory),
  \item functions |even|, |odd|, |first|, |last|,
  \item |\xintdefvar A3:=3.1415;| for variable definitions (non expandable,
    naturally), usable in subsequent expressions; variable names may contain
    letters, digits, underscores. They should not start with a digit, the
    |@| is reserved, and single lowercase and uppercase Latin letters are
    predefined to work as dummy variables (see next),
   \item generation of comma separated lists |a..b|, |a..[d]..b|,
   \item Python syntax-like list extractors |[list][n:]|, |[list][:n]|, |[list][a:b]|
     and |[list][n]| (|n=0| for the number of list items), the step is always
     |+1|,
   \item function |reversed|, to reverse the order of list items,
   \item itemwise sequence operations |a*[list]|, etc.., on both sides |a*[list]^b|,
   \item dummy variables in |add| and |mul|: |add(x(x+1)(x-1), x=-10..10)|,
   \item variable substitutions with |subs|: |subs(subs(add(x^2+y^2,x=1..y),y=t),t=20)|,
   \item sequence generation using |seq| with a dummy variable: |seq(x^3, x=-10..10)|,
   \item simple recursive sequences with |rseq|, with |@| given the last value,
     |rseq(1;2@+1,i=1..10)|,
   \item higher recursion with |rrseq|, |@1|, |@2|, |@3|, |@4|, and |@@(n)|
     for earlier values, up to |n=K| where |K| is the number of terms of the
     initial stretch |rrseq(0,1;@1+@2,i=2..100)|,
   \item iteration with |iter| which is like |rrseq| but outputs only the
     last |K| terms, where |K| was the number of initial terms,
   \item inside |seq|, |rseq|, |rrseq|, |iter|, possibility to use |omit|,
     |abort| and |break| to control termination,
   \item |n++| potentially infinite index generation for |seq|, |rseq|,
     |rrseq|, and |iter|, it is advised to use |abort| or |break(..)| at
     some point,
   \item the |add|, |mul|, |seq|, ... are nestable,
   \item |\xintthecoords| converts a comma separated list of an even number
     of items to the format as expected by the |TikZ| |coordinates| syntax,
   \item completely rewritten |\xintNewExpr|, new |protect| function to handle
     external macros. However not all constructs are compatible with
     |\xintNewExpr|.
\end{itemize}

% \subsection{Examples with the \texorpdfstring{\unexpanded{\unexpanded{|v1.1|}}}{v1.1} \csh{xintexpr}}

And now some examples:

\begin{itemize}[parsep=0pt, listparindent=.5\leftmargini]
\item One can define variables (the definition itself is a non expandable
  step). The allowed names are composed of letters, digits, and underscores.
  The variable should not start with a digit and single letters |a..z|, |A..Z|
  are predefined for use as dummy variables --- see below. The |@| is
  reserved.
\begin{everbatim*}
\begingroup
   \xintdefvar a_1 := 3.14159;\xintdefvar a2 := 2.71828;\xinttheiexpr [5] a_1+a2\relax
\endgroup
\end{everbatim*}
\item |add| and |mul| have a new syntax requiring a dummy variable:
\begin{everbatim*}
\xinttheexpr add(x, x=1,3,19), mul(x^2, x=1,3,19), add(x(x+1), x= 1,3,19)\relax
\end{everbatim*}

Use |`+`| and |`*`| (left ticks mandatory) for syntax without dummy variables:
\begin{everbatim*}
\xinttheexpr `+`(1,3,19), `*`(1^2,3^2,19^2), `+`(1*2,3*4,19*20)\relax
\end{everbatim*}
\item The |seq| function generates sequences according to a given formula:
\begin{everbatim*}
\xinttheexpr seq(x(x+1)(x+2), x=1,3,19), `+`(seq(x(x+1)(x+2), x=1,3,19)), 
             add(x(x+1)(x+2), x=1,3,19)\relax
\end{everbatim*}
\begin{everbatim*}
And this is nestable!
\xinttheexpr seq(seq(x^y, y=1..5),x=1..5), add(mul(x^y,y=1..5), x=1..5), 
             add(x^15, x=1..5)\relax % 15 = 1+2+3+4+5
\end{everbatim*}

One should use parentheses appropriately. The \csbxint{expr} parser in normal
operation is not bad at identifying missing or extra opening or closing
parentheses, but when it handles |seq|, |add|, |mul| or similar constructs it
switches to another mode of operation (it starts using delimited macros,
something which is almost non-existent in all its other operations) and ill-formed
expressions are much more likely to let the parser fetch tokens from beyond the
mandatory ending |\relax|. Thus, in case of a missing parenthesis in such
circumstances the error message from \TeX{} might be very cryptic, even for
the seasoned \xintname user.

\item As seen in the last example |a..b| constructs the integers from |a| to
  |b|. This is (small) integer only. A more general |a..[d]..b| works with big
  integers, or fractions, from |a| to |b| with step |d|.
\begin{everbatim*}
\xinttheexpr seq(2x+1, x=1..10, 100..110, 3/5..[1/5]..7/5)\relax
\end{everbatim*}
\item itemwise operations on lists are possible, as well as item extractions:
\begin{everbatim*}
\xinttheiiexpr 2*[1,10,100]^3, 5+[2*[1,10,100]^3]*100 \relax
\end{everbatim*}
\begin{everbatim*}
\xinttheiiexpr 1+[seq(3^j, j=1..10, 21..30)][17], 1+3^27\relax
\end{everbatim*}

We used the |[list][n]| construct which gives the nth item from the list. In
this context there are also the functions |last| and |first|. There is no real
concept of a list object, nor list operations, although itemwise manipulation
are made possible as shown above via the |[..]| constructor. The list
manipulation utilities are so far a bit limited. There is no notion of an
``nuple'' object. The variable |nil| is reserved, it represents an empty list.

\item |subs| is similar to  |seq| in syntax but is for variable substitution:
\begin{everbatim*}
\xinttheexpr subs(100*subs(10*subs(3*x+5,x=y+50)+2,y=z^2),z=10)\relax % 100(10(3*150+5)+2)
\end{everbatim*}
\begin{everbatim*}
\xinttheexpr subs(subs(add(x^2+y^2,x=1..y),y=t),t=20)\relax
\end{everbatim*}

The substituted variable may be a comma separated list (this is impossible
with |seq| which will always pick one item after the other of a list).
\begin{everbatim*}
\xinttheexpr subs([x]^2,x=-123,17,32)\relax
\end{everbatim*}

\item last but not least, |seq| has variants |rseq| and |rrseq| which allow
  recursive definitions. They start with at least one initial value, then a
  semi-colon, then the formula, then the list of indices to iterate over. |@|
  (or |@1|) evaluates to the last computed item, and |rrseq| keeps the memory
  of the |K| last results, where |K| was the number of initial terms. One
  accesses them via |@1, @2, @3, @4| and |@@(N)| for |N>4|. It is even
  possible to nest them and use |@@@| to access the values of the master
  recursion...
\begin{everbatim*}
\xinttheiiexpr rseq(1; 2*@, i=1..10), `+`(rseq(1; 2*@, i=1..10))\relax
\end{everbatim*}
\begin{everbatim*}
\xinttheiiexpr rseq(2; @(@+1)/2, i=1..5)\relax
\end{everbatim*}
\begin{everbatim*}
\xinttheiiexpr rseq(0,1; (@1,add(x,x=@1)), y=2..10)\relax
\end{everbatim*}

Some Fibonacci fun
\begin{everbatim*}
\xinttheiiexpr rrseq(0,1; @1+@2, x=2..10), last(rrseq(0,1; @1+@2, x=2..100))\relax
\end{everbatim*}
\begin{everbatim*}
Sum of previous last three: \xinttheiiexpr rrseq(0,0,1; @1+@2+@3, i=1..20)\relax
\end{everbatim*}
\begin{everbatim*}
Big numbers: \printnumber{\xinttheexpr rseq(1; @(@+1), j=1..10)\relax }
\end{everbatim*}

Nested recursion often quickly leads to gigantic outputs. This is an
experimental feature, susceptible to be removed or altered in the future.
\begin{everbatim*}
\xinttheexpr rrseq(1; `+`(rrseq(0,1; @@@(1)+@1+@2, i=1..10)), j=1..5)\relax
\end{everbatim*}

\item The special keywords |omit|, |abort| and |break(..)| are available
  inside |seq|, |rseq|, |rrseq|, as well as the |n++| for a potentially
  infinite iteration. The |n++| construct in conjunction with an |abort| or
  |break| is often more efficient, because in other cases the list to iterate
  over is first completely constructed.
\begin{everbatim*}
First Fibonacci number at least |2^31| and its index
\xinttheiiexpr iter(0,1; (@1>=2^31)?{break(i)}{@2+@1}, i=1++)\relax
\end{everbatim*}

\begin{everbatim*}
Prime numbers are always cool
\xinttheiiexpr seq((seq((subs((x/:m)?{(m*m>x)?{1}{0}}{-1},m=2n+1))
                        ??{break(0)}{omit}{break(1)},n=1++))?{x}{omit},
               x=10001..[2]..10200)\relax
\end{everbatim*}

The syntax in this last example may look a bit involved. First |x/:m| computes
|x modulo m| (this is the modulo with respect to truncated division, which
here for positive arguments is like Euclidean division; in
|\xintexpr...\relax|, |a/:b| is such that |a = b*(a//b)+a/:b|, with |a//b| the
algebraic quotient |a/b| truncated to an integer.). The |(x)?{yes}{no}|
construct checks if |x| (which \emph{must} be within parentheses) is true or
false, i.e. non zero or zero. It then executes either the |yes| or the |no|
branch, the non chosen branch is \emph{not} evaluated. Thus if |m| divides |x|
we are in the second (``false'') branch. This gives a |-1|. This |-1| is the
argument to a |??| branch which is of the type |(y)??{y<0}{y=0}{y>0}|, thus here
the |y<0|, i.e., |break(0)| is chosen. This |0| is thus given to another |?|
which consequently chooses |omit|, hence the number is not kept in the list.
The numbers which survive are the prime numbers.

% A006877 In the `3x+1' problem, these values for the starting value set new
% records for number of steps to reach 1. (Formerly M0748) 14 1, 2, 3, 6, 7,
% 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161,
% 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239,
% 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935,
% 626331, 837799

\item The |iter| function is like |rrseq| but does not leave a trace of earlier items,
  it starts with |K| initial values, then it iterates: either a fixed number of times,
  or until aborting or breaking. And ultimately it prints |K| final values.
\begin{everbatim*}
The first Fibonacci number beyond the \TeX{} bound is
\xinttheiiexpr subs(iter(0,1;(@1>N)?{break(i)}{@1+@2},i=1++),N=2^31)\relax{}
and the previous number was its index.
\end{everbatim*}But this was a bit too easy, what is the smallest Fibonacci number not representable on 64 bits?
\begin{everbatim*}
The first Fibonacci number beyond |2^64| bound is
\xinttheiiexpr subs(iter(0,1;(@1>N)?{break(i)}{@1+@2},i=1++),N=2^64)\relax{}
and the previous number was its index.
\end{everbatim*}

One more recursion:
\begin{everbatim*}
\def\syr #1{\xinttheiiexpr rseq(#1; (@<=1)?{break(i)}{odd(@)?{3@+1}{@//2}},i=0++)\relax}
The 3x+1 problem: \syr{231}\par
\end{everbatim*}

Ok, a final one:
\begin{everbatim*}
\def\syrMax #1{\xinttheiiexpr iter(#1,#1;even(i)?
                                       {(@2<=1)?{break(i/2)}{odd(@2)?{3@2+1}{@2//2}}}
                                       {(@1>@2)?{@1}{@2}},i=0++)\relax }
With initial value 1161, the maximal number attained is \syrMax{1161} and that latter
number is the number of steps which was needed to reach 1.\par
\end{everbatim*}

Well, one more:

\begin{everbatim*}
\newcommand\GCD [2]{\xinttheiiexpr rrseq(#1,#2; (@1=0)?{abort}{@2/:@1}, i=1++)\relax }
\GCD {13^10*17^5*29^5}{2^5*3^6*17^2}
\end{everbatim*}

and the ultimate:

\begin{everbatim*}
\newcommand\Factors [1]{\xinttheiiexpr
    subs(seq((i/:3=2)?{omit}{[L][i]},i=1..([L][0])),
    L=rseq(#1;(p^2>[@][1])?{([@][1]>1)?{break(1,[@][1],1)}{abort}}
                          {(([@][1])/:p)?{omit}
    {iter(([@][1])//p; (@/:p)?{break(@,p,e)}{@//p},e=1++)}},p=2++))\relax }
\Factors {41^4*59^2*29^3*13^5*17^8*29^2*59^4*37^6}
\end{everbatim*}

This might look a bit scary, I admit. \xintexprname has minimal tools and
is obstinate about doing everything expandably! We are hampered by absence of a
notion of ``nuple''. The algorithm divides |N| by |2| until no more possible,
then by |3|, then by |4| (which is silly), then by |5|, then by |6| (silly
again), \dots. 

The variable |L=rseq(#1;...)| expands, if one follows the steps, to a comma
separated list starting with the initial (evaluated) |N=#1| and then
pseudo-triplets where the first item is |N| trimmed of small primes, the
second item is the last prime divisor found, and the third item is its
exponent in original |N|.

The algorithm needs to keep handy the last computed quotient by prime powers,
hence all of them, but at the very end it will be cleaner to get rid of them
(this corresponds to the first line in the code above). This is achieved in a
cumbersome inefficient way; indeed each item extraction |[L][i]| is costly: it
is not like accessing an array stored in memory, due to expandability, nothing
can be stored in memory! Nevertheless, this step could be done here in a far
less inefficient manner if there was a variant of |seq| which, in the spirit
of \csbxint{iloopindex}, would know how many steps it had been through so far.
This is a feature to be added to |\xintexpr|! (as well as a |++| construct
allowing a non unit step).

Notice that in |iter(([@][1])//p;| the |@| refers to the previous triplet (or
in the first step to |N|), but the latter |@| showing up in |(@/:p)?| refers
to the previous value computed by |iter|. 

\begin{snugframed}
  Parentheses are essential in |..([y][0])| else the parser will see |..[| and
  end up in ultimate confusion, and also in |([@][1])/:p| else the parser will
  see the itemwise operator |]/| on lists and again be very confused (I could
  implement a |]/:| on lists, but in this situation this would also be very
  confusing to the parser.)
\end{snugframed}

For comparison, here is an \fexpan dable macro expanding to the same result,
but coded directly with the \xintname macros. Here |#1| can not be itself an
expression, naturally. But at least we let |\Factorize| \fexpan d its
argument.
\begin{everbatim}
\makeatletter
\newcommand\Factorize [1]
      {\romannumeral0\expandafter\factorize\expandafter{\romannumeral-`0#1}}%
\newcommand\factorize [1]{\xintiiifOne{#1}{ 1}{\factors@a #1.{#1};}}%
\def\factors@a #1.{\xintiiifOdd{#1}
   {\factors@c 3.#1.}%
   {\expandafter\factors@b \expandafter1\expandafter.\romannumeral0\xinthalf{#1}.}}%
\def\factors@b #1.#2.{\xintiiifOne{#2}
   {\factors@end {2, #1}}%
   {\xintiiifOdd{#2}{\factors@c 3.#2.{2, #1}}%
                     {\expandafter\factors@b \the\numexpr #1+\@ne\expandafter.%
                         \romannumeral0\xinthalf{#2}.}}%
}%
\def\factors@c #1.#2.{%
    \expandafter\factors@d\romannumeral0\xintiidivision {#2}{#1}{#1}{#2}%
}%
\def\factors@d #1#2#3#4{\xintiiifNotZero{#2}
   {\xintiiifGt{#3}{#1}
        {\factors@end {#4, 1}}% ultimate quotient is a prime with power 1
        {\expandafter\factors@c\the\numexpr #3+\tw@.#4.}}%
   {\factors@e 1.#3.#1.}%
}%
\def\factors@e #1.#2.#3.{\xintiiifOne{#3}
   {\factors@end {#2, #1}}%
   {\expandafter\factors@f\romannumeral0\xintiidivision {#3}{#2}{#1}{#2}{#3}}%
}%
\def\factors@f #1#2#3#4#5{\xintiiifNotZero{#2}
   {\expandafter\factors@c\the\numexpr #4+\tw@.#5.{#4, #3}}%
   {\expandafter\factors@e\the\numexpr #3+\@ne.#4.#1.}%
}%
\def\factors@end #1;{\xintlistwithsep{, }{\xintRevWithBraces {#1}}}%
\makeatother
\end{everbatim}

The macro |\Factorize| puts a little stress on the input save stack in order
not be bothered with previously gathered things. I timed it to be about eight
times faster than |\Factors| in test cases such as
|16246355912554185673266068721806243461403654781833| and others. Among the
various things explaining the speed-up, there is fact that we step by
increments of two, not one, and also that we divide only once, obtaining
quotient and remainder in one go. These two things already make for a speed-up
factor of about four. Thus, our earlier |\Factors| was not completely
inefficient, and was quite easier to come up with than |\Factorize|.

If we only considered small integers, we could write pure |\numexpr| methods
which would be very much faster (especially if we had a table of small primes
prepared first) but still ridiculously slow compared to any non expandable
implementation, not to mention use of programming languages directly accessing
the CPU registers\dots
\end{itemize}

%\phantomsection
\phantomsection\label{sec:expr11coords}

To conclude with this overview of the new features in \xintexprname |1.1|, I
will mention {\bfseries |\xintthecoords|} which converts a comma separated
list as produced by |\xintfloatexpr| or |\xintiexpr [d]| to the format
expected by the |TikZ| |coordinates| syntax.
\begin{everbatim*}
{\centering\begin{tikzpicture}[scale=10]\xintDigits:=8;
  \clip (-1.1,-.25) rectangle (.3,.25);
 \draw [blue] (-1.1,0)--(1,0);
 \draw [blue] (0,-1)--(0,+1);
  \draw plot[smooth] coordinates {\xintthecoords
    % converts into (x1, y1) (x2, y2)... format 
    \xintfloatexpr seq((x^2-1,mul(x-t,t=-1+[0..4]/2)),x=-1.2..[0.1]..+1.2) \relax };
\end{tikzpicture}\par }
\end{everbatim*}

% Notice: if x goes no take exactly value 1 or -1, the origin appears slightly
% off the curve, not MY fault!!!

\csbxint{thecoords} should be followed immediately by \csbxint{floatexpr} or
\csbxint{iexpr} or \csbxint{iiexpr}, but not |\xintthefloatexpr|, etc\dots 

Besides, as |TikZ| will not understand the |A/B[N]| format which is used on
output by |\xintexpr|, |\xintexpr| is not really usable with |\xintthecoords|
for a |TikZ| picture, but one may use it on its own, and the reason for the
spaces in and between coordinate pairs is to allow if necessary to print on
the page for examination with about correct line-breaks.

\begin{everbatim*}
\oodef\x{\xintthecoords \xintexpr rrseq(1/2,1/3; @1+@2, x=1..20)\relax }
\meaning\x +++
\end{everbatim*}

\etocdepthtag.toc {commands}
\indescriptionfalse
\addtocontents{toc}{\gdef\string\sectioncouleur{{joli}}}

\renewcommand{\etocaftertochook}{\addvspace{\bigskipamount}}


\section{Commands of the \xintkernelname package}
\label{sec:kernel}

\localtableofcontents

The \xintkernelname package contains mainly the common code base for handling
the load-order of the bundle packages, the management of catcodes at loading
time, definition of common constants and macro utilities which are used
throughout the code etc ... it is automatically loaded by all packages of the
bundle.

It provides a few macros possibly useful in other contexts.

\subsection{\csbh{odef}, \csbh{oodef}, \csbh{fdef}}
\label{odef}
\label{oodef}
\label{fdef}

\csa{oodef}|\controlsequence {<stuff>}| does
\everb|@
    \expandafter\expandafter\expandafter\def
    \expandafter\expandafter\expandafter\controlsequence
    \expandafter\expandafter\expandafter{<stuff>}
|

This works only for a single
|\controlsequence|, with no parameter text, even without parameters. An
alternative would be:
\everb|@
\def\oodef #1#{\def\oodefparametertext{#1}%
               \expandafter\expandafter\expandafter\expandafter
               \expandafter\expandafter\expandafter\def
               \expandafter\expandafter\expandafter\oodefparametertext
               \expandafter\expandafter\expandafter }
|

\noindent
but it does not allow |\global| as prefix, and, besides, would have anyhow its
use (almost) limited to parameter texts without macro parameter tokens
(except if the expanded thing does not see them, or is designed to deal with
them).

There is a similar macro |\odef| with only one expansion of the replacement text
|<stuff>|, and |\fdef| which expands fully |<stuff>| using |\romannumeral-`0|.

These tools are provided as it is sometimes wasteful (from the point of view
of running time) to do an |\edef| when one knows that the contents expand in
only two steps for example, as is the case with all (except \csbxint{loop} and
\csbxint{iloop}) the expandable macros of the \xintname packages. Each will be
defined only if \xintkernelname finds them currently undefined. They can be
prefixed with |\global|.


\subsection{\csbh{xintReverseOrder}}\label{xintReverseOrder}

\csa{xintReverseOrder}\marg{list}\etype{n} does not do any expansion of its
argument and just reverses the order of the tokens in the \meta{list}. Braces
are removed once and the enclosed material, now unbraced, does not get
reversed. Unprotected spaces (of any character code) are gobbled.
%
\leftedline{|\xintReverseOrder{\xintDigitsOf\xintiPow {2}{100}\to\Stuff}|}
%
\leftedline{gives:
  \ttfamily{\string\Stuff\string\to1002\string\xintiPow\string\xintDigitsOf}}

\subsection{\csbh{xintLength}}\label{xintLength}

\csa{xintLength}\marg{list}\etype{n} does not do \emph{any} expansion of its
argument and just counts how many tokens there are (possibly none). So to use
it to count things in the replacement text of a macro one should do
|\expandafter\xintLength\expandafter{\x}|. One may also use it inside macros
as |\xintLength{#1}|. Things enclosed in braces count as one. Blanks between
tokens are not counted. See \csbxint{NthElt}|{0}| (from \xinttoolsname) for a
variant which first \fexpan ds its argument.
%
\leftedline{|\xintLength {\xintiPow {2}{100}}|\dtt{=\xintLength
    {\xintiPow{2}{100}}}}
%
\leftedline{${}\neq{}$|\xintLen {\xintiPow {2}{100}}|\dtt{=\xintLen
    {\xintiPow{2}{100}}}}

\section{Commands of the \xinttoolsname package}
\label{sec:tools}

\localtableofcontents

\def\n{|{N}|}
\def\m{|{M}|}
\def\x{|{x}|}

These utilities used to be provided within the \xintname package; since |1.09g|
(|2013/11/22|) they have been moved to an independently usable package
\xinttoolsname, which has none of the \xintname facilities regarding big
numbers. Whenever relevant release |1.09h| has made the macros |\long| so they
accept |\par| tokens on input.

First the  completely expandable utilities up to \csbxint{iloop}, then the non
expandable utilities.

This section contains various concrete examples and ends with a
\hyperref[ssec:quicksort]{completely expandable implementation of the Quick Sort
  algorithm} together with a graphical illustration of its action.

See also \ref{xintReverseOrder} and \ref{xintLength} which come with package
\xintkernelname, automatically loaded by \xinttoolsname.

\subsection{\csbh{xintRevWithBraces}}\label{xintRevWithBraces}

%{\small New in release |1.06|.\par}

\edef\X{\xintRevWithBraces{12345}}
\edef\y{\xintRevWithBraces\X}
\expandafter\def\expandafter\w\expandafter
     {\romannumeral0\xintrevwithbraces{{\A}{\B}{\C}{\D}{\E}}}

%
\csa{xintRevWithBraces}\marg{list}\etype{f} first does the \fexpan sion of its
argument then it reverses the order of the tokens, or braced material, it
encounters, maintaining existing braces and adding a brace pair around each
naked token encountered. Space tokens (in-between top level braces or naked
tokens) are gobbled. This macro is mainly thought out for use on a \meta{list}
of such braced material; with such a list as argument the \fexpan sion will only
hit against the first opening brace, hence do nothing, and the braced stuff may
thus be macros one does not want to expand.
%
\leftedline{|\edef\x{\xintRevWithBraces{12345}}|}
%
\leftedline{|\meaning\x:|\dtt{\meaning\X}}
%
\leftedline{|\edef\y{\xintRevWithBraces\x}|}
%
\leftedline{|\meaning\y:|\dtt{\meaning\y}} 
%
The examples above could be defined with |\edef|'s because the braced material
did not contain macros. Alternatively:
%
\leftedline{|\expandafter\def\expandafter\w\expandafter|}
%
\leftedline{|{\romannumeral0\xintrevwithbraces{{\A}{\B}{\C}{\D}{\E}}}|}
%
\leftedline{|\meaning\w:|\dtt{\meaning\w}} 
%
The macro \csa{xintReverseWithBracesNoExpand}\etype{n} does the same job
without the initial expansion of its argument.


\subsection{\csbh{xintZapFirstSpaces}, \csbh{xintZapLastSpaces}, \csbh{xintZapSpaces}, \csbh{xintZapSpacesB}}
\label{xintZapFirstSpaces}
\label{xintZapLastSpaces}
\label{xintZapSpaces}
\label{xintZapSpacesB}
%{\small New with release |1.09f|.\par}

\csa{xintZapFirstSpaces}\marg{stuff}\etype{n} does not do \emph{any} expansion
of its argument, nor brace removal of any sort, nor does it alter \meta{stuff}
in anyway apart from stripping away all \emph{leading} spaces.

This macro will be mostly of interest to programmers who will know what I will
now be talking about. \emph{The essential points, naturally, are the complete
  expandability and the fact that no brace removal nor any other alteration is
  done to the input.}

\TeX's input scanner already converts consecutive blanks into single space
tokens, but |\xintZapFirstSpaces| handles successfully also inputs with
consecutive multiple space tokens.
However, it is assumed that \meta{stuff} does not contain (except inside braced
sub-material) space tokens of character code distinct from $32$.

It expands in two steps, and if the goal is to apply it to the
expansion text of |\x| to define |\y|, then one should do:
|\expandafter\def\expandafter\y\expandafter
        {\romannumeral0\expandafter\xintzapfirstspaces\expandafter{\x}}|.

Other use case: inside a macro as |\edef\x{\xintZapFirstSpaces {#1}}| assuming
naturally that |#1| is compatible with such an |\edef| once the leading spaces
have been stripped.

\begingroup
\def\x {  \a {  \X } {  \b  \Y }  }
%
\leftedline{|\xintZapFirstSpaces {  \a {  \X } {  \b  \Y }  }->|%
\dtt{\color{magenta}{}\expandafter\detokenize\expandafter
{\romannumeral0\expandafter\xintzapfirstspaces\expandafter{\x}}}+++}
\endgroup

\medskip

\noindent\csbxint{ZapLastSpaces}\marg{stuff}\etype{n}  does not do \emph{any} expansion of
its argument, nor brace removal of any sort, nor does it alter \meta{stuff} in
anyway apart from stripping away all \emph{ending} spaces. The same remarks as
for \csbxint{ZapFirstSpaces} apply.

% ATTENTION à l'\ignorespaces fait par \color!
\begingroup
\def\x {  \a {  \X } {  \b  \Y }  }
%
\leftedline{|\xintZapLastSpaces {  \a {  \X } {  \b  \Y }  }->|%
\dtt{\color{magenta}{}\expandafter\detokenize\expandafter
{\romannumeral0\expandafter\xintzaplastspaces\expandafter{\x}}}+++}
\endgroup

\medskip

\noindent\csbxint{ZapSpaces}\marg{stuff}\etype{n}  does not do \emph{any}
expansion of its
argument, nor brace removal of any sort, nor does it alter \meta{stuff} in
anyway apart from stripping away all \emph{leading} and all \emph{ending}
spaces. The same remarks as for \csbxint{ZapFirstSpaces} apply.

\begingroup
\def\x {  \a {  \X } {  \b  \Y }  }
%
\leftedline{|\xintZapSpaces {  \a {  \X } {  \b  \Y }  }->|%
\dtt{\color{magenta}{}\expandafter\detokenize\expandafter
{\romannumeral0\expandafter\xintzapspaces\expandafter{\x}}}+++}
\endgroup

\medskip

\noindent\csbxint{ZapSpacesB}\marg{stuff}\etype{n}  does not do \emph{any}
expansion of
its argument, nor does it alter \meta{stuff} in anyway apart from stripping away
all leading and all ending spaces and possibly removing one level of braces if
\meta{stuff} had the shape |<spaces>{braced}<spaces>|. The same remarks as for
\csbxint{ZapFirstSpaces} apply.

\begingroup
\def\x {  \a {  \X } {  \b  \Y }  }
%
\leftedline{|\xintZapSpacesB {  \a {  \X } {  \b  \Y }  }->|%
\dtt{\color{magenta}{}\expandafter\detokenize\expandafter
{\romannumeral0\expandafter\xintzapspacesb\expandafter{\x}}}+++}
\def\x {  { \a {  \X } {  \b  \Y } }  }
%
\leftedline{|\xintZapSpacesB {  { \a {  \X } {  \b  \Y } }  }->|%
\dtt{\color{magenta}{}\expandafter\detokenize\expandafter
{\romannumeral0\expandafter\xintzapspacesb\expandafter{\x}}}+++}
\endgroup
 The spaces here at the start and end of the output come from the braced
 material, and are not removed (one would need a second application for that;
 recall though that the \xintname zapping macros do not expand their argument).

\subsection{\csbh{xintCSVtoList}}
\label{xintCSVtoList}
\label{xintCSVtoListNoExpand}

% {\small New with release |1.06|. Starting with |1.09f|, \fbox{\emph{removes
%     spaces around commas}!}\par}

\csa{xintCSVtoList}|{a,b,c...,z}|\etype{f}  returns |{a}{b}{c}...{z}|. A
\emph{list} is by
convention in this manual simply a succession of tokens, where each braced thing
will count as one item (``items'' are defined according to the rules of \TeX{}
for fetching undelimited parameters of a macro, which are exactly the same rules
as for \LaTeX{} and command arguments [they are the same things]). The word
`list' in `comma separated list of items' has its usual linguistic meaning,
and then an ``item'' is what is delimited by commas.

So \csa{xintCSVtoList} takes on input a `comma separated list of items' and
converts it into a `\TeX{} list of braced items'. The argument to
|\xintCSVtoList| may be a macro: it will first be
\hyperref[sec:expansions]{\fexpan ded}. Hence the item before the first comma,
if it is itself a macro, will be expanded which may or may not be a good thing.
A space inserted at the start of the first item serves to stop that expansion
(and disappears). The macro \csbxint{CSVtoListNoExpand}\etype{n} does the same
job without
the initial expansion of the list argument.

Apart from that no expansion of the items is done and the list items may thus be
completely arbitrary (and even contain perilous stuff such as unmatched |\if|
and |\fi| tokens).

Contiguous spaces and tab characters, are collapsed by \TeX{}
into single spaces. All such spaces around commas%
%
\footnote{and multiple space tokens are not a problem; but those at the
  top level (not hidden inside braces) \emph{must} be of character code
  |32|.}
%
\fbox{are removed}, as well as
the spaces at the start and the spaces at the end of the list.%
%
\footnote{let us recall that this is all done completely expandably...
  There is absolutely no alteration of any sort of the item apart from
  the stripping of initial and final space tokens (of character code
  |32|) and brace removal if and only if the item apart from intial and
  final spaces (or more generally multiple |char 32| space tokens) is
  braced.}
%
The items may contain explicit |\par|'s or
empty lines (converted by the \TeX{} input parsing into |\par| tokens).

\begingroup

\edef\X{\xintCSVtoList { 1 ,{ 2 , 3 , 4 , 5 }, a , {b,T} U , { c , d } , { {x ,
        y} } }}

%
\leftedline{|\xintCSVtoList { 1 ,{ 2 , 3 , 4 , 5 }, a , {b,T} U , { c , d } ,
    { {x , y} } }|}
%
\leftedline{|->|%
{\makeatletter\dtt{\expandafter\strip@prefix\meaning\X}}}

One sees on this example how braces protect commas from
sub-lists to be perceived as delimiters of the top list. Braces around an entire
item are removed, even when surrounded by spaces before and/or after. Braces for
sub-parts of an item are not removed.

We observe also that there is a slight difference regarding the brace stripping
of an item: if the braces were not surrounded by spaces, also the initial and
final (but no other) spaces of the \emph{enclosed} material are removed. This is
the only situation where spaces protected by braces are nevertheless removed.

From the rules above: for an empty argument (only spaces, no braces, no comma)
the output is
\dtt{\expandafter\detokenize\expandafter{\romannumeral0\xintcsvtolist { }}}
(a list with one empty item),
for ``|<opt. spaces>{}<opt.
spaces>|'' the output is
\dtt{\expandafter\detokenize\expandafter
   {\romannumeral0\xintcsvtolist { {} }}}
(again a list with one empty item, the braces were removed),
for ``|{ }|'' the output is
\dtt{\expandafter\detokenize\expandafter
 {\romannumeral0\xintcsvtolist {{ }}}}
(again a list with one empty item, the braces were removed and then
the inner space was removed),
for ``| { }|'' the output is
\dtt{\expandafter\detokenize\expandafter
{\romannumeral0\xintcsvtolist { { }}}} (again a list with one empty item, the initial space served only to stop the expansion, so this was like ``|{ }|'' as input, the braces were removed and the inner space was stripped),
for ``\texttt{\ \{\ \ \}\ }'' the output is
\dtt{\expandafter\detokenize\expandafter
{\romannumeral0\xintcsvtolist { {  } }}} (this time the ending space of the first
item meant that after brace removal the inner spaces were kept; recall though
that \TeX{} collapses on input consecutive blanks into one space token),
for ``|,|'' the output consists of two consecutive
empty items
\dtt{\expandafter\detokenize\expandafter{\romannumeral0\xintcsvtolist
    {,}}}. Recall that on output everything is braced, a |{}| is an ``empty''
item.
%
Most of the above is mainly irrelevant for every day use, apart perhaps from the
fact to be noted that an empty input does not give an empty output but a
one-empty-item list (it is as if an ending comma was always added at the end of
the input).

\def\y { \a,\b,\c,\d,\e}
\expandafter\def\expandafter\Y\expandafter{\romannumeral0\xintcsvtolist{\y}}
\def\t {{\if},\ifnum,\ifx,\ifdim,\ifcat,\ifmmode}
\expandafter\def\expandafter\T\expandafter{\romannumeral0\xintcsvtolist{\t}}

%
\leftedline{|\def\y{ \a,\b,\c,\d,\e} \xintCSVtoList\y->|%
  {\makeatletter\dtt{\expandafter\strip@prefix\meaning\Y}}}
%
\leftedline{|\def\t {{\if},\ifnum,\ifx,\ifdim,\ifcat,\ifmmode}|} 
%
\leftedline
{|\xintCSVtoList\t->|\makeatletter\dtt{\expandafter\strip@prefix\meaning\T}}
%
The results above were automatically displayed using \TeX's primitive
\csa{meaning}, which adds a space after each control sequence name. These spaces
are not in the actual braced items of the produced lists. The first items |\a|
and |\if| were either preceded by a space or braced to prevent expansion. The
macro \csa{xintCSVtoListNoExpand} would have done the same job without the
initial expansion of the list argument, hence no need for such protection but if
|\y| is defined as |\def\y{\a,\b,\c,\d,\e}| we then must do:
%
\leftedline{|\expandafter\xintCSVtoListNoExpand\expandafter {\y}|} Else, we
may have direct use: %
%
\leftedline{|\xintCSVtoListNoExpand
 {\if,\ifnum,\ifx,\ifdim,\ifcat,\ifmmode}|}
%
% ATTENTION 18 novembre TEST DE newtxtt 1.05 PAS POSSIBLE \textsc DANS \dtt
% mais on peut avec \scshape. Finalement je n'utilise pas les old style figures.
\leftedline{|->|\dtt{\expandafter\detokenize\expandafter
    {\romannumeral0\xintcsvtolistnoexpand
      {\if,\ifnum,\ifx,\ifdim,\ifcat,\ifmmode}}}}
%
Again these spaces are an artefact from the use in the source of the document of
\csa{meaning} (or rather here, \csa{detokenize}) to display the result of using
\csa{xintCSVtoListNoExpand} (which is done for real in this document
source).

For the similar conversion from comma separated list to braced items list, but
without removal of spaces around the commas, there is
\csa{xintCSVtoListNonStripped}\etype{f} and
\csa{xintCSVtoListNonStrippedNoExpand}\etype{n}.

\endgroup

\subsection{\csbh{xintNthElt}}\label{xintNthElt}

% {\small New in release |1.06|. With |1.09b| negative indices count from the tail.\par}

\def\macro #1{\the\numexpr 9-#1\relax}

\csa{xintNthElt\x}\marg{list}\etype{\numx f} gets (expandably) the |x|th braced
item of the \meta{list}. An unbraced item token will be returned as is. The list
itself may be a macro which is first \fexpan ded.

\leftedline{|\xintNthElt {3}{{agh}\u{zzz}\v{Z}}| is
    \texttt{\xintNthElt {3}{{agh}\u{zzz}\v{Z}}}}
%
\leftedline{|\xintNthElt {3}{{agh}\u{{zzz}}\v{Z}}| is
    \texttt{\expandafter\expandafter\expandafter
      \detokenize\expandafter\expandafter\expandafter {\xintNthElt
        {3}{{agh}\u{{zzz}}\v{Z}}}}}
%
\leftedline{|\xintNthElt {2}{{agh}\u{{zzz}}\v{Z}}| is
    \texttt{\expandafter\expandafter\expandafter
      \detokenize\expandafter\expandafter\expandafter {\xintNthElt
        {2}{{agh}\u{{zzz}}\v{Z}}}}}
%
\leftedline{|\xintNthElt {37}{\xintFac {100}}|\dtt{=\xintNthElt
      {37}{\xintFac {100}}} is the thirty-seventh digit of $100!$.}
%
\leftedline{|\xintNthElt {10}{\xintFtoCv
      {566827/208524}}|\dtt{=\xintNthElt {10}{\xintFtoCv
        {566827/208524}}}}
\leftedline{is the tenth convergent of $566827/208524$ (uses \xintcfracname
  package).} 
%
\leftedline{|\xintNthElt {7}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}|%
    \dtt{=\xintNthElt {7}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}}}
%
\leftedline{|\xintNthElt {0}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}|%
    \dtt{=\xintNthElt {0}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}}}
%
\leftedline{|\xintNthElt {-3}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}|%
    \dtt{=\xintNthElt {-3}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}}}

If |x=0|,
the macro returns the \emph{length} of the expanded list: this is not equivalent
to \csbxint{Length} which does no pre-expansion. And it is different from
\csbxint{Len} which is to be used only on integers or fractions.

If |x<0|, the macro returns the \verb+|x|+th element from the end of the list.
%
\leftedline {|\xintNthElt {-5}{{{agh}}\u{zzz}\v{Z}}| is
  \texttt{\expandafter\expandafter\expandafter \detokenize
  \expandafter\expandafter\expandafter{\xintNthElt {-5}{{{agh}}\u{zzz}\v{Z}}}}}

The macro \csa{xintNthEltNoExpand}\etype{\numx n} does the same job but without
first expanding the list argument: |\xintNthEltNoExpand {-4}{\u\v\w T\x\y\z}| is
\xintNthEltNoExpand {-4}{\a\b\c\u\v\w T\x\y\z}.

In cases where |x| is larger (in absolute value) than the length of the list
then |\xintNthElt| returns nothing.

\subsection{\csbh{xintKeep}}\label{xintKeep}

\csa{xintKeep\x}\marg{list}\etype{\numx f} expands the list argument and returns
a new list containing only the first |x| elements.\NewWith {1.09m} If |x<0| the
macro returns the last \verb+|x|+ elements (in the same order as in the initial
list). If \verb+|x|+ equals or exceeds the length of the list, the list (as
arising from expansion of the second argument) is returned. For |x=0| the empty
list is returned.

Naked (non space) tokens from the original count each as one item and they end
up braced in the output (if present there): if one later wants to remove all
brace pairs (either added to a naked token, or initially present), one may use
\csbxint {ListWithSep} with an empty separator.

\csa{xintKeepNoExpand} does the same without first \fexpan ding its list
argument.
%
\begin{everbatim*}
\oodef\test {\xintKeep {17}{\xintKeep {-69}{\xintSeq {1}{100}}}}\meaning\test
\end{everbatim*}
%

\subsection{\csbh{xintTrim}}\label{xintTrim}

\csa{xintTrim\x}\marg{list}\etype{\numx f} expands the list argument and
gobbles its first |x| elements. If |x<0| the macro gobbles the last
\verb+|x|+ elements.\NewWith {1.09m} If \verb+|x|+ equals or exceeds
the length of the list, the empty list is returned. For |x=0| the full
list is returned.

Naked (non space) tokens from the original count each as one item and they end
up braced in the output (if present there).

\csa{xintTrimNoExpand} does the same without first \fexpan ding its list
argument.
\begin{everbatim*}
\oodef\test {\xintTrim {17}{\xintTrim {-69}{\xintSeq {1}{100}}}}\meaning\test
\end{everbatim*}

\subsection{\csbh{xintListWithSep}}\label{xintListWithSep}

%{\small New with release |1.04|.\par}

\def\macro #1{\the\numexpr 9-#1\relax}

\csa{xintListWithSep}|{sep}|\marg{list}\etype{nf} inserts the given separator
|sep| in-between all items of the given list of braced items: this separator may
be a macro (or multiple tokens) but will not be expanded. The second argument
also may be itself a macro: it is \fexpan ded. Applying \csa{xintListWithSep}
removes the braces from the list items (for example |{1}{2}{3}| turns into
\dtt{\xintListWithSep,{123}} via |\xintListWithSep{,}{{1}{2}{3}}|). An
empty input gives an empty output, a singleton gives a singleton, the separator
is used starting with at least two elements. Using an empty separator has the
net effect of unbracing the braced items constituting the \meta{list} (in such
cases the new list may thus be longer than the original).
%
\leftedline{|\xintListWithSep{:}{\xintFac
    {20}}|\dtt{=\xintListWithSep{:}{\xintFac {20}}}}

The  macro \csa{xintListWithSepNoExpand}\etype{nn} does the same
job without the initial expansion.

\subsection{\csbh{xintApply}}\label{xintApply}

%{\small New with release |1.04|.\par}

\def\macro #1{\the\numexpr 9-#1\relax}

\csa{xintApply}|{\macro}|\marg{list}\etype{ff} expandably applies the one
parameter command |\macro| to each item in the \meta{list} given as second
argument and returns a new list with these outputs: each item is given one after
the other as parameter to |\macro| which is expanded at that time (as usual,
\emph{i.e.} fully for what comes first), the results are braced and output
together as a succession of braced items (if |\macro| is defined to start with a
space, the space will be gobbled and the |\macro| will not be expanded; it is
allowed to have its own arguments, the list items serve as last arguments to
|\macro|). Hence |\xintApply{\macro}{{1}{2}{3}}| returns
|{\macro{1}}{\macro{2}}{\macro{3}}| where all instances of |\macro| have been
already \fexpan ded.

Being expandable, |\xintApply| is useful for example inside alignments where
implicit groups make standard loops constructs usually fail. In such situation
it is often not wished that the new list elements be braced, see
\csbxint{ApplyUnbraced}. The |\macro| does not have to be expandable:
|\xintApply| will try to expand it, the expansion may remain partial.

The \meta{list} may
itself be some macro expanding (in the previously described way) to the list of
tokens to which the command |\macro| will be applied. For example, if the
\meta{list} expands to some positive number, then each digit will be replaced by
the result of applying |\macro| on it. %
%
\leftedline{|\def\macro #1{\the\numexpr
    9-#1\relax}|} %
%
\leftedline{|\xintApply\macro{\xintFac
 {20}}|\dtt{=\xintApply\macro{\xintFac {20}}}}

The macro \csa{xintApplyNoExpand}\etype{fn} does the same job without the first
initial expansion which gave the \meta{list} of braced tokens to which |\macro|
is applied.

\subsection{\csbh{xintApplyUnbraced}}\label{xintApplyUnbraced}

%{\small New in release |1.06b|.\par}

\csa{xintApplyUnbraced}|{\macro}|\marg{list}\etype{ff} is like \csbxint{Apply}.
The difference is that after having expanded its list argument, and applied
|\macro| in turn to each item from the list, it reassembles the outputs without
enclosing them in braces. The net effect is the same as doing
%
\leftedline{|\xintListWithSep {}{\xintApply {\macro}|\marg{list}|}|} This is
useful for preparing a macro which will itself define some other macros or make
assignments, as the scope will not be limited by brace pairs.
%
\begin{everbatim*}
\def\macro #1{\expandafter\def\csname myself#1\endcsname {#1}}
\xintApplyUnbraced\macro{{elta}{eltb}{eltc}}
\begin{enumerate}[nosep,label=(\arabic{*})]
\item \meaning\myselfelta
\item \meaning\myselfeltb
\item \meaning\myselfeltc
\end{enumerate}
\end{everbatim*}

%
The macro \csa{xintApplyUnbracedNoExpand}\etype{fn} does the same job without
the first initial expansion which gave the \meta{list} of braced tokens to which
|\macro| is applied.

\subsection{\csbh{xintSeq}}\label{xintSeq}
%{\small New with release |1.09c|.\par}

\csa{xintSeq}|[d]{x}{y}|\etype{{{\upshape[\numx]}}\numx\numx} generates
expandably |{x}{x+d}...| up to and possibly including |{y}| if |d>0| or
down to and including |{y}| if |d<0|. Naturally |{y}| is omitted if
|y-x| is not a multiple of |d|. If |d=0| the macro returns |{x}|. If
|y-x| and |d| have opposite signs, the macro returns nothing. If the
optional argument |d| is omitted it is taken to be the sign of |y-x|
(beware that |\xintSeq {1}{0}| is thus not empty but |{1}{0}|, use
|\xintSeq [1]{1}{N}| if you want an empty sequence for |N| zero or
negative).

The current implementation is only for (short) integers; possibly, a future
variant could allow big integers and fractions, although one already has
access to similar
functionality using \csbxint{Apply} to get any arithmetic sequence of long
integers. Currently thus, |x| and |y| are expanded inside a
|\numexpr| so they may be count registers or a \LaTeX{} |\value{countername}|,
or arithmetic with such things.

%
\begin{everbatim*}
\xintListWithSep{,\hskip2pt plus 1pt minus 1pt }{\xintSeq {12}{-25}}
\end{everbatim*}
%
\begin{everbatim*}
\xintiiSum{\xintSeq [3]{1}{1000}}
\end{everbatim*}

\textbf{Important:} for reasons of efficiency, this macro, when not given the
optional argument |d|, works backwards, leaving in the token stream the already
constructed integers, from the tail down (or up). But this will provoke a
failure of \IMPORTANT{} the |tex| run if the number of such items exceeds the
input stack
limit; on my installation this limit is at $5000$.

However, when given the optional argument |d| (which may be $+1$ or
$-1$), the macro proceeds differently and does not put stress on the input stack
(but is significantly slower for sequences with thousands of integers,
especially if they are somewhat big). For
example: |\xintSeq [1]{0}{5000}| works and |\xintiiSum{\xintSeq [1]{0}{5000}}|
returns the correct value \dtt{\xintHalf{\xintiMul{5000}{5001}}}.

The produced integers are with explicit litteral digits, so if used in |\ifnum|
or other tests they should be properly terminated%
%
\footnote{a \csa{space} will
  stop the \TeX{} scanning of a number and be gobbled in the process,
  maintaining expandability if this is required; the \csa{relax} stops the
  scanning but is not gobbled and remains afterwards as a token.}.

\subsection{Completely expandable prime test}\label{ssec:primesI}

Let us now construct a completely expandable macro which returns $1$ if its
given input is prime and $0$ if not:
\everb|@
\def\remainder #1#2{\the\numexpr #1-(#1/#2)*#2\relax }
\def\IsPrime #1%
 {\xintANDof {\xintApply {\remainder {#1}}{\xintSeq {2}{\xintiSqrt{#1}}}}}
|

This uses \csbxint{iSqrt} and assumes its input is at least $5$. Rather than
\xintname's own \csbxint{iRem} we used a quicker |\numexpr| expression as we
are dealing with short integers. Also we used \csbxint{ANDof} which will
return $1$ only if all the items are non-zero. The macro is a bit
silly with an even input, ok, let's enhance it to detect an even input:
\everb|@
\def\IsPrime #1%
   {\xintifOdd {#1}
        {\xintANDof % odd case
            {\xintApply {\remainder {#1}}
                        {\xintSeq [2]{3}{\xintiSqrt{#1}}}%
            }%
        }
        {\xintifEq {#1}{2}{1}{0}}%
   }
|

We used the \xintname provided expandable tests (on big integers or fractions)
in oder for |\IsPrime| to be \fexpan dable.

Our integers are short, but without |\expandafter|'s with
%\makeatletter|\@firstoftwo|\catcode`@ \active, 
|\@firstoftwo|, % @ n'est plus actif dans le dtx 1.1 !
or some other related techniques,
direct use of |\ifnum..\fi| tests is dangerous. So to make the macro more
efficient we are going to use the expandable tests provided by the package
\href{http://ctan.org/pkg/etoolbox}{etoolbox}%
%
\footnote{\url{http://ctan.org/pkg/etoolbox}}.
%
The macro becomes:
%
\everb|@
\def\IsPrime #1%
   {\ifnumodd {#1}
    {\xintANDof % odd case
     {\xintApply {\remainder {#1}}{\xintSeq [2]{3}{\xintiSqrt{#1}}}}}
    {\ifnumequal {#1}{2}{1}{0}}}
|

In the odd case however we have to assume the integer is at least $7$, as
|\xintSeq| generates an empty list if |#1=3| or |5|, and |\xintANDof| returns
$1$ when supplied an empty list. Let us ease up a bit |\xintANDof|'s work by
letting it work on only $0$'s and $1$'s. We could use:
%
\everb|@
\def\IsNotDivisibleBy #1#2%
  {\ifnum\numexpr #1-(#1/#2)*#2=0 \expandafter 0\else \expandafter1\fi}
|
\noindent
where the |\expandafter|'s are crucial for this macro to be \fexpan dable and
hence work within the applied \csbxint{ANDof}. Anyhow, now that we have loaded
\href{http://ctan.org/pkg/etoolbox}{etoolbox}, we might as well use:
%
\everb|@
\newcommand{\IsNotDivisibleBy}[2]{\ifnumequal{#1-(#1/#2)*#2}{0}{0}{1}}
|
\noindent
Let us enhance our prime macro to work also on the small primes:
\everb|@
\newcommand{\IsPrime}[1] % returns 1 if #1 is prime, and 0 if not
  {\ifnumodd {#1}
    {\ifnumless {#1}{8}
      {\ifnumequal{#1}{1}{0}{1}}% 3,5,7 are primes
      {\xintANDof
         {\xintApply
        { \IsNotDivisibleBy {#1}}{\xintSeq [2]{3}{\xintiSqrt{#1}}}}%
        }}% END OF THE ODD BRANCH
    {\ifnumequal {#1}{2}{1}{0}}% EVEN BRANCH
}
|

The input is still assumed positive. There is a deliberate blank before
\csa{IsNotDivisibleBy} to use this feature of \csbxint{Apply}: a space stops the
expansion of the applied macro (and disappears). This expansion will be done by
\csbxint{ANDof}, which has been designed to skip everything as soon as it finds
a false (i.e. zero) input. This way, the efficiency is considerably improved.

We did generate via the \csbxint{Seq} too many potential divisors though. Later
sections give two variants: one with \csbxint{iloop} (\autoref{ssec:primesII})
which is still expandable and another one (\autoref{ssec:primesIII}) which is a
close variant of the |\IsPrime| code above but with the \csbxint{For} loop, thus
breaking expandability. The \hyperref[ssec:primesII]{xintiloop variant} does not
first evaluate the integer square root, the \hyperref[ssec:primesIII]{xintFor
  variant} still does. I did not compare their efficiencies.

% Hmm, if one really needs to compute primes fast, sure I do applaud using
% \xintname, but, well, there is some slight
% overhead\MyMarginNoteWithBrace{funny private joke} in using \TeX{} for these
% things (something like a factor $1000$? not tested\dots) compared to accessing
% to the |CPU| ressources via standard compiled code from a standard programming
% language\dots

Let us construct with this expandable primality test a table of the prime
numbers up to $1000$. We need to count how many we have in order to know how
many tab stops one shoud add in the last row.%
%
\footnote{although a tabular row may have less tabs than in the
  preamble, there is a problem with the \char`\|\space\space vertical
  rule, if one does that.}
%
There is some subtlety for this
last row. Turns out to be better to insert a |\\| only when we know for sure we
are starting a new row; this is how we have designed the |\OneCell| macro. And
for the last row, there are many ways, we use again |\xintApplyUnbraced| but
with a macro which gobbles its argument and replaces it with a tabulation
character. The \csbxint{For*} macro would be more elegant here.
%
\everb?@
\newcounter{primecount}
\newcounter{cellcount}
\newcommand{\NbOfColumns}{13}
\newcommand{\OneCell}[1]{%
    \ifnumequal{\IsPrime{#1}}{1}
     {\stepcounter{primecount}
      \ifnumequal{\value{cellcount}}{\NbOfColumns}
       {\\\setcounter{cellcount}{1}#1}
       {&\stepcounter{cellcount}#1}%
     } % was prime
  {}% not a prime, nothing to do
}
\newcommand{\OneTab}[1]{&}
\begin{tabular}{|*{\NbOfColumns}{r}|}
\hline
2  \setcounter{cellcount}{1}\setcounter{primecount}{1}%
   \xintApplyUnbraced \OneCell {\xintSeq [2]{3}{999}}%
   \xintApplyUnbraced \OneTab
      {\xintSeq [1]{1}{\the\numexpr\NbOfColumns-\value{cellcount}\relax}}%
    \\
\hline
\end{tabular}
There are \arabic{primecount} prime numbers up to 1000.
?

The table has been put in \hyperref[primesupto1000]{float} which appears
\vpageref{primesupto1000}.
We had to be careful to use in the last row \csbxint{Seq} with its optional
argument |[1]| so as to not generate a decreasing sequence from |1| to |0|, but
really an empty sequence in case the row turns out to already have all its
cells (which doesn't happen here but would with a number of columns dividing
$168$).
%
\newcommand{\IsNotDivisibleBy}[2]{\ifnumequal{#1-(#1/#2)*#2}{0}{0}{1}}

\newcommand{\IsPrime}[1]
   {\ifnumodd {#1}
        {\ifnumless {#1}{8}
          {\ifnumequal{#1}{1}{0}{1}}% 3,5,7 are primes
          {\xintANDof
             {\xintApply
                { \IsNotDivisibleBy {#1}}{\xintSeq [2]{3}{\xintiSqrt{#1}}}}%
            }}% END OF THE ODD BRANCH
        {\ifnumequal {#1}{2}{1}{0}}% EVEN BRANCH
}

\newcounter{primecount}
\newcounter{cellcount}
\newcommand{\NbOfColumns}{13}
\newcommand{\OneCell}[1]
     {\ifnumequal{\IsPrime{#1}}{1}
        {\stepcounter{primecount}
         \ifnumequal{\value{cellcount}}{\NbOfColumns}
            {\\\setcounter{cellcount}{1}#1}
            {&\stepcounter{cellcount}#1}%
        } % was prime
        {}% not a prime nothing to do
}
\newcommand{\OneTab}[1]{&}
\begin{figure*}[ht!]
  \centering
  \phantomsection\label{primesupto1000}
  \begin{tabular}{|*{\NbOfColumns}{r}|}
    \hline
    2\setcounter{cellcount}{1}\setcounter{primecount}{1}%
    \xintApplyUnbraced \OneCell {\xintSeq [2]{3}{999}}%
    \xintApplyUnbraced \OneTab
    {\xintSeq [1]{1}{\the\numexpr\NbOfColumns-\value{cellcount}\relax}}%
    \\
    \hline
  \end{tabular}
\smallskip
\centeredline{There are \arabic{primecount} prime numbers up to 1000.}
\end{figure*}

\subsection{\csbh{xintloop}, \csbh{xintbreakloop}, \csbh{xintbreakloopanddo}, \csbh{xintloopskiptonext}}
\label{xintloop}
\label{xintbreakloop}
\label{xintbreakloopanddo}
\label{xintloopskiptonext}
% {\small New with release |1.09g|. Release |1.09h|
%   makes them long macros.\par}

|\xintloop|\meta{stuff}|\if<test>...\repeat|\retype{} is an expandable loop
compatible with nesting. However to break out of the loop one almost always need
some un-expandable step. The cousin \csbxint{iloop} is \csbxint{loop} with an
embedded expandable mechanism allowing to exit from the loop. The iterated
commands may contain |\par| tokens or empty lines.

If a sub-loop is to be used all the material from the start of the main loop and
up to the end of the entire subloop should be braced; these braces will be
removed and do not create a group. The simplest to allow the nesting of one or
more sub-loops is to brace everything between \csa{xintloop} and \csa{repeat},
being careful not to leave a space between the closing brace and |\repeat|.

As this loop and \csbxint{iloop} will primarily be of interest to experienced
\TeX{} macro programmers, my description will assume that the user is
knowledgeable enough. Some examples in this document will be perhaps more
illustrative than my attemps at explanation of use.

One can abort the loop with \csbxint{breakloop}; this should not be used inside
the final test, and one should expand the |\fi| from the corresponding test
before. One has also \csbxint{breakloopanddo} whose first argument will be
inserted in the token stream after the loop; one may need a macro such as
|\xint_afterfi| to move the whole thing after the |\fi|, as a simple
|\expandafter| will not be enough.

One will usually employ some count registers to manage the exit test from the
loop; this breaks expandability, see \csbxint{iloop} for an expandable integer
indexed loop. Use in alignments will be complicated by the fact that cells
create groups, and also from the fact that any encountered unexpandable material
will cause the \TeX{} input scanner to insert |\endtemplate| on each encountered
|&| or |\cr|; thus |\xintbreakloop| may not work as expected, but the situation
can be resolved via |\xint_firstofone{&}| or use of |\TAB| with |\def\TAB{&}|.
It is thus simpler for alignments to use rather than \csbxint{loop} either the
expandable \csbxint{ApplyUnbraced} or the non-expandable but alignment
compatible \csbxint{ApplyInline}, \csbxint{For} or \csbxint{For*}.

As an example, let us suppose we have two macros |\A|\marg{i}\marg{j} and
|\B|\marg{i}\marg{j} behaving like (small) integer valued matrix entries, and we
want to define a macro |\C|\marg{i}\marg{j} giving the matrix product (|i| and
|j| may be count registers). We will assume that |\A[I]| expands to the number
of rows, |\A[J]| to the number of columns and want the produced |\C| to act in
the same manner. The code is very dispendious in use of |\count| registers, not
optimized in any way, not made very robust (the defined macro can not have the
same name as the first two matrices for example), we just wanted to quickly
illustrate use of the nesting capabilities of |\xintloop|.%
%
\footnote{for a more sophisticated implementation of matrix
  multiplication, inclusive of determinants, inverses, and display
  utilities, with entries big integers or decimal numbers or even
  fractions see \url{http://tex.stackexchange.com/a/143035/4686} from
  November 11, 2013.}
%

%\def\everbhook {\makeatother }

\begin{everbatim*}
\newcount\rowmax   \newcount\colmax   \newcount\summax
\newcount\rowindex \newcount\colindex \newcount\sumindex
\newcount\tmpcount
\makeatletter
\def\MatrixMultiplication #1#2#3{%
    \rowmax #1[I]\relax
    \colmax #2[J]\relax
    \summax #1[J]\relax
    \rowindex 1
    \xintloop % loop over row index i
    {\colindex 1
     \xintloop % loop over col index k
     {\tmpcount 0
      \sumindex 1
      \xintloop % loop over intermediate index j
      \advance\tmpcount \numexpr #1\rowindex\sumindex*#2\sumindex\colindex\relax
      \ifnum\sumindex<\summax
         \advance\sumindex 1
      \repeat }%
     \expandafter\edef\csname\string#3{\the\rowindex.\the\colindex}\endcsname
      {\the\tmpcount}%
     \ifnum\colindex<\colmax
         \advance\colindex 1
     \repeat }%
    \ifnum\rowindex<\rowmax
    \advance\rowindex 1
    \repeat
    \expandafter\edef\csname\string#3{I}\endcsname{\the\rowmax}%
    \expandafter\edef\csname\string#3{J}\endcsname{\the\colmax}%
    \def #3##1{\ifx[##1\expandafter\Matrix@helper@size
                    \else\expandafter\Matrix@helper@entry\fi #3{##1}}%
}%
\def\Matrix@helper@size #1#2#3]{\csname\string#1{#3}\endcsname }%
\def\Matrix@helper@entry #1#2#3%
   {\csname\string#1{\the\numexpr#2.\the\numexpr#3}\endcsname }%
\def\A #1{\ifx[#1\expandafter\A@size
            \else\expandafter\A@entry\fi {#1}}%
\def\A@size #1#2]{\ifx I#23\else4\fi}% 3rows, 4columns
\def\A@entry #1#2{\the\numexpr #1+#2-1\relax}% not pre-computed...
\def\B #1{\ifx[#1\expandafter\B@size
            \else\expandafter\B@entry\fi {#1}}%
\def\B@size #1#2]{\ifx I#24\else3\fi}% 4rows, 3columns
\def\B@entry #1#2{\the\numexpr #1-#2\relax}% not pre-computed...
\makeatother
\MatrixMultiplication\A\B\C \MatrixMultiplication\C\C\D 
\MatrixMultiplication\C\D\E \MatrixMultiplication\C\E\F
\begin{multicols}2
  \[\begin{pmatrix}
    \A11&\A12&\A13&\A14\\
    \A21&\A22&\A23&\A24\\
    \A31&\A32&\A33&\A34
  \end{pmatrix}
  \times
  \begin{pmatrix}
    \B11&\B12&\B13\\
    \B21&\B22&\B23\\
    \B31&\B32&\B33\\
    \B41&\B42&\B43
  \end{pmatrix}
  =
  \begin{pmatrix}
    \C11&\C12&\C13\\
    \C21&\C22&\C23\\
    \C31&\C32&\C33
  \end{pmatrix}\]
  \[\begin{pmatrix}
    \C11&\C12&\C13\\
    \C21&\C22&\C23\\
    \C31&\C32&\C33
  \end{pmatrix}^2 = \begin{pmatrix}
    \D11&\D12&\D13\\
    \D21&\D22&\D23\\
    \D31&\D32&\D33
  \end{pmatrix}\]
  \[\begin{pmatrix}
    \C11&\C12&\C13\\
    \C21&\C22&\C23\\
    \C31&\C32&\C33
  \end{pmatrix}^3 = \begin{pmatrix}
    \E11&\E12&\E13\\
    \E21&\E22&\E23\\
    \E31&\E32&\E33
  \end{pmatrix}\]
  \[\begin{pmatrix}
    \C11&\C12&\C13\\
    \C21&\C22&\C23\\
    \C31&\C32&\C33
  \end{pmatrix}^4 = \begin{pmatrix}
    \F11&\F12&\F13\\
    \F21&\F22&\F23\\
    \F31&\F32&\F33
  \end{pmatrix}\]
\end{multicols}
\end{everbatim*}

% \restoreeverbhook

\subsection{\csbh{xintiloop}, \csbh{xintiloopindex}, \csbh{xintouteriloopindex},
  \csbh{xintbreakiloop}, \csbh{xintbreakiloopanddo}, \csbh{xintiloopskiptonext},
\csbh{xintiloopskipandredo}}
\label{xintiloop}
\label{xintbreakiloop}
\label{xintbreakiloopanddo}
\label{xintiloopskiptonext}
\label{xintiloopskipandredo}
\label{xintiloopindex}
\label{xintouteriloopindex}
%{\small New with release |1.09g|.\par}

\csa{xintiloop}|[start+delta]|\meta{stuff}|\if<test> ... \repeat|\retype{} is a
completely expandable nestable loop. complete expandability depends naturally on
the actual iterated contents, and complete expansion will not be achievable
under a sole \fexpan sion, as is indicated by the hollow star in the margin;
thus the loop can be used inside an |\edef| but not inside arguments to the
package macros. It can be used inside an |\xintexpr..\relax|.

This loop benefits via \csbxint{iloopindex} to (a limited access to) the integer
index of the iteration. The starting value |start| (which may be a |\count|) and
increment |delta| (\emph{id.}) are mandatory arguments. A space after the
closing square bracket is not significant, it will be ignored. Spaces inside the
square brackets will also be ignored as the two arguments are first given to a
|\numexpr...\relax|. Empty lines and explicit |\par| tokens are accepted.

As with \csbxint{loop}, this tool will mostly be of interest to advanced users.
For nesting, one puts inside braces all the
material from the start (immediately after |[start+delta]|) and up to and
inclusive of the inner loop, these braces will be removed and do not create a
loop. In case of nesting, \csbxint{outeriloopindex} gives access to the index of
the outer loop. If needed one could write on its model a macro giving access to
the index of the outer outer loop (or even to the |nth| outer loop).

The \csa{xintiloopindex} and \csa{xintouteriloopindex} can not be used inside
braces, and generally speaking this means they should be expanded first when
given as argument to a macro, and that this macro receives them as delimited
arguments, not braced ones. Or, but naturally this will break expandability, one
can assign the value of \csa{xintiloopindex} to some |\count|. Both
\csa{xintiloopindex} and \csa{xintouteriloopindex} extend to the litteral
representation of the index, thus in |\ifnum| tests, if it comes last one has to
correctly end the macro with a |\space|, or encapsulate it in a
|\numexpr..\relax|.

When the repeat-test of the loop is, for example, |\ifnum\xintiloopindex<10
\repeat|, this means that the last iteration will be with |\xintiloopindex=10|
(assuming |delta=1|). There is also |\ifnum\xintiloopindex=10 \else\repeat| to
get the last iteration to be the one with |\xintiloopindex=10|.

One has \csbxint{breakiloop} and \csbxint{breakiloopanddo} to abort the loop.
The syntax of |\xintbreakiloopanddo| is a bit surprising, the sequence of tokens
to be executed after breaking the loop is not within braces but is delimited by
a dot as in:
%
\leftedline{|\xintbreakiloopanddo <afterloop>.etc.. etc... \repeat|}
%
The reason is that one may wish to use the then current value of
|\xintiloopindex| in |<afterloop>| but it can't be within braces at the time it
is evaluated. However, it is not that easy as |\xintiloopindex| must be expanded
before, so one ends up with code like this:
%
\leftedline
{|\expandafter\xintbreakiloopanddo\expandafter\macro\xintiloopindex.%|}
%
\leftedline{|etc.. etc.. \repeat|}
%
As moreover the |\fi| from the test leading to the decision of breaking out of
the loop must be cleared out of the way, the above should be
a branch of an expandable conditional test, else one needs something such
as:
%
\leftedline
{|\xint_afterfi{\expandafter\xintbreakiloopanddo\expandafter\macro\xintiloopindex.}%|}
%
\leftedline{|\fi etc..etc.. \repeat|}

There is \csbxint{iloopskiptonext} to abort the current iteration and skip to
the next, \hyperref[xintiloopskipandredo]{\ttfamily\hyphenchar\font45 \char92
  xintiloopskip\-and\-redo} to skip to the end of the current iteration and redo
it with the same value of the index (something else will have to change for this
not to become an eternal loop\dots ).

Inside alignments, if the looped-over text contains a |&| or a |\cr|, any
un-expandable material before a \csbxint{iloopindex} will make it fail because
of |\endtemplate|; in such cases one can always either replace |&| by a macro
expanding to it or replace it by a suitable |\firstofone{&}|, and similarly for
|\cr|.

\phantomsection\label{edefprimes}
As an example, let us construct an |\edef\z{...}| which will define |\z| to be a
list of prime numbers:
\begin{everbatim*}
\begingroup
\edef\z
{\xintiloop [10001+2]
  {\xintiloop [3+2]
   \ifnum\xintouteriloopindex<\numexpr\xintiloopindex*\xintiloopindex\relax
          \xintouteriloopindex,
          \expandafter\xintbreakiloop
   \fi
   \ifnum\xintouteriloopindex=\numexpr
        (\xintouteriloopindex/\xintiloopindex)*\xintiloopindex\relax
   \else
   \repeat
  }% no space here
 \ifnum \xintiloopindex < 10999 \repeat }%
\meaning\z\endgroup
\end{everbatim*}and we should have taken
some steps to not have a trailing comma, but 
the point was to show that one can do that in an |\edef|\,! See also
\autoref{ssec:primesII} which extracts from this code its way of testing
primality.

Let us create an alignment where each row will contain all divisors of its
first entry.
Here is the output, thus obtained without any count register:
\begin{everbatim*}
\begin{multicols}2
\tabskip1ex \normalcolor
\halign{&\hfil#\hfil\cr
    \xintiloop [1+1]
    {\expandafter\bfseries\xintiloopindex &
     \xintiloop [1+1]
     \ifnum\xintouteriloopindex=\numexpr
           (\xintouteriloopindex/\xintiloopindex)*\xintiloopindex\relax
     \xintiloopindex&\fi
     \ifnum\xintiloopindex<\xintouteriloopindex\space % CRUCIAL \space HERE
     \repeat \cr }%
    \ifnum\xintiloopindex<30
    \repeat
}
\end{multicols}
\end{everbatim*}
We wanted this first entry in bold face, but |\bfseries| leads to
unexpandable tokens, so the |\expandafter| was necessary for |\xintiloopindex|
and |\xintouteriloopindex| not to be confronted with a hard to digest
|\endtemplate|. An alternative way of coding:
%
\begin{everbatim}
\tabskip1ex
\def\firstofone #1{#1}%
\halign{&\hfil#\hfil\cr
  \xintiloop [1+1]
    {\bfseries\xintiloopindex\firstofone{&}%
    \xintiloop [1+1] \ifnum\xintouteriloopindex=\numexpr
    (\xintouteriloopindex/\xintiloopindex)*\xintiloopindex\relax
    \xintiloopindex\firstofone{&}\fi
    \ifnum\xintiloopindex<\xintouteriloopindex\space % \space is CRUCIAL
    \repeat \firstofone{\cr}}%
  \ifnum\xintiloopindex<30 \repeat }
\end{everbatim}


\subsection{Another completely expandable prime test}\label{ssec:primesII}

The |\IsPrime| macro from \autoref{ssec:primesI} checked expandably if a (short)
integer was prime, here is a partial rewrite using \csbxint{iloop}. We use the
|etoolbox| expandable conditionals for convenience, but not everywhere as
|\xintiloopindex| can not be evaluated while being braced. This is also the
reason why |\xintbreakiloopanddo| is delimited, and the next macro
|\SmallestFactor| which returns the smallest prime factor examplifies that. One
could write more efficient completely expandable routines, the aim here was only
to illustrate use of the general purpose \csbxint{iloop}. A little table giving
the first values of |\SmallestFactor| follows, its coding uses \csbxint{For},
which is described later; none of this uses count registers.
%
%\tracingmacros1

\begin{everbatim*}
\let\IsPrime\undefined \let\SmallestFactor\undefined % clean up possible previous mess
\newcommand{\IsPrime}[1] % returns 1 if #1 is prime, and 0 if not
  {\ifnumodd {#1}
    {\ifnumless {#1}{8}
      {\ifnumequal{#1}{1}{0}{1}}% 3,5,7 are primes
      {\if
       \xintiloop [3+2]
       \ifnum#1<\numexpr\xintiloopindex*\xintiloopindex\relax
           \expandafter\xintbreakiloopanddo\expandafter1\expandafter.%
       \fi
       \ifnum#1=\numexpr (#1/\xintiloopindex)*\xintiloopindex\relax
       \else
       \repeat 00\expandafter0\else\expandafter1\fi
      }%
    }% END OF THE ODD BRANCH
    {\ifnumequal {#1}{2}{1}{0}}% EVEN BRANCH
}%
\catcode`_ 11
\newcommand{\SmallestFactor}[1] % returns the smallest prime factor of #1>1
  {\ifnumodd {#1}
    {\ifnumless {#1}{8}
      {#1}% 3,5,7 are primes
      {\xintiloop [3+2]
       \ifnum#1<\numexpr\xintiloopindex*\xintiloopindex\relax
           \xint_afterfi{\xintbreakiloopanddo#1.}%
       \fi
       \ifnum#1=\numexpr (#1/\xintiloopindex)*\xintiloopindex\relax
           \xint_afterfi{\expandafter\xintbreakiloopanddo\xintiloopindex.}%
       \fi
       \iftrue\repeat
      }%
     }% END OF THE ODD BRANCH
   {2}% EVEN BRANCH
}%
\catcode`_ 8
{\centering
  \begin{tabular}{|c|*{10}c|}
    \hline
    \xintFor #1 in {0,1,2,3,4,5,6,7,8,9}\do {&\bfseries #1}\\
    \hline
    \bfseries 0&--&--&2&3&2&5&2&7&2&3\\
    \xintFor #1 in {1,2,3,4,5,6,7,8,9}\do
    {\bfseries #1%
      \xintFor #2 in {0,1,2,3,4,5,6,7,8,9}\do
      {&\SmallestFactor{#1#2}}\\}%
    \hline
  \end{tabular}\par
}
\end{everbatim*}

\subsection{A table of factorizations}
\label{ssec:factorizationtable}

As one more example with \csbxint{iloop} let us use an alignment to display the
factorization of some numbers. The loop will actually only play a minor r\^ole
here, just handling the row index, the row contents being almost entirely
produced via a macro |\factorize|. The factorizing macro does not use
|\xintiloop| as it didn't appear to be the convenient tool. As |\factorize| will
have to be used on |\xintiloopindex|, it has been defined as a delimited macro.

To spare some fractions of a second in the compilation time of this document
(which has many many other things to do), \number"7FFFFFED{} and
\number"7FFFFFFF, which turn out to be prime numbers, are not given to
|factorize| but just typeset directly; this illustrates use of
\csbxint{iloopskiptonext}.

The code next generates a \hyperref[floatfactorize]{table} which has
been made into a float appearing \vpageref{floatfactorize}. Here is now
the code for factorization; the conditionals use the package provided
|\xint_firstoftwo| and |\xint_secondoftwo|, one could have employed
rather \LaTeX{}'s own |\@firstoftwo| and |\@secondoftwo|, or, simpler
still in \LaTeX{} context, the |\ifnumequal|, |\ifnumless| \dots,
utilities from the package |etoolbox| which do exactly that under the
hood. Only \TeX{} acceptable numbers are treated here, but it would be
easy to make a translation and use the \xintname macros, thus extending
the scope to big numbers; naturally up to a cost in speed.

The reason for some strange looking expressions is to avoid arithmetic overflow.

\begin{everbatim*}
\catcode`_ 11
\def\abortfactorize #1\xint_secondoftwo\fi #2#3{\fi}

\def\factorize #1.{\ifnum#1=1 \abortfactorize\fi
          \ifnum\numexpr #1-2=\numexpr ((#1/2)-1)*2\relax
               \expandafter\xint_firstoftwo
          \else\expandafter\xint_secondoftwo
          \fi
         {2&\expandafter\factorize\the\numexpr#1/2.}%
         {\factorize_b #1.3.}}%

\def\factorize_b #1.#2.{\ifnum#1=1 \abortfactorize\fi
         \ifnum\numexpr #1-(#2-1)*#2<#2
                 #1\abortfactorize
         \fi
         \ifnum \numexpr #1-#2=\numexpr ((#1/#2)-1)*#2\relax
              \expandafter\xint_firstoftwo
         \else\expandafter\xint_secondoftwo
         \fi
         {#2&\expandafter\factorize_b\the\numexpr#1/#2.#2.}%
         {\expandafter\factorize_b\the\numexpr #1\expandafter.%
                                  \the\numexpr #2+2.}}%
\catcode`_ 8
\begin{figure*}[ht!]
\centering\phantomsection\label{floatfactorize}\normalcolor
\tabskip1ex
\centeredline{\vbox{\halign {\hfil\strut#\hfil&&\hfil#\hfil\cr\noalign{\hrule}
         \xintiloop ["7FFFFFE0+1]
         \expandafter\bfseries\xintiloopindex &
         \ifnum\xintiloopindex="7FFFFFED
              \number"7FFFFFED\cr\noalign{\hrule}
         \expandafter\xintiloopskiptonext
         \fi
         \expandafter\factorize\xintiloopindex.\cr\noalign{\hrule}
         \ifnum\xintiloopindex<"7FFFFFFE
         \repeat
         \bfseries \number"7FFFFFFF&\number "7FFFFFFF\cr\noalign{\hrule}
}}}
\centeredline{A table of factorizations}
\end{figure*}
\end{everbatim*}

\begin{framed}
  The next utilities are not compatible with expansion-only context.
\end{framed}

\subsection{\csbh{xintApplyInline}}\label{xintApplyInline}

% {\small |1.09a|, enhanced in |1.09c| to be usable within alignments, and
%   corrected in |1.09d| for a problem related to spaces at the very end of the
%   list parameter.\par}

\csa{xintApplyInline}|{\macro}|\marg{list}\ntype{o{\lowast f}} works non
expandably. It applies the one-parameter |\macro| to the first element of the
expanded list (|\macro| may have itself some arguments, the list item will be
appended as last argument), and is then re-inserted in the input stream after
the tokens resulting from this first expansion of |\macro|. The next item is
then handled.

This is to be used in situations where one needs to do some repetitive
things. It is not expandable and can not be completely expanded inside a
macro definition, to prepare material for later execution, contrarily to what
\csbxint{Apply} or \csbxint{ApplyUnbraced} achieve.

\begin{everbatim*}
\def\Macro #1{\advance\cnta #1 , \the\cnta}
\cnta 0
0\xintApplyInline\Macro {3141592653}.
\end{everbatim*}
The first argument |\macro| does not have to be an expandable macro.

\csa{xintApplyInline} submits its second, token list parameter to an
\hyperref[sec:expansions]{\fexpan
sion}. Then, each \emph{unbraced} item will also be \fexpan ded. This provides
an easy way to insert one list inside another. \emph{Braced} items are not
expanded. Spaces in-between items are gobbled (as well as those at the start
or the end of the list), but not the spaces \emph{inside} the braced items.

\csa{xintApplyInline}, despite being non-expandable, does survive to
contexts where the executed |\macro| closes groups, as happens inside
alignments with the tabulation character |&|.
This tabular provides an example:\par
\begin{everbatim*}
\centerline{\normalcolor\begin{tabular}{ccc}
     $N$ & $N^2$ & $N^3$ \\ \hline
     \def\Row #1{ #1 & \xintiiSqr {#1} & \xintiiPow {#1}{3} \\ \hline }%
     \xintApplyInline \Row {\xintCSVtoList{17,28,39,50,61}}
\end{tabular}}\medskip
\end{everbatim*}

We see that despite the fact that the first encountered tabulation character in
the first row close a group and thus erases |\Row| from \TeX's memory,
|\xintApplyInline| knows how to deal with this.

Using \csbxint{ApplyUnbraced} is an alternative: the difference is that
this would have prepared all rows first and only put them back into the
token stream once they are all assembled, whereas with |\xintApplyInline|
each row is constructed and immediately fed back into the token stream: when
one does things with numbers having hundreds of digits, one learns that
keeping on hold and shuffling around hundreds of tokens has an impact on
\TeX{}'s speed (make this ``thousands of tokens'' for the impact to be
noticeable).

One may nest various |\xintApplyInline|'s. For example (see the
\hyperref[float]{table} \vpageref{float}):\par
\begin{everbatim*}
\begin{figure*}[ht!]
  \centering\phantomsection\label{float}
  \def\Row #1{#1:\xintApplyInline {\Item {#1}}{0123456789}\\ }%
  \def\Item #1#2{&\xintiPow {#1}{#2}}%
  \centeredline {\begin{tabular}{ccccccccccc} &0&1&2&3&4&5&6&7&8&9\\ \hline
      \xintApplyInline \Row {0123456789}
    \end{tabular}}
\end{figure*}
\end{everbatim*}

One could not move the definition of |\Item| inside the tabular,
as it would get lost after the first |&|. But this
works:
\everb|@
\begin{tabular}{ccccccccccc}
    &0&1&2&3&4&5&6&7&8&9\\ \hline
    \def\Row #1{#1:\xintApplyInline {&\xintiPow {#1}}{0123456789}\\ }%
    \xintApplyInline \Row {0123456789}
\end{tabular}
|

A limitation is that, contrarily to what one may have expected, the
|\macro| for an |\xintApplyInline| can not be used to define
the |\macro| for a nested sub-|\xintApplyInline|. For example,
this does not work:\par
\everb|@
  \def\Row #1{#1:\def\Item ##1{&\xintiPow {#1}{##1}}%
                 \xintApplyInline \Item {0123456789}\\ }%
  \xintApplyInline \Row {0123456789} % does not work
|
\noindent But see \csbxint{For}.

\subsection{\csbh{xintFor}, \csbh{xintFor*}}\label{xintFor}\label{xintFor*}
% {\small New with |1.09c|. Extended in |1.09e| (\csbxint{BreakFor},
%   \csbxint{integers}, \dots). |1.09f| version handles all macro parameters up
%   to
%   |#9| and removes spaces around commas.\par}

\csbxint{For}\ntype{on} is a new kind of for loop. Rather than using macros
for encapsulating list items, its behavior is more like a macro with parameters:
|#1|, |#2|, \dots, |#9| are used to represent the items for up to nine levels of
nested loops. Here is an example:
%
\everb|@
\xintFor #9 in {1,2,3} \do {%
  \xintFor #1 in {4,5,6} \do {%
    \xintFor #3 in {7,8,9} \do {%
      \xintFor #2 in {10,11,12} \do {%
      $$#9\times#1\times#3\times#2=\xintiiPrd{{#1}{#2}{#3}{#9}}$$}}}}
|
\noindent This example illustrates that one does not have to use |#1| as the
first one: 
the order is arbitrary. But each level of nesting should have its specific macro
parameter. Nine levels of nesting is presumably overkill, but I did not know
where it was reasonable to stop. |\par| tokens are accepted in both the comma
separated list and the replacement text.

\begin{framed}
  A macro |\macro| whose definition uses internally an \csbxint{For} loop may be
  used inside another \csbxint{For} loop even if the two loops both use the same
  macro parameter. Note: the loop definition inside |\macro| must double
  the character |#| as is the general rule in \TeX{} with definitions done
  inside macros.

  The macros \csa{xintFor} and \csa{xintFor*} are not expandable, one can not
  use them inside an |\edef|. But they may be used inside alignments (such as a
  \LaTeX{} |tabular|), as will be shown in examples.
\end{framed}

The spaces between the various declarative elements are all optional;
furthermore spaces around the commas or at the start and end of the list
argument are allowed, they will be removed. If an item must contain itself
commas, it should be braced to prevent these commas from being misinterpreted as
list separator. These braces will be removed during processing. The list
argument may be a macro |\MyList| expanding in one step to the comma separated
list (if it has no arguments, it does not have to be braced). It
will be expanded (only once) to reveal its comma separated items for processing,
comma separated items will not be expanded before being fed into the replacement
text as |#1|, or |#2|, etc\dots, only leading and trailing spaces are removed.

A starred variant \csbxint{For*}\ntype{{\lowast f}n} deals with lists of braced
items, rather than comma separated items. It has also a distinct expansion
policy, which is detailed below.

Contrarily to what happens in loops where the item is represented by a macro,
here it is truly exactly as when defining (in \LaTeX{}) a ``command'' with
parameters |#1|, etc... This may avoid the user quite a few troubles with
|\expandafter|s or other |\edef/\noexpand|s which one encounters at times when
trying to do things with \LaTeX's {\makeatother|\@for|} or other loops
which encapsulate the item in a macro expanding to that item.

\begin{framed}
  The non-starred variant \csbxint{For} deals with comma separated values
  (\emph{spaces before and after the commas are removed}) and the comma
  separated list may be a macro which is only expanded once (to prevent
  expansion of the first item |\x| in a list directly input as |\x,\y,...| it
  should be input as |{\x},\y,..| or |<space>\x,\y,..|, naturally all of that
  within the mandatory braces of the \csa{xintFor \#n in \{list\}} syntax). The
  items are not expanded, if the input is |<stuff>,\x,<stuff>| then |#1| will be
  at some point |\x| not its expansion (and not either a macro with |\x| as
  replacement text, just the token |\x|). Input such as |<stuff>,,<stuff>|
  creates an empty |#1|, the iteration is not skipped. An empty list does lead
  to the use of the replacement text, once, with an empty |#1| (or |#n|). Except
  if the entire list is represented as a single macro with no parameters,
  \fbox{it must be braced.}
\end{framed}

\begin{framed}
  The starred variant \csbxint{For*} deals with token lists (\emph{spaces
    between braced items or single tokens are not significant}) and
  \hyperref[sec:expansions]{\fexpan ds} each \emph{unbraced} list item. This
  makes it easy to simulate concatenation of various list macros |\x|, |\y|, ...
  If |\x| expands to |{1}{2}{3}| and |\y| expands to |{4}{5}{6}| then |{\x\y}|
  as argument to |\xintFor*| has the same effect as |{{1}{2}{3}{4}{5}{6}}|%
  \stepcounter{footnote}%
  \makeatletter\hbox {\@textsuperscript {\normalfont \thefootnote
    }}\makeatother. Spaces at the start, end, or in-between items are gobbled
  (but naturally not the spaces which may be inside \emph{braced} items). Except
  if the list argument is a single macro with no parameters, \fbox{it must be
    braced.} Each item which is not braced will be fully expanded (as the |\x|
  and |\y| in the example above). An empty list leads to an empty result.
\end{framed}
\begingroup\makeatletter
\def\@footnotetext #1{\insert\footins {\reset@font \footnotesize \interlinepenalty \interfootnotelinepenalty \splittopskip \footnotesep \splitmaxdepth \dp \strutbox \floatingpenalty \@MM \hsize \columnwidth \@parboxrestore \color@begingroup \@makefntext {\rule \z@ \footnotesep \ignorespaces #1\@finalstrut \strutbox }\color@endgroup }}
\addtocounter{footnote}{-1}
\edef\@thefnmark {\thefootnote}
\@footnotetext{braces around single token items
    are optional so this is the same as \texttt{\{123456\}}.}
% \stepcounter{footnote}
% \edef\@thefnmark {\thefootnote}
% \@footnotetext{the \csa{space} will stop the \TeX{} scanning of a number and be
%   gobbled in the process; the \csa{relax} stops the scanning but is not
%   gobbled. Or one may do \csa{numexpr}\texttt{\#1}\csa{relax}, and then the
%   \csa{relax} is gobbled.}
\endgroup
%\addtocounter{Hfootnote}{2}
\addtocounter{Hfootnote}{1}

The macro \csbxint{Seq} which generates arithmetic sequences may only be used
with \csbxint{For*} (numbers from output of |\xintSeq| are braced, not separated
by commas). %
%
\leftedline{|\xintFor* #1 in {\xintSeq [+2]{-7}{+2}}\do {stuff
    with #1}|} will have |#1=-7,-5,-3,-1, and 1|. The |#1| as issued from the
list produced by \csbxint{Seq} is the litteral representation as would be
produced by |\arabic| on a \LaTeX{} counter, it is not a count register. When
used in |\ifnum| tests or other contexts where \TeX{} looks for a number it
should thus be postfixed with |\relax| or |\space|.

When nesting \csa{xintFor*} loops, using \csa{xintSeq} in the inner loops is
inefficient, as the arithmetic sequence will be re-created each time. A more
efficient style is:
%
\begin{everbatim}
    \edef\innersequence {\xintSeq[+2]{-50}{50}}%
    \xintFor* #1 in {\xintSeq {13}{27}} \do
        {\xintFor* #2 in \innersequence \do {stuff with #1 and #2}%
         .. some other macros .. }
\end{everbatim}

This is a general remark applying for any nesting of loops, one should avoid
recreating the inner lists of arguments at each iteration of the outer loop.
However, in the example above, if the |.. some other macros ..| part
closes a group which was opened before the |\edef\innersequence|, then
this definition will be lost. An alternative to |\edef|, also efficient,
exists when dealing with arithmetic sequences: it is to use the
\csbxint{integers} keyword (described later) which simulates infinite
arithmetic sequences; the loops will then be terminated via a test |#1|
(or |#2| etc\dots) and subsequent use of \csbxint{BreakFor}.

The \csbxint{For} loops are not completely expandable; but they may be nested
and used inside alignments or other contexts where the replacement text closes
groups. Here is an example (still using \LaTeX's tabular):

\begin{everbatim*}
\begin{tabular}{rccccc}
    \xintFor #7 in {A,B,C} \do {%
      #7:\xintFor* #3 in {abcde} \do {&($ #3 \to #7 $)}\\ }%
\end{tabular}
\end{everbatim*}

When
inserted inside a macro for later execution the |#| characters must be
doubled.%
%
\footnote{sometimes what seems to be a macro argument isn't really; in
  \csa{raisebox\{1cm\}\{}\csa{xintFor \#1 in \{a,b,c\} }\csa{do
    \{\#1\}\}} no doubling should be done.}
%
For example:
%
\begin{everbatim*}
\def\T{\def\z {}%
  \xintFor* ##1 in {{u}{v}{w}} \do {%
    \xintFor ##2 in {x,y,z} \do {%
      \expandafter\def\expandafter\z\expandafter {\z\sep (##1,##2)} }%
  }%
}%
\T\def\sep {\def\sep{, }}\z 
\end{everbatim*}

Similarly when the replacement text
of |\xintFor| defines a macro with parameters, the macro character |#| must be
doubled.

It is licit to use inside an \csbxint{For} a |\macro| which itself has
been defined to use internally some other \csbxint{For}. The same macro
parameter |#1| can be used with no conflict (as mentioned above, in the
definition of |\macro| the |#| used in the \csbxint{For} declaration must be
doubled, as is the general rule in \TeX{} with things defined inside other
things).

The iterated commands as well as the list items are allowed to contain explicit
|\par| tokens. Neither \csbxint{For} nor \csbxint{For*} create groups. The
effect is like piling up the iterated commands with each time |#1| (or |#2| ...)
replaced by an item of the list. However, contrarily to the completely
expandable \csbxint{ApplyUnbraced}, but similarly to the non completely
expandable \csbxint{ApplyInline} each iteration is executed first before looking
at the next |#1|%
%
\footnote{to be completely honest, both \csbxint{For} and
  \csbxint{For*} initially scoop up both the list and the iterated
  commands; \csbxint{For} scoops up a second time the entire comma
  separated list in order to feed it to \csbxint{CSVtoList}. The starred
  variant \csbxint{For*} which does not need this step will thus be a
  bit faster on equivalent inputs.}
%
(and
the starred variant \csbxint{For*} keeps on expanding each unbraced item it
finds, gobbling spaces).

\subsection{\csbh{xintifForFirst}, \csbh{xintifForLast}}
\label{xintifForFirst}\label{xintifForLast}
% {\small New in |1.09e|.\par}

\csbxint{ifForFirst}\,\texttt{\{YES branch\}\{NO branch\}}\etype{nn}
 and \csbxint{ifForLast}\,\texttt{\{YES
  branch\}\hskip 0pt plus 0.2em \{NO branch\}}\etype{nn} execute the |YES| or
|NO| branch
if the
\csbxint{For}
or \csbxint{For*} loop is currently in its first, respectively last, iteration.

Designed to work as expected under nesting. Don't forget an empty brace pair
|{}| if a branch is to do nothing. May be used multiple times in the replacement
text of the loop.

There is no such thing as an iteration counter provided by the \csa{xintFor}
loops; the user is invited to define if needed his own count register or
\LaTeX{} counter, for example with a suitable |\stepcounter| inside the
replacement text of the loop to update it.

\begin{framed}
  It is a known feature of these conditionals that they cease to function if
  put at a location of the |\xintFor| replacement text which has closed a
  group, for example in the last cell of an alignment created by the loop,
  assuming the replacement text of the |\xintFor| loop creates a row. The
  conditional must be used before the first cell is closed. This is not likely
  to change in future versions. It is not an intrinsic limitation as the
  branches of the conditional can be the complete rows, inclusive of all |&|'s
  and the tabular newline |\\|.
\end{framed}

\subsection{ \csbh{xintBreakFor}, \csbh{xintBreakForAndDo}}
\label{xintBreakFor}\label{xintBreakForAndDo}
%{\small New in |1.09e|.\par}

One may immediately terminate an \csbxint{For} or \csbxint{For*} loop with
\csbxint{BreakFor}. As the criterion for breaking will be decided on a
basis of some test, it is recommended to use for this test the syntax of
\href{http://ctan.org/pkg/ifthen}{ifthen}%
%
\footnote{\url{http://ctan.org/pkg/ifthen}}
%
or
\href{http://ctan.org/pkg/etoolbox}{etoolbox}%
%
\footnote{\url{http://ctan.org/pkg/etoolbox}}
%
or the \xintname own conditionals, rather than one of the various
|\if...\fi| of \TeX{}. Else (and this is without even mentioning all the various
pecularities of the
|\if...\fi| constructs), one has to carefully move the break after the closing
of
the conditional, typically with |\expandafter\xintBreakFor\fi|.%
%
\footnote{the difficulties here are similar to those mentioned in
  \autoref{sec:ifcase}, although less severe, as complete expandability
  is not to be maintained; hence the allowed use of
  \href{http://ctan.org/pkg/ifthen}{ifthen}.}

There is also \csbxint{BreakForAndDo}. Both are illustrated by various examples
in the next section which is devoted to ``forever'' loops.

\subsection{\csbh{xintintegers}, \csbh{xintdimensions}, \csbh{xintrationals}}
\label{xintegers}\label{xintintegers}
\label{xintdimensions}\label{xintrationals}
%{\small New in |1.09e|.\par}

If the list argument to \csbxint{For} (or \csbxint{For*}, both are equivalent in
this context) is \csbxint{integers} (equivalently \csbxint{egers}) or more
generally \csbxint{integers}|[||start|\allowbreak|+|\allowbreak|delta||]|
(\emph{the whole within braces}!)%
%
\footnote{the |start+delta| optional specification may have extra spaces
  around the plus sign of near the square brackets, such spaces are
  removed. The same applies with \csa{xintdimensions} and
  \csa{xintrationals}.},
%
then \csbxint{For} does an infinite iteration where
|#1| (or |#2|, \dots, |#9|) will run through the arithmetic sequence of (short)
integers with initial value |start| and increment |delta| (default values:
|start=1|, |delta=1|; if the optional argument is present it must contains both
of them, and they may be explicit integers, or macros or count registers). The
|#1| (or |#2|, \dots, |#9|) will stand for |\numexpr <opt sign><digits>\relax|,
and the litteral representation as a string of digits can thus be obtained as
\fbox{\csa{the\#1}} or |\number#1|. Such a |#1| can be used in an |\ifnum| test
with no need to be postfixed with a space or a |\relax| and one should
\emph{not} add them.

If the list argument is \csbxint{dimensions} or more generally
\csbxint{dimensions}|[||start|\allowbreak|+|\allowbreak|delta||]|  (\emph{within
  braces}!), then
\csbxint{For} does an infinite iteration where |#1| (or |#2|, \dots, |#9|) will
run through the arithmetic sequence of dimensions with initial value
|start| and increment |delta|. Default values: |start=0pt|, |delta=1pt|; if
the optional argument is present it must contain both of them, and they may
be explicit specifications, or macros, or dimen registers, or length commands
in \LaTeX{} (the stretch and shrink components will be discarded). The |#1|
will be |\dimexpr <opt sign><digits>sp\relax|, from which one can get the
litteral (approximate) representation in points via |\the#1|. So |#1| can be
used anywhere \TeX{} expects a dimension (and there is no need in conditionals
to insert a |\relax|, and one should \emph{not} do it), and to print its value
one uses \fbox{\csa{the\#1}}. The chosen representation guarantees exact
incrementation with no rounding errors accumulating from converting into
points at each step.

% original definitions, a bit slow.
%\def\DimToNum #1{\number\dimexpr #1\relax }
% cube
%\xintNewIExpr \FA [2] {protect(\DimToNum {#2})^3/protect(\DimToNum{#1})^2}
% square root
%\xintNewIExpr \FB [2] {sqrt (protect(\DimToNum {#2})*protect(\DimToNum {#1}))}
%\xintNewExpr \Ratio [2] {trunc(protect(\DimToNum {#2})/protect(\DimToNum{#1}),3)}

% improved faster code (4 four times faster)

\def\DimToNum #1{\the\numexpr \dimexpr#1\relax/10000\relax }
\def\FA #1#2{\xintDSH{-4}{\xintiQuo {\xintiPow {\DimToNum {#2}}{3}}{\xintiSqr
{\DimToNum{#1}}}}}
\def\FB #1#2{\xintDSH {-4}{\xintiSqrt
                          {\xintiMul {\DimToNum {#2}}{\DimToNum{#1}}}}}
\def\Ratio #1#2{\xintTrunc {2}{\DimToNum {#2}/\DimToNum{#1}}}

% a further 2.5 gain is made through using .25pt as horizontal step.
\begin{figure*}[ht!]
\phantomsection\hypertarget{graphic}{}%
\centeredline{%
\raisebox{-1cm}{\xintFor #1 in {\xintdimensions [0pt+.25pt]} \do
 {\ifdim #1>2cm \expandafter\xintBreakFor\fi
  {\color [rgb]{\Ratio {2cm}{#1},0,0}%
  \vrule width .25pt height \FB {2cm}{#1}sp depth -\FA {2cm}{#1}sp }%
  }% end of For iterated text
}%
\hspace{.5cm}%
\scriptsize\baselineskip8pt\relax
\begin{minipage}{\dimexpr\linewidth-2.5cm-\parindent\relax}\def\everbatimindent{0pt }%
\begin{everbatim}
\def\DimToNum #1{\number\dimexpr #1\relax }
\xintNewIExpr \FA [2] {protect(\DimToNum {#2})^3/protect(\DimToNum{#1})^2}     %cube
\xintNewIExpr \FB [2] {sqrt (protect(\DimToNum {#2})*protect(\DimToNum {#1}))} %sqrt
\xintNewExpr \Ratio [2] {trunc(protect(\DimToNum {#2})/protect(\DimToNum{#1}),3)}
\xintFor #1 in {\xintdimensions [0pt+.1pt]} \do
 {\ifdim #1>2cm \expandafter\xintBreakFor\fi
  {\color [rgb]{\Ratio {2cm}{#1},0,0}%
  \vrule width .1pt height \FB {2cm}{#1}sp depth -\FA {2cm}{#1}sp }%
 }% end of For iterated text
\end{everbatim}
\end{minipage}}
\end{figure*}

The\xintNewIExpr \FA [2] {protect(\DimToNum {#2})^3/protect(\DimToNum{#1})^2}
\hyperlink{graphic}{graphic}, with the code on its right%
%
\footnote{see \autoref{sssec:protect} for the significance of the |protect|'s:
  they are needed because the expression has macro parameters inside macros,
  and not only functions from the \csbxint{expr} syntax. The \csa{FA} turns
  out to have meaning \texttt{\meaning\FA}. The \csa{romannumeral} part is
  only to ensure it expands in only two steps, and could be removed. The
  \expandafter|\string\xintRound::csv| and
  \expandafter|\string\xintSPRaw::csv| commands are used internally by
  \csbxint{iexpr} to round and pretty print its result (or comma separated
  results). See also the next footnote.},
%
is for illustration only, not
only because of pdf rendering artefacts when displaying adjacent rules (which do
\emph{not} show in |dvi| output as rendered by |xdvi|, and depend from your
viewer), but because not using anything but rules it is quite inefficient and
must do lots of computations to not confer a too ragged look to the borders.
With a width of |.5pt| rather than |.1pt| for the rules, one speeds up the
drawing by a factor of five, but the boundary is then visibly ragged.
\newbox\codebox
\begingroup\makeatletter
\def\x{%
     \parindent0pt
     \def\par{\@@par\leavevmode\null}%
     \let\do\do@noligs \verbatim@nolig@list
     \let\do\@makeother \dospecials
     \catcode`\@ 14 \makestarlowast
     \ttfamily \scriptsize\baselineskip 8pt \obeylines \@vobeyspaces
     \catcode`\|\active
     \lccode`\~`\|\lowercase{\let~\egroup}}%
\global\setbox\codebox \vbox\bgroup\x
\def\DimToNum #1{\the\numexpr \dimexpr#1\relax/10000\relax } % no need to be more precise!
\def\FA #1#2{\xintDSH {-4}{\xintiQuo {\xintiPow {\DimToNum {#2}}{3}}{\xintiSqr {\DimToNum{#1}}}}}
\def\FB #1#2{\xintDSH {-4}{\xintiSqrt {\xintiMul {\DimToNum {#2}}{\DimToNum{#1}}}}}
\def\Ratio #1#2{\xintTrunc {2}{\DimToNum {#2}/\DimToNum{#1}}}
\xintFor #1 in {\xintdimensions [0pt+.25pt]} \do
 {\ifdim #1>2cm \expandafter\xintBreakFor\fi
  {\color [rgb]{\Ratio {2cm}{#1},0,0}%
  \vrule width .25pt height \FB {2cm}{#1}sp depth -\FA {2cm}{#1}sp }%
 }% end of For iterated text
|%
\endgroup
\footnote{to tell the whole truth we cheated and divided by |10| the
  computation time through using the following definitions, together with a
  horizontal step of |.25pt| rather than |.1pt|. The displayed original code
  would make the slowest computation of all those done in this document using
  the \xintname bundle macros!\par\smallskip 
  \noindent\box \codebox\par }

If the list argument to \csbxint{For} (or \csbxint{For*}) is \csbxint{rationals}
or more generally
\csbxint{rationals}|[||start|\allowbreak|+|\allowbreak|delta||]| (\emph{within
  braces}!), then \csbxint{For} does an infinite iteration where |#1| (or |#2|,
\dots, |#9|) will run through the arithmetic sequence of \xintfracname fractions
with initial value |start| and increment |delta| (default values: |start=1/1|,
|delta=1/1|). This loop works \emph{only with \xintfracname loaded}. if the
optional argument is present it must contain both of them, and they may be given
in any of the formats recognized by \xintfracname (fractions, decimal
numbers, numbers in scientific notations, numerators and denominators in
scientific notation, etc...) , or as macros or count registers (if they are
short integers). The |#1| (or |#2|, \dots, |#9|) will be an |a/b| fraction
(without a |[n]| part), where
the denominator |b| is the product of the denominators of
|start| and |delta| (for reasons of speed |#1| is not reduced to irreducible
form, and for another reason explained later  |start| and |delta| are not put
either into irreducible form; the input may use explicitely \csa{xintIrr} to
achieve that).
\begin{everbatim*}
\begingroup\small
\noindent\parbox{\dimexpr\linewidth-3em}{\color[named]{OrangeRed}%
\xintFor #1 in {\xintrationals [10/21+1/21]} \do
{#1=\xintifInt {#1}
    {\textcolor{blue}{\xintTrunc{10}{#1}}}
    {\xintTrunc{10}{#1}}% display in blue if an integer
    \xintifGt {#1}{1.123}{\xintBreakFor}{, }%
  }}
\endgroup\smallskip
\end{everbatim*}

\smallskip The example above confirms that computations are done exactly, and
illustrates that the two initial (reduced) denominators are not multiplied when
they are found to be equal.  It is thus recommended to input |start| and |delta|
with a common smallest possible denominator, or as fixed point numbers with the
same numbers of digits after the decimal mark;  and this is also the reason why
|start| and |delta| are not by default made irreducible. As internally the
computations are done with numerators and denominators completely expanded, one
should be careful not to input numbers in scientific notation with exponents in
the hundreds, as they will get converted into as many zeroes.

\begin{everbatim*}
\noindent\parbox{\dimexpr.7\linewidth}{\raggedright
\xintFor #1 in {\xintrationals [0.000+0.125]} \do
{\edef\tmp{\xintTrunc{3}{#1}}%
 \xintifInt {#1}
    {\textcolor{blue}{\tmp}}
    {\tmp}%
    \xintifGt {#1}{2}{\xintBreakFor}{, }%
  }}\smallskip
\end{everbatim*}

We see here that \csbxint{Trunc} outputs (deliberately) zero as $0$, not (here)
$0.000$, the idea being not to lose the information that the truncated thing was
truly zero. Perhaps this behavior should be changed? or made optional? Anyhow
printing of fixed points numbers should be dealt with via dedicated packages
such as |numprint| or |siunitx|.\par

\subsection{Another table of primes}\label{ssec:primesIII}

As a further example, let us dynamically generate a tabular with the first $50$
prime numbers after $12345$. First we need a macro to test if a (short) number
is prime. Such a completely expandable macro was given in \autoref{xintSeq},
here we consider a variant which will be slightly more efficient. This new
|\IsPrime| has two parameters. The first one is a macro which it redefines to
expand to the result of the primality test applied to the second argument. For
convenience we use the \href{http://ctan.org/pkg/etoolbox}{etoolbox} wrappers to
various |\ifnum| tests, although here there isn't anymore the constraint of
complete expandability (but using explicit |\if..\fi| in tabulars has its
quirks); equivalent tests are provided by \xintname, but they have some overhead
as they are able to deal with arbitrarily big integers.

\def\IsPrime #1#2%
{\edef\TheNumber {\the\numexpr #2}% positive integer
 \ifnumodd {\TheNumber}
 {\ifnumgreater {\TheNumber}{1}
  {\edef\ItsSquareRoot{\xintiSqrt \TheNumber}%
    \xintFor ##1 in {\xintintegers [3+2]}\do
    {\ifnumgreater {##1}{\ItsSquareRoot}
               {\def#1{1}\xintBreakFor}
               {}%
     \ifnumequal {\TheNumber}{(\TheNumber/##1)*##1}
                 {\def#1{0}\xintBreakFor }
                 {}%
    }}
  {\def#1{0}}}% 1 is not prime
 {\ifnumequal {\TheNumber}{2}{\def#1{1}}{\def#1{0}}}%
}%

\everb|@
\def\IsPrime #1#2% """color[named]{PineGreen}#1=\Result, #2=tested number (assumed >0).;!
{\edef\TheNumber {\the\numexpr #2}%"""color[named]{PineGreen} hence #2 may be a count or \numexpr.;!
 \ifnumodd {\TheNumber}
 {\ifnumgreater {\TheNumber}{1}
  {\edef\ItsSquareRoot{\xintiSqrt \TheNumber}%
    \xintFor """color{red}##1;! in {"""color{red}\xintintegers;! [3+2]}\do
    {\ifnumgreater {"""color{red}##1;!}{\ItsSquareRoot} """color[named]{PineGreen}% "textcolor{red}{##1} is a \numexpr.;!
               {\def#1{1}\xintBreakFor}
               {}%
     \ifnumequal {\TheNumber}{(\TheNumber/##1)*##1}
                 {\def#1{0}\xintBreakFor }
                 {}%
    }}
  {\def#1{0}}}% 1 is not prime
 {\ifnumequal {\TheNumber}{2}{\def#1{1}}{\def#1{0}}}%
}
|

%\newcounter{primecount}
%\newcounter{cellcount}

As we used \csbxint{For} inside a macro we had to double the |#| in its |#1|
parameter. Here is now the code which creates the prime table (the table has
been put in a \hyperref[primes]{float}, which should be found on page
\pageref{primes}):

\everb?@
\newcounter{primecount}
\newcounter{cellcount}
\begin{figure*}[ht!]
  \centering
  \begin{tabular}{|*{7}c|}
  \hline
  \setcounter{primecount}{0}\setcounter{cellcount}{0}%
  \xintFor """color{red}#1;! in {"""color{red}\xintintegers;! [12345+2]} \do
"""color[named]{PineGreen}% "textcolor{red}{#1} is a \numexpr.;!
  {\IsPrime\Result{#1}%
   \ifnumgreater{\Result}{0}
   {\stepcounter{primecount}%
    \stepcounter{cellcount}%
    \ifnumequal {\value{cellcount}}{7}
       {"""color{red}\the#1;! \\\setcounter{cellcount}{0}}
       {"""color{red}\the#1;! &}}
   {}%
    \ifnumequal {\value{primecount}}{50}
     {\xintBreakForAndDo
      {\multicolumn {6}{l|}{These are the first 50 primes after 12345.}\\}}
     {}%
  }\hline
\end{tabular}
\end{figure*}
?

\begin{figure*}[ht!]
  \centering\phantomsection\label{primes}
  \begin{tabular}{|*{7}c|}
  \hline
  \setcounter{primecount}{0}\setcounter{cellcount}{0}%
  \xintFor #1 in {\xintintegers [12345+2]} \do
  {\IsPrime\Result{#1}%
   \ifnumgreater{\Result}{0}
   {\stepcounter{primecount}%
    \stepcounter{cellcount}%
    \ifnumequal {\value{cellcount}}{7}
       {\the#1 \\\setcounter{cellcount}{0}}
       {\the#1 &}}
   {}%
    \ifnumequal {\value{primecount}}{50}
     {\xintBreakForAndDo
      {\multicolumn {6}{l|}{These are the first 50 primes after 12345.}\\}}
     {}%
  }\hline
\end{tabular}
\end{figure*}

\subsection{Some arithmetic with Fibonacci numbers}
\label{ssec:fibonacci}

Here is the code employed on the title page to compute (expandably, of
course!) the 1250th Fibonacci number:

\begin{everbatim*}
\catcode`_ 11
\def\Fibonacci #1{%  \Fibonacci{N} computes F(N) with F(0)=0, F(1)=1.
    \expandafter\Fibonacci_a\expandafter
        {\the\numexpr #1\expandafter}\expandafter
        {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter
        {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter
        {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter
        {\romannumeral0\xintiieval 0\relax}}
%
\def\Fibonacci_a #1{%
    \ifcase #1
          \expandafter\Fibonacci_end_i
    \or
          \expandafter\Fibonacci_end_ii
    \else
          \ifodd #1
              \expandafter\expandafter\expandafter\Fibonacci_b_ii
          \else
              \expandafter\expandafter\expandafter\Fibonacci_b_i
          \fi
    \fi {#1}%
}% * signs are omitted from the next macros, tacit multiplications
\def\Fibonacci_b_i #1#2#3{\expandafter\Fibonacci_a\expandafter
  {\the\numexpr #1/2\expandafter}\expandafter
  {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter
  {\romannumeral0\xintiieval (2#2-#3)#3\relax}%
}% end of Fibonacci_b_i
\def\Fibonacci_b_ii #1#2#3#4#5{\expandafter\Fibonacci_a\expandafter
  {\the\numexpr (#1-1)/2\expandafter}\expandafter
  {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter
  {\romannumeral0\xintiieval (2#2-#3)#3\expandafter\relax\expandafter}\expandafter
  {\romannumeral0\xintiieval #2#4+#3#5\expandafter\relax\expandafter}\expandafter
  {\romannumeral0\xintiieval #2#5+#3(#4-#5)\relax}%
}% end of Fibonacci_b_ii
%         code as used on title page:
%\def\Fibonacci_end_i  #1#2#3#4#5{\xintthe#5}
%\def\Fibonacci_end_ii #1#2#3#4#5{\xinttheiiexpr #2#5+#3(#4-#5)\relax}
%         new definitions:
\def\Fibonacci_end_i  #1#2#3#4#5{{#4}{#5}}% {F(N+1)}{F(N)} in \xintexpr format
\def\Fibonacci_end_ii #1#2#3#4#5%
    {\expandafter
     {\romannumeral0\xintiieval #2#4+#3#5\expandafter\relax
      \expandafter}\expandafter
     {\romannumeral0\xintiieval #2#5+#3(#4-#5)\relax}}% idem.
% \FibonacciN returns F(N) (in encapsulated format: needs \xintthe for printing)
\def\FibonacciN {\expandafter\xint_secondoftwo\romannumeral-`0\Fibonacci }%
\catcode`_ 8
\end{everbatim*}



% ok
% \def\Fibo #1.{\xintthe\FibonacciN {#1}}% to use \xintiloopindex...
% \message{\xintiloop [0+1]
%          \expandafter\Fibo\xintiloopindex.,
%          \ifnum\xintiloopindex<49 \repeat \xintthe\FibonacciN{50}.}

I have modified the ending: we want not only one specific value |F(N)| but
a pair of successive values which can serve as starting point of another routine
devoted to compute a whole sequence |F(N), F(N+1), F(N+2),....|. This pair is,
for efficiency, kept in the encapsulated internal \xintexprname format.
|\FibonacciN| outputs the single |F(N)|, also as an |\xintexpr|-ession, and
printing it will thus need the |\xintthe| prefix.

\begingroup\footnotesize\sffamily\baselineskip 10pt
Here a code snippet which
checks the routine via a \string\message\ of the first $51$ Fibonacci
numbers (this is not an efficient way to generate a sequence of such
numbers, it is only for validating \csa{FibonacciN}).
%
\begin{everbatim}
\def\Fibo #1.{\xintthe\FibonacciN {#1}}%
\message{\xintiloop [0+1] \expandafter\Fibo\xintiloopindex., 
                          \ifnum\xintiloopindex<49 \repeat \xintthe\FibonacciN{50}.}
\end{everbatim}
\endgroup

The various |\romannumeral0\xintiieval| could very well all have been
|\xintiiexpr|'s but then we would have needed more |\expandafter|'s.
Indeed the order of expansion must be controlled for the whole thing to work,
and |\romannumeral0\xintiieval| is the first expanded form of |\xintiiexpr|.

The way we use |\expandafter|'s to chain successive |\xintexpr| evaluations is
exactly analogous to well-known expandable techniques made possible by
|\numexpr|.

\begin{framed}
  There is a difference though: |\numexpr| is \emph{NOT} expandable, and to
  force its expansion we must prefix it with |\the| or |\number|. On the other
  hand |\xintexpr|, |\xintiexpr|, ..., (or |\xinteval|, |\xintieval|, ...)
  expand fully when prefixed by |\romannumeral-`0|: the computation is fully
  executed and its result encapsulated in a private format.

  Using |\xintthe| as prefix is necessary to print the result (this is like
  |\the| for |\numexpr|), but it is not necessary to get the computation done
  (contrarily to the situation with |\numexpr|).

  And, starting with release |1.09j|, it is also allowed to expand a non
  |\xintthe| prefixed |\xintexpr|-ession inside an |\edef|: the private format
  is now protected, hence the error message complaining about a missing
  |\xintthe| will not be executed, and the integrity of the format will be
  preserved.

  This new possibility brings some efficiency gain, when one writes
  non-expandable algorithms using \xintexprname. If |\xintthe| is
  employed inside |\edef| the number or fraction will be un-locked into
  its possibly hundreds of digits and all these tokens will possibly
  weigh on the upcoming shuffling of (braced) tokens. The private
  encapsulated format has only a few tokens, hence expansion will
  proceed a bit faster.

  \indent see footnote\footnotemark
\end{framed}

\footnotetext{To be completely honest the examination by \TeX{} of all
  successive digits was not avoided, as it occurs already in the
  locking-up of the result, what is avoided is to spend time un-locking,
  and then have the macros shuffle around possibly hundreds of digit
  tokens rather than a few control words.}

Our |\Fibonacci| expands completely under \fexpan sion,
so we can use \hyperref[fdef]{\ttfamily\char92fdef} rather than |\edef| in a
situation such as %
\leftedline {|\fdef \X {\FibonacciN {100}}|} but for the
reasons explained above, it is as efficient to employ |\edef|. And if we want
%
\leftedline{|\edef \Y {(\FibonacciN{100},\FibonacciN{200})}|,} then |\edef| is
necessary.

Allright, so let's now give the code to generate a sequence of braced Fibonacci
numbers |{F(N)}{F(N+1)}{F(N+2)}...|, using |\Fibonacci| for the first
two and then using the standard recursion |F(N+2)=F(N+1)+F(N)|:

\catcode`_ 11
\def\FibonacciSeq #1#2{%#1=starting index, #2>#1=ending index
    \expandafter\Fibonacci_Seq\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2-1}%
}%
\def\Fibonacci_Seq #1#2{%
     \expandafter\Fibonacci_Seq_loop\expandafter
                {\the\numexpr #1\expandafter}\romannumeral0\Fibonacci {#1}{#2}%
}%
\def\Fibonacci_Seq_loop #1#2#3#4{% standard Fibonacci recursion
    {#3}\unless\ifnum #1<#4  \Fibonacci_Seq_end\fi
        \expandafter\Fibonacci_Seq_loop\expandafter
        {\the\numexpr #1+1\expandafter}\expandafter
        {\romannumeral0\xintiieval #2+#3\relax}{#2}{#4}%
}%
\def\Fibonacci_Seq_end\fi\expandafter\Fibonacci_Seq_loop\expandafter
    #1\expandafter #2#3#4{\fi {#3}}%
\catcode`_ 8

\begingroup\footnotesize\baselineskip10pt
\everb|@
\catcode`_ 11
\def\FibonacciSeq #1#2{%#1=starting index, #2>#1=ending index
    \expandafter\Fibonacci_Seq\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2-1}%
}%
\def\Fibonacci_Seq #1#2{%
     \expandafter\Fibonacci_Seq_loop\expandafter
                {\the\numexpr #1\expandafter}\romannumeral0\Fibonacci {#1}{#2}%
}%
\def\Fibonacci_Seq_loop #1#2#3#4{% standard Fibonacci recursion
    {#3}\unless\ifnum #1<#4  \Fibonacci_Seq_end\fi
        \expandafter\Fibonacci_Seq_loop\expandafter
        {\the\numexpr #1+1\expandafter}\expandafter
        {\romannumeral0\xintiieval #2+#3\relax}{#2}{#4}%
}%
\def\Fibonacci_Seq_end\fi\expandafter\Fibonacci_Seq_loop\expandafter
    #1\expandafter #2#3#4{\fi {#3}}%
\catcode`_ 8
|
\endgroup

Deliberately and for optimization, this |\FibonacciSeq| macro is
completely expandable but not \fexpan dable. It would be easy to modify
it to be so. But I wanted to check that the \csbxint{For*} does apply
full expansion to what comes next each time it fetches an item from its
list argument. Thus, there is no need to generate lists of braced
Fibonacci numbers beforehand, as \csbxint{For*}, without using any
|\edef|, still manages to generate the list via iterated full expansion.

I initially used only one |\halign| in a three-column |multicols|
environment, but |multicols| only knows to divide the page horizontally
evenly, thus I employed in the end one |\halign| for each column (I
could have then used a |tabular| as no column break was then needed).

\begin{figure*}[ht!]
  \phantomsection\label{fibonacci}
  \newcounter{index}
  \fdef\Fibxxx{\FibonacciN {30}}%
  \setcounter{index}{30}%
\centeredline{\tabskip 1ex
\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr
  \xintFor* #1 in {\FibonacciSeq {30}{59}}\do
  {\theindex &\xintthe#1 &
    \xintiRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}%
}\vrule
\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr
  \xintFor* #1 in {\FibonacciSeq {60}{89}}\do
  {\theindex &\xintthe#1 &
    \xintiRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}%
}\vrule
\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr
  \xintFor* #1 in {\FibonacciSeq {90}{119}}\do
  {\theindex &\xintthe#1 &
   \xintiRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}%
}}%
%
\centeredline{Some Fibonacci numbers together with their residues modulo
  |F(30)|\dtt{=\xintthe\Fibxxx}}
\end{figure*}

\begingroup\footnotesize\baselineskip10pt
\everb|@
\newcounter{index}
\tabskip 1ex
  \fdef\Fibxxx{\FibonacciN {30}}%
  \setcounter{index}{30}%
\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr
  \xintFor* #1 in {\FibonacciSeq {30}{59}}\do
  {\theindex &\xintthe#1 &
    \xintiRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}%
}\vrule
\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr
  \xintFor* #1 in {\FibonacciSeq {60}{89}}\do
  {\theindex &\xintthe#1 &
    \xintiRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}%
}\vrule
\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr
  \xintFor* #1 in {\FibonacciSeq {90}{119}}\do
  {\theindex &\xintthe#1 &
   \xintiRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}%
}%
|
\endgroup

This produces the Fibonacci numbers from |F(30)| to |F(119)|, and
computes also  all the
congruence classes modulo |F(30)|.  The output has
been put in a \hyperref[fibonacci]{float}, which appears
\vpageref[above]{fibonacci}. I leave to the mathematically inclined
readers the task to explain the visible patterns\dots |;-)|.

\subsection{\csbh{xintForpair}, \csbh{xintForthree}, \csbh{xintForfour}}\label{xintForpair}\label{xintForthree}\label{xintForfour}
% {\small New in |1.09c|. The \csa{xintifForFirst}
%   |1.09e| mechanism was missing and has been added for |1.09f|. The |1.09f|
%   version handles better spaces and admits all (consecutive) macro
%   parameters.\par}

The syntax\ntype{on} is illustrated in this
example. The notation is the usual one for |n|-uples, with parentheses and
commas. Spaces around commas and parentheses are ignored.
%
\begin{everbatim*}
{\centering\begin{tabular}{cccc}
    \xintForpair #1#2 in { ( A , a ) , ( B , b ) , ( C , c ) } \do {%
      \xintForpair #3#4 in { ( X , x ) , ( Y , y ) , ( Z , z ) } \do {%
        $\Biggl($\begin{tabular}{cc}
          -#1- & -#3-\\
          -#4- & -#2-\\
        \end{tabular}$\Biggr)$&}\\\noalign{\vskip1\jot}}%
\end{tabular}\\}
\end{everbatim*}

Only |#1#2|, |#2#3|, |#3#4|, \dots, |#8#9| are valid (no error check
is done on the input syntax, |#1#3| or similar all end up in errors).
One can nest with \csbxint{For}, for disjoint sets of macro parameters. There is
also \csa{xintForthree} (from |#1#2#3| to |#7#8#9|) and \csa{xintForfour} (from
|#1#2#3#4| to |#6#7#8#9|). |\par| tokens are accepted in both the comma
separated list and the replacement text.

% These three macros |\xintForpair|, |\xintForthree| and |\xintForfour| are to
% be considered in experimental status, and may be removed, replaced or
% substantially modified at some later stage.

\subsection{\csbh{xintAssign}}\label{xintAssign}
%\small{ |1.09i| adds optional parameter. |1.09j| has default optional
% parameter |[]| rather than |[e]|\par}

\csa{xintAssign}\meta{braced things}\csa{to}%
\meta{as many cs as they are things} %\ntype{{(f$\to$\lowast [x)}{\lowast N}}
%
defines (without checking if something gets overwritten) the control sequences
on the right of \csa{to} to expand to the successive tokens or braced items
found one after the otehr on the on the left of \csa{to}. It is not expandable.

A `full' expansion is first applied to the material in front of
\csa{xintAssign}, which may thus be a macro expanding to a list of braced items.

\xintAssign \xintiiPow {7}{13}\to\SevenToThePowerThirteen
\xintAssign \xintiiDivision{1000000000000}{133333333}\to\Q\R

Special case: if after this initial expansion no brace is found immediately
after \csa{xintAssign}, it is assumed that there is only one control sequence
following |\to|, and this control sequence is then defined via
|\def| to expand to the material between
\csa{xintAssign} and \csa{to}. Other types of expansions are specified through
an optional parameter to \csa{xintAssign}, see \emph{infra}.
%
\leftedline{|\xintAssign \xintiiDivision{1000000000000}{133333333}\to\Q\R|}
%
\leftedline{|\meaning\Q: |\dtt{\meaning\Q}, |\meaning\R:|
  \dtt{\meaning\R}} %
%
\leftedline{|\xintAssign \xintiiPow
  {7}{13}\to\SevenToThePowerThirteen|}
%
\leftedline{|\SevenToThePowerThirteen|\dtt{=\SevenToThePowerThirteen}}
%
\leftedline{(same as |\edef\SevenToThePowerThirteen{\xintiPow {7}{13}}|)}

\noindent\csa{xintAssign} admits since |1.09i| an
optional parameter, for example |\xintAssign [e]...| or |\xintAssign [oo]
...|. The latter means that the definitions of the macros initially on the
right of |\to| will be made with \hyperref[oodef]{\ttfamily\char92oodef} which
expands twice the replacement text. The default is simply to make the
definitions with |\def|, corresponding to an empty optional paramter |[]|.
Possibilities: |[], [g], [e], [x], [o], [go], [oo], [goo], [f], [gf]|.

In all cases, recall that |\xintAssign| starts with an \fexpan sion of what
comes next; this produces some list of tokens or braced items, and the
optional parameter only intervenes to decide the expansion type to be applied
then to each one of these items.

\emph{Note:} prior to release |1.09j|, |\xintAssign| did an |\edef| by
default, but it now does |\def|. Use the optional parameter |[e]| to force use
of |\edef|.

\begin{framed}
  It is known that \csa{xintAssign} is very picky and does not want a space
  before the |\to| or will give a surprising result if the material prior to
  |\to| does not start with a brace but does contain brace pairs. Next
  release of \xinttoolsname will presumably modify these ``features''.
\end{framed}

%  This
% macro uses various \csa{edef}'s, thus is incompatible with expansion-only
% contexts.

\subsection{\csbh{xintAssignArray}}\label{xintAssignArray}
% {\small Changed in release |1.06| to let the defined macro pass its
%   argument through a |\numexpr...\relax|. |1.09i| adds optional
%   parameter. \par}

\xintAssignArray \xintBezout {1000}{113}\to\Bez

\csa{xintAssignArray}\meta{braced
  things}\csa{to}\csa{myArray} %\ntype{{(f$\to$\lowast x)}N}
%
first expands fully what comes immediately after |\xintAssignArray| and
expects to find a list of braced things |{A}{B}...| (or tokens). It then
defines \csa{myArray} as a macro with one parameter, such that \csa{myArray\x}
expands to give the |x|th braced thing of this original
list (the argument \texttt{\x} itself is fed to a |\numexpr| by |\myArray|,
and |\myArray| expands in two steps to its output). With |0| as parameter,
\csa{myArray}|{0}| returns the number |M| of elements of the array so that the
successive elements are \csa{myArray}|{1}|, \dots, \csa{myArray}|{M}|.
%
\leftedline{|\xintAssignArray \xintBezout {1000}{113}\to\Bez|} will set
|\Bez{0}| to \dtt{\Bez0}, |\Bez{1}| to \dtt{\Bez1}, |\Bez{2}| to
\dtt{\Bez2}, |\Bez{3}| to \dtt{\Bez3}, |\Bez{4}| to
\dtt{\Bez4}, and |\Bez{5}| to \dtt{\Bez5}:
\dtt{(\Bez3)${}\times{}$\Bez1${}-{}$(\Bez4)${}\times{}$\Bez2${}={}$\Bez5.}
This macro is incompatible with expansion-only contexts.

\csa{xintAssignArray} admits now an optional
parameter, for example |\xintAssignArray [e]...|. This means that the
definitions of the macros will be made with |\edef|. The default is
|[]|, which makes the definitions with |\def|. Other possibilities: |[],
[o], [oo], [f]|. Contrarily to \csbxint{Assign} one can not use the |g|
here to make the definitions global. For this, one should rather do
|\xintAssignArray| within a group starting with |\globaldefs 1|.

Note that prior to release |1.09j| each item (token or braced material) was
submitted to an |\edef|, but the default is now to use |\def|.

\subsection{\csbh{xintRelaxArray}}\label{xintRelaxArray}

\csa{xintRelaxArray}\csa{myArray} %\ntype{N}
%
(globally) sets to \csa{relax} all macros which were defined by the previous
\csa{xintAssignArray} with \csa{myArray} as array macro.


\subsection{The Quick Sort algorithm illustrated}\label{ssec:quicksort}

First a completely expandable macro which sorts a list of numbers. The |\QSfull|
macro expands its list argument, which may thus be a macro; its items must
expand to possibly big integers (or also decimal numbers or fractions if using
\xintfracname), but if an item is expressed as a computation, this computation
will be redone each time the item is considered! If the numbers have many digits
(i.e. hundreds of digits...), the expansion of |\QSfull| is fastest if each
number, rather than being explicitely given, is represented as a single token
which expands to it in one step.

If the interest is only in \TeX{} integers, then one should replace the macros
|\QSMore|, |QSEqual|, |QSLess| with versions using the
\href{http://ctan.org/pkg/etoolbox}{etoolbox} (\LaTeX{} only) |\ifnumgreater|,
|\ifnumequal| and |\ifnumless| conditionals rather than \csbxint{ifGt},
\csbxint{ifEq}, \csbxint{ifLt}.

%% \makeatletter\let\check@percent\relax lorsque je faisais avec verbatim 
%% ne pas changer la taille dans \MacroFont
%% \def\MacroFont{\ttbfamily \small }
%% anciennement avec \dverb, puis \everb
%% \everb|"makeatletter"@gobble

\begin{everbatim*}
% THE QUICK SORT ALGORITHM EXPANDABLY
% \usepackage{xintfrac} in the preamble (latex), or \input xintfrac.sty (Plain)
\catcode`@ 11 % = \makeatletter
\def\QSMore  #1#2{\xintifGt {#2}{#1}{{#2}}{ }}
% the spaces stop the \romannumeral-`0 done by \xintapplyunbraced each time
% it applies its macro argument to an item
\def\QSEqual #1#2{\xintifEq {#2}{#1}{{#2}}{ }}
\def\QSLess  #1#2{\xintifLt {#2}{#1}{{#2}}{ }}
%
\def\QSfull {\romannumeral0\qsfull }
\def\qsfull   #1{\expandafter\qsfull@a\expandafter{\romannumeral-`0#1}}
\def\qsfull@a #1{\expandafter\qsfull@b\expandafter {\xintLength {#1}}{#1}}
\def\qsfull@b #1{\ifcase #1
                    \expandafter\qsfull@empty
                 \or\expandafter\qsfull@single
                 \else\expandafter\qsfull@c
                 \fi }
\def\qsfull@empty  #1{ }% the space stops the \QSfull \romannumeral0
\def\qsfull@single #1{ #1}
\def\qsfull@c #1{\qsfull@ci #1\undef {#1}}% we pick up the first as Pivot
\def\qsfull@ci #1#2\undef {\qsfull@d {#1}}
\def\qsfull@d #1#2{\expandafter\qsfull@e\expandafter
                   {\romannumeral0\qsfull {\xintApplyUnbraced {\QSMore {#1}}{#2}}}%
                   {\romannumeral0\xintapplyunbraced {\QSEqual {#1}}{#2}}%
                   {\romannumeral0\qsfull {\xintApplyUnbraced {\QSLess {#1}}{#2}}}%
}
\def\qsfull@e #1#2#3{\expandafter\qsfull@f\expandafter {#2}{#3}{#1}}
\def\qsfull@f #1#2#3{\expandafter\space #2#1#3}
\catcode`@ 12 % = \makeatother
% EXAMPLE
\begingroup
\edef\z {\QSfull {{1.0}{0.5}{0.3}{1.5}{1.8}{2.0}{1.7}{0.4}{1.2}{1.4}%
               {1.3}{1.1}{0.7}{1.6}{0.6}{0.9}{0.8}{0.2}{0.1}{1.9}}}
\printnumber{\meaning\z}

\def\a {3.123456789123456789}\def\b {3.123456789123456788}
\def\c {3.123456789123456790}\def\d {3.123456789123456787}
\expandafter\def\expandafter\z\expandafter
  {\romannumeral0\qsfull {{\a}\b\c\d}}% \a is braced to not be expanded
\printnumber{\meaning\z}
\endgroup
\end{everbatim*}

We then turn to a graphical illustration of the algorithm. For simplicity the
pivot is always chosen to be the first list item. We also show later how to
illustrate the  variant which picks up the last item of each unsorted
chunk as pivot.

\begin{everbatim*}
% in LaTeX preamble:
% \usepackage{xintfrac}
% \usepackage{color}
% or, when using Plain TeX:
% \input xintfrac.sty
% \input color.tex
%
% Color definitions
\definecolor{LEFT}{RGB}{216,195,88}
\definecolor{RIGHT}{RGB}{208,231,153}
\definecolor{INERT}{RGB}{199,200,194}
\definecolor{PIVOT}{RGB}{109,8,57}
% Start of macro defintions
\catcode`@ 11 % = \makeatletter in latex
\def\QSMore  #1#2{\xintifGt {#2}{#1}{{#2}}{ }}% space will be gobbled
\def\QSEqual #1#2{\xintifEq {#2}{#1}{{#2}}{ }}
\def\QSLess  #1#2{\xintifLt {#2}{#1}{{#2}}{ }}
%
\def\QS@a  #1{\expandafter \QS@b \expandafter {\xintLength {#1}}{#1}}
\def\QS@b  #1{\ifcase #1
                      \expandafter\QS@empty
                   \or\expandafter\QS@single
                 \else\expandafter\QS@c
                 \fi }
\def\QS@empty  #1{}
\def\QS@single #1{\QSIr {#1}}
\def\QS@c #1{\QS@d #1!{#1}}    % we pick up the first as pivot.
\def\QS@d #1#2!{\QS@e {#1}}    % #1 = first element, #3 = list
\def\QS@e #1#2{\expandafter\QS@f\expandafter
                   {\romannumeral0\xintapplyunbraced {\QSMore  {#1}}{#2}}%
                   {\romannumeral0\xintapplyunbraced {\QSEqual {#1}}{#2}}%
                   {\romannumeral0\xintapplyunbraced {\QSLess  {#1}}{#2}}}
\def\QS@f #1#2#3{\expandafter\QS@g\expandafter {#2}{#3}{#1}}
% #2= elements < pivot, #1 = elements = pivot, #3 = elements > pivot
% Here \QSLr, \QSIr, \QSr have been let to \relax, so expansion stops.
\def\QS@g #1#2#3{\QSLr {#2}\QSIr {#1}\QSRr {#3}}
%
\def\DecoLEFT   #1{\xintFor* ##1 in {#1} \do {\colorbox{LEFT}{##1}}}
\def\DecoINERT  #1{\xintFor* ##1 in {#1} \do {\colorbox{INERT}{##1}}}
\def\DecoRIGHT  #1{\xintFor* ##1 in {#1} \do {\colorbox{RIGHT}{##1}}}
\def\DecoPivot  #1{\begingroup\color{PIVOT}\advance\fboxsep-\fboxrule\fbox{#1}\endgroup}
\def\DecoLEFTwithPivot #1{%
     \xintFor* ##1 in {#1} \do {\xintifForFirst {\DecoPivot {##1}}{\colorbox{LEFT}{##1}}}}
\def\DecoRIGHTwithPivot #1{%
     \xintFor* ##1 in {#1} \do {\xintifForFirst {\DecoPivot {##1}}{\colorbox{RIGHT}{##1}}}}
%
\def\QSinitialize #1{\def\QS@list{\QSRr {#1}}\let\QSRr\DecoRIGHT
                     \par\centerline{\QS@list}}
\def\QSoneStep {\let\QSLr\DecoLEFTwithPivot \let\QSIr\DecoINERT \let\QSRr\DecoRIGHTwithPivot
    \centerline{\QS@list}%
                \def\QSLr {\noexpand\QS@a}\let\QSIr\relax\def\QSRr {\noexpand\QS@a}%
                    \edef\QS@list{\QS@list}%
                \let\QSLr\relax\let\QSRr\relax
                    \edef\QS@list{\QS@list}%
                \let\QSLr\DecoLEFT \let\QSIr\DecoINERT \let\QSRr\DecoRIGHT
    \centerline{\QS@list}}
\catcode`@ 12 % = \makeatother in latex
% End of macro definitions.
% The next line is for xint.pdf use only. 
\normalcolor\phantomsection\label{quicksort}
% Start of Example
\begingroup\offinterlineskip
\small
\QSinitialize {{1.0}{0.5}{0.3}{1.5}{1.8}{2.0}{1.7}{0.4}{1.2}{1.4}%
               {1.3}{1.1}{0.7}{1.6}{0.6}{0.9}{0.8}{0.2}{0.1}{1.9}}
\QSoneStep\QSoneStep\QSoneStep\QSoneStep\QSoneStep
\endgroup
\end{everbatim*}

If one wants rather to have the pivot from the end of the yet to sort chunks,
then one should use the following variants:

\begin{everbatim*}
\normalcolor
\makeatletter
\def\QS@c #1{\expandafter\QS@e\expandafter {\romannumeral0\xintnthelt {-1}{#1}}{#1}}
\def\DecoLEFTwithPivot #1{%
    \xintFor* ##1 in {#1} \do {\xintifForLast {\DecoPivot {##1}}{\colorbox{LEFT}{##1}}}}
\def\DecoRIGHTwithPivot #1{%
    \xintFor* ##1 in {#1} \do{\xintifForLast {\DecoPivot {##1}}{\colorbox{RIGHT}{##1}}}}
\def\QSinitialize #1{\def\QS@list{\QSLr {#1}}\let\QSLr\DecoLEFT\par\centerline{\QS@list}}
\makeatother
\begingroup\offinterlineskip
\small
\QSinitialize {{1.0}{0.5}{0.3}{1.5}{1.8}{2.0}{1.7}{0.4}{1.2}{1.4}%
               {1.3}{1.1}{0.7}{1.6}{0.6}{0.9}{0.8}{0.2}{0.1}{1.9}}
\QSoneStep\QSoneStep\QSoneStep\QSoneStep\QSoneStep
\QSoneStep\QSoneStep\QSoneStep\QSoneStep\QSoneStep
\endgroup
\end{everbatim*}

It is possible to modify this code to let it do \csa{QSonestep} repeatedly and
stop automatically when the sort is finished.%
%
\footnote{\url{http://tex.stackexchange.com/a/142634/4686}}

\section{Commands of the \xintcorename package}
\label{sec:core}

\localtableofcontents

Prior to release |1.1| the macros which are now included in the separate
package \xintcorename were part of \xintname. Package \xintcorename is
automatically loaded by \xintname.\IMPORTANT\

\xintcorename provides the five basic arithmetic operations on big integers:
addition, subtraction, multiplication, division and powers. Division may be
either rounded (\csbxint{iiDivRound}) (the rounding of |0.5| is |1| and the
one of |-0.5| is |-1|) or Euclidean (\csbxint{iiQuo}) (which for positive
operands is the same as truncated division), or truncated (\csbxint{iiDivTrunc}).

In the description of the macros the \texttt{\n} and \texttt{\m} symbols stand
for explicit (big) integers within braces or more generally any control
sequence (possibly within braces) \hyperref[sec:expansions]{\fexpan ding} to
such a big integer.

The macros with a single |i| in their names parse their arguments
automatically through \hyperref[xintiNum]{\string\xintNum}. This type of
expansion applied to an argument is signaled by a
\textcolor[named]{PineGreen}{\Numf} in the margin. The accepted input format
is then a sequence of plus and minus signs, followed by some string of zeroes,
followed by digits. 

If \xintfracname additionally to \xintcorename is loaded, \csbxint{Num}
becomes a synonym to \csbxint{TTrunc}; this means that
arbitrary\inmarg{Changed} fractions will be accepted as arguments of the
macros with a single |i| in their names, but get truncated to integers before
further processing. The format of the output will be as with only \xintname
loaded. The only extension is in allowing a wider variety of inputs.

The macros with |ii| in their names have arguments which will only be \fexpan
ded, but will not be parsed via \hyperref[xintiNum]{\string\xintNum}.
Arguments of this type are signaled by the margin annotation
\textcolor[named]{PineGreen}{\emph{f}}. For such big integers only one minus
sign and no plus sign, nor leading zeros, are accepted. |-0| is not valid in
this strict input format. Loading \xintfracname does not bring any
modification to these macros whether for input or output.

The letter \texttt{x} (with margin annotation
\textcolor[named]{PineGreen}{\numx}) stands for something which will be
inserted in-between a |\numexpr| and a |\relax|. It will thus be completely
expanded and must give an integer obeying the \TeX{} bounds. Thus, it may be
for example a count register, or itself a \csa{numexpr} expression, or just a
number written explicitely with digits or something like |4*\count 255 + 17|,
etc...

For the rules regarding direct use of count registers or \csa{numexpr}
expression, in the arguments to the package macros, see the
\hyperref[sec:useofcount]{Use of count} section.

\begin{framed}
  \xintcorename also provides macros |\xintAdd|, |\xintMul|,\dots as synonyms
  to |\xintiAdd|, |\xintiMul|,\dots. Their usage is
  \fbox{deprecated}\IMPORTANT{} for the following reason: with \xintfracname
  loaded they become the routines dealing fully with fractions on input. But
  this means that they now use fraction format on output, even with integer
  arguments.

  Due to this variability of the output format on whether the document uses
  only \xintname or loads additionally \xintfracname, code using these macros
  is fragile, because loading at some later date a package which itself loads
  \xintfracname or \xintexprname will modify their output format, and this is
  catastrophic for example in locations expanded by |\ifnum|, or even in
  arguments to those other macros of \xintname with |ii| in their names.

  Prefer thus, when writing code loading only \xintcorename or \xintname, to
  use the macros \csbxint{iAdd}, \csbxint{iMul}, \dots, or \csbxint{iiAdd},
  \csbxint{iiMul}, \dots. They are guaranteed to always output an integer
  without a trailing |/B[n]|. The latter have the less overhead, and the
  former do not complain, if \xintfracname is loaded, even if used with true
  fractions, as they will then truncate their arguments to
  integers.\MyMarginNote[\kern\dimexpr\FrameSep+\FrameRule\relax]{Changed}
 
  It was an error for the \xintname package (now \xintcorename) to provide
  macros |\xintAdd|, |\xintMul|, |\xintSub| \dots. They should be used only
  with \xintfracname loaded.
\end{framed}

The {\color[named]{PineGreen}$\star$}'s in the margin are there to remind of
the complete expandability, even \fexpan dability of the macros, as discussed
in \autoref{sec:expansions}.


\subsection{\csbh{xintNum}}\label{xintiNum}

|\xintNum|\n\etype{f} removes chains of plus or minus signs, followed by
zeroes. %
%
\leftedline{|\xintNum{+---++----+--000000000367941789479}|\dtt
 {=\xintNum{+---++----+--000000000367941789479}}} 

All \xintname macros with a single |i| in their names, such as \csbxint{iAdd},
\csbxint{iMul} apply \csbxint{Num} to their arguments.

When \xintfracname is loaded, \csbxint{Num} becomes a synonym to
\csbxint{TTrunc}.\inmarg{Changed}

\subsection{\csbh{xintSgn}, \csbh{xintiiSgn}}\label{xintiiSgn}

|\xintiiSgn|\n\etype{f} returns 1 if the number is positive, 0 if it is zero
and -1 if it is negative. It skips the \csbxint{Num} overhead.

\csbxint{Sgn}\etype{\Numf} is the variant using \csbxint{Num} and getting
extended by \xintfracname to fractions.

\subsection{\csbh{xintiOpp}, \csbh{xintiiOpp}}\label{xintiOpp}\label{xintiiOpp}

|\xintiOpp|\n\etype{\Numf} return the opposite |-N| of the number |N|.
\csbxint{Opp} is initially a synonym but gets extended by \xintfracname to
fractions (its output format will be a fraction even if the argument is an
integer) and \csa{xintiiOpp} is the strict integer-only variant which skips
the \csbxint{Num} overhead.\etype{f}

\subsection{\csbh{xintiAbs}, \csbh{xintiiAbs}}\label{xintiAbs}\label{xintiiAbs}

|\xintiAbs|\n\etype{\Numf} returns the absolute value of the number.
\csbxint{Abs} is a synonym but gets modified by \xintfracname. \csa{xintiiAbs}
skips the \csbxint{Num} overhead.\etype{f}

\subsection{\csbh{xintiiFDg}}\label{xintFDg}\label{xintiiFDg}

|\xintiiFDg|\n\etype{f} returns the first digit (most significant) of the
decimal expansion. It skips the overhead of parsing via \csbxint{Num}. The
variant \csa{xintFDg}\etype{\Numf} uses |\xintNum| and gets extended by
\xintfracname. 

\subsection{\csbh{xintiiLDg}}\label{xintLDg}\label{xintiiLDg}

|\xintiiLDg|\n\etype{f} returns the least significant digit. When the number
is positive, this is the same as the remainder in the euclidean division by
ten. It skips the overhead of parsing via \csbxint{Num}. The variant
\csa{xintLDg}\etype{\Numf} uses |\xintNum| and gets extended by \xintfracname.

\subsection{\csbh{xintDouble}, \csbh{xintHalf}}
\label{xintDouble}
\label{xintHalf}
%{\small New with |1.08|.\par}

|\xintDouble|\n\etype{f} returns |2N| and |\xintHalf|\n is |N/2| rounded
towards zero. These macros remain integer-only, even with \xintfracname loaded.

\subsection{\csbh{xintInc}, \csbh{xintDec}}
\label{xintInc}
\label{xintDec}
%{\small New with |1.08|.\par}

|\xintInc|\n\etype{f} is |N+1| and |\xintDec|\n{} is |N-1|. These macros
remain integer-only, even with \xintfracname loaded. They skip the overhead
of parsing via \csbxint{Num}.

\subsection{\csbh{xintiAdd}, \csbh{xintiiAdd}}\label{xintiAdd}\label{xintiiAdd}

|\xintiAdd|\n\m\etype{\Numf\Numf} returns the sum of the two numbers.
\csbxint{Add} is initially a synonym but gets extended by \xintfracname.
\csa{xintiiAdd} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintiSub}, \csbh{xintiiSub}}\label{xintiSub}\label{xintiiSub}

|\xintiSub|\n\m\etype{\Numf\Numf} returns the difference |N-M|. \csbxint{Sub}
is initially a synonym but gets extended by \xintfracname.
\csa{xintiiSub} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintiMul}, \csbh{xintiiMul}}\label{xintiMul}\label{xintiiMul}
%{\small Modified in release |1.03|.\par}

|\xintiMul|\n\m\etype{\Numf\Numf} returns the product of the two numbers.
\csbxint{Mul} is the initial synonym modified by \xintfracname, and
\csa{xintiiMul} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintiSqr}, \csbh{xintiiSqr}}\label{xintiSqr}\label{xintiiSqr}

|\xintiSqr|\n\etype{\Numf} returns the square. \csbxint{Sqr} is the initial
synonym extended by \xintfracname to fractions. \csa{xintiiSqr} skips the
\csbxint{Num} overhead.\etype{f}

\subsection{\csbh{xintiPow}, \csbh{xintiiPow}}\label{xintiPow}\label{xintiiPow}

|\xintiPow|\n\x\etype{\Numf\numx} returns |N^x|. When |x| is zero, this is 1.
If |N=0| and |x<0|, if \verb+|N|>1+ and |x<0|, or if \verb+|N|>1+
and |x>100000|, then an error is raised. 

The |x>100000| condition should perhaps be made more strict: as it stands it
allows launching operations taking hours to complete. Indeed, observe that
|2^50000| already has \dtt{\xintLen{\xintFloatPow [1]{2}{50000}}} digits; as
it turns out each exact multiplication done via \csbxint{iiMul} of two
numbers with one thousand digits each already takes of the order of seconds,
and it would take hours for arguments each with circa $15000$ digits. Perhaps
some completely expandable but not \fexpan dable variants could fare better?

\csa{xintiiPow} is an integer only variant skipping the \csbxint{Num}
overhead\etype{f\numx}, it produces the same result as \csa{xintiPow} with
stricter assumptions on the inputs, and is thus a tiny bit faster.

\csbxint{Pow} is the initial synonym of \csa{xintiPow} which gets extended by
\xintfracname to fractions (see 
also \csbxint{FloatPow} for which the exponent must still obey the \TeX{} bound
and \csbxint{FloatPower} which has no restriction at all on the size of the
exponent). Negative exponents do not then raise errors anymore. The float
version is able to deal with things such as |2^999999999| without any problem.
For example |\xintFloatPow[4]{2}{50000}|\dtt{=\xintFloatPow[4]{2}{50000}}
and |\xintFloatPow[4]{2}{999999999}|
\dtt{=\xintFloatPow[4]{2}{999999999}}.%
%
\footnote{On my laptop \texttt{\detokenize{\xintiiPow {2}{9999}}}
  obtains all |3010| digits in about ten or eleven seconds. In contrast,
  the float versions for |8|, |16|, |24|, or even more significant
  figures, do their jobs in less than one hundredth of a second
  (|1.09j|; we used in the text only four significant digits only for
  reasons of space, not time.) This is done without |log|/|exp| which
  are not (yet?) implemented in \xintfracname. The \LaTeX3
  \href{http://www.ctan.org/pkg/l3kernel}{l3fp} package does this with
  |log|/|exp| and is ten times faster, but allows only |16| significant
  figures and the (exactly represented) floating point numbers must have
  their exponents limited to $\pm$\dtt{9999}.}

Within an \csbxint{iiexpr}|..\relax| the infix operator |^| is mapped to
\csa{xintiiPow}; within an \csbxint{expr}-ession it is mapped to \csbxint{Pow}
(as extended by \xintfracname); in \csbxint{floatexpr}, it is mapped to
\csbxint{FloatPower}.

\subsection{\csbh{xintiDivision},
  \csbh{xintiiDivision}}\label{xintiDivision}\label{xintiiDivision}

% 17 octobre 2014: je supprime \xintDivision, seulement \xintiDivision.

|\xintiiDivision|\n\m\etype{ff} returns |{quotient Q}{remainder R}|. This is
euclidean division: |N = QM + R|, |0|${}\leq{}$\verb+R < |M|+. So the
remainder is always non-negative and the formula |N = QM + R| always holds
independently of the signs of |N| or |M|. Division by zero is an error (even
if |N| vanishes) and returns |{0}{0}|. It skips the overhead of parsing via
\csbxint{Num}.

|\xintiDivision|\etype{\Numf\Numf} submits its arguments to \csbxint{Num} and
is extended by \xintfracname to accept fractions on input, which it truncates
first, and is not to be confused with the \xintfracname macro \csbxint{Div}
which divides one fraction by another.

Note: |\xintDivision| was the former name of |\xintiDivision|. Its use is
deprecated since release |1.1|.\inmarg{Changed}

\subsection{\csbh{xintiQuo}, \csbh{xintiiQuo}}\label{xintiQuo}\label{xintiiQuo}

|\xintiiQuo|\n\m\etype{ff} returns the quotient from the euclidean division.
It skips the overhead of parsing via \csbxint{Num}. 

|\xintiQuo|\etype{\Numf\Numf}  submits its arguments to \csbxint{Num} and
is extended by \xintfracname to accept fractions on input, which it truncates
first.

Note: |\xintQuo| is the former name of |\xintiQuo|. Its use is deprecated.
\inmarg{Changed}

\subsection{\csbh{xintiRem}, \csbh{xintiiRem}}\label{xintiRem}\label{xintiiRem}

|\xintiiRem|\n\m\etype{ff} returns the remainder from the euclidean
division. It skips the overhead of parsing via \csbxint{Num}.

|\xintiRem|\etype{\Numf\Numf}  submits its arguments to \csbxint{Num} and
is extended by \xintfracname to accept fractions on input, which it truncates
first.

Note: |\xintRem| is the former name of |\xintiRem|. Its use is deprecated.
\inmarg{Changed}

\subsection{\csbh{xintiDivRound}, \csbh{xintiiDivRound}}
\label{xintiDivRound}\label{xintiiDivRound}

|\xintiiDivRound|\n\m\etype{ff} returns the rounded value of the algebraic
quotient $N/M$ of two big integers. The rounding of half integers is towards
the nearest integer of bigger absolute value. The macro skips the overhead of
parsing via \csbxint{Num}. The rounding is away from zero.

|\xintiDivRound|\etype{\Numf\Numf} submits its arguments to \csbxint{Num}. It
is extended by \xintfracname to accept fractions on input, which it truncates
first before computing the rounded quotient.

\subsection{\csbh{xintiDivTrunc}, \csbh{xintiiDivTrunc}}
\label{xintiDivTrunc}\label{xintiiDivTrunc}

|\xintiiDivTrunc|\n\m\etype{ff} computes the truncation towards zero of the
algebraic quotient $N/M$. It skips the overhead of parsing the operands with
\csbxint{Num}. For $M>0$ it is the same as \csbxint{iiQuo}.
\begin{everbatim*}
$\xintiiQuo {1000}{-57}, \xintiiDivRound {1000}{-57}, \xintiiDivTrunc {1000}{-57}$
\end{everbatim*}

|\xintiDivTrunc|\etype{\Numf\Numf} submits first its arguments to \csbxint{Num}.

\subsection{\csbh{xintiMod}, \csbh{xintiiMod}}
\label{xintiMod}\label{xintiiMod}

|\xintiiMod|\n\m\etype{ff} computes $N - M*t(N/M)$, where $t(N/M)$ is the
algebraic quotient truncated towards zero . The macro skips the overhead of parsing
the operands with \csbxint{Num}. For $M>0$ it is the same as \csbxint{iiRem}.
\begin{everbatim*}
$\xintiiRem {1000}{-57}, \xintiiMod {1000}{-57}, 
 \xintiiRem {-1000}{57}, \xintiiMod {-1000}{57}$
\end{everbatim*}

|\xintiMod|\etype{\Numf\Numf} submits first its arguments to \csbxint{Num}.

\section{Commands of the \xintname package}
\label{sec:xint}

\localtableofcontents

Version |1.0| was released |2013/03/28|. This is \texttt{\xintbndlversion} of
\texttt{\xintbndldate}. The core arithmetic macros have been
moved\inmarg{Changed} to separate package \xintcorename, which is
automatically loaded by \xintname. 

See the documentation of \xintcorename or \autoref{sec:expansions} for the
significance of the \textcolor[named]{PineGreen}{\Numf},
\textcolor[named]{PineGreen}{\emph{f}}, \textcolor[named]{PineGreen}{\numx}
and \textcolor[named]{PineGreen}{$\star$} margin annotations and some
important background information.

\subsection{\csbh{xintRev}} \label{xintRev}

|\xintRev|\n\etype{f} will reverse the order of the digits of the number,
keeping the optional sign. Leading zeroes
resulting from the operation are not removed (see the
\csa{xintNum} macro for this). This macro and all other
macros dealing with numbers first expand `fully' their arguments.
%
\leftedline{|\xintRev{-123000}|\dtt{=\xintRev{-123000}}}
%
\leftedline{|\xintNum{\xintRev{-123000}}|%
               \dtt{=\xintNum{\xintRev{-123000}}}}

\subsection{\csbh{xintLen}}\label{xintiLen}

|\xintLen|\n\etype{\Numf} returns the length of the number, not counting the
sign. %
%
\leftedline{|\xintLen{-12345678901234567890123456789}|\dtt
 {=\xintLen{-12345678901234567890123456789}}} Extended by \xintfracname to
fractions: the length of |A/B[n]| is the length of |A| plus the
length of |B| plus the absolute value of |n| and minus one (an integer input as
|N| is internally represented in a form equivalent to |N/1[0]| so the minus one
means that the extended \csa{xintLen} behaves the same as the original for
integers). %
%
\leftedline{|\xintLen{-1e3/5.425}|\dtt
 {=\xintLen{-1e3/5.425}}} The length is computed on the |A/B[n]| which would
have been returned by \csbxint{Raw}: |\xintRaw {-1e3/5.425}|\dtt{=\xintRaw
  {-1e3/5.425}}.

Let's point out that the whole thing should sum up to
less than circa $2^{31}$, but this is a bit theoretical.

|\xintLen| is only for numbers or fractions. See \csbxint{Length} for counting
tokens (or rather braced groups), more generally.

\subsection{\csbh{xintDigitsOf}}\label{xintDigitsOf}

This is a synonym for \csbxint{AssignArray},\ntype{fN} to be used to define
an array giving all the digits of a given (positive, else the minus sign will
be treated as first item) number.
\begingroup\xintDigitsOf\xintiPow {7}{500}\to\digits
%
\leftedline{|\xintDigitsOf\xintiPow {7}{500}\to\digits|}
\noindent $7^{500}$ has |\digits{0}=|\digits{0} digits, and the 123rd among them
(starting from the most significant) is
|\digits{123}=|\digits{123}.
\endgroup

\subsection{\csbh{xintCmp}, \csbh{xintiiCmp}}

|\xintCmp|\n\m\etype{\Numf\Numf} returns \dtt{1} if |N>M|, \dtt{0} if |N=M|,
and \dtt{-1} if |N<M|. Extended by \xintfracname to fractions (its output
naturally still being either |1|, |0|, or |-1|).

\csa{xintiiCmp} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintEq}, \csbh{xintiiEq}}\label{xintEq}
%{\small New with release |1.09a|.\par}

|\xintEq|\n\m\etype{\Numf\Numf} returns 1 if |N=M|, 0 otherwise. Extended
by \xintfracname to fractions.

\csa{xintiiEq} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintNeq}, \csbh{xintiiNeq}}
%{\small New with release |1.09a|.\par}

|\xintNeq|\n\m\etype{\Numf\Numf} returns 0 if |N=M|, 1 otherwise. Extended
by \xintfracname to fractions.

\csa{xintiiNeq} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintGt}, \csbh{xintiiGt}}\label{xintGt}
%{\small New with release |1.09a|.\par}

|\xintGt|\n\m\etype{\Numf\Numf} returns 1 if |N|$>$|M|, 0 otherwise.
Extended by \xintfracname to fractions.

\csa{xintiiGt} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintLt}, \csbh{xintiiLt}}\label{xintLt}
%{\small New with release |1.09a|.\par}

|\xintLt|\n\m\etype{\Numf\Numf} returns 1 if |N|$<$|M|, 0 otherwise.
Extended by \xintfracname to fractions.

\csa{xintiiLt} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintLtorEq}, \csbh{xintiiLtorEq}}

|\xintLtorEq|\n\m\etype{\Numf\Numf} returns 1 if |N|$\leqslant$|M|, 0 otherwise.
Extended by \xintfracname to fractions.

\csa{xintiiLtorEq} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintGtorEq}, \csbh{xintiiGtorEq}}

|\xintGtorEq|\n\m\etype{\Numf\Numf} returns 1 if |N|$\geqslant$|M|, 0 otherwise.
Extended by \xintfracname to fractions.

\csa{xintiiGtorEq} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintIsZero}, \csbh{xintiiIsZero}}\label{xintIsZero}
%{\small New with release |1.09a|.\par}

|\xintIsZero|\n\etype{\Numf} returns 1 if |N=0|, 0 otherwise.
Extended by \xintfracname to fractions.

\csa{xintiiIsZero} skips the \csbxint{Num} overhead.\etype{f}

\subsection{\csbh{xintNot}}\label{xintNot}
%{\small New with release |1.09c|.\par}

\csa{xintNot}\etype{\Numf} is a synonym for \csa{xintIsZero}.

\subsection{\csbh{xintIsNotZero}, \csbh{xintiiIsNotZero}}\label{xintIsNotZero}
%{\small New with release |1.09a|.\par}

|\xintIsNotZero|\n\etype{\Numf} returns 1 if |N<>0|, 0 otherwise.
Extended by \xintfracname to fractions.

\csa{xintiiIsNotZero} skips the \csbxint{Num} overhead.\etype{f}

\subsection{\csbh{xintIsOne},
  \csbh{xintiiIsOne}}\label{xintIsOne}\label{xintiiIsOne} 
%{\small New with release |1.09a|.\par}

|\xintIsOne|\n\etype{\Numf} returns 1 if |N=1|, 0 otherwise.
Extended by \xintfracname to fractions.

\csa{xintiiIsOne} skips the \csbxint{Num} overhead.\etype{f}

\subsection{\csbh{xintAND}}\label{xintAND}
%{\small New with release |1.09a|.\par}

|\xintAND|\n\m\etype{\Numf\Numf} returns 1 if |N<>0| and |M<>0| and zero
otherwise. Extended by \xintfracname to fractions.

\subsection{\csbh{xintOR}}\label{xintOR}
%{\small New with release |1.09a|.\par}

|\xintOR|\n\m\etype{\Numf\Numf} returns 1 if |N<>0| or |M<>0| and zero
otherwise. Extended by \xintfracname to fractions.

\subsection{\csbh{xintXOR}}\label{xintXOR}
%{\small New with release |1.09a|.\par}

|\xintXOR|\n\m\etype{\Numf\Numf} returns 1 if exactly one of |N| or |M|
is true (i.e. non-zero). Extended by \xintfracname to fractions.

\subsection{\csbh{xintANDof}}\label{xintANDof}
%{\small New with release |1.09a|.\par}

\csa{xintANDof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns 1 if all
are true (i.e. non zero) and zero otherwise. The list argument may be a macro,
it (or rather its first token) is \fexpan ded first (each item also is \fexpan
ded). Extended by \xintfracname to fractions.

\subsection{\csbh{xintORof}}\label{xintORof}
%{\small New with release |1.09a|.\par}

\csa{xintORof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns 1 if at
least one is true (i.e. does not vanish). The list argument may be a macro, it
is \fexpan ded first. Extended by \xintfracname to fractions.

\subsection{\csbh{xintXORof}}\label{xintXORof}
%{\small New with release |1.09a|.\par}

\csa{xintXORof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns 1 if an odd
number of them are true (i.e. does not vanish). The list argument may be a
macro, it is \fexpan ded first. Extended by \xintfracname to fractions.

\subsection{\csbh{xintGeq}}\label{xintiGeq}

|\xintGeq|\n\m\etype{\Numf\Numf} returns 1 if the \emph{absolute value}
of the first number is at least equal to the absolute value of the second
number. If \verb+|N|<|M|+ it returns 0. Extended by \xintfracname to fractions.
%(starting with release |1.07|)
Important: the macro compares \emph{absolute values}.

\subsection{\csbh{xintiMax}, \csbh{xintiiMax}}\label{xintiMax}\label{xintiiMax}

|\xintiMax|\n\m\etype{\Numf\Numf} returns the largest of the two in the sense
of the order structure on the relative integers (\emph{i.e.} the right-most
number if they are put on a line with positive numbers on the right):
|\xintiMax {-5}{-6}|\dtt{=\xintiMax{-5}{-6}}. The initial synonym \csbxint{Max}
gets modified by \xintfracname which extends it to fractions.  Its usage when
only \xintname is loaded is discouraged.

The |\xintiiMax| macro skips the overhead of parsing the operands with
\csbxint{Num}.\etype{ff}

\subsection{\csbh{xintiMin}, \csbh{xintiiMin}}\label{xintiMin}\label{xintiiMin}

|\xintiMin|\n\m\etype{\Numf\Numf} returns the smallest of the two in the
sense of the order structure on the relative integers (\emph{i.e.} the left-most
number if they are put on a line with positive numbers on the right): |\xintiMin
{-5}{-6}|\dtt{=\xintiMin{-5}{-6}}. The initial synonym \csbxint{Min}
gets modified by \xintfracname which extends it to fractions. Its usage when
only \xintname is loaded is discouraged.

The |\xintiiMin| macro skips the overhead of parsing the operands with
\csbxint{Num}.\etype{ff}

\subsection{\csbh{xintiMaxof}}\label{xintiMaxof}
%{\small New with release |1.09a|.\par}

\csa{xintiMaxof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns the
maximum. The list argument may be a macro, it is \fexpan ded first. Each item
is submitted to |\xintNum| normalization. \csbxint{Maxof} is the initial
synonym which gets extended by \xintfracname to fractions. Do not use it when
only \xintname is loaded.

\subsection{\csbh{xintiMinof}}\label{xintiMinof}
%{\small New with release |1.09a|.\par}

\csa{xintiMinof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns the
minimum. The list argument may be a macro, it is \fexpan ded first. Each item
is submitted to |\xintNum| normalization. \csbxint{Minof} is the initial
synonym which gets extended by \xintfracname to fractions. Do not use it when
only \xintname is loaded.

\subsection{\csbh{xintiiSum}}\label{xintiiSum}

\csa{xintiiSum}\marg{braced things}\etype{{\lowast f}} after expanding its
argument expects to find a sequence of tokens (or braced material). Each is
expanded (with the usual meaning), and the sum of all these numbers is returned.
Note: the summands are \emph{not} parsed by \csbxint{Num}.

\csbxint{Sum} is initially a synonym, it gets extended by \xintfracname to
fractions.

%
\leftedline{%
  \csa{xintiiSum}|{{123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}}|%
  \dtt{=\xintiiSum{{123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}}}}
%
\leftedline{\csa{xintiiSum}|{1234567890}|\dtt{=\xintiiSum{1234567890}}}
An empty sum is no error and returns zero: |\xintiiSum
{}|\dtt{=\xintiiSum {}}. A sum with only one term returns that
number: |\xintiiSum {{-1234}}|\dtt{=\xintiiSum {{-1234}}}.
Attention that |\xintiiSum {-1234}| is not legal input and will make the
\TeX{} run fail. On the other hand |\xintiiSum
{1234}|\dtt{=\xintiiSum{1234}}.

% retiré de la doc le 22 octobre 2013
% \subsection{\csbh{xintSumExpr}}\label{xintiiSumExpr}

\subsection{\csbh{xintiiPrd}}\label{xintiiPrd}

\csa{xintiiPrd}\marg{braced things}\etype{{\lowast f}} after expanding its
argument expects to find a sequence of (of braced items or unbraced
single tokens). Each is
expanded (with the usual meaning), and the product of all these numbers is
returned. Note: the operands are \emph{not} parsed by \csbxint{Num}.
%
\leftedline{\csa{xintiiPrd}|{{-9876}{\xintFac{7}}{\xintiMul{3347}{591}}}|%
  \dtt{=%
    \xintiiPrd{{-9876}{\xintFac{7}}{\xintiMul{3347}{591}}}}}
%
\leftedline{\csa{xintiiPrd}|{123456789123456789}|\dtt{=%
    \xintiiPrd{123456789123456789}}} An empty product is no error and returns 1:
|\xintiiPrd {}|\dtt{=\xintiiPrd {}}. A product reduced to a single term
returns this number: |\xintiiPrd {{-1234}}=|\dtt{\xintiiPrd {{-1234}}}.
Attention that |\xintiiPrd {-1234}| is not legal input and will make the \TeX{}
compilation fail. On the other hand |\xintiiPrd {1234}|\dtt{=\xintiiPrd
  {1234}}. %
%
\begin{everbatim*}
$2^{200}3^{100}7^{100}=\printnumber
       {\xintiiPrd {{\xintiPow {2}{200}}{\xintiPow {3}{100}}{\xintiPow {7}{100}}}}$
\end{everbatim*}

With \xintexprname, this would be easier:
%
\leftedline {|\xinttheiiexpr 2^200*3^100*7^100\relax |}

The initial synonym \csbxint{Prd} is extended by \xintfracname to fractions.

% \subsection{\csbh{xintPrdExpr}}\label{xintiiPrdExpr}


\subsection{\csbh{xintSgnFork}}\label{xintSgnFork}
%{\small New with release |1.07|. See also \csbxint{ifSgn}.\par}

\csa{xintSgnFork}\verb+{-1|0|1}+\marg{A}\marg{B}\marg{C}\etype{xnnn}
expandably chooses to execute either the \meta{A}, \meta{B} or \meta{C} code,
depending on its first argument. This first argument should be anything
expanding to either |-1|, |0| or |1| in a non self-delimiting way (i.e. a
count register must be prefixed by |\the| and a |\numexpr...\relax| also must
be prefixed by |\the|). This utility is provided to help construct expandable
macros choosing depending on a condition which one of the package macros to
use, or which values to confer to their arguments.

\subsection{\csbh{xintifSgn}, \csbh{xintiiifSgn}}\label{xintifSgn}
%{\small New with release |1.09a|.\par}

Similar to \csa{xintSgnFork}\etype{\Numf nnn} except that the first argument may
expand to a (big) integer (or a fraction if \xintfracname is loaded), and it is
its sign which decides which of the three branches is taken. Furthermore this
first argument may be a count register, with no |\the| or |\number| prefix.

\csa{xintiiifSgn} skips the \csbxint{Num} overhead.\etype{f}

\subsection{\csbh{xintifZero}, \csbh{xintiiifZero}}\label{xintifZero}
%{\small New with release |1.09a|.\par}

\csa{xintifZero}\marg{N}\marg{IsZero}\marg{IsNotZero}\etype{\Numf nn} expandably
checks if the first mandatory argument |N| (a number, possibly a fraction if
\xintfracname is loaded, or a macro expanding to one such) is zero or not. It
then either executes the first or the second branch. Beware that both branches
must be present.

\csa{xintiiifZero} skips the \csbxint{Num} overhead.\etype{f}


\subsection{\csbh{xintifNotZero}, \csbh{xintiiifNotZero}}\label{xintifNotZero}
%{\small New with release |1.09a|.\par}

\csa{xintifNotZero}\marg{N}\marg{IsNotZero}\marg{IsZero}\etype{\Numf nn}
expandably checks if the first mandatory argument |N| (a number, possibly a
fraction if \xintfracname is loaded, or a macro expanding to one such) is not
zero or is zero. It then either executes the first or the second branch. Beware
that both branches must be present.

\csa{xintiiifNotZero} skips the \csbxint{Num} overhead.\etype{f}

\subsection{\csbh{xintifOne}, \csbh{xintiiifOne}}\label{xintifOne}
%{\small New with release |1.09i|.\par}

\csa{xintifOne}\marg{N}\marg{IsOne}\marg{IsNotOne}\etype{\Numf nn} expandably
checks if the first mandatory argument |N| (a number, possibly a fraction if
\xintfracname is loaded, or a macro expanding to one such) is one or not. It
then either executes the first or the second branch. Beware that both branches
must be present.

\csa{xintiiifOne} skips the \csbxint{Num} overhead.\etype{f}

\subsection{\csbh{xintifTrueAelseB}, \csbh{xintifFalseAelseB}}
\label{xintifTrueAelseB}
\label{xintifFalseAelseB}

%\label{xintifFalseTrue}
%{\small New with release |1.09c|, renamed in |1.09e|.\par}

\csa{xintifTrueAelseB}\marg{N}\marg{true branch}\marg{false branch}\etype{\Numf
  nn} is a synonym for \csbxint{ifNotZero}.

{\small
\noindent 1. with |1.09i|, the synonyms |\xintifTrueFalse| and |\xintifTrue| are
  deprecated
  and will be removed in next release.\par
\noindent 2. These macros have no lowercase versions, use |\xintifzero|,
|\xintifnotzero|.\par }

\csa{xintifFalseAelseB}\marg{N}\marg{false branch}\marg{true branch}\etype{\Numf
  nn} is a synonym for \csbxint{ifZero}.

\subsection{\csbh{xintifCmp}, \csbh{xintiiifCmp}}\label{xintifCmp}
%{\small New with release |1.09e|.\par}

\csa{xintifCmp}\marg{A}\marg{B}\marg{if A<B}\marg{if A=B}\marg{if
  A>B}\etype{\Numf\Numf nnn} compares
its arguments and chooses accordingly the correct branch.

\csa{xintiiifCmp} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintifEq}, \csbh{xintiiifEq}}\label{xintifEq}
%{\small New with release |1.09a|.\par}

\csa{xintifEq}\marg{A}\marg{B}\marg{YES}\marg{NO}\etype{\Numf\Numf nn}
checks equality of its two first arguments (numbers, or fractions if
\xintfracname is loaded) and does the |YES| or the |NO| branch.

\csa{xintiiifEq} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintifGt}, \csbh{xintiiifGt}}\label{xintifGt}
%{\small New with release |1.09a|.\par}

\csa{xintifGt}\marg{A}\marg{B}\marg{YES}\marg{NO}\etype{\Numf\Numf nn} checks if
$A>B$ and in that case executes the |YES| branch. Extended to fractions (in
particular decimal numbers) by \xintfracname.

\csa{xintiiifGt} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintifLt}, \csbh{xintiiifLt}}\label{xintifLt}
%{\small New with release |1.09a|.\par}

\csa{xintifLt}\marg{A}\marg{B}\marg{YES}\marg{NO}\etype{\Numf\Numf nn}
checks if $A<B$ and in that case executes the |YES| branch. Extended to
fractions (in particular decimal numbers) by \xintfracname.

\csa{xintiiifLt} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintifOdd}, \csbh{xintiiifOdd}}\label{xintifOdd}
%{\small New with release |1.09e|.\par}

\csa{xintifOdd}\marg{A}\marg{YES}\marg{NO}\etype{\Numf nn} checks if $A$ is and
odd integer and in that case executes the |YES| branch.

\csa{xintiiifOdd} skips the \csbxint{Num} overhead.\etype{f}

\begin{framed}
  The macros described next are all integer-only on input. Those with |ii| in
  their names skip the \csbxint{Num} parsing. The others, with \xintfracname
  loaded, can have fractions as arguments, which will get truncated to
  integers via \csbxint{TTrunc}. On output, the macros here always produce
  integers (with no |/B[N]|).
\end{framed}

\subsection{\csbh{xintiFac}}\label{xintiFac}

|\xintiFac|\x\etype{\numx} returns the factorial. It is an error if the
argument is negative or at least $10^5$.% avant 1.09j c'était 1000000.

|\xintFac| is a variant using |\xintNum| and thus, when \xintfracname is
loaded, accepting a fraction on input (but it truncates it first).

% temps obsolètes, mettre à jour
% On my laptop $1000!$ (2568 digits)
% is computed in a little less than ten seconds, $2000!$ (5736
% digits) is computed in a little less than one hundred seconds, and
% $3000!$ (which has 9131 digits) needs close to seven minutes\dots
% I have no idea how much time $10000!$ would need (do rather
% $9999!$ if you can, the algorithm has some overhead at the
% transition from $N=9999$ to $10000$ and higher; $10000!$ has 35660
% digits). Not to mention $100000!$ which, from the Stirling formula,
% should have 456574 digits.

\subsection{\csbh{xintiiMON}, \csbh{xintiiMMON}}
\label{xintMON}\label{xintMMON}\label{xintiiMON}\label{xintiiMMON}
%{\small New in version |1.03|.\par}

|\xintiiMON|\n\etype{f} returns |(-1)^N| and |\xintiiMMON|\n{} returns
|(-1)^{N-1}|. They skip the overhead of parsing via \csbxint{Num}.
%
\leftedline{|\xintiiMON {-280914019374101929}|\dtt{=\xintiiMON
    {280914019374101929}}, |\xintiiMMON
  {-280914019374101929}|\dtt{=\xintiiMMON {280914019374101929}}} 

The variants
\csa{xintMON}\etype{\Numf} and \csa{xintMMON} use |\xintNum| and get extended
to fractions by \xintfracname.

\subsection{\csbh{xintiiOdd}}\label{xintOdd}\label{xintiiOdd}

|\xintiiOdd|\n\etype{f} is 1 if the number is odd and 0 otherwise. It skips
the overhead of parsing via \csbxint{Num}. \csa{xintOdd}\etype{\Numf} is the
variant using |\xintNum| and extended to fractions by \xintfracname.

\subsection{\csbh{xintiiEven}}\label{xintEven}\label{xintiiEven}

|\xintiiEven|\n\etype{f} is 1 if the number is even and 0 otherwise. It skips
the overhead of parsing via \csbxint{Num}. \csa{xintEven}\etype{\Numf} is the
variant using |\xintNum| and extended to fractions by \xintfracname.

\subsection{\csbh{xintiSqrt}, \csbh{xintiiSqrt}, \csbh{xintiiSqrtR}, \csbh{xintiSquareRoot},
  \csbh{xintiiSquareRoot}}\label{xintiSqrt}\label{xintiiSqrt}\label{xintiiSqrtR}
\label{xintiSquareRoot}\label{xintiiSquareRoot}
%{\small New with |1.08|.\par}

\noindent|\xintiSqrt|\n\etype{\Numf} returns the largest integer whose square
is at most equal to |N|. |\xintiiSqrt| is the variant skipping the |\xintNum|
overhead.\etype{f} |\xintiiSqrtR| also skips the |\xintNum| overhead and it
returns the rounded, not truncated, square root.\etype{f}
\begin{everbatim*}
\begin{itemize}[nosep]
\item \xintiiSqrt  {3000000000000000000000000000000000000}
\item \xintiiSqrtR {3000000000000000000000000000000000000}
\item \xintiiSqrt  {\xintiiE {3}{100}}
\end{itemize}
\end{everbatim*}

|\xintiSquareRoot|\n\etype{\Numf} returns |{M}{d}| with |d>0|, |M^2-d=N| and
|M| smallest (hence |=1+\xintiSqrt{N}|). 

|\xintiiSquareRoot|\etype{f} is the variant  skipping the |\xintNum| overhead.

\begin{everbatim*}
\xintAssign\xintiiSquareRoot {17000000000000000000000000}\to\A\B
\xintiiSub{\xintiiSqr\A}\B=\A\string^2-\B
\end{everbatim*}

A rational approximation to $\sqrt{|N|}$ is $|M|-\frac{|d|}{|2M|}$ (this is a
majorant and the error is at most |1/2M|; if |N| is a perfect square |k^2|
then |M=k+1| and this gives |k+1/(2k+2)|, not |k|).

Package \xintfracname has \csbxint{FloatSqrt} for square
roots of floating point numbers.

\begin{framed}
  The macros described next are strictly for integer-only arguments. These
  arguments are \emph{not} filtered via \csbxint{Num}. The macros are not
  usable with fractions, even with \xintfracname loaded.
\end{framed}

\subsection{\csbh{xintDSL}}\label{xintDSL}

|\xintDSL|\n\etype{f} is decimal shift left, \emph{i.e.} multiplication by
ten.

\subsection{\csbh{xintDSR}}\label{xintDSR}

|\xintDSR|\n\etype{f} is decimal shift right, \emph{i.e.} it removes the last
digit (keeping the sign), equivalently it is the closest integer to |N/10| when
starting at zero.

\subsection{\csbh{xintDSH}}\label{xintDSH}

|\xintDSH|\x\n\etype{\numx f} is parametrized decimal shift. When |x| is
negative, it is like iterating \csa{xintDSL} \verb+|x|+ times (\emph{i.e.}
multiplication by $10^{-x}$). When |x| positive, it is like iterating
\csa{DSR} |x| times (and is more efficient), and for a non-negative |N| this is
thus the same as the quotient from the euclidean division by |10^x|.

\subsection{\csbh{xintDSHr}, \csbh{xintDSx}}\label{xintDSHr}\label{xintDSx}
%{\small New in release |1.01|.\par}

|\xintDSHr|\x\n\etype{\numx f} expects |x| to be zero or positive and it
returns then a value |R| which is correlated to the value |Q| returned by
|\xintDSH|\x\n{} in the following manner:
\begin{itemize}
\item if |N| is
  positive or zero, |Q| and |R| are the quotient and remainder in
  the euclidean division by |10^x| (obtained in a more efficient
  manner than using \csa{xintiDivision}),
\item if |N| is negative let
  |Q1| and |R1| be the quotient and remainder in the euclidean
  division by |10^x| of the absolute value of |N|. If |Q1|
  does not vanish, then |Q=-Q1| and |R=R1|. If |Q1| vanishes, then
  |Q=0| and |R=-R1|.
\item for |x=0|, |Q=N| and |R=0|.
\end{itemize}
So one has |N = 10^x Q + R| if |Q| turns out to be zero or
positive, and |N = 10^x Q - R| if |Q| turns out to be negative,
which is exactly the case when |N| is at most |-10^x|.

|\xintDSx|\x\n\etype{\numx f} for |x| negative is exactly as
|\xintDSH|\x\n, \emph{i.e.} multiplication by $10^{-|x|}$. For |x| zero or
positive it returns the two numbers |{Q}{R}| described above, each one within
braces. So |Q| is |\xintDSH|\x\n, and |R| is |\xintDSHr|\x\n, but computed
simultaneously.

    \xintAssign\xintDSx {-1}{-123456789}\to\M
\leftedline{|\xintAssign\xintDSx {-1}{-123456789}\to\M|}
\leftedline{|\meaning\M: |\dtt{\meaning\M}.}
    \xintAssign\xintDSx {-20}{1234567689}\to\M
\leftedline{|\xintAssign\xintDSx {-20}{123456789}\to\M|}
\leftedline{|\meaning\M: |\dtt{\meaning\M}.}
    \xintAssign\xintDSx{0}{-123004321}\to\Q\R
\leftedline{|\xintAssign\xintDSx {0}{-123004321}\to\Q\R|}
\leftedline{|\meaning\Q: |\dtt{\meaning\Q}, |\meaning\R:|\dtt{\meaning\R.}}
\leftedline{|\xintDSH {0}{-123004321}|\dtt{=\xintDSH {0}{-123004321}},
|\xintDSHr {0}{-123004321}|\dtt{=\xintDSHr {0}{-123004321}}}
    \xintAssign\xintDSx {6}{-123004321}\to\Q\R
\leftedline{|\xintAssign\xintDSx {6}{-123004321}\to\Q\R|}
\leftedline{|\meaning\Q: |\dtt{\meaning\Q},|\meaning\R: |\dtt{\meaning\R.}}
\leftedline{|\xintDSH {6}{-123004321}|\dtt{=\xintDSH {6}{-123004321}},
|\xintDSHr {6}{-123004321}|\dtt{=\xintDSHr {6}{-123004321}}}
    \xintAssign\xintDSx {8}{-123004321}\to\Q\R
\leftedline{|\xintAssign\xintDSx {8}{-123004321}\to\Q\R|}
\leftedline{|\meaning\Q: |\dtt{\meaning\Q},|\meaning\R: |\dtt{\meaning\R.}}
\leftedline{|\xintDSH {8}{-123004321}|\dtt{=\xintDSH {8}{-123004321}},
|\xintDSHr {8}{-123004321}|\dtt{=\xintDSHr {8}{-123004321}}}
    \xintAssign\xintDSx {9}{-123004321}\to\Q\R
\leftedline{|\xintAssign\xintDSx {9}{-123004321}\to\Q\R|}
\leftedline{|\meaning\Q: |\dtt{\meaning\Q},|\meaning\R: |\dtt{\meaning\R.}}
\leftedline{|\xintDSH {9}{-123004321}|\dtt{=\xintDSH {9}{-123004321}},
|\xintDSHr {9}{-123004321}|\dtt{=\xintDSHr {9}{-123004321}}}

\subsection{\csbh{xintDecSplit}}\label{xintDecSplit}

%{\small This has been modified in release |1.01|.\par}

|\xintDecSplit|\x\n\etype{\numx f} cuts the number into two pieces (each one
within a pair of enclosing braces). First the sign if present is \emph{removed}.
Then, for |x| positive or null, the second piece contains the |x| least
significant digits (\emph{empty} if |x=0|) and the first piece the remaining
digits (\emph{empty} when |x| equals or exceeds the length of |N|). Leading
zeroes in the second piece are not removed. When |x| is negative the first piece
contains the \verb+|x|+ most significant digits and the second piece the
remaining digits (\emph{empty} if $|x|$ equals or exceeds the length of |N|).
Leading zeroes in this second piece are not removed. So the absolute value of the
original number is always the concatenation of the first and second piece.

{\footnotesize This macro's behavior for |N| non-negative is final and will not
  change. I am still hesitant about what to do with the sign of a
  negative |N|.\par}

\xintAssign\xintDecSplit {0}{-123004321}\to\L\R
\leftedline{|\xintAssign\xintDecSplit {0}{-123004321}\to\L\R|}
\leftedline{|\meaning\L: |\dtt{\meaning\L}, |\meaning\R: |\dtt{\meaning\R.}}
\xintAssign\xintDecSplit {5}{-123004321}\to\L\R
\leftedline{|\xintAssign\xintDecSplit {5}{-123004321}\to\L\R|}
\leftedline{|\meaning\L: |\dtt{\meaning\L}, |\meaning\R: |\dtt{\meaning\R.}}
\xintAssign\xintDecSplit {9}{-123004321}\to\L\R
\leftedline{|\xintAssign\xintDecSplit {9}{-123004321}\to\L\R|}
\leftedline{|\meaning\L: |\dtt{\meaning\L}, |\meaning\R: |\dtt{\meaning\R.}}
\xintAssign\xintDecSplit {10}{-123004321}\to\L\R
\leftedline{|\xintAssign\xintDecSplit {10}{-123004321}\to\L\R|}
\leftedline{|\meaning\L: |\dtt{\meaning\L}, |\meaning\R: |\dtt{\meaning\R.}}
\xintAssign\xintDecSplit {-5}{-12300004321}\to\L\R
\leftedline{|\xintAssign\xintDecSplit {-5}{-12300004321}\to\L\R|}
\leftedline{|\meaning\L: |\dtt{\meaning\L}, |\meaning\R: |\dtt{\meaning\R.}}
\xintAssign\xintDecSplit {-11}{-12300004321}\to\L\R
\leftedline{|\xintAssign\xintDecSplit {-11}{-12300004321}\to\L\R|}
\leftedline{|\meaning\L: |\dtt{\meaning\L}, |\meaning\R: |\dtt{\meaning\R.}}
\xintAssign\xintDecSplit {-15}{-12300004321}\to\L\R
\leftedline{|\xintAssign\xintDecSplit {-15}{-12300004321}\to\L\R|}
\leftedline{|\meaning\L: |\dtt{\meaning\L}, |\meaning\R: |\dtt{\meaning\R.}}

\subsection{\csbh{xintDecSplitL}}\label{xintDecSplitL}

|\xintDecSplitL|\x\n\etype{\numx f} returns the first piece after the action
of \csa{xintDecSplit}.

\subsection{\csbh{xintDecSplitR}}\label{xintDecSplitR}

|\xintDecSplitR|\x\n\etype{\numx f} returns the second piece after the action
of \csa{xintDecSplit}.

\subsection{\csbh{xintiiE}}\label{xintiiE}

|\xintiiE|\n\x\etype{f\numx } serves to add zeros to the right of |N|.
\begin{everbatim*}
\xintiiE {123}{89}
\end{everbatim*}


\section{Commands of the \xintfracname package}
\label{sec:frac}

\localtableofcontents

\def\x{|{x}|}

This package was first included in release |1.03| (|2013/04/14|) of the
\xintname bundle. The general rule of the bundle that each macro first expands
(what comes first, fully) each one of its arguments applies.

|f|\ntype{\Ff} stands for an integer or a fraction (see \autoref{sec:inputs}
for the accepted input formats) or something which expands to an integer or
fraction. It is possible to use in the numerator or the denominator of |f| count
registers and even expressions with infix arithmetic operators, under some rules
which are explained in the previous \hyperref[sec:useofcount]{Use of count
  registers} section.

As in the \hyperref[sec:xint]{xint.sty} documentation, |x|\ntype{\numx}
stands for something which will internally be embedded in a \csa{numexpr}.
It
may thus be a count register or something like |4*\count 255 + 17|, etc..., but
must expand to an integer obeying the \TeX{} bound.

The fraction format on output is the scientific notation for the `float' macros,
and the |A/B[n]| format for all other fraction macros, with the exception of
\csbxint{Trunc}, {\color{blue}\string\xint\-Round} (which produce decimal
numbers) and \csbxint{Irr}, \csbxint{Jrr}, \csbxint{RawWithZeros} (which returns
an |A/B| with no trailing |[n]|, and prints the |B| even if it is |1|), and
\csbxint{PRaw} which does not print the |[n]| if |n=0| or the |B| if |B=1|.

To be certain to print an integer output without trailing |[n]| nor fraction
slash, one should use either |\xintPRaw {\xintIrr {f}}| or |\xintNum {f}| when
it is already known that |f| evaluates to a (big) integer. For example
|\xintPRaw {\xintAdd {2/5}{3/5}}| gives a perhaps disappointing
\dtt{\xintPRaw {\xintAdd {2/5}{3/5}}}%
%
%
%
whereas |\xintPRaw {\xintIrr {\xintAdd
    {2/5}{3/5}}}| returns \dtt{\xintPRaw {\xintIrr {\xintAdd
    {2/5}{3/5}}}}. As we knew the result was an integer we could have used
|\xintNum {\xintAdd {2/5}{3/5}}=|\xintNum {\xintAdd {2/5}{3/5}}.

Some macros (such as \csbxint{iTrunc},
\csbxint{iRound}, and \csbxint{Fac}) always produce directly integers on output.

\subsection{\csbh{xintNum}}\label{xintNum}

The macro\etype{f} from \xintname is made a synonym to \csbxint{TTrunc}.
(\textcolor[named]{PineGreen}{Changed!})\footnote{In earlier releases than
  |1.1|, \csbxint{Num} did \csbxint{Irr} and then complained if the
  denominator was not |1|, else, it silently removed the denominator.}

The original (which
normalizes big integers to strict format) is still available as
\csbxint{iNum}.
It is imprudent to apply \csa{xintNum} to numbers with a large
power of ten given either in scientific notation or with the |[n]| notation,
as the macro will according to its definition add all the needed zeroes to
produce an explicit integer in strict format.

\subsection{\csbh{xintifInt}}\label{xintifInt}
%{\small New with release |1.09e|.\par}

\csa{xintifInt}|{f}{YES branch}{NO branch}|\etype{\Ff nn} expandably chooses
the |YES| branch if |f| reveals itself after expansion and simplification to
be an integer. As with the other \xintname conditionals, both branches must be
present although one of the two (or both, but why then?) may well be an empty
brace pair |{}|. Spaces in-between the braced things do not matter, but a
space after the closing brace of the |NO| branch is significant.

\subsection{\csbh{xintLen}}\label{xintLen}

The original macro\etype{\Ff} is extended to accept a fraction on input.
%
\leftedline {|\xintLen {201710/298219}|\dtt{=\xintLen {201710/298219}},
|\xintLen {1234/1}|\dtt{=\xintLen {1234/1}}, |\xintLen {1234}|%
                    \dtt{=\xintLen {1234}}}

\subsection{\csbh{xintRaw}}\label{xintRaw}
%{\small New with release |1.04|.\par}
%{\small \color{red}MODIFIED IN |1.07|.\par}

This macro `prints' the\etype{\Ff}
fraction |f| as it is received by the package after its parsing and
expansion, in a form |A/B[n]| equivalent to the internal
representation: the denominator |B| is always strictly positive and is
printed even if it has value |1|.
%
\leftedline{|\xintRaw{\the\numexpr 571*987\relax.123e-10/\the\numexpr
    -201+59\relax e-7}=|}
%
\leftedline{\dtt{\xintRaw{\the\numexpr
      571*987\relax.123e-10/\the\numexpr -201+59\relax e-7}}}

\subsection{\csbh{xintPRaw}}\label{xintPRaw}
%{\small New in |1.09b|.\par}

|PRaw|\etype{\Ff} stands for ``pretty raw''. It does \emph{not} show the |[n]|
if |n=0| and does \emph{not} show the |B| if |B=1|. 
% %
%
\leftedline{|\xintPRaw {123e10/321e10}=|\dtt{\xintPRaw {123e10/321e10}}, %
|\xintPRaw {123e9/321e10}=|\dtt{\xintPRaw {123e9/321e10}}} 
% %
%
\leftedline{|\xintPRaw {\xintIrr{861/123}}=|\dtt{\xintPRaw{\xintIrr{861/123}}}\ vz.\ 
  |\xintIrr{861/123}=|\dtt{\xintIrr{861/123}}} 
% %
See also \csbxint{Frac} (or \csbxint{FwOver}) for math mode. As is examplified
above the \csbxint{Irr} macro which puts the fraction into irreducible form
does not remove the |/1| if the fraction is an integer. One can use
|\xintNum{f}| or |\xintPRaw{\xintIrr{f}}| which produces the same output only
if |f| is an integer (after simplication).

\subsection{\csbh{xintNumerator}}\label{xintNumerator}

This returns\etype{\Ff} the numerator corresponding to the internal
representation of a fraction, with positive powers of ten converted into zeroes
of this numerator: %
%
\leftedline{|\xintNumerator
 {178000/25600000[17]}|\dtt{=\xintNumerator {178000/25600000[17]}}}
%
\leftedline{|\xintNumerator {312.289001/20198.27}|%
  \dtt{=\xintNumerator {312.289001/20198.27}}}
%
\leftedline{|\xintNumerator {178000e-3/256e5}|\dtt{=\xintNumerator
    {178000e-3/256e5}}} %
%
\leftedline{|\xintNumerator
 {178.000/25600000}|\dtt{=\xintNumerator {178.000/25600000}}} As shown by
the examples, no simplification of the input is done. For a result uniquely
associated to the value of the fraction first apply \csa{xintIrr}.

\subsection{\csbh{xintDenominator}}\label{xintDenominator}

This returns\etype{\Ff} the denominator corresponding to the internal
representation of the fraction:%
%
\footnote{recall that the |[]| construct excludes
  presence of a decimal point.} 
%
\leftedline{|\xintDenominator
 {178000/25600000[17]}|\dtt{=\xintDenominator {178000/25600000[17]}}}
%
\leftedline{|\xintDenominator {312.289001/20198.27}|%
  \dtt{=\xintDenominator {312.289001/20198.27}}}
%
\leftedline{|\xintDenominator {178000e-3/256e5}|\dtt{=\xintDenominator
    {178000e-3/256e5}}} %
%
\leftedline{|\xintDenominator
 {178.000/25600000}|\dtt{=\xintDenominator {178.000/25600000}}} As shown
by the examples, no simplification of the input is done. The denominator looks
wrong in the last example, but the numerator was tacitly multiplied by $1000$
through the removal of the decimal point. For a result uniquely associated to
the value of the fraction first apply \csa{xintIrr}.

\subsection{\csbh{xintRawWithZeros}}\label{xintRawWithZeros}
%{\small New name in |1.07| (former name |\xintRaw|).\par}

This macro `prints'\etype{\Ff} the
fraction |f| (after its parsing and expansion) in |A/B| form, with |A|
as returned by \csa{xintNumerator}|{f}| and |B| as returned by
\csa{xintDenominator}|{f}|.
%
\leftedline{|\xintRawWithZeros{\the\numexpr 571*987\relax.123e-10/\the\numexpr
    -201+59\relax e-7}=|}
%
\leftedline{\dtt{\xintRawWithZeros{\the\numexpr
      571*987\relax.123e-10/\the\numexpr -201+59\relax e-7}}}

\subsection{\csbh{xintREZ}}\label{xintREZ}

This command\etype{\Ff} normalizes a fraction by removing the powers of ten from
its numerator and denominator: %
%
\leftedline{|\xintREZ
 {178000/25600000[17]}|\dtt{=\xintREZ {178000/25600000[17]}}}
%
\leftedline{|\xintREZ {1780000000000e30/2560000000000e15}|\dtt{=\xintREZ
    {1780000000000e30/2560000000000e15}}} As shown by the example, it does not
otherwise simplify the fraction.

\subsection{\csbh{xintFrac}}\label{xintFrac}

This is a \LaTeX{} only command,\etype{\Ff} to be used in math mode only. It
will print a fraction, internally represented as something equivalent to
|A/B[n]| as |\frac {A}{B}10^n|. The power of ten is omitted when |n=0|, the
denominator is omitted when it has value one, the number being separated from
the power of ten by a |\cdot|. |$\xintFrac {178.000/25600000}$| gives $\xintFrac
{178.000/25600000}$, |$\xintFrac {178.000/1}$| gives $\xintFrac {178.000/1}$,
|$\xintFrac {3.5/5.7}$| gives $\xintFrac {3.5/5.7}$, and |$\xintFrac {\xintNum
  {\xintFac{10}/|\allowbreak|\xintiSqr{\xintFac {5}}}}$| gives $\xintFrac
{\xintNum {\xintFac{10}/\xintiSqr{\xintFac {5}}}}$. As shown by the examples,
simplification of the input (apart from removing the decimal points and moving
the minus sign to the numerator) is not done automatically and must be the
result of macros such as |\xintIrr|, |\xintREZ|, or |\xintNum| (for fractions
being in fact integers.)

\subsection{\csbh{xintSignedFrac}}\label{xintSignedFrac}

%{\small New with release |1.04|.\par}

This is as \csbxint{Frac}\etype{\Ff} except that a negative fraction has the
sign put in front, not in the numerator. %
%
\leftedline{|\[\xintFrac
 {-355/113}=\xintSignedFrac {-355/113}\]|}
\[\xintFrac {-355/113}=\xintSignedFrac {-355/113}\]

\subsection{\csbh{xintFwOver}}\label{xintFwOver}

This does the same as \csa{xintFrac}\etype{\Ff} except that the \csa{over}
primitive is used for the fraction (in case the denominator is not one; and a
pair of braces contains the |A\over B| part). |$\xintFwOver {178.000/25600000}$|
gives $\xintFwOver {178.000/25600000}$, |$\xintFwOver {178.000/1}$| gives
$\xintFwOver {178.000/1}$, |$\xintFwOver {3.5/5.7}$| gives $\xintFwOver
{3.5/5.7}$, and |$\xintFwOver {\xintNum {\xintFac{10}/\xintiSqr{\xintFac
      {5}}}}$| gives $\xintFwOver {\xintNum {\xintFac{10}/\xintiSqr{\xintFac
      {5}}}}$.

\subsection{\csbh{xintSignedFwOver}}\label{xintSignedFwOver}

%{\small New with release |1.04|.\par}

This is as \csbxint{FwOver}\etype{\Ff} except that a negative fraction has the
sign put in front, not in the numerator. %
%
\leftedline{|\[\xintFwOver
 {-355/113}=\xintSignedFwOver {-355/113}\]|}
\[\xintFwOver {-355/113}=\xintSignedFwOver {-355/113}\]

\subsection{\csbh{xintIrr}}\label{xintIrr}

This puts the fraction\etype{\Ff} into its unique irreducible form:
%
\leftedline{|\xintIrr {178.256/256.178}|%
  \dtt{=\xintIrr {178.256/256.178}}${}=\xintFrac{\xintIrr
    {178.256/256.178}[0]}$}
%
Note that the current implementation does not cleverly first factor powers of 2
and 5, so input such as |\xintIrr {2/3[100]}| will make \xintfracname do the
Euclidean division of |2|\raisebox{.5ex}{|.|}|10^{100}| by |3|, which is a bit
stupid.

Starting with release |1.08|, \csa{xintIrr} does not remove the trailing |/1|
when the output is an integer. This was deemed better for various (stupid?)
reasons and thus the output format is now \emph{always} |A/B| with |B>0|. Use
\csbxint{PRaw} on top of \csa{xintIrr} if it is needed to get rid of a possible
trailing |/1|. For display in math mode, use rather |\xintFrac{\xintIrr {f}}| or
|\xintFwOver{\xintIrr {f}}|.

\subsection{\csbh{xintJrr}}\label{xintJrr}

This also puts the fraction\etype{\Ff} into its unique irreducible form:
%
\leftedline{|\xintJrr {178.256/256.178}|%
  \dtt{=\xintJrr {178.256/256.178}}}
%
This is faster than \csa{xintIrr} for fractions having some big common
factor in the numerator and the denominator.\par
{\centering |\xintJrr {\xintiPow{\xintFac {15}}{3}/\xintiiPrdExpr
{\xintFac{10}}{\xintFac{30}}{\xintFac{5}}\relax }|\dtt{=%
 \xintJrr {\xintiPow{\xintFac {15}}{3}/\xintiiPrdExpr
{\xintFac{10}}{\xintFac{30}}{\xintFac{5}}\relax }}\par} But to notice the
difference one would need computations with much bigger numbers than in this
example.
Starting with release |1.08|, \csa{xintJrr} does not remove the trailing |/1|
when the output is an integer.

\subsection{\csbh{xintTrunc}}\label{xintTrunc}

\csa{xintTrunc}|{x}{f}|\etype{\numx\Ff} returns the integral part, a dot, and
then the first |x| digits  of the decimal
expansion of the fraction |f|. The
argument |x| should be non-negative.

In the special case when |f| evaluates to $0$, the output is $0$ with no decimal
point nor decimal digits, else the post decimal mark digits are always printed.
A non-zero negative |f| which is smaller in absolute value than |10^{-x}| will
give $-0.000...$.
%
\leftedline{|\xintTrunc
  {16}{-803.2028/20905.298}|\dtt{=\xintTrunc {16}{-803.2028/20905.298}}}
%
\leftedline{|\xintTrunc {20}{-803.2028/20905.298}|\dtt{=\xintTrunc
    {20}{-803.2028/20905.298}}}
%
\leftedline{|\xintTrunc {10}{\xintPow {-11}{-11}}|\dtt{=\xintTrunc
    {10}{\xintPow {-11}{-11}}}}
%
\leftedline{|\xintTrunc {12}{\xintPow {-11}{-11}}|\dtt{=\xintTrunc
    {12}{\xintPow {-11}{-11}}}}
%
\leftedline{|\xintTrunc {12}{\xintAdd {-1/3}{3/9}}|\dtt{=\xintTrunc
    {12}{\xintAdd {-1/3}{3/9}}}} The digits printed are exact up to and
including the last one.

% The identity |\xintTrunc {x}{-f}=-\xintTrunc {x}{f}|
% holds.%
%
% \footnote{Recall that |-\string\macro| is not valid as argument to any
%   package macro, one must use |\string\xintOpp\string{\string\macro\string}| or
%   |\string\xintiOpp\string{\string\macro\string}|, except inside
%   |\string\xinttheexpr...\string\relax|.}

\subsection{\csbh{xintiTrunc}}\label{xintiTrunc}

\csa{xintiTrunc}|{x}{f}|\etype{\numx\Ff} returns the integer equal to |10^x|
times what \csa{xintTrunc}|{x}{f}| would produce.
%
\leftedline{|\xintiTrunc
  {16}{-803.2028/20905.298}|\dtt{=\xintiTrunc {16}{-803.2028/20905.298}}}
%
\leftedline{|\xintiTrunc {10}{\xintPow {-11}{-11}}|\dtt{=\xintiTrunc
    {10}{\xintPow {-11}{-11}}}}
%
\leftedline{|\xintiTrunc {12}{\xintPow {-11}{-11}}|\dtt{=\xintiTrunc
    {12}{\xintPow {-11}{-11}}}}
%
The difference between \csa{xintTrunc}|{0}{f}| and \csa{xintiTrunc}|{0}{f}| is
that the latter never has the decimal mark always present in the former except
for |f=0|. And \csa{xintTrunc}|{0}{-0.5}| returns ``\dtt{\xintTrunc
  0{-0.5}}'' whereas \csa{xintiTrunc}|{0}{-0.5}| simply returns
``\dtt{\xintiTrunc 0{-0.5}}''.

\subsection{\csbh{xintTTrunc}}\label{xintTTrunc}

\csa{xintTTrunc}|{f}|\etype{\Ff} truncates to an integer (truncation towards
zero; \textcolor[named]{PineGreen}{New}). This is the same as |\xintiTrunc
{0}{f}| and as \csbxint{Num}.

\subsection{\csbh{xintXTrunc}}\label{xintXTrunc}

%{\small New with release |1.09j|.\par}

\csa{xintXTrunc}|{x}{f}|\retype{\numx\Ff} is completely expandable but not
\fexpan dable, as is indicated by the hollow star in the margin. It can not be
used as argument to the other package macros, but is designed to be used inside
an |\edef|, or rather a |\write|. Here is an example session where the user
after some warming up checks that $1/66049=1/257^2$ has period $257*256=65792$
(it is also checked here that this is indeed the smallest period).
%
\begingroup\small
\everb|@
xxx:_xint $ etex -jobname worksheet-66049
This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)
 restricted \write18 enabled.
**\relax
entering extended mode

*\input xintfrac.sty
(./xintfrac.sty (./xint.sty (./xinttools.sty)))
*\message{\xintTrunc {100}{1/71}}% Warming up!

0.01408450704225352112676056338028169014084507042253521126760563380281690140845
07042253521126760563380
*\message{\xintTrunc {350}{1/71}}% period is 35

0.01408450704225352112676056338028169014084507042253521126760563380281690140845
0704225352112676056338028169014084507042253521126760563380281690140845070422535
2112676056338028169014084507042253521126760563380281690140845070422535211267605
6338028169014084507042253521126760563380281690140845070422535211267605633802816
901408450704225352112676056338028169
*\edef\Z {\xintXTrunc  {65792}{1/66049}}% getting serious...

*\def\trim 0.{}\oodef\Z {\expandafter\trim\Z}% removing 0.

*\edef\W {\xintXTrunc {131584}{1/66049}}% a few seconds

*\oodef\W {\expandafter\trim\W}

*\oodef\ZZ {\expandafter\Z\Z}% doubling the period

*\ifx\W\ZZ \message{YES!}\else\message{BUG!}\fi % xint never has bugs...
YES!
*\message{\xintTrunc {260}{1/66049}}% check visually that 256 is not a period

0.00001514027464458205271843631243470756559523989765174340262532362337052794137
6856576178291874214598252812306015231116292449545034746930309315810988811337037
6538630410755651107511090251177156353616254598858423291798513225029902042423049
5541189117170585474420505
*\edef\X {\xintXTrunc {257*128}{1/66049}}% infix here ok, less than 8 tokens

*\oodef\X {\expandafter\trim\X}% we now have the first 257*128 digits

*\oodef\XX {\expandafter\X\X}% was 257*128 a period?

*\ifx\XX\Z \message{257*128 is a period}\else \message{257 * 128 not a period}\fi
257 * 128 not a period
*\immediate\write-1 {1/66049=0.\Z... (repeat)}

*\oodef\ZA {\xintNum {\Z}}% we remove the 0000, or we could use next \xintiMul

*\immediate\write-1 {10\string^65792-1=\xintiiMul {\ZA}{66049}}

*% This was slow :( I should write a multiplication, still completely

*% expandable, but not f-expandable, which could be much faster on such cases.

*\bye
No pages of output.
Transcript written on worksheet-66049.log.
xxx:_xint $ 
|
\endgroup

Using |\xintTrunc| rather than |\xintXTrunc| would be hopeless on such long
outputs (and even |\xintXTrunc| needed of the order of seconds to complete
here). But it is not worth it to use |\xintXTrunc| for less than hundreds of
digits.

Fraction arguments to |\xintXTrunc| corresponding to a |A/B[N]| with a negative
|N| are treated somewhat less efficiently (additional memory impact) than for positive or zero |N|. This is because the algorithm tries to work with the
smallest denominator hence does not extend |B| with zeroes, and technical
reasons lead to the use of some tricks.%
%
\footnote{Technical note: I do not provide an |\xintXFloat|
  because this would almost certainly mean having to clone the entire
  core division routines into a ``long division'' variant. But this
  could have given another approach to the implementation of 
  |\xintXTrunc|, especially for the case of a negative |N|. Doing these
  things with \TeX{} is an effort. Besides an |\xintXFloat|
  would be interesting only if also for example the square root routine
  was provided in an |X| version (I have not given thought to that). If
  feasible |X| routines would be interesting in the |\xintexpr|
  context where things are expanded inside |\csname..\endcsname|.}

Contrarily to \csbxint{Trunc}, in the case of the second argument revealing
itself to be exactly zero, \csbxint{XTrunc} will output $0.000...$, not $0$.
Also, the first argument must be at least $1$.

\subsection{\csbh{xintRound}}\label{xintRound}

%{\small New with release |1.04|.\par}

\csa{xintRound}|{x}{f}|\etype{\numx\Ff} returns the start of the decimal
expansion of the fraction |f|, rounded to |x| digits precision after the decimal
point. The argument |x| should be non-negative. Only when |f| evaluates exactly
to zero does \csa{xintRound} return |0| without decimal point. When |f| is not
zero, its sign is given in the output, also when the digits printed are all
zero. %
%
\leftedline{|\xintRound {16}{-803.2028/20905.298}|\dtt{=\xintRound
    {16}{-803.2028/20905.298}}}
%
\leftedline{|\xintRound {20}{-803.2028/20905.298}|\dtt{=\xintRound
    {20}{-803.2028/20905.298}}}
%
\leftedline{|\xintRound {10}{\xintPow {-11}{-11}}|\dtt{=\xintRound
    {10}{\xintPow {-11}{-11}}}}
%
\leftedline{|\xintRound {12}{\xintPow {-11}{-11}}|\dtt{=\xintRound
    {12}{\xintPow {-11}{-11}}}}
%
\leftedline{|\xintRound {12}{\xintAdd {-1/3}{3/9}}|\dtt{=\xintRound
    {12}{\xintAdd {-1/3}{3/9}}}} The identity |\xintRound {x}{-f}=-\xintRound
{x}{f}| holds. And regarding $(-11)^{-11}$ here is some more of its expansion:
%
\leftedline{\dtt{\xintTrunc {50}{\xintPow {-11}{-11}}\dots}}

\subsection{\csbh{xintiRound}}\label{xintiRound}

%{\small New with release |1.04|.\par}

\csa{xintiRound}|{x}{f}|\etype{\numx\Ff} returns the integer equal to |10^x|
times what \csa{xintRound}|{x}{f}| would return. %
%
\leftedline{|\xintiRound
 {16}{-803.2028/20905.298}|\dtt{=\xintiRound {16}{-803.2028/20905.298}}}
%
\leftedline{|\xintiRound {10}{\xintPow {-11}{-11}}|\dtt{=\xintiRound
    {10}{\xintPow {-11}{-11}}}}
%
Differences between \csa{xintRound}|{0}{f}| and \csa{xintiRound}|{0}{f}|: the
former cannot be used inside integer-only macros, and the latter removes the
decimal point, and never returns |-0| (and removes all superfluous leading
zeroes.)

\subsection{\csbh{xintFloor}, \csbh{xintiFloor}}
\label{xintFloor}\label{xintiFloor}
%{\small New with release |1.09a|.\par}

|\xintFloor {f}|\etype{\Ff} returns the largest relative integer |N| with
|N|${}\leqslant{}$|f|. %
%
\leftedline{|\xintFloor {-2.13}|\dtt{=\xintFloor
    {-2.13}}, |\xintFloor {-2}|\dtt{=\xintFloor {-2}}, |\xintFloor
  {2.13}|\dtt{=\xintFloor {2.13}}
%
}

|\xintiFloor {f}|\etype{\Ff} does the same but without adding the
|/1[0]|.\inmarg{New} 
%
\leftedline{|\xintiFloor {-2.13}|\dtt{=\xintiFloor
    {-2.13}}, |\xintiFloor {-2}|\dtt{=\xintiFloor {-2}}, |\xintiFloor
  {2.13}|\dtt{=\xintiFloor {2.13}}}

\subsection{\csbh{xintCeil}, \csbh{xintiCeil}}
\label{xintCeil}\label{xintiCeil}
%{\small New with release |1.09a|.\par}

|\xintCeil {f}|\etype{\Ff} returns the smallest relative integer |N| with
|N|${}>{}$|f|. %
%
\leftedline{|\xintCeil {-2.13}|\dtt{=\xintCeil {-2.13}},
  |\xintCeil {-2}|\dtt{=\xintCeil {-2}}, |\xintCeil
  {2.13}|\dtt{=\xintCeil {2.13}}
%
}

|\xintiCeil {f}|\etype{\Ff} does the same but without adding the
|/1[0]|.\inmarg{New}


\subsection{\csbh{xintTFrac}}\label{xintTFrac}

\csa{xintTFrac}|{f}|\etype{\Ff} returns the fractional part,
|f=trunc(f)+frac(f)|.
The |T| stands for `Trunc', and there could similar macros associated to
`Round', `Floor', and `Ceil'. Inside |\xintexpr..\relax|, the function |frac| is
mapped to \csa{xintTFrac}. Inside |\xint|\-|floatexpr..\relax|, |frac| first
applies
\csa{xintTFrac} to its argument (which may be in float format, or
an exact fraction), and only next makes the float conversion.
%
\leftedline{|\xintTFrac {1235/97}|\dtt{=\xintTFrac {1235/97}}\quad
              |\xintTFrac {-1235/97}|\dtt{=\xintTFrac {-1235/97}}}
%
\leftedline{|\xintTFrac {1235.973}|\dtt{=\xintTFrac {1235.973}}\quad
              |\xintTFrac {-1235.973}|\dtt{=\xintTFrac {-1235.973}}}
%
\leftedline{|\xintTFrac {1.122435727e5}|%
       \dtt{=\xintTFrac {1.122435727e5}}}

\subsection{\csbh{xintE}}\label{xintE}
%{\small New with |1.07|.}

|\xintE {f}{x}|\etype{\Ff\numx} multiplies the fraction |f| by $10^x$. The
\emph{second} argument |x| must obey the \TeX{} bounds. Example:
%
\leftedline{|\count 255 123456789 \xintE {10}{\count 255}|\dtt{->\count
    255 123456789 \xintE {10}{\count 255}}} Be careful that for obvious reasons
such gigantic numbers should not be given to \csbxint{Num}, or added to
something with a widely different order of magnitude, as the package always
works to get the \emph{exact} result. There is \emph{no problem} using them for
\emph{float} operations:%
%
\leftedline{|\xintFloatAdd
 {1e1234567890}{1}|\dtt{=\xintFloatAdd {1e1234567890}{1}}}

\subsection{\csbh{xintFloatE}}\label{xintFloatE}
%{\small New with |1.097|.}

|\xintFloatE [P]{f}{x}|\etype{{\upshape[\numx]}\Ff\numx} multiplies the input
|f| by $10^x$, and
converts it to float format according to the optional first argument or current
value of |\xintDigits|.
%
\leftedline{|\xintFloatE {1.23e37}{53}|\dtt{=\xintFloatE {1.23e37}{53}}}

\subsection{\csbh{xintDigits}, \csbh{xinttheDigits}}\label{xintDigits}

%{\small New with release |1.07|.\par}

The syntax |\xintDigits := D;| (where spaces do not matter) assigns the
value of |D| to the number of digits to be used by floating point
operations. The default is |16|. The maximal value is |32767|. The macro
|\xinttheDigits|\etype{} serves to print the current value.

\subsection{\csbh{xintFloat}}\label{xintFloat}

%{\small New with release |1.07|.\par}

The macro |\xintFloat [P]{f}|\etype{{\upshape[\numx]}\Ff} has an optional argument |P| which replaces
the current value of |\xintDigits|. The (rounded truncation of the) fraction
|f| is then printed in scientific form, with |P| digits,
a lowercase |e| and an exponent |N|.  The first digit is from |1| to |9|, it is
preceded by an optional minus sign and
is followed by a dot and |P-1| digits, the trailing zeroes
are not trimmed. In the exceptional case where the
rounding went to the next power of ten, the output is |10.0...0eN|
(with a sign, perhaps). The sole exception is for a zero value, which then gets
output as |0.e0| (in an \csa{xintCmp} test it is the only possible output of
\csa{xintFloat} or one of the `Float' macros which will test positive for
equality with zero).
%
\leftedline{|\xintFloat[32]{1234567/7654321}|%
               \dtt{=\xintFloat[32]{1234567/7654321}}}
% \pdfresettimer
%
\leftedline{|\xintFloat[32]{1/\xintFac{100}}|%
               \dtt{=\xintFloat[32]{1/\xintFac{100}}}}
% \the\pdfelapsedtime
% 992: plus rapide que ce que j'aurais cru..

The argument to \csa{xintFloat} may be an |\xinttheexpr|-ession, like the
other macros; only its final evaluation is submitted to \csa{xintFloat}: the
inner evaluations of chained arguments are not at all done in `floating'
mode. For this one must use |\xintthefloatexpr|.

\subsection{\csbh{xintPFloat}}\label{xintPFloat}

The macro |\xintPFloat [P]{f}|\etype{{\upshape[\numx]}\Ff} is like
\csbxint{Float} but ``pretty-prints'' the output, in the sense of dropping the
scientific notation if possible. Here are the rules:
\begin{enumerate}
\item if it is possible to drop the scientific part and express the number as
  a decimal number with the same number of digits as in the significand and a
  decimal mark, it is done so,
\item if the number is less than one and at most four zeros need be inserted
  after the decimal mark to express it without scientific part, it is done
  so,
\item if the number is zero it is printed as \dtt{\xintPFloat{0}}. All other
  cases have either a decimal mark or a scientific part or both.
\item trailing zeros are not trimmed.
\end{enumerate}
\begin{everbatim*}
\begin{itemize}[noitemsep]
\item \xintPFloat {0}
\item \xintPFloat {123}
\item \xintPFloat {0.00004567}
\item \xintPFloat {0.000004567}
\item \xintPFloat {12345678e-12}
\item \xintPFloat {12345678e-13}
\item \xintPFloat {12345678.12345678}
\item \xintPFloat {123456789.123456789}
\item \xintPFloat {123456789123456789}
\item \xintPFloat {1234567891234567}
\end{itemize}
\end{everbatim*}

\subsection{\csbh{xintAdd}}\label{xintAdd}

Computes the addition\etype{\Ff\Ff} of two fractions. To keep for integers the
integer format on output use \csbxint{iAdd}.

Checks if one denominator is a multiple of the other. Else multiplies the
denominators.

\subsection{\csbh{xintFloatAdd}}\label{xintFloatAdd}

%{\small New with release |1.07|.\par}

|\xintFloatAdd [P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Ff} first replaces |f| and
|g| with their float approximations, with 2 safety digits. It then adds exactly
and outputs in float format with precision |P| (which is optional) or
|\xintDigits| if |P| was absent, the result of this computation.

\subsection{\csbh{xintSub}}\label{xintSub}

Computes the difference\etype{\Ff\Ff} of two fractions (|\xintSub{F}{G}|
computes |F-G|). To keep for integers the integer format on output use
\csbxint{iSub}.

Checks if one denominator is a multiple of the other. Else multiplies the
denominators.

\subsection{\csbh{xintFloatSub}}\label{xintFloatSub}

%{\small New with release |1.07|.\par}

|\xintFloatSub [P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Ff} first replaces |f| and
|g| with their float approximations, with 2 safety digits. It then subtracts
exactly and outputs in float format with precision |P| (which is optional), or
|\xintDigits| if |P| was absent, the result of this computation.

\subsection{\csbh{xintMul}}\label{xintMul}

Computes the product\etype{\Ff\Ff} of two fractions. To keep for integers the
integer format on output use \csbxint{iMul}.

No reduction attempted.

\subsection{\csbh{xintSqr}}\label{xintSqr}

Computes the square\etype{\Ff} of one fraction. To maintain for integer input
an integer format on output use \csbxint{iSqr}.

\subsection{\csbh{xintFloatMul}}\label{xintFloatMul}

%{\small New with release |1.07|.\par}

|\xintFloatMul [P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Ff} first replaces |f| and
|g| with their float approximations, with 2 safety digits. It then multiplies
exactly and outputs in float format with precision |P| (which is optional), or
|\xintDigits| if |P| was absent, the result of this computation.

\subsection{\csbh{xintDiv}}\label{xintDiv}

Computes the algebraic quotient \etype{\Ff\Ff} of two fractions.
(|\xintDiv{F}{G}| computes |F/G|). To keep for integers the integer format on
output use \csbxint{iMul}.

No reduction attempted.

\subsection{\csbh{xintFloatDiv}}\label{xintFloatDiv}

%{\small New with release |1.07|.\par}

|\xintFloatDiv [P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Ff} first replaces |f| and
|g| with their float approximations, with 2 safety digits. It then divides
exactly and outputs in float format with precision |P| (which is optional), or
|\xintDigits| if |P| was absent, the result of this computation.

\subsection{\csbh{xintFac}}\label{xintFac}
%{\small Modified in |1.08b| (to allow fractions on input).\par}

The original\etype{\Numf} is extended to allow a fraction |f| which will be
truncated first to an integer |n| (non negative and at most |999999|, but
already |100000!| is prohibitively time-costly). On output |n!| (with no
trailing |/1[0]|). 

The original macro\etype{\numx} (which parses its input via |\numexpr|) is
still available as \csbxint{iFac}.

\subsection{\csbh{xintPow}}\label{xintPow}

\csa{xintPow}{|{f}{g}|}:\etype{\Ff\Numf} computes |f^g| with |f| a fraction
and |g| possibly also, but |g| will first get truncated to an integer.

The output will now always be in the form |A/B[n]| (even when the exponent
vanishes: |\xintPow {2/3}{0}|\dtt{=\xintPow{2/3}{0}}).

The original
is available as \csbxint{iPow}.

The exponent (after truncation to an integer) will be checked to not exceed
|100000|. Indeed |2^50000| already has \dtt{\xintLen {\xintFloatPow
    [1]{2}{50000}}} digits, and squaring such a number would take hours (I
think) with the expandable routine of \xintname.

\subsection{\csbh{xintFloatPow}}\label{xintFloatPow}
%{\small New with |1.07|.\par}

|\xintFloatPow [P]{f}{x}|\etype{{\upshape[\numx]}\Ff\numx} uses either the
optional argument |P| or the value of |\xintDigits|. It computes a floating
approximation to |f^x|. The precision |P| must be at least |1|, naturally.

The exponent |x| will be fed to a |\numexpr|, hence count registers are accepted
on input for this |x|. And the absolute value \verb+|x|+ must obey the \TeX{}
bound. For larger exponents use the slightly slower routine \csbxint{FloatPower}
which allows the exponent to be a fraction simplifying to an integer and does
not limit its size. This slightly slower routine is the one to which |^| is
mapped inside |\xintthefloatexpr...\relax|.

The macro |\xintFloatPow| chooses dynamically an appropriate number of
digits for the intermediate computations, large enough to achieve the desired
accuracy (hopefully).

%
\leftedline{|\xintFloatPow [8]{3.1415}{1234567890}|%
               \dtt{=\xintFloatPow [8]{3.1415}{1234567890}}}

\subsection{\csbh{xintFloatPower}}\label{xintFloatPower}
%{\small New with |1.07|.\par}

\csa{xintFloatPower}|[P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Numf} computes a
floating point value |f^g| where the exponent |g| is not constrained to be at
most the \TeX{} bound \texttt{\number "7FFFFFFF}. It may even be a fraction
|A/B| but must simplify to a (possibly big) integer.
%
\leftedline{|\xintFloatPower [8]{1.000000000001}{1e12}|%
  \dtt{=\xintFloatPower [8]{1.000000000001}{1e12}}}
%
\leftedline{|\xintFloatPower [8]{3.1415}{3e9}|%
  \dtt{=\xintFloatPower [8]{3.1415}{3e9}}} Note that |3e9>2^31|. But the
number following |e| in the output must at any rate obey the \TeX{}
\dtt{\number"7FFFFFFF} bound.

Inside an |\xintfloatexpr|-ession, \csa{xintFloatPower} is the function to which
|^| is mapped. The exponent may then be something like |(144/3/(1.3-.5)-37)|
which is, in disguise, an integer.

The intermediate multiplications are done with a higher precision than
|\xintDigits| or the optional |P| argument, in order for the
final result to hopefully have the desired accuracy.

\subsection{\csbh{xintFloatSqrt}}\label{xintFloatSqrt}
%{\small New with |1.08|.\par}

\csa{xintFloatSqrt}|[P]{f}|\etype{{\upshape[\numx]}\Ff} computes a floating
point approximation of $\sqrt{|f|}$, either using the optional precision |P| or
the value of |\xintDigits|. The computation is done for a precision of at least
17 figures (and the output is rounded if the asked-for precision was smaller).
%
\leftedline{|\xintFloatSqrt [50]{12.3456789e12}|}
%
\leftedline{${}\approx{}$\dtt{\xintFloatSqrt [50]{12.3456789e12}}}
%
\leftedline{|\xintDigits:=50;\xintFloatSqrt {\xintFloatSqrt {2}}|}
%
\leftedline{${}\approx{}$\xintDigits:=50;\dtt{\xintFloatSqrt {\xintFloatSqrt
      {2}}}}

% maple: 0.351364182864446216166582311675807703715914271812431919843183 1O^7
%         3.5136418286444621616658231167580770371591427181243e6
% maple: 1.18920711500272106671749997056047591529297209246381741301900
%        1.1892071150027210667174999705604759152929720924638e0

\xintDigits:=16;

% removed from doc october 22

% \subsection{\csbh{xintSum}, \csbh{xintSumExpr}}\label{xintSum}
% \label{xintSumExpr}

\subsection{\csbh{xintSum}}\label{xintSum}\label{xintSumExpr}

% The original commands are extended to accept fractions on input and produce
% fractions on output. Their outputs will now always be in the form |A/B[n]|. The
% originals are available as \csa{xintiiSum} and \csa{xintiiSumExpr}.

This\etype{f{$\to$}{\lowast\Ff}} computes the sum of fractions. The output
will now always be in the form |A/B[n]|. The original, for big integers only
(in strict format), is available as \csa{xintiiSum}.

\begin{everbatim*}
\xintSum {{1282/2196921}{-281710/291927}{4028/28612}}
\end{everbatim*}

No simplification attempted. 

% \subsection{\csbh{xintPrd}, \csbh{xintPrdExpr}}\label{xintPrd}\label{xintPrdExpr}

\subsection{\csbh{xintPrd}}\label{xintPrd}\label{xintPrdExpr}

TThis\etype{f{$\to$}{\lowast\Ff}} computes the product of fractions. The output
will now always be in the form |A/B[n]|. The original, for big integers only
(in strict format), is available as \csa{xintiiPrd}.

\begin{everbatim*}
\xintPrd {{1282/2196921}{-281710/291927}{4028/28612}}
\end{everbatim*}

No simplification attempted. 

\subsection{\csbh{xintCmp}}\label{xintCmp}
%{\small Rewritten in |1.08a|.\par}

This\etype{\Ff\Ff} compares two fractions |F| and |G| and produces 
|-1|, |0|, or |1| according to |F<G|, |F=G|, |F>G|.

For choosing branches according to the result of comparing |f| and |g|, the
following syntax is recommended: |\xintSgnFork{\xintCmp{f}{g}}{code for
  f<g}{code for f=g}{code for f>g}|.

% Note that since release |1.08a| using this macro on inputs with large powers of
% tens does not take a quasi-infinite time, contrarily to the earlier, somewhat
% dumb version (the earlier version indirectly led to the creation of giant chains
% of zeroes in certain circumstances, causing a serious efficiency impact).

\subsection{\csbh{xintIsOne}}

This\etype{\Ff} returns |1| if the fraction is |1| and |0| if not.

\begin{everbatim*}
\xintIsOne {21921379213/21921379213} but \xintIsOne {1.00000000000000000000000000000001}
\end{everbatim*}

\subsection{\csbh{xintGeq}}\label{xintGeq}
%{\small Rewritten in |1.08a|.\par}

This\etype{\Ff\Ff} compares the \emph{absolute values} of two
fractions.|\xintGeq{f}{g}| returns |1| if {\catcode`| 12 $|f|\geqslant|g|$} and |0|
if not.

May be used for expandably branching as:
\verb+\xintSgnFork{\xintGeq{f}{g}}{}{code for |f|<|g|}{code for
  |f|+$\geqslant$\verb+|g|}+

\subsection{\csbh{xintMax}}\label{xintMax}
%{\small Rewritten in |1.08a|.\par}

The maximum of two fractions.\etype{\Ff\Ff} But now |\xintMax {2}{3}|
returns \dtt{\xintMax {2}{3}}. The original, for use with (possibly big)
integers only with no need of normalization, is available as \csbxint{iiMax}:
|\xintiiMax {2}{3}=|\dtt{\xintiMax {2}{3}}.\etype{ff}

There is also \csbxint{iMax}\etype{\Numf\Numf} which works with fractions but
first truncates them to integers.

\begin{everbatim*}
\xintMax {2.5}{7.2} but \xintiMax {2.5}{7.2}
\end{everbatim*}

\subsection{\csbh{xintMin}}\label{xintMin}
%{\small Rewritten in |1.08a|.\par}

The maximum of two fractions.\etype{\Ff\Ff}  The original, for use with (possibly big)
integers only with no need of normalization, is available as \csbxint{iiMin}:
|\xintiiMin {2}{3}=|\dtt{\xintiMin {2}{3}}.\etype{ff}

There is also \csbxint{iMin}\etype{\Numf\Numf} which works with fractions but first
truncates them to integers.

\begin{everbatim*}
\xintMin {2.5}{7.2} but \xintiMin {2.5}{7.2}
\end{everbatim*}

\subsection{\csbh{xintMaxof}}\label{xintMaxof}

The maximum of any number of fractions, each within braces, and the whole
thing within braces. \etype{f{$\to$}{\lowast\Ff}}

\begin{everbatim*}
\xintMaxof {{1.23}{1.2299}{1.2301}} and \xintMaxof {{-1.23}{-1.2299}{-1.2301}}
\end{everbatim*}

\subsection{\csbh{xintMinof}}\label{xintMinof}

The minimum of any number of fractions, each within braces, and the whole
thing within braces. \etype{f{$\to$}{\lowast\Ff}}

\begin{everbatim*}
\xintMinof {{1.23}{1.2299}{1.2301}} and \xintMinof {{-1.23}{-1.2299}{-1.2301}}
\end{everbatim*}

\subsection{\csbh{xintAbs}}\label{xintAbs}

The absolute value\etype{\Ff}. Note that |\xintAbs {-2}|\dtt{=\xintAbs {-2}}
whereas |\xintiAbs {-2}|\dtt{=\xintiAbs {-2}}.

\subsection{\csbh{xintSgn}}\label{xintSgn}

The sign of a fraction.\etype{\Ff} 

\subsection{\csbh{xintOpp}}\label{xintOpp}

The opposite of a fraction. Note that |\xintOpp {3}| now outputs \dtt{\xintOpp
  {3}} whereas |\xintiOpp {3}| returns \dtt{\xintiOpp {3}}.

\subsection{\csbh{xintiDivision}, \csbh{xintiQuo}, \csbh{xintiRem},
  \csbh{xintFDg}, \csbh{xintLDg}, \csbh{xintMON}, \csbh{xintMMON},
  \csbh{xintOdd}}

These macros\etype{\Ff\Ff} accept a fraction (or two) on input but will
truncate it (them) to an integer using \csbxint{Num} (which is the same as
\csbxint{TTrunc}). On output they produce integers without |/| nor |[N]|.

All have variants from package \xintname whose names start with |xintii|
rather than |xint|; these variants accept on input only integers in the strict
format (they do not use \csbxint{Num}). They thus have less overhead, and may
be used when one is dealing exclusively with (big) integers.

%
\leftedline{|\xintNum {1e80}|}
%
\leftedline{\dtt{\xintNum{1e80}}}

%\etocdepthtag.toc {xintexpr}

\section{Commands of the \xintexprname package}%
\label{sec:expr}

\localtableofcontents

The \xintexprname package was first released with version |1.07|
(|2013/05/25|) of the \xintname bundle. The package loads automatically
\xintfracname and \xinttoolsname (it is now the only arithmetic package from the \xintname
bundle which loads \xinttoolsname). 
\begin{itemize}
\item for using the |gcd| and |lcm| functions, it is necessary to load package
  \xintgcdname.
\begin{everbatim*}
\xinttheexpr lcm (2^5*7*13^10*17^5,2^3*13^15*19^3,7^3*13*23^2)\relax
\end{everbatim*}
\item for allowing hexadecimal numbers (uppercase letters) on input, it is necessary
  to load package \xintbinhexname.
  \begin{everbatim*}
\xinttheexpr "A*"B*"C*"D*"D*"F, "FF.FF, reduce("FF.FFF + 16^-3)\relax
\end{everbatim*}
\end{itemize}

Release |1.1| has brought many changes to \xintexprname.
See \autoref{sec:expr11} if you are already familiar with the earlier versions.

\subsection{The \csbh{xintexpr} expressions}\label{xintexpr}%
\label{xinttheexpr}\label{xintthe}

An \xintexprname{}ession is a construct
\csbxint{expr}\meta{expandable\_expression}|\relax|\etype{x} where the expandable
expression is read and completely expanded from left to right.

During this parsing, braced sub-content may be serving as usual as a macro
parameter, or as a branch to the |?| two-way and |??| three-way operators.
Prior to release |1.1|, there were also some other usage, but this has been
removed. It was mainly needed because there was no other way to feed the
number parser directtly with fractions in the |A/B[N]| notation which is the
output format of the \xintfracname macros. There was no real need to use such
macros anyhow. If one really wants to, one can now directly:
\begin{everbatim*}
\xinttheexpr \xintAdd{3/5[2]}{7/13[2]}+199/13[1]\relax
\end{everbatim*}

Notice in passing that the expressions benefit from the improved handling of
denominators by \csbxint{Add} and \csbxint{Sub} from \xintfracname, which are
the macros to which naturally |+| and |-| are mapped.

An |\xintexpr..\relax| \emph{must} end in a |\relax| (which will be absorbed).
Like a |\numexpr| expression, it is not printable as is, nor can it be directly
employed as argument to the other package macros. For this one must use one
of the two equivalent forms:
\begin{itemize}
\item \csbxint{theexpr}\meta{expandable\_expression}|\relax|\etype{x}, or
\item \csbxint{the}|\xintexpr|\meta{expandable\_expression}|\relax|.\etype{x}
\end{itemize}

The computations are done \emph{exactly}, and with no simplification of the
result. The result can be modified via the functions |round|, |trunc|, |float|,
or |reduce|.%
%
\footnote{In |round| and |trunc| the second optional parameter is the
  number of digits of the fractional part; in |float| it is the total
  number of digits of the mantissa.}
%
Here are some examples\par
\leftedline{|\xinttheexpr 1/5!-1/7!-1/9!\relax|%
    \dtt{=\xinttheexpr 1/5!-1/7!-1/9!\relax}}
\leftedline{|\xinttheexpr round(1/5!-1/7!-1/9!,18)\relax|%
    \dtt{=\xinttheexpr round(1/5!-1/7!-1/9!,18)\relax}}
\leftedline{|\xinttheexpr float(1/5!-1/7!-1/9!,18)\relax|%
    \dtt{=\xinttheexpr float(1/5!-1/7!-1/9!,18)\relax}}
\leftedline{|\xinttheexpr reduce(1/5!-1/7!-1/9!)\relax|%
    \dtt{=\xinttheexpr reduce(1/5!-1/7!-1/9!)\relax}}
\leftedline{|\xinttheexpr 1.99^-2 - 2.01^-2 \relax|%
    \dtt{=\xinttheexpr 1.99^-2 - 2.01^-2 \relax}}
\leftedline{|\xinttheexpr round(1.99^-2 - 2.01^-2, 10)\relax|%
    \dtt{=\xinttheexpr round(1.99^-2 - 2.01^-2, 10) \relax}}\par

With |1.1| on has:
\leftedline{|\xinttheiexpr [10] 1.99^-2 - 2.01^-2\relax|%
    \dtt{=\xinttheiexpr [10] 1.99^-2 - 2.01^-2\relax}}


\begin{itemize}
\item the expression may contain arbitrarily many levels of nested parenthesized
  sub-expressions.
\item to let sub-contents evaluate as a sub-unit it should be either
   \begin{enumerate}
   \item parenthesized,
   \item or a sub-expression |\xintexpr...\relax|.
   \end{enumerate}
   When the parser scans a number and hits against either an opening
   parenthesis or a sub-expression it inserts tacitly a |*|.
 \item to either give an expression as argument to the other package macros,
   or more generally to macros which expand their arguments, one must use the
   |\xinttheexpr...\relax| or |\xintthe\xintexpr...\relax| forms. Similarly,
   printing the result itself must be done with these forms.
 \item one should not use |\xinttheexpr...\relax| as a sub-constituent of an
   |\xintexpr...\relax| but rather the |\xintexpr...\relax| form which will be
   more efficient.
 \item each \xintexprname{}ession, whether prefixed or not with |\xintthe|, is
   completely expandable and obtains its result in two expansion steps.
\end{itemize}

In an algorithm implemented non-expandably, one may define macros to
expand to infix expressions to be used within others. One then has the
choice between parentheses or |\xintexpr...\relax|: |\def\x {(\a+\b)}|
or |\def\x {\xintexpr \a+\b\relax}|. The latter is the better choice as
it allows also to be prefixed with |\xintthe|. Furthemore, if |\a| and
|\b| are already defined |\edef\x {\xintexpr \a+\b\relax}| will do the
computation on the spot. Rather than |\edef| one can use |\oodef|.

\subsection{The syntax}

An expression is enclosed between either \csbxint{expr}, or \csbxint{iexpr},
or \csbxint{iiexpr}, or \csbxint{floatexpr}, or \csbxint{boolexpr} and a
\emph{mandatory} ending |\relax|. An expression may be a sub-unit of another
one.

Apart from \csbxint{floatexpr} the evaluations of algebraic operations are
\emph{exact}. The variant \csbxint{iiexpr} does not know fractions and is
provided for integer-only calculations. The variant \csbxint{iexpr} is exactly
like \csbxint{expr} except that it either rounds the final result to an
integer, or in case of an optional parameter |[d]|, rounds to a fixed point
number with |d| digits after decimal mark. The variant \csbxint{floatexpr}
does float calculations according to the current value of the precision set by
|\xintDigits|.

The whole expression should be prefixed by |\xintthe| when it is destined to
be printed on the typeset page, or given as argument to a macro (assuming this
macro systematically expands its argument). As a shortcut to
|\xintthe\xintexpr| there is |\xinttheexpr|. One gets used to not forget the
two |t|'s.

|\xintexpr|-essions and |\xinttheexpr|-essions are completely expandable, in two steps.

\begin{itemize}
\item An expression is built the standard way with opening and closing
  parentheses, infix operators, and (big) numbers, with possibly a fractional
  part, and/or scientific notation (except for \csbxint{iiexpr} which only
  admits big integers). All variants work with comma separated expressions. On
  output each comma will be followed by a space.

\item as everything gets expanded, the characters |.|, |+|, |-|, |*|, |/|, |^|,
  |!|, |&|, \verb+|+, |?|, |:|, |<|, |>|, |=|, |(|, |)|, |"|, |]|, |[|, |@|
  and the comma |,| should not (if used in the expression) be active. For
  example, the French language in |Babel| system, for pdf\LaTeX, activates |!|,
  |?|, |;| and |:|. Turn off the activity before the expressions.

  Alternatively the command \csbxint{exprSafeCatcodes} resets all
  characters potentially needed by \csbxint{expr} to their standard catcodes
  and \csbxint{exprRestoreCatcodes} restores the status prevailing at the time
  of the previous \csa{xintexprSafeCatcodes}.

\item The infix operators are |+|, |-|, |*|, |/|, |^| (or |**|) for exact or
  floating point algebra (only integer exponents for power operations), |&&|
  and \verb+||+ \footnote{with releases earlier than |1.1|, only single
    character operators |&| and \verb+|+ were available, because the parser
    did not handle multi-character operators. Their usage in this rôle is now
    deprecated,\IMPORTANT{} as they may be assigned some new meaning in the
    future.} for combining ``true'' (non zero) and ``false'' (zero)
  conditions, as can be formed for example with the |=| (or |==|), |<|, |>|,
  |<=|, |>=|, |!=| comparison operators.

\item The |!| is either a
  function (the logical not) requiring an argument within parentheses, or a
  post-fix operator which does the factorial (so far, no float version).

\item The |?| may serve either as a function (the truth value) requiring an
  argument within parentheses), or as two-way post-fix branching operator
  |(cond)?{YES}{NO}|. The false branch will \emph{not} be evaluated. 

\item There is also |??| which branches according to the scheme
  |(x)??{<0}{=0}{>0}|.

\item Comma separated lists may be generated with |a..b| and |a..[d]..b| and
  they may be
  manipulated to some extent once put into brackets:
  \begin{itemize}
  \item |a..b| constructs the \textbf{small} integers from $\lceil a\rceil$ to
    $\lfloor b\rfloor$ (possibly a decreasing sequence),
\begin{everbatim*}
\xinttheexpr 1.5..11.23\relax
\end{everbatim*}
  
   \item |a..[d]..b| allows big integers, or fractions, it proceeds by step of |d|.
\begin{everbatim*}
\xinttheexpr 1.5..[0.97]..11.23\relax
\end{everbatim*}

  \item |[list][n]| extracts the |n|th element, or give the number of items if
    |n=0|. If |n<0| it extracts from the tail.
\begin{everbatim*}
\xinttheiexpr \empty[1..10][6], [1..10][0], [1..10][-1], [1..10][23*18-22*19]\relax\
(and 23*18-22*19 has value \the\numexpr 23*18-22*19\relax).
\end{everbatim*}

See the frame coming next for the |\empty| token.
As shown, it is perfectly legal to do operations in the index parameter, which
will be handled by the parser as everything else. The same remark applies to
the next items. 

  \item |[list][:n]| extracts the first |n| elements if |n>0|, or suppresses
    the last \verb+|n|+ elements if |n<0|.
\begin{everbatim*}
\xinttheiiexpr [1..10][:6]\relax\ and \xinttheiiexpr [1..10][:-6]\relax
\end{everbatim*}
  \item |[list][n:]| suppresses the first |n| elements if |n>0|, or extracts
    the last \verb+|n|+ elements if |n<0|.
\begin{everbatim*}
\xinttheiiexpr [1..10][6:]\relax\ and \xinttheiiexpr [1..10][-6:]\relax
\end{everbatim*}
\item More generally, |[list][a:b]| works according to the Python ``slicing''
  rules (inclusive of negative indices). Notice though that there is no
  optional third argument for the step, which always defaults to |+1|.
\begin{everbatim*}
\xinttheiiexpr [1..20][6:13]\relax\ = \xinttheiiexpr [1..20][6-20:13-20]\relax
\end{everbatim*}
\item It is naturally possible to nest these things:
\begin{everbatim*}
\xinttheexpr [[1..50][13:37]][10:-10]\relax
\end{everbatim*}
\item itemwise operations either on the left or the right are possible:
\begin{everbatim*}
\xinttheiiexpr 123*[1..10]^2\relax
\end{everbatim*}

\begin{snugframed}
  As list operations are implemented using square brackets, it is
  necessary in |\xintiexpr| and |\xintfloatexpr| to insert something before
  the first bracket if it belongs to a list, to avoid confusion with the
  bracket of an optional parameter. We need something expandable which does
  not leave a trace: the |\empty| does the trick.\IMPORTANT{}

\begin{everbatim*}
\xinttheiexpr \empty [1,3,6,99,100,200][2:4]\relax
\end{everbatim*}

   An alternative is to use parentheses
\begin{everbatim*}
\xinttheiexpr ([1,3,6,99,100,200][2:4])\relax
\end{everbatim*}

   Notice though that |([1,3,6,99,100,200])[2:4]| would not work. It is
   mandatory for |][| and |][:| not to be interspersed with parentheses. On
   the other hand spaces are perfectly legal:
\begin{everbatim*}
\xinttheiiexpr [1..10 ]   [  :  7  ]\relax
\end{everbatim*}

Similarly all the |+[|, |*[|, \dots and |]**|, |]/|, \dots operators admit
spaces but nothing else in-between their constituent characters.
\begin{everbatim*}
\xinttheiiexpr [ 1 . . 1 0 ]  * *  1 1 \relax
\end{everbatim*}

  In an other vein, the parser will be confused by |1..[list][3]|, one must
  write |1..([list][3])|. Also things such as |[100,300,500,700][2]//11| will
  be confusing because the |]/| is an operator with higher priority than the
  |][|, and then there will a dangling |/11| which does not make sense. In
  fact even |[100,300,500,700][2]/11| is a syntax error: one must write
  |([100,300,500,700][2])//11|.
\end{snugframed}

\end{itemize}

\item count registers and |\numexpr|-essions are accepted (LaTeX{}'s counters
  can be inserted using |\value|) natively without |\the| or |\number| as prefix.
  Also dimen registers and control sequences, skip registers and control
  sequences (\LaTeX{}'s lengths), |\dimexpr|-essions, |\glueexpr|-essions are
  automatically unpacked using |\number|, discarding the stretch and shrink
  components and giving the dimension value in |sp| units ($1/65536$th of a
  \TeX{} point). Furthermore, tacit multiplication is implied, when the
  register, variable, or expression is immediately prefixed by a (decimal)
  number.

\item tacit multiplication (the parser inserts a |*|) applies when the parser is
  currently scanning the digits of a number (or its decimal part or scientific
  part), or is looking
  for an infix operator, and: (1.)~\emph{encounters a register,
  variable or \eTeX{} expression (as described in the previous item)},
  (2.)~\emph{encounters a sub-\csa{xintexpr}-ession}, or
  (3.)~\emph{encounters an opening parenthesis.}

\item when defining a macro to expand to an expression either via
  %
  \leftedline{|\def\x {\xintexpr \a + \b \relax}| or |\edef\x {\xintexpr
      \a+\b\relax}|}
  %
  one may then do |\xintthe\x|, either for printing the result on the page or
  to use it in some other macros expanding their arguments. The |\edef| does
  the computation but keeps it in an internal private format. Naturally, the
  |\edef| is only possible if |\a| and |\b| are already defined, either as
  macros expanding to legal syntax like |37^23*17| or themselves in the same
  way |\x| above was defined. Indeed in both cases (the `yet-to-be computed'
  and the `already computed') |\x| can then be inserted in other expressions,
  as for example
  %
  \leftedline {|\edef\y {\xintexpr \x^3\relax}|}
  %
  This would have worked also with |\x| defined as |\def\x {(\a+\b)}| but
  |\edef\x| would not have been an option then, and |\x| could have been used
  only inside an |\xintexpr|-ession, whereas the previous |\x| can also be
  used as |\xintthe\x| in any context triggering the expansion of |\xintthe|.

\item there is also \csbxint{boolexpr}| ... \relax| and
  \csbxint{theboolexpr}| ... \relax|. Same as |\xintexpr| with the final
  result converted to $1$ if it is not zero. See also
  \csbxint{ifboolexpr} (\autoref{xintifboolexpr}) and the
  \hyperlink{item:bool}{discussion} of the |bool| and |togl| functions
  in \autoref{sec:expr}. Here is an example:
\catcode`| 12 % \xintNewBoolExpr le fait mais ça sera trop tard
                      % après le \scantokens qui aura redonné à | son
                      % caractère actif... avant j'utilisais ici \everb
                      % avec délimiteur !
\begin{everbatim*}
\xintNewBoolExpr \AssertionA[3]{ #1 && (#2|#3) }
\xintNewBoolExpr \AssertionB[3]{ #1 || (#2&#3) }
\xintNewBoolExpr \AssertionC[3]{ xor(#1,#2,#3) }
{\centering\normalcolor\xintFor #1 in {0,1} \do {%
  \xintFor #2 in {0,1} \do {%
    \xintFor #3 in {0,1} \do {%
    #1 AND (#2 OR #3) is \textcolor[named]{OrangeRed}{\AssertionA {#1}{#2}{#3}}\hfil
    #1 OR (#2 AND #3) is \textcolor[named]{OrangeRed}{\AssertionB {#1}{#2}{#3}}\hfil
    #1 XOR #2 XOR #3  is \textcolor[named]{OrangeRed}{\AssertionC {#1}{#2}{#3}}\\}}}}
\end{everbatim*}\catcode`| 13

   This example used for efficiency \csbxint{NewBoolExpr}. See also the
   \autoref{xintNewExpr}. 

\item there is  \csbxint{floatexpr}| ... \relax| where the algebra is done
  in floating point approximation (also for each intermediate result). Use the
  syntax |\xintDigits:=N;| to set the precision. Default: $16$ digits.
  %
  \leftedline{|\xintthefloatexpr 2^100000\relax:| \dtt{\xintthefloatexpr
      2^100000\relax }} 
  %
  The square-root operation can be used in |\xintexpr|, it is computed
  as a float with the precision set by |\xintDigits| or by the optional
  second argument:
  %
\begin{everbatim*}
\xinttheexpr sqrt(2,60)\relax

Here the [60] is to avoid truncation to |\xintDigits| of precision on output.
\printnumber{\xintthefloatexpr [60] sqrt(2,60)\relax}
\end{everbatim*}
  
\item 
  Floats are quickly indispensable when using the power function (which
  can only have an integer exponent), as exact results will easily have
  hundreds, if not thousands, of digits.
  %
\begin{everbatim*}
\xintDigits:=48;\xintthefloatexpr 2^100000\relax
\end{everbatim*}

\item hexadecimal \TeX{} number denotations (\emph{i.e.}, with a |"| prefix)
  are recognized by the |\xintexpr| parser and its variants. \fbox{This
    requires \xintbinhexname}. Except in |\xintiiexpr|, a (possibly empty)
  fractional part with the dot |.| as ``hexadecimal'' mark is allowed.
  %
  \leftedline{|\xinttheexpr "FEDCBA9876543210\relax|$\to$\dtt{\xinttheexpr
    "FEDCBA9876543210\relax}}
  %
  \leftedline{|\xinttheiexpr
    16^5-("F75DE.0A8B9+"8A21.F5746+16^-5)\relax|$\to$\dtt{\xinttheiexpr
      16^5-("F75DE.0A8B9+"8A21.F5746+16^-5)\relax}}
  %
  Letters must be uppercased, as with standard \TeX{} hexadecimal
  denotations.
\end{itemize}


Note that |2^-10| is perfectly accepted input, no need for parentheses;
operators of power |^|, division |/|, and subtraction |-| are all
left-associative: |2^4^8| is evaluated as |(2^4)^8|. The minus sign as prefix
has various precedence levels: |\xintexpr -3-4*-5^-7\relax| evaluates as
|(-3)-(4*(-(5^(-7))))| and |-3^-4*-5-7| as |(-((3^(-4))*(-5)))-7|.

If one uses directly macros within |\xintexpr..\relax|, rather than the
operators or the functions which are described next, one should obviously take
into account that the parser will \emph{not} see the macro arguments.

Here is, listed from the highest priority to the lowest, the complete list of
operators and functions. 

% \ctexttt is a remnant of 1.09n manual, don't have time to get rid of it now.

\newcommand\ctexttt [1]{\begingroup\color[named]{DarkOrchid}%\bfseries
                        #1\endgroup}

\begin{itemize}
\item
  Functions are at the same top level of priority. All functions even
  |?| and |!| (as prefix) require parentheses around their arguments.

    \begin{snugframed}
      \ctexttt{num, reduce, abs, sgn, frac, floor, ceil, sqr, sqrt, sqrtr, float,
        round, trunc, mod, quo, rem, gcd, lcm, max, min, `+`, `*`, ?, !, not,
        all, any, xor, if, ifsgn, even, odd, first, last, reversed, bool,
        togl, add, mul, seq, subs, rseq, rrseq, iter}

      |quo|, |rem|, |even|, |odd|, |gcd| and |lcm| will first truncate their
      arguments to integers; the latter two require package \xintgcdname;
      |togl| requires the |etoolbox| package; |all|, |any|, |xor|, |`+`|,
      |`*`|, |max| and |min| are functions with arbitrarily many comma
      separated arguments.

      |bool| and |togl| use delimited macros to fetch their argument whose
      closing parenthesis thus must be explicit, not arising from expansion.

      Similarly |add|, |mul|, |subs|, |seq|, |rseq|, |rrseq|, |iter| use at
      some stages delimited macros. They work with \emph{dummy variables},
      represented as one Latin letter (lowercase or uppercase) followed by a
      mandatory |=| sign, then a comma separated list of values to assign in
      turn to the dummy variable, which will be substituted in the expression
      which was parsed as the first, comma delimited, argument to the
      function; additionally |rseq|, |rrseq| and |iter| have a mandatory
      initial comma separated list which is separated by a semi-colon from the
      expression to evaluate iteratively. |seq|, |rseq|, |rrseq|, |iter| but
      not |add|, |mul|, |subs| admit the |omit|, |abort|, and |break(..)|
      keywords, possibly but not necessarily in combination with a potentially
      infinite list generated by a |n++| expression.

      They may be nested.
    \end{snugframed}

\begin{description}
  \item[functions with a single (numeric) argument] 
\noindent\par
\begin{description}
  \item[num] truncates to the nearest integer (truncation towards zero)
\begin{everbatim*}
\xinttheexpr num(3.1415^20)\relax
\end{everbatim*}

  \item[reduce] reduces a fraction to smallest terms
\begin{everbatim*}
\xinttheexpr reduce(50!/20!/20!/10!)\relax
\end{everbatim*}

Recall that this is NOT done automatically, for example when adding fractions.
  \item[abs] absolute value
  \item[sgn] sign
  \item[frac] fractional part
\begin{everbatim*}
\xinttheexpr frac(-355/113), frac(-1129.218921791279)\relax
\end{everbatim*}

  \item[floor] floor function
  \item[ceil]  ceil function
  \item[sqr]   square
  \item[sqrt]  in |\xintiiexpr|, truncated square root; in |\xintexpr| or
    |\xintfloatexpr| this is the floating point square root, and there is an
    optional second argument for the precision. 
  \item[sqrtr] in |\xintiiexpr| only, rounded square root.
  \item[?] |?(x)| is the truth value, $1$ if non zero, $0$ if zero. Must use parentheses.
  \item[!] |!(x)| is logical not, $0$ if non zero, $1$ if zero. Must use parentheses.
  \item[not] logical not
  \item[even] evenness of the truncation
\begin{everbatim*}
\xinttheexpr seq((x,even(x)), x=-5/2..[1/3]..+5/2)\relax
\end{everbatim*}

  \item[odd] oddness of the truncation
\begin{everbatim*}
\xinttheexpr seq((x,odd(x)), x=-5/2..[1/3]..+5/2)\relax
\end{everbatim*}
\end{description}

\item[functions with an alphabetical argument]
\noindent\par
    \hypertarget{item:bool} {\ctexttt{bool,togl}}. |bool(name)| returns
    $1$ if the \TeX{} conditional |\ifname| would act as |\iftrue| and
    $0$ otherwise. This works with conditionals defined by |\newif| (in
    \TeX{} or \LaTeX{}) or with primitive conditionals such as
    |\ifmmode|. For example:
    %
    \leftedline{|\xintifboolexpr{25*4-if(bool(mmode),100,75)}{YES}{NO}|}
    %
    will return $\xintifboolexpr{25*4-if(bool(mmode),100,75)}{YES}{NO}$
    if executed in math mode (the computation is then $100-100=0$) and
    \xintifboolexpr{25*4-if(bool(mmode),100,75)}{YES}{NO} if not (the
    \ctexttt{if} conditional is described below; the
    \csbxint{ifboolexpr} test automatically encapsulates its first
    argument in an |\xintexpr| and follows the first branch if the
    result is non-zero (see \autoref{xintifboolexpr})).

    The alternative syntax |25*4-\ifmmode100\else75\fi| could have been
    used here, the usefulness of |bool(name)| lies in the availability
    in the |\xintexpr| syntax of the logic operators of conjunction
    |&&|, inclusive disjunction \verb+||+, negation |!| (or |not|), of
    the multi-operands functions |all|, |any|, |xor|, of the two
    branching operators |if| and |ifsgn| (see also |?| and |??|), which
    allow arbitrarily complicated combinations of various |bool(name)|.

    Similarly |togl(name)| returns $1$ if the \LaTeX{} package
    \href{http://www.ctan.org/pkg/etoolbox}{etoolbox}%
    %
    %
%
\footnote{\url{http://www.ctan.org/pkg/etoolbox}}
    %
    has been used to define a toggle named |name|, and this toggle is
    currently set to |true|. Using |togl| in an |\xintexpr..\relax|
    without having loaded
    \href{http://www.ctan.org/pkg/etoolbox}{etoolbox} will result in an
    error from |\iftoggle| being a non-defined macro. If |etoolbox| is
    loaded but |togl| is used on a name not recognized by |etoolbox|
    the error message will be of the type ``ERROR: Missing |\endcsname|
    inserted.'', with further information saying that |\protect| should
    have not been encountered (this |\protect| comes from the expansion
    of the non-expandable |etoolbox| error message).

    When |bool| or |togl| is encountered by the |\xintexpr| parser, the
    argument enclosed in a parenthesis pair is expanded as usual from
    left to right, token by token, until the closing parenthesis is
    found, but everything is taken literally, no computations are
    performed. For example |togl(2+3)| will test the value of a toggle
    declared to |etoolbox| with name |2+3|, and not |5|. Spaces are
    gobbled in this process. It is impossible to use |togl| on such
    names containing spaces, but |\iftoggle{name with spaces}{1}{0}|
    will work, naturally, as its expansion will pre-empt the
    |\xintexpr| scanner.

    There isn't in |\xintexpr...| a |test| function available analogous
    to the |test{\ifsometest}| construct from the |etoolbox| package;
    but any \emph{expandable} |\ifsometest| can be inserted directly in
    an |\xintexpr|-ession as |\ifsometest10| (or |\ifsometest{1}{0}|),
    for example |if(\ifsometest{1}{0},YES,NO)| (see the |if| operator
    below) works.

    A straight |\ifsometest{YES}{NO}| would do the same more
    efficiently, the point of |\ifsometest10| is to allow arbitrary
    boolean combinations using the (described later) \verb+&+ and
    \verb+|+ logic operators:
    \verb+\ifsometest10 & \ifsomeothertest10 | \ifsomethirdtest10+,
    etc... |YES| or |NO| above stand for material compatible with the
    |\xintexpr| parser syntax.

    See  also \csbxint{ifboolexpr}, in this context.

\item[functions with one mandatory and a second but optional argument]
\noindent\par
  \begin{description}
  \item[round] For example
    |round(-2^9/3^5,12)=|\dtt{\xinttheexpr round(-2^9/3^5,12)\relax.} 
  \item[trunc] For example
    |trunc(-2^9/3^5,12)=|\dtt{\xinttheexpr trunc(-2^9/3^5,12)\relax.} 
  \item[float] For example
    |float(-20^9/3^5,12)=|\dtt{\xinttheexpr float(-20^9/3^5,12)\relax.} 
  \item [sqrt] in \csbxint{expr} and \csbxint{floatexpr}, uses the float evaluation
    with the precision given by the optional second argument.
\begin{everbatim*}
\xinttheexpr sqrt(2,31)\relax\ and \xinttheiiexpr sqrt(num(2e60))\relax
\end{everbatim*}
  \end{description}

  \item[functions with two arguments] 
\noindent\par
  \begin{description}
  \item[quo] first truncates the arguments then computes the Euclidean quotient.
  \item[rem] first truncates the arguments then computes the Euclidean remainder.
  \item[mod] computes the modulo associated to the truncated division, same as
    |/:| infix operator
\begin{everbatim*}
\xinttheexpr mod(11/7,1/13), reduce(((11/7)//(1/13))*1/13+mod(11/7,1/13)),
mod(11/7,1/13)- (11/7)/:(1/13), (11/7)//(1/13)\relax
\end{everbatim*}
  \end{description}

  \item[the if conditional (twofold way)] 
\noindent\par
\ctexttt{if}|(cond,yes,no)|
    checks if |cond| is true or false and takes the corresponding
    branch. Any non zero number or fraction is logical true. The zero
    value is logical false. Both ``branches'' are evaluated (they are
    not really branches but just numbers). See also the |?| operator.

  \item[the ifsgn conditional (threefold way)]
\noindent\par
    \ctexttt{ifsgn}|(cond,<0,=0,>0)| checks the sign of |cond| and
    proceeds correspondingly. All three are evaluated. See also the |??|
    operator.

  \item[functions with an arbitrary number of arguments]
\noindent\par
This argument may well be generated by one or many |a..b| or |a..[d]..b|
constructs, separated by commas.
  \begin{description}
\item[all] inserts a logical |AND| in between arguments and evaluates,
\item[any] inserts a logical |OR| in between all arguments and evaluates,
\item[xor] inserts a logical |XOR| in between all arguments and evaluates,
\item[|`+`|] adds (left ticks mandatory),
\item[|`*`|] multiplies (left ticks mandatory),
\item[max] maximum,
\item[min] minimum,
\item[gcd] first truncates to integers then computes the |GCD|, requires \xintgcdname,
\item[lcm] first truncates to integers then computes the |LCM|, requires \xintgcdname,
\item[first] first among comma separated items, |first(list)| is like |[list][:1]|.
\begin{everbatim*}
\xinttheiiexpr first(-7..3), [-7..3][:1]\relax
\end{everbatim*}
\item[last] last among comma separated items, |last(list)| is like |[list][-1:]|.
\begin{everbatim*}
\xinttheiiexpr last(-7..3), [-7..3][-1:]\relax
\end{everbatim*}
\item[reversed] reverses the order
\begin{everbatim*}
\xinttheiiexpr reversed(123..150)\relax
\end{everbatim*}
  \end{description}

\item[functions using dummy variables]
\noindent\par
They are nestable to arbitrary depth if suitably parenthesized.
\begin{description}
\item [subs] for variable substitution, useful to get something evaluated only
  once
\begin{everbatim*}
% ATTENTION that xz generates an error, 'unknown variable xz', one must use x*z
\xinttheexpr subs(subs(seq(x*z,x=1..10),z=y^2),y=10)\relax
\end{everbatim*}

\item[add] addition
\begin{everbatim*}
\xinttheiiexpr add(x^3,x=1..50)\relax
\end{everbatim*}

\item[mul] multiplication
\begin{everbatim*}
\xinttheiiexpr subs(mul(2n+1,n=1..N),N=30)\relax
\end{everbatim*}

\item[seq] comma separated values generated according to a formula
\begin{everbatim*}
\xinttheiiexpr seq(x(x+1)(x+2)(x+3),x=1..10)\relax
\end{everbatim*}
\begin{everbatim*}
\xinttheiiexpr seq(seq(i^2+j^2, i=0..j), j=0..10)\relax
\end{everbatim*}
\end{description}

\item[rseq] recursive sequence, |@| for the previous value.
\begin{everbatim*}
% ATTENTION y/2@ would give (y/2)@, that is (y/2)*@ !!
\printnumber {\xintthefloatexpr subs(rseq (1; @/2+y/(2@), i=1..10),y=1000)\relax }
\end{everbatim*}

In case the initial stretch is a comma separated list, |@| refers at the first
iteration to the whole list. Use parentheses at each iteration to maintain
this ``nuple''.
\begin{everbatim*}
\printnumber{\xintthefloatexpr rseq(1,10^6;
             (sqrt([@][1]*[@][2]),([@][1]+[@][2])/2), i=1..10)\relax }
\end{everbatim*}

\item[rrseq] recursive sequence with multiple initial terms. Say, there are
  |K| of them. Then |@1|, ..., |@4| and then |@@(n)| up to |n=K| refer to the
  last |K| values. Notice the difference with |rseq| for which |@| refers to
  the complete list of all initial terms (if there are more than one).
\begin{everbatim*}
\xinttheiiexpr rrseq(0,1; @1+@2, i=2..30)\relax
\end{everbatim*}

\begin{everbatim*}
\xinttheiiexpr rrseq(0,1,2,3,4,5; @1+@2+@3+@4+@@(5)+@@(6), i=1..20)\relax
\end{everbatim*}

I implemented an |Rseq| which at all times keeps the memory of \emph{all}
previous items, but decided to drop it as the package was becoming big.

\item[iter] same as |rrseq| but does not print any value until the last |K|.
\begin{everbatim*}
\xinttheiiexpr iter(0,1; @1+@2, i=2..5, 6..10)\relax 
% the iterated over list is allowed to have disjoint defining parts.
\end{everbatim*}
\end{description}

Recursions may be nested, with |@@@(n)| giving access to the values of the
outer recursion\dots and there is even |@@@@(n)| but I never tried it!

With |seq|, |rseq|,
|rrseq|, |iter|, but not with |subs|, |add|, |mul|, one has:
\begin{description}
\item[abort] stop here and now.
\item[omit] omit this value.
\item[break] |break(stuff)| to abort and have |stuff| as last value.
\item[n++] serves to generate a potentially infinite list
\begin{everbatim*}
\xinttheiiexpr iter(1;(@>10^40)?{break(@)}{2@},i=1++)\relax 
% this would not work with i=1,2,3++. Only n++ syntax, nothing before.
\end{everbatim*}
\end{description}

Refer to \autoref{sec:expr11} for more examples.

\item The postfix operators \ctexttt{!} and the branching conditionals \ctexttt{?, ??}.
  \begin{description}
  \item[{\color[named]{DarkOrchid}!}] computes the factorial of an integer.
    This is the exact factorial even when used inside |\xintfloatexpr|.

  \item[{\color[named]{DarkOrchid}?}] is used as |(cond)?{yes}{no}|. It
    evaluates the (numerical) condition (any non-zero value counts as
    |true|, zero counts as |false|). It then acts as a macro with two
    mandatory arguments within braces (hence this escapes from the
    parser scope, the braces can not be hidden in a macro), chooses the
    correct branch \emph{without evaluating the wrong one}. Once the
    braces are removed, the parser scans and expands the uncovered
    material so for example
    %
    \leftedline{|\xinttheiexpr (3>2)?{5+6}{7-1}2^3\relax|} 
    %
    is legal and computes
    |5+62^3=|\dtt{\xinttheiexpr(3>2)?{5+(6}{7-(1}2^3)\relax}. Note
    though that it would be better practice to include here the |2^3|
    inside the branches. The contents of the branches may be arbitrary
    as long as once glued to what is next the syntax is respected:
    {|\xintexpr (3>2)?{5+(6}{7-(1}2^3)\relax| also works.} Differs thus
    from the |if| conditional in two ways: the false branch is not at
    all computed, and the number scanner is still active on exit, more
    digits may follow.

  \item[{\color[named]{DarkOrchid}??}] is used as |(cond)??{<0}{=0}{>0}|.
    |cond| is anything, its sign is evaluated and depending on the sign the
    correct branch is un-braced, the two others are swallowed. The un-braced
    branch will then be parsed as usual. Differs from the |ifsgn| conditional
    as the two false branches are not evaluated and furthermore the number
    scanner is still active on exit.
    %
    \leftedline{|\def\x{0.33}\def\y{1/3}|}
    %
    \leftedline{|\xinttheexpr (\x-\y)??{sqrt}{0}{1/}(\y-\x)\relax|%
      \dtt{=\def\x{0.33}\def\y{1/3}%
      \xinttheexpr (\x-\y)??{sqrt}{0}{1/}(\y-\x)\relax }}
    %
  \end{description}

\item \def\MicroFont{\color[named]{DarkOrchid}\ttfamily}The
  |.| as decimal mark; the number scanner treats it as an inherent,
  optional and unique component of a being formed number. One can do
  things such as
  %
  \leftedline{\restoreMicroFont|\xinttheexpr .^2+2^.\relax|}
  %
  which is |0^2+2^0| and produces \dtt{\xinttheexpr .^2+2^.\relax}.

\item The |e| and |E| for scientific notation.\inmarg{Changed} They are parsed
  like the decimal mark is. Thus |1e(3+2)| is no legal syntax anymore, one
  must use |10^(3+2)| in such cases.
\begingroup
\restoreMicroFont |1e3^2| is \dtt{\xinttheexpr 1e3^2\relax}
\endgroup

\item The |"| for hexadecimal numbers: it is treated with highest
  priority, allowed only at locations where the parser expects to start
  forming a numeric operand, once encountered it triggers the
  hexadecimal scanner which looks for successive hexadecimal digits (as
  usual skipping spaces and expanding forward everything) possibly a
  unique optional dot (allowed directly in front) and then an optional
  (possibly empty) fractional part. The dot and fractional part are not
  allowed in {\restoreMicroFont|\xintiiexpr..\relax|}. The |"|
  functionality \fbox{requires package \xintbinhexname} (there is
  no warning, but an ``undefined control sequence'' error will
  naturally results if the package has not been loaded).
\begingroup
  \restoreMicroFont |"A*"A^"A| is \dtt{\xinttheexpr "A*"A^"A\relax}.
\endgroup


\item The power operator |^|, or |**|. It is left associative:
  {\restoreMicroFont|\xinttheiexpr 2^2^3\relax|} evaluates to \xinttheiexpr
    2^2^3\relax, not \xinttheiexpr 2^(2^3)\relax. Note that if the float
    precision is too low, iterated powers within |\xintfloatexpr..\relax| may
    fail: for example with the default setting |(1+1e-8)^(12^16)| will be
    computed with |12^16| approximated from its $16$ most significant digits
    but it has $18$ digits (\dtt{={\xintiiPow{12}{16}}}), hence the result is
    wrong:
  % REVOIR CECI
  \begingroup
  %
  \leftedline{$\np{\xintthefloatexpr (1+1e-8)^(12^16)\relax }$}
  %
  One should code
  %
  \leftedline{\restoreMicroFont|\xintthe\xintfloatexpr (1+1e-8)^\xintiiexpr 12^16\relax
    \relax|}
  %
  to obtain the correct floating point evaluation
  % REVOIR CECI
  % 
  \leftedline{$\np{1.00000001}^{12^{16}}\approx\np{\xintthefloatexpr
      (1+1e-8)^\xintiiexpr 12^16\relax\relax }$}
  %
  \endgroup

\item Multiplication and division \raisebox{-.3\height}{|*|}, |/|. The
  division is left associative, too: 
  %
  \begingroup\restoreMicroFont
  %
  |\xinttheiexpr 100/50/2\relax| evaluates to \xinttheiexpr 100/50/2\relax,
  not \xinttheiexpr 100/(50/2)\relax.
  %
  \endgroup
  Inside \csbxint{iiexpr}, |/| does \emph{rounded} division.\inmarg{Changed}

\item Truncated division |//| and modulo |/:| (equivalently |'mod'|, quotes
  mandatory) are at the same level of priority than multiplication and
  division, thus left-associative with them. Apply parentheses for
  disambiguation.
\begin{everbatim*}
\xinttheexpr 100000//13, 100000/:13, 100000 'mod' 13, trunc(100000/13,10),
            trunc(100000/:13/13,10)\relax
\end{everbatim*}

\item The list itemwise operators |*[|, |/[|, |^[|, |**[|, |]*|, |]/|, |]^|,
  |]**| are at the same precedence level as, respectively, |*| and |/| or |^|
  and |**|.

\item Addition and subtraction |+|, |-|. Again, |-| is left
  associative: 
  %
  \begingroup\restoreMicroFont 
  %
  |\xinttheiexpr 100-50-2\relax| evaluates to \xinttheiexpr 100-50-2\relax,
  not \xinttheiexpr 100-(50-2)\relax.
  %
  \endgroup

\item The list itemwise operators |+[|, |-[|, |]+|, |]-|,  are at
  the same precedence level as |+| and |-|, 

\item Comparison operators |<|, |>|, |=| (same as |==|), |<=|, |>=|, |!=| all
  at the same level of precedence, use parentheses for disambiguation.

\item Conjunction (logical and): |&&| or equivalently
  |'and'| (quotes mandatory).

\item Inclusive disjunction (logical or): \verb+||+
  and equivalently |'or'| (quotes mandatory).

\item XOR: |'xor'| with mandatory quotes is at the same level of precedence
  as \verb+||+.

\item The comma:
\restoreMicroFont With |\xinttheexpr 2^3,3^4,5^6\relax| 
one obtains as output \xinttheexpr 2^3,3^4,5^6\relax{}.

\item The parentheses.
\end{itemize}

% See \autoref{ssec:countinexpr} for count and dimen registers and variables.

\subsection{\texorpdfstring{\texttt{\protect\string\numexpr}}{\textbackslash
    numexpr} or \texorpdfstring{\texttt{\protect\string\dimexpr}}{\textbackslash
    dimexpr} expressions, count and dimension registers and variables}
\label{ssec:countinexpr}

Count registers, count control sequences, dimen registers,
dimen control sequences, skips and skip control sequences, |\numexpr|,
|\dimexpr|, |\glueexpr| can be inserted directly, they will be unpacked using
|\number| (which gives the internal value in terms of scaled points for the
dimensional variables: $1$\,|pt|${}=65536$\,|sp|; stretch and shrink
components are thus discarded). Tacit multiplication is implied, when a
number or decimal number prefixes such a register or control sequence.

\LaTeX{} lengths are skip control sequences and \LaTeX{} counters should be
inserted using |\value|.

In the case of numbered registers like |\count255| or |\dimen0|, the resulting
digits will be re-parsed, so for example |\count255 0| is like |100| if
|\the\count255| would give |10|. Control sequences define complete numbers, thus
cannot be extended that way with more digits, on the other hand they are more
efficient as they avoid the re-parsing of their unpacked contents.

A token list variable must be prefixed by |\the|, it will not be unpacked
automatically (the parser will actually try |\number|, and thus fail). Do not
use |\the| but only |\number| with a dimen or skip, as the |\xintexpr| parser
doesn't understand |pt| and its presence is a syntax error. To use a dimension
expressed in terms of points or other \TeX{} recognized units, incorporate it in
|\dimexpr...\relax|.

% REVOIR
% If one needs to optimize, |1.72\dimexpr 3.2pt\relax| is less efficient
% than |1.72*{\number\dimexpr 3.2pt\relax}| as the latter avoids re-parsing the
% digits of the representation of the dimension as scaled points.
% \centeredline{|\xinttheexpr 1.72\dimexpr 3.2pt\relax/2.71828\relax=|}
% \centeredline{|\xinttheexpr 1.72*{\number\dimexpr 3.2pt\relax}/2.71828\relax|}
% \centeredline{\dtt{\xinttheexpr 1.72\dimexpr
%     3.2pt\relax/2.71828\relax=\xinttheexpr 1.72*{\number\dimexpr
%       3.2pt\relax}/2.71828\relax}}
Regarding how dimensional expressions are converted by \TeX{} into scaled points
see \autoref{sec:Dimensions}.

\subsection{Catcodes and spaces}

Active characters may (and will) break the functioning of \csbxint{expr}.
Inside an expression one may prefix, for example a |:| with |\string|. Or, for
a more radical way, there is \csbxint{exprSafeCatcodes}. This is a
non-expandable step as it changes catcodes.

\subsubsection{\csbh{xintexprSafeCatcodes}}
\label{xintexprSafeCatcodes}
%{\small New with release |1.09a|.\par}

This command sets the catcodes of
the relevant characters to safe values. This is used internally by
\csbxint{NewExpr} (restoring the catcodes on exit), hence \csbxint{NewExpr} does
not have to be protected against active characters.

\subsubsection{\csbh{xintexprRestoreCatcodes}}\label{xintexprRestoreCatcodes}
%{\small New with release |1.09a|.\par}

Restores the catcodes to the earlier state. 

\bigskip

Spaces inside an |\xinttheexpr...\relax| should mostly be
innocuous (except inside macro arguments).

|\xintexpr| and |\xinttheexpr| are for the most part agnostic regarding
catcodes: (unbraced) digits, binary operators, minus and plus signs as
prefixes, dot as decimal mark, parentheses, may be indifferently of catcode
letter or other or subscript or superscript, ..., it doesn't matter.%
%
\footnote{Furthermore, although \csbxint{expr} uses \csa{string}, it is
  (we hope) escape-char agnostic.}

The characters |+|, |-|, |*|, |/|, |^|, |!|, |&|, \verb+|+, |?|, |:|, |<|, |>|,
|=|, |(|, |)|, |"|, |[|, |]|, |;|, the dot and the comma should not be active if
in the expression, as everything is expanded along the way. If one of them is
active, it should be prefixed with |\string|.

The |!| as either logical negation or postfix factorial operator must be a
standard (\emph{i.e.} catcode $12$) |!|, more precisely a catcode $11$
exclamation point |!| must be avoided as it is used internally by |\xintexpr|
for various special purposes.

Digits, slash, square brackets, minus sign, in the output from an |\xinttheexpr|
are all of catcode 12. For |\xintthefloatexpr| the `e' in the output has its standard catcode ``letter''.

A macro with arguments will expand and grab its arguments before the
parser may get a chance to see them, so the situation with catcodes and spaces
is not the same within such macro arguments.

\subsection{Expandability, \csh{xinteval}}

As is the case with all other package macros |\xintexpr| \fexpan ds (in two
steps) to its final (non-printable) result; and |\xinttheexpr| \fexpan ds (in
two steps) to the chain of digits (and possibly minus sign |-|, decimal mark
|.|, fraction slash |/|, scientific |e|, square brackets |[|, |]|) representing
the result.

Starting with |1.09j|, an |\xintexpr..\relax| can be inserted without
|\xintthe| prefix inside an |\edef|, or a |\write|. It expands to a private
more compact representation (five tokens) than |\xinttheexpr| or
|\xintthe\xintexpr|.

The material between |\xintexpr| and |\relax| should contain only expandable
material.

The once expanded |\xintexpr| is |\romannumeral0\xinteval|. And there is
similarly |\xintieval|, |\xintiieval|, and |\xintfloateval|. For the other
cases one can use |\romannumeral-`0| as prefix. For an example of expandable
algorithms making use of chains of |\xinteval|-uations connected via
|\expandafter| see \autoref{ssec:fibonacci}.

An expression can only be legally finished by a |\relax| token, which
will be absorbed.

It is quite possible to nest expressions among themselves; for example, if one
needs inside an |\xintiiexpr...\relax| to do some computations with fractions,
rounding the final result to an integer, on just has to insert
|\xintiexpr...\relax|. The functioning of the infix operators will not be in
the least affected from the fact that the surrounding ``environment'' is the
|\xintiiexpr| one.

\subsection{Memory considerations}

The parser creates an undefined control sequence for each intermediate
computation evaluation: addition, subtraction, etc\dots Thus, a moderately sized
expression might create 10, or 20 such control sequences. On my \TeX{}
installation, the memory available for such things is of circa \np{200000}
multi-letter control words. So this means that a document containing hundreds,
perhaps even thousands of expressions will compile with no problem.

Besides the hash table, also \TeX{} main memory is impacted. Thus, if
\xintexprname is used for computing plots%
%
\footnote{this is not very probable as so far \xintname does not include
  a mathematical library with floating point calculations, but provides
  only the basic operations of algebra.}%
%
, this may cause a problem.

There is a (partial) solution.%
%
\footnote{which convinced me that I could stick with the parser
  implementation despite its potential impact on the hash-table and
  other parts of \TeX{}'s memory.}

A
document can possibly do tens of thousands of evaluations only
if some formulas are being used repeatedly, for example inside loops, with
counters being incremented, or with data being fetched from a file. So it is the
same formula used again and again with varying numbers inside.

With the \csbxint{NewExpr} command, it is possible to convert once and
for all an expression containing parameters into an expandable macro
with parameters. Only this initial definition of this macro actually
activates the \csbxint{expr} parser and will (very moderately) impact
the hash-table: once this unique parsing is done, a macro with
parameters is produced which is built-up recursively from the
\csbxint{Add}, \csbxint{Mul}, etc... macros, exactly as it would be
necessary to do without the facilities of the \xintexprname package.

\subsection{The \csbh{xintNewExpr} command}\label{xintNewExpr}

% This allows to define a completely expandable macro with parameters, expanding
% in two steps to its final evaluation, and corresponding to the given
% \xintname{}expression where the parameters are input using the usual
% macro-parameter: |#1|, ..., |#9|.

The command is used as:
%
\leftedline{|\xintNewExpr{\myformula}[n]|\marg{stuff}, where}
\begin{itemize}
\item \meta{stuff} will be inserted inside |\xinttheexpr . . . \relax|,
\item |n| is an integer between zero and nine, inclusive, and tells how many
  parameters will |\myformula| have (it is \emph{mandatory} even if
  |n=0|%
%
\footnote{there is some use for \csa{xintNewExpr}|[0]| compared to an
    \csa{edef} as \csa{xintNewExpr} has some built-in catcode protection.})
%
\item the placeholders |#1|, |#2|, ..., |#n| are used inside \meta{stuff} in
  their usual r\^ole.\footnote{such \#'s may be of category code other or have
    their usual catcode.}
\end{itemize}

The macro |\myformula| is defined without checking if it
already exists, \LaTeX{} users might prefer to do first |\newcommand*\myformula
{}| to get a reasonable error message in case |\myformula| already exists.

The definition of |\myformula| made by |\xintNewExpr| is global (i.e. it does
not obey the scope of environments). The protection against active characters
is done automatically.

It will be a completely expandable macro entirely built-up using |\xintAdd|,
|\xintSub|, |\xintMul|, |\xintDiv|, |\xintPow|, etc\dots as corresponds to the
expression written with the infix operators.
Macros created by |\xintNewExpr| can thus be nested.

\begin{everbatim*}
    \xintNewFloatExpr \FA [2]{(#1+#2)^10}
    \xintNewFloatExpr \FB [2]{sqrt(#1*#2)}
\begin{enumerate}[nosep]
    \item \FA {5}{5}
    \item \FB {30}{10}
    \item \FA {\FB {30}{10}}{\FB {40}{20}}
\end{enumerate}
\end{everbatim*}

\begin{framed}
  The whole point of using \csbxint{NewExpr} is to produce a macro whose
  execution will not populate the hash table contrarily to what |\xintexpr|
  does to some extent (this impact on some of \TeX{}'s memory gets noticeable
  only if many of thousands of \csbxint{expr}-essions are evaluated).

  A ``formula'' created by |\xintNewExpr| is thus a macro whose parameters are
  given to a possibly very complicated combination of the various macros of
  \xintname and \xintfracname. Consequently, one can not use at all any infix
  notation there, but only the input format recognized by the \xintfracname
  macros. 

The situation is thus quite distinct from a macro with parameters defined via
a simple |\def| such as:
  %
  \leftedline{|\def\myformula #1{\xinttheexpr (#1)^3\relax}|}
\end{framed}

|\xintNewExpr| tries to do as many evaluations as are possible at the time the
macro parameters are still parameters. Let's see a few examples. For this I
will use |\meaning| which reveals the contents of a macro. We will thus see
some private macros of the \xintname bundle, which should not be directly
used. If the things look a bit complicated, it is because they have to cater
for many possibilities. With |\meaning| we will see what has already been
evaluated.

\emph{Remark:} In these examples we sometimes use |\printnumber| to avoid for
the meaning to go into the right margin, but this zaps all spaces originally
in the output from |\meaning|.

\begin{everbatim*}
\xintNewIIExpr\FA [1]{13*25*78*#1+2826*292}\meaning\FA
\end{everbatim*}
\smallskip

\begin{everbatim*}
\xintNewIExpr\FA [2]{(3/5*9/7*13/11*#1-#2)*3^7}
\printnumber{\meaning\FA}
\end{everbatim*}

\smallskip

\begin{everbatim*}
% an example with optional parameter
\xintNewIExpr\FA [3]{[24] (#1+#2)/(#1-#2)^#3}
\printnumber{\meaning\FA}
\end{everbatim*}

\smallskip

\begin{everbatim*}
\xintNewFloatExpr\FA [2]{[12] 3.1415^3*#1-#2^5}
\printnumber{\meaning\FA}
\end{everbatim*}

\smallskip

\begin{everbatim*}
\xintNewExpr\DET[9]{ #1*#5*#9+#2*#6*#7+#3*#4*#8-#1*#6*#8-#2*#4*#9-#3*#5*#7 }
\printnumber{\meaning\DET}
\end{everbatim*}

\smallskip

\begin{everbatim*}
\xintNewExpr\FA[3]{ #1*#1+#2*#2+#3*#3-(#1*#2+#2*#3+#3*#1) }
\printnumber{\meaning\FA }
\end{everbatim*}


One can even do some quite daring things:
\begin{everbatim*}
\xintNewExpr\FA[5]{[#1..[#2]..#3][#4:#5]}
And this works:
\begin{itemize}[nosep]
\item \FA{1}{3}{90}{20}{30}
\item \FA{1}{3}{90}{-40}{-15}
\item \FA{1.234}{-0.123}{-10}{3}{7}
\end{itemize}
\oodef\test {\FA {0}{10}{100}{3}{6}}\meaning\test +++
\end{everbatim*}

In the last example though, do not hope to use empty |#4| or |#5|: this is
possible in an expression, because the parser identifies |][:| or |:]| and
handles them appropriately. Here the macro |\FA| is built with idea that there
is something non-empty as it found the place holders |#4| and |#5|.


\subsubsection {Conditional operators and \csbh{NewExpr}}

The |?| and |??| conditional operators cannot be parsed by |\xintNewExpr| when
they contain macro parameters |#1|,\dots, |#9| within their scope. However
replacing them with the functions |if| and, respectively |ifsgn|, the parsing
should succeed. And the created macro will \emph{not evaluate the branches to
  be skipped}, thus behaving exactly like |?| and |??| would have in the
|\xintexpr|.

\begin{everbatim*}
\xintNewExpr\Formula [3]{ if((#1>#2) && (#2>#3), sqrt(#1-#2)*sqrt(#2-#3),  #1^2+#3/#2) }%
\printnumber{\meaning\Formula }
\end{everbatim*}

This formula (with its |\xintiiifNotZero|) will gobble the false branch without
evaluating it when used with given arguments.

Remark: the meaning above reveals some of the private macros used by the
package. They are not for direct use.

Another example

\begin{everbatim*}
\xintNewExpr\myformula[3]{ ifsgn(#1,#2/#3,#2-#3,#2*#3) }%
\meaning\myformula
\end{everbatim*}

Again,  this macro gobbles the false branches, as would have the operator |??|
inside an |\xintexpr|-ession.

\subsubsection{External macros and \csbh{NewExpr}; the protect function}
\label{sssec:protect}

For macros within such a created \xintname-formula command, there
are two cases:\inmarg{Changed}
\begin{itemize}
\item the macro does not involve the numbered parameters in its arguments: it
  may then be left as is, and will be evaluated once during the construction of
  the formula,
\item it does involve at least one of the macro parameters as argument. Then:
  \begin{snugframed}
    the whole thing (macro + argument) should be |protect|-ed, not in the
    \LaTeX{} sense (!), but in the following way: |protect(\macro {#1})|.\IMPORTANT
  \end{snugframed}
\end{itemize}

Here is a silly example illustrating the general principle: the macros here have
equivalent functional forms which are more convenient; but some of the more
obscure package macros of \xintname dealing with integers do not have functions
pre-defined to be in correspondance with them, use this mechanism could be
applied to them.

\begin{everbatim*}
\xintNewExpr\myformI[2]{protect(\xintRound{#1}{#2}) - protect(\xintTrunc{#1}{#2})}%
\meaning\myformI

\xintNewIIExpr\formula [3]{rem(#1,quo(protect(\the\numexpr #2\relax),#3))}%
\noindent\meaning\formula
\end{everbatim*}

Only macros involving the |#1|, |#2|, etc\dots should be protected in this
way; the |+|, |*|, etc\dots symbols, the functions from the \csbxint{expr}
syntax, none should ever be included in a protected string.


\subsubsection{Limitations of \csbxint{NewExpr}}

All depends on where the macro parameters arise |#1|, |#2|, ... we went to some
effort to allow many things but not everything goes through. |\xintNewExpr|
tries to evaluate completely as many things as possible which do not involve the
macro parameters. A somewhat elaborate scheme allows to handle also
complicated situations with list operations:

\begin{everbatim*}
\xintNewIExpr \FA [3] {[3] `+`([1.5..[3.5+#1]..#2]*#3)}
\begin{itemize}[nosep]
\item \FA {3.5}{50}{100} (cf. \xinttheiexpr [3] 1.5..[7]..50\relax)
\item \FA {-15}{-100}{20} (cf. \xinttheiexpr [3] 1.5..[-11.5]..-100\relax)
\item \FA {0}{20}{1} (cf. \xinttheiexpr [3] 1.5..[3.5]..20\relax)
\end{itemize}
\end{everbatim*}

Some things are definitely expected not to work therein: particularly the
|seq|, |rseq|, |rrseq|, |iter| with |omit|, |abort|,
|break|. Also, but this is quite anecdotical, |first| and |last| should not
work (I did not try; actually I did not try the functions with dummy letters
either, because each time I think about compatibility with \csbxint{NewExpr},
my head starts spinning.)

Also, using sub-|\xintexpr|-essions (including some of the macro parameters)
inside something given to |\xintNewExpr| will probably not work.

Naturally, it is always possible to use, after the macro has been constructed,
|\xinttheexpr...\relax| among the arguments.

\subsection{\csbh{xintiexpr}, \csbh{xinttheiexpr}}
\label{xintiexpr}\label{xinttheiexpr}
% \label{xintnumexpr}\label{xintthenumexpr}

Equivalent\etype{x} to doing |\xintexpr round(...)\relax|. Thus, \emph{only} the
final result is rounded to an integer. Half integers are rounded towards
$+\infty$ for positive numbers and towards $-\infty$ for negative ones. Comma
separated lists of expressions are allowed. 

An optional parameter within brackets is allowed: if strictly positive it
instructs the expression to do its final rounding to the nearest value with
that many digits after the decimal mark.\inmarg{New}

% Le temps est venu pour leur obsolescence

% Initially baptized |\xintnumexpr|,
% |\xintthenumexpr| but
% I am not too happy about this choice of name; one should keep in mind that
% |\numexpr|'s integer division rounds, whereas in |\xintiexpr|, the |/| is an
% exact fractional operation, and only the final result is rounded to an integer.

% So |\xintnumexpr|, |\xintthenumexpr| are deprecated, and although still provided
% for the time being this might change in the future.

\subsection{\csbh{xintiiexpr}, \csbh{xinttheiiexpr}}
\label{xintiiexpr}\label{xinttheiiexpr}

This variant\etype{x} does not know fractions. It deals almost only with long
integers. Comma separated lists of expressions are allowed.

\begin{framed}
  It maps |/| to the \emph{rounded} quotient.\inmarg{Changed} The operator
  |//| is, like in |\xintexpr...\relax|, mapped to \emph{truncated} division.
  The euclidean quotient (which for positive operands is like the truncated
  quotient) was, prior to release |1.1|, associated to |/|. The function
  |quo(a,b)| can still be employed.
\end{framed}

The \csbxint{iiexpr}-essions use the `ii' macros for addition, subtraction,
multiplication, power, square, sums, products, euclidean quotient and
remainder. 

The |round|, |trunc|, |floor|, |ceil| functions are still available, and are
about the only places where fractions can be used, but |/| can not be used!
This dilemma is solved using |protect|.\inmarg{Changed} For understanding the
next example, recall that |round| and |trunc| have a second (non negative)
optional argument. In a normal \csbxint{expr}-essions, |round| and |trunc| are
mapped to \csbxint{Round} and \csbxint{Trunc}, in \csbxint{iiexpr}-essions,
they are mapped to \csbxint{iRound} and \csbxint{iTrunc}.


\begin{everbatim*}
\xinttheiiexpr 5/3, round(5/3,3), trunc(5/3,3), trunc(\xintDiv {5}{3},3), 
trunc(\xintRaw {5/3},3)\relax{},
but \xinttheiiexpr 5/3, round(protect(5/3),3), trunc(protect(5/3),3), floor(protect(5/3)), 
ceil(protect(5/3))\relax{} works!

\noindent And with negative numbers:  \xinttheiiexpr -5/3, round(protect(-5/3),3),  
trunc(protect(-5/3),3), floor(protect(-5/3)), ceil(protect(-5/3))\relax.
\end{everbatim*}


Decimal numbers and numbers using scientific notations must be given as
arguments to one of the |num|, or |round|, or etc\dots functions, which will truncate
them to an integer.%
\inmarg{Changed} Internally the number will be represented with as many zeros
as is necessary, thus one does not want to do |num(1e100000)| for example!

%
\begin{everbatim*}
\xinttheiiexpr num(13.4567e3)+num(10000123e-3)\relax % should compute 13456+10000
\end{everbatim*}
%

The |reduce| function is not available and will raise un error. The |frac|
function also. The |sqrt| function is mapped to \csbxint{iiSqrt} which gives
a truncated square root. The |sqrtr| function is mapped to \csbxint{iiSqrtR}
which gives a rounded square root.

One can use the Float macros if one is careful to use |num|, or |round| on
their output,

\begin{everbatim*}
\xinttheiiexpr \xintFloatSqrt [20]{2}, \xintFloatSqrt [20]{3}\relax % no operations

\noindent The next example requires the |round|, and one could not put the |+| inside it:

\xinttheiiexpr round(\xintFloatSqrt [20]{2},19)+round(\xintFloatSqrt [20]{3},19)\relax

(the second argument of |round| and |trunc| tells how many digits from after the 
decimal mark one should keep.)
\end{everbatim*}

The whole point of \csbxint{iiexpr} is to gain some speed in \emph{integer-only}
algorithms, and the above explanations related to how to nevertheless use
fractions therein are a bit peripheral. We observed of the order of
$30$\% speed gain when dealing with numbers with circa one hundred digits, but this
gain decreases the longer the manipulated numbers become and becomes negligible
for numbers with thousand digits: the overhead from parsing fraction format is
little compared to other expensive aspects of the expandable shuffling of
tokens.

\subsection{\csbh{xintboolexpr},
  \csbh{xinttheboolexpr}}\label{xintboolexpr}\label{xinttheboolexpr}
%{\small New in |1.09c|.\par}

Equivalent\etype{x} to doing |\xintexpr ...\relax| and returning $1$ if the
result does not vanish, and $0$ is the result is zero. As |\xintexpr|, this
can be used on comma separated lists of expressions, and will return a
comma separated list of $0$'s and $1$'s.

\subsection{\csbh{xintfloatexpr},
  \csbh{xintthefloatexpr}}\label{xintfloatexpr}\label{xintthefloatexpr}

\csbxint{floatexpr}|...\relax|\etype{x} is exactly like |\xintexpr...\relax|
but with the four binary operations and the power function mapped to
\csa{xintFloatAdd}, \csa{xintFloatSub}, \csa{xintFloatMul}, \csa{xintFloatDiv}
and \csa{xintFloatPower}. The precision for the computation is from the
current setting of |\xintDigits|. Comma separated lists of expressions are
allowed. 

An optional parameter within brackets is allowed: the final float will have
that many digits of precision. This is provided to get rid of non-relevant
last digits.\inmarg{New}

Currently, the factorial function hasn't yet a float version; so inside
|\xintthefloatexpr . . . \relax|, |n!| will be computed exactly. Perhaps this
will be improved in a future release.

\xintDigits:= 9;

Note that |1.000000001| and |(1+1e-9)| will not be equivalent for
|D=\xinttheDigits| set to nine or less. Indeed the addition implicit in |1+1e-9|
(and executed when the closing parenthesis is found) will provoke the rounding
to |1|. Whereas |1.000000001|, when found as operand of one of the four
elementary operations is kept with |D+2| digits, and even more for the power
function.
% REVOIR ceci
%
\leftedline{|\xintDigits:= 9; \xintthefloatexpr
  (1+1e-9)-1\relax|\dtt{=\xintthefloatexpr (1+1e-9)-1\relax}}
%
\leftedline{|\xintDigits:= 9; \xintthefloatexpr
  1.000000001-1\relax|\dtt{=\xintthefloatexpr 1.000000001-1\relax}}

For the fun of it:\xintDigits:=20; |\xintDigits:=20;|%
%
\leftedline{|\xintthefloatexpr (1+1e-7)^1e7\relax|%
       \dtt{=\xintthefloatexpr (1+1e-7)^1e7\relax}}

|\xintDigits:=36;|\xintDigits:=36;
%
\leftedline{|\xintthefloatexpr
  ((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax|}
%
\leftedline{\dtt{\xintthefloatexpr
  ((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax}}
%
\leftedline{|\xintFloat{\xinttheexpr
  ((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax}|}
%
\leftedline{\dtt{\xintFloat
  {\xinttheexpr((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax}}}

\xintDigits := 16;

The latter result is the rounding of the exact result. The previous one has
rounding errors coming from the various roundings done for each
sub-expression. It was a bit funny  to discover that |maple|, configured with
|Digits:=36;| and with decimal dots everywhere to let it input the numbers as
floats, gives exactly the same result with the same rounding errors
as does |\xintthefloatexpr|!

Using |\xintthefloatexpr| only pays off compared to using |\xinttheexpr|
followed with |\xintFloat| if the computations turn out to involve hundreds of
digits. For elementary calculations with hand written numbers (not using the
scientific notation with exponents differing greatly) it will generally be more
efficient to use |\xinttheexpr|. The situation is quickly otherwise if one
starts using the Power function. Then, |\xintthefloat| is often useful; and
sometimes indispensable to achieve the (approximate) computation in reasonable
time.

We can try some crazy things:
%
\leftedline{|\xintDigits:=12;\xintthefloatexpr 1.000000000000001^1e15\relax|}
%
\leftedline{\xintDigits:=12;%
  \dtt{\xintthefloatexpr 1.000000000000001^1e15\relax}}
%
Contrarily to some professional computing sofware which are our concurrents on
this market, the \dtt{1.000000000000001} wasn't rounded to |1| despite the
setting of \csa{xintDigits}; it would have been if we had input it as
|(1+1e-15)|.

% \xintDigits:=12;
% \pdfresettimer
% \edef\z{\xintthefloatexpr 1.000000000000001^1e15\relax}%
% \edef\temps{\the\pdfelapsedtime}%
% \xintRound {5}{\temps/65536}s\endgraf

\xintDigits := 16; % mais en fait \centeredline crée un groupe.

\subsection{\csbh{xintifboolexpr}}\label{xintifboolexpr}
%{\small New in |1.09c|.\par}

\csh{xintifboolexpr}|{<expr>}{YES}{NO}|\etype{xnn} does |\xinttheexpr
<expr>\relax| and then executes the |YES| or the |NO| branch depending on
whether the outcome was non-zero or zero. |<expr>| can involve various |&| and
\verb+|+, parentheses, |all|, |any|, |xor|, the |bool| or |togl| operators, but
is not limited to them: the most general computation can be done, the test is on
whether the outcome of the computation vanishes or not.

Will not work on an expression composed of comma separated sub-expressions.

\subsection{\csbh{xintifboolfloatexpr}}\label{xintifboolfloatexpr}
%{\small New in |1.09c|.\par}

\csh{xintifboolfloatexpr}|{<expr>}{YES}{NO}|\etype{xnn} does |\xintthefloatexpr
<expr>\relax| and then executes the |YES| or the |NO| branch depending on
whether the outcome was non zero or zero.

\subsection{\csbh{xintifbooliiexpr}}\label{xintifbooliiexpr}
%{\small New in |1.09i|.\par}

\csh{xintifbooliiexpr}|{<expr>}{YES}{NO}|\etype{xnn} does |\xinttheiiexpr
<expr>\relax| and then executes the |YES| or the |NO| branch depending on
whether the outcome was non zero or zero.

\subsection{\csbh{xintNewFloatExpr}}\label{xintNewFloatExpr}

This is exactly like \csbxint{NewExpr} except that the created formulas are
set-up to use |\xintthefloatexpr|. The precision used for the computation will
be the one given by |\xintDigits| at the time of use of the created formulas.
However, the numbers hard-wired in the original expression will have been
evaluated with the then current setting for |\xintDigits|.

\begin{everbatim*}
\xintNewFloatExpr \f [1] {sqrt(#1)}
\f {2} (with \xinttheDigits{} of precision).

{\xintDigits := 32;\f {2} (with \xinttheDigits{} of precision).}

\xintNewFloatExpr \f [1] {sqrt(#1)*sqrt(2)}
\f {2} (with \xinttheDigits {} of precision).

{\xintDigits := 32;\f {2} (?? we thought we had a higher precision.Explanation next)}

The sqrt(2) in the second formula was computed with only \xinttheDigits{} of
precision. Setting |\xintDigits| to a higher value at the time of definition will
confirm that the result above is from a mismatch of the precision for |sqrt(2)| at
the time of its evaluation and the precision for the new |sqrt(2)| with |#1=2| at
the time of use.

{\xintDigits := 32;\xintNewFloatExpr \f [1] {sqrt(#1)*sqrt(2)}
\f {2} (with \xinttheDigits {} of precision)}
\end{everbatim*}

\subsection{\csbh{xintNewIExpr}}\label{xintNewIExpr}
%{\small New in |1.09c|.\par }

Like \csbxint{NewExpr} but using |\xinttheiexpr|. 

%Former denomination was
%|\xintNewNumExpr| which is deprecated and should not be used.

\subsection{\csbh{xintNewIIExpr}}\label{xintNewIIExpr}
%{\small New in |1.09i|.\par }

Like \csbxint{NewExpr} but using |\xinttheiiexpr|.

\subsection{\csbh{xintNewBoolExpr}}\label{xintNewBoolExpr}
%{\small New in |1.09c|.\par }

Like \csbxint{NewExpr} but using |\xinttheboolexpr|.

\xintDigits:= 16;

\subsection{\csbh{xintthecoords}}\label{xintthecoords}

From a comma separated output of an even number of items as output by
|\xintfloatexpr [P] ...\relax| or by |\xintiexpr [D] ...\relax|, creates
coordinate pairs for |TikZ|. See \hyperref[sec:expr11coords]{an example} on
page \pageref{sec:expr11coords}.


\subsection{Technicalities}

As already mentioned \csa{xintNewExpr}|\myformula[n]| does not check the prior
existence of a macro |\myformula|. And the number of parameters |n| given as
mandatory argument within square brackets should be (at least) equal
to the number of parameters in the expression.

Obviously I should mention that \csa{xintNewExpr} itself can not be used in an
expansion-only context, as it creates a macro.

The |\escapechar| setting may be arbitrary when using |\xintexpr|.

The format of the output  of
|\xintexpr|\meta{stuff}|\relax| is a |!| (with catcode 11) followed by various things:
\begin{everbatim*}
\oodef\f {\xintexpr 1.23^10\relax }\meaning\f
\end{everbatim*}

\begin{framed}
  Note that |\xintexpr| is thus compatible with complete expansion, contrarily
  to |\numexpr| which is non-expandable, if not prefixed by |\the| or |\number|,
  and away from contexts where \TeX{} is building a number. See
  \autoref{ssec:fibonacci} for some illustration.
%  pour mémoire:
%  \MyMarginNote[\kern\dimexpr\FrameSep+\FrameRule\relax]{New with 1.09j!}
\end{framed}

I decided to put all intermediate results (from each evaluation of an infix
operators, or of a parenthesized subpart of the expression, or from application
of the minus as prefix, or of the exclamation sign as postfix, or any
encountered braced material) inside |\csname...\endcsname|, as this can be done
expandably and encapsulates an arbitrarily long fraction in a single token (left
with undefined meaning), thus providing tremendous relief to the programmer in
his/her expansion control.

\begin{framed}
  As the |\xintexpr| computations corresponding to functions and infix
  or postfix operators are done inside |\csname...\endcsname|, the
  \fexpan dability could possibly be dropped and one could imagine
  implementing the basic operations with expandable but not \fexpan
  dable macros (as \csbxint{XTrunc}.) I have not investigated that
  possibility.
\end{framed}

% \begin{framed}
%   This implementation and user interface are still to be considered
%   \emph{experimental}.
% \end{framed}

Syntax errors in the input such as using a one-argument function with two
arguments will generate low-level \TeX{} processing unrecoverable errors, with
cryptic accompanying message. 

Some other problems will give rise to `error messages' macros giving some
indication on the location and nature of the problem. Mainly, an attempt has
been made to handle gracefully missing or extraneous parentheses.

However, this mechanism is completely inoperant for parentheses involved in
the syntax of the |seq|, |add|, |mul|, |subs|, |rseq| and |rrseq| functions.

% When the scanner is looking for a number and finds something else not otherwise
% treated, it assumes it is the start of the function name and will expand forward
% in the hope of hitting an opening parenthesis; if none is found at least it
% should stop when encountering the |\relax| marking the end of the expressions.

Note that |\relax| is \emph{mandatory} (contrarily to a |\numexpr|).

\subsection{Acknowledgements}

I was greatly helped in my preparatory thinking, prior to producing such an
expandable parser, by the commented source of the
\href{http://www.ctan.org/pkg/l3kernel}{l3fp} package, specifically the
|l3fp-parse.dtx| file (in the version of April-May 2013). Also the source of the
|calc| package was instructive, despite the fact that here for |\xintexpr| the
principles are necessarily different due to the aim of achieving expandability.

%\etocdepthtag.toc {commandsB}

\section{Commands of the \xintbinhexname package}
\label{sec:binhex}

\localtableofcontents

This package was first included in the |1.08| (|2013/06/07|) release of
\xintname. It provides expandable conversions of arbitrarily big integers to and
from binary and hexadecimal.

The argument is first \fexpan ded. It then may start with an optional minus
sign (unique, of category code other), followed with optional leading zeroes
(arbitrarily many, category code other) and then ``digits'' (hexadecimal
letters may be of category code letter or other, and must be
uppercased). The optional (unique) minus sign (plus sign is not allowed) is
kept in the output. Leading zeroes are allowed, and stripped. The
hexadecimal letters on output are of category code letter, and
uppercased.

% \clearpage

\subsection{\csbh{xintDecToHex}}\label{xintDecToHex}

Converts from decimal to hexadecimal.\etype{f}

\texttt{\string\xintDecToHex \string{\printnumber{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}\string}}\endgraf\noindent\dtt{->\printnumber{\xintDecToHex{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}}}

\subsection{\csbh{xintDecToBin}}\label{xintDecToBin}

Converts from decimal to binary.\etype{f}

\texttt{\string\xintDecToBin \string{\printnumber{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}\string}}\endgraf\noindent\dtt{->\printnumber{\xintDecToBin{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}}}

\subsection{\csbh{xintHexToDec}}\label{xintHexToDec}

Converts from hexadecimal to decimal.\etype{f}

\texttt{\string\xintHexToDec
  \string{\printnumber{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}\string}}\endgraf\noindent
\dtt{->\printnumber{\xintHexToDec{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}}}

\subsection{\csbh{xintBinToDec}}\label{xintBinToDec}

Converts from binary to decimal.\etype{f}

\texttt{\string\xintBinToDec
  \string{\printnumber{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}\string}}\endgraf\noindent
\dtt{->\printnumber{\xintBinToDec{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}}}

\subsection{\csbh{xintBinToHex}}\label{xintBinToHex}

Converts from binary to hexadecimal.\etype{f}

\texttt{\string\xintBinToHex
  \string{\printnumber{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}\string}}\endgraf\noindent
\dtt{->\printnumber{\xintBinToHex{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}}}

\subsection{\csbh{xintHexToBin}}\label{xintHexToBin}

Converts from hexadecimal to binary.\etype{f}

\texttt{\string\xintHexToBin
  \string{\printnumber{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}\string}}\endgraf\noindent
\dtt{->\printnumber{\xintHexToBin{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}}}

\subsection{\csbh{xintCHexToBin}}\label{xintCHexToBin}

Also converts from hexadecimal to binary.\etype{f} Faster on inputs with at
least one hundred hexadecimal digits.

\texttt{\string\xintCHexToBin
  \string{\printnumber{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}\string}}\endgraf\noindent
\dtt{->\printnumber{\xintCHexToBin{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}}}

\section{Commands of the \xintgcdname package}
\label{sec:gcd}

\localtableofcontents

This package was included in the original release |1.0| (|2013/03/28|) of the
\xintname bundle.

Since release |1.09a| the macros filter their inputs through the \csbxint{Num}
macro, so one can use count registers, or fractions as long as they reduce to
integers.

Since release |1.1|, the two ``|typeset|'' macros require the explicit
loading by the user of package \xinttoolsname.\inmarg{Changed}


%% \clearpage

\subsection{\csbh{xintGCD}, \csbh{xintiiGCD}}\label{xintGCD}\label{xintiiGCD}

|\xintGCD|\n\m\etype{\Numf\Numf} computes the greatest common divisor. It is
positive, except when both |N| and |M| vanish, in which case the macro returns
zero.
%
\leftedline{\csa{xintGCD}|{10000}{1113}|\dtt{=\xintGCD{10000}{1113}}}
%
\leftedline{|\xintiiGCD{123456789012345}{9876543210321}=|\dtt
  {\xintiiGCD{123456789012345}{9876543210321}}}

\csa{xintiiGCD} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintGCDof}}\label{xintGCDof}
%{\small New with release |1.09a|.\par}

\csa{xintGCDof}|{{a}{b}{c}...}|\etype{f{$\to$}{\lowast\Numf}} computes the greatest common divisor of all
integers |a|, |b|, \dots{}  The list argument
may be a macro, it is \fexpan ded first and must contain at least one item.

\subsection{\csbh{xintLCM}, \csbh{xintiiLCM}}\label{xintLCM}\label{xintiiLCM}
%{\small New with release |1.09a|.\par}

|\xintGCD|\n\m\etype{\Numf\Numf} computes the least common multiple. It is
|0| if one of the two integers vanishes.

\csa{xintiiLCM} skips the \csbxint{Num} overhead.\etype{ff}

\subsection{\csbh{xintLCMof}}\label{xintLCMof}
%{\small New with release |1.09a|.\par}

\csa{xintLCMof}|{{a}{b}{c}...}|\etype{f{$\to$}{\lowast\Numf}} computes the least
common multiple of all integers |a|, |b|, \dots{} The list argument may be a
macro, it is \fexpan ded first and must contain at least one item.

\subsection{\csbh{xintBezout}}\label{xintBezout}

\xintAssign{{\xintBezout {10000}{1113}}}\to\X
\xintAssign {\xintBezout {10000}{1113}}\to\A\B\U\V\D

|\xintBezout|\n\m\etype{\Numf\Numf} returns five numbers |A|, |B|, |U|, |V|,
|D| within braces. |A| is the first (expanded, as usual) input number, |B| the
second, |D| is the GCD, and \dtt{UA - VB = D}. %
%
\leftedline{|\xintAssign
 {{\xintBezout {10000}{1113}}}\to\X|} %
%
\leftedline{|\meaning\X:
 |\dtt{\meaning\X }.}
\noindent{|\xintAssign {\xintBezout {10000}{1113}}\to\A\B\U\V\D|}\\
|\A: |\dtt{\A },
|\B: |\dtt{\B },
|\U: |\dtt{\U },
|\V: |\dtt{\V },
|\D: |\dtt{\D }.\\
\xintAssign {\xintBezout {123456789012345}{9876543210321}}\to\A\B\U\V\D
\noindent{|\xintAssign {\xintBezout {123456789012345}{9876543210321}}\to\A\B\U\V\D
|}\\
|\A: |\dtt{\A },
|\B: |\dtt{\B },
|\U: |\dtt{\U },
|\V: |\dtt{\V },
|\D: |\dtt{\D }.

\subsection{\csbh{xintEuclideAlgorithm}}\label{xintEuclideAlgorithm}

\xintAssign {{\xintEuclideAlgorithm {10000}{1113}}}\to\X

\def\restorebracecatcodes
   {\catcode`\{=1 \catcode`\}=2 }

\def\allowlistsplit
   {\catcode`\{=12 \catcode`\}=12 \allowlistsplita }

\def\allowlistsplitx {\futurelet\listnext\allowlistsplitxx }

\def\allowlistsplitxx {\ifx\listnext\relax \restorebracecatcodes
                        \else \expandafter\allowlistsplitxxx \fi }
\begingroup
\catcode`\[=1
\catcode`\]=2
\catcode`\{=12
\catcode`\}=12
\gdef\allowlistsplita #1{[#1\allowlistsplitx {]
\gdef\allowlistsplitxxx {#1}%
     [{#1}\hskip 0pt plus 1pt \allowlistsplitx ]
\endgroup

|\xintEuclideAlgorithm|\n\m\etype{\Numf\Numf} applies the Euclide algorithm
and keeps a copy of all quotients and remainders. %
%
\leftedline{|\xintAssign {{\xintEuclideAlgorithm {10000}{1113}}}\to\X|}

|\meaning\X: |\dtt{\expandafter\allowlistsplit\meaning\X\relax .}

The first token is the number of steps, the second is |N|, the
third is the GCD, the fourth is |M| then the first quotient and
remainder, the second quotient and remainder, \dots until the
final quotient and last (zero) remainder.

\subsection{\csbh{xintBezoutAlgorithm}}\label{xintBezoutAlgorithm}

\xintAssign {{\xintBezoutAlgorithm {10000}{1113}}}\to\X

|\xintBezoutAlgorithm|\n\m\etype{\Numf\Numf} applies the Euclide algorithm
and keeps a copy of all quotients and remainders. Furthermore it computes the
entries of the successive products of the 2 by 2 matrices
$\left(\vcenter{\halign {\,#&\,#\cr q & 1 \cr 1 & 0 \cr}}\right)$ formed from
the quotients arising in the algorithm. %
%
\leftedline{|\xintAssign {{\xintBezoutAlgorithm {10000}{1113}}}\to\X|}

|\meaning\X: |\dtt{\expandafter\allowlistsplit\meaning\X \relax .}

The first token is the number of steps, the second is |N|, then
|0|, |1|, the GCD, |M|, |1|, |0|, the first quotient, the first
remainder, the top left entry of the first matrix, the bottom left
entry, and then these four things at each step until the end.

\subsection{\csbh{xintTypesetEuclideAlgorithm}}\label{xintTypesetEuclideAlgorithm}

% Requires explicit loading by the user of package \xinttoolsname.\inmarg{Changed}

This macro is just an example of how to organize the data returned by
\csa{xintEuclideAlgorithm}.\ntype{\Numf\Numf} Copy the source code to a new
macro and modify it to what is needed.
%
\leftedline{|\xintTypesetEuclideAlgorithm {123456789012345}{9876543210321}|}
\xintTypesetEuclideAlgorithm {123456789012345}{9876543210321}

\subsection{\csbh{xintTypesetBezoutAlgorithm}}%
\label{xintTypesetBezoutAlgorithm}

% Requires explicit loading by the user of package \xinttoolsname.\inmarg{Changed}

This macro is just an example of how to organize the data returned by
\csa{xintBezoutAlgorithm}.\ntype{\Numf\Numf} Copy the source code to a new
macro and modify it to what is needed.
%
\leftedline{|\xintTypesetBezoutAlgorithm {10000}{1113}|}
\xintTypesetBezoutAlgorithm {10000}{1113}


% problème de color stack overflow avec dvipdfmx
% 14 octobre puis 28 octobre 2014
% 7 mars 2015

\def\everbatimxprehook  {}
\def\everbatimxposthook {}

\def\digitstt #1{\begingroup #1\endgroup}
\let\dtt\digitstt

\section{Commands of the \xintseriesname package}
\label{sec:series}

\localtableofcontents

This package was first released with version |1.03| (|2013/04/14|) of the
\xintname bundle.

The \Ff{} expansion type of various macro arguments is only a \Numf{} if only
\xintname but not \xintfracname is loaded. The macro \csbxint{iSeries} is
special and expects summing big integers obeying the strict format, even if
\xintfracname is loaded.

The arguments serving as indices are of the \numx{} expansion type.

In some cases one or two of the macro arguments are only expanded at a later
stage not immediately.

%% \clearpage

\subsection{\csbh{xintSeries}}\label{xintSeries}

\csa{xintSeries}|{A}{B}{\coeff}|\etype{\numx\numx\Ff} computes
$\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|$. The initial and final indices
must obey the |\numexpr| constraint of expanding to numbers at most |2^31-1|.
The |\coeff| macro must be a one-parameter \fexpan dable command, taking on
input an explicit number |n| and producing some number or fraction |\coeff{n}|;
it is expanded at the time it is
needed.%
%
\footnote{\label{fn:xintiiMON}\csbxint{iiMON} is like \csbxint{MON} but
  does not parse its argument through \csbxint{Num}, for efficiency;
  other macros of this type are \csbxint{iiAdd}, \csbxint{iiMul},
  \csbxint{iiSum}, \csbxint{iiPrd}, \csbxint{iiMMON}, \csbxint{iiLDg},
  \csbxint{iiFDg}, \csbxint{iiOdd}, \dots}

\begin{everbatim*}
\def\coeff #1{\xintiiMON{#1}/#1.5} % (-1)^n/(n+1/2)
\oodef\w {\xintSeries {0}{50}{\coeff}} % we want to re-use it
\oodef\z {\xintJrr {\w}[0]} % the [0] for a microsecond gain.
% \xintJrr preferred to \xintIrr: a big common factor is suspected.
% But numbers much bigger would be needed to show the greater efficiency.
\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} = \xintFrac\z \]
\end{everbatim*}

The definition of |\coeff| as |\xintiiMON{#1}/#1.5| is quite suboptimal. It
allows |#1| to be a big integer, but anyhow only small integers are accepted
as initial and final indices (they are of the \numx{} type). Second, when the
\xintfracname parser sees the |#1.5| it will remove the dot hence create a
denominator with one digit more. For example |1/3.5| turns internally into
|10/35| whereas it would be more efficient to have |2/7|. For info here is the
non-reduced |\w|:
\[\xintFrac\w\]
It would have been bigger still in releases earlier than |1.1|: now, the
\xintfracname \csbxint{Add} routine does not multiply blindly denominators
anymore, it checks if one is a multiple of the other. However it does not
practice systematic reduction to lowest terms.

A more efficient way to code |\coeff| is illustrated next.
\begin{everbatim*}
\def\coeff #1{\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}%
% The [0] in \coeff is a tiny optimization: in its presence the \xintfracname parser 
% sees something which is already in internal format.
\oodef\w {\xintSeries {0}{50}{\coeff}}
\[\sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12}=\xintFrac\w\]
\end{everbatim*}
The reduced form |\z| as displayed above only differs from this one by a
factor of \dtt{\xintNum {\xintDenominator\w/\xintDenominator\z}}.

\setlength{\columnsep}{0pt}
\everb|@
\def\coeffleibnitz #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]}
\cnta 1
\loop  % in this loop we recompute from scratch each partial sum!
% we can afford that, as \xintSeries is fast enough.
\noindent\hbox to 2em{\hfil\texttt{\the\cnta.} }%
         \xintTrunc {12}{\xintSeries {1}{\cnta}{\coeffleibnitz}}\dots
\endgraf
\ifnum\cnta < 30 \advance\cnta 1 \repeat
|

\begin{multicols}{3}
  \def\coeffleibnitz #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]} \cnta 1
  \loop
  \noindent\hbox to 2em{\hfil\dtt{\the\cnta.} }%
  \xintTrunc {12}{\xintSeries {1}{\cnta}{\coeffleibnitz}}\dots
    \endgraf
    \ifnum\cnta < 30 \advance\cnta 1 \repeat
\end{multicols}

\subsection{\csbh{xintiSeries}}\label{xintiSeries}

\def\coeff #1{\xintiTrunc {40}
   {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}}%

 \csa{xintiSeries}|{A}{B}{\coeff}|\etype{\numx\numx f} computes
 $\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|$ where |\coeff{n}|
 must \fexpan d to a (possibly long) integer in the strict format.
\everb|@
\def\coeff #1{\xintiTrunc {40}{\xintMON{#1}/#1.5}}%
% better:
\def\coeff #1{\xintiTrunc {40}
   {\the\numexpr 2*\xintiiMON{#1}\relax/\the\numexpr 2*#1+1\relax [0]}}%
% better still:
\def\coeff #1{\xintiTrunc {40}
 {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}}%
% (-1)^n/(n+1/2) times 10^40, truncated to an integer.
\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx
        \xintTrunc {40}{\xintiSeries {0}{50}{\coeff}[-40]}\dots\]
|

\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx \xintTrunc
{40}{\xintiSeries {0}{50}{\coeff}[-40]}\] 

We should have cut out at
least the last two digits: truncating errors originating with the first
coefficients of the sum will never go away, and each truncation
introduces an uncertainty in the last digit, so as we have 40 terms, we
should trash the last two digits, or at least round at 38 digits. It is
interesting to compare with the computation where rounding rather than
truncation is used, and with the decimal
expansion of the exactly computed partial sum of the series:
\everb|@
\def\coeff #1{\xintiRound {40} % rounding at 40
  {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}}%
% (-1)^n/(n+1/2) times 10^40, rounded to an integer.
\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx
        \xintTrunc {40}{\xintiSeries {0}{50}{\coeff}[-40]}\]
\def\exactcoeff #1%
  {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}%
\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12}
   = \xintTrunc {50}{\xintSeries {0}{50}{\exactcoeff}}\dots\]
|

\def\coeff #1{\xintiRound {40}
   {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}}%
% (-1)^n/(n+1/2) times 10^40, rounded to an integer.
\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx
        \xintTrunc {40}{\xintiSeries {0}{50}{\coeff}[-40]}\]
\def\exactcoeff #1%
  {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}%
\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12}
   = \xintTrunc {50}{\xintSeries {0}{50}{\exactcoeff}}\dots\]
This shows indeed that our sum of truncated terms
estimated wrongly the 39th and 40th digits of the exact result%
%
\footnote{as the series is alternating, we can roughly expect an error
  of $\sqrt{40}$ and the last two digits are off by 4 units, which is
  not contradictory to our expectations.}
%
and that the sum of rounded terms fared a bit better.

\subsection{\csbh{xintRationalSeries}}\label{xintRationalSeries}

%{\small \hspace*{\parindent}New with release |1.04|.\par}

\noindent \csa{xintRationalSeries}|{A}{B}{f}{\ratio}|\etype{\numx\numx\Ff\Ff}
evaluates $\sum_{\text{|n=A|}}^{\text{|n=B|}}|F(n)|$, where |F(n)| is specified
indirectly via the data of |f=F(A)| and the one-parameter macro |\ratio| which
must be such that |\macro{n}| expands to |F(n)/F(n-1)|. The name indicates that
\csa{xintRationalSeries} was designed to be useful in the cases where
|F(n)/F(n-1)| is a rational function of |n| but it may be anything expanding to
a fraction. The macro |\ratio| must be an expandable-only compatible command and
expand to its value after iterated full expansion of its first token. |A| and
|B| are fed to a |\numexpr| hence may be count registers or arithmetic
expressions built with such; they must obey the \TeX{} bound. The initial term
|f| may be a macro |\f|, it will be expanded to its value representing |F(A)|.


% 7 mars 2015, problème de color stack overflow, je dois supprimer textcolor
% j'essaie en supprimant \normalcolor aussi
% et aussi dans le multicols plus bas après
% «Here is a slightly more complicated evaluation»

\begin{everbatim*}
\def\ratio #1{2/#1[0]}% 2/n, to compute exp(2)
\cnta 0 % previously declared count
\begin{quote}
\loop \oodef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}%
\noindent$\sum_{n=0}^{\the\cnta} \frac{2^n}{n!}=
           \xintTrunc{12}\z\dots=
           \xintFrac\z=\xintFrac{\xintIrr\z}$\vtop to 5pt{}\par
\ifnum\cnta<20 \advance\cnta 1 \repeat 
\end{quote}
\end{everbatim*}

\begin{everbatim*}
\def\ratio #1{-1/#1[0]}% -1/n, comes from the series of exp(-1)
\cnta 0 % previously declared count
\begin{quote}
\loop
\oodef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}%
\noindent$\sum_{n=0}^{\the\cnta} \frac{(-1)^n}{n!}=
  \xintTrunc{20}\z\dots=\xintFrac{\z}=\xintFrac{\xintIrr\z}$%
         \vtop to 5pt{}\par
\ifnum\cnta<20 \advance\cnta 1 \repeat
\end{quote}
\end{everbatim*}


 \def\ratioexp #1#2{\xintDiv{#1}{#2}}% #1/#2

\medskip We can incorporate an indeterminate if we define |\ratio| to be
a macro with two parameters: |\def\ratioexp
  #1#2{\xintDiv{#1}{#2}}|\texttt{\%}| x/n: x=#1, n=#2|.
Then, if |\x| expands to some fraction |x|, the
command %
%
\leftedline{|\xintRationalSeries {0}{b}{1}{\ratioexp{\x}}|}
will compute $\sum_{n=0}^{n=b} x^n/n!$:\par
\begin{everbatim*}
\cnta 0
\def\ratioexp #1#2{\xintDiv{#1}{#2}}% #1/#2
\loop
\noindent
$\sum_{n=0}^{\the\cnta} (.57)^n/n! = \xintTrunc {50}
     {\xintRationalSeries {0}{\cnta}{1}{\ratioexp{.57}}}\dots$
     \vtop to 5pt {}\endgraf
\ifnum\cnta<50 \advance\cnta 10 \repeat
\end{everbatim*}

Observe that in this last example the |x| was directly inserted; if it
had been a more complicated explicit fraction it would have been
worthwile to use |\ratioexp\x| with |\x| defined to expand to its value.
In the further situation where this fraction |x| is not explicit but
itself defined via a complicated, and time-costly, formula, it should be
noted that \csa{xintRationalSeries} will do again the evaluation of |\x|
for each term of the partial sum. The easiest is thus when |x| can be
defined as an |\edef|. If however, you are in an expandable-only context
and cannot store in a macro like |\x| the value to be used, a variant of
\csa{xintRationalSeries} is needed which will first evaluate this |\x| and then
use this result without recomputing it. This is \csbxint{RationalSeriesX},
documented next.

Here is a slightly more complicated evaluation:
\begin{everbatim*}
\cnta 1
\begin{multicols}{2}
\loop \oodef\z {\xintRationalSeries
                   {\cnta}
                   {2*\cnta-1}
                   {\xintiPow {\the\cnta}{\cnta}/\xintFac{\cnta}}
                   {\ratioexp{\the\cnta}}}%
\oodef\w {\xintRationalSeries {0}{2*\cnta-1}{1}{\ratioexp{\the\cnta}}}%
\noindent
$\sum_{n=\the\cnta}^{\the\numexpr 2*\cnta-1\relax} \frac{\the\cnta^n}{n!}/%
          \sum_{n=0}^{\the\numexpr 2*\cnta-1\relax} \frac{\the\cnta^n}{n!} =
          \xintTrunc{8}{\xintDiv\z\w}\dots$ \vtop to 5pt{}\endgraf
\ifnum\cnta<20 \advance\cnta 1 \repeat
\end{multicols}
\end{everbatim*}


\subsection{\csbh{xintRationalSeriesX}}\label{xintRationalSeriesX}

%{\small \hspace*{\parindent}New with release |1.04|.\par}

\noindent\csa{xintRationalSeriesX}|{A}{B}{\first}{\ratio}{\g}|%
\etype{\numx\numx\Ff\Ff f} is a parametrized version of \csa{xintRationalSeries}
where |\first| is now a one-parameter macro such that |\first{\g}| gives the
initial term and |\ratio| is a two-parameter macro such that |\ratio{n}{\g}|
represents the ratio of one term to the previous one. The parameter |\g| is
evaluated only once at the beginning of the computation, and can thus itself be
the yet unevaluated result of a previous computation.

Let |\ratio| be such a two-parameter macro; note the subtle differences
between%
%
\leftedline{|\xintRationalSeries {A}{B}{\first}{\ratio{\g}}|}
%
\leftedline{and |\xintRationalSeriesX {A}{B}{\first}{\ratio}{\g}|.} First the
location of braces differ... then, in the former case |\first| is a
\emph{no-parameter} macro expanding to a fractional number, and in the latter,
it is a
\emph{one-parameter} macro which will use |\g|. Furthermore the |X| variant
will expand |\g| at the very beginning whereas the former non-|X| former variant
will evaluate it each time it needs it (which is bad if this
evaluation is time-costly, but good if |\g| is a big explicit fraction
encapsulated in a macro).

The example will use the macro \csbxint{PowerSeries} which computes
efficiently exact partial sums of power series, and is discussed in the
next section.
\begin{everbatim*}
\def\firstterm #1{1[0]}% first term of the exponential series
% although it is the constant 1, here it must be defined as a
% one-parameter macro. Next comes the ratio function for exp:
\def\ratioexp  #1#2{\xintDiv {#1}{#2}}% x/n
% These are the (-1)^{n-1}/n of the log(1+h) series:
\def\coefflog #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]}%
% Let L(h) be the first 10 terms of the log(1+h) series and
% let E(t) be the first 10 terms of the exp(t) series.
% The following computes E(L(a/10)) for a=1,...,12.
\begin{multicols}{3}\raggedcolumns
\cnta 0
\loop
\noindent\xintTrunc {18}{%
     \xintRationalSeriesX {0}{9}{\firstterm}{\ratioexp}
         {\xintPowerSeries{1}{10}{\coefflog}{\the\cnta[-1]}}}\dots
\endgraf
\ifnum\cnta < 12 \advance \cnta 1 \repeat
\end{multicols}
\end{everbatim*}


These completely exact operations rapidly create numbers with many digits. Let
us print in full the raw fractions created by the operation illustrated above:

\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm}
{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1[-1]}}}

|E(L(1[-1]))=|\dtt{\printnumber{\z}} (length of numerator:
\xintLen {\xintNumerator \z})

\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm}
{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{12[-2]}}}

|E(L(12[-2]))=|\dtt{\printnumber{\z}} (length of numerator:
\xintLen {\xintNumerator \z})

\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm}
{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{123[-3]}}}

|E(L(123[-3]))=|\dtt{\printnumber{\z}} (length of numerator:
\xintLen {\xintNumerator \z})

We see that the denominators here remain the same, as our input only had various
powers of ten as denominators, and \xintfracname efficiently assemble (some
only, as we can see) powers of ten. Notice that 1 more digit in an input
denominator seems to mean 90 more in the raw output. We can check that with some
other test cases:

\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm}
{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/7}}}

|E(L(1/7))=|\dtt{\printnumber{\z}} (length of numerator:
\xintLen {\xintNumerator \z}; length of denominator:
\xintLen {\xintDenominator \z})

\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm}
{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/71}}}

|E(L(1/71))=|\dtt{\printnumber{\z}} (length of numerator:
\xintLen {\xintNumerator \z}; length of denominator:
\xintLen {\xintDenominator \z})

\oodef\z{\xintRationalSeriesX {0}{9}{\firstterm}
{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/712}}}

|E(L(1/712))=|\dtt{\printnumber{\z}} (length of numerator:
\xintLen {\xintNumerator \z}; length of denominator:
\xintLen {\xintDenominator \z})

% \pdfresettimer
% \edef\w{\xintDenominator{\xintIrr{\z}}}
% \the\pdfelapsedtime

Thus
decimal numbers such as |0.123| (equivalently
|123[-3]|) give less computing intensive tasks than fractions such as |1/712|:
in the case of decimal numbers the (raw) denominators originate in the
coefficients of the series themselves, powers of ten of the input within
brackets being treated separately. And even then the
numerators will grow with the size of the input in a sort of linear way, the
coefficient being given by the order of series: here 10 from the log and 9 from
the exp, so 90. One more digit in the input means 90 more digits in the
numerator of the output: obviously we can not go on composing such partial sums
of series and hope that \xintname will joyfully do all at the speed of light!

Hence, truncating the output (or better, rounding) is the only way to go if one
needs a general calculus of special functions. This is why the package
\xintseriesname provides, besides \csbxint{Series}, \csbxint{RationalSeries}, or
\csbxint{PowerSeries} which compute \emph{exact} sums, 
\csbxint{FxPtPowerSeries} for fixed-point computations and a (tentative naive)
\csbxint{FloatPowerSeries}.

\subsection{\csbh{xintPowerSeries}}\label{xintPowerSeries}

\csa{xintPowerSeries}|{A}{B}{\coeff}{f}|\etype{\numx\numx\Ff\Ff}
evaluates the sum
$\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|\cdot |f|^{\text{|n|}}$. The
initial and final indices are given to a |\numexpr| expression. The |\coeff|
macro (which, as argument to \csa{xintPowerSeries} is expanded only at the time
|\coeff{n}| is needed) should be defined as a one-parameter expandable command,
its input will be an explicit number.

The |f| can be either a fraction directly input or a macro |\f| expanding to
such a fraction. It is actually more efficient to encapsulate an explicit
fraction |f| in such a macro, if it has big numerators and denominators (`big'
means hundreds of digits) as it will then take less space in the processing
until being (repeatedly) used.

This macro computes the \emph{exact} result (one can use it also for
polynomial evaluation), using a Horner scheme which helps avoiding a
denominator build-up (this problem however,  even if using a naive additive
approach, is much less acute since release |1.1| and its new policy regarding
\csbxint{Add}).

\begin{everbatim*}
\def\geom #1{1[0]} % the geometric series
\def\f {5/17[0]}
\[ \sum_{n=0}^{n=20} \Bigl(\frac 5{17}\Bigr)^n
 =\xintFrac{\xintIrr{\xintPowerSeries {0}{20}{\geom}{\f}}}
 =\xintFrac{\xinttheexpr (17^21-5^21)/12/17^20\relax}\]
\end{everbatim*}

\begin{everbatim*}
\def\coefflog #1{1/#1[0]}% 1/n
\def\f {1/2[0]}%
\[ \log 2 \approx \sum_{n=1}^{20} \frac1{n\cdot 2^n}
    = \xintFrac {\xintIrr {\xintPowerSeries {1}{20}{\coefflog}{\f}}}\]
\[ \log 2 \approx \sum_{n=1}^{50} \frac1{n\cdot 2^n}
    = \xintFrac {\xintIrr {\xintPowerSeries {1}{50}{\coefflog}{\f}}}\]
\end{everbatim*}


\begin{everbatim*}
\setlength{\columnsep}{0pt}
\begin{multicols}{3}
\cnta 1 % previously declared count
\loop   % in this loop we recompute from scratch each partial sum!
% we can afford that, as \xintPowerSeries is fast enough.
\noindent\hbox to 2em{\hfil\texttt{\the\cnta.} }%
         \xintTrunc {12}
             {\xintPowerSeries {1}{\cnta}{\coefflog}{\f}}\dots
\endgraf
\ifnum \cnta < 30 \advance\cnta 1 \repeat
\end{multicols}
\end{everbatim*}


\begin{everbatim*}
\def\coeffarctg  #1{1/\the\numexpr\ifodd #1 -2*#1-1\else2*#1+1\fi\relax }%
% the above gives (-1)^n/(2n+1). The sign being in the denominator,
%             **** no [0] should be added ****,
% else nothing is guaranteed to work (even if it could by sheer luck)
% Notice in passing this aspect of \numexpr:
%         ****  \numexpr -(1)\relax is ilegal !!! ****
\def\f {1/25[0]}% 1/5^2
\[\mathrm{Arctg}(\frac15)\approx \frac15\sum_{n=0}^{15} \frac{(-1)^n}{(2n+1)25^n}
= \xintFrac{\xintIrr {\xintDiv {\xintPowerSeries {0}{15}{\coeffarctg}{\f}}{5}}}\]
\end{everbatim*}


\subsection{\csbh{xintPowerSeriesX}}\label{xintPowerSeriesX}

%{\small\hspace*{\parindent}New with release |1.04|.\par}

\noindent This is the same as \csbxint{PowerSeries}\ntype{\numx\numx\Ff\Ff}
apart
from the fact that the last parameter |f| is expanded once and for all before
being then used repeatedly. If the |f| parameter is to be an explicit big
fraction with many (dozens) digits, rather than using it directly it is slightly
better to have some macro |\g| defined to expand to the explicit fraction and
then use \csbxint{PowerSeries} with |\g|; but if |f| has not yet been evaluated
and will be the output of a complicated expansion of some |\f|, and if, due to
an expanding only context, doing |\edef\g{\f}| is no option, then
\csa{xintPowerSeriesX} should be used with |\f| as last parameter.
%
\begin{everbatim*}
\def\ratioexp #1#2{\xintDiv {#1}{#2}}% x/n
% These are the (-1)^{n-1}/n of the log(1+h) series:
\def\coefflog #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]}%
% Let L(h) be the first 10 terms of the log(1+h) series and
% let E(t) be the first 10 terms of the exp(t) series.
% The following computes L(E(a/10)-1) for a=1,..., 12.
\begin{multicols}{3}\raggedcolumns
\cnta 1
\loop
\noindent\xintTrunc {18}{%
   \xintPowerSeriesX {1}{10}{\coefflog}
  {\xintSub
      {\xintRationalSeries {0}{9}{1[0]}{\ratioexp{\the\cnta[-1]}}}
      {1}}}\dots
\endgraf
\ifnum\cnta < 12 \advance \cnta 1 \repeat
\end{multicols}
\end{everbatim*}


\subsection{\csbh{xintFxPtPowerSeries}}\label{xintFxPtPowerSeries}

\csa{xintFxPtPowerSeries}|{A}{B}{\coeff}{f}{D}|\etype{\numx\numx}
computes
$\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|\cdot |f|^{\,\text{|n|}}$ with each
    term of the series truncated to |D| digits\etype{\Ff\Ff\numx}
 after the decimal point. As
    usual, |A| and |B| are completely expanded through their inclusion in a
    |\numexpr| expression. Regarding |D| it will be similarly be expanded each
    time it is used inside an \csa{xintTrunc}. The one-parameter macro |\coeff|
    is similarly  expanded at the time it is used inside the
    computations. Idem for |f|. If |f| itself is some complicated macro it is
    thus better to use the variant \csbxint{FxPtPowerSeriesX} which expands it
    first and then uses the result of that expansion.

The current (|1.04|) implementation is: the first power |f^A| is
computed exactly, then \emph{truncated}. Then each successive power is
obtained from the previous one by multiplication by the exact value of
|f|, and truncated. And |\coeff{n}|\raisebox{.5ex}{|.|}|f^n| is obtained
from that by multiplying by |\coeff{n}| (untruncated) and then
truncating. Finally the sum is computed exactly. Apart from that
\csa{xintFxPtPowerSeries} (where |FxPt| means `fixed-point') is like
\csa{xintPowerSeries}.

There should be a variant for things of the type $\sum c_n \frac {f^n}{n!}$ to
avoid having to compute the factorial from scratch at each coefficient, the same
way \csa{xintFxPtPowerSeries} does not compute |f^n| from scratch at each |n|.
Perhaps in the next package release.

\def\coeffexp #1{1/\xintFac {#1}[0]}% [0] for faster parsing
\def\f {-1/2[0]}%
\newcount\cnta

\setlength{\multicolsep}{0pt}

\begin{multicols}{3}[%
\centeredline{$e^{-\frac12}\approx{}$}]%
\cnta 0
\noindent\loop
$\xintFxPtPowerSeries {0}{\cnta}{\coeffexp}{\f}{20}$\\
\ifnum\cnta<19
\advance\cnta 1
\repeat\par
\end{multicols}
\everb|@
\def\coeffexp #1{1/\xintFac {#1}[0]}% 1/n!
\def\f {-1/2[0]}% [0] for faster input parsing
\cnta 0 % previously declared \count register
\noindent\loop
$\xintFxPtPowerSeries {0}{\cnta}{\coeffexp}{\f}{20}$\\
\ifnum\cnta<19 \advance\cnta 1 \repeat\par
% One should **not** trust the final digits, as the potential truncation
% errors of up to 10^{-20} per term accumulate and never disappear! (the
% effect is attenuated by the alternating signs in the series). We  can
% confirm that the last two digits (of our evaluation of the nineteenth
% partial sum) are wrong via the evaluation with more digits:   
|

%
\leftedline{|\xintFxPtPowerSeries {0}{19}{\coeffexp}{\f}{25}=|
\dtt{\xintFxPtPowerSeries {0}{19}{\coeffexp}{\f}{25}}}
\oodef\z{\xintIrr {\xintPowerSeries {0}{19}{\coeffexp}{\f}}}
%

\texttt{\hyphenchar\font45 }%
It is no difficulty for \xintfracname to compute exactly, with the help
of \csa{xintPowerSeries}, the nineteenth partial sum, and to then give
(the start of) its exact decimal expansion:
%
\leftedline{|\xintPowerSeries {0}{19}{\coeffexp}{\f}| ${}=
  \displaystyle\xintFrac{\z}$%
  \vphantom{\vrule height 20pt depth 12pt}}%
%
\leftedline{${}=\xintTrunc {30}{\z}\dots$} Thus, one should always
estimate a priori how many ending digits are not reliable: if there are
|N| terms and |N| has |k| digits, then digits up to but excluding the
last |k| may usually be trusted. If we are optimistic and the series is
alternating we may even replace |N| with $\sqrt{|N|}$ to get the number |k|
of digits possibly of dubious significance.

\subsection{\csbh{xintFxPtPowerSeriesX}}\label{xintFxPtPowerSeriesX}

%{\small\hspace*{\parindent}New with release |1.04|.\par}

\noindent\csa{xintFxPtPowerSeriesX}|{A}{B}{\coeff}{\f}{D}|%
\ntype{\numx\numx}
computes, exactly as
\csa{xintFxPtPowerSeries}, the sum of
|\coeff{n}|\raisebox{.5ex}{|.|}|\f^n|\etype{\Ff\Ff\numx} from |n=A| to |n=B| with each term
of the series being \emph{truncated} to |D| digits after the decimal
point. The sole difference is that |\f| is first expanded and it
is the result of this which is used in the computations.

% Let us illustrate this on the computation of |(1+y)^{5/3}| where
% |1+y=(1+x)^{3/5}| and each of the two binomial series is evaluated with ten
% terms, the results being computed with |8| digits after the decimal point, and
% $|f|<1/10$.

Let us illustrate this on the numerical exploration of the identity
%
\leftedline{|log(1+x) = -log(1/(1+x))|}
%
Let |L(h)=log(1+h)|, and |D(h)=L(h)+L(-h/(1+h))|. Theoretically thus,
|D(h)=0| but we shall evaluate |L(h)| and |-h/(1+h)| keeping only 10
terms of their respective series. We will assume $|h|<0.5$. With only
ten terms kept in the power series we do not have quite 3 digits
precision as $2^{10}=1024$. So it wouldn't make sense to evaluate things
more precisely than, say circa 5 digits after the decimal points.
\begin{everbatim*}
\cnta 0
\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}% (-1)^{n-1}/n
\def\coeffalt #1{\the\numexpr\ifodd#1 -1\else1\fi\relax [0]}%   (-1)^n
\begin{multicols}2
\loop
\noindent \hbox to 2.5cm {\hss\texttt{D(\the\cnta/100): }}%
\xintAdd {\xintFxPtPowerSeriesX {1}{10}{\coefflog}{\the\cnta [-2]}{5}}
         {\xintFxPtPowerSeriesX {1}{10}{\coefflog}
             {\xintFxPtPowerSeriesX {1}{10}{\coeffalt}{\the\cnta [-2]}{5}}
          {5}}\endgraf
\ifnum\cnta < 49 \advance\cnta 7 \repeat
\end{multicols}
\end{everbatim*}


Let's say we evaluate functions on |[-1/2,+1/2]| with values more or less also
in |[-1/2,+1/2]| and we want to keep 4 digits of precision. So, roughly we need
at least 14 terms in series like the geometric or log series. Let's make this
15. Then it doesn't make sense to compute intermediate summands with more than 6
digits precision. So we compute with 6 digits
precision but return only 4 digits (rounded) after the decimal point.
This result with 4 post-decimal points precision is then used as input
to the next evaluation.
\begin{everbatim*}
\begin{multicols}2
\loop
\noindent \hbox to 2.5cm {\hss\texttt{D(\the\cnta/100): }}%
\dtt{\xintRound{4}
 {\xintAdd {\xintFxPtPowerSeriesX {1}{15}{\coefflog}{\the\cnta [-2]}{6}}
           {\xintFxPtPowerSeriesX {1}{15}{\coefflog}
                  {\xintRound {4}{\xintFxPtPowerSeriesX {1}{15}{\coeffalt}
                                 {\the\cnta [-2]}{6}}}
            {6}}%
 }}\endgraf
\ifnum\cnta < 49 \advance\cnta 7 \repeat
\end{multicols}
\end{everbatim*}

Not bad... I have cheated a bit: the `four-digits precise' numeric
evaluations were left unrounded in the final addition. However the inner
rounding to four digits worked fine and made the next step faster than
it would have been with longer inputs. The morale is that one should not
use the raw results of \csa{xintFxPtPowerSeriesX} with the |D| digits
with which it was computed, as the last are to be considered garbage.
Rather, one should keep from the output only some smaller number of
digits. This will make further computations faster and not less precise.
I guess there should be some command to do this final truncating, or
better, rounding, at a given number |D'<D| of digits. Maybe for the next
release.

\subsection{\csbh{xintFloatPowerSeries}}\label{xintFloatPowerSeries}

%{\small\hspace*{\parindent}New with |1.08a|.\par}

\noindent\csa{xintFloatPowerSeries}|[P]{A}{B}{\coeff}{f}|%
\ntype{{\upshape[\numx]}\numx\numx}
 computes
$\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|\cdot |f|^{\,\text{|n|}}$
with a floating point
precision given by the optional parameter |P| or by the current setting of
|\xintDigits|.\etype{\Ff\Ff}

In the current, preliminary, version, no attempt has been made to try to
guarantee to the final result the precision |P|. Rather, |P| is used for all
intermediate floating point evaluations. So
rounding errors will make some of the last printed digits invalid. The
operations done are first the evaluation of |f^A| using \csa{xintFloatPow}, then
each successive power is obtained from this first one by multiplication by |f|
using \csa{xintFloatMul}, then again with \csa{xintFloatMul} this is multiplied
with |\coeff{n}|, and the sum is done adding one term at a time with
\csa{xintFloatAdd}. To sum up, this is just the naive transformation of
\csa{xintFxPtPowerSeries} from fixed point to floating point.

\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}%

\everb+@
\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}%
\xintFloatPowerSeries [8]{1}{30}{\coefflog}{-1/2[0]}
+

%
\leftedline{\dtt{\xintFloatPowerSeries [8]{1}{30}{\coefflog}{-1/2[0]}}}

\subsection{\csbh{xintFloatPowerSeriesX}}\label{xintFloatPowerSeriesX}

%{\small\hspace*{\parindent}New with |1.08a|.\par}

\noindent\csa{xintFloatPowerSeriesX}|[P]{A}{B}{\coeff}{f}|%
\ntype{{\upshape[\numx]}\numx\numx}
is like
\csa{xintFloatPowerSeries} with the difference that |f| is
expanded once\etype{\Ff\Ff}
and for all at the start of the computation, thus allowing
efficient chaining of such series evaluations.
\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}%

\everb+@
\def\coeffexp #1{1/\xintFac {#1}[0]}% 1/n! (exact, not float)
\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}%
\xintFloatPowerSeriesX [8]{0}{30}{\coeffexp}
    {\xintFloatPowerSeries [8]{1}{30}{\coefflog}{-1/2[0]}}
+

%
\leftedline{\dtt{\xintFloatPowerSeriesX [8]{0}{30}{\coeffexp}
    {\xintFloatPowerSeries [8]{1}{30}{\coefflog}{-1/2[0]}}}}

\subsection{Computing \texorpdfstring{$\log 2$}{log(2)} and \texorpdfstring{$\pi$}{pi}}\label{ssec:Machin}

In this final section, the use of \csbxint{FxPtPowerSeries} (and
\csbxint{PowerSeries}) will be
illustrated on the (expandable... why make things simple when it is so easy to
make them difficult!) computations of the first digits of the decimal expansion
of the familiar constants $\log 2$ and $\pi$.

Let us start with $\log 2$. We will get it from this formula (which is
left as an exercise): %
%
\leftedline{\dtt{log(2)=-2\,log(1-13/256)-%
 5\,log(1-1/9)}}
%
The number of terms to be kept in the log series, for a desired
precision of |10^{-D}| was roughly estimated without much theoretical
analysis. Computing exactly the partial sums with \csa{xintPowerSeries}
and then printing the truncated values, from |D=0| up to |D=100| showed
that it worked in terms of quality of the approximation. Because of
possible strings of zeroes or nines in the exact decimal expansion (in
the present case of $\log 2$, strings of zeroes around the fourtieth and
the sixtieth decimals), this
does not mean though that all digits printed were always exact. In
the end one always end up having to compute at some higher level of
desired precision to validate the earlier result.

Then we tried with \csa{xintFxPtPowerSeries}: this is worthwile only for
|D|'s at least 50, as the exact evaluations are faster (with these
short-length |f|'s) for a lower
number of digits. And as expected the degradation in the quality of
approximation was in this range of the order of two or three digits.
This meant roughly that the 3+1=4 ending digits were wrong. Again, we ended
up having to compute with five more digits and compare with the earlier
value to validate it. We use truncation rather than rounding because our
goal is not to obtain the correct rounded decimal expansion but the
correct exact truncated one.

% 693147180559945309417232121458176568075500134360255254120680009493

\begin{everbatim*}
\def\coefflog #1{1/#1[0]}% 1/n
\def\xa {13/256[0]}%  we will compute log(1-13/256)
\def\xb {1/9[0]}%     we will compute log(1-1/9)
\def\LogTwo #1%
%  get log(2)=-2log(1-13/256)- 5log(1-1/9)
{% we want to use \printnumber, hence need something expanding in two steps
 % only, so we use here the \romannumeral0 method
  \romannumeral0\expandafter\LogTwoDoIt \expandafter
    % Nb Terms for 1/9:
  {\the\numexpr #1*150/143\expandafter}\expandafter
    % Nb Terms for 13/256:
  {\the\numexpr #1*100/129\expandafter}\expandafter
    % We print #1 digits, but we know the ending ones are garbage
  {\the\numexpr #1\relax}% allows #1 to be a count register
}%
\def\LogTwoDoIt #1#2#3%
%  #1=nb of terms for 1/9, #2=nb of terms for 13/256,
{% #3=nb of digits for computations, also used for printing
 \xinttrunc {#3} % lowercase form to stop the \romannumeral0 expansion!
 {\xintAdd
  {\xintMul {2}{\xintFxPtPowerSeries {1}{#2}{\coefflog}{\xa}{#3}}}
  {\xintMul {5}{\xintFxPtPowerSeries {1}{#1}{\coefflog}{\xb}{#3}}}%
 }%
}%
\noindent $\log 2 \approx \LogTwo {60}\dots$\endgraf
\noindent\phantom{$\log 2$}${}\approx{}$\printnumber{\LogTwo {65}}\dots\endgraf
\noindent\phantom{$\log 2$}${}\approx{}$\printnumber{\LogTwo {70}}\dots\endgraf
\end{everbatim*}

Here is the code doing an exact evaluation of the partial sums. We have
added a |+1| to the number of digits for estimating the number of terms
to keep from the log series: we experimented that this gets exactly the
first |D| digits, for all values from |D=0| to |D=100|, except in one
case (|D=40|) where the last digit is wrong. For values of |D|
higher than |100| it is more efficient to use the code using
\csa{xintFxPtPowerSeries}.
\everb|@
\def\LogTwo #1% get log(2)=-2log(1-13/256)- 5log(1-1/9)
{%
    \romannumeral0\expandafter\LogTwoDoIt \expandafter
    {\the\numexpr (#1+1)*150/143\expandafter}\expandafter
    {\the\numexpr (#1+1)*100/129\expandafter}\expandafter
    {\the\numexpr #1\relax}%
}%
\def\LogTwoDoIt #1#2#3%
{%   #3=nb of digits for truncating an EXACT partial sum
  \xinttrunc {#3}
    {\xintAdd
      {\xintMul {2}{\xintPowerSeries {1}{#2}{\coefflog}{\xa}}}
      {\xintMul {5}{\xintPowerSeries {1}{#1}{\coefflog}{\xb}}}%
    }%
}%
|

Let us turn now to Pi, computed with the Machin formula. Again the numbers of
terms to keep in the two |arctg| series were roughly estimated, and some
experimentations showed that removing the last three digits was enough (at least
for |D=0-100| range). And the algorithm does print the correct digits when used
with |D=1000| (to be convinced of that one needs to run it for |D=1000| and
again, say for |D=1010|.) A theoretical analysis could help confirm that this
algorithm always gets better than |10^{-D}| precision, but again, strings of
zeroes or nines encountered in the decimal expansion may falsify the ending
digits, nines may be zeroes (and the last non-nine one should be increased) and
zeroes may be nine (and the last non-zero one should be decreased).

\hypertarget{MachinCode}{}
\begin{everbatim*}
\def\coeffarctg #1{\the\numexpr\ifodd#1 -1\else1\fi\relax/%
                                       \the\numexpr 2*#1+1\relax [0]}%
%\def\coeffarctg #1{\romannumeral0\xintmon{#1}/\the\numexpr 2*#1+1\relax }%
\def\xa {1/25[0]}%      1/5^2, the [0] for faster parsing
\def\xb {1/57121[0]}% 1/239^2, the [0] for faster parsing
\def\Machin #1{% #1 may be a count register, \Machin {\mycount} is allowed
    \romannumeral0\expandafter\MachinA \expandafter
     % number of terms for arctg(1/5):
    {\the\numexpr (#1+3)*5/7\expandafter}\expandafter
     % number of terms for arctg(1/239):
    {\the\numexpr (#1+3)*10/45\expandafter}\expandafter
     % do the computations with 3 additional digits:
    {\the\numexpr #1+3\expandafter}\expandafter
     % allow #1 to be a count register:
    {\the\numexpr #1\relax }}%
\def\MachinA #1#2#3#4%
{\xinttrunc {#4}
 {\xintSub
  {\xintMul {16/5}{\xintFxPtPowerSeries {0}{#1}{\coeffarctg}{\xa}{#3}}}
  {\xintMul{4/239}{\xintFxPtPowerSeries {0}{#2}{\coeffarctg}{\xb}{#3}}}%
 }}%
\begin{framed}
  \[ \pi = \Machin {60}\dots \]
\end{framed}
\end{everbatim*}

Here is a variant|\MachinBis|,
which evaluates the partial sums \emph{exactly} using
\csa{xintPowerSeries}, before their final truncation. No need for a
``|+3|'' then.
\begin{everbatim*}
\def\MachinBis #1{% #1 may be a count register,
% the final result will be truncated to #1 digits post decimal point
    \romannumeral0\expandafter\MachinBisA \expandafter
     % number of terms for arctg(1/5):
    {\the\numexpr #1*5/7\expandafter}\expandafter
     % number of terms for arctg(1/239):
    {\the\numexpr #1*10/45\expandafter}\expandafter
      % allow #1 to be a count register:
    {\the\numexpr #1\relax }}%
\def\MachinBisA #1#2#3%
{\xinttrunc {#3} %
 {\xintSub
   {\xintMul {16/5}{\xintPowerSeries {0}{#1}{\coeffarctg}{\xa}}}
   {\xintMul{4/239}{\xintPowerSeries {0}{#2}{\coeffarctg}{\xb}}}%
}}%
\end{everbatim*}

Let us use this variant for a loop showing the build-up of digits:
\begin{everbatim*}
\begin{multicols}{2}
  \cnta 0 % previously declared \count register
  \loop \noindent
        \centeredline{\dtt{\MachinBis{\cnta}}}%
  \ifnum\cnta < 30
  \advance\cnta 1 \repeat
\end{multicols}
\end{everbatim*}

\hypertarget{Machin1000}{}
%
You want more digits and have some time? compile this copy of the
\hyperlink{MachinCode}{|\Machin|} with |etex| (or |pdftex|):
%
\everb|@
% Compile with e-TeX extensions enabled (etex, pdftex, ...)
\input xintfrac.sty
\input xintseries.sty
% pi = 16 Arctg(1/5) - 4 Arctg(1/239) (John Machin's formula)
\def\coeffarctg #1{\the\numexpr\ifodd#1 -1\else1\fi\relax/%
                                       \the\numexpr 2*#1+1\relax [0]}%
\def\xa {1/25[0]}%
\def\xb {1/57121[0]}%
\def\Machin #1{%
    \romannumeral0\expandafter\MachinA \expandafter
    {\the\numexpr (#1+3)*5/7\expandafter}\expandafter
    {\the\numexpr (#1+3)*10/45\expandafter}\expandafter
    {\the\numexpr #1+3\expandafter}\expandafter
    {\the\numexpr #1\relax }}%
\def\MachinA #1#2#3#4%
{\xinttrunc {#4}
 {\xintSub
  {\xintMul {16/5}{\xintFxPtPowerSeries {0}{#1}{\coeffarctg}{\xa}{#3}}}
  {\xintMul {4/239}{\xintFxPtPowerSeries {0}{#2}{\coeffarctg}{\xb}{#3}}}%
}}%
\pdfresettimer
\oodef\Z {\Machin {1000}}
\odef\W {\the\pdfelapsedtime}
\message{\Z}
\message{computed in \xintRound {2}{\W/65536} seconds.}
\bye 
|

This will log the first 1000 digits of $\pi$ after the decimal point. On my
laptop (a 2012 model) this took about $16$ seconds last time I tried.%
%
\footnote{With \texttt{1.09i} and earlier \xintname releases, this used
  to be \dtt{42} seconds; the \texttt{1.09j} division is much faster
  with small denominators as occurs here with \dtt{\char92xa=1/25}, and
  I believe this to be the main explanation for the speed gain.}
%
As mentioned in the
introduction, the file \href{http://www.ctan.org/pkg/pi}{pi.tex} by \textsc{D.
  Roegel} shows that orders of magnitude faster computations are possible within
\TeX{}, but recall our constraints of complete expandability and be merciful,
please.

\textbf{Why truncating rather than rounding?} One of our main competitors
on the market of scientific computing, a canadian product (not
encumbered with expandability constraints, and having barely ever heard
of \TeX{} ;-), prints numbers rounded in the last digit. Why didn't we
follow suit in the macros \csa{xintFxPtPowerSeries} and
\csa{xintFxPtPowerSeriesX}? To round at |D| digits, and excluding a
rewrite or cloning of the division algorithm which anyhow would add to
it some overhead in its final steps, \xintfracname needs to truncate at
|D+1|, then round. And rounding loses information! So, with more time
spent, we obtain a worst result than the one truncated at |D+1| (one
could imagine that additions and so on, done with only |D| digits, cost
less; true, but this is a negligeable effect per summand compared to the
additional cost for this term of having been truncated at |D+1| then
rounded). Rounding is the way to go when setting up algorithms to
evaluate functions destined to be composed one after the other: exact
algebraic operations with many summands and an |f| variable which is a
fraction are costly and create an even bigger fraction; replacing |f|
with a reasonable rounding, and rounding the result, is necessary to
allow arbitrary chaining.

But, for the
computation of a single constant, we are really interested in the exact
decimal expansion, so we truncate and compute more terms until the
earlier result gets validated. Finally if we do want the rounding we can
always do it on a value computed with |D+1| truncation.

%  \clearpage

\section{Commands of the \xintcfracname package}
\label{sec:cfrac}

\localtableofcontents

This package was first included in release |1.04| (|2013/04/25|) of the
\xintname bundle. It was kept almost unchanged until |1.09m| of |2014/02/26|
which brings some new macros: \csbxint{FtoC}, \csbxint{CtoF}, \csbxint{CtoCv},
dealing with sequences of braced partial quotients rather than comma separated
ones, \csbxint{FGtoC} which is to produce ``guaranteed'' coefficients of some
real number known approximately, and \csbxint{GGCFrac} for displaying arbitrary
material as a continued fraction; also, some changes to existing macros:
\csbxint{FtoCs} and \csbxint{CntoCs} insert spaces after the commas,
\csbxint{CstoF} and \csbxint{CstoCv} authorize spaces in the input also before
the commas.

This section contains:
\begin{enumerate}
\item an \hyperref[ssec:cfracoverview]{overview} of the package functionalities,
\item a description of each one of the package macros,
\item further illustration of their use via the study of the
  \hyperref[ssec:e-convergents]{convergents of $e$}.
\end{enumerate}

\subsection{Package overview}\label{ssec:cfracoverview}

The package computes partial quotients and convergents of a fraction, or
conversely start from coefficients and obtain the corresponding fraction; three
macros \csbxint {CFrac}, \csbxint {GCFrac} and \csbxint {GGCFrac} are
for typesetting (the first two assume that the coefficients are numeric
quantities acceptable by the \xintfracname \csbxint{Frac} macro, the
last one will display arbitrary material), the others
can be nested (if applicable) or see their outputs further processed by other
macros from the \xintname bundle, particularly the macros of \xinttoolsname
dealing with sequences of braced items or comma separated lists.

A \emph{simple} continued fraction has coefficients
|[c0,c1,...,cN]| (usually called partial quotients, but I
dislike this entrenched terminology), where |c0| is a positive or
negative integer and the others are positive integers.

Typesetting is usually done via the |amsmath| macro |\cfrac|:
\begin{everbatim*}
\[ c_0 + \cfrac{1}{c_1+\cfrac1{c_2+\cfrac1{c_3+\cfrac1{\ddots}}}}\]
\end{everbatim*}

Here is a concrete example:
\begin{everbatim*}
\[ \xintFrac {208341/66317}=\xintCFrac {208341/66317}\]%
\end{everbatim*}
But it is the command \csbxint{CFrac} which did all the work of \emph{computing}
the continued fraction \emph{and} using |\cfrac| from |amsmath| to typeset
it.

A \emph{generalized} continued fraction has the same structure but the
numerators are not restricted to be $1$, and numbers used in the continued
fraction may be arbitrary, also fractions, irrationals, complex,
indeterminates.%
%
\footnote{\xintcfracname may be used with indeterminates,
  for basic conversions from one inline format to another, but not for
  actual computations. See \csbxint{GGCFrac}.}
%
The \emph{centered} continued fraction is an
example:
\begin{everbatim*}
\[ \xintFrac {915286/188421}=\xintGCFrac {5+-1/7+1/39+-1/53+-1/13}
  =\xintCFrac {915286/188421}\]
\end{everbatim*}

The command \csbxint{GCFrac}, contrarily to
\csbxint{CFrac}, does not compute anything, it just typesets starting from a
generalized continued fraction in inline format, which in this example
was input literally.  We also used \csa{xintCFrac}
for comparison of the two types of continued fractions.

To let \TeX{} compute the centered continued fraction of |f| there is
\csbxint{FtoCC}:
\begin{everbatim*}
\[\xintFrac {915286/188421}\to\xintFtoCC {915286/188421}\]
\end{everbatim*}
The package macros are expandable and may be nested (naturally \csa{xintCFrac}
and \csa{xintGCFrac} must be at the top level, as they deal with typesetting).
\begin{everbatim*}
\[\xintGCFrac {\xintFtoCC{915286/188421}}\]
\end{everbatim*}

The `inline' format expected on input by \csbxint{GCFrac} is
%
\leftedline{$a_0+b_0/a_1+b_1/a_2+b_2/a_3+\cdots+b_{n-2}/a_{n-1}+b_{n-1}/a_n$}
%
Fractions among the coefficients are allowed but they must be enclosed
within braces. Signed integers may be left without braces (but the |+|
signs are mandatory). No spaces are allowed around the plus and fraction
symbols. The coefficients may themselves be macros, as long as these
macros are \fexpan dable.
\begin{everbatim*}
\[ \xintFrac{\xintGCtoF {1+-1/57+\xintPow {-3}{7}/\xintiQuo {132}{25}}}
    = \xintGCFrac {1+-1/57+\xintPow {-3}{7}/\xintiQuo {132}{25}}\]
\end{everbatim*}
To compute the actual fraction one has \csbxint{GCtoF}:
\begin{everbatim*}
\[\xintFrac{\xintGCtoF {1+-1/57+\xintPow {-3}{7}/\xintiQuo {132}{25}}}\]
\end{everbatim*}
For non-numeric input  there is \csbxint{GGCFrac}.
\begin{everbatim*}
\[\xintGGCFrac {a_0+b_0/a_1+b_1/a_2+b_2/\ddots+\ddots/a_{n-1}+b_{n-1}/a_n}\]
\end{everbatim*}
For regular continued fractions, there is a simpler comma separated format:
\begin{everbatim*}
\[-7,6,19,1,33\to\xintFrac{\xintCstoF{-7,6,19,1,33}}=\xintCFrac{\xintCstoF{-7,6,19,1,33}}\]
\end{everbatim*}
The command \csbxint{FtoCs} produces from a fraction |f| the comma separated
list of its coefficients.
\begin{everbatim*}
\[\xintFrac{1084483/398959}=[\xintFtoCs{1084483/398959}]\]
\end{everbatim*}
If one prefers other separators, one can use the two arguments macros
\csbxint{FtoCx} whose first argument is the separator (which may consist of more
than one token) which is to be used.
\begin{everbatim*}
\[\xintFrac{2721/1001}=\xintFtoCx {+1/(}{2721/1001})\cdots)\]
\end{everbatim*}
This allows under Plain  \TeX{} with |amstex| to obtain the same effect
as with \LaTeX{}+|\amsmath|+\csbxint{CFrac}:
%
\leftedline{|$$\xintFwOver{2721/1001}=\xintFtoCx {+\cfrac1\\ }{2721/1001}\endcfrac$$|}

As a shortcut to \csa{xintFtoCx} with separator |1+/|, there is
\csbxint{FtoGC}:
\begin{everbatim*}
2721/1001=\xintFtoGC {2721/1001}
\end{everbatim*}
Let us compare in that case with the output of \csbxint{FtoCC}:
\begin{everbatim*}
2721/1001=\xintFtoCC {2721/1001}
\end{everbatim*}
To obtain the coefficients as a sequence of braced numbers, there is
\csbxint{FtoC} (this is a shortcut for |\xintFtoCx {}|). This list
(sequence) may then be manipulated using the various macros of \xinttoolsname
such as the non-expandable macro \csbxint{AssignArray} or the expandable
\csbxint{Apply} and \csbxint{ListWithSep}.

Conversely to go from such a sequence of braced coefficients to the
corresponding fraction there is \csbxint{CtoF}.

The `|\printnumber|' (\autoref{ssec:printnumber}) macro which we use in this
document to print long numbers can also be useful on long continued fractions.
%
\begin{everbatim*}
\printnumber{\xintFtoCC {35037018906350720204351049/244241737886197404558180}}
\end{everbatim*}
%
If we apply \csbxint{GCtoF} to this generalized continued fraction, we
discover that the original fraction was reducible:
%
\leftedline{|\xintGCtoF
  {143+1/2+...+-1/9}|\dtt{=\xintGCtoF{143+1/2+1/5+-1/4+-1/4+-1/4+-1/3+1/2+1/2+1/6+-1/22+1/2+1/10+-1/5+-1/11+-1/3+1/4+-1/2+1/2+1/4+-1/2+1/23+1/3+1/8+-1/6+-1/9}}}

\def\mymacro #1{$\xintFrac{#1}=[\xintFtoCs{#1}]$\vtop to 6pt{}}

\begingroup
\catcode`^\active
\def^#1^{\hbox{#1}}%

When a generalized continued fraction is built with integers, and
numerators are only |1|'s or |-1|'s, the produced fraction is
irreducible. And if we compute it again with the last sub-fraction
omitted we get another irreducible fraction related to the bigger one by
a Bezout identity. Doing this here we get:
%
\leftedline{|\xintGCtoF {143+1/2+...+-1/6}|\dtt{=\xintGCtoF{143+1/2+1/5+-1/4+-1/4+-1/4+-1/3+1/2+1/2+1/6+-1/22+1/2+1/10+-1/5+-1/11+-1/3+1/4+-1/2+1/2+1/4+-1/2+1/23+1/3+1/8+-1/6}}}
and indeed:
\[\begin{vmatrix}
    ^2897319801297630107^ & ^328124887710626729^\\
      ^20197107104701740^ & ^2287346221788023^
   \end{vmatrix} = \mbox{\dtt{\xintiSub {\xintiMul {2897319801297630107}{2287346221788023}}{\xintiMul{20197107104701740}{328124887710626729}}}}\]

\endgroup

The various fractions obtained from the truncation of a continued fraction to
its initial terms are called the convergents. The commands of \xintcfracname
such as \csbxint{FtoCv}, \csbxint{FtoCCv}, and others which compute such
convergents, return them as a list of braced items, with no separator (as does
\csbxint {FtoC} for the partial quotients). Here is an example:

\begin{everbatim*}
\[\xintFrac{915286/188421}\to 
  \xintListWithSep{,}{\xintApply\xintFrac{\xintFtoCv{915286/188421}}}\]
\end{everbatim*}
\begin{everbatim*}
\[\xintFrac{915286/188421}\to 
  \xintListWithSep{,}{\xintApply\xintFrac{\xintFtoCCv{915286/188421}}}\]
\end{everbatim*}
%
We thus see that the `centered convergents' obtained with \csbxint{FtoCCv} are
among the fuller list of convergents as returned by \csbxint{FtoCv}.

Here is a more complicated use of \csa{xintApply}
and \csa{xintListWithSep}. We first define a macro which will be applied to each
convergent:%
%
\leftedline{|\newcommand{\mymacro}[1]{$\xintFrac{#1}=[\xintFtoCs{#1}]$\vtop to 6pt{}}|}
%
Next, we use the following code:
%
\leftedline{|$\xintFrac{49171/18089}\to{}$|}
%
\leftedline{|\xintListWithSep {,
  }{\xintApply{\mymacro}{\xintFtoCv{49171/18089}}}|}
It produces:\par
\noindent$ \xintFrac{49171/18089}\to {}$\xintListWithSep {,
  }{\xintApply{\mymacro}{\xintFtoCv{49171/18089}}}.

The macro \csbxint{CntoF} allows to specify the coefficients as a function given
by a one-parameter macro. The produced values do not have to be integers.
\begin{everbatim*}
\def\cn #1{\xintiiPow {2}{#1}}% 2^n
  \[\xintFrac{\xintCntoF {6}{\cn}}=\xintCFrac [l]{\xintCntoF {6}{\cn}}\]
\end{everbatim*}

% Mardi 14 octobre 2014 à 23:10:16
% color stack overflow dans cette zone avec dvipdfmx commit b988dcb45
% (pas avec pdflatex)

Notice  the use of the optional argument |[l]| to \csa{xintCFrac}. Other
possibilities are |[r]| and (default) |[c]|.
\begin{everbatim*}
\def\cn #1{\xintPow {2}{-#1}}%
  \[\xintFrac{\xintCntoF {6}{\cn}}=\xintGCFrac [r]{\xintCntoGC {6}{\cn}}=
      [\xintFtoCs {\xintCntoF {6}{\cn}}]\]
\end{everbatim*}
We used \csbxint{CntoGC} as we wanted to display also the continued fraction and
not only the fraction returned by \csa{xintCntoF}.

There are also \csbxint{GCntoF} and \csbxint{GCntoGC} which allow the same for
generalized fractions. An initial portion of a generalized continued
fraction for $\pi$ is obtained like this
\begin{everbatim*}
\def\an #1{\the\numexpr 2*#1+1\relax }%
\def\bn #1{\the\numexpr (#1+1)*(#1+1)\relax }%
\[\xintFrac{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}} =
        \cfrac{4}{\xintGCFrac{\xintGCntoGC {5}{\an}{\bn}}} =
                  \xintTrunc {10}{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}}\dots\]
\end{everbatim*}

We see that the quality of approximation is not fantastic compared to the simple
continued fraction of $\pi$ with about as many terms:
\begin{everbatim*}
\[\xintFrac{\xintCstoF{3,7,15,1,292,1,1}}=
             \xintGCFrac{3+1/7+1/15+1/1+1/292+1/1+1/1}=
                       \xintTrunc{10}{\xintCstoF{3,7,15,1,292,1,1}}\dots\]
\end{everbatim*}

When studying the continued fraction of some real number, there is always
some doubt about how many terms are valid, when computed starting from some
approximation. If $f\leqslant x\leqslant g$ and $f, g$ both have the
same first $K$ partial quotients, then $x$ also has the same first $K$ quotients
and convergents. The macro \csbxint{FGtoC} outputs as a sequence of braced items
the common partial quotients of its two arguments. We can thus use it to produce
a sure list of valid convergents of $\pi$ for example, starting from some proven
lower and upper bound:
\begin{everbatim*}
$$\pi\to [\xintListWithSep{,}
         {\xintFGtoC {3.14159265358979323}{3.14159265358979324}}, \dots]$$
\noindent$\pi\to\xintListWithSep{,\allowbreak\;}
   {\xintApply{\xintFrac}
   {\xintCtoCv{\xintFGtoC {3.14159265358979323}{3.14159265358979324}}}}, \dots$
\end{everbatim*}


\subsection{\csbh{xintCFrac}}\label{xintCFrac}

\csa{xintCFrac}|{f}|\ntype{\Ff} is a math-mode only, \LaTeX{} with |amsmath|
only, macro which first computes then displays with the help of |\cfrac| the
simple continued fraction corresponding to the given fraction. It admits an
optional argument which may be |[l]|, |[r]| or (the default) |[c]| to specify
the location of the one's in the numerators of the sub-fractions. Each
coefficient is typeset using the \csbxint{Frac} macro from the \xintfracname
package. This macro is \fexpan dable in the sense that it prepares expandably
the whole expression with the multiple |\cfrac|'s, but it is not completely
expandable naturally as |\cfrac| isn't.

\subsection{\csbh{xintGCFrac}}\label{xintGCFrac}

\csa{xintGCFrac}|{a+b/c+d/e+f/g+h/...+x/y}|\ntype{f} uses similarly |\cfrac|
to prepare the typesetting with the |amsmath| |\cfrac| (\LaTeX{}) of a
generalized continued fraction given in inline format (or as macro which
will \fexpan d to it). It admits the
same optional argument as \csa{xintCFrac}. Plain \TeX{} with |amstex|
users, see \csbxint{GCtoGCx}.
\begin{everbatim*}
\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}\]
\end{everbatim*}
This is mostly a typesetting macro, although it does provoke the
expansion of the coefficients. See \csbxint{GCtoF} if you are impatient
to see this specific fraction computed.

It admits an optional argument within square brackets which may be
either |[l]|, |[c]| or |[r]|. Default is |[c]| (numerators are centered).

Numerators and denominators are made arguments to the \csbxint{Frac}
macro. This allows them to be themselves fractions or anything \fexpan
dable giving numbers or fractions, but also means however that they can
not be arbitrary material, they can not contain color changing commands
for example. One of the reasons is that \csa{xintGCFrac} tries to
determine the signs of the numerators and chooses accordingly to use
$+$ or $-$.

\subsection{\csbh{xintGGCFrac}}\label{xintGGCFrac}

\csa{xintGGCFrac}|{a+b/c+d/e+f/g+h/...+x/y}|\ntype{f} is a clone of
\csbxint{GCFrac}, hence again \LaTeX{} specific with package
|amsmath|.\NewWith {1.09m}
It does not assume the coefficients to be numbers as understood by
\xintfracname. The macro can be used for displaying arbitrary content as
a continued fraction with |\cfrac|, using only plus signs though. Note
though that it will first \fexpan d its argument, which may be thus be
one of the \xintcfracname macros producing a (general) continued
fraction in inline format, see \csbxint{FtoCx} for an example. If this
expansion is not wished, it is enough to start the argument with a
space.
\begin{everbatim*}
\[\xintGGCFrac {1+q/1+q^2/1+q^3/1+q^4/1+q^5/\ddots}\]
\end{everbatim*}

\subsection{\csbh{xintGCtoGCx}}\label{xintGCtoGCx}
%{\small New with release |1.05|.\par}

\csa{xintGCtoGCx}|{sepa}{sepb}{a+b/c+d/e+f/...+x/y}|\etype{nnf} returns the list
of the coefficients of the generalized continued fraction of |f|, each one
within a pair of braces, and separated with the help of |sepa| and |sepb|. Thus
%
\leftedline{|\xintGCtoGCx :;{1+2/3+4/5+6/7}| gives \xintGCtoGCx
  :;{1+2/3+4/5+6/7}}
%
The following can be used byt Plain \TeX{}+|amstex| users to obtain an
output similar as the ones produced by \csbxint{GCFrac} and
\csbxint{GGCFrac}:\par
\everb|@
$$\xintGCtoGCx {+\cfrac}{\\}{a+b/...}\endcfrac$$
$$\xintGCtoGCx {+\cfrac\xintFwOver}{\\\xintFwOver}{a+b/...}\endcfrac$$
|

\subsection{\csbh{xintFtoC}}\label{xintFtoC}

\csa{xintFtoC}|{f}|\etype{\Ff} computes the
coefficients of the simple continued fraction of |f| and returns them as a list
(sequence) of braced items.\NewWith {1.09m}

\begin{everbatim*}
\oodef\test{\xintFtoC{-5262046/89233}}\texttt{\meaning\test}
\end{everbatim*}

\subsection{\csbh{xintFtoCs}}\label{xintFtoCs}

\csa{xintFtoCs}|{f}|\etype{\Ff} returns the comma separated list of the
coefficients of the simple continued fraction of |f|. Notice that starting with
|1.09m| a space follows each comma (mainly for usage in text mode, as in math
mode spaces are produced in the typeset output by \TeX{} itself).
\begin{everbatim*}
\[ \xintSignedFrac{-5262046/89233} \to [\xintFtoCs{-5262046/89233}]\]
\end{everbatim*}

\subsection{\csbh{xintFtoCx}}\label{xintFtoCx}

\csa{xintFtoCx}|{sep}{f}|\etype{n\Ff} returns the list of the
coefficients of the simple continued fraction of |f| separated with the
help of |sep|, which may be anything (and is kept unexpanded). For
example, with Plain \TeX{} and |amstex|,
%
\leftedline{|$$\xintFtoCx {+\cfrac1\\ }{-5262046/89233}\endcfrac$$|}
%
will display the continued fraction using
|\cfrac|. Each coefficient is inside a brace pair \hbox{|{ }|}, allowing
a macro to end the separator and fetch it as argument,
for example, again with Plain \TeX{} and |amstex|:
\everb|@
  \def\highlight #1{\ifnum #1>200 \textcolor{red}{#1}\else #1\fi}
  $$\xintFtoCx {+\cfrac1\\ \highlight}{104348/33215}\endcfrac$$
|

Due to the different and extremely cumbersome syntax of |\cfrac| under
\LaTeX{} it proves a bit tortuous to obtain there the same effect.
Actually, it is partly for this purpose that |1.09m| added \csbxint
{GGCFrac}. We thus use \csa{xintFtoCx} with a suitable separator, and\;
then the whole thing as argument to \csbxint{GGCFrac}:
\begin{everbatim*}
\def\highlight #1{\ifnum #1>200 \fcolorbox{blue}{white}{\boldmath\color{red}$#1$}%
    \else #1\fi}
\[\xintGGCFrac {\xintFtoCx {+1/\highlight}{208341/66317}}\]
\end{everbatim*}

\subsection{\csbh{xintFtoGC}}\label{xintFtoGC}

\csa{xintFtoGC}|{f}|\etype{\Ff} does the same as \csa{xintFtoCx}|{+1/}{f}|. Its
output may thus be used in the package macros expecting such an `inline
format'.
% This continued fraction is a \emph{simple} one, not a
% \emph{generalized} one, but as it is produced in the format used for
% user input of generalized continued fractions, the macro was called
% \csa{xintFtoGC} rather than \csa{xintFtoC} for example.
%
\begin{everbatim*}
566827/208524=\xintFtoGC {566827/208524}
\end{everbatim*}

\subsection{\csbh{xintFGtoC}}\label{xintFGtoC}

\csa{xintFGtoC}|{f}{g}|\etype{\Ff\Ff} computes the common initial coefficients
to
two given fractions |f| and |g|. Notice\NewWith {1.09m} that any real number |f<x<g| or |f>x>g|
will then necessarily share with |f| and |g| these common initial coefficients
for its regular continued fraction. The coefficients are output as a sequence of
braced numbers. This list can then be manipulated via macros from
\xinttoolsname, or other macros of \xintcfracname.

\begin{everbatim*}
\oodef\test{\xintFGtoC{-5262046/89233}{-5314647/90125}}\texttt{\meaning\test}
\end{everbatim*}
\begin{everbatim*}
\oodef\test{\xintFGtoC{3.141592653}{3.141592654}}\texttt{\meaning\test}
\end{everbatim*}
\begin{everbatim*}
\oodef\test{\xintFGtoC{3.1415926535897932384}{3.1415926535897932385}}\meaning\test
\end{everbatim*}
\begin{everbatim*}
\xintRound {30}{\xintCstoF{\xintListWithSep{,}{\test}}}
\end{everbatim*}
\begin{everbatim*}
\xintRound {30}{\xintCtoF{\test}}
\end{everbatim*}
\begin{everbatim*}
\oodef\test{\xintFGtoC{1.41421356237309}{1.4142135623731}}\meaning\test
\end{everbatim*}

\subsection{\csbh{xintFtoCC}}\label{xintFtoCC}

\csa{xintFtoCC}|{f}|\etype{\Ff} returns the `centered' continued fraction of
|f|, in `inline format'. %
\begin{everbatim*}
566827/208524=\xintFtoCC {566827/208524}
\end{everbatim*}
\begin{everbatim*}
\[\xintFrac{566827/208524} = \xintGCFrac{\xintFtoCC{566827/208524}}\]
\end{everbatim*}

\subsection{\csbh{xintCstoF}}\label{xintCstoF}

\csa{xintCstoF}|{a,b,c,d,...,z}|\etype{f} computes the fraction corresponding to
the coefficients, which may be fractions or even macros expanding to such
fractions. The final fraction may then be highly reducible. Starting with
release |1.09m| spaces before commas are allowed and trimmed automatically
(spaces after commas were already silently handled in earlier releases).
\begin{everbatim*}
\[\xintGCFrac {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}=
  \xintSignedFrac{\xintCstoF {-1,3,-5,7,-9,11,-13}}=\xintSignedFrac{\xintGCtoF
    {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}}\]
\end{everbatim*}
\begin{everbatim*}
\[\xintGCFrac{{1/2}+1/{1/3}+1/{1/4}+1/{1/5}}=\xintFrac{\xintCstoF {1/2,1/3,1/4,1/5}}\]
\end{everbatim*}
%
A generalized continued fraction may produce a reducible fraction
(\csa{xintCstoF} tries its best not to accumulate in a silly way superfluous
factors but will not do simplifications which would be obvious to a human, like
simplification by 3 in the result above).

\subsection{\csbh{xintCtoF}}\label{xintCtoF}

\csa{xintCtoF}|{{a}{b}{c}...{z}}|\etype{f} computes the fraction corresponding
to the coefficients, which may be fractions or even macros.\NewWith {1.09m}
\begin{everbatim*}
\xintCtoF {\xintApply {\xintiiPow 3}{\xintSeq {1}{5}}}
\end{everbatim*}
\begin{everbatim*}
\[ \xintFrac{14946960/4805083}=\xintCFrac {14946960/4805083}\]
\end{everbatim*}
In the example above the power of $3$ was already pre-computed via the expansion
done by |\xintApply|, but if we try with |\xintApply { \xintiiPow 3}| where the
space will stop this expansion, we can check that |\xintCtoF| will itself
provoke the needed coefficient expansion.% ok

\subsection{\csbh{xintGCtoF}}\label{xintGCtoF}

\csa{xintGCtoF}|{a+b/c+d/e+f/g+......+v/w+x/y}|\etype{f} computes the fraction
defined by the inline generalized continued fraction. Coefficients may be
fractions but must then be put within braces. They can be macros. The plus signs
are mandatory. 
\begin{everbatim*}
\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}} =
\xintFrac{\xintGCtoF {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}} =
\xintFrac{\xintIrr{\xintGCtoF
                  {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}}}\]
\end{everbatim*}

\begin{everbatim*}
\[ \xintGCFrac{{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}} =
   \xintFrac{\xintGCtoF {{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}}} \]
\end{everbatim*}

The macro tries its best not to accumulate superfluous factor in the
denominators, but doesn't reduce the fraction to irreducible form before
returning it and does not do simplifications which would be obvious to a human.

\subsection{\csbh{xintCstoCv}}\label{xintCstoCv}

\csa{xintCstoCv}|{a,b,c,d,...,z}|\etype{f} returns the sequence of the
corresponding convergents, each one within braces.

It is allowed to use fractions as coefficients (the computed
convergents have then no reason to be the real convergents of the final
fraction). When the coefficients are integers, the convergents are irreducible
fractions, but otherwise it is not necessarily the case.
\begin{everbatim*}
\xintListWithSep:{\xintCstoCv{1,2,3,4,5,6}}
\end{everbatim*}
\begin{everbatim*}
\xintListWithSep:{\xintCstoCv{1,1/2,1/3,1/4,1/5,1/6}}
\end{everbatim*}
\begin{everbatim*}
\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintCstoCv {\xintPow
        {-.3}{-5},7.3/4.57,\xintCstoF{3/4,9,-1/3}}}}\]
\end{everbatim*}

\subsection{\csbh{xintCtoCv}}\label{xintCtoCv}

\csa{xintCtoCv}|{{a}{b}{c}...{z}}|\etype{f} returns the sequence of the
corresponding convergents, each one within braces.\NewWith {1.09m}
\begin{everbatim*}
\oodef\test{\xintCtoCv {11111111111}}\texttt{\meaning\test}
\end{everbatim*}

\subsection{\csbh{xintGCtoCv}}\label{xintGCtoCv}

\csa{xintGCtoCv}|{a+b/c+d/e+f/g+......+v/w+x/y}|\etype{f} returns the list of
the corresponding convergents. The coefficients may be fractions, but must then
be inside braces. Or they may be macros, too.

The convergents will in the general case be reducible. To put them into
irreducible form, one needs one more step, for example it can be done
with |\xintApply\xintIrr|.
\begin{everbatim*}
\[\xintListWithSep{,}{\xintApply\xintFrac
                {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}\]
\[\xintListWithSep{,}{\xintApply\xintFrac{\xintApply\xintIrr
                {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}}\]
\end{everbatim*}


\subsection{\csbh{xintFtoCv}}\label{xintFtoCv}

\csa{xintFtoCv}|{f}|\etype{\Ff} returns the list of the (braced) convergents of
|f|, with no separator. To be treated with \csbxint{AssignArray} or
\csbxint{ListWithSep}.
\begin{everbatim*}
\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCv{5211/3748}}}\]
\end{everbatim*}

\subsection{\csbh{xintFtoCCv}}\label{xintFtoCCv}

\csa{xintFtoCCv}|{f}|\etype{\Ff} returns the list of the (braced) centered
convergents of |f|, with no separator. To be treated with \csbxint{AssignArray}
or \csbxint{ListWithSep}.
\begin{everbatim*}
\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCCv{5211/3748}}}\]
\end{everbatim*}

\subsection{\csbh{xintCntoF}}\label{xintCntoF}


\csa{xintCntoF}|{N}{\macro}|\etype{\numx f} computes the fraction |f| having
coefficients |c(j)=\macro{j}| for |j=0,1,...,N|. The |N| parameter is given to a
|\numexpr|. The values of the coefficients, as returned by |\macro| do not have
to be positive, nor integers, and it is thus not necessarily the case that the
original |c(j)| are the true coefficients of the final |f|.
\begin{everbatim*}
\def\macro #1{\the\numexpr 1+#1*#1\relax} \xintCntoF {5}{\macro}
\end{everbatim*}

This example shows that the fraction is output with a trailing number in square
brackets (representing a power of ten), this is for consistency with what do
most macros of \xintfracname, and does not have to be always this annoying |[0]|
as the coefficients may for example be numbers in scientific notation. To avoid
these trailing square brackets, for example if the coefficients are known to be integers, there is always the possibility to filter the output via
\csbxint{PRaw}, or \csbxint{Irr} (the latter is overkill in the case of integer
coefficients, as the fraction is guaranteed to be irreducible then).

\subsection{\csbh{xintGCntoF}}\label{xintGCntoF}

\csa{xintGCntoF}|{N}{\macroA}{\macroB}|\etype{\numx ff} returns the fraction |f|
corresponding to the inline generalized continued fraction
|a0+b0/a1+b1/a2+....+b(N-1)/aN|, with |a(j)=\macroA{j}| and |b(j)=\macroB{j}|.
The |N| parameter is given to a |\numexpr|.
\begin{everbatim*}
\def\coeffA #1{\the\numexpr #1+4-3*((#1+2)/3)\relax }%
\def\coeffB #1{\the\numexpr \ifodd #1 -\fi 1\relax }% (-1)^n
\[\xintGCFrac{\xintGCntoGC {6}{\coeffA}{\coeffB}} = 
                                \xintFrac{\xintGCntoF {6}{\coeffA}{\coeffB}}\]
\end{everbatim*}
There is also \csbxint{GCntoGC} to get the `inline format' continued
fraction.

\subsection{\csbh{xintCntoCs}}\label{xintCntoCs}

\csa{xintCntoCs}|{N}{\macro}|\etype{\numx f} produces the comma separated list
of the corresponding coefficients, from |n=0| to |n=N|. The |N| is given to a
|\numexpr|. %
\begin{everbatim*}
\xintCntoCs {5}{\macro}
\end{everbatim*}
\begin{everbatim*}
\[ \xintFrac{\xintCntoF{5}{\macro}}=\xintCFrac{\xintCntoF {5}{\macro}}\]
\end{everbatim*}

\subsection{\csbh{xintCntoGC}}\label{xintCntoGC}

%
\csa{xintCntoGC}|{N}{\macro}|\etype{\numx f} evaluates the |c(j)=\macro{j}| from
|j=0| to |j=N| and returns a continued fraction written in inline format:
|{c(0)}+1/{c(1)}+1/...+1/{c(N)}|. The parameter |N| is given to a |\numexpr|.
The coefficients, after expansion, are, as shown, being enclosed in an added
pair of braces, they may thus be fractions.
\begin{everbatim*}
\def\macro #1{\the\numexpr\ifodd#1 -1-#1\else1+#1\fi\relax/\the\numexpr 1+#1*#1\relax} 
\oodef\x{\xintCntoGC {5}{\macro}}\meaning\x
\[\xintGCFrac{\xintCntoGC {5}{\macro}}\]
\end{everbatim*}

\subsection{\csbh{xintGCntoGC}}\label{xintGCntoGC}

\csa{xintGCntoGC}|{N}{\macroA}{\macroB}|\etype{\numx ff} evaluates the
coefficients and then returns the corresponding
|{a0}+{b0}/{a1}+{b1}/{a2}+...+{b(N-1)}/{aN}| inline generalized fraction. |N| is
givent to a |\numexpr|. The coefficients are enclosed into pairs
of braces, and may thus be fractions, the fraction slash will not be
confused in further processing by the continued fraction slashes.
%
\begin{everbatim*}
\def\an #1{\the\numexpr  #1*#1*#1+1\relax}%
\def\bn #1{\the\numexpr \ifodd#1 -\fi 1*(#1+1)\relax}%
$\xintGCntoGC {5}{\an}{\bn}=\xintGCFrac {\xintGCntoGC {5}{\an}{\bn}} =
\displaystyle\xintFrac {\xintGCntoF {5}{\an}{\bn}}$\par
\end{everbatim*}

\subsection{\csbh{xintCstoGC}}\label{xintCstoGC}

\csa{xintCstoGC}|{a,b,..,z}|\etype{f} transforms a comma separated list (or
something expanding to such a list) into an `inline format' continued fraction
|{a}+1/{b}+1/...+1/{z}|. The coefficients are just copied and put within braces,
without expansion. The output can then be used in \csbxint{GCFrac} for example.
\begin{everbatim*}
\[\xintGCFrac {\xintCstoGC {-1,1/2,-1/3,1/4,-1/5}}=\xintSignedFrac{\xintCstoF {-1,1/2,-1/3,1/4,-1/5}}\]
\end{everbatim*}
\subsection{\csbh{xintiCstoF}, \csbh{xintiGCtoF}, \csbh{xintiCstoCv}, \csbh{xintiGCtoCv}}\label{xintiCstoF}
\label{xintiGCtoF}
\label{xintiCstoCv}
\label{xintiGCtoCv}

Essentially\etype{f} the same as the corresponding macros without the
`i', but for integer-only input. Infinitesimally faster, mainly for
internal use by the package.

\subsection{\csbh{xintGCtoGC}}\label{xintGCtoGC}

\csa{xintGCtoGC}|{a+b/c+d/e+f/g+......+v/w+x/y}|\etype{f} expands (with the
usual meaning) each one of the coefficients and returns an inline continued
fraction of the same type, each expanded coefficient being enclosed within
braces.
%
\begin{everbatim*}
\oodef\x {\xintGCtoGC {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/%
                        \xintFac {6}+\xintCstoF {2,-7,-5}/16}} \meaning\x
\end{everbatim*}

To be honest I have forgotten for which purpose I wrote this macro in the first
place.

\subsection{Euler's number \texorpdfstring{$e$}{e}}\label{ssec:e-convergents}

Let us explore
the convergents of Euler's number $e$.
\smallskip The volume of computation is kept minimal by the following steps:
\begin{itemize}
\item a comma separated list of the first 36 coefficients is produced by
  \csbxint{CntoCs},
\item this is then given to \csbxint{iCstoCv} which produces the list of the
  convergents (there is also \csbxint{CstoCv}, but our
  coefficients being integers we used the infinitesimally
  faster \csbxint{iCstoCv}),
\item then the whole list was converted into a sequence of one-line paragraphs,
  each convergent becomes the argument to a  macro printing it
  together with its decimal expansion with 30 digits after the decimal point.
\item A count register |\cnta| was used to give a line count serving as a visual
  aid: we could also have done that in an expandable way, but well, let's relax
  from time to time\dots
\end{itemize}

\begin{everbatim*}
\def\cn #1{\the\numexpr\ifcase \numexpr #1+3-3*((#1+2)/3)\relax
                           1\or1\or2*(#1/3)\fi\relax }
% produces the pattern 1,1,2,1,1,4,1,1,6,1,1,8,... which are the
% coefficients of the simple continued fraction of e-1.
\cnta 0
\def\mymacro #1{\advance\cnta by 1
                \noindent
                \hbox to 3em {\hfil\small\dtt{\the\cnta.} }%
                $\xintTrunc {30}{\xintAdd {1[0]}{#1}}\dots=
                 \xintFrac{\xintAdd {1[0]}{#1}}$}%
\xintListWithSep{\vtop to 6pt{}\vbox to 12pt{}\par}
    {\xintApply\mymacro{\xintiCstoCv{\xintCntoCs {35}{\cn}}}}
\end{everbatim*}

% \def\testmacro #1{\xintTrunc {30}{\xintAdd {1[0]}{#1}}\xintAdd {1[0]}{#1}}
% \pdfresettimer
% \oodef\z{\xintApply\testmacro{\xintiCstoCv{\xintCntoCs {35}{\cn}}}}
% (\the\pdfelapsedtime)

\smallskip

% The actual computation of the list of all 36 convergents accounts for
% only 8\% of the total time (total time equal to about 5 hundredths of a second
% in my testing, on my laptop): another 80\% is occupied with the computation of
% the truncated decimal expansions (and the addition of 1 to everything as the
% formula gives the continued fraction of $e-1$).

One can with no problem compute
much bigger convergents. Let's get the 200th convergent. It turns out to
have the same first 268 digits after the decimal point as $e-1$. Higher
convergents get more and more digits in proportion to their index: the 500th
convergent already gets 799 digits correct! To allow speedy compilation of the
source of this document when the need arises, I limit here to the 200th
convergent.
%  (getting the 500th took about 1.2s on my laptop last time I tried,
% and the 200th convergent is obtained ten times faster).
\begin{everbatim*}
\oodef\z {\xintCntoF {199}{\cn}}%
\begingroup\parindent 0pt \leftskip 2.5cm
\indent\llap {Numerator = }\printnumber{\xintNumerator\z}\par
\indent\llap {Denominator = }\printnumber{\xintDenominator\z}\par
\indent\llap {Expansion = }\printnumber{\xintTrunc{268}\z}\dots\par\endgroup
\end{everbatim*}


One can also use a centered continued fraction: we get more digits but there are
also more computations as the numerators may be either
$1$ or $-1$.

\ifnum\NoSourceCode=1
\bigskip
\begin{framed}
  \small This documentation has been compiled without the source code,
  which is available in the separate file:
  %
  \centeredline{|sourcexint.pdf|,}
  %
  which should be among the candidates proposed by |texdoc --list xint|. To
  produce a single file including both the user documentation and the 
  source code, run |tex xint.dtx| to generate |xint.tex| (if not already
  available), then edit |xint.tex| to set the |\NoSourceCode| toggle to |0|,
  then run thrice |latex| on |xint.tex| and finally |dvipdfmx| on |xint.dvi|.
  Alternatively, run |pdflatex| either directly on |xint.dtx|, or on
  |xint.tex| with |\NoSourceCode| set to |0|.
\end{framed}
\fi

% Mercredi 08 octobre 2014 à 22:09:54
\ifnum\dosourcexint=1
+fi
+catcode`\ 0
\catcode`\+ 12
\etocignoredepthtags
\etocsetnexttocdepth{section}
\tableofcontents
\makeatletter
\@gobble\fi

\StopEventually{\end{document}\endinput}

\ifnum\dosourcexint=1
\renewcommand{\etocaftertochook}{\addvspace{\bigskipamount}}
\etocsettocstyle {}{}
\else
\clearpage
% \newgeometry{%hmarginratio=4:3,
%              hscale=0.75,vscale=0.75}% ATTENTION \newgeometry fait
%                                 % un reset de vscale si on ne le
%                                 % précise pas ici !!!
\fi

\def\MARGEPAGENO{1.25em}

\etocdepthtag.toc {implementation}
\addtocontents{toc}{\gdef\string\sectioncouleur{[named]{RoyalPurple}}}

\makeatletter
\def\storedlinecounts {}
\def\StoreCodelineNo #1{\edef\storedlinecounts{%
                        \unexpanded\expandafter{\storedlinecounts}%
                        {{#1}{\the\c@CodelineNo}}}\c@CodelineNo\z@ }

% Pour le code des macros 
% On va redéfinir \macro@font 
% Pas de couleur, et 0 slashed 

\def\macro@font {\ttbfamily }

% Pour \lverb

\def\MicroFont {\ttzfamily\color[named]{Purple}\makestarlowast }

\makeatother

\MakePercentIgnore
%
% \catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
% \let</dtx>\relax
% \def<*xintkernel>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</dtx>
%<*xintkernel>
%
% \bigskip
% This is \expandafter|\xintbndlversion| of \expandafter|\xintbndldate|.
%
% Extensive changes in release |1.1| of |2014/10/28| were located in
% \xintexprnameimp. Also with that release,
% packages \xintkernelnameimp and \xintcorenameimp were extracted from
% \xinttoolsnameimp and \xintnameimp, and |\xintAdd| was modified to not
% multiply denominators blindly.
% 
% \smallskip
% \noindent\fbox{\xinttoolsnameimp is not loaded anymore by
% \xintnameimp, nor by \xintfracnameimp. It is loaded by \xintexprnameimp.}
%
% \section {Package \xintkernelnameimp implementation}
% \label{sec:kernelimp}
%
% \localtableofcontents
%
% This package provides the common minimal code base for loading management
% and catcode control and also a few programming utilities. It is loaded by both
% |xintcore.sty| and |xinttools.sty| hence by all other packages.
% 
% First appeared as a separate package with release |1.1|.
%
% \subsection{Catcodes, \protect\eTeX{} and reload detection}
%
% The code for reload detection was initially copied from \textsc{Heiko
% Oberdiek}'s packages, then modified.
%
% The method for catcodes was also initially directly inspired by these
% packages.
%
% Starting with version |1.06| of the package, also |`| must be
% catcode-protected, because we replace everywhere in the code the
% twice-expansion done with |\expandafter| by the systematic use of
% |\romannumeral-`0|.
%
% Starting with version |1.06b| I decide that I suffer from an indigestion of @
% signs, so I replace them all with underscores |_|, \`a la \LaTeX 3.
%
% Release |1.09b| is more economical: some macros are defined already in
% |xint.sty| (now in |xintkernel.sty|) and re-used in other modules. All catcode
% changes have been unified and \csa{XINT_storecatcodes} will be used by each
% module to redefine |\XINT_restorecatcodes_endinput| in case catcodes have
% changed in-between the loading of |xint.sty| (now |xintkernel.sty|) and the
% module (not very probable but...).
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \catcode95=11   % _
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
    \ifx\csname numexpr\endcsname\relax
     \y{xintkernel}{\numexpr not available, aborting input}%
     \aftergroup\endinput
    \else
    \expandafter
     \ifx\csname XINTsetupcatcodes\endcsname\relax
      \else
       \y{xintkernel}{I was already loaded, aborting input}%
       \aftergroup\endinput
      \fi
    \fi
  \def\SetCatcodesIfInputNotAborted
  {%
      \endgroup
      \def\XINT_restorecatcodes
      {% takes care of all, to allow more economical code in modules
           \catcode59=\the\catcode59   % ; xintexpr
           \catcode126=\the\catcode126 % ~ xintexpr
           \catcode39=\the\catcode39   % ' xintexpr
           \catcode34=\the\catcode34   % " xintbinhex, and xintexpr
           \catcode63=\the\catcode63   % ? xintexpr
           \catcode124=\the\catcode124 % | xintexpr
           \catcode38=\the\catcode38   % & xintexpr
           \catcode64=\the\catcode64   % @ xintexpr
           \catcode33=\the\catcode33   % ! xintexpr
           \catcode93=\the\catcode93   % ] -, xintfrac, xintseries, xintcfrac
           \catcode91=\the\catcode91   % [ -, xintfrac, xintseries, xintcfrac
           \catcode36=\the\catcode36   % $ xintgcd only
           \catcode94=\the\catcode94   % ^
           \catcode96=\the\catcode96   % `
           \catcode47=\the\catcode47   % /
           \catcode41=\the\catcode41   % )
           \catcode40=\the\catcode40   % (
           \catcode42=\the\catcode42   % *
           \catcode43=\the\catcode43   % +
           \catcode62=\the\catcode62   % >
           \catcode60=\the\catcode60   % <
           \catcode58=\the\catcode58   % :
           \catcode46=\the\catcode46   % .
           \catcode45=\the\catcode45   % -
           \catcode44=\the\catcode44   % ,
           \catcode35=\the\catcode35   % #
           \catcode95=\the\catcode95   % _
           \catcode125=\the\catcode125 % }
           \catcode123=\the\catcode123 % {
           \endlinechar=\the\endlinechar
           \catcode13=\the\catcode13   % ^^M
           \catcode32=\the\catcode32   %
           \catcode61=\the\catcode61\relax   % =
      }%
      \edef\XINT_restorecatcodes_endinput
      {%
           \XINT_restorecatcodes\noexpand\endinput %
      }%
      \def\XINT_setcatcodes
      {%
        \catcode61=12   % =
        \catcode32=10   % space
        \catcode13=5    % ^^M
        \endlinechar=13 %
        \catcode123=1   % {
        \catcode125=2   % }
        \catcode95=11   % _ LETTER
        \catcode35=6    % #
        \catcode44=12   % ,
        \catcode45=12   % -
        \catcode46=12   % .
        \catcode58=11   % : LETTER
        \catcode60=12   % <
        \catcode62=12   % >
        \catcode43=12   % +
        \catcode42=12   % *
        \catcode40=12   % (
        \catcode41=12   % )
        \catcode47=12   % /
        \catcode96=12   % `
        \catcode94=11   % ^ LETTER
        \catcode36=3   % $
        \catcode91=12  % [
        \catcode93=12  % ]
        \catcode33=11  % ! LETTER
        \catcode64=11  % @ LETTER
        \catcode38=12  % &
        \catcode124=12 % |
        \catcode63=11  % ? LETTER
        \catcode34=12  % "
        \catcode39=12  % '
        \catcode126=3  % ~
        \catcode59=12  % ;
      }%
      \XINT_setcatcodes
  }%
\SetCatcodesIfInputNotAborted
%    \end{macrocode}
% Other modules could possibly be loaded under a different catcode regime.
%    \begin{macrocode}
\def\XINTsetupcatcodes {% for use by other modules
      \edef\XINT_restorecatcodes_endinput
      {%
           \XINT_restorecatcodes\noexpand\endinput %
      }%
      \XINT_setcatcodes
}%
%    \end{macrocode}
% \subsection{Package identification}
%
% Inspired from \textsc{Heiko Oberdiek}'s packages. Modified in |1.09b| to allow
% re-use in the other modules. Also I assume now that if |\ProvidesPackage|
% exists it then does define |\ver@<pkgname>.sty|, code of |HO| for some reason
% escaping me  (compatibility with LaTeX 2.09 or other things ??) seems to set
% extra precautions.
%
% |1.09c| uses e-\TeX{} |\ifdefined|.
%    \begin{macrocode}
\ifdefined\ProvidesPackage
  \let\XINT_providespackage\relax
\else
  \def\XINT_providespackage #1#2[#3]%
              {\immediate\write-1{Package: #2 #3}%
               \expandafter\xdef\csname ver@#2.sty\endcsname{#3}}%
\fi
\XINT_providespackage
\ProvidesPackage {xintkernel}%
  [2015/08/31 v1.1b Paraphernalia for the xint packages (jfB)]%
%    \end{macrocode}
% \subsection{Token management utilities}
%    \begin{macrocode}
\long\def\xint_gobble_     {}%
\long\def\xint_gobble_i    #1{}%
\long\def\xint_gobble_ii   #1#2{}%
\long\def\xint_gobble_iii  #1#2#3{}%
\long\def\xint_gobble_iv   #1#2#3#4{}%
\long\def\xint_gobble_v    #1#2#3#4#5{}%
\long\def\xint_gobble_vi   #1#2#3#4#5#6{}%
\long\def\xint_gobble_vii  #1#2#3#4#5#6#7{}%
\long\def\xint_gobble_viii #1#2#3#4#5#6#7#8{}%
\long\def\xint_firstofone  #1{#1}%
\long\def\xint_firstoftwo  #1#2{#1}%
\long\def\xint_secondoftwo #1#2{#2}%
\long\def\xint_firstofone_thenstop  #1{ #1}%
\long\def\xint_firstoftwo_thenstop  #1#2{ #1}%
\long\def\xint_secondoftwo_thenstop #1#2{ #2}%
%    \end{macrocode}
% \subsection{gob til macros and UD style fork}
%    \begin{macrocode}
\def\xint_gob_til_zero #10{}%
\def\xint_UDzerominusfork #10-#2#3\krof {#2}%
\long\def\xint_gob_til_R #1\R {}%
\long\def\xint_gob_til_W #1\W {}%
\long\def\xint_gob_til_Z #1\Z {}%
\let\xint_relax\relax
\def\xint_brelax {\xint_relax }%
\long\def\xint_gob_til_xint_relax #1\xint_relax {}%
%    \end{macrocode}
% \subsection{\csh{xint_afterfi}}
%    \begin{macrocode}
\long\def\xint_afterfi #1#2\fi {\fi #1}%
%    \end{macrocode}
% \subsection{\csh{xint_bye}}
%    \begin{macrocode}
\long\def\xint_bye #1\xint_bye {}%
%    \end{macrocode}
% \subsection{\csh{xint_dothis}, \csh{xint_orthat}}
% \lverb|New with 1.1. Used as \if..\xint_dothis{..}\fi <multiple times>
% followed by \xint_orthat{...}. To be used with less probable things first.|
%    \begin{macrocode}
\long\def\xint_dothis #1#2\xint_orthat #3{\fi #1}% v1.1
\let\xint_orthat \xint_firstofone
%    \end{macrocode}
% \subsection{\csh{xint_zapspaces}}
% \lverb|1.1. Zaps leading, intermediate, trailing, spaces in completely
% expanding context (\edef, \csname...\endcsname). To be used as
%
% \xint_zapspaces foo \xint_gobble_i % notice the mandatory space after foo
%
% Will remove some brace pairs (but not spaces inside them). By the way the
% \zap@spaces of LaTeX2e handles unexpectedly things such as \zap@spaces 1
% {22} 3 4 \@empty (spaces are not all removed). This does not happen with
% \xint_zapspaces.
%
% Explanation: if there are leading spaces, then the first #1 will be empty,
% and the first #2 being undelimited will be stripped from all the remaining
% leading spaces, if there was more than one to start with. Of course
% brace-stripping may occur. And this iterates: each time a #2 is removed,
% either we then have spaces and next #1 will be empty, or we have no spaces
% and #1 will end at the first space. Ultimately #2 will be \xint_gobble_i.
% 
% Code comment from 1.1 release said to do:
%
% \xint_zapspaces foo \xint_bye\xint_bye 
%
% perhaps because it was pretty. It works also, but \xint_gobble_i is one
% token less. Compatible with an empty foo.|
%    \begin{macrocode}
\def\xint_zapspaces #1 #2{#1#2\xint_zapspaces }% v1.1
%    \end{macrocode}
% \subsection{Constants}
%    \begin{macrocode}
\chardef\xint_c_     0
\chardef\xint_c_i    1
\chardef\xint_c_ii   2
\chardef\xint_c_iii  3
\chardef\xint_c_iv   4
\chardef\xint_c_v    5
\chardef\xint_c_vi   6
\chardef\xint_c_vii  7
\chardef\xint_c_viii 8
%    \end{macrocode}
% \subsection{\csh{odef}, \csh{oodef}, \csh{fdef}}
% \lverb|May be prefixed with \global. No parameter text.|
%    \begin{macrocode}
\def\xintodef #1{\expandafter\def\expandafter#1\expandafter }%
\def\xintoodef #1{\expandafter\expandafter\expandafter\def
                  \expandafter\expandafter\expandafter#1%
                  \expandafter\expandafter\expandafter }%
\def\xintfdef #1#2{\expandafter\def\expandafter#1\expandafter
                           {\romannumeral-`0#2}}%
\ifdefined\odef\else\let\odef\xintodef\fi
\ifdefined\oodef\else\let\oodef\xintoodef\fi
\ifdefined\fdef\else\let\fdef\xintfdef\fi
%    \end{macrocode}
% \subsection{\csh{xintReverseOrder}}
% \lverb|\xintReverseOrder: does NOT expand its argument.|
%    \begin{macrocode}
\def\xintReverseOrder {\romannumeral0\xintreverseorder }%
\long\def\xintreverseorder #1%
{%
    \XINT_rord_main {}#1%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
}%
\long\def\XINT_rord_main #1#2#3#4#5#6#7#8#9%
{%
    \xint_bye #9\XINT_rord_cleanup\xint_bye
    \XINT_rord_main {#9#8#7#6#5#4#3#2#1}%
}%
\long\edef\XINT_rord_cleanup\xint_bye\XINT_rord_main #1#2\xint_relax
{%
    \noexpand\expandafter\space\noexpand\xint_gob_til_xint_relax #1%
}%
%    \end{macrocode}
% \subsection{\csh{xintLength}}
% \lverb|\xintLength does NOT expand its argument.|
%    \begin{macrocode}
\def\xintLength {\romannumeral0\xintlength }%
\long\def\xintlength #1%
{%
    \XINT_length_loop
    0.#1\xint_relax\xint_relax\xint_relax\xint_relax
         \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\long\def\XINT_length_loop #1.#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_xint_relax #9\XINT_length_finish_a\xint_relax
    \expandafter\XINT_length_loop\the\numexpr #1+\xint_c_viii.%
}%
\def\XINT_length_finish_a\xint_relax\expandafter\XINT_length_loop
    \the\numexpr #1+\xint_c_viii.#2\xint_bye
{%
    \XINT_length_finish_b #2\W\W\W\W\W\W\W\Z {#1}%
}%
\def\XINT_length_finish_b #1#2#3#4#5#6#7#8\Z
{%
    \xint_gob_til_W
            #1\XINT_length_finish_c \xint_c_
            #2\XINT_length_finish_c \xint_c_i
            #3\XINT_length_finish_c \xint_c_ii
            #4\XINT_length_finish_c \xint_c_iii
            #5\XINT_length_finish_c \xint_c_iv
            #6\XINT_length_finish_c \xint_c_v
            #7\XINT_length_finish_c \xint_c_vi
            \W\XINT_length_finish_c \xint_c_vii\Z
}%
\edef\XINT_length_finish_c #1#2\Z #3%
   {\noexpand\expandafter\space\noexpand\the\numexpr #3+#1\relax}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xintkernel>\relax
%\def<*xinttools>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xintkernel>
%<*xinttools>
%
% \StoreCodelineNo {xintkernel}
%
% \section{Package \xinttoolsnameimp implementation}
% \label{sec:toolsimp}
%
% \localtableofcontents
%
% Release |1.09g| of |2013/11/22| splits off |xinttools.sty| from |xint.sty|.
% Starting with |1.1|, \xinttoolsnameimp ceases being loaded automatically by
% \xintnameimp.
%
% \subsection{Catcodes, \protect\eTeX{} and reload detection}
%
% The code for reload detection was initially copied from \textsc{Heiko
% Oberdiek}'s packages, then modified.
%
% The method for catcodes was also initially directly inspired by these
% packages.
%
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode64=11   % @
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \let\z\endgroup
  \expandafter\let\expandafter\x\csname ver@xinttools.sty\endcsname
  \expandafter\let\expandafter\w\csname ver@xintkernel.sty\endcsname
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
  \ifx\csname numexpr\endcsname\relax
     \y{xinttools}{\numexpr not available, aborting input}%
     \aftergroup\endinput
  \else
    \ifx\x\relax   % plain-TeX, first loading of xinttools.sty
      \ifx\w\relax % but xintkernel.sty not yet loaded.
         \def\z{\endgroup\input xintkernel.sty\relax}%
      \fi
    \else
      \def\empty {}%
      \ifx\x\empty % LaTeX, first loading,
      % variable is initialized, but \ProvidesPackage not yet seen
          \ifx\w\relax % xintkernel.sty not yet loaded.
            \def\z{\endgroup\RequirePackage{xintkernel}}%
          \fi
      \else
        \aftergroup\endinput % xinttools already loaded.
      \fi
    \fi
  \fi
\z%
\XINTsetupcatcodes% defined in xintkernel.sty
%    \end{macrocode}
% \subsection{Package identification}
%    \begin{macrocode}
\XINT_providespackage
\ProvidesPackage{xinttools}%
  [2015/08/31 v1.1b Expandable and non-expandable utilities (jfB)]%
%    \end{macrocode}
% \lverb|\XINT_toks is used in macros such as \xintFor. It is not used
% elsewhere in the xint bundle.|
%    \begin{macrocode}
\newtoks\XINT_toks
\xint_firstofone{\let\XINT_sptoken= } %<- space here!
%    \end{macrocode}
% \subsection{\csh{xintgodef}, \csh{xintgoodef}, \csh{xintgfdef}}
% \lverb|1.09i. For use in \xintAssign.|
%    \begin{macrocode}
\def\xintgodef  {\global\xintodef }%
\def\xintgoodef {\global\xintoodef }%
\def\xintgfdef  {\global\xintfdef }%
%    \end{macrocode}
% \subsection{\csh{xintRevWithBraces}}
% \lverb|New with 1.06. Makes the expansion of its argument and then reverses
% the resulting tokens or braced tokens, adding a pair of braces to each (thus,
% maintaining it when it was already there.) The reason for
% \xint_relax, here and in other locations, is in case #1 expands to nothing,
% the \romannumeral-`0 must be stopped|
%    \begin{macrocode}
\def\xintRevWithBraces         {\romannumeral0\xintrevwithbraces }%
\def\xintRevWithBracesNoExpand {\romannumeral0\xintrevwithbracesnoexpand }%
\long\def\xintrevwithbraces #1%
{%
    \expandafter\XINT_revwbr_loop\expandafter{\expandafter}%
    \romannumeral-`0#1\xint_relax\xint_relax\xint_relax\xint_relax
                      \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\long\def\xintrevwithbracesnoexpand #1%
{%
    \XINT_revwbr_loop {}%
    #1\xint_relax\xint_relax\xint_relax\xint_relax
      \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\long\def\XINT_revwbr_loop #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_xint_relax #9\XINT_revwbr_finish_a\xint_relax
    \XINT_revwbr_loop {{#9}{#8}{#7}{#6}{#5}{#4}{#3}{#2}#1}%
}%
\long\def\XINT_revwbr_finish_a\xint_relax\XINT_revwbr_loop #1#2\xint_bye
{%
    \XINT_revwbr_finish_b #2\R\R\R\R\R\R\R\Z #1%
}%
\def\XINT_revwbr_finish_b #1#2#3#4#5#6#7#8\Z
{%
    \xint_gob_til_R
            #1\XINT_revwbr_finish_c \xint_c_viii
            #2\XINT_revwbr_finish_c \xint_c_vii
            #3\XINT_revwbr_finish_c \xint_c_vi
            #4\XINT_revwbr_finish_c \xint_c_v
            #5\XINT_revwbr_finish_c \xint_c_iv
            #6\XINT_revwbr_finish_c \xint_c_iii
            #7\XINT_revwbr_finish_c \xint_c_ii
            \R\XINT_revwbr_finish_c \xint_c_i\Z
}%
\def\XINT_revwbr_finish_c #1#2\Z
{%
    \expandafter\expandafter\expandafter
        \space
    \csname xint_gobble_\romannumeral #1\endcsname
}%
%    \end{macrocode}
% \subsection{\csh{xintZapFirstSpaces}}
% \lverb|1.09f, written [2013/11/01]. Modified (2014/10/21) for release 1.1 to
% correct the bug in case of an empty argument, or argument containing only
% spaces, which had been forgotten in first version. New version is simpler than
% the initial one. This macro does NOT expand its argument.|
%    \begin{macrocode}
\def\xintZapFirstSpaces {\romannumeral0\xintzapfirstspaces }%
%    \end{macrocode}
% \lverb|defined via an \edef in order to inject space tokens inside.|
%    \begin{macrocode}
\long\edef\xintzapfirstspaces #1%
  {\noexpand\XINT_zapbsp_a \space #1\xint_relax \space\space\xint_relax }%            
\xint_firstofone {\long\edef\XINT_zapbsp_a #1 } %<- space token here
{%
%    \end{macrocode}
% \lverb|If the original #1 started with a space, the grabbed #1 is empty. Thus
% _again? will see #1=\xint_bye, and hand over control to _again which will loop
% back into \XINT_zapbsp_a, with one initial space less. If the original #1 did
% not start with a space, or was empty, then the #1 below will be a <sptoken>,
% then an extract of the original #1, not empty and not starting with a space,
% which contains what was up to the first <sp><sp> present in original #1, or,
% if none preexisted, <sptoken> and all of #1 (possibly empty) plus an ending
% \xint_relax. The added initial space will stop later the \romannumeral0. No
% brace stripping is possible. Control is handed over to \XINT_zapbsp_b which
% strips out the ending \xint_relax<sp><sp>\xint_relax|
%    \begin{macrocode}
  \noexpand\XINT_zapbsp_again? #1\noexpand\xint_bye\noexpand\XINT_zapbsp_b #1\space\space
}%
\long\def\XINT_zapbsp_again? #1{\xint_bye #1\XINT_zapbsp_again }%
\xint_firstofone{\def\XINT_zapbsp_again\XINT_zapbsp_b} {\XINT_zapbsp_a }%
\long\def\XINT_zapbsp_b #1\xint_relax #2\xint_relax {#1}%
%    \end{macrocode}
% \subsection{\csh{xintZapLastSpaces}}
% \lverb+1.09f, written [2013/11/01]. +
%    \begin{macrocode}
\def\xintZapLastSpaces {\romannumeral0\xintzaplastspaces }%
%    \end{macrocode}
% \lverb|Next macro is defined via an \edef for the space tokens.|
%    \begin{macrocode}
\long\edef\xintzaplastspaces #1{\noexpand\XINT_zapesp_a {}\noexpand\empty#1%
                                \space\space\noexpand\xint_bye\xint_relax}%
%    \end{macrocode}
% \lverb|The \empty from \xintzaplastspaces is to prevent brace removal in the
% #2 below. The \expandafter chain removes it.|
%    \begin{macrocode}
\xint_firstofone {\long\def\XINT_zapesp_a #1#2 } %<- second space here
    {\expandafter\XINT_zapesp_b\expandafter{#2}{#1}}%
%    \end{macrocode}
% \lverb|Notice again an \empty added here. This is in preparation for possibly looping
% back to \XINT_zapesp_a. If the initial #1 had no <sp><sp>, the stuff however
% will not loop, because #3 will already be <some spaces>\xint_bye. Notice
% that this macro fetches all way to the ending \xint_relax. This looks not
% very efficient, but how often do we have to strip ending spaces from
% something which also has inner stretches of _multiple_ space tokens ?;-). |
%    \begin{macrocode}
\long\def\XINT_zapesp_b #1#2#3\xint_relax
    {\XINT_zapesp_end? #3\XINT_zapesp_e {#2#1}\empty #3\xint_relax }%
%    \end{macrocode}
% \lverb|When we have been over all possible <sp><sp> things, we reach the
% ending space tokens, and #3 will be a bunch of spaces (possibly none)
% followed by \xint_bye. So the #1 in _end? will be \xint_bye. In all other cases
% #1 can not be \xint_bye (assuming naturally this token does nor arise in
% original input), hence control falls back to \XINT_zapesp_e which will loop back
% to \XINT_zapesp_a.|
%    \begin{macrocode}
\long\def\XINT_zapesp_end? #1{\xint_bye #1\XINT_zapesp_end }%
%    \end{macrocode}
% \lverb|We are done. The #1 here has accumulated all the previous material,
% and is stripped of its ending spaces, if any.|
%    \begin{macrocode}
\long\def\XINT_zapesp_end\XINT_zapesp_e #1#2\xint_relax { #1}%
%    \end{macrocode}
% \lverb|We haven't yet reached the end, so we need to re-inject two space
% tokens after what we have gotten so far. Then we loop.|
%    \begin{macrocode}
\long\edef\XINT_zapesp_e #1{\noexpand \XINT_zapesp_a {#1\space\space}}%
%    \end{macrocode}
% \subsection{\csh{xintZapSpaces}}
% \lverb+1.09f, written [2013/11/01]. Modified for 1.1, 2014/10/21 as it has the
% same bug as \xintZapFirstSpaces. We in effect do first \xintZapFirstSpaces,
% then \xintZapLastSpaces.+
%    \begin{macrocode}
\def\xintZapSpaces {\romannumeral0\xintzapspaces }%
\long\edef\xintzapspaces #1% like \xintZapFirstSpaces.
                   {\noexpand\XINT_zapsp_a \space #1\xint_relax \space\space\xint_relax }%
\xint_firstofone {\long\edef\XINT_zapsp_a #1 } %
  {\noexpand\XINT_zapsp_again? #1\noexpand\xint_bye\noexpand\XINT_zapsp_b #1\space\space}%
\long\def\XINT_zapsp_again? #1{\xint_bye #1\XINT_zapsp_again }%
\xint_firstofone{\def\XINT_zapsp_again\XINT_zapsp_b} {\XINT_zapsp_a }%
\xint_firstofone{\def\XINT_zapsp_b} {\XINT_zapsp_c }%
\long\edef\XINT_zapsp_c #1\xint_relax #2\xint_relax {\noexpand\XINT_zapesp_a
    {}\noexpand \empty #1\space\space\noexpand\xint_bye\xint_relax }%
%    \end{macrocode}
% \subsection{\csh{xintZapSpacesB}}
% \lverb+1.09f, written [2013/11/01]. Strips up to one pair of braces (but then
% does not strip spaces inside).+
%    \begin{macrocode}
\def\xintZapSpacesB {\romannumeral0\xintzapspacesb }%
\long\def\xintzapspacesb #1{\XINT_zapspb_one? #1\xint_relax\xint_relax
                         \xint_bye\xintzapspaces {#1}}%
\long\def\XINT_zapspb_one? #1#2%
   {\xint_gob_til_xint_relax #1\XINT_zapspb_onlyspaces\xint_relax
    \xint_gob_til_xint_relax #2\XINT_zapspb_bracedorone\xint_relax
    \xint_bye {#1}}%
\def\XINT_zapspb_onlyspaces\xint_relax
    \xint_gob_til_xint_relax\xint_relax\XINT_zapspb_bracedorone\xint_relax
    \xint_bye #1\xint_bye\xintzapspaces #2{ }%
\long\def\XINT_zapspb_bracedorone\xint_relax
    \xint_bye #1\xint_relax\xint_bye\xintzapspaces #2{ #1}%
%    \end{macrocode}
% \subsection{\csh{xintCSVtoList}, \csh{xintCSVtoListNonStripped}}
% \lverb|\xintCSVtoList transforms a,b,..,z into {a}{b}...{z}. The comma
% separated list may be a macro which is first f-expanded. First included in
% release 1.06. Here, use of \Z (and \R) perfectly safe.
%
% [2013/11/02]: Starting with 1.09f, automatically filters items with
% \xintZapSpacesB to strip away all spaces around commas, and spaces at the start
% and end of the list. The original is kept as \xintCSVtoListNonStripped, and is
% faster. But ... it doesn't strip spaces.|
%    \begin{macrocode}
\def\xintCSVtoList {\romannumeral0\xintcsvtolist }%
\long\def\xintcsvtolist #1{\expandafter\xintApply
           \expandafter\xintzapspacesb
           \expandafter{\romannumeral0\xintcsvtolistnonstripped{#1}}}%
\def\xintCSVtoListNoExpand {\romannumeral0\xintcsvtolistnoexpand }%
\long\def\xintcsvtolistnoexpand #1{\expandafter\xintApply
           \expandafter\xintzapspacesb
           \expandafter{\romannumeral0\xintcsvtolistnonstrippednoexpand{#1}}}%
\def\xintCSVtoListNonStripped {\romannumeral0\xintcsvtolistnonstripped }%
\def\xintCSVtoListNonStrippedNoExpand
         {\romannumeral0\xintcsvtolistnonstrippednoexpand }%
\long\def\xintcsvtolistnonstripped #1%
{%
    \expandafter\XINT_csvtol_loop_a\expandafter
    {\expandafter}\romannumeral-`0#1%
        ,\xint_bye,\xint_bye,\xint_bye,\xint_bye
        ,\xint_bye,\xint_bye,\xint_bye,\xint_bye,\Z
}%
\long\def\xintcsvtolistnonstrippednoexpand #1%
{%
    \XINT_csvtol_loop_a
    {}#1,\xint_bye,\xint_bye,\xint_bye,\xint_bye
        ,\xint_bye,\xint_bye,\xint_bye,\xint_bye,\Z
}%
\long\def\XINT_csvtol_loop_a #1#2,#3,#4,#5,#6,#7,#8,#9,%
{%
    \xint_bye #9\XINT_csvtol_finish_a\xint_bye
    \XINT_csvtol_loop_b {#1}{{#2}{#3}{#4}{#5}{#6}{#7}{#8}{#9}}%
}%
\long\def\XINT_csvtol_loop_b #1#2{\XINT_csvtol_loop_a {#1#2}}%
\long\def\XINT_csvtol_finish_a\xint_bye\XINT_csvtol_loop_b #1#2#3\Z
{%
    \XINT_csvtol_finish_b #3\R,\R,\R,\R,\R,\R,\R,\Z #2{#1}%
}%
\def\XINT_csvtol_finish_b #1,#2,#3,#4,#5,#6,#7,#8\Z
{%
    \xint_gob_til_R
            #1\XINT_csvtol_finish_c \xint_c_viii
            #2\XINT_csvtol_finish_c \xint_c_vii
            #3\XINT_csvtol_finish_c \xint_c_vi
            #4\XINT_csvtol_finish_c \xint_c_v
            #5\XINT_csvtol_finish_c \xint_c_iv
            #6\XINT_csvtol_finish_c \xint_c_iii
            #7\XINT_csvtol_finish_c \xint_c_ii
            \R\XINT_csvtol_finish_c \xint_c_i\Z
}%
\def\XINT_csvtol_finish_c #1#2\Z
{%
    \csname XINT_csvtol_finish_d\romannumeral #1\endcsname
}%
\long\def\XINT_csvtol_finish_dviii #1#2#3#4#5#6#7#8#9{ #9}%
\long\def\XINT_csvtol_finish_dvii  #1#2#3#4#5#6#7#8#9{ #9{#1}}%
\long\def\XINT_csvtol_finish_dvi   #1#2#3#4#5#6#7#8#9{ #9{#1}{#2}}%
\long\def\XINT_csvtol_finish_dv    #1#2#3#4#5#6#7#8#9{ #9{#1}{#2}{#3}}%
\long\def\XINT_csvtol_finish_div   #1#2#3#4#5#6#7#8#9{ #9{#1}{#2}{#3}{#4}}%
\long\def\XINT_csvtol_finish_diii  #1#2#3#4#5#6#7#8#9{ #9{#1}{#2}{#3}{#4}{#5}}%
\long\def\XINT_csvtol_finish_dii   #1#2#3#4#5#6#7#8#9%
                                            { #9{#1}{#2}{#3}{#4}{#5}{#6}}%
\long\def\XINT_csvtol_finish_di    #1#2#3#4#5#6#7#8#9%
                                            { #9{#1}{#2}{#3}{#4}{#5}{#6}{#7}}%
%    \end{macrocode}
% \subsection{\csh{xintListWithSep}}
% \lverb|1.04.
% \xintListWithSep {\sep}{{a}{b}...{z}} returns a \sep b \sep ....\sep z. It
% f-expands its second argument. The 'sep' may be \par's: the macro
% \xintlistwithsep etc... are all declared long. 'sep' does not have to be a
% single token. It is not expanded.|
%    \begin{macrocode}
\def\xintListWithSep {\romannumeral0\xintlistwithsep }%
\def\xintListWithSepNoExpand {\romannumeral0\xintlistwithsepnoexpand }%
\long\def\xintlistwithsep #1#2%
    {\expandafter\XINT_lws\expandafter {\romannumeral-`0#2}{#1}}%
\long\def\XINT_lws #1#2{\XINT_lws_start {#2}#1\xint_bye }%
\long\def\xintlistwithsepnoexpand #1#2{\XINT_lws_start {#1}#2\xint_bye }%
\long\def\XINT_lws_start #1#2%
{%
    \xint_bye #2\XINT_lws_dont\xint_bye
    \XINT_lws_loop_a {#2}{#1}%
}%
\long\def\XINT_lws_dont\xint_bye\XINT_lws_loop_a #1#2{ }%
\long\def\XINT_lws_loop_a #1#2#3%
{%
    \xint_bye #3\XINT_lws_end\xint_bye
    \XINT_lws_loop_b {#1}{#2#3}{#2}%
}%
\long\def\XINT_lws_loop_b #1#2{\XINT_lws_loop_a {#1#2}}%
\long\def\XINT_lws_end\xint_bye\XINT_lws_loop_b #1#2#3{ #1}%
%    \end{macrocode}
% \subsection{\csh{xintNthElt}}
% \lverb+First included in release 1.06.
%
% \xintNthElt {i}{stuff f-expanding to {a}{b}...{z}} (or `tokens'
% abcd...z)returns the i th element (one pair of braces removed). The list is
% first f-expanded. The \xintNthEltNoExpand does no expansion of its second
% argument. Both variants expand the first argument inside \numexpr.
%
% With i = 0, the number of items is returned. This is different from \xintLen
% which is only for numbers (particularly, it checks the sign) and different
% from \xintLength which does not f-expand its argument.
%
% Negative values return the |i|th element from the end. Release 1.09m
% rewrote the initial bits of the code (which checked the sign of #1 and
% expanded or not #2), ome `improvements' made earlier in 1.09c were quite
% sub-efficient. Now uses \xint_UD$-zero$-minus$-fork, moved from xint.sty.+
%    \begin{macrocode}
\def\xintNthElt         {\romannumeral0\xintnthelt }%
\def\xintNthEltNoExpand {\romannumeral0\xintntheltnoexpand }%
\def\xintnthelt #1#2%
{%
    \expandafter\XINT_nthelt_a\the\numexpr #1\expandafter.%
    \expandafter{\romannumeral-`0#2}%
}%
\def\xintntheltnoexpand #1%
{%
    \expandafter\XINT_nthelt_a\the\numexpr #1.%
}%
\def\XINT_nthelt_a #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\XINT_nthelt_bzero}%
        0#1{\XINT_nthelt_bneg {#2}}%
         0-{\XINT_nthelt_bpos {#1#2}}%
    \krof
}%
\long\def\XINT_nthelt_bzero #1%
{%
     \XINT_length_loop 0.#1\xint_relax\xint_relax\xint_relax\xint_relax
                 \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\long\def\XINT_nthelt_bneg #1#2%
{%
     \expandafter\XINT_nthelt_loop_a\expandafter {\the\numexpr #1\expandafter}%
     \romannumeral0\xintrevwithbracesnoexpand {#2}%
          \xint_relax\xint_relax\xint_relax\xint_relax
          \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\long\def\XINT_nthelt_bpos #1#2%
{%
     \XINT_nthelt_loop_a {#1}#2\xint_relax\xint_relax\xint_relax\xint_relax
                    \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\def\XINT_nthelt_loop_a #1%
{%
    \ifnum #1>\xint_c_viii
        \expandafter\XINT_nthelt_loop_b
    \else
        \XINT_nthelt_getit
    \fi
    {#1}%
}%
\long\def\XINT_nthelt_loop_b #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_xint_relax #9\XINT_nthelt_silentend\xint_relax
    \expandafter\XINT_nthelt_loop_a\expandafter{\the\numexpr #1-\xint_c_viii}%
}%
\def\XINT_nthelt_silentend #1\xint_bye { }%
\def\XINT_nthelt_getit\fi #1%
{%
    \fi\expandafter\expandafter\expandafter\XINT_nthelt_finish
    \csname xint_gobble_\romannumeral\numexpr#1-\xint_c_i\endcsname
}%
\long\edef\XINT_nthelt_finish #1#2\xint_bye
   {\noexpand\xint_gob_til_xint_relax #1\noexpand\expandafter\space
                               \noexpand\xint_gobble_ii\xint_relax\space #1}%
%    \end{macrocode}
% \subsection{\csh{xintKeep}}
% \lverb+First included in release 1.09m.
%
% \xintKeep {i}{stuff f-expanding to {a}{b}...{z}} (or `tokens' abcd...z,
% but each naked token ends up braced in the output) returns (in two
% expansion steps) the first i elements from the list, which is first
% f-expanded. The i is expanded inside \numexpr. The variant
% \xintKeepNoExpand does not expand the list argument.
%
% With i = 0, the empty sequence is returned.
%
% With i<0, the last |i| elements are returned (in the same order as in
% the original list).
%
% With |i| equal to or bigger than the length of the (f-expanded) list,
% the full list is returned.+
%    \begin{macrocode}
\def\xintKeep         {\romannumeral0\xintkeep }%
\def\xintKeepNoExpand {\romannumeral0\xintkeepnoexpand }%
\def\xintkeep #1#2%
{%
    \expandafter\XINT_keep_a\the\numexpr #1\expandafter.%
    \expandafter{\romannumeral-`0#2}%
}%
\def\xintkeepnoexpand #1%
{%
    \expandafter\XINT_keep_a\the\numexpr #1.%
}%
\def\XINT_keep_a #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\expandafter\space\xint_gobble_i }%
        0#1{\XINT_keep_bneg_a {#2}}%
         0-{\XINT_keep_bpos {#1#2}}%
    \krof
}%
\long\def\XINT_keep_bneg_a #1#2%
{%
    \expandafter\XINT_keep_bneg_b \the\numexpr \xintLength{#2}-#1.{#2}%
}%
\def\XINT_keep_bneg_b #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\xint_firstofone_thenstop }%
        0#1{\xint_firstofone_thenstop }%
         0-{\XINT_trim_bpos {#1#2}}%
    \krof
}%
\long\def\XINT_keep_bpos #1#2%
{%
     \XINT_keep_loop_a {#1}{}#2\xint_relax\xint_relax\xint_relax\xint_relax
          \xint_relax\xint_relax\xint_relax\xint_bye
}%
\def\XINT_keep_loop_a #1%
{%
    \ifnum #1>\xint_c_vi
        \expandafter\XINT_keep_loop_b
    \else
        \XINT_keep_finish
    \fi
    {#1}%
}%
\long\def\XINT_keep_loop_b #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_xint_relax #9\XINT_keep_enda\xint_relax
    \expandafter\XINT_keep_loop_c\expandafter{\the\numexpr #1-\xint_c_vii}%
    {{#3}{#4}{#5}{#6}{#7}{#8}{#9}}{#2}%
}%
\long\def\XINT_keep_loop_c #1#2#3{\XINT_keep_loop_a {#1}{#3#2}}%
\long\def\XINT_keep_enda\xint_relax
     \expandafter\XINT_keep_loop_c\expandafter #1#2#3#4\xint_bye
{%
     \XINT_keep_endb #4\W\W\W\W\W\W\Z #2{#3}%
}%
\def\XINT_keep_endb #1#2#3#4#5#6#7\Z
{%
    \xint_gob_til_W
    #1\XINT_keep_endc_
    #2\XINT_keep_endc_i
    #3\XINT_keep_endc_ii
    #4\XINT_keep_endc_iii
    #5\XINT_keep_endc_iv
    #6\XINT_keep_endc_v
    \W\XINT_keep_endc_vi\Z
}%
\long\def\XINT_keep_endc_    #1\Z #2#3#4#5#6#7#8#9{ #9}%
\long\def\XINT_keep_endc_i   #1\Z #2#3#4#5#6#7#8#9{ #9{#2}}%
\long\def\XINT_keep_endc_ii  #1\Z #2#3#4#5#6#7#8#9{ #9{#2}{#3}}%
\long\def\XINT_keep_endc_iii #1\Z #2#3#4#5#6#7#8#9{ #9{#2}{#3}{#4}}%
\long\def\XINT_keep_endc_iv  #1\Z #2#3#4#5#6#7#8#9{ #9{#2}{#3}{#4}{#5}}%
\long\def\XINT_keep_endc_v   #1\Z #2#3#4#5#6#7#8#9{ #9{#2}{#3}{#4}{#5}{#6}}%
\long\def\XINT_keep_endc_vi\Z #1#2#3#4#5#6#7#8{ #8{#1}{#2}{#3}{#4}{#5}{#6}}%
\long\def\XINT_keep_finish\fi #1#2#3#4#5#6#7#8#9\xint_bye
{%
    \fi\XINT_keep_finish_loop_a {#1}{}{#3}{#4}{#5}{#6}{#7}{#8}\Z {#2}%
}%
\def\XINT_keep_finish_loop_a #1%
{%
    \xint_gob_til_zero #1\XINT_keep_finish_z0%
    \expandafter\XINT_keep_finish_loop_b\expandafter
       {\the\numexpr #1-\xint_c_i}%
}%
\long\def\XINT_keep_finish_z0%
     \expandafter\XINT_keep_finish_loop_b\expandafter #1#2#3\Z #4{ #4#2}%
\long\def\XINT_keep_finish_loop_b #1#2#3%
{%
    \xint_gob_til_xint_relax #3\XINT_keep_finish_exit\xint_relax
    \XINT_keep_finish_loop_c {#1}{#2}{#3}%
}%
\long\def\XINT_keep_finish_exit\xint_relax
         \XINT_keep_finish_loop_c #1#2#3\Z #4{ #4#2}%
\long\def\XINT_keep_finish_loop_c #1#2#3%
        {\XINT_keep_finish_loop_a {#1}{#2{#3}}}%
%    \end{macrocode}
% \subsection{\csh{xintTrim}}
% \lverb+First included in release 1.09m.
%
% \xintTrim {i}{stuff f-expanding to {a}{b}...{z}} (or `tokens' abcd...z,
% but each naked token ends up braced in the output) returns (in two
% expansion steps) the sequence with the first i elements omitted. The
% list is first f-expanded. The i is expanded inside \numexpr. Variant
% \xintTrimNoExpand does not expand the list argument.
%
% With i = 0, the original (expanded) list is returned.
%
% With i<0, the last |i| elements from the tail are suppressed.
%
% With |i| equal to or bigger than the length of the (f-expanded) list,
% the empty list is returned.+
%    \begin{macrocode}
\def\xintTrim         {\romannumeral0\xinttrim }%
\def\xintTrimNoExpand {\romannumeral0\xinttrimnoexpand }%
\def\xinttrim #1#2%
{%
    \expandafter\XINT_trim_a\the\numexpr #1\expandafter.%
    \expandafter{\romannumeral-`0#2}%
}%
\def\xinttrimnoexpand #1%
{%
    \expandafter\XINT_trim_a\the\numexpr #1.%
}%
\def\XINT_trim_a #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\xint_firstofone_thenstop }%
        0#1{\XINT_trim_bneg_a {#2}}%
         0-{\XINT_trim_bpos {#1#2}}%
    \krof
}%
\long\def\XINT_trim_bneg_a #1#2%
{%
    \expandafter\XINT_trim_bneg_b \the\numexpr \xintLength{#2}-#1.{#2}%
}%
\def\XINT_trim_bneg_b #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\expandafter\space\xint_gobble_i }%
        0#1{\expandafter\space\xint_gobble_i }%
         0-{\XINT_keep_bpos {#1#2}}%
    \krof
}%
\long\def\XINT_trim_bpos #1#2%
{%
     \XINT_trim_loop_a {#1}#2\xint_relax\xint_relax\xint_relax\xint_relax
                    \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\def\XINT_trim_loop_a #1%
{%
    \ifnum #1>\xint_c_vii
        \expandafter\XINT_trim_loop_b
    \else
        \XINT_trim_finish
    \fi
    {#1}%
}%
\long\def\XINT_trim_loop_b #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_xint_relax #9\XINT_trim_silentend\xint_relax
    \expandafter\XINT_trim_loop_a\expandafter{\the\numexpr #1-\xint_c_viii}%
}%
\def\XINT_trim_silentend #1\xint_bye { }%
\def\XINT_trim_finish\fi #1%
{%
    \fi\expandafter\expandafter\expandafter\XINT_trim_finish_a
    \expandafter\expandafter\expandafter\space % avoids brace removal
    \csname xint_gobble_\romannumeral\numexpr#1\endcsname
}%
\long\def\XINT_trim_finish_a #1\xint_relax #2\xint_bye {#1}%
%    \end{macrocode}
% \subsection{\csh{xintApply}}
% \lverb|\xintApply {\macro}{{a}{b}...{z}} returns {\macro{a}}...{\macro{b}}
% where each instance of \macro is f-expanded. The list itself is first
% f-expanded and may thus be a macro. Introduced with release 1.04.|
%    \begin{macrocode}
\def\xintApply         {\romannumeral0\xintapply }%
\def\xintApplyNoExpand {\romannumeral0\xintapplynoexpand }%
\long\def\xintapply #1#2%
{%
    \expandafter\XINT_apply\expandafter {\romannumeral-`0#2}%
    {#1}%
}%
\long\def\XINT_apply #1#2{\XINT_apply_loop_a {}{#2}#1\xint_bye }%
\long\def\xintapplynoexpand #1#2{\XINT_apply_loop_a {}{#1}#2\xint_bye }%
\long\def\XINT_apply_loop_a #1#2#3%
{%
    \xint_bye #3\XINT_apply_end\xint_bye
    \expandafter
    \XINT_apply_loop_b
    \expandafter {\romannumeral-`0#2{#3}}{#1}{#2}%
}%
\long\def\XINT_apply_loop_b #1#2{\XINT_apply_loop_a {#2{#1}}}%
\long\def\XINT_apply_end\xint_bye\expandafter\XINT_apply_loop_b
    \expandafter #1#2#3{ #2}%
%    \end{macrocode}
% \subsection{\csh{xintApplyUnbraced}}
% \lverb|\xintApplyUnbraced {\macro}{{a}{b}...{z}} returns \macro{a}...\macro{z}
% where each instance of \macro is f-expanded using \romannumeral-`0. The second
% argument may be a macro as it is itself also f-expanded. No braces
% are added: this allows for example a non-expandable \def in \macro, without
% having to do \gdef. Introduced with release 1.06b.|
%    \begin{macrocode}
\def\xintApplyUnbraced {\romannumeral0\xintapplyunbraced }%
\def\xintApplyUnbracedNoExpand {\romannumeral0\xintapplyunbracednoexpand }%
\long\def\xintapplyunbraced #1#2%
{%
    \expandafter\XINT_applyunbr\expandafter {\romannumeral-`0#2}%
    {#1}%
}%
\long\def\XINT_applyunbr #1#2{\XINT_applyunbr_loop_a {}{#2}#1\xint_bye }%
\long\def\xintapplyunbracednoexpand #1#2%
   {\XINT_applyunbr_loop_a {}{#1}#2\xint_bye }%
\long\def\XINT_applyunbr_loop_a #1#2#3%
{%
    \xint_bye #3\XINT_applyunbr_end\xint_bye
    \expandafter\XINT_applyunbr_loop_b
    \expandafter {\romannumeral-`0#2{#3}}{#1}{#2}%
}%
\long\def\XINT_applyunbr_loop_b #1#2{\XINT_applyunbr_loop_a {#2#1}}%
\long\def\XINT_applyunbr_end\xint_bye\expandafter\XINT_applyunbr_loop_b
    \expandafter #1#2#3{ #2}%
%    \end{macrocode}
% \subsection{\csh{xintSeq}}
% \lverb|1.09c. Without the optional argument puts stress on the input stack,
% should not be used to generated thousands of terms then.|
%    \begin{macrocode}
\def\xintSeq {\romannumeral0\xintseq }%
\def\xintseq #1{\XINT_seq_chkopt  #1\xint_bye }%
\def\XINT_seq_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_seq_opt
       \else\expandafter\XINT_seq_noopt
    \fi  #1%
}%
\def\XINT_seq_noopt #1\xint_bye #2%
{%
    \expandafter\XINT_seq\expandafter
       {\the\numexpr#1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_seq #1#2%
{%
   \ifcase\ifnum #1=#2 0\else\ifnum #2>#1 1\else -1\fi\fi\space
      \expandafter\xint_firstoftwo_thenstop
   \or
      \expandafter\XINT_seq_p
   \else
      \expandafter\XINT_seq_n
   \fi
   {#2}{#1}%
}%
\def\XINT_seq_p #1#2%
{%
    \ifnum #1>#2
      \expandafter\expandafter\expandafter\XINT_seq_p
    \else
      \expandafter\XINT_seq_e
    \fi
    \expandafter{\the\numexpr #1-\xint_c_i}{#2}{#1}%
}%
\def\XINT_seq_n #1#2%
{%
    \ifnum #1<#2
      \expandafter\expandafter\expandafter\XINT_seq_n
    \else
      \expandafter\XINT_seq_e
    \fi
     \expandafter{\the\numexpr #1+\xint_c_i}{#2}{#1}%
}%
\def\XINT_seq_e #1#2#3{ }%
\def\XINT_seq_opt [\xint_bye #1]#2#3%
{%
    \expandafter\XINT_seqo\expandafter
    {\the\numexpr #2\expandafter}\expandafter
    {\the\numexpr #3\expandafter}\expandafter
    {\the\numexpr #1}%
}%
\def\XINT_seqo #1#2%
{%
   \ifcase\ifnum #1=#2 0\else\ifnum #2>#1 1\else -1\fi\fi\space
      \expandafter\XINT_seqo_a
   \or
      \expandafter\XINT_seqo_pa
   \else
      \expandafter\XINT_seqo_na
   \fi
   {#1}{#2}%
}%
\def\XINT_seqo_a #1#2#3{ {#1}}%
\def\XINT_seqo_o #1#2#3#4{ #4}%
\def\XINT_seqo_pa #1#2#3%
{%
    \ifcase\ifnum #3=\xint_c_ 0\else\ifnum #3>\xint_c_ 1\else -1\fi\fi\space
           \expandafter\XINT_seqo_o
    \or
           \expandafter\XINT_seqo_pb
    \else
           \xint_afterfi{\expandafter\space\xint_gobble_iv}%
    \fi
    {#1}{#2}{#3}{{#1}}%
}%
\def\XINT_seqo_pb #1#2#3%
{%
    \expandafter\XINT_seqo_pc\expandafter{\the\numexpr #1+#3}{#2}{#3}%
}%
\def\XINT_seqo_pc #1#2%
{%
    \ifnum #1>#2
        \expandafter\XINT_seqo_o
    \else
        \expandafter\XINT_seqo_pd
    \fi
    {#1}{#2}%
}%
\def\XINT_seqo_pd #1#2#3#4{\XINT_seqo_pb {#1}{#2}{#3}{#4{#1}}}%
\def\XINT_seqo_na #1#2#3%
{%
    \ifcase\ifnum #3=\xint_c_ 0\else\ifnum #3>\xint_c_ 1\else -1\fi\fi\space
        \expandafter\XINT_seqo_o
    \or
        \xint_afterfi{\expandafter\space\xint_gobble_iv}%
    \else
        \expandafter\XINT_seqo_nb
    \fi
    {#1}{#2}{#3}{{#1}}%
}%
\def\XINT_seqo_nb #1#2#3%
{%
    \expandafter\XINT_seqo_nc\expandafter{\the\numexpr #1+#3}{#2}{#3}%
}%
\def\XINT_seqo_nc #1#2%
{%
    \ifnum #1<#2
        \expandafter\XINT_seqo_o
    \else
        \expandafter\XINT_seqo_nd
    \fi
    {#1}{#2}%
}%
\def\XINT_seqo_nd #1#2#3#4{\XINT_seqo_nb {#1}{#2}{#3}{#4{#1}}}%
%    \end{macrocode}
%\subsection{\csh{xintloop}, \csh{xintbreakloop}, \csh{xintbreakloopanddo},
% \csh{xintloopskiptonext}}
% \lverb|1.09g [2013/11/22]. Made long with 1.09h.|
%    \begin{macrocode}
\long\def\xintloop #1#2\repeat {#1#2\xintloop_again\fi\xint_gobble_i {#1#2}}%
\long\def\xintloop_again\fi\xint_gobble_i #1{\fi
                             #1\xintloop_again\fi\xint_gobble_i {#1}}%
\long\def\xintbreakloop #1\xintloop_again\fi\xint_gobble_i #2{}%
\long\def\xintbreakloopanddo #1#2\xintloop_again\fi\xint_gobble_i #3{#1}%
\long\def\xintloopskiptonext #1\xintloop_again\fi\xint_gobble_i #2{%
                        #2\xintloop_again\fi\xint_gobble_i {#2}}%
%    \end{macrocode}
% \subsection{\csh{xintiloop}, \csh{xintiloopindex}, \csh{xintouteriloopindex},
%   \csh{xintbreakiloop}, \csh{xintbreakiloopanddo}, \csh{xintiloopskiptonext},
%  \csh{xintiloopskipandredo}}
% \lverb|1.09g [2013/11/22]. Made long with 1.09h.|
%    \begin{macrocode}
\def\xintiloop [#1+#2]{%
    \expandafter\xintiloop_a\the\numexpr #1\expandafter.\the\numexpr #2.}%
\long\def\xintiloop_a #1.#2.#3#4\repeat{%
    #3#4\xintiloop_again\fi\xint_gobble_iii {#1}{#2}{#3#4}}%
\def\xintiloop_again\fi\xint_gobble_iii #1#2{%
    \fi\expandafter\xintiloop_again_b\the\numexpr#1+#2.#2.}%
\long\def\xintiloop_again_b #1.#2.#3{%
    #3\xintiloop_again\fi\xint_gobble_iii {#1}{#2}{#3}}%
\long\def\xintbreakiloop #1\xintiloop_again\fi\xint_gobble_iii #2#3#4{}%
\long\def\xintbreakiloopanddo
     #1.#2\xintiloop_again\fi\xint_gobble_iii #3#4#5{#1}%
\long\def\xintiloopindex #1\xintiloop_again\fi\xint_gobble_iii #2%
                {#2#1\xintiloop_again\fi\xint_gobble_iii {#2}}%
\long\def\xintouteriloopindex #1\xintiloop_again
                         #2\xintiloop_again\fi\xint_gobble_iii #3%
   {#3#1\xintiloop_again #2\xintiloop_again\fi\xint_gobble_iii {#3}}%
\long\def\xintiloopskiptonext #1\xintiloop_again\fi\xint_gobble_iii #2#3{%
    \expandafter\xintiloop_again_b \the\numexpr#2+#3.#3.}%
\long\def\xintiloopskipandredo #1\xintiloop_again\fi\xint_gobble_iii #2#3#4{%
    #4\xintiloop_again\fi\xint_gobble_iii {#2}{#3}{#4}}%
%    \end{macrocode}
% \subsection{\csh{XINT_xflet}}
% \lverb|1.09e [2013/10/29]: we f-expand unbraced tokens and swallow arising
% space tokens until the dust settles.|
%    \begin{macrocode}
\def\XINT_xflet #1%
{%
    \def\XINT_xflet_macro {#1}\XINT_xflet_zapsp
}%
\def\XINT_xflet_zapsp
{%
    \expandafter\futurelet\expandafter\XINT_token
    \expandafter\XINT_xflet_sp?\romannumeral-`0%
}%
\def\XINT_xflet_sp?
{%
    \ifx\XINT_token\XINT_sptoken
         \expandafter\XINT_xflet_zapsp
    \else\expandafter\XINT_xflet_zapspB
    \fi
}%
\def\XINT_xflet_zapspB
{%
    \expandafter\futurelet\expandafter\XINT_tokenB
    \expandafter\XINT_xflet_spB?\romannumeral-`0%
}%
\def\XINT_xflet_spB?
{%
    \ifx\XINT_tokenB\XINT_sptoken
         \expandafter\XINT_xflet_zapspB
    \else\expandafter\XINT_xflet_eq?
    \fi
}%
\def\XINT_xflet_eq?
{%
    \ifx\XINT_token\XINT_tokenB
         \expandafter\XINT_xflet_macro
    \else\expandafter\XINT_xflet_zapsp
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintApplyInline}}
% \lverb|1.09a: \xintApplyInline\macro{{a}{b}...{z}} has the same effect as
% executing \macro{a} and then applying again \xintApplyInline to the shortened
% list {{b}...{z}} until nothing is left. This is a non-expandable command
% which will result in quicker code than using \xintApplyUnbraced. It f-expands
% its second (list) argument first, which may thus be encapsulated in a macro.
%
% Rewritten in 1.09c. Nota bene: uses catcode 3 Z as privated list terminator.|
%    \begin{macrocode}
\catcode`Z 3
\long\def\xintApplyInline #1#2%
{%
  \long\expandafter\def\expandafter\XINT_inline_macro
  \expandafter ##\expandafter 1\expandafter {#1{##1}}%
  \XINT_xflet\XINT_inline_b #2Z% this Z has catcode 3
}%
\def\XINT_inline_b
{%
    \ifx\XINT_token Z\expandafter\xint_gobble_i
    \else\expandafter\XINT_inline_d\fi
}%
\long\def\XINT_inline_d #1%
{%
  \long\def\XINT_item{{#1}}\XINT_xflet\XINT_inline_e
}%
\def\XINT_inline_e
{%
    \ifx\XINT_token Z\expandafter\XINT_inline_w
    \else\expandafter\XINT_inline_f\fi
}%
\def\XINT_inline_f
{%
  \expandafter\XINT_inline_g\expandafter{\XINT_inline_macro {##1}}%
}%
\long\def\XINT_inline_g #1%
{%
   \expandafter\XINT_inline_macro\XINT_item
   \long\def\XINT_inline_macro ##1{#1}\XINT_inline_d
}%
\def\XINT_inline_w #1%
{%
   \expandafter\XINT_inline_macro\XINT_item
}%
%    \end{macrocode}
% \subsection{\csh{xintFor}, \csh{xintFor*}, \csh{xintBreakFor}, \csh{xintBreakForAndDo}}
% \lverb|1.09c [2013/10/09]: a new kind of loop which uses macro parameters
% #1, #2, #3, #4 rather than macros; while not expandable it survives executing
% code closing groups, like what happens in an alignment with the $& character.
% When inserted in a macro for later use, the # character must be doubled.
%
% The non-star variant works on a csv list, which it expands once, the
% star variant works on a token list, which it (repeatedly) f-expands.
%
% 1.09e adds \XINT_forever with \xintintegers, \xintdimensions, \xintrationals
% and \xintBreakFor, \xintBreakForAndDo, \xintifForFirst, \xintifForLast. On
% this occasion \xint_firstoftwo and \xint_secondoftwo are made long.
%
% 1.09f: rewrites large parts of \xintFor code in order to filter the comma
% separated list via \xintCSVtoList which gets rid of spaces. The #1 in
% \XINT_for_forever? has an initial space token which serves two purposes:
% preventing brace stripping, and stopping the expansion made by \xintcsvtolist.
% If the \XINT_forever branch is taken, the added space will not be a problem
% there.
%
% 1.09f rewrites (2013/11/03) the code which now allows all macro parameters
% from #1 to #9 in \xintFor, \xintFor*, and \XINT_forever.|
%    \begin{macrocode}
\def\XINT_tmpa #1#2{\ifnum #2<#1 \xint_afterfi {{#########2}}\fi}%
\def\XINT_tmpb #1#2{\ifnum #1<#2 \xint_afterfi {{#########2}}\fi}%
\def\XINT_tmpc #1%
{%
    \expandafter\edef \csname XINT_for_left#1\endcsname
               {\xintApplyUnbraced {\XINT_tmpa #1}{123456789}}%
    \expandafter\edef \csname XINT_for_right#1\endcsname
               {\xintApplyUnbraced {\XINT_tmpb #1}{123456789}}%
}%
\xintApplyInline \XINT_tmpc {123456789}%
\long\def\xintBreakFor      #1Z{}%
\long\def\xintBreakForAndDo #1#2Z{#1}%
\def\xintFor {\let\xintifForFirst\xint_firstoftwo
              \futurelet\XINT_token\XINT_for_ifstar }%
\def\XINT_for_ifstar {\ifx\XINT_token*\expandafter\XINT_forx
                                 \else\expandafter\XINT_for \fi }%
\catcode`U 3 % with numexpr
\catcode`V 3 % with xintfrac.sty (xint.sty not enough)
\catcode`D 3 % with dimexpr
\def\XINT_flet_zapsp
{%
    \futurelet\XINT_token\XINT_flet_sp?
}%
\def\XINT_flet_sp?
{%
    \ifx\XINT_token\XINT_sptoken
         \xint_afterfi{\expandafter\XINT_flet_zapsp\romannumeral0}%
    \else\expandafter\XINT_flet_macro
    \fi
}%
\long\def\XINT_for #1#2in#3#4#5%
{%
    \expandafter\XINT_toks\expandafter
        {\expandafter\XINT_for_d\the\numexpr #2\relax {#5}}%
    \def\XINT_flet_macro {\expandafter\XINT_for_forever?\space}%
    \expandafter\XINT_flet_zapsp #3Z%
}%
\def\XINT_for_forever? #1Z%
{%
    \ifx\XINT_token U\XINT_to_forever\fi
    \ifx\XINT_token V\XINT_to_forever\fi
    \ifx\XINT_token D\XINT_to_forever\fi
    \expandafter\the\expandafter\XINT_toks\romannumeral0\xintcsvtolist {#1}Z%
}%
\def\XINT_to_forever\fi #1\xintcsvtolist #2{\fi \XINT_forever #2}%
\long\def\XINT_forx *#1#2in#3#4#5%
{%
    \expandafter\XINT_toks\expandafter
       {\expandafter\XINT_forx_d\the\numexpr #2\relax {#5}}%
    \XINT_xflet\XINT_forx_forever? #3Z%
}%
\def\XINT_forx_forever?
{%
    \ifx\XINT_token U\XINT_to_forxever\fi
    \ifx\XINT_token V\XINT_to_forxever\fi
    \ifx\XINT_token D\XINT_to_forxever\fi
    \XINT_forx_empty?
}%
\def\XINT_to_forxever\fi #1\XINT_forx_empty? {\fi \XINT_forever }%
\catcode`U 11
\catcode`D 11
\catcode`V 11
\def\XINT_forx_empty?
{%
    \ifx\XINT_token Z\expandafter\xintBreakFor\fi
    \the\XINT_toks
}%
\long\def\XINT_for_d #1#2#3%
{%
  \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}%
  \XINT_toks {{#3}}%
  \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname
                     \the\XINT_toks   \csname XINT_for_right#1\endcsname }%
  \XINT_toks {\XINT_x\let\xintifForFirst\xint_secondoftwo\XINT_for_d #1{#2}}%
  \futurelet\XINT_token\XINT_for_last?
}%
\long\def\XINT_forx_d #1#2#3%
{%
  \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}%
  \XINT_toks {{#3}}%
  \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname
                     \the\XINT_toks   \csname XINT_for_right#1\endcsname }%
  \XINT_toks {\XINT_x\let\xintifForFirst\xint_secondoftwo\XINT_forx_d #1{#2}}%
  \XINT_xflet\XINT_for_last?
}%
\def\XINT_for_last?
{%
    \let\xintifForLast\xint_secondoftwo
    \ifx\XINT_token Z\let\xintifForLast\xint_firstoftwo
        \xint_afterfi{\xintBreakForAndDo{\XINT_x\xint_gobble_i Z}}\fi
    \the\XINT_toks
}%
%    \end{macrocode}
% \subsection{\csh{XINT_forever}, \csh{xintintegers}, \csh{xintdimensions}, \csh{xintrationals}}
% \lverb|New with 1.09e. But this used inadvertently \xintiadd/\xintimul which
% have the unnecessary \xintnum overhead. Changed in 1.09f to use
% \xintiiadd/\xintiimul which do not have this overhead. Also 1.09f uses
% \xintZapSpacesB for the \xintrationals case to get rid of leading and ending
% spaces in the #4 and #5 delimited parameters of \XINT_forever_opt_a
% (for \xintintegers and \xintdimensions this is not necessary, due to the use
% of \numexpr resp. \dimexpr in \XINT_?expr_Ua, resp.\XINT_?expr_Da).|
%    \begin{macrocode}
\catcode`U 3
\catcode`D 3
\catcode`V 3
\let\xintegers      U%
\let\xintintegers   U%
\let\xintdimensions D%
\let\xintrationals  V%
\def\XINT_forever #1%
{%
  \expandafter\XINT_forever_a
  \csname XINT_?expr_\ifx#1UU\else\ifx#1DD\else V\fi\fi a\expandafter\endcsname
  \csname XINT_?expr_\ifx#1UU\else\ifx#1DD\else V\fi\fi i\expandafter\endcsname
  \csname XINT_?expr_\ifx#1UU\else\ifx#1DD\else V\fi\fi \endcsname
}%
\catcode`U 11
\catcode`D 11
\catcode`V 11
\def\XINT_?expr_Ua #1#2%
   {\expandafter{\expandafter\numexpr\the\numexpr #1\expandafter\relax
                              \expandafter\relax\expandafter}%
    \expandafter{\the\numexpr #2}}%
\def\XINT_?expr_Da #1#2%
   {\expandafter{\expandafter\dimexpr\number\dimexpr #1\expandafter\relax
                 \expandafter s\expandafter p\expandafter\relax\expandafter}%
    \expandafter{\number\dimexpr #2}}%
\catcode`Z 11
\def\XINT_?expr_Va #1#2%
{%
    \expandafter\XINT_?expr_Vb\expandafter
          {\romannumeral-`0\xintrawwithzeros{\xintZapSpacesB{#2}}}%
          {\romannumeral-`0\xintrawwithzeros{\xintZapSpacesB{#1}}}%
}%
\catcode`Z 3
\def\XINT_?expr_Vb #1#2{\expandafter\XINT_?expr_Vc #2.#1.}%
\def\XINT_?expr_Vc #1/#2.#3/#4.%
{%
     \xintifEq {#2}{#4}%
       {\XINT_?expr_Vf {#3}{#1}{#2}}%
       {\expandafter\XINT_?expr_Vd\expandafter
        {\romannumeral0\xintiimul {#2}{#4}}%
        {\romannumeral0\xintiimul {#1}{#4}}%
        {\romannumeral0\xintiimul {#2}{#3}}%
       }%
}%
\def\XINT_?expr_Vd #1#2#3{\expandafter\XINT_?expr_Ve\expandafter {#2}{#3}{#1}}%
\def\XINT_?expr_Ve #1#2{\expandafter\XINT_?expr_Vf\expandafter {#2}{#1}}%
\def\XINT_?expr_Vf #1#2#3{{#2/#3}{{0}{#1}{#2}{#3}}}%
\def\XINT_?expr_Ui {{\numexpr 1\relax}{1}}%
\def\XINT_?expr_Di {{\dimexpr 0pt\relax}{65536}}%
\def\XINT_?expr_Vi {{1/1}{0111}}%
\def\XINT_?expr_U #1#2%
   {\expandafter{\expandafter\numexpr\the\numexpr #1+#2\relax\relax}{#2}}%
\def\XINT_?expr_D #1#2%
   {\expandafter{\expandafter\dimexpr\the\numexpr #1+#2\relax sp\relax}{#2}}%
\def\XINT_?expr_V #1#2{\XINT_?expr_Vx #2}%
\def\XINT_?expr_Vx #1#2%
{%
     \expandafter\XINT_?expr_Vy\expandafter
        {\romannumeral0\xintiiadd {#1}{#2}}{#2}%
}%
\def\XINT_?expr_Vy #1#2#3#4%
{%
     \expandafter{\romannumeral0\xintiiadd {#3}{#1}/#4}{{#1}{#2}{#3}{#4}}%
}%
\def\XINT_forever_a #1#2#3#4%
{%
    \ifx #4[\expandafter\XINT_forever_opt_a
       \else\expandafter\XINT_forever_b
    \fi #1#2#3#4%
}%
\def\XINT_forever_b #1#2#3Z{\expandafter\XINT_forever_c\the\XINT_toks #2#3}%
\long\def\XINT_forever_c #1#2#3#4#5%
    {\expandafter\XINT_forever_d\expandafter #2#4#5{#3}Z}%
\def\XINT_forever_opt_a #1#2#3[#4+#5]#6Z%
{%
    \expandafter\expandafter\expandafter
    \XINT_forever_opt_c\expandafter\the\expandafter\XINT_toks
    \romannumeral-`0#1{#4}{#5}#3%
}%
\long\def\XINT_forever_opt_c #1#2#3#4#5#6{\XINT_forever_d #2{#4}{#5}#6{#3}Z}%
\long\def\XINT_forever_d #1#2#3#4#5%
{%
  \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#5}%
  \XINT_toks {{#2}}%
  \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname
                     \the\XINT_toks   \csname XINT_for_right#1\endcsname }%
  \XINT_x
  \let\xintifForFirst\xint_secondoftwo
  \expandafter\XINT_forever_d\expandafter #1\romannumeral-`0#4{#2}{#3}#4{#5}%
}%
%    \end{macrocode}
% \subsection{\csh{xintForpair}, \csh{xintForthree}, \csh{xintForfour}}
% \lverb|1.09c.
% 
% [2013/11/02] 1.09f \xintForpair delegate to \xintCSVtoList and its
% \xintZapSpacesB the handling of spaces. Does not share code with \xintFor
% anymore.
%
% [2013/11/03] 1.09f: \xintForpair extended to accept #1#2, #2#3 etc... up to
% #8#9, \xintForthree, #1#2#3 up to #7#8#9, \xintForfour id. |
%    \begin{macrocode}
\catcode`j 3
\long\def\xintForpair #1#2#3in#4#5#6%
{%
    \let\xintifForFirst\xint_firstoftwo
    \XINT_toks  {\XINT_forpair_d #2{#6}}%
    \expandafter\the\expandafter\XINT_toks #4jZ%
}%
\long\def\XINT_forpair_d #1#2#3(#4)#5%
{%
  \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}%
  \XINT_toks \expandafter{\romannumeral0\xintcsvtolist{ #4}}%
  \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname
      \the\XINT_toks \csname XINT_for_right\the\numexpr#1+\xint_c_i\endcsname}%
  \let\xintifForLast\xint_secondoftwo
  \ifx #5j\expandafter\xint_firstoftwo
     \else\expandafter\xint_secondoftwo
  \fi
  {\let\xintifForLast\xint_firstoftwo
   \xintBreakForAndDo {\XINT_x \xint_gobble_i Z}}%
  \XINT_x
  \let\xintifForFirst\xint_secondoftwo\XINT_forpair_d #1{#2}%
}%
\long\def\xintForthree #1#2#3in#4#5#6%
{%
    \let\xintifForFirst\xint_firstoftwo
    \XINT_toks  {\XINT_forthree_d #2{#6}}%
    \expandafter\the\expandafter\XINT_toks #4jZ%
}%
\long\def\XINT_forthree_d #1#2#3(#4)#5%
{%
  \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}%
  \XINT_toks \expandafter{\romannumeral0\xintcsvtolist{ #4}}%
  \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname
      \the\XINT_toks \csname XINT_for_right\the\numexpr#1+\xint_c_ii\endcsname}%
  \let\xintifForLast\xint_secondoftwo
  \ifx #5j\expandafter\xint_firstoftwo
     \else\expandafter\xint_secondoftwo
  \fi
  {\let\xintifForLast\xint_firstoftwo
   \xintBreakForAndDo {\XINT_x \xint_gobble_i Z}}%
  \XINT_x
  \let\xintifForFirst\xint_secondoftwo\XINT_forthree_d #1{#2}%
}%
\long\def\xintForfour #1#2#3in#4#5#6%
{%
    \let\xintifForFirst\xint_firstoftwo
    \XINT_toks  {\XINT_forfour_d #2{#6}}%
    \expandafter\the\expandafter\XINT_toks #4jZ%
}%
\long\def\XINT_forfour_d #1#2#3(#4)#5%
{%
  \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}%
  \XINT_toks \expandafter{\romannumeral0\xintcsvtolist{ #4}}%
  \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname
     \the\XINT_toks \csname XINT_for_right\the\numexpr#1+\xint_c_iii\endcsname}%
  \let\xintifForLast\xint_secondoftwo
  \ifx #5j\expandafter\xint_firstoftwo
     \else\expandafter\xint_secondoftwo
  \fi
  {\let\xintifForLast\xint_firstoftwo
   \xintBreakForAndDo {\XINT_x \xint_gobble_i Z}}%
  \XINT_x
  \let\xintifForFirst\xint_secondoftwo\XINT_forfour_d #1{#2}%
}%
\catcode`Z 11
\catcode`j 11
%    \end{macrocode}
% \subsection{\csh{xintAssign}, \csh{xintAssignArray}, \csh{xintDigitsOf}}
% \lverb|\xintAssign {a}{b}..{z}\to\A\B...\Z or \xintAssignArray {a}{b}..{z}\to\U|
%    \begin{macrocode}
\def\xintAssign{\def\XINT_flet_macro {\XINT_assign_fork}\XINT_flet_zapsp }%
\def\XINT_assign_fork
{%
    \let\XINT_assign_def\def
    \ifx\XINT_token[\expandafter\XINT_assign_opt
               \else\expandafter\XINT_assign_a
    \fi
}%
\def\XINT_assign_opt [#1]%
{%
    \ifcsname #1def\endcsname
      \expandafter\let\expandafter\XINT_assign_def \csname #1def\endcsname
    \else
      \expandafter\let\expandafter\XINT_assign_def \csname xint#1def\endcsname
    \fi
    \XINT_assign_a
}%
\long\def\XINT_assign_a #1\to
{%
    \expandafter\XINT_assign_b\romannumeral-`0#1{}\to
}%
\long\def\XINT_assign_b #1% attention to the # at the beginning of next line
#{%
    \def\xint_temp {#1}%
    \ifx\empty\xint_temp
        \expandafter\XINT_assign_c
    \else
        \expandafter\XINT_assign_d
    \fi
}%
\long\def\XINT_assign_c #1#2\to #3%
{%
    \XINT_assign_def #3{#1}%
    \def\xint_temp {#2}%
    \unless\ifx\empty\xint_temp\xint_afterfi{\XINT_assign_b #2\to }\fi
}%
\def\XINT_assign_d #1\to #2% normally #1 is {} here.
{%
    \expandafter\XINT_assign_def\expandafter #2\expandafter{\xint_temp}%
}%
\def\xintRelaxArray #1%
{%
    \edef\XINT_restoreescapechar {\escapechar\the\escapechar\relax}%
    \escapechar -1
    \expandafter\def\expandafter\xint_arrayname\expandafter {\string #1}%
    \XINT_restoreescapechar
    \xintiloop [\csname\xint_arrayname 0\endcsname+-1]
      \global
      \expandafter\let\csname\xint_arrayname\xintiloopindex\endcsname\relax
      \ifnum \xintiloopindex > \xint_c_
    \repeat
    \global\expandafter\let\csname\xint_arrayname 00\endcsname\relax
    \global\let #1\relax
}%
\def\xintAssignArray{\def\XINT_flet_macro {\XINT_assignarray_fork}%
                     \XINT_flet_zapsp }%
\def\XINT_assignarray_fork
{%
    \let\XINT_assignarray_def\def
    \ifx\XINT_token[\expandafter\XINT_assignarray_opt
               \else\expandafter\XINT_assignarray
    \fi
}%
\def\XINT_assignarray_opt [#1]%
{%
    \ifcsname #1def\endcsname
      \expandafter\let\expandafter\XINT_assignarray_def \csname #1def\endcsname
    \else
      \expandafter\let\expandafter\XINT_assignarray_def
                                  \csname xint#1def\endcsname
    \fi
    \XINT_assignarray
}%
\long\def\XINT_assignarray #1\to #2%
{%
    \edef\XINT_restoreescapechar {\escapechar\the\escapechar\relax }%
    \escapechar -1
    \expandafter\def\expandafter\xint_arrayname\expandafter {\string #2}%
    \XINT_restoreescapechar
    \def\xint_itemcount {0}%
    \expandafter\XINT_assignarray_loop \romannumeral-`0#1\xint_relax
    \csname\xint_arrayname 00\expandafter\endcsname
    \csname\xint_arrayname 0\expandafter\endcsname
    \expandafter {\xint_arrayname}#2%
}%
\long\def\XINT_assignarray_loop #1%
{%
    \def\xint_temp {#1}%
    \ifx\xint_brelax\xint_temp
       \expandafter\def\csname\xint_arrayname 0\expandafter\endcsname
                   \expandafter{\the\numexpr\xint_itemcount}%
       \expandafter\expandafter\expandafter\XINT_assignarray_end
    \else
       \expandafter\def\expandafter\xint_itemcount\expandafter
                   {\the\numexpr\xint_itemcount+\xint_c_i}%
       \expandafter\XINT_assignarray_def
          \csname\xint_arrayname\xint_itemcount\expandafter\endcsname
             \expandafter{\xint_temp }%
       \expandafter\XINT_assignarray_loop
    \fi
}%
\def\XINT_assignarray_end #1#2#3#4%
{%
    \def #4##1%
    {%
        \romannumeral0\expandafter #1\expandafter{\the\numexpr ##1}%
    }%
    \def #1##1%
    {%
        \ifnum ##1<\xint_c_
            \xint_afterfi {\xintError:ArrayIndexIsNegative\space }%
        \else
        \xint_afterfi {%
              \ifnum ##1>#2
                  \xint_afterfi {\xintError:ArrayIndexBeyondLimit\space }%
              \else\xint_afterfi
       {\expandafter\expandafter\expandafter\space\csname #3##1\endcsname}%
              \fi}%
        \fi
     }%
}%
\let\xintDigitsOf\xintAssignArray
\let\XINT_tmpa\relax \let\XINT_tmpb\relax \let\XINT_tmpc\relax
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xinttools>\relax
%\def<*xintcore>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xinttools>
%<*xintcore>
%
% \StoreCodelineNo {xinttools}
%
% \section{Package \xintcorenameimp implementation}
% \label{sec:coreimp}
%
% \localtableofcontents
%
% Got split off from \xintnameimp with release |1.1| (macros |\XINT_SQ|,
% |\xintLDg|, |\xintHalf| which are dependencies of |\XINT_div_prepare| were
% forgotten and they were added to the package only later with |1.1b|).
% Release |1.1| also added the new macro |\xintiiDivRound|. The package does
% not load \xinttoolsnameimp.
%
% Since release |xint 1.09a| the macros doing arithmetic operations
% apply systematically |\xintnum| to their arguments; this adds a little
% overhead but this is more convenient for using count registers even
% with infix notation; also this is what |xintfrac.sty| did all along.
% It simplifies the discussion in the documentation too.
%
% \subsection{Catcodes, \protect\eTeX{} and reload detection}
%
% The code for reload detection was initially copied from \textsc{Heiko
% Oberdiek}'s packages, then modified.
%
% The method for catcodes was also initially directly inspired by these
% packages.
%
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode64=11   % @
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \let\z\endgroup
  \expandafter\let\expandafter\x\csname ver@xintcore.sty\endcsname
  \expandafter\let\expandafter\w\csname ver@xintkernel.sty\endcsname
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
  \ifx\csname numexpr\endcsname\relax
     \y{xintcore}{\numexpr not available, aborting input}%
     \aftergroup\endinput
  \else
    \ifx\x\relax   % plain-TeX, first loading of xintcore.sty
      \ifx\w\relax % but xintkernel.sty not yet loaded.
         \def\z{\endgroup\input xintkernel.sty\relax}%
      \fi
    \else
      \def\empty {}%
      \ifx\x\empty % LaTeX, first loading,
      % variable is initialized, but \ProvidesPackage not yet seen
          \ifx\w\relax % xintkernel.sty not yet loaded.
            \def\z{\endgroup\RequirePackage{xintkernel}}%
          \fi
      \else
        \aftergroup\endinput % xintkernel already loaded.
      \fi
    \fi
  \fi
\z%
\XINTsetupcatcodes% defined in xintkernel.sty
%    \end{macrocode}
% \subsection{Package identification}
%    \begin{macrocode}
\XINT_providespackage
\ProvidesPackage{xintcore}%
  [2015/08/31 v1.1b Expandable arithmetic on big integers (jfB)]%
%    \end{macrocode}
% \subsection{More token management, constants}
%    \begin{macrocode}
\def\xint_minus_thenstop { -}%
\def\xint_gob_til_zeros_iii #1000{}%
\def\xint_gob_til_zeros_iv  #10000{}%
\def\xint_gob_til_one   #11{}%
\def\xint_gob_til_G     #1G{}%
\def\xint_gob_til_minus #1-{}%
\def\xint_gob_til_relax #1\relax {}%
\def\xint_exchangetwo_keepbraces          #1#2{{#2}{#1}}%
\def\xint_exchangetwo_keepbraces_thenstop #1#2{ {#2}{#1}}%
\def\xint_UDzerofork       #10#2#3\krof {#2}%
\def\xint_UDsignfork       #1-#2#3\krof {#2}%
\def\xint_UDwfork         #1\W#2#3\krof {#2}%
\def\xint_UDzerosfork     #100#2#3\krof {#2}%
\def\xint_UDonezerofork   #110#2#3\krof {#2}%
\def\xint_UDsignsfork     #1--#2#3\krof {#2}%
\chardef\xint_c_ix     9
\chardef\xint_c_x     10
\chardef\xint_c_ii^v  32 % not used in xint, common to xintfrac and xintbinhex
\chardef\xint_c_ii^vi 64
\mathchardef\xint_c_ixixixix 9999
\mathchardef\xint_c_x^iv    10000
\newcount\xint_c_x^viii  \xint_c_x^viii 100000000
%    \end{macrocode}
% \subsection{\csh{XINT_RQ}}
% \lverb|Cette macro renverse et ajoute le nombre minimal de zéros à
% la fin pour que la longueur soit alors multiple de 4$\ 
% \romannumeral0\XINT_RQ {}<le truc à renverser>\R\R\R\R\R\R\R\R\Z$\ 
% Attention, ceci n'est utilisé que pour des chaînes de chiffres, et donc le
% comportement avec des {..} ou autres espaces n'a fait l'objet d'aucune
% attention.|
%    \begin{macrocode}
\def\XINT_RQ #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_R #9\XINT_RQ_end_a\R\XINT_RQ {#9#8#7#6#5#4#3#2#1}%
}%
\def\XINT_RQ_end_a\R\XINT_RQ #1#2\Z
{%
    \XINT_RQ_end_b #1\Z
}%
\def\XINT_RQ_end_b #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_R
            #8\XINT_RQ_end_viii
            #7\XINT_RQ_end_vii
            #6\XINT_RQ_end_vi
            #5\XINT_RQ_end_v
            #4\XINT_RQ_end_iv
            #3\XINT_RQ_end_iii
            #2\XINT_RQ_end_ii
            \R\XINT_RQ_end_i
            \Z #2#3#4#5#6#7#8%
}%
\def\XINT_RQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}%
\def\XINT_RQ_end_vii  #1\Z #2#3#4#5#6#7#8#9\Z { #8#9000}%
\def\XINT_RQ_end_vi   #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#900}%
\def\XINT_RQ_end_v    #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#90}%
\def\XINT_RQ_end_iv   #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#9}%
\def\XINT_RQ_end_iii  #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}%
\def\XINT_RQ_end_ii   #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}%
\def\XINT_RQ_end_i      \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}%
%    \end{macrocode}
% \subsection{\csh{XINT_OQ}}
%    \begin{macrocode}
\def\XINT_OQ #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_R #9\XINT_OQ_end_a\R\XINT_OQ {#9#8#7#6#5#4#3#2#1}%
}%
\def\XINT_OQ_end_a\R\XINT_OQ #1#2\Z
{%
    \XINT_OQ_end_b #1\Z
}%
\def\XINT_OQ_end_b #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_R
            #8\XINT_OQ_end_viii
            #7\XINT_OQ_end_vii
            #6\XINT_OQ_end_vi
            #5\XINT_OQ_end_v
            #4\XINT_OQ_end_iv
            #3\XINT_OQ_end_iii
            #2\XINT_OQ_end_ii
            \R\XINT_OQ_end_i
            \Z #2#3#4#5#6#7#8%
}%
\def\XINT_OQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}%
\def\XINT_OQ_end_vii  #1\Z #2#3#4#5#6#7#8#9\Z { #8#90000000}%
\def\XINT_OQ_end_vi   #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#9000000}%
\def\XINT_OQ_end_v    #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#900000}%
\def\XINT_OQ_end_iv   #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#90000}%
\def\XINT_OQ_end_iii  #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}%
\def\XINT_OQ_end_ii   #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}%
\def\XINT_OQ_end_i      \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}%
%    \end{macrocode}
% \subsection{\csh{XINT_SQ}}
%    \begin{macrocode}
\def\XINT_SQ #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_R #8\XINT_SQ_end_a\R\XINT_SQ {#8#7#6#5#4#3#2#1}%
}%
\def\XINT_SQ_end_a\R\XINT_SQ #1#2\Z
{%
    \XINT_SQ_end_b #1\Z
}%
\def\XINT_SQ_end_b #1#2#3#4#5#6#7%
{%
    \xint_gob_til_R
            #7\XINT_SQ_end_vii
            #6\XINT_SQ_end_vi
            #5\XINT_SQ_end_v
            #4\XINT_SQ_end_iv
            #3\XINT_SQ_end_iii
            #2\XINT_SQ_end_ii
            \R\XINT_SQ_end_i
            \Z #2#3#4#5#6#7%
}%
\def\XINT_SQ_end_vii  #1\Z #2#3#4#5#6#7#8\Z { #8}%
\def\XINT_SQ_end_vi   #1\Z #2#3#4#5#6#7#8\Z { #7#8000000}%
\def\XINT_SQ_end_v    #1\Z #2#3#4#5#6#7#8\Z { #6#7#800000}%
\def\XINT_SQ_end_iv   #1\Z #2#3#4#5#6#7#8\Z { #5#6#7#80000}%
\def\XINT_SQ_end_iii  #1\Z #2#3#4#5#6#7#8\Z { #4#5#6#7#8000}%
\def\XINT_SQ_end_ii   #1\Z #2#3#4#5#6#7#8\Z { #3#4#5#6#7#800}%
\def\XINT_SQ_end_i      \Z #1#2#3#4#5#6#7\Z { #1#2#3#4#5#6#70}%
%    \end{macrocode}
% \subsection{\csh{XINT_cuz}}
%    \begin{macrocode}
\edef\xint_cleanupzeros_andstop #1#2#3#4%
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax
}%
\def\xint_cleanupzeros_nostop #1#2#3#4%
{%
    \the\numexpr #1#2#3#4\relax
}%
\def\XINT_rev_andcuz #1%
{%
    \expandafter\xint_cleanupzeros_andstop
    \romannumeral0\XINT_rord_main {}#1%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
}%
%    \end{macrocode}
% \lverb|&
% routine CleanUpZeros. Utilisée en particulier par la
% soustraction.$\ 
% INPUT:  longueur **multiple de 4**  (<-- ATTENTION)$\ 
% OUTPUT: on a retiré tous les leading zéros, on n'est **plus*
% nécessairement de longueur 4n$\ 
% Délimiteur pour _main: \W\W\W\W\W\W\W\Z avec SEPT \W|
%    \begin{macrocode}
\def\XINT_cuz #1%
{%
    \XINT_cuz_loop #1\W\W\W\W\W\W\W\Z%
}%
\def\XINT_cuz_loop #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_W #8\xint_cuz_end_a\W
    \xint_gob_til_Z #8\xint_cuz_end_A\Z
    \XINT_cuz_check_a {#1#2#3#4#5#6#7#8}%
}%
\def\xint_cuz_end_a #1\XINT_cuz_check_a #2%
{%
    \xint_cuz_end_b #2%
}%
\edef\xint_cuz_end_b #1#2#3#4#5\Z
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax
}%
\def\xint_cuz_end_A \Z\XINT_cuz_check_a #1{ 0}%
\def\XINT_cuz_check_a #1%
{%
    \expandafter\XINT_cuz_check_b\the\numexpr #1\relax
}%
\def\XINT_cuz_check_b #1%
{%
    \xint_gob_til_zero #1\xint_cuz_backtoloop 0\XINT_cuz_stop #1%
}%
\def\XINT_cuz_stop #1\W #2\Z{ #1}%
\def\xint_cuz_backtoloop 0\XINT_cuz_stop 0{\XINT_cuz_loop }%
%    \end{macrocode}
% \subsection{\csh{xintNum}}
% \lverb|&
% For example \xintNum {----+-+++---+----000000000000003}$\ 
% 1.05 defines \xintiNum, which allows redefinition of \xintNum by xintfrac.sty
% Slightly modified in 1.06b (\R->\xint_relax) to avoid initial re-scan of
% input stack (while still allowing empty #1). In versions earlier than 1.09a
% it was entirely up to the user to apply \xintnum; starting with 1.09a
% arithmetic
% macros of xint.sty (like earlier already xintfrac.sty with its own \xintnum)
% make use of \xintnum. This allows arguments to
% be count registers, or even \numexpr arbitrary long expressions (with the
% trick of braces, see the user documentation).
%
% Note (22/06/14): \xintiNum jamais utilisé sous ce nom, le supprimer?
% \XINT_num maintenant utilisé par le parseur de xintexpr.|
%    \begin{macrocode}
\def\xintiNum {\romannumeral0\xintinum }%
\def\xintinum #1%
{%
    \expandafter\XINT_num_loop
    \romannumeral-`0#1\xint_relax\xint_relax\xint_relax\xint_relax
                      \xint_relax\xint_relax\xint_relax\xint_relax\Z
}%
\let\xintNum\xintiNum \let\xintnum\xintinum
\def\XINT_num #1%
{%
    \XINT_num_loop #1\xint_relax\xint_relax\xint_relax\xint_relax
                     \xint_relax\xint_relax\xint_relax\xint_relax\Z
}%
\def\XINT_num_loop #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_xint_relax #8\XINT_num_end\xint_relax
    \XINT_num_NumEight #1#2#3#4#5#6#7#8%
}%
\edef\XINT_num_end\xint_relax\XINT_num_NumEight #1\xint_relax #2\Z
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1+\xint_c_\relax
}%
\def\XINT_num_NumEight #1#2#3#4#5#6#7#8%
{%
    \ifnum \numexpr #1#2#3#4#5#6#7#8+\xint_c_= \xint_c_
      \xint_afterfi {\expandafter\XINT_num_keepsign_a
                     \the\numexpr #1#2#3#4#5#6#7#81\relax}%
    \else
      \xint_afterfi {\expandafter\XINT_num_finish
                     \the\numexpr #1#2#3#4#5#6#7#8\relax}%
    \fi
}%
\def\XINT_num_keepsign_a #1%
{%
    \xint_gob_til_one#1\XINT_num_gobacktoloop 1\XINT_num_keepsign_b
}%
\def\XINT_num_gobacktoloop 1\XINT_num_keepsign_b {\XINT_num_loop }%
\def\XINT_num_keepsign_b #1{\XINT_num_loop -}%
\def\XINT_num_finish #1\xint_relax #2\Z { #1}%
%    \end{macrocode}
% \subsection{\csh{xintSgn}, \csh{xintiiSgn}, \csh{XINT_Sgn}, \csh{XINT_cntSgn}}
% \lverb|&
% Changed in 1.05. Earlier code was unnecessarily strange. 1.09a with \xintnum
%
% 1.09i defines \XINT_Sgn and \XINT_cntSgn (was \XINT__Sgn in 1.09i) for reasons
% of internal optimizations.
%
% xintfrac.sty will overwrite \xintsgn with use of \xintraw rather than
% \xintnum, naturally.|
%    \begin{macrocode}
\def\xintiiSgn {\romannumeral0\xintiisgn }%
\def\xintiisgn #1%
{%
    \expandafter\XINT_sgn \romannumeral-`0#1\Z%
}%
\def\xintSgn {\romannumeral0\xintsgn }%
\def\xintsgn #1%
{%
    \expandafter\XINT_sgn \romannumeral0\xintnum{#1}\Z%
}%
\def\XINT_sgn #1#2\Z
{%
    \xint_UDzerominusfork
      #1-{ 0}%
      0#1{ -1}%
       0-{ 1}%
    \krof
}%
\def\XINT_Sgn #1#2\Z
{%
    \xint_UDzerominusfork
      #1-{0}%
      0#1{-1}%
       0-{1}%
    \krof
}%
\def\XINT_cntSgn #1#2\Z
{%
    \xint_UDzerominusfork
      #1-\xint_c_
      0#1\m@ne % I will not allocate a count only for -1?
       0-\xint_c_i
    \krof
}%
%    \end{macrocode}
% \subsection{\csh{xintiOpp}}
% \lverb|\xintnum added in 1.09a|
%    \begin{macrocode}
\def\xintiiOpp {\romannumeral0\xintiiopp }%
\def\xintiiopp #1%
{%
    \expandafter\XINT_opp \romannumeral-`0#1%
}%
\def\xintiOpp {\romannumeral0\xintiopp }%
\def\xintiopp #1%
{%
    \expandafter\XINT_opp \romannumeral0\xintnum{#1}%
}%
\let\xintOpp\xintiOpp \let\xintopp\xintiopp
\def\XINT_Opp #1{\romannumeral0\XINT_opp #1}%
\def\XINT_opp #1%
{%
    \xint_UDzerominusfork
      #1-{ 0}%      zero
      0#1{ }%     negative
       0-{ -#1}%  positive
    \krof
}%
%    \end{macrocode}
% \subsection{\csh{xintiAbs}, \csh{xintiiAbs}}
% \lverb|Release 1.09a has now \xintiabs which does \xintnum and this is
% inherited by DecSplit, by Sqr, and macros of xintgcd.sty. Attention, car ces
% macros de toute façon doivent passer à la valeur absolue et donc en profite
% pour faire le \xintnum, mais pour optimisation sans overhead il vaut mieux
% utiliser \xintiiAbs ou autre point d'accès.|
%    \begin{macrocode}
\def\xintiiAbs {\romannumeral0\xintiiabs }%
\def\xintiiabs #1%
{%
    \expandafter\XINT_abs \romannumeral-`0#1%
}%
\def\xintiAbs {\romannumeral0\xintiabs }%
\def\xintiabs #1%
{%
    \expandafter\XINT_abs \romannumeral0\xintnum{#1}%
}%
\let\xintAbs\xintiAbs \let\xintabs\xintiabs
\def\XINT_Abs #1{\romannumeral0\XINT_abs #1}%
\def\XINT_abs #1%
{%
    \xint_UDsignfork
      #1{ }%
       -{ #1}%
    \krof
}%
%    \end{macrocode}
% \subsection{\csh{xintFDg}, \csh{xintiiFDg}}
% \lverb|&
% FIRST DIGIT. Code simplified in 1.05.
% And prepared for redefinition by xintfrac to parse through \xintNum. Version
% 1.09a inserts the \xintnum already here.|
%    \begin{macrocode}
\def\xintiiFDg {\romannumeral0\xintiifdg }%
\def\xintiifdg #1%
{%
    \expandafter\XINT_fdg \romannumeral-`0#1\W\Z
}%
\def\xintFDg {\romannumeral0\xintfdg }%
\def\xintfdg #1%
{%
    \expandafter\XINT_fdg \romannumeral0\xintnum{#1}\W\Z
}%
\def\XINT_FDg #1{\romannumeral0\XINT_fdg #1\W\Z }%
\def\XINT_fdg #1#2#3\Z
{%
    \xint_UDzerominusfork
      #1-{ 0}%   zero
      0#1{ #2}%  negative
       0-{ #1}%  positive
    \krof
}%
%    \end{macrocode}
% \subsection{\csh{xintLDg}, \csh{xintiiLDg}}
% \lverb|&
% LAST DIGIT. Simplified in 1.05. And prepared for extension by xintfrac
%    to parse through \xintNum. Release 1.09a adds the \xintnum already here,
%    and this propagates to \xintOdd, etc... 1.09e The \xintiiLDg is for
%    defining \xintiiOdd which is used once (currently) elsewhere .
%
%    bug fix (1.1b): \xintiiLDg is needed by the division macros next, thus
%    it needs to be in the xintcore.sty|
%    \begin{macrocode}
\def\xintiiLDg {\romannumeral0\xintiildg }%
\def\xintiildg #1%
{%
    \expandafter\XINT_ldg\expandafter {\romannumeral-`0#1}%
}%
\def\xintLDg {\romannumeral0\xintldg }%
\def\xintldg #1%
{%
    \expandafter\XINT_ldg\expandafter {\romannumeral0\xintnum{#1}}%
}%
\def\XINT_LDg #1{\romannumeral0\XINT_ldg {#1}}%
\def\XINT_ldg #1%
{%
    \expandafter\XINT_ldg_\romannumeral0\xintreverseorder {#1}\Z
}%
\def\XINT_ldg_ #1#2\Z{ #1}%
%    \end{macrocode}
% \subsection{\csh{xintDouble}}
% \lverb|v1.08|
%    \begin{macrocode}
\def\xintDouble {\romannumeral0\xintdouble }%
\def\xintdouble #1%
{%
     \expandafter\XINT_dbl\romannumeral-`0#1%
     \R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W
}%
\def\XINT_dbl #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_dbl_zero
      0#1\XINT_dbl_neg
       0-{\XINT_dbl_pos #1}%
    \krof
}%
\def\XINT_dbl_zero #1\Z \W\W\W\W\W\W\W { 0}%
\def\XINT_dbl_neg
   {\expandafter\xint_minus_thenstop\romannumeral0\XINT_dbl_pos }%
\def\XINT_dbl_pos
{%
    \expandafter\XINT_dbl_a \expandafter{\expandafter}\expandafter 0%
    \romannumeral0\XINT_SQ {}%
}%
\def\XINT_dbl_a #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_W #9\XINT_dbl_end_a\W
    \expandafter\XINT_dbl_b
    \the\numexpr \xint_c_x^viii+#2+\xint_c_ii*#9#8#7#6#5#4#3\relax {#1}%
}%
\def\XINT_dbl_b 1#1#2#3#4#5#6#7#8#9%
{%
    \XINT_dbl_a {#2#3#4#5#6#7#8#9}{#1}%
}%
\def\XINT_dbl_end_a #1+#2+#3\relax #4%
{%
    \expandafter\XINT_dbl_end_b #2#4%
}%
\edef\XINT_dbl_end_b #1#2#3#4#5#6#7#8%
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax
}%
%    \end{macrocode}
% \subsection{\csh{xintHalf}}
% \lverb|v1.08. Relase 1.1 left it in xint.sty, but it is needed by the 
% division routines included in xintcore.sty. Thus moved here for bugfix
% release 1.1b.
% Also \XINT_SQ which it uses. Moved here \xintDouble as well by sympathy.|
%    \begin{macrocode}
\def\xintHalf {\romannumeral0\xinthalf }%
\def\xinthalf #1%
{%
     \expandafter\XINT_half\romannumeral-`0#1%
     \R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W
}%
\def\XINT_half #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_half_zero
      0#1\XINT_half_neg
       0-{\XINT_half_pos #1}%
    \krof
}%
\def\XINT_half_zero #1\Z \W\W\W\W\W\W\W { 0}%
\def\XINT_half_neg {\expandafter\XINT_opp\romannumeral0\XINT_half_pos }%
\def\XINT_half_pos {\expandafter\XINT_half_a\romannumeral0\XINT_SQ {}}%
\def\XINT_half_a #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_W #8\XINT_half_dont\W
    \expandafter\XINT_half_b
    \the\numexpr \xint_c_x^viii+\xint_c_v*#7#6#5#4#3#2#1\relax #8%
}%
\edef\XINT_half_dont\W\expandafter\XINT_half_b
    \the\numexpr \xint_c_x^viii+\xint_c_v*#1#2#3#4#5#6#7\relax \W\W\W\W\W\W\W
{%
    \noexpand\expandafter\space
    \noexpand\the\numexpr (#1#2#3#4#5#6#7+\xint_c_i)/\xint_c_ii-\xint_c_i \relax
}%
\def\XINT_half_b 1#1#2#3#4#5#6#7#8%
{%
    \XINT_half_c {#2#3#4#5#6#7}{#1}%
}%
\def\XINT_half_c #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_W #3\XINT_half_end_a #2\W
    \expandafter\XINT_half_d
    \the\numexpr \xint_c_x^viii+\xint_c_v*#9#8#7#6#5#4#3+#2\relax {#1}%
}%
\def\XINT_half_d 1#1#2#3#4#5#6#7#8#9%
{%
    \XINT_half_c {#2#3#4#5#6#7#8#9}{#1}%
}%
\def\XINT_half_end_a #1\W #2\relax #3%
{%
    \xint_gob_til_zero #1\XINT_half_end_b 0\space #1#3%
}%
\edef\XINT_half_end_b 0\space 0#1#2#3#4#5#6#7%
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7\relax
}%
%    \end{macrocode}
% \subsection{\csh{xintDec}}
% \lverb!v1.08!
%    \begin{macrocode}
\def\xintDec {\romannumeral0\xintdec }%
\def\xintdec #1%
{%
     \expandafter\XINT_dec\romannumeral-`0#1%
     \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W
}%
\def\XINT_dec #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_dec_zero
      0#1\XINT_dec_neg
       0-{\XINT_dec_pos #1}%
    \krof
}%
\def\XINT_dec_zero #1\W\W\W\W\W\W\W\W { -1}%
\def\XINT_dec_neg
   {\expandafter\xint_minus_thenstop\romannumeral0\XINT_inc_pos }%
\def\XINT_dec_pos
{%
    \expandafter\XINT_dec_a \expandafter{\expandafter}%
    \romannumeral0\XINT_OQ {}%
}%
\def\XINT_dec_a #1#2#3#4#5#6#7#8#9%
{%
    \expandafter\XINT_dec_b
    \the\numexpr 11#9#8#7#6#5#4#3#2-\xint_c_i\relax {#1}%
}%
\def\XINT_dec_b 1#1%
{%
    \xint_gob_til_one #1\XINT_dec_A 1\XINT_dec_c
}%
\def\XINT_dec_c #1#2#3#4#5#6#7#8#9{\XINT_dec_a {#1#2#3#4#5#6#7#8#9}}%
\def\XINT_dec_A 1\XINT_dec_c #1#2#3#4#5#6#7#8#9%
   {\XINT_dec_B {#1#2#3#4#5#6#7#8#9}}%
\def\XINT_dec_B #1#2\W\W\W\W\W\W\W\W
{%
    \expandafter\XINT_dec_cleanup
    \romannumeral0\XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
    #1%
}%
\edef\XINT_dec_cleanup #1#2#3#4#5#6#7#8%
    {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax }%
%    \end{macrocode}
% \subsection{\csh{xintInc}}
% \lverb!v1.08!
%    \begin{macrocode}
\def\xintInc {\romannumeral0\xintinc }%
\def\xintinc #1%
{%
     \expandafter\XINT_inc\romannumeral-`0#1%
     \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W
}%
\def\XINT_inc #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_inc_zero
      0#1\XINT_inc_neg
       0-{\XINT_inc_pos #1}%
    \krof
}%
\def\XINT_inc_zero #1\W\W\W\W\W\W\W\W { 1}%
\def\XINT_inc_neg {\expandafter\XINT_opp\romannumeral0\XINT_dec_pos }%
\def\XINT_inc_pos
{%
    \expandafter\XINT_inc_a \expandafter{\expandafter}%
    \romannumeral0\XINT_OQ {}%
}%
\def\XINT_inc_a #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_W #9\XINT_inc_end\W
    \expandafter\XINT_inc_b
    \the\numexpr 10#9#8#7#6#5#4#3#2+\xint_c_i\relax {#1}%
}%
\def\XINT_inc_b 1#1%
{%
    \xint_gob_til_zero #1\XINT_inc_A 0\XINT_inc_c
}%
\def\XINT_inc_c #1#2#3#4#5#6#7#8#9{\XINT_inc_a {#1#2#3#4#5#6#7#8#9}}%
\def\XINT_inc_A 0\XINT_inc_c #1#2#3#4#5#6#7#8#9%
               {\XINT_dec_B {#1#2#3#4#5#6#7#8#9}}%
\def\XINT_inc_end\W #1\relax #2{ 1#2}%
%    \end{macrocode}
% \lverb|&
% -----------------------------------------------------------------$\ 
% -----------------------------------------------------------------$\ 
% ARITHMETIC OPERATIONS: ADDITION, SUBTRACTION, SUMS,
% MULTIPLICATION, PRODUCTS, FACTORIAL, POWERS, EUCLIDEAN DIVISION.
%
% Release 1.03 re-organizes sub-routines to facilitate future developments: the
% diverse variants of addition, with diverse conditions on inputs and output are
% first listed; they will be used in multiplication, or in the summation, or in
% the power routines. I am aware that the commenting is close to non-existent,
% sorry about that.
%
% ADDITION I: \XINT_add_A
%
% INPUT:$\ 
% \romannumeral0\XINT_add_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z$\ 
%     1.    <N1> et <N2> renversés $\ 
%     2.    de longueur 4n (avec des leading zéros éventuels)$\ 
%     3.    l'un des deux ne doit pas se terminer par 0000$\$relax
% [Donc on peut avoir 0000 comme input si l'autre est >0 et ne se termine pas en
% 0000 bien sûr]. On peut avoir l'un des deux vides. Mais alors l'autre ne doit
% être ni vide ni 0000.
%
% OUTPUT: la somme <N1>+<N2>, ordre normal, plus sur 4n, pas de leading zeros
% La procédure est plus rapide lorsque <N1> est le plus court des deux.$\ 
% Nota bene: (30 avril 2013). J'ai une version qui est deux fois plus rapide sur
% des nombres d'environ 1000 chiffres chacun, et qui commence à être avantageuse
% pour des nombres d'au moins 200 chiffres. Cependant il serait vraiment
% compliqué d'en étendre l'utilisation aux emplois de l'addition dans les
% autres routines, comme celle de multiplication ou celle de division; et son
% implémentation ajouterait au minimum la mesure de la longueur des summands.|
%    \begin{macrocode}
\def\XINT_add_A #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_add_az\W
    \XINT_add_AB #1{#3#4#5#6}{#2}%
}%
\def\xint_add_az\W\XINT_add_AB #1#2%
{%
    \XINT_add_AC_checkcarry #1%
}%
%    \end{macrocode}
% \lverb|&
% ici #2 est prévu pour l'addition, mais attention il devra être renversé
% pour \numexpr. #3 = résultat partiel. #4 = chiffres qui restent. On vérifie si
% le deuxième nombre s'arrête.|
%    \begin{macrocode}
\def\XINT_add_AB #1#2#3#4\W\X\Y\Z #5#6#7#8%
{%
    \xint_gob_til_W #5\xint_add_bz\W
    \XINT_add_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z
}%
\def\XINT_add_ABE #1#2#3#4#5#6%
{%
    \expandafter\XINT_add_ABEA\the\numexpr #1+10#5#4#3#2+#6.%
}%
\def\XINT_add_ABEA #1#2#3.#4%
{%
    \XINT_add_A  #2{#3#4}%
}%
%    \end{macrocode}
% \lverb|&
% ici le deuxième nombre est fini
% #6 part à la poubelle, #2#3#4#5 est le #2 dans \XINT_add_AB
% on ne vérifie pas la retenue cette fois, mais les fois suivantes|
%    \begin{macrocode}
\def\xint_add_bz\W\XINT_add_ABE #1#2#3#4#5#6%
{%
    \expandafter\XINT_add_CC\the\numexpr #1+10#5#4#3#2.%
}%
\def\XINT_add_CC #1#2#3.#4%
{%
    \XINT_add_AC_checkcarry  #2{#3#4}% on va examiner et \'eliminer #2
}%
%    \end{macrocode}
% \lverb|&
% retenue plus chiffres qui restent de l'un des deux nombres.
% #2 = résultat partiel
% #3#4#5#6 = summand, avec plus significatif à droite|
%    \begin{macrocode}
\def\XINT_add_AC_checkcarry #1%
{%
    \xint_gob_til_zero #1\xint_add_AC_nocarry 0\XINT_add_C
}%
\def\xint_add_AC_nocarry 0\XINT_add_C #1#2\W\X\Y\Z
{%
    \expandafter
    \xint_cleanupzeros_andstop
    \romannumeral0%
    \XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
    #1%
}%
\def\XINT_add_C #1#2#3#4#5%
{%
    \xint_gob_til_W #2\xint_add_cz\W
    \XINT_add_CD {#5#4#3#2}{#1}%
}%
\def\XINT_add_CD #1%
{%
    \expandafter\XINT_add_CC\the\numexpr 1+10#1.%
}%
\def\xint_add_cz\W\XINT_add_CD #1#2{ 1#2}%
%    \end{macrocode}
% \lverb|Addition II: \XINT_addr_A.$\ 
% INPUT: \romannumeral0\XINT_addr_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z
%
% Comme \XINT_add_A, la différence principale c'est qu'elle donne son résultat
% aussi *sur 4n*, renversé. De plus cette variante accepte que l'un ou même les
% deux inputs soient vides. Utilisé par la sommation et par la division (pour
% les quotients). Et aussi par la multiplication d'ailleurs.$\ 
% INPUT: comme pour \XINT_add_A$\ 
%     1.    <N1> et <N2> renversés $\ 
%     2.    de longueur 4n (avec des leading zéros éventuels)$\ 
%     3.    l'un des deux ne doit pas se terminer par 0000$\ 
% OUTPUT: la somme <N1>+<N2>, *aussi renversée* et *sur 4n*|
%    \begin{macrocode}
\def\XINT_addr_A #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_addr_az\W
    \XINT_addr_B #1{#3#4#5#6}{#2}%
}%
\def\xint_addr_az\W\XINT_addr_B #1#2%
{%
    \XINT_addr_AC_checkcarry #1%
}%
\def\XINT_addr_B #1#2#3#4\W\X\Y\Z #5#6#7#8%
{%
    \xint_gob_til_W #5\xint_addr_bz\W
    \XINT_addr_E #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z
}%
\def\XINT_addr_E #1#2#3#4#5#6%
{%
    \expandafter\XINT_addr_ABEA\the\numexpr #1+10#5#4#3#2+#6\relax
}%
\def\XINT_addr_ABEA #1#2#3#4#5#6#7%
{%
    \XINT_addr_A  #2{#7#6#5#4#3}%
}%
\def\xint_addr_bz\W\XINT_addr_E #1#2#3#4#5#6%
{%
    \expandafter\XINT_addr_CC\the\numexpr #1+10#5#4#3#2\relax
}%
\def\XINT_addr_CC #1#2#3#4#5#6#7%
{%
    \XINT_addr_AC_checkcarry  #2{#7#6#5#4#3}%
}%
\def\XINT_addr_AC_checkcarry #1%
{%
    \xint_gob_til_zero #1\xint_addr_AC_nocarry 0\XINT_addr_C
}%
\def\xint_addr_AC_nocarry 0\XINT_addr_C #1#2\W\X\Y\Z { #1#2}%
\def\XINT_addr_C #1#2#3#4#5%
{%
    \xint_gob_til_W #2\xint_addr_cz\W
    \XINT_addr_D {#5#4#3#2}{#1}%
}%
\def\XINT_addr_D #1%
{%
    \expandafter\XINT_addr_CC\the\numexpr 1+10#1\relax
}%
\def\xint_addr_cz\W\XINT_addr_D #1#2{ #21000}%
%    \end{macrocode}
% \lverb|ADDITION III, \XINT_addm_A$\ 
% INPUT:\romannumeral0\XINT_addm_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z$\ 
%     1.    <N1> et <N2> renversés$\ 
%     2.    <N1> de longueur 4n ; <N2> non$\ 
%     3.    <N2> est *garanti au moins aussi long* que <N1>$\ 
% OUTPUT: la somme <N1>+<N2>, ordre normal, pas sur 4n, leading zeros retirés.
% Utilisé par la multiplication.|
%    \begin{macrocode}
\def\XINT_addm_A #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_addm_az\W
    \XINT_addm_AB #1{#3#4#5#6}{#2}%
}%
\def\xint_addm_az\W\XINT_addm_AB #1#2%
{%
    \XINT_addm_AC_checkcarry #1%
}%
\def\XINT_addm_AB #1#2#3#4\W\X\Y\Z #5#6#7#8%
{%
    \XINT_addm_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z
}%
\def\XINT_addm_ABE #1#2#3#4#5#6%
{%
    \expandafter\XINT_addm_ABEA\the\numexpr #1+10#5#4#3#2+#6.%
}%
\def\XINT_addm_ABEA #1#2#3.#4%
{%
    \XINT_addm_A  #2{#3#4}%
}%
\def\XINT_addm_AC_checkcarry #1%
{%
    \xint_gob_til_zero #1\xint_addm_AC_nocarry 0\XINT_addm_C
}%
\def\xint_addm_AC_nocarry 0\XINT_addm_C #1#2\W\X\Y\Z
{%
    \expandafter
    \xint_cleanupzeros_andstop
    \romannumeral0%
    \XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
    #1%
}%
\def\XINT_addm_C #1#2#3#4#5%
{%
    \xint_gob_til_W
    #5\xint_addm_cw
    #4\xint_addm_cx
    #3\xint_addm_cy
    #2\xint_addm_cz
    \W\XINT_addm_CD {#5#4#3#2}{#1}%
}%
\def\XINT_addm_CD #1%
{%
    \expandafter\XINT_addm_CC\the\numexpr 1+10#1.%
}%
\def\XINT_addm_CC #1#2#3.#4%
{%
    \XINT_addm_AC_checkcarry  #2{#3#4}%
}%
\def\xint_addm_cw
    #1\xint_addm_cx
    #2\xint_addm_cy
    #3\xint_addm_cz
    \W\XINT_addm_CD
{%
    \expandafter\XINT_addm_CDw\the\numexpr 1+#1#2#3.%
}%
\def\XINT_addm_CDw #1.#2#3\X\Y\Z
{%
    \XINT_addm_end #1#3%
}%
\def\xint_addm_cx
    #1\xint_addm_cy
    #2\xint_addm_cz
    \W\XINT_addm_CD
{%
    \expandafter\XINT_addm_CDx\the\numexpr 1+#1#2.%
}%
\def\XINT_addm_CDx #1.#2#3\Y\Z
{%
    \XINT_addm_end #1#3%
}%
\def\xint_addm_cy
    #1\xint_addm_cz
    \W\XINT_addm_CD
{%
    \expandafter\XINT_addm_CDy\the\numexpr 1+#1.%
}%
\def\XINT_addm_CDy  #1.#2#3\Z
{%
    \XINT_addm_end #1#3%
}%
\def\xint_addm_cz\W\XINT_addm_CD #1#2#3{\XINT_addm_end #1#3}%
\edef\XINT_addm_end #1#2#3#4#5%
    {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5\relax}%
%    \end{macrocode}
% \lverb|ADDITION IV, variante \XINT_addp_A$\ 
% INPUT:
% \romannumeral0\XINT_addp_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z$\ 
%     1.    <N1> et <N2> renversés$\ 
%     2.    <N1> de longueur 4n ; <N2> non$\ 
%     3.    <N2> est *garanti au moins aussi long* que <N1>$\ 
% OUTPUT: la somme <N1>+<N2>, dans l'ordre renversé, sur 4n, et en faisant
% attention de ne pas terminer en 0000.
% Utilisé par la multiplication servant pour le calcul des puissances.|
%    \begin{macrocode}
\def\XINT_addp_A #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_addp_az\W
    \XINT_addp_AB #1{#3#4#5#6}{#2}%
}%
\def\xint_addp_az\W\XINT_addp_AB #1#2%
{%
    \XINT_addp_AC_checkcarry #1%
}%
\def\XINT_addp_AC_checkcarry #1%
{%
    \xint_gob_til_zero #1\xint_addp_AC_nocarry 0\XINT_addp_C
}%
\def\xint_addp_AC_nocarry 0\XINT_addp_C
{%
    \XINT_addp_F
}%
\def\XINT_addp_AB #1#2#3#4\W\X\Y\Z #5#6#7#8%
{%
    \XINT_addp_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z
}%
\def\XINT_addp_ABE #1#2#3#4#5#6%
{%
    \expandafter\XINT_addp_ABEA\the\numexpr #1+10#5#4#3#2+#6\relax
}%
\def\XINT_addp_ABEA #1#2#3#4#5#6#7%
{%
   \XINT_addp_A  #2{#7#6#5#4#3}%<-- attention on met donc \`a droite
}%
\def\XINT_addp_C #1#2#3#4#5%
{%
    \xint_gob_til_W
    #5\xint_addp_cw
    #4\xint_addp_cx
    #3\xint_addp_cy
    #2\xint_addp_cz
    \W\XINT_addp_CD   {#5#4#3#2}{#1}%
}%
\def\XINT_addp_CD #1%
{%
    \expandafter\XINT_addp_CC\the\numexpr 1+10#1\relax
}%
\def\XINT_addp_CC #1#2#3#4#5#6#7%
{%
    \XINT_addp_AC_checkcarry  #2{#7#6#5#4#3}%
}%
\def\xint_addp_cw
    #1\xint_addp_cx
    #2\xint_addp_cy
    #3\xint_addp_cz
    \W\XINT_addp_CD
{%
    \expandafter\XINT_addp_CDw\the\numexpr \xint_c_i+10#1#2#3\relax
}%
\def\XINT_addp_CDw #1#2#3#4#5#6%
{%
    \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDw_zeros
                          0000\XINT_addp_endDw #2#3#4#5%
}%
\def\XINT_addp_endDw_zeros 0000\XINT_addp_endDw 0000#1\X\Y\Z{ #1}%
\def\XINT_addp_endDw #1#2#3#4#5\X\Y\Z{ #5#4#3#2#1}%
\def\xint_addp_cx
    #1\xint_addp_cy
    #2\xint_addp_cz
    \W\XINT_addp_CD
{%
    \expandafter\XINT_addp_CDx\the\numexpr \xint_c_i+100#1#2\relax
}%
\def\XINT_addp_CDx #1#2#3#4#5#6%
{%
    \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDx_zeros
                          0000\XINT_addp_endDx #2#3#4#5%
}%
\def\XINT_addp_endDx_zeros 0000\XINT_addp_endDx 0000#1\Y\Z{ #1}%
\def\XINT_addp_endDx #1#2#3#4#5\Y\Z{ #5#4#3#2#1}%
\def\xint_addp_cy #1\xint_addp_cz\W\XINT_addp_CD
{%
    \expandafter\XINT_addp_CDy\the\numexpr \xint_c_i+1000#1\relax
}%
\def\XINT_addp_CDy #1#2#3#4#5#6%
{%
    \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDy_zeros
                          0000\XINT_addp_endDy #2#3#4#5%
}%
\def\XINT_addp_endDy_zeros 0000\XINT_addp_endDy 0000#1\Z{ #1}%
\def\XINT_addp_endDy #1#2#3#4#5\Z{ #5#4#3#2#1}%
\def\xint_addp_cz\W\XINT_addp_CD #1#2{ #21000}%
\def\XINT_addp_F #1#2#3#4#5%
{%
    \xint_gob_til_W
    #5\xint_addp_Gw
    #4\xint_addp_Gx
    #3\xint_addp_Gy
    #2\xint_addp_Gz
    \W\XINT_addp_G   {#2#3#4#5}{#1}%
}%
\def\XINT_addp_G #1#2%
{%
    \XINT_addp_F {#2#1}%
}%
\def\xint_addp_Gw
    #1\xint_addp_Gx
    #2\xint_addp_Gy
    #3\xint_addp_Gz
    \W\XINT_addp_G #4%
{%
    \xint_gob_til_zeros_iv  #3#2#10\XINT_addp_endGw_zeros
                          0000\XINT_addp_endGw #3#2#10%
}%
\def\XINT_addp_endGw_zeros 0000\XINT_addp_endGw 0000#1\X\Y\Z{ #1}%
\def\XINT_addp_endGw #1#2#3#4#5\X\Y\Z{ #5#1#2#3#4}%
\def\xint_addp_Gx
    #1\xint_addp_Gy
    #2\xint_addp_Gz
    \W\XINT_addp_G #3%
{%
    \xint_gob_til_zeros_iv  #2#100\XINT_addp_endGx_zeros
                         0000\XINT_addp_endGx #2#100%
}%
\def\XINT_addp_endGx_zeros 0000\XINT_addp_endGx 0000#1\Y\Z{ #1}%
\def\XINT_addp_endGx #1#2#3#4#5\Y\Z{ #5#1#2#3#4}%
\def\xint_addp_Gy
    #1\xint_addp_Gz
    \W\XINT_addp_G #2%
{%
    \xint_gob_til_zeros_iv   #1000\XINT_addp_endGy_zeros
                         0000\XINT_addp_endGy #1000%
}%
\def\XINT_addp_endGy_zeros 0000\XINT_addp_endGy 0000#1\Z{ #1}%
\def\XINT_addp_endGy #1#2#3#4#5\Z{ #5#1#2#3#4}%
\def\xint_addp_Gz\W\XINT_addp_G #1#2{ #2}%
%    \end{macrocode}
% \subsection{\csh{xintiAdd}, \csh{xintiiAdd}}
% \lverb|ADDITION
% [algo plus efficace lorsque le premier argument plus long que le second]
%
% Note (octobre 2014, pendant la préparation de la sortie de 1.1)
%
% Je n'aurais pas dû l'appeler \xintAdd, mais seulement \xintiAdd. Le format
% de sortie de \xintAdd est modifié par xintfrac.sty, celui de \xintiAdd ne
% bouge pas, et \xintiiAdd reste la version stricte.|
%    \begin{macrocode}
\def\xintiiAdd {\romannumeral0\xintiiadd }%
\def\xintiiadd #1{\expandafter\xint_iiadd\romannumeral-`0#1\Z  }%
\def\xint_iiadd #1#2\Z #3%
{%
    \expandafter\XINT_add_fork\expandafter #1\romannumeral-`0#3\Z #2\Z
}%
\def\xintiAdd {\romannumeral0\xintiadd }%
\def\xintiadd #1%
{%
    \expandafter\xint_add\romannumeral0\xintnum{#1}\Z
}%
\def\xint_add #1#2\Z #3%
{%
    \expandafter\XINT_add_fork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z
}%
\let\xintAdd\xintiAdd \let\xintadd\xintiadd
\def\XINT_add_fork #1#2%
{%
    \xint_UDzerofork
      #1\XINT_add_firstiszero
      #2\XINT_add_secondiszero
       0{}%
    \krof
    \xint_UDsignsfork
          #1#2\XINT_add_minusminus
           #1-\XINT_add_minusplus
           #2-\XINT_add_plusminus
            --\XINT_add_plusplus
    \krof #1#2%
}%
\def\XINT_add_firstiszero  #1\krof #2#3\Z #4\Z { #3}%
\def\XINT_add_secondiszero #1\krof #2#3\Z #4\Z { #2#4}%
\def\XINT_add_plusplus     #1#2#3\Z #4\Z {\XINT_add_pre {#1#4}{#2#3}}%
\def\XINT_add_minusminus   #1#2#3\Z #4\Z
   {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pre {#4}{#3}}%
\def\XINT_add_minusplus    #1#2#3\Z #4\Z {\XINT_sub_pre {#2#3}{#4}}%
\def\XINT_add_plusminus    #1#2#3\Z #4\Z {\XINT_sub_pre {#1#4}{#3}}%
%    \end{macrocode}
% \lverb|positive summands|
%    \begin{macrocode}
\def\XINT_add_pre #1%
{%
  \expandafter\XINT_add_pre_b\expandafter
  {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }%
}%
\def\XINT_add_pre_b #1#2%
{%
    \expandafter\XINT_add_A
        \expandafter0\expandafter{\expandafter}%
    \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z
        \W\X\Y\Z #1\W\X\Y\Z
}%
%    \end{macrocode}
% \subsection{\csh{xintiSub}, \csh{xintiiSub}}
% \lverb|Release 1.09a has \xintnum added into \xintiSub.|
%    \begin{macrocode}
\def\xintiiSub   {\romannumeral0\xintiisub }%
\def\xintiisub #1{\expandafter\xint_iisub\romannumeral-`0#1\Z  }%
\def\xint_iisub #1#2\Z #3%
{%
    \expandafter\XINT_sub_fork\expandafter #1\romannumeral-`0#3\Z #2\Z
}%
\def\xintiSub {\romannumeral0\xintisub }%
\def\xintisub #1%
{%
    \expandafter\xint_sub\romannumeral0\xintnum{#1}\Z
}%
\def\xint_sub #1#2\Z #3%
{%
    \expandafter\XINT_sub_fork\expandafter #1\romannumeral0\xintnum{#3}\Z #2\Z
}%
\let\xintSub\xintiSub \let\xintsub\xintisub
\def\XINT_sub_fork #1#2%
{%
    \xint_UDzerofork
      #1\XINT_sub_firstiszero
      #2\XINT_sub_secondiszero
       0{}%
    \krof
    \xint_UDsignsfork
          #1#2\XINT_sub_minusminus
           #1-\XINT_sub_minusplus
           #2-\XINT_sub_plusminus
            --\XINT_sub_plusplus
    \krof #1#2%
}%
\def\XINT_sub_firstiszero  #1\krof #2#3\Z #4\Z {\XINT_opp #3}%
\def\XINT_sub_secondiszero #1\krof #2#3\Z #4\Z { #2#4}%
\def\XINT_sub_plusplus     #1#2#3\Z #4\Z {\XINT_sub_pre {#1#4}{#2#3}}%
\def\XINT_sub_minusminus   #1#2#3\Z #4\Z {\XINT_sub_pre {#3}{#4}}%
\def\XINT_sub_minusplus    #1#2#3\Z #4\Z
   {\expandafter\xint_minus_thenstop\romannumeral0\XINT_add_pre {#4}{#2#3}}%
\def\XINT_sub_plusminus    #1#2#3\Z #4\Z {\XINT_add_pre {#1#4}{#3}}%
%    \end{macrocode}
% \lverb|SOUSTRACTION A-B avec A premier argument, B second argument de
% \xintSub et ensuite \XINT_sub_pre ici|
%    \begin{macrocode}
\def\XINT_sub_pre #1%
{%
  \expandafter\XINT_sub_pre_b\expandafter
  {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }%
}%
\def\XINT_sub_pre_b #1#2%
{%
    \expandafter\XINT_sub_A
        \expandafter1\expandafter{\expandafter}%
    \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z
        \W\X\Y\Z #1 \W\X\Y\Z
}%
%    \end{macrocode}
% \lverb|&
% \romannumeral0\XINT_sub_A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z$\ 
% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS
% POUR QUE LEURS LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS
% AUCUN NE SE TERMINE EN 0000.$\ output: N2 - N1$\ 
% Elle donne le résultat dans le **bon ordre**, avec le bon signe,
% et sans zéros superflus.|
%    \begin{macrocode}
\def\XINT_sub_A #1#2#3\W\X\Y\Z #4#5#6#7%
{%
    \xint_gob_til_W
    #4\xint_sub_az
    \W\XINT_sub_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z
}%
\def\XINT_sub_B #1#2#3#4#5#6#7%
{%
    \xint_gob_til_W
    #4\xint_sub_bz
    \W\XINT_sub_onestep #1#2{#7#6#5#4}{#3}%
}%
%    \end{macrocode}
% \lverb|&
% d'abord la branche principale
% #6 = 4 chiffres de N1, plus significatif en *premier*,
% #2#3#4#5 chiffres de N2, plus significatif en *dernier*
% On veut N2 - N1.|
%    \begin{macrocode}
\def\XINT_sub_onestep #1#2#3#4#5#6%
{%
    \expandafter\XINT_sub_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.%
}%
%    \end{macrocode}
% \lverb|ON PRODUIT LE RÉSULTAT DANS LE BON ORDRE|
%    \begin{macrocode}
\def\XINT_sub_backtoA #1#2#3.#4%
{%
    \XINT_sub_A #2{#3#4}%
}%
\def\xint_sub_bz
    \W\XINT_sub_onestep #1#2#3#4#5#6#7%
{%
    \xint_UDzerofork
      #1\XINT_sub_C   % une retenue
       0\XINT_sub_D   % pas de retenue
    \krof
    {#7}#2#3#4#5%
}%
\def\XINT_sub_D #1#2\W\X\Y\Z
{%
    \expandafter
    \xint_cleanupzeros_andstop
    \romannumeral0%
    \XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
    #1%
}%
\def\XINT_sub_C #1#2#3#4#5%
{%
    \xint_gob_til_W
    #2\xint_sub_cz
    \W\XINT_sub_AC_onestep {#5#4#3#2}{#1}%
}%
\def\XINT_sub_AC_onestep #1%
{%
    \expandafter\XINT_sub_backtoC\the\numexpr 11#1-\xint_c_i.%
}%
\def\XINT_sub_backtoC #1#2#3.#4%
{%
    \XINT_sub_AC_checkcarry #2{#3#4}% la retenue va \^etre examin\'ee
}%
\def\XINT_sub_AC_checkcarry #1%
{%
    \xint_gob_til_one #1\xint_sub_AC_nocarry 1\XINT_sub_C
}%
\def\xint_sub_AC_nocarry 1\XINT_sub_C #1#2\W\X\Y\Z
{%
    \expandafter
    \XINT_cuz_loop
    \romannumeral0%
    \XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
    #1\W\W\W\W\W\W\W\Z
}%
\def\xint_sub_cz\W\XINT_sub_AC_onestep #1%
{%
    \XINT_cuz
}%
\def\xint_sub_az\W\XINT_sub_B #1#2#3#4#5#6#7%
{%
    \xint_gob_til_W
    #4\xint_sub_ez
    \W\XINT_sub_Eenter #1{#3}#4#5#6#7%
}%
%    \end{macrocode}
% \lverb|le premier nombre continue, le résultat sera < 0.|
%    \begin{macrocode}
\def\XINT_sub_Eenter #1#2%
{%
    \expandafter
    \XINT_sub_E\expandafter1\expandafter{\expandafter}%
    \romannumeral0%
    \XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
    \W\X\Y\Z #1%
}%
\def\XINT_sub_E #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_sub_F\W
    \XINT_sub_Eonestep #1{#6#5#4#3}{#2}%
}%
\def\XINT_sub_Eonestep #1#2%
{%
    \expandafter\XINT_sub_backtoE\the\numexpr 109999-#2+#1.%
}%
\def\XINT_sub_backtoE #1#2#3.#4%
{%
    \XINT_sub_E #2{#3#4}%
}%
\def\xint_sub_F\W\XINT_sub_Eonestep #1#2#3#4%
{%
    \xint_UDonezerofork
      #4#1{\XINT_sub_Fdec 0}% soustraire  1. Et faire signe -
      #1#4{\XINT_sub_Finc 1}% additionner 1. Et faire signe -
        10\XINT_sub_DD     % terminer. Mais avec signe -
    \krof
    {#3}%
}%
\def\XINT_sub_DD {\expandafter\xint_minus_thenstop\romannumeral0\XINT_sub_D }%
\def\XINT_sub_Fdec #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_sub_Fdec_finish\W
    \XINT_sub_Fdec_onestep #1{#6#5#4#3}{#2}%
}%
\def\XINT_sub_Fdec_onestep #1#2%
{%
    \expandafter\XINT_sub_backtoFdec\the\numexpr 11#2+#1-\xint_c_i.%
}%
\def\XINT_sub_backtoFdec #1#2#3.#4%
{%
    \XINT_sub_Fdec #2{#3#4}%
}%
\def\xint_sub_Fdec_finish\W\XINT_sub_Fdec_onestep #1#2%
{%
    \expandafter\xint_minus_thenstop\romannumeral0\XINT_cuz
}%
\def\XINT_sub_Finc #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_sub_Finc_finish\W
    \XINT_sub_Finc_onestep #1{#6#5#4#3}{#2}%
}%
\def\XINT_sub_Finc_onestep #1#2%
{%
    \expandafter\XINT_sub_backtoFinc\the\numexpr 10#2+#1.%
}%
\def\XINT_sub_backtoFinc #1#2#3.#4%
{%
    \XINT_sub_Finc #2{#3#4}%
}%
\def\xint_sub_Finc_finish\W\XINT_sub_Finc_onestep #1#2#3%
{%
    \xint_UDzerofork
     #1{\expandafter\expandafter\expandafter
        \xint_minus_thenstop\xint_cleanupzeros_nostop}%
      0{ -1}%
    \krof
    #3%
}%
\def\xint_sub_ez\W\XINT_sub_Eenter #1%
{%
    \xint_UDzerofork
      #1\XINT_sub_K % il y a une retenue
       0\XINT_sub_L % pas de retenue
    \krof
}%
\def\XINT_sub_L #1\W\X\Y\Z {\XINT_cuz_loop #1\W\W\W\W\W\W\W\Z }%
\def\XINT_sub_K #1%
{%
    \expandafter
    \XINT_sub_KK\expandafter1\expandafter{\expandafter}%
    \romannumeral0%
    \XINT_rord_main {}#1%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
}%
\def\XINT_sub_KK #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_sub_KK_finish\W
    \XINT_sub_KK_onestep #1{#6#5#4#3}{#2}%
}%
\def\XINT_sub_KK_onestep #1#2%
{%
    \expandafter\XINT_sub_backtoKK\the\numexpr 109999-#2+#1.%
}%
\def\XINT_sub_backtoKK #1#2#3.#4%
{%
    \XINT_sub_KK #2{#3#4}%
}%
\def\xint_sub_KK_finish\W\XINT_sub_KK_onestep #1#2#3%
{%
    \expandafter\xint_minus_thenstop
    \romannumeral0\XINT_cuz_loop #3\W\W\W\W\W\W\W\Z
}%
%    \end{macrocode}
% \subsection{\csh{xintiMul}, \csh{xintiiMul}}
% \lverb|1.09a adds \xintnum|
%    \begin{macrocode}
\def\xintiiMul {\romannumeral0\xintiimul }%
\def\xintiimul #1%
{%
    \expandafter\xint_iimul\expandafter {\romannumeral-`0#1}%
}%
\def\xint_iimul #1#2%
{%
    \expandafter\XINT_mul_fork \romannumeral-`0#2\Z #1\Z
}%
\def\xintiMul {\romannumeral0\xintimul }%
\def\xintimul #1%
{%
    \expandafter\xint_mul\expandafter {\romannumeral0\xintnum{#1}}%
}%
\def\xint_mul #1#2%
{%
    \expandafter\XINT_mul_fork \romannumeral0\xintnum{#2}\Z #1\Z
}%
\let\xintMul\xintiMul \let\xintmul\xintimul
\def\XINT_Mul #1#2{\romannumeral0\XINT_mul_fork #2\Z #1\Z }%
%    \end{macrocode}
% \lverb|&
% MULTIPLICATION$\ 
% Ici #1#2 = 2e input et #3#4 = 1er input $\ 
% Release 1.03 adds some overhead to first compute and compare the
% lengths of the two inputs. The algorithm is asymmetrical and whether
% the first input is the longest or the shortest sometimes has a strong
% impact. 50 digits times 1000 digits used to be 5 times faster
% than 1000 digits times 50 digits. With the new code, the user input
% order does not matter as it is decided by the routine what is best.
% This is important for  the extension to fractions, as there is no way
% then to generally control or guess the most frequent sizes of the
% inputs besides actually computing their lengths.  |
%    \begin{macrocode}
\def\XINT_mul_fork #1#2\Z #3#4\Z
{%
    \xint_UDzerofork
      #1\XINT_mul_zero
      #3\XINT_mul_zero
       0{}%
    \krof
    \xint_UDsignsfork
          #1#3\XINT_mul_minusminus          % #1 = #3 = -
           #1-{\XINT_mul_minusplus #3}%          % #1 = -
           #3-{\XINT_mul_plusminus #1}%          % #3 = -
            --{\XINT_mul_plusplus #1#3}%
    \krof
    {#2}{#4}%
}%
\def\XINT_mul_zero #1\krof #2#3{ 0}%
\def\XINT_mul_minusminus #1#2%
{%
      \expandafter\XINT_mul_choice_a
      \expandafter{\romannumeral0\xintlength {#2}}%
      {\romannumeral0\xintlength {#1}}{#1}{#2}%
}%
\def\XINT_mul_minusplus #1#2#3%
{%
    \expandafter\xint_minus_thenstop\romannumeral0\expandafter
    \XINT_mul_choice_a
    \expandafter{\romannumeral0\xintlength {#1#3}}%
    {\romannumeral0\xintlength {#2}}{#2}{#1#3}%
}%
\def\XINT_mul_plusminus #1#2#3%
{%
    \expandafter\xint_minus_thenstop\romannumeral0\expandafter
    \XINT_mul_choice_a
    \expandafter{\romannumeral0\xintlength {#3}}%
    {\romannumeral0\xintlength {#1#2}}{#1#2}{#3}%
}%
\def\XINT_mul_plusplus #1#2#3#4%
{%
    \expandafter\XINT_mul_choice_a
    \expandafter{\romannumeral0\xintlength {#2#4}}%
    {\romannumeral0\xintlength {#1#3}}{#1#3}{#2#4}%
}%
\def\XINT_mul_choice_a #1#2%
{%
    \expandafter\XINT_mul_choice_b\expandafter{#2}{#1}%
}%
\def\XINT_mul_choice_b #1#2%
{%
    \ifnum #1<\xint_c_v
       \expandafter\XINT_mul_choice_littlebyfirst
    \else
    \ifnum #2<\xint_c_v
      \expandafter\expandafter\expandafter\XINT_mul_choice_littlebysecond
      \else
      \expandafter\expandafter\expandafter\XINT_mul_choice_compare
      \fi
    \fi
    {#1}{#2}%
}%
\def\XINT_mul_choice_littlebyfirst #1#2#3#4%
{%
    \expandafter\XINT_mul_M
    \expandafter{\the\numexpr #3\expandafter}%
    \romannumeral0\XINT_RQ {}#4\R\R\R\R\R\R\R\R\Z \Z\Z\Z\Z
}%
\def\XINT_mul_choice_littlebysecond #1#2#3#4%
{%
    \expandafter\XINT_mul_M
    \expandafter{\the\numexpr #4\expandafter}%
    \romannumeral0\XINT_RQ {}#3\R\R\R\R\R\R\R\R\Z \Z\Z\Z\Z
}%
\def\XINT_mul_choice_compare #1#2%
{%
    \ifnum #1>#2
        \expandafter \XINT_mul_choice_i
    \else
        \expandafter \XINT_mul_choice_ii
    \fi
    {#1}{#2}%
}%
\def\XINT_mul_choice_i #1#2%
{%
   \ifnum #1<\numexpr\ifcase \numexpr (#2-\xint_c_iii)/\xint_c_iv\relax
                     \or 330\or 168\or 109\or 80\or 66\or 52\else 0\fi\relax
       \expandafter\XINT_mul_choice_same
   \else
       \expandafter\XINT_mul_choice_permute
   \fi
}%
\def\XINT_mul_choice_ii #1#2%
{%
   \ifnum #2<\numexpr\ifcase \numexpr (#1-\xint_c_iii)/\xint_c_iv\relax
                     \or 330\or 168\or 109\or 80\or 66\or 52\else 0\fi\relax
       \expandafter\XINT_mul_choice_permute
   \else
       \expandafter\XINT_mul_choice_same
   \fi
}%
\def\XINT_mul_choice_same #1#2%
{%
    \expandafter\XINT_mul_enter
    \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z
    \Z\Z\Z\Z #2\W\W\W\W
}%
\def\XINT_mul_choice_permute #1#2%
{%
    \expandafter\XINT_mul_enter
    \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z
    \Z\Z\Z\Z #1\W\W\W\W
}%
%    \end{macrocode}
% \lverb|&
% Cette portion de routine d'addition se branche directement sur _addr_
% lorsque
% le premier nombre est épuisé, ce qui est garanti arriver avant le second
% nombre. Elle produit son résultat toujours sur 4n, renversé. Ses deux inputs
% sont garantis sur 4n.|
%    \begin{macrocode}
\def\XINT_mul_Ar #1#2#3#4#5#6%
{%
    \xint_gob_til_Z #6\xint_mul_br\Z\XINT_mul_Br #1{#6#5#4#3}{#2}%
}%
\def\xint_mul_br\Z\XINT_mul_Br #1#2%
{%
     \XINT_addr_AC_checkcarry #1%
}%
\def\XINT_mul_Br #1#2#3#4\W\X\Y\Z #5#6#7#8%
{%
    \expandafter\XINT_mul_ABEAr
    \the\numexpr #1+10#2+#8#7#6#5.{#3}#4\W\X\Y\Z
}%
\def\XINT_mul_ABEAr #1#2#3#4#5#6.#7%
{%
    \XINT_mul_Ar #2{#7#6#5#4#3}%
}%
%    \end{macrocode}
% \lverb|&
% << Petite >> multiplication.
% mul_Mr renvoie le résultat *à l'envers*, sur *4n*$\ 
% \romannumeral0\XINT_mul_Mr {<n>}<N>\Z\Z\Z\Z$\ 
% Fait la multiplication de <N> par <n>, qui est < 10000.
% <N> est présenté *à l'envers*, sur *4n*. Lorsque <n> vaut 0, donne 0000.|
%    \begin{macrocode}
\def\XINT_mul_Mr #1%
{%
    \expandafter\XINT_mul_Mr_checkifzeroorone\expandafter{\the\numexpr #1}%
}%
\def\XINT_mul_Mr_checkifzeroorone #1%
{%
    \ifcase #1
      \expandafter\XINT_mul_Mr_zero
    \or
      \expandafter\XINT_mul_Mr_one
    \else
      \expandafter\XINT_mul_Nr
    \fi
    {0000}{}{#1}%
}%
\def\XINT_mul_Mr_zero #1\Z\Z\Z\Z { 0000}%
\def\XINT_mul_Mr_one #1#2#3#4\Z\Z\Z\Z { #4}%
\def\XINT_mul_Nr #1#2#3#4#5#6#7%
{%
    \xint_gob_til_Z #4\xint_mul_pr\Z\XINT_mul_Pr {#1}{#3}{#7#6#5#4}{#2}{#3}%
}%
\def\XINT_mul_Pr #1#2#3%
{%
    \expandafter\XINT_mul_Lr\the\numexpr \xint_c_x^viii+#1+#2*#3\relax
}%
\def\XINT_mul_Lr 1#1#2#3#4#5#6#7#8#9%
{%
    \XINT_mul_Nr {#1#2#3#4}{#9#8#7#6#5}%
}%
\def\xint_mul_pr\Z\XINT_mul_Pr #1#2#3#4#5%
{%
    \xint_gob_til_zeros_iv #1\XINT_mul_Mr_end_nocarry 0000%
    \XINT_mul_Mr_end_carry #1{#4}%
}%
\def\XINT_mul_Mr_end_nocarry 0000\XINT_mul_Mr_end_carry 0000#1{ #1}%
\def\XINT_mul_Mr_end_carry #1#2#3#4#5{ #5#4#3#2#1}%
%    \end{macrocode}
% \lverb|&
% << Petite >> multiplication.
% renvoie le résultat *à l'endroit*, avec *nettoyage des leading zéros*.$\ 
% \romannumeral0\XINT_mul_M  {<n>}<N>\Z\Z\Z\Z$\ 
% Fait la multiplication de <N> par <n>, qui est < 10000.
% <N> est présenté *à l'envers*, sur *4n*. |
%    \begin{macrocode}
\def\XINT_mul_M #1%
{%
    \expandafter\XINT_mul_M_checkifzeroorone\expandafter{\the\numexpr #1}%
}%
\def\XINT_mul_M_checkifzeroorone #1%
{%
    \ifcase #1
      \expandafter\XINT_mul_M_zero
    \or
      \expandafter\XINT_mul_M_one
    \else
      \expandafter\XINT_mul_N
    \fi
    {0000}{}{#1}%
}%
\def\XINT_mul_M_zero #1\Z\Z\Z\Z { 0}%
\def\XINT_mul_M_one #1#2#3#4\Z\Z\Z\Z
{%
    \expandafter\xint_cleanupzeros_andstop\romannumeral0\xintreverseorder{#4}%
}%
\def\XINT_mul_N #1#2#3#4#5#6#7%
{%
    \xint_gob_til_Z #4\xint_mul_p\Z\XINT_mul_P {#1}{#3}{#7#6#5#4}{#2}{#3}%
}%
\def\XINT_mul_P #1#2#3%
{%
    \expandafter\XINT_mul_L\the\numexpr \xint_c_x^viii+#1+#2*#3\relax
}%
\def\XINT_mul_L 1#1#2#3#4#5#6#7#8#9%
{%
    \XINT_mul_N {#1#2#3#4}{#5#6#7#8#9}%
}%
\def\xint_mul_p\Z\XINT_mul_P #1#2#3#4#5%
{%
    \XINT_mul_M_end #1#4%
}%
\edef\XINT_mul_M_end #1#2#3#4#5#6#7#8%
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax
}%
%    \end{macrocode}
% \lverb|&
% Routine de multiplication principale
% (attention délimiteurs modifiés pour 1.08)$\ 
% Le résultat partiel est toujours maintenu avec significatif à
% droite et il a un nombre multiple de 4 de chiffres$\ 
% \romannumeral0\XINT_mul_enter <N1>\Z\Z\Z\Z <N2>\W\W\W\W$\ 
% avec <N1> *renversé*, *longueur 4n* (zéros éventuellement ajoutés
% au-delà du chiffre le plus significatif)
% et <N2> dans l'ordre *normal*, et pas forcément longueur 4n.
% pas de signes.$\ 
% Pour 1.08: dans \XINT_mul_enter et les modifs de 1.03
% qui filtrent les courts, on pourrait croire  que le
% second opérande a au moins quatre chiffres; mais le problème c'est que
% ceci est appelé par \XINT_sqr. Et de plus \XINT_sqr est utilisé dans
% la nouvelle routine d'extraction de racine carrée: je ne veux pas
% rajouter l'overhead à \XINT_sqr de voir si a longueur est au moins 4.
% Dilemme donc. Il ne semble pas y avoir d'autres accès
% directs (celui de big fac n'est pas un problème). J'ai presque été
% tenté de faire du 5x4, mais si on veut maintenir les résultats
% intermédiaires sur 4n, il y a des complications. Par ailleurs,
% je modifie aussi un petit peu la façon de coder la suite, compte tenu
% du style que j'ai développé ultérieurement. Attention terminaison
% modifiée pour le deuxième opérande.|
%    \begin{macrocode}
\def\XINT_mul_enter #1\Z\Z\Z\Z #2#3#4#5%
{%
    \xint_gob_til_W #5\XINT_mul_exit_a\W
    \XINT_mul_start {#2#3#4#5}#1\Z\Z\Z\Z
}%
\def\XINT_mul_exit_a\W\XINT_mul_start #1%
{%
    \XINT_mul_exit_b #1%
}%
\def\XINT_mul_exit_b #1#2#3#4%
{%
    \xint_gob_til_W
      #2\XINT_mul_exit_ci
      #3\XINT_mul_exit_cii
      \W\XINT_mul_exit_ciii #1#2#3#4%
}%
\def\XINT_mul_exit_ciii #1\W #2\Z\Z\Z\Z \W\W\W
{%
    \XINT_mul_M {#1}#2\Z\Z\Z\Z
}%
\def\XINT_mul_exit_cii\W\XINT_mul_exit_ciii #1\W\W #2\Z\Z\Z\Z \W\W
{%
    \XINT_mul_M {#1}#2\Z\Z\Z\Z
}%
\def\XINT_mul_exit_ci\W\XINT_mul_exit_cii
                     \W\XINT_mul_exit_ciii #1\W\W\W #2\Z\Z\Z\Z \W
{%
    \XINT_mul_M {#1}#2\Z\Z\Z\Z
}%
\def\XINT_mul_start #1#2\Z\Z\Z\Z
{%
    \expandafter\XINT_mul_main\expandafter
    {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z}#2\Z\Z\Z\Z
}%
\def\XINT_mul_main #1#2\Z\Z\Z\Z #3#4#5#6%
{%
    \xint_gob_til_W #6\XINT_mul_finish_a\W
    \XINT_mul_compute {#3#4#5#6}{#1}#2\Z\Z\Z\Z
}%
\def\XINT_mul_compute #1#2#3\Z\Z\Z\Z
{%
    \expandafter\XINT_mul_main\expandafter
    {\romannumeral0\expandafter
     \XINT_mul_Ar\expandafter0\expandafter{\expandafter}%
     \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z
     \W\X\Y\Z 0000#2\W\X\Y\Z }#3\Z\Z\Z\Z
}%
%    \end{macrocode}
% \lverb|&
% Ici, le deuxième nombre se termine. Fin du calcul. On utilise la variante
% \XINT_addm_A de l'addition car on sait que le deuxième terme est au moins
% aussi long que le premier. Lorsque le multiplicateur avait longueur 4n, la
% dernière addition a fourni le résultat à l'envers, il faut donc encore le
% renverser. |
%    \begin{macrocode}
\def\XINT_mul_finish_a\W\XINT_mul_compute #1%
{%
    \XINT_mul_finish_b #1%
}%
\def\XINT_mul_finish_b #1#2#3#4%
{%
    \xint_gob_til_W
      #1\XINT_mul_finish_c
      #2\XINT_mul_finish_ci
      #3\XINT_mul_finish_cii
      \W\XINT_mul_finish_ciii #1#2#3#4%
}%
\def\XINT_mul_finish_ciii #1\W #2#3\Z\Z\Z\Z \W\W\W
{%
    \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}%
    \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 000#2\W\X\Y\Z
}%
\def\XINT_mul_finish_cii
    \W\XINT_mul_finish_ciii #1\W\W #2#3\Z\Z\Z\Z \W\W
{%
    \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}%
    \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 00#2\W\X\Y\Z
}%
\def\XINT_mul_finish_ci #1\XINT_mul_finish_ciii #2\W\W\W #3#4\Z\Z\Z\Z \W
{%
    \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}%
    \romannumeral0\XINT_mul_Mr {#2}#4\Z\Z\Z\Z \W\X\Y\Z 0#3\W\X\Y\Z
}%
\def\XINT_mul_finish_c #1\XINT_mul_finish_ciii \W\W\W\W #2#3\Z\Z\Z\Z
{%
    \expandafter\xint_cleanupzeros_andstop\romannumeral0\xintreverseorder{#2}%
}%
%    \end{macrocode}
% \lverb|&
% Variante de la Multiplication$\ 
%   \romannumeral0\XINT_mulr_enter <N1>\Z\Z\Z\Z <N2>\W\W\W\W $\ 
%   Ici <N1> est à l'envers sur 4n, et <N2> est à l'endroit, pas sur 4n, comme
%   dans \XINT_mul_enter, mais le résultat est lui-même fourni *à l'envers*, sur
%   *4n* (en faisant attention de ne pas avoir 0000 à la fin).$\ 
% Utilisé par le calcul des puissances. J'ai modifié dans 1.08 sur le
% modèle de la nouvelle version de \XINT_mul_enter. Je pourrais économiser des
% macros et fusionner \XINT_mul_enter et \XINT_mulr_enter. Une autre fois.|
%    \begin{macrocode}
\def\XINT_mulr_enter #1\Z\Z\Z\Z #2#3#4#5%
{%
    \xint_gob_til_W #5\XINT_mulr_exit_a\W
    \XINT_mulr_start {#2#3#4#5}#1\Z\Z\Z\Z
}%
\def\XINT_mulr_exit_a\W\XINT_mulr_start #1%
{%
    \XINT_mulr_exit_b #1%
}%
\def\XINT_mulr_exit_b #1#2#3#4%
{%
    \xint_gob_til_W
      #2\XINT_mulr_exit_ci
      #3\XINT_mulr_exit_cii
      \W\XINT_mulr_exit_ciii #1#2#3#4%
}%
\def\XINT_mulr_exit_ciii #1\W #2\Z\Z\Z\Z \W\W\W
{%
    \XINT_mul_Mr {#1}#2\Z\Z\Z\Z
}%
\def\XINT_mulr_exit_cii\W\XINT_mulr_exit_ciii #1\W\W #2\Z\Z\Z\Z \W\W
{%
    \XINT_mul_Mr {#1}#2\Z\Z\Z\Z
}%
\def\XINT_mulr_exit_ci\W\XINT_mulr_exit_cii
                     \W\XINT_mulr_exit_ciii #1\W\W\W #2\Z\Z\Z\Z \W
{%
    \XINT_mul_Mr {#1}#2\Z\Z\Z\Z
}%
\def\XINT_mulr_start #1#2\Z\Z\Z\Z
{%
    \expandafter\XINT_mulr_main\expandafter
    {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z}#2\Z\Z\Z\Z
}%
\def\XINT_mulr_main #1#2\Z\Z\Z\Z #3#4#5#6%
{%
    \xint_gob_til_W #6\XINT_mulr_finish_a\W
    \XINT_mulr_compute {#3#4#5#6}{#1}#2\Z\Z\Z\Z
}%
\def\XINT_mulr_compute #1#2#3\Z\Z\Z\Z
{%
    \expandafter\XINT_mulr_main\expandafter
    {\romannumeral0\expandafter
     \XINT_mul_Ar\expandafter0\expandafter{\expandafter}%
     \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z
     \W\X\Y\Z 0000#2\W\X\Y\Z }#3\Z\Z\Z\Z
}%
\def\XINT_mulr_finish_a\W\XINT_mulr_compute #1%
{%
    \XINT_mulr_finish_b #1%
}%
\def\XINT_mulr_finish_b #1#2#3#4%
{%
    \xint_gob_til_W
      #1\XINT_mulr_finish_c
      #2\XINT_mulr_finish_ci
      #3\XINT_mulr_finish_cii
      \W\XINT_mulr_finish_ciii #1#2#3#4%
}%
\def\XINT_mulr_finish_ciii #1\W #2#3\Z\Z\Z\Z \W\W\W
{%
    \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}%
    \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 000#2\W\X\Y\Z
}%
\def\XINT_mulr_finish_cii
    \W\XINT_mulr_finish_ciii #1\W\W #2#3\Z\Z\Z\Z \W\W
{%
    \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}%
    \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 00#2\W\X\Y\Z
}%
\def\XINT_mulr_finish_ci #1\XINT_mulr_finish_ciii #2\W\W\W #3#4\Z\Z\Z\Z \W
{%
    \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}%
    \romannumeral0\XINT_mul_Mr {#2}#4\Z\Z\Z\Z \W\X\Y\Z 0#3\W\X\Y\Z
}%
\def\XINT_mulr_finish_c #1\XINT_mulr_finish_ciii \W\W\W\W #2#3\Z\Z\Z\Z { #2}%
%    \end{macrocode}
% \subsection{\csh{xintiSqr}, \csh{xintiiSqr}}
%    \begin{macrocode}
\def\xintiiSqr {\romannumeral0\xintiisqr }%
\def\xintiisqr #1%
{%
    \expandafter\XINT_sqr\expandafter {\romannumeral0\xintiiabs{#1}}%
}%
\def\xintiSqr {\romannumeral0\xintisqr }%
\def\xintisqr #1%
{%
    \expandafter\XINT_sqr\expandafter {\romannumeral0\xintiabs{#1}}%
}%
\let\xintSqr\xintiSqr \let\xintsqr\xintisqr
\def\XINT_sqr #1%
{%
    \expandafter\XINT_mul_enter
          \romannumeral0%
          \XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z
          \Z\Z\Z\Z #1\W\W\W\W
}%
%    \end{macrocode}
% \subsection{\csh{xintiPow}, \csh{xintiiPow}}
% \lverb|1.02 modified the \XINT_posprod routine, the was renamed
% \XINT_pow_posprod and moved here, as it was well adapted for computing powers.
% Then 1.03 moved the special variants of multiplication (hence of addition)
% which were needed to earlier in this style file.
%
% Modified in 1.06, the exponent is given to a \numexpr rather than twice
% expanded. \xintnum added in 1.09a.
%
% \XINT_pow_posprod: Routine de produit servant pour le calcul des
% puissances. Chaque nouveau terme est plus grand que ce qui a déjà été calculé.
% Par conséquent on a intérêt à le conserver en second dans la routine de
% multiplication, donc le précédent calcul a intérêt à avoir été donné sur 4n, à
% l'envers. Il faut donc modifier la multiplication pour qu'elle fasse cela. Ce
% qui oblige à utiliser une version spéciale de l'addition également.
%
% 1.09j has reorganized the main loop, the described above \XINT_pow_posprod
% routine has been removed, intermediate multiplications are done
% immediately. Also, the maximal accepted exponent is now 100000 (no such
% restriction in \xintFloatPow, which accepts any exponent less than 2^31, and
% in \xintFloatPower which accepts long integers as exponent).
%
% 2^100000=9.990020930143845e30102 and multiplication of two numbers
% with 30000 digits would take hours on my laptop (seconds for 1000 digits).|
%    \begin{macrocode}
\def\xintiiPow {\romannumeral0\xintiipow }%
\def\xintiipow #1%
{%
    \expandafter\xint_pow\romannumeral-`0#1\Z%
}%
\def\xintiPow {\romannumeral0\xintipow }%
\def\xintipow #1%
{%
    \expandafter\xint_pow\romannumeral0\xintnum{#1}\Z%
}%
\let\xintPow\xintiPow \let\xintpow\xintipow
\def\xint_pow #1#2\Z
{%
    \xint_UDsignfork
      #1\XINT_pow_Aneg
       -\XINT_pow_Anonneg
    \krof
       #1{#2}%
}%
\def\XINT_pow_Aneg #1#2#3%
{%
   \expandafter\XINT_pow_Aneg_\expandafter{\the\numexpr #3}#2\Z
}%
\def\XINT_pow_Aneg_ #1%
{%
   \ifodd #1
       \expandafter\XINT_pow_Aneg_Bodd
   \fi
   \XINT_pow_Anonneg_ {#1}%
}%
\def\XINT_pow_Aneg_Bodd #1%
{%
    \expandafter\XINT_opp\romannumeral0\XINT_pow_Anonneg_
}%
%    \end{macrocode}
% \lverb|B = #3, faire le xpxp. Modified with 1.06: use of \numexpr.|
%    \begin{macrocode}
\def\XINT_pow_Anonneg #1#2#3%
{%
   \expandafter\XINT_pow_Anonneg_\expandafter {\the\numexpr #3}#1#2\Z
}%
%    \end{macrocode}
% \lverb+#1 = B, #2 = |A|. Modifié pour v1.1, car utilisait \XINT_Cmp, ce qui
% d'ailleurs n'était sans doute pas super efficace, et m'obligeait à mettre
% \xintCmp dans xintcore. Donc ici A est déjà #2#3 et il y a un \Z après.+
%    \begin{macrocode}
\def\XINT_pow_Anonneg_ #1#2#3\Z
{%
    \if\relax #3\relax\xint_dothis
                      {\ifcase #2 \expandafter\XINT_pow_AisZero
                               \or\expandafter\XINT_pow_AisOne
                             \else\expandafter\XINT_pow_AatleastTwo
                       \fi }\fi
    \xint_orthat \XINT_pow_AatleastTwo {#1}{#2#3}%
}%
\def\XINT_pow_AisOne #1#2{ 1}%
%    \end{macrocode}
% \lverb|#1 = B|
%    \begin{macrocode}
\def\XINT_pow_AisZero #1#2%
{%
     \ifcase\XINT_cntSgn #1\Z
         \xint_afterfi { 1}%
     \or
         \xint_afterfi { 0}%
     \else
         \xint_afterfi {\xintError:DivisionByZero\space 0}%
     \fi
}%
\def\XINT_pow_AatleastTwo #1%
{%
    \ifcase\XINT_cntSgn #1\Z
        \expandafter\XINT_pow_BisZero
    \or
        \expandafter\XINT_pow_checkBsize
    \else
        \expandafter\XINT_pow_BisNegative
    \fi
    {#1}%
}%
\edef\XINT_pow_BisNegative #1#2%
    {\noexpand\xintError:FractionRoundedToZero\space 0}%
\def\XINT_pow_BisZero #1#2{ 1}%
%    \end{macrocode}
% \lverb|B = #1 > 0, A = #2 > 1. With 1.05, I replace \xintiLen{#1}>9 by
% direct use of \numexpr [to generate an error message if the exponent is too
% large] 1.06: \numexpr was already used above.|
%    \begin{macrocode}
\def\XINT_pow_checkBsize #1%
{%
    \ifnum #1>100000
        \expandafter\XINT_pow_BtooBig
    \else
        \expandafter\XINT_pow_loopI
    \fi
    {#1}%
}%
\edef\XINT_pow_BtooBig #1#2{\noexpand\xintError:ExponentTooBig\space 0}%
\def\XINT_pow_loopI #1%
{%
    \ifnum #1=\xint_c_i\XINT_pow_Iend\fi
    \ifodd #1
       \expandafter\XINT_pow_loopI_odd
    \else
       \expandafter\XINT_pow_loopI_even
    \fi
    {#1}%
}%
\edef\XINT_pow_Iend\fi #1\fi #2#3{\noexpand\fi\space #3}%
\def\XINT_pow_loopI_even #1#2%
{%
    \expandafter\XINT_pow_loopI\expandafter
    {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter
    {\romannumeral0\xintiisqr {#2}}%
}%
\def\XINT_pow_loopI_odd #1#2%
{%
    \expandafter\XINT_pow_loopI_odda\expandafter
    {\romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z }{#1}{#2}%
}%
\def\XINT_pow_loopI_odda #1#2#3%
{%
    \expandafter\XINT_pow_loopII\expandafter
    {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter
    {\romannumeral0\xintiisqr {#3}}{#1}%
}%
\def\XINT_pow_loopII #1%
{%
    \ifnum #1 = \xint_c_i\XINT_pow_IIend\fi
    \ifodd #1
       \expandafter\XINT_pow_loopII_odd
    \else
       \expandafter\XINT_pow_loopII_even
    \fi
    {#1}%
}%
\def\XINT_pow_loopII_even #1#2%
{%
    \expandafter\XINT_pow_loopII\expandafter
    {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter
    {\romannumeral0\xintiisqr {#2}}%
}%
\def\XINT_pow_loopII_odd #1#2#3%
{%
    \expandafter\XINT_pow_loopII_odda\expandafter
    {\romannumeral0\XINT_mulr_enter #3\Z\Z\Z\Z #2\W\W\W\W}{#1}{#2}%
}%
\def\XINT_pow_loopII_odda #1#2#3%
{%
    \expandafter\XINT_pow_loopII\expandafter
    {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter
    {\romannumeral0\xintiisqr {#3}}{#1}%
}%
\def\XINT_pow_IIend\fi #1\fi #2#3#4%
{%
    \fi\XINT_mul_enter #4\Z\Z\Z\Z #3\W\W\W\W
}%
%    \end{macrocode}
% \subsection{\csh{xintiDivision}, \csh{xintiQuo}, \csh{xintiRem},
% \csh{xintiiDivision}, \csh{xintiiQuo}, \csh{xintiiRem}}
% \lverb|The 1.09a release inserted the use of \xintnum. The \xintiiDivision
% etc... are the ones which do only \romannumeral-`0.
%
% January 5, 2014: Naturally, addition, subtraction, multiplication and division
% are the first things I did and since then I had left the division
% untouched. So in preparation of release 1.09j, I started revisiting the
% division, I did various minor improvements obtaining roughly
% 10$% efficiency gain. Then I decided I
% should deliberately impact the input save stack, with the hope to gain more
% speed from removing tokens and leaving them upstream.
%
% For this however I had to modify the underlying mathematical algorithm. The
% initial one is a bit unusual I guess, and, I trust, rather efficient, but it
% does not produce the quotient digits (in base 10000) one by one; at any given
% time it is possible that some correction will be made, which means it is not
% an appropriate algorithm for a TeX implementation which will abandon the
% quotient upstream. Thus I now have with 1.09j a new underlying mathematical
% algorithm, presumably much more standard. It is a bit complicated to implement
% expandably these things, but in the end I had regained the already mentioned
% 10$% efficiency and even more for
% small to medium sized inputs (up to 30$% perhaps). And in passing I did a
% special routine for divisors < 10000, which is 5 to 10 times faster still.
%
% But, I then tested a variant of my new implementation which again did
% not impact the input save stack and, for sizes of up to 200 digits, it
% is not much worse, indeed it is perhaps actually better than the one
% abandoning the quotient digits upstream (and in the end putting them
% in the correct order). So, finally, I re-incorporated the produced
% quotient digits within a tail recursion. Hence \xintiDivision, like all
% other routines in xint (except \xintSeq without optional parameter)
% does not impact the input save stack. One can have a produced
% quotient longer than 4x5000=20000 digits, and no need to worry about
% consequences propagating to \xintTrunc, \xintRound, \xintFloat,
% \xintFloatSqrt, etc... and all other places using the division. See
% also \xintXTrunc in this context.|
%    \begin{macrocode}
\def\xintiiQuo {\romannumeral0\xintiiquo }%
\def\xintiiRem {\romannumeral0\xintiirem }%
\def\xintiiquo {\expandafter\xint_firstoftwo_thenstop\romannumeral0\xintiidivision }%
\def\xintiirem {\expandafter\xint_secondoftwo_thenstop\romannumeral0\xintiidivision }%
\def\xintiQuo {\romannumeral0\xintiquo }%
\def\xintiRem {\romannumeral0\xintirem }%
\def\xintiquo {\expandafter\xint_firstoftwo_thenstop\romannumeral0\xintidivision }%
\def\xintirem {\expandafter\xint_secondoftwo_thenstop\romannumeral0\xintidivision }%
\let\xintQuo\xintiQuo\let\xintquo\xintiquo % deprecated (1.1) 
\let\xintRem\xintiRem\let\xintrem\xintirem % deprecated (1.1)
%    \end{macrocode}
% \lverb-#1 = A, #2 = B. On calcule le quotient et le reste dans la division
% euclidienne de A par B: A=BQ+R, 0<= R < |B|.-
%    \begin{macrocode}
\def\xintiDivision {\romannumeral0\xintidivision }%
\def\xintidivision #1{\expandafter\XINT_division\romannumeral0\xintnum{#1}\Z }%
\let\xintDivision\xintiDivision \let\xintdivision\xintidivision % deprecated
\def\XINT_division #1#2\Z #3{\expandafter\XINT_iidivision_a\expandafter #1%
                             \romannumeral0\xintnum{#3}\Z #2\Z }%
\def\xintiiDivision   {\romannumeral0\xintiidivision }%
\def\xintiidivision  #1{\expandafter\XINT_iidivision \romannumeral-`0#1\Z }%
\def\XINT_iidivision #1#2\Z #3{\expandafter\XINT_iidivision_a\expandafter #1%
                             \romannumeral-`0#3\Z #2\Z }%
\def\XINT_iidivision_a #1#2% #1 de A, #2 de B.
{%
    \if0#2\xint_dothis\XINT_iidivision_divbyzero\fi
    \if0#1\xint_dothis\XINT_iidivision_aiszero\fi
    \if-#2\xint_dothis{\expandafter\XINT_iidivision_bneg
                       \romannumeral0\XINT_iidivision_bpos #1}\fi
    \xint_orthat{\XINT_iidivision_bpos #1#2}%
}%
\def\XINT_iidivision_divbyzero #1\Z #2\Z {\xintError:DivisionByZero\space {0}{0}}%
\def\XINT_iidivision_aiszero #1\Z #2\Z   { {0}{0}}%
\def\XINT_iidivision_bneg #1% q->-q, r unchanged
   {\expandafter\space\expandafter{\romannumeral0\XINT_opp #1}}%
\def\XINT_iidivision_bpos #1%
{%
    \xint_UDsignfork
            #1\XINT_iidivision_aneg
             -{\XINT_iidivision_apos #1}%
    \krof
}%
\def\XINT_iidivision_apos #1#2\Z #3\Z{\XINT_div_prepare {#2}{#1#3}}%
\def\XINT_iidivision_aneg #1\Z #2\Z
   {\expandafter
    \XINT_iidivision_aneg_b\romannumeral0\XINT_div_prepare {#1}{#2}{#1}}%
\def\XINT_iidivision_aneg_b #1#2{\if0\XINT_Sgn #2\Z
                              \expandafter\XINT_iidivision_aneg_rzero
                           \else
                              \expandafter\XINT_iidivision_aneg_rpos
                           \fi {#1}{#2}}%
\def\XINT_iidivision_aneg_rzero #1#2#3{ {-#1}{0}}% necessarily q was >0
\def\XINT_iidivision_aneg_rpos #1%
{%
    \expandafter\XINT_iidivision_aneg_end\expandafter
               {\expandafter-\romannumeral0\xintinc {#1}}% q-> -(1+q)
}%
\def\XINT_iidivision_aneg_end #1#2#3%
{%
     \expandafter\xint_exchangetwo_keepbraces_thenstop
     \expandafter{\romannumeral0\XINT_sub_pre {#3}{#2}}{#1}% r-> b-r
}%
%    \end{macrocode}
% \lverb|&
% Pour la suite A et B sont > 0.
% #1 = B. Pour le moment à l'endroit.
% Calcul du plus petit K = 4n >= longueur de B|
%    \begin{macrocode}
\def\XINT_div_prepare #1%
{%
    \expandafter \XINT_div_prepareB_aa \expandafter
        {\romannumeral0\xintlength {#1}}{#1}% B > 0 ici
}%
\def\XINT_div_prepareB_aa #1%
{%
    \ifnum #1=\xint_c_i
      \expandafter\XINT_div_prepareB_onedigit
    \else
      \expandafter\XINT_div_prepareB_a
    \fi
    {#1}%
}%
\def\XINT_div_prepareB_a #1%
{%
  \expandafter\XINT_div_prepareB_c\expandafter
  {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}%
}%
%    \end{macrocode}
% \lverb|B=1 and B=2 treated specially.|
%    \begin{macrocode}
\def\XINT_div_prepareB_onedigit #1#2%
{%
    \ifcase#2
    \or\expandafter\XINT_div_BisOne
    \or\expandafter\XINT_div_BisTwo
    \else\expandafter\XINT_div_prepareB_e
    \fi {000}{0}{4}{#2}%
}%
\def\XINT_div_BisOne #1#2#3#4#5{ {#5}{0}}%
\def\XINT_div_BisTwo #1#2#3#4#5%
{%
    \expandafter\expandafter\expandafter\XINT_div_BisTwo_a
    \ifodd\xintiiLDg{#5} \expandafter1\else \expandafter0\fi {#5}%
}%
\edef\XINT_div_BisTwo_a #1#2%
{%
    \noexpand\expandafter\space\noexpand\expandafter
    {\noexpand\romannumeral0\noexpand\xinthalf {#2}}{#1}%
}%
%    \end{macrocode}
% \lverb|#1 = K. 1.09j uses \csname, earlier versions did it with
% \ifcase.|
%    \begin{macrocode}
\def\XINT_div_prepareB_c #1#2%
{%
    \csname XINT_div_prepareB_d\romannumeral\numexpr#1-#2\endcsname
    {#1}%
}%
\def\XINT_div_prepareB_d    {\XINT_div_prepareB_e {}{0000}}%
\def\XINT_div_prepareB_di   {\XINT_div_prepareB_e {0}{000}}%
\def\XINT_div_prepareB_dii  {\XINT_div_prepareB_e {00}{00}}%
\def\XINT_div_prepareB_diii {\XINT_div_prepareB_e {000}{0}}%
\def\XINT_div_cleanR        #10000.{{#1}}%
%    \end{macrocode}
% \lverb|#1 = zéros à rajouter à B, #2=c [modifié dans 1.09j, ce sont maintenant
% des zéros explicites en nombre 4 - ancien c, et on utilisera
% \XINT_div_cleanR et non plus \XINT_dsh_checksignx pour nettoyer à la fin
% des zéros en excès dans le Reste; in all comments next, «c» stands now {0} or
% {00} or {000} or {0000} rather than a digit as in earlier versions], #3=K, #4
% = B|
%    \begin{macrocode}
\def\XINT_div_prepareB_e #1#2#3#4%
{%
    \ifnum#3=\xint_c_iv\expandafter\XINT_div_prepareLittleB_f
                   \else\expandafter\XINT_div_prepareB_f
    \fi
    #4#1{#3}{#2}{#1}%
}%
%    \end{macrocode}
% \lverb|x = #1#2#3#4 = 4 premiers chiffres de B. #1 est non nul. B is reversed.
% With 1.09j or latter x+1 and (x+1)/2 are pre-computed. Si K=4 on ne renverse
% pas B, et donc B=x dans la suite. De plus pour K=4 on ne travaille pas avec
% x+1 et (x+1)/2 mais avec x et x/2.|
%    \begin{macrocode}
\def\XINT_div_prepareB_f #1#2#3#4#5#{%
    \expandafter\XINT_div_prepareB_g
     \the\numexpr  #1#2#3#4+\xint_c_i\expandafter
    .\the\numexpr (#1#2#3#4+\xint_c_i)/\xint_c_ii\expandafter
    .\romannumeral0\xintreverseorder {#1#2#3#4#5}.{#1#2#3#4}%
}%
\def\XINT_div_prepareLittleB_f #1#{%
    \expandafter\XINT_div_prepareB_g \the\numexpr  #1/\xint_c_ii.{}.{}.{#1}%
}%
%    \end{macrocode}
% \lverb|&
% #1 = x' = x+1= 1+quatre premiers chiffres de B, #2 = y = (x+1)/2 précalculé
% #3 = B préparé et maintenant renversé, #4=x,
% #5 = K, #6 = «c», #7= {} ou {0} ou {00} ou {000}, #8 = A initial
% On multiplie aussi A par 10^c. -> AK{x'yx}B«c». Par contre dans le
% cas little on a #1=y=(x/2), #2={}, #3={}, #4=x, donc cela donne
% ->AK{y{}x}{}«c», il n'y a pas de B.|
%    \begin{macrocode}
\def\XINT_div_prepareB_g #1.#2.#3.#4#5#6#7#8%
{%
    \XINT_div_prepareA_a {#8#7}{#5}{{#1}{#2}{#4}}{#3}{#6}%
}%
%    \end{macrocode}
% \lverb|A, K, {x'yx}, B«c» |
%    \begin{macrocode}
\def\XINT_div_prepareA_a #1%
{%
    \expandafter\XINT_div_prepareA_b\expandafter
       {\romannumeral0\xintlength {#1}}{#1}%
}%
%    \end{macrocode}
% \lverb|L0, A, K, {x'yx}, B«c»|
%    \begin{macrocode}
\def\XINT_div_prepareA_b #1%
{%
  \expandafter\XINT_div_prepareA_c\expandafter
    {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}%
}%
%    \end{macrocode}
% \lverb|L, L0, A, K, {x'yx}, B, «c»|
%    \begin{macrocode}
\def\XINT_div_prepareA_c #1#2%
{%
    \csname XINT_div_prepareA_d\romannumeral\numexpr #1-#2\endcsname
    {#1}%
}%
\def\XINT_div_prepareA_d     {\XINT_div_prepareA_e {}}%
\def\XINT_div_prepareA_di    {\XINT_div_prepareA_e {0}}%
\def\XINT_div_prepareA_dii   {\XINT_div_prepareA_e {00}}%
\def\XINT_div_prepareA_diii  {\XINT_div_prepareA_e {000}}%
%    \end{macrocode}
% \lverb|#1#3 = A préparé, #2 = longueur de ce A préparé, #4=K, #5={x'yx}->
% LKAx'yxB«c»|
%    \begin{macrocode}
\def\XINT_div_prepareA_e #1#2#3#4#5%
{%
    \XINT_div_start_a {#2}{#4}{#1#3}#5%
}%
%    \end{macrocode}
% \lverb|L, K, A, x',y,x, B, «c» (avec y{}x{} au lieu de x'yxB dans la
% variante little)|
%    \begin{macrocode}
\def\XINT_div_start_a #1#2%
{%
    \ifnum #2=\xint_c_iv \expandafter\XINT_div_little_b
    \else
      \ifnum #1 < #2
        \expandafter\expandafter\expandafter\XINT_div_III_aa
      \else
        \expandafter\expandafter\expandafter\XINT_div_start_b
      \fi
    \fi
    {#1}{#2}%
}%
%    \end{macrocode}
% \lverb|L, K, A, x',y,x, B, «c».|
%    \begin{macrocode}
\def\XINT_div_III_aa #1#2#3#4#5#6#7%
{%
    \expandafter\expandafter\expandafter
    \XINT_div_III_b\xint_cleanupzeros_nostop #3.{0000}%
}%
%    \end{macrocode}
% \lverb|R.Q«c».|
%    \begin{macrocode}
\def\XINT_div_III_b #1%
{%
    \if0#1%
       \expandafter\XINT_div_III_bRzero
    \else
       \expandafter\XINT_div_III_bRpos
    \fi
    #1%
}%
\def\XINT_div_III_bRzero 0.#1#2%
{%
    \expandafter\space\expandafter
   {\romannumeral0\XINT_cuz_loop #1\W\W\W\W\W\W\W\Z}{0}%
}%
\def\XINT_div_III_bRpos #1.#2#3%
{%
    \expandafter\XINT_div_III_c \XINT_div_cleanR #1#3.{#2}%
}%
\def\XINT_div_III_c #1#2%
{%
    \expandafter\space\expandafter
   {\romannumeral0\XINT_cuz_loop #2\W\W\W\W\W\W\W\Z}{#1}%
}%
%    \end{macrocode}
% \lverb|L, K, A, x',y,x, B, «c»->K.A.x{LK{x'y}x}B«c»|
%    \begin{macrocode}
\def\XINT_div_start_b #1#2#3#4#5#6%
{%
    \XINT_div_start_c {#2}.#3.{#6}{{#1}{#2}{{#4}{#5}}{#6}}%
}%
%    \end{macrocode}
% \lverb|Kalpha.A.x{LK{x'y}x}, B, «c», au début #2=alpha est vide|
%    \begin{macrocode}
\def\XINT_div_start_c #1#2.#3#4#5#6%
{%
    \ifnum #1=\xint_c_iv\XINT_div_start_ca\fi
    \expandafter\XINT_div_start_c\expandafter
         {\the\numexpr #1-\xint_c_iv}#2#3#4#5#6.%
}%
\def\XINT_div_start_ca\fi\expandafter\XINT_div_start_c\expandafter
    #1#2#3#4#5{\fi\XINT_div_start_d {#2#3#4#5}#2#3#4#5}%
%    \end{macrocode}
% \lverb|#1=a, #2=alpha (de longueur K, à l'endroit).#3=reste de A.#4=x,
% #5={LK{x'y}x},#6=B,«c» -> a, x, alpha, B, {0000}, L, K, {x'y},x,
% alpha'=reste de A, B{}«c». Pour K=4 on a en fait B=x, faudra revoir après.|
%    \begin{macrocode}
\def\XINT_div_start_d #1#2.#3.#4#5#6%
{%
    \XINT_div_I_a {#1}{#4}{#2}{#6}{0000}#5{#3}{#6}{}%
}%
%    \end{macrocode}
% \lverb|Ceci est le point de retour de la boucle principale. a, x, alpha, B,
% q0, L, K, {x'y}, x, alpha', BQ«c» |
%    \begin{macrocode}
\def\XINT_div_I_a #1#2%
{%
    \expandafter\XINT_div_I_b\the\numexpr #1/#2.{#1}{#2}%
}%
\def\XINT_div_I_b #1%
{%
    \xint_gob_til_zero #1\XINT_div_I_czero 0\XINT_div_I_c #1%
}%
%    \end{macrocode}
% \lverb|On intercepte quotient nul: #1=a, x, alpha, B, #5=q0, L, K, {x'y}, x,
% alpha', BQ«c» -> q{alpha} L, K, {x'y}, x, alpha', BQ«c»|
%    \begin{macrocode}
\def\XINT_div_I_czero 0%
    \XINT_div_I_c 0.#1#2#3#4#5{\XINT_div_I_g {#5}{#3}}%
\def\XINT_div_I_c #1.#2#3%
{%
    \expandafter\XINT_div_I_da\the\numexpr #2-#1*#3.#1.%
}%
%    \end{macrocode}
% \lverb|r.q.alpha, B, q0, L, K, {x'y}, x, alpha', BQ«c»|
%    \begin{macrocode}
\def\XINT_div_I_da #1.%
{%
    \ifnum #1>\xint_c_ix
       \expandafter\XINT_div_I_dP
    \else
       \ifnum #1<\xint_c_
        \expandafter\expandafter\expandafter\XINT_div_I_dN
       \else
        \expandafter\expandafter\expandafter\XINT_div_I_db
       \fi
    \fi
}%
\def\XINT_div_I_dN #1.%
{%
    \expandafter\XINT_div_I_dP\the\numexpr #1-\xint_c_i.%
}%
\def\XINT_div_I_db #1.#2#3% #1=q=un chiffre, #2=alpha, #3=B
{%
    \expandafter\XINT_div_I_dc\expandafter
    {\romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter
       {\romannumeral0\xintreverseorder{#2}}%
       {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}%
    #1{#2}{#3}%
}%
\def\XINT_div_I_dc #1#2%
{%
    \if-#1% s'arranger pour que si n\'egatif on ait renvoy\'e alpha=-.
         \expandafter\xint_firstoftwo
    \else\expandafter\xint_secondoftwo\fi
    {\expandafter\XINT_div_I_dP\the\numexpr #2-\xint_c_i.}%
    {\XINT_div_I_e {#1}#2}%
}%
%    \end{macrocode}
% \lverb|alpha,q,ancien alpha,B, q0->1nouveauq.alpha, L, K, {x'y},x, alpha',
% BQ«c»|
%    \begin{macrocode}
\def\XINT_div_I_e #1#2#3#4#5%
{%
    \expandafter\XINT_div_I_f \the\numexpr \xint_c_x^iv+#2+#5{#1}%
}%
%    \end{macrocode}
% \lverb|q.alpha, B, q0, L, K, {x'y},x, alpha'BQ«c» (intercepter q=0?)
%        -> 1nouveauq.nouvel alpha, L, K, {x'y}, x, alpha',BQ«c»|
%    \begin{macrocode}
\def\XINT_div_I_dP #1.#2#3#4%
{%
    \expandafter \XINT_div_I_f \the\numexpr \xint_c_x^iv+#1+#4\expandafter
    {\romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter
      {\romannumeral0\xintreverseorder{#2}}%
      {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}%
}%
%    \end{macrocode}
% \lverb|1#1#2#3#4=nouveau q, nouvel alpha, L, K, {x'y},x,alpha', BQ«c»|
%    \begin{macrocode}
\def\XINT_div_I_f 1#1#2#3#4{\XINT_div_I_g {#1#2#3#4}}%
%    \end{macrocode}
% \lverb|#1=q,#2=nouvel alpha,#3=L, #4=K, #5={x'y}, #6=x, #7= alpha',#8=B,
% #9=Q«c» -> {x'y}alpha.alpha'.{{x'y}xKL}B{Qq}«c»|
%    \begin{macrocode}
\def\XINT_div_I_g #1#2#3#4#5#6#7#8#9%
{%
     \ifnum#3=#4
          \expandafter\XINT_div_III_ab
     \else
          \expandafter\XINT_div_I_h
     \fi
     {#5}#2.#7.{{#5}{#6}{#4}{#3}}{#8}{#9#1}%
}%
%    \end{macrocode}
% \lverb|{x'y}alpha.alpha'.{{x'y}xKL}B{Qq}«c» -> R sans leading zeros.{Qq}«c»|
%    \begin{macrocode}
\def\XINT_div_III_ab #1#2.#3.#4#5%
{%
    \expandafter\XINT_div_III_b
    \romannumeral0\XINT_cuz_loop #2#3\W\W\W\W\W\W\W\Z.%
}%
%    \end{macrocode}
% \lverb|#1={x'y}alpha.#2#3#4#5#6=reste de A.
% #7={{x'y},x,K,L},#8=B,nouveauQ«c» devient {x'y},alpha sur K+4 chiffres.B,
% {{x'y},x,K,L}, #6= nouvel alpha',B,nouveauQ«c»|
%    \begin{macrocode}
\def\XINT_div_I_h #1.#2#3#4#5#6.#7#8%
{%
    \XINT_div_II_b #1#2#3#4#5.{#8}{#7}{#6}{#8}%
}%
%    \end{macrocode}
% \lverb|{x'y}alpha.B, {{x'y},x,K,L}, nouveau alpha',B, Q«c» On
% intercepte la situation avec alpha débutant par 0000 qui est la seule qui
% pourrait donner un q1 nul. Donc q1 est non nul et la soustraction spéciale
% recevra un q1*B de longueur K ou K+4 et jamais 0000. Ensuite un q2 éventuel
% s'il est calculé est nécessairement non nul lui aussi. Comme dans la phase I
% on a aussi intercepté un q nul, la soustraction spéciale ne reçoit donc jamais
% un qB nul. Note: j'ai testé plusieurs fois que ma technique de gob_til_zeros
% est plus rapide que d'utiliser un \ifnum |
%    \begin{macrocode}
\def\XINT_div_II_b #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_zeros_iv #2#3#4#5\XINT_div_II_skipc 0000%
    \XINT_div_II_c #1{#2#3#4#5}{#6#7#8#9}%
}%
%    \end{macrocode}
% \lverb|x'y{0000}{4chiffres}reste de alpha.#6=B,#7={{x'y},x,K,L}, alpha',B,
% Q«c» -> {x'y}x,K,L (à diminuer de 4), {alpha sur
% K}B{q1=0000}{alpha'}B,Q«c»|
%    \begin{macrocode}
\def\XINT_div_II_skipc 0000\XINT_div_II_c #1#2#3#4#5.#6#7%
{%
    \XINT_div_II_k #7{#4#5}{#6}{0000}%
}%
%    \end{macrocode}
% \lverb|x'ya->1qx'yalpha.B, {{x'y},x,K,L}, nouveau alpha',B, Q«c»|
%    \begin{macrocode}
\def\XINT_div_II_c #1#2#3#4%
{%
     \expandafter\XINT_div_II_d\the\numexpr (#3#4+#2)/#1+\xint_c_ixixixix\relax
     {#1}{#2}#3#4%
}%
%    \end{macrocode}
% \lverb|1 suivi de q1 sur quatre chiffres, #5=x', #6=y, #7=alpha.#8=B,
% {{x'y},x,K,L}, alpha', B, Q«c» --> nouvel alpha.x',y,B,q1,{{x'y},x,K,L},
% alpha', B, Q«c» |
%    \begin{macrocode}
\def\XINT_div_II_d 1#1#2#3#4#5#6#7.#8%
{%
    \expandafter\XINT_div_II_e
    \romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter
      {\romannumeral0\xintreverseorder{#7}}%
      {\romannumeral0\XINT_mul_Mr {#1#2#3#4}#8\Z\Z\Z\Z }.%
    {#5}{#6}{#8}{#1#2#3#4}%
}%
%    \end{macrocode}
% \lverb|alpha.x',y,B,q1, {{x'y},x,K,L}, alpha', B, Q«c»|
%    \begin{macrocode}
\def\XINT_div_II_e #1#2#3#4%
{%
    \xint_gob_til_zeros_iv #1#2#3#4\XINT_div_II_skipf 0000%
    \XINT_div_II_f #1#2#3#4%
}%
%    \end{macrocode}
% \lverb|0000alpha sur K chiffres.#2=x',#3=y,#4=B,#5=q1, #6={{x'y},x,K,L},
% #7=alpha',BQ«c» -> {x'y}x,K,L (à diminuer de 4),
% {alpha sur K}B{q1}{alpha'}BQ«c»|
%    \begin{macrocode}
\def\XINT_div_II_skipf 0000\XINT_div_II_f 0000#1.#2#3#4#5#6%
{%
    \XINT_div_II_k #6{#1}{#4}{#5}%
}%
%    \end{macrocode}
% \lverb|a1 (huit chiffres), alpha (sur K+4), x', y, B, q1, {{x'y},x,K,L},
% alpha', B,Q«c»|
%    \begin{macrocode}
\def\XINT_div_II_f #1#2#3#4#5#6#7#8#9.%
{%
    \XINT_div_II_fa {#1#2#3#4#5#6#7#8}{#1#2#3#4#5#6#7#8#9}%
}%
\def\XINT_div_II_fa #1#2#3#4%
{%
    \expandafter\XINT_div_II_g\expandafter
                                {\the\numexpr (#1+#4)/#3-\xint_c_i}{#2}%
}%
%    \end{macrocode}
% \lverb|#1=q, #2=alpha (K+4), #3=B, #4=q1, {{x'y},x,K,L}, alpha', BQ«c»
%        -> 1 puis nouveau q sur 4 chiffres, nouvel alpha sur K chiffres,
%        B, {{x'y},x,K,L}, alpha',BQ«c» |
%    \begin{macrocode}
\def\XINT_div_II_g #1#2#3#4%
{%
    \expandafter \XINT_div_II_h
    \the\numexpr #4+#1+\xint_c_x^iv\expandafter\expandafter\expandafter
    {\expandafter\xint_gobble_iv
    \romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter
       {\romannumeral0\xintreverseorder{#2}}%
       {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}{#3}%
}%
%    \end{macrocode}
% \lverb|1 puis nouveau q sur 4 chiffres, #5=nouvel alpha sur K chiffres,
% #6=B, #7={{x'y},x,K,L} avec L à ajuster,  alpha', BQ«c»
% -> {x'y}x,K,L à diminuer de 4, {alpha}B{q}, alpha', BQ«c»|
%    \begin{macrocode}
\def\XINT_div_II_h 1#1#2#3#4#5#6#7%
{%
    \XINT_div_II_k #7{#5}{#6}{#1#2#3#4}%
}%
%    \end{macrocode}
% \lverb|{x'y}x,K,L à diminuer de 4, alpha, B{q}alpha',BQ«c»
%        ->nouveau L.K,x',y,x,alpha.B,q,alpha',B,Q«c»
%        ->{LK{x'y}x},x,a,alpha.B,q,alpha',B,Q«c»|
%    \begin{macrocode}
\def\XINT_div_II_k #1#2#3#4#5%
{%
    \expandafter\XINT_div_II_l \the\numexpr #4-\xint_c_iv.{#3}#1{#2}#5.%
}%
\def\XINT_div_II_l #1.#2#3#4#5#6#7#8#9%
{%
    \XINT_div_II_m {{#1}{#2}{{#3}{#4}}{#5}}{#5}{#6#7#8#9}#6#7#8#9%
}%
%    \end{macrocode}
% \lverb|{LK{x'y}x},x,a,alpha.B{q}alpha'BQ -> a, x, alpha, B, q,
% L, K, {x'y}, x, alpha', BQ«c» |
%    \begin{macrocode}
\def\XINT_div_II_m #1#2#3#4.#5#6%
{%
     \XINT_div_I_a {#3}{#2}{#4}{#5}{#6}#1%
}%
%    \end{macrocode}
% \lverb|L, K, A, y,{},x, {},«c»->A.{yx}L{}«c» Comme ici K=4, dans
% la phase I on n'a pas besoin de alpha, car a = alpha. De plus on a maintenu B
% dans l'ordre qui est donc la même chose que x. Par ailleurs la phase I est
% simplifiée, il s'agit simplement de la division euclidienne de a par x, et de
% plus on n'a à la faire qu'une unique fois et ensuite la phase II peut boucler
% sur elle-même au lieu de revenir en phase I, par conséquent il n'y a pas non
% plus de q0 ici. Enfin, le y est (x/2) pas ((x+1)/2) il n'y a pas de x'=x+1|
%    \begin{macrocode}
\def\XINT_div_little_b #1#2#3#4#5#6#7%
{%
    \XINT_div_little_c #3.{{#4}{#6}}{#1}%
}%
%    \end{macrocode}
% \lverb|#1#2#3#4=a, #5=alpha'=reste de A.#6={yx}, #7=L, «c» -> a,
% y, x, L, alpha'=reste de A, «c».|
%    \begin{macrocode}
\def\XINT_div_little_c #1#2#3#4#5.#6#7%
{%
    \XINT_div_littleI_a {#1#2#3#4}#6{#7}{#5}%
}%
%    \end{macrocode}
% \lverb|a, y, x, L, alpha',«c» On calcule ici (contrairement à la
% phase I générale) le vrai quotient euclidien de a par x=B, c'est donc un
% chiffre de 0 à 9. De plus on n'a à faire cela qu'une unique fois.|
%    \begin{macrocode}
\def\XINT_div_littleI_a #1#2#3%
{%
    \expandafter\XINT_div_littleI_b
    \the\numexpr (#1+#2)/#3-\xint_c_i{#1}{#2}{#3}%
}%
%    \end{macrocode}
% \lverb|On intercepte quotient nul: [est-ce vraiment utile? ou n'est-ce pas
% plutôt une perte de temps en moyenne? il faudrait tester]  q=0#1=a,
% #2=y, x, L, alpha', «c» ->
% II_a avec L{alpha}alpha'.{yx}{0000}«c». Et en cas de quotient non nul on
% procède avec littleI_c avec #1=q, #2=a, #3=y, #4=x ->  {nouvel alpha sur 4
% chiffres}q{yx},L,alpha',«c».|
%    \begin{macrocode}
\def\XINT_div_littleI_b #1%
{%
    \xint_gob_til_zero #1\XINT_div_littleI_skip 0\XINT_div_littleI_c #1%
}%
\def\XINT_div_littleI_skip 0\XINT_div_littleI_c 0#1#2#3#4#5%
   {\XINT_div_littleII_a {#4}{#1}#5.{{#2}{#3}}{0000}}%
\def\XINT_div_littleI_c #1#2#3#4%
{%
    \expandafter\expandafter\expandafter\XINT_div_littleI_e
    \expandafter\expandafter\expandafter
    {\expandafter\xint_gobble_i\the\numexpr \xint_c_x^iv+#2-#1*#4}#1{{#3}{#4}}%
}%
%    \end{macrocode}
% \lverb|#1=nouvel alpha sur 4 chiffres#2=q,#3={yx}, #4=L, #5=alpha',«c» ->
% L{alpha}alpha'.{yx}{000q}«c» point d'entrée de la boucle principale|
%    \begin{macrocode}
\def\XINT_div_littleI_e #1#2#3#4#5%
   {\XINT_div_littleII_a {#4}{#1}#5.{#3}{000#2}}%
%    \end{macrocode}
% \lverb|L{alpha}alpha'.{yx}Q«c» et c'est là qu'on boucle|
%    \begin{macrocode}
\def\XINT_div_littleII_a #1%
{%
     \ifnum#1=\xint_c_iv
          \expandafter\XINT_div_littleIII_ab
     \else
          \expandafter\XINT_div_littleII_b
     \fi {#1}%
}%
%    \end{macrocode}
% \lverb|L{alpha}alpha'.{yx}Q«c» -> (en fait #3 est vide normalement ici) R
% sans leading zeros.Q«c»|
%    \begin{macrocode}
\def\XINT_div_littleIII_ab #1#2#3.#4%
{%
    \expandafter\XINT_div_III_b\the\numexpr #2#3.%
}%
%    \end{macrocode}
% \lverb|L{alpha}alpha'.{yx}Q«c». On diminue L de quatre, comme cela c'est
% fait.|
%    \begin{macrocode}
\def\XINT_div_littleII_b #1%
{%
   \expandafter\XINT_div_littleII_c\expandafter {\the\numexpr #1-\xint_c_iv}%
}%
%    \end{macrocode}
% \lverb|{nouveauL}{alpha}alpha'.{yx}Q«c». On prélève 4 chiffres de alpha' ->
% {nouvel alpha sur huit chiffres}yx{nouveau L}{nouvel alpha'}Q«c». Regarder
% si l'ancien alpha était 0000 n'avancerait à rien car obligerait à refaire une
% chose comme la phase I, donc on ne perd pas de temps avec ça, on reste en
% permanence en phase II.|
%    \begin{macrocode}
\def\XINT_div_littleII_c #1#2#3#4#5#6#7.#8%
{%
    \XINT_div_littleII_d {#2#3#4#5#6}#8{#1}{#7}%
}%
\def\XINT_div_littleII_d #1#2#3%
{%
    \expandafter\XINT_div_littleII_e\the\numexpr (#1+#2)/#3+\xint_c_ixixixix.%
    {#1}{#2}{#3}%
}%
%    \end{macrocode}
% \lverb|1 suivi de #1=q1 sur quatre chiffres.#2=alpha, #3=y, #4=x,
% L, alpha', Q«c» --> nouvel alpha sur 4.{q1}{yx},L,alpha', Q«c» |
%    \begin{macrocode}
\def\XINT_div_littleII_e 1#1.#2#3#4%
{%
    \expandafter\expandafter\expandafter\XINT_div_littleII_f
    \expandafter\xint_gobble_i\the\numexpr \xint_c_x^iv+#2-#1*#4.%
    {#1}{{#3}{#4}}%
}%
%    \end{macrocode}
% \lverb|alpha.q,{yx},L,alpha',Q«c»->L{alpha}alpha'.{yx}{Qq}«c»|
%    \begin{macrocode}
\def\XINT_div_littleII_f #1.#2#3#4#5#6%
{%
    \XINT_div_littleII_a {#4}{#1}#5.{#3}{#6#2}%
}%
%    \end{macrocode}
% \lverb|La soustraction spéciale. Dans 1.09j, elle fait A-qB, pour A (en fait
% alpha dans mes dénominations des commentaires du code) et qB chacun de
% longueur K ou K+4, avec K au moins huit multiple de quatre, qB a ses quatre
% chiffres significatifs (qui sont à droite) non nuls. Si A-qB<0 il suffit de
% renvoyer -, le résultat n'importe pas. On est sûr que qB est non nul. On le
% met dans cette version en premier pour tester plus facilement le cas avec qB
% de longueur K+4 et A de longueur seulement K. Lorsque la longueur de qB est
% inférieure ou égale à celle de A, on va jusqu'à la fin de A et donc c'est la
% retenue finale qui décide du cas négatif éventuel. Le résultat non négatif est
% toujours donc renvoyé avec la même longueur que A, et il est dans l'ordre.
% J'ai fait une implémentation des phases I et II en maintenant alpha toujours à
% l'envers afin d'éviter le reverse order systématique fait sur A (ou plutôt
% alpha), mais alors il fallait que la soustraction ici s'arrange pour repérer
% les huit chiffres les plus significatifs, au final ce n'était pas plus rapide,
% et même pénalisant pour de gros inputs. Dans les versions 1.09i et antérieures
% (en fait je pense qu'ici rien quasiment n'avait bougé depuis la première
% implémentation), la soustraction spéciale n'était pratiquée que dans des cas
% avec certainement A-qB positif ou nul. De plus on n'excluait pas q=0, donc il
% fallait aussi faire un éventuel reverseorder sur ce qui était encore non
% traité. Les cas avec q=0 sont maintenant interceptés en amont et comme A et qB
% ont toujours quasiment la même longueur on ne s'embarrasse pas de
% complications pour la fin.|
%    \begin{macrocode}
\def\XINT_div_sub_xpxp #1#2% #1=alpha d\'ej\`a renvers\'e, #2 se d\'eveloppe en qB
{%
    \expandafter\XINT_div_sub_xpxp_b #2\W\X\Y\Z #1\W\X\Y\Z
}%
\def\XINT_div_sub_xpxp_b
{%
    \XINT_div_sub_A 1{}%
}%
\def\XINT_div_sub_A #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_div_sub_az\W
    \XINT_div_sub_B #1{#3#4#5#6}{#2}%
}%
\def\XINT_div_sub_B #1#2#3#4\W\X\Y\Z #5#6#7#8%
{%
    \xint_gob_til_W #5\xint_div_sub_bz\W
    \XINT_div_sub_onestep #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z
}%
\def\XINT_div_sub_onestep #1#2#3#4#5#6%
{%
    \expandafter\XINT_div_sub_backtoA
    \the\numexpr 11#6-#5#4#3#2+#1-\xint_c_i.%
}%
\def\XINT_div_sub_backtoA #1#2#3.#4%
{%
    \XINT_div_sub_A #2{#3#4}%
}%
%    \end{macrocode}
% \lverb|si on arrive en sub_bz c'est que qB était de longueur K+4 et A
% seulement de longueur K, le résultat est donc < 0, renvoyer juste -|
%    \begin{macrocode}
\def\xint_div_sub_bz\W\XINT_div_sub_onestep #1\Z { -}%
%    \end{macrocode}
% \lverb|si on arrive en sub_az c'est que qB était de longueur inférieure ou
% égale à celle de A, donc on continue jusqu'à la fin de A, et on vérifiera la
% retenue à la fin.|
%    \begin{macrocode}
\def\xint_div_sub_az\W\XINT_div_sub_B #1#2{\XINT_div_sub_C #1}%
\def\XINT_div_sub_C #1#2#3#4#5#6%
{%
    \xint_gob_til_W #3\xint_div_sub_cz\W
    \XINT_div_sub_C_onestep #1{#6#5#4#3}{#2}%
}%
\def\XINT_div_sub_C_onestep #1#2%
{%
    \expandafter\XINT_div_sub_backtoC \the\numexpr 11#2+#1-\xint_c_i.%
}%
\def\XINT_div_sub_backtoC #1#2#3.#4%
{%
    \XINT_div_sub_C #2{#3#4}%
}%
%    \end{macrocode}
% \lverb|une fois arrivé en sub_cz on teste la retenue pour voir si le résultat
% final est en fait négatif, dans ce cas on renvoie seulement -|
%    \begin{macrocode}
\def\xint_div_sub_cz\W\XINT_div_sub_C_onestep #1#2%
{%
    \if#10% retenue
         \expandafter\xint_div_sub_neg
    \else\expandafter\xint_div_sub_ok
    \fi
}%
\def\xint_div_sub_neg #1{ -}%
\def\xint_div_sub_ok  #1{ #1}%
%    \end{macrocode}
% \subsection{\csh{xintiDivRound}, \csh{xintiiDivRound}}
% \lverb|v1.1, transferred from first release of bnumexpr.|
%    \begin{macrocode}
\def\xintiDivRound    {\romannumeral0\xintidivround }%
\def\xintidivround  #1{\expandafter\XINT_iidivround\romannumeral0\xintnum{#1}\Z }%
\def\xintiiDivRound   {\romannumeral0\xintiidivround }%
\def\xintiidivround #1{\expandafter\XINT_iidivround \romannumeral-`0#1\Z }%
\def\XINT_iidivround #1#2\Z #3{\expandafter\XINT_iidivround_a\expandafter #1%
                             \romannumeral-`0#3\Z #2\Z }%
\def\XINT_iidivround_a #1#2% #1 de A, #2 de B.
{%
    \if0#2\xint_dothis\XINT_iidivround_divbyzero\fi
    \if0#1\xint_dothis\XINT_iidivround_aiszero\fi
    \if-#2\xint_dothis{\XINT_iidivround_bneg #1}\fi
          \xint_orthat{\XINT_iidivround_bpos #1#2}%
}%
\def\XINT_iidivround_divbyzero #1\Z #2\Z {\xintError:DivisionByZero\space 0}%
\def\XINT_iidivround_aiszero   #1\Z #2\Z { 0}%
\def\XINT_iidivround_bpos #1%
{%
    \xint_UDsignfork
            #1{\xintiiopp\XINT_iidivround_pos {}}%
             -{\XINT_iidivround_pos #1}%
    \krof
}%
\def\XINT_iidivround_bneg #1%
{%
    \xint_UDsignfork
            #1{\XINT_iidivround_pos {}}%
             -{\xintiiopp\XINT_iidivround_pos #1}%
    \krof
}%
\def\XINT_iidivround_pos #1#2\Z #3\Z{\expandafter\XINT_iidivround_pos_a
                          \romannumeral0\XINT_div_prepare {#2}{#1#30}}%
\def\XINT_iidivround_pos_a #1#2{\xintReverseOrder {#1\XINT_iidivround_pos_b}\Z }%
\def\XINT_iidivround_pos_b #1#2{\xint_gob_til_Z #2\XINT_iidivround_pos_small\Z
                        \XINT_iidivround_pos_c #1#2}%
\def\XINT_iidivround_pos_c #1#2\Z {\ifnum #1>\xint_c_iv
                              \expandafter\XINT_iidivround_pos_up
                      \else   \expandafter\xintreverseorder
                      \fi {#2}}%
\def\XINT_iidivround_pos_up #1{\xintinc {\xintReverseOrder{#1}}}%
\def\XINT_iidivround_pos_small\Z\XINT_iidivround_pos_c #1#2%
    {\ifnum #1>\xint_c_iv\expandafter\xint_secondoftwo\else\expandafter
                                  \xint_firstoftwo\fi { 0}{ 1}}%
%    \end{macrocode}
% \subsection{\csh{xintiDivTrunc}, \csh{xintiiDivTrunc}}
%    \begin{macrocode}
\def\xintiDivTrunc    {\romannumeral0\xintidivtrunc }%
\def\xintidivtrunc  #1{\expandafter\XINT_iidivtrunc\romannumeral0\xintnum{#1}\Z }%
\def\xintiiDivTrunc   {\romannumeral0\xintiidivtrunc }%
\def\xintiidivtrunc #1{\expandafter\XINT_iidivtrunc \romannumeral-`0#1\Z }%
\def\XINT_iidivtrunc #1#2\Z #3{\expandafter\XINT_iidivtrunc_a\expandafter #1%
                             \romannumeral-`0#3\Z #2\Z }%
\def\XINT_iidivtrunc_a #1#2% #1 de A, #2 de B.
{%
    \if0#2\xint_dothis\XINT_iidivround_divbyzero\fi
    \if0#1\xint_dothis\XINT_iidivround_aiszero\fi
    \if-#2\xint_dothis{\XINT_iidivtrunc_bneg #1}\fi
          \xint_orthat{\XINT_iidivtrunc_bpos #1#2}%
}%
\def\XINT_iidivtrunc_bpos #1%
{%
    \xint_UDsignfork
            #1{\xintiiopp\XINT_iidivtrunc_pos {}}%
             -{\XINT_iidivtrunc_pos #1}%
    \krof
}%
\def\XINT_iidivtrunc_bneg #1%
{%
    \xint_UDsignfork
            #1{\XINT_iidivtrunc_pos {}}%
             -{\xintiiopp\XINT_iidivtrunc_pos #1}%
    \krof
}%
\def\XINT_iidivtrunc_pos #1#2\Z #3\Z%
    {\expandafter\xint_firstoftwo_thenstop\romannumeral0\XINT_div_prepare {#2}{#1#3}}%
%    \end{macrocode}
% \subsection{\csh{xintiMod}, \csh{xintiiMod}}
%    \begin{macrocode}
\def\xintiMod    {\romannumeral0\xintimod }%
\def\xintimod  #1{\expandafter\XINT_iimod\romannumeral0\xintnum{#1}\Z }%
\def\xintiiMod   {\romannumeral0\xintiimod }%
\def\xintiimod #1{\expandafter\XINT_iimod \romannumeral-`0#1\Z }%
\def\XINT_iimod #1#2\Z #3{\expandafter\XINT_iimod_a\expandafter #1%
                             \romannumeral-`0#3\Z #2\Z }%
\def\XINT_iimod_a #1#2% #1 de A, #2 de B.
{%
    \if0#2\xint_dothis\XINT_iidivround_divbyzero\fi
    \if0#1\xint_dothis\XINT_iidivround_aiszero\fi
    \if-#2\xint_dothis{\XINT_iimod_bneg #1}\fi
          \xint_orthat{\XINT_iimod_bpos #1#2}%
}%
\def\XINT_iimod_bpos #1%
{%
    \xint_UDsignfork
            #1{\xintiiopp\XINT_iimod_pos {}}%
             -{\XINT_iimod_pos #1}%
    \krof
}%
\def\XINT_iimod_bneg #1%
{%
    \xint_UDsignfork
            #1{\xintiiopp\XINT_iimod_pos {}}%
             -{\XINT_iimod_pos #1}%
    \krof
}%
\def\XINT_iimod_pos #1#2\Z #3\Z%
    {\expandafter\xint_secondoftwo_thenstop\romannumeral0\XINT_div_prepare {#2}{#1#3}}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xintcore>\relax
%\def<*xint>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xintcore>
%<*xint>
% \StoreCodelineNo {xintcore}
%
% \section{Package \xintnameimp implementation}
% \label{sec:xintimp}
%
% \localtableofcontents
% 
% The basic arithmetic routines |\xintiiAdd|, |\xintiiSub|,
% |\xintiiMul|, |\xintiiQuo| and |\xintiiPow| have been moved to new
% package \xintcorenameimp.
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode64=11   % @
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \let\z\endgroup
  \expandafter\let\expandafter\x\csname ver@xint.sty\endcsname
  \expandafter\let\expandafter\w\csname ver@xintcore.sty\endcsname
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
  \ifx\csname numexpr\endcsname\relax
     \y{xint}{\numexpr not available, aborting input}%
     \aftergroup\endinput
  \else
    \ifx\x\relax   % plain-TeX, first loading of xintcore.sty
      \ifx\w\relax % but xintkernel.sty not yet loaded.
         \def\z{\endgroup\input xintcore.sty\relax}%
      \fi
    \else
      \def\empty {}%
      \ifx\x\empty % LaTeX, first loading,
      % variable is initialized, but \ProvidesPackage not yet seen
          \ifx\w\relax % xintcore.sty not yet loaded.
            \def\z{\endgroup\RequirePackage{xintcore}}%
          \fi
      \else
        \aftergroup\endinput % xint already loaded.
      \fi
    \fi
  \fi
\z%
\XINTsetupcatcodes% defined in xintkernel.sty (loaded by xintcore.sty)
%    \end{macrocode}
% \subsection{Package identification}
%    \begin{macrocode}
\XINT_providespackage
\ProvidesPackage{xint}%
  [2015/08/31 v1.1b Expandable operations on big integers (jfB)]%
%    \end{macrocode}
% \subsection{More token management}
%    \begin{macrocode}
\long\def\xint_firstofthree  #1#2#3{#1}%
\long\def\xint_secondofthree #1#2#3{#2}%
\long\def\xint_thirdofthree  #1#2#3{#3}%
\long\def\xint_firstofthree_thenstop  #1#2#3{ #1}% 1.09i
\long\def\xint_secondofthree_thenstop #1#2#3{ #2}%
\long\def\xint_thirdofthree_thenstop  #1#2#3{ #3}%
%    \end{macrocode}
% \subsection{\csh{xintSgnFork}}
% \lverb|Expandable three-way fork added in 1.07. The argument #1 must expand
% to non-self-ending -1,0 or 1. 1.09i with _thenstop.|
%    \begin{macrocode}
\def\xintSgnFork {\romannumeral0\xintsgnfork }%
\def\xintsgnfork #1%
{%
    \ifcase #1 \expandafter\xint_secondofthree_thenstop
            \or\expandafter\xint_thirdofthree_thenstop
          \else\expandafter\xint_firstofthree_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintIsOne}, \csh{xintiiIsOne}}
% \lverb|Added in 1.03. 1.09a defines \xintIsOne. 1.1a adds \xintiiIsOne.|
%    \begin{macrocode}
\def\xintiiIsOne   {\romannumeral0\xintiiisone }%
\def\xintiiisone #1{\expandafter\XINT_isone\romannumeral-`0#1\W\Z }%
\def\xintIsOne   {\romannumeral0\xintisone }%
\def\xintisone #1{\expandafter\XINT_isone\romannumeral0\xintnum{#1}\W\Z }%
\def\XINT_isOne #1{\romannumeral0\XINT_isone #1\W\Z }%
\def\XINT_isone #1#2%
{%
    \xint_gob_til_one #1\XINT_isone_b 1%
    \expandafter\space\expandafter 0\xint_gob_til_Z #2%
}%
\def\XINT_isone_b #1\xint_gob_til_Z #2%
{%
    \xint_gob_til_W #2\XINT_isone_yes \W
    \expandafter\space\expandafter 0\xint_gob_til_Z
}%
\def\XINT_isone_yes #1\Z { 1}%
%    \end{macrocode}
% \subsection{\csh{xintRev}}
% \lverb|&
% \xintRev: expands fully its argument \romannumeral-`0, and checks the sign.
% However this last aspect does not appear like a very useful thing. And despite
% the fact that a special check is made for a sign, actually the input is not
% given to \xintnum, contrarily to \xintLen. This is all a bit incoherent.
% Should be fixed.|
%    \begin{macrocode}
\def\xintRev {\romannumeral0\xintrev }%
\def\xintrev #1%
{%
    \expandafter\XINT_rev_fork
    \romannumeral-`0#1\xint_relax % empty #1 ok, \xint_relax stops expansion
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
    \xint_relax
}%
\def\XINT_rev_fork #1%
{%
    \xint_UDsignfork
    #1{\expandafter\xint_minus_thenstop\romannumeral0\XINT_rord_main {}}%
     -{\XINT_rord_main {}#1}%
    \krof
}%
%    \end{macrocode}
% \subsection{\csh{xintLen}}
% \lverb|\xintLen is ONLY for (possibly long) integers. Gets extended to
% fractions by xintfrac.sty|
%    \begin{macrocode}
\def\xintLen {\romannumeral0\xintlen }%
\def\xintlen #1%
{%
    \expandafter\XINT_len_fork
    \romannumeral0\xintnum{#1}\xint_relax\xint_relax\xint_relax\xint_relax
                      \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\def\XINT_Len #1% variant which does not expand via \xintnum.
{%
    \romannumeral0\XINT_len_fork
    #1\xint_relax\xint_relax\xint_relax\xint_relax
      \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\def\XINT_len_fork #1%
{%
    \expandafter\XINT_length_loop
    \xint_UDsignfork
      #1{0.}%
       -{0.#1}%
    \krof
}%
%    \end{macrocode}
% \subsection{\csh{xintBool}, \csh{xintToggle}}
% \lverb|1.09c|
%    \begin{macrocode}
\def\xintBool #1{\romannumeral-`0%
                 \csname if#1\endcsname\expandafter1\else\expandafter0\fi }%
\def\xintToggle #1{\romannumeral-`0\iftoggle{#1}{1}{0}}%
%    \end{macrocode}
% \subsection{\csh{xintifSgn}, \csh{xintiiifSgn}}
% \lverb|Expandable three-way fork added in 1.09a. Branches expandably
% depending on whether <0, =0, >0. Choice of branch guaranteed in two steps.
%
% 1.09i has \xint_firstofthreeafterstop (now _thenstop) etc for faster
% expansion.
%
% 1.1 adds \xintiiifSgn for optimization in xintexpr-essions. Should I move
% them to xintcore? (for bnumexpr)|
%    \begin{macrocode}
\def\xintifSgn {\romannumeral0\xintifsgn }%
\def\xintifsgn #1%
{%
    \ifcase \xintSgn{#1}
               \expandafter\xint_secondofthree_thenstop
            \or\expandafter\xint_thirdofthree_thenstop
          \else\expandafter\xint_firstofthree_thenstop
    \fi
}%
\def\xintiiifSgn {\romannumeral0\xintiiifsgn }%
\def\xintiiifsgn #1%
{%
    \ifcase \xintiiSgn{#1}
               \expandafter\xint_secondofthree_thenstop
            \or\expandafter\xint_thirdofthree_thenstop
          \else\expandafter\xint_firstofthree_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintifZero}, \csh{xintifNotZero}, \csh{xintiiifZero}, \csh{xintiiifNotZero}}
% \lverb|Expandable two-way fork added in 1.09a. Branches expandably depending on
% whether the argument is zero (branch A) or not (branch B). 1.09i restyling. By
% the way it appears (not thoroughly tested, though) that \if tests are faster
% than \ifnum tests. 1.1 adds ii  versions.|
%    \begin{macrocode}
\def\xintifZero {\romannumeral0\xintifzero }%
\def\xintifzero #1%
{%
    \if0\xintSgn{#1}%
       \expandafter\xint_firstoftwo_thenstop
    \else
       \expandafter\xint_secondoftwo_thenstop
    \fi
}%
\def\xintifNotZero {\romannumeral0\xintifnotzero }%
\def\xintifnotzero #1%
{%
    \if0\xintSgn{#1}%
       \expandafter\xint_secondoftwo_thenstop
    \else
       \expandafter\xint_firstoftwo_thenstop
    \fi
}%
\def\xintiiifZero {\romannumeral0\xintiiifzero }%
\def\xintiiifzero #1%
{%
    \if0\xintiiSgn{#1}%
       \expandafter\xint_firstoftwo_thenstop
    \else
       \expandafter\xint_secondoftwo_thenstop
    \fi
}%
\def\xintiiifNotZero {\romannumeral0\xintiiifnotzero }%
\def\xintiiifnotzero #1%
{%
    \if0\xintiiSgn{#1}%
       \expandafter\xint_secondoftwo_thenstop
    \else
       \expandafter\xint_firstoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintifOne},\csh{xintiiifOne}}
% \lverb|added in 1.09i. 1.1a adds \xintiiifOne.|
%    \begin{macrocode}
\def\xintiiifOne {\romannumeral0\xintiiifone }%
\def\xintiiifone #1%
{%
    \if1\xintiiIsOne{#1}%
       \expandafter\xint_firstoftwo_thenstop
    \else
       \expandafter\xint_secondoftwo_thenstop
    \fi
}%
\def\xintifOne {\romannumeral0\xintifone }%
\def\xintifone #1%
{%
    \if1\xintIsOne{#1}%
       \expandafter\xint_firstoftwo_thenstop
    \else
       \expandafter\xint_secondoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintifTrueAelseB}, \csh{xintifFalseAelseB}}
% \lverb|1.09i. Warning, \xintifTrueFalse, \xintifTrue deprecated, to be
% removed|
%    \begin{macrocode}
\let\xintifTrueAelseB\xintifNotZero
\let\xintifFalseAelseB\xintifZero
\let\xintifTrue\xintifNotZero
\let\xintifTrueFalse\xintifNotZero
%    \end{macrocode}
% \subsection{\csh{xintifCmp}, \csh{xintiiifCmp}}
% \lverb|1.09e
% \xintifCmp {n}{m}{if n<m}{if n=m}{if n>m}. 1.1a adds ii variant|
%    \begin{macrocode}
\def\xintifCmp {\romannumeral0\xintifcmp }%
\def\xintifcmp #1#2%
{%
    \ifcase\xintCmp {#1}{#2}
               \expandafter\xint_secondofthree_thenstop
            \or\expandafter\xint_thirdofthree_thenstop
          \else\expandafter\xint_firstofthree_thenstop
    \fi
}%
\def\xintiiifCmp {\romannumeral0\xintiiifcmp }%
\def\xintiiifcmp #1#2%
{%
    \ifcase\xintiiCmp {#1}{#2}
               \expandafter\xint_secondofthree_thenstop
            \or\expandafter\xint_thirdofthree_thenstop
          \else\expandafter\xint_firstofthree_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintifEq}, \csh{xintiiifEq}}
% \lverb|1.09a \xintifEq {n}{m}{YES if n=m}{NO if n<>m}. 1.1a adds ii variant|
%    \begin{macrocode}
\def\xintifEq {\romannumeral0\xintifeq }%
\def\xintifeq #1#2%
{%
    \if0\xintCmp{#1}{#2}%
               \expandafter\xint_firstoftwo_thenstop
          \else\expandafter\xint_secondoftwo_thenstop
    \fi
}%
\def\xintiiifEq {\romannumeral0\xintiiifeq }%
\def\xintiiifeq #1#2%
{%
    \if0\xintiiCmp{#1}{#2}%
               \expandafter\xint_firstoftwo_thenstop
          \else\expandafter\xint_secondoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintifGt}, \csh{xintiiifGt}}
% \lverb|1.09a \xintifGt {n}{m}{YES if n>m}{NO if n<=m}. 1.1a adds ii variant|
%    \begin{macrocode}
\def\xintifGt {\romannumeral0\xintifgt }%
\def\xintifgt #1#2%
{%
    \if1\xintCmp{#1}{#2}%
               \expandafter\xint_firstoftwo_thenstop
          \else\expandafter\xint_secondoftwo_thenstop
    \fi
}%
\def\xintiiifGt {\romannumeral0\xintiiifgt }%
\def\xintiiifgt #1#2%
{%
    \if1\xintiiCmp{#1}{#2}%
               \expandafter\xint_firstoftwo_thenstop
          \else\expandafter\xint_secondoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintifLt}, \csh{xintiiifLt}}
% \lverb|1.09a \xintifLt {n}{m}{YES if n<m}{NO if n>=m}. Restyled in 1.09i.
% 1.1a adds ii variant|
%    \begin{macrocode}
\def\xintifLt {\romannumeral0\xintiflt }%
\def\xintiflt #1#2%
{%
    \ifnum\xintCmp{#1}{#2}<\xint_c_
          \expandafter\xint_firstoftwo_thenstop
    \else \expandafter\xint_secondoftwo_thenstop
    \fi
}%
\def\xintiiifLt {\romannumeral0\xintiiiflt }%
\def\xintiiiflt #1#2%
{%
    \ifnum\xintiiCmp{#1}{#2}<\xint_c_
          \expandafter\xint_firstoftwo_thenstop
    \else \expandafter\xint_secondoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintifOdd}, \csh{xintiiifOdd}}
% \lverb|1.09e. Restyled in 1.09i. 1.1a adds \xintiiifOdd.|
%    \begin{macrocode}
\def\xintiiifOdd {\romannumeral0\xintiiifodd }%
\def\xintiiifodd #1%
{%
    \if\xintiiOdd{#1}1%
       \expandafter\xint_firstoftwo_thenstop
    \else
       \expandafter\xint_secondoftwo_thenstop
    \fi
}%
\def\xintifOdd {\romannumeral0\xintifodd }%
\def\xintifodd #1%
{%
    \if\xintOdd{#1}1%
       \expandafter\xint_firstoftwo_thenstop
    \else
       \expandafter\xint_secondoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintCmp}, \csh{xintiiCmp}}
% \lverb|Release 1.09a has \xintnum inserted into \xintCmp. Unnecessary
% \xintiCmp suppressed in 1.09f. And 1.1a does \xintiiCmp, for
% optimization in \xintiiexpr. (not needed before, because \XINT_cmp_fork was
% directly used, or \XINT_Cmp)|
%    \begin{macrocode}
\def\xintCmp {\romannumeral0\xintcmp }%
\def\xintcmp #1%
{%
    \expandafter\xint_cmp\expandafter{\romannumeral0\xintnum{#1}}%
}%
\def\xint_cmp #1#2%
{%
    \expandafter\XINT_cmp_fork \romannumeral0\xintnum{#2}\Z #1\Z
}%
\def\xintiiCmp {\romannumeral0\xintiicmp }%
\def\xintiicmp #1%
{%
    \expandafter\xint_iicmp\expandafter{\romannumeral-`0#1}%
}%
\def\xint_iicmp #1#2%
{%
    \expandafter\XINT_cmp_fork \romannumeral-`0#2\Z #1\Z
}%
\def\XINT_Cmp #1#2{\romannumeral0\XINT_cmp_fork #2\Z #1\Z }%
%    \end{macrocode}
% \lverb|&
% COMPARAISON $\ 
% 1 si #3#4>#1#2, 0 si #3#4=#1#2, -1 si #3#4<#1#2$\ 
% #3#4 vient du *premier*,$
% #1#2 vient du *second*|
%    \begin{macrocode}
\def\XINT_cmp_fork #1#2\Z #3#4\Z
{%
    \xint_UDsignsfork
          #1#3\XINT_cmp_minusminus
           #1-\XINT_cmp_minusplus
           #3-\XINT_cmp_plusminus
            --{\xint_UDzerosfork
                      #1#3\XINT_cmp_zerozero
                       #10\XINT_cmp_zeroplus
                       #30\XINT_cmp_pluszero
                        00\XINT_cmp_plusplus
                      \krof }%
    \krof
    {#2}{#4}#1#3%
}%
\def\XINT_cmp_minusplus #1#2#3#4{ 1}%
\def\XINT_cmp_plusminus #1#2#3#4{ -1}%
\def\XINT_cmp_zerozero  #1#2#3#4{ 0}%
\def\XINT_cmp_zeroplus  #1#2#3#4{ 1}%
\def\XINT_cmp_pluszero  #1#2#3#4{ -1}%
\def\XINT_cmp_plusplus #1#2#3#4%
{%
    \XINT_cmp_pre {#4#2}{#3#1}%
}%
\def\XINT_cmp_minusminus #1#2#3#4%
{%
    \XINT_cmp_pre {#1}{#2}%
}%
\def\XINT_cmp_pre #1%
{%
  \expandafter\XINT_cmp_pre_b\expandafter
  {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }%
}%
\def\XINT_cmp_pre_b #1#2%
{%
    \expandafter\XINT_cmp_A
    \expandafter1\expandafter{\expandafter}%
    \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z
        \W\X\Y\Z #1\W\X\Y\Z
}%
%    \end{macrocode}
% \lverb|&
% COMPARAISON$\ 
% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS
% POUR QUE LEUR LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS
% AUCUN NE SE TERMINE EN 0000.
% routine appelée via$\ 
% \XINT_cmp_A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z$\ 
% ATTENTION RENVOIE 1 SI N1 < N2, 0 si N1 = N2, -1 si N1 > N2|
%    \begin{macrocode}
\def\XINT_cmp_A #1#2#3\W\X\Y\Z #4#5#6#7%
{%
    \xint_gob_til_W #4\xint_cmp_az\W
    \XINT_cmp_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z
}%
\def\XINT_cmp_B #1#2#3#4#5#6#7%
{%
    \xint_gob_til_W#4\xint_cmp_bz\W
    \XINT_cmp_onestep #1#2{#7#6#5#4}{#3}%
}%
\def\XINT_cmp_onestep #1#2#3#4#5#6%
{%
    \expandafter\XINT_cmp_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.%
}%
\def\XINT_cmp_backtoA #1#2#3.#4%
{%
    \XINT_cmp_A #2{#3#4}%
}%
\def\xint_cmp_bz\W\XINT_cmp_onestep #1\Z { 1}%
\def\xint_cmp_az\W\XINT_cmp_B #1#2#3#4#5#6#7%
{%
    \xint_gob_til_W #4\xint_cmp_ez\W
    \XINT_cmp_Eenter #1{#3}#4#5#6#7%
}%
\def\XINT_cmp_Eenter #1\Z { -1}%
\def\xint_cmp_ez\W\XINT_cmp_Eenter #1%
{%
    \xint_UDzerofork
      #1\XINT_cmp_K             %     il y a une retenue
       0\XINT_cmp_L             %     pas de retenue
    \krof
}%
\def\XINT_cmp_K #1\Z { -1}%
\def\XINT_cmp_L #1{\XINT_OneIfPositive_main #1}%
\def\XINT_OneIfPositive #1%
{%
    \XINT_OneIfPositive_main #1\W\X\Y\Z%
}%
\def\XINT_OneIfPositive_main #1#2#3#4%
{%
    \xint_gob_til_Z #4\xint_OneIfPositive_terminated\Z
    \XINT_OneIfPositive_onestep #1#2#3#4%
}%
\def\xint_OneIfPositive_terminated\Z\XINT_OneIfPositive_onestep\W\X\Y\Z { 0}%
\def\XINT_OneIfPositive_onestep #1#2#3#4%
{%
    \expandafter\XINT_OneIfPositive_check\the\numexpr #1#2#3#4\relax
}%
\def\XINT_OneIfPositive_check #1%
{%
    \xint_gob_til_zero #1\xint_OneIfPositive_backtomain 0%
    \XINT_OneIfPositive_finish #1%
}%
\def\XINT_OneIfPositive_finish #1\W\X\Y\Z{ 1}%
\def\xint_OneIfPositive_backtomain 0\XINT_OneIfPositive_finish 0%
                   {\XINT_OneIfPositive_main }%
%    \end{macrocode}
% \subsection{\csh{xintEq}, \csh{xintGt}, \csh{xintLt}}
% \lverb|1.09a.|
%    \begin{macrocode}
\def\xintEq {\romannumeral0\xinteq }\def\xinteq #1#2{\xintifeq{#1}{#2}{1}{0}}%
\def\xintGt {\romannumeral0\xintgt }\def\xintgt #1#2{\xintifgt{#1}{#2}{1}{0}}%
\def\xintLt {\romannumeral0\xintlt }\def\xintlt #1#2{\xintiflt{#1}{#2}{1}{0}}%
%    \end{macrocode}
% \subsection{\csh{xintNeq}, \csh{xintGtorEq}, \csh{xintLtorEq}}
% \lverb|1.1. Pour xintexpr. No lowercase macros|
%    \begin{macrocode}
\def\xintLtorEq #1#2{\romannumeral0\xintifgt {#1}{#2}{0}{1}}%
\def\xintGtorEq #1#2{\romannumeral0\xintiflt {#1}{#2}{0}{1}}%
\def\xintNeq    #1#2{\romannumeral0\xintifeq {#1}{#2}{0}{1}}%
%    \end{macrocode}
% \subsection{\csh{xintiiEq}, \csh{xintiiGt}, \csh{xintiiLt}}
% \lverb|1.1a Pour \xintiiexpr. No lowercase macros.|
%    \begin{macrocode}
\def\xintiiEq #1#2{\romannumeral0\xintiiifeq{#1}{#2}{1}{0}}%
\def\xintiiGt #1#2{\romannumeral0\xintiiifgt{#1}{#2}{1}{0}}%
\def\xintiiLt #1#2{\romannumeral0\xintiiiflt{#1}{#2}{1}{0}}%
%    \end{macrocode}
% \subsection{\csh{xintiiNeq}, \csh{xintiiGtorEq}, \csh{xintiiLtorEq}}
% \lverb|1.1a. Pour \xintiiexpr. No lowercase macros.|
%    \begin{macrocode}
\def\xintiiLtorEq #1#2{\romannumeral0\xintiiifgt {#1}{#2}{0}{1}}%
\def\xintiiGtorEq #1#2{\romannumeral0\xintiiiflt {#1}{#2}{0}{1}}%
\def\xintiiNeq    #1#2{\romannumeral0\xintiiifeq {#1}{#2}{0}{1}}%
%    \end{macrocode}
% \subsection{\csh{xintIsZero}, \csh{xintIsNotZero}, \csh{xintiiIsZero},
% \csh{xintiiIsNotZero}}
% \lverb|1.09a. restyled in 1.09i. 1.1 adds \xintiiIsZero, etc... for
% optimization in \xintexpr|
%    \begin{macrocode}
\def\xintIsZero {\romannumeral0\xintiszero }%
\def\xintiszero #1{\if0\xintSgn{#1}\xint_afterfi{ 1}\else\xint_afterfi{ 0}\fi}%
\def\xintIsNotZero {\romannumeral0\xintisnotzero }%
\def\xintisnotzero
          #1{\if0\xintSgn{#1}\xint_afterfi{ 0}\else\xint_afterfi{ 1}\fi}%
\def\xintiiIsZero {\romannumeral0\xintiiiszero }%
\def\xintiiiszero #1{\if0\xintiiSgn{#1}\xint_afterfi{ 1}\else\xint_afterfi{ 0}\fi}%
\def\xintiiIsNotZero {\romannumeral0\xintiiisnotzero }%
\def\xintiiisnotzero
          #1{\if0\xintiiSgn{#1}\xint_afterfi{ 0}\else\xint_afterfi{ 1}\fi}%
%    \end{macrocode}
% \subsection{\csh{xintIsTrue}, \csh{xintNot}, \csh{xintIsFalse}}
% \lverb|1.09c|
%    \begin{macrocode}
\let\xintIsTrue\xintIsNotZero
\let\xintNot\xintIsZero
\let\xintIsFalse\xintIsZero
%    \end{macrocode}
% \subsection{\csh{xintAND}, \csh{xintOR}, \csh{xintXOR}}
% \lverb|1.09a. Embarrasing bugs in \xintAND and \xintOR which inserted a space
% token corrected in 1.09i. \xintxor restyled with \if (faster) in 1.09i|
%    \begin{macrocode}
\def\xintAND {\romannumeral0\xintand }%
\def\xintand #1#2{\if0\xintSgn{#1}\expandafter\xint_firstoftwo
                             \else\expandafter\xint_secondoftwo\fi
                  { 0}{\xintisnotzero{#2}}}%
\def\xintOR {\romannumeral0\xintor }%
\def\xintor #1#2{\if0\xintSgn{#1}\expandafter\xint_firstoftwo
                            \else\expandafter\xint_secondoftwo\fi
                 {\xintisnotzero{#2}}{ 1}}%
\def\xintXOR {\romannumeral0\xintxor }%
\def\xintxor #1#2{\if\xintIsZero{#1}\xintIsZero{#2}%
                     \xint_afterfi{ 0}\else\xint_afterfi{ 1}\fi }%
%    \end{macrocode}
% \subsection{\csh{xintANDof}}
% \lverb|New with 1.09a. \xintANDof works also with an empty list.|
%    \begin{macrocode}
\def\xintANDof      {\romannumeral0\xintandof }%
\def\xintandof    #1{\expandafter\XINT_andof_a\romannumeral-`0#1\relax }%
\def\XINT_andof_a #1{\expandafter\XINT_andof_b\romannumeral-`0#1\Z }%
\def\XINT_andof_b #1%
           {\xint_gob_til_relax #1\XINT_andof_e\relax\XINT_andof_c #1}%
\def\XINT_andof_c #1\Z
           {\xintifTrueAelseB {#1}{\XINT_andof_a}{\XINT_andof_no}}%
\def\XINT_andof_no #1\relax { 0}%
\def\XINT_andof_e #1\Z { 1}%
%    \end{macrocode}
% \subsection{\csh{xintORof}}
% \lverb|New with 1.09a. Works also with an empty list.|
%    \begin{macrocode}
\def\xintORof      {\romannumeral0\xintorof }%
\def\xintorof    #1{\expandafter\XINT_orof_a\romannumeral-`0#1\relax }%
\def\XINT_orof_a #1{\expandafter\XINT_orof_b\romannumeral-`0#1\Z }%
\def\XINT_orof_b #1%
           {\xint_gob_til_relax #1\XINT_orof_e\relax\XINT_orof_c #1}%
\def\XINT_orof_c #1\Z
           {\xintifTrueAelseB {#1}{\XINT_orof_yes}{\XINT_orof_a}}%
\def\XINT_orof_yes #1\relax { 1}%
\def\XINT_orof_e #1\Z { 0}%
%    \end{macrocode}
% \subsection{\csh{xintXORof}}
% \lverb|New with 1.09a. Works with an empty list, too. \XINT_xorof_c more
% efficient in 1.09i|
%    \begin{macrocode}
\def\xintXORof      {\romannumeral0\xintxorof }%
\def\xintxorof    #1{\expandafter\XINT_xorof_a\expandafter
                     0\romannumeral-`0#1\relax }%
\def\XINT_xorof_a #1#2{\expandafter\XINT_xorof_b\romannumeral-`0#2\Z #1}%
\def\XINT_xorof_b #1%
           {\xint_gob_til_relax #1\XINT_xorof_e\relax\XINT_xorof_c #1}%
\def\XINT_xorof_c #1\Z #2%
           {\xintifTrueAelseB {#1}{\if #20\xint_afterfi{\XINT_xorof_a 1}%
                                   \else\xint_afterfi{\XINT_xorof_a 0}\fi}%
                                  {\XINT_xorof_a #2}%
           }%
\def\XINT_xorof_e #1\Z #2{ #2}%
%    \end{macrocode}
% \subsection{\csh{xintGeq}}
% \lverb|&
% Release 1.09a has \xintnum added into \xintGeq.
% PLUS GRAND OU ÉGAL
% attention compare les **valeurs absolues**|
%    \begin{macrocode}
\def\xintGeq {\romannumeral0\xintgeq }%
\def\xintgeq #1%
{%
    \expandafter\xint_geq\expandafter {\romannumeral0\xintnum{#1}}%
}%
\def\xint_geq #1#2%
{%
    \expandafter\XINT_geq_fork \romannumeral0\xintnum{#2}\Z #1\Z
}%
\def\XINT_Geq #1#2{\romannumeral0\XINT_geq_fork #2\Z #1\Z }%
%    \end{macrocode}
% \lverb|&
% PLUS GRAND OU ÉGAL
% ATTENTION, TESTE les VALEURS ABSOLUES|
%    \begin{macrocode}
\def\XINT_geq_fork #1#2\Z #3#4\Z
{%
    \xint_UDzerofork
      #1\XINT_geq_secondiszero % |#1#2|=0
      #3\XINT_geq_firstiszero % |#1#2|>0
       0{\xint_UDsignsfork
                  #1#3\XINT_geq_minusminus
                   #1-\XINT_geq_minusplus
                   #3-\XINT_geq_plusminus
                    --\XINT_geq_plusplus
                \krof }%
    \krof
    {#2}{#4}#1#3%
}%
\def\XINT_geq_secondiszero     #1#2#3#4{ 1}%
\def\XINT_geq_firstiszero      #1#2#3#4{ 0}%
\def\XINT_geq_plusplus   #1#2#3#4{\XINT_geq_pre {#4#2}{#3#1}}%
\def\XINT_geq_minusminus #1#2#3#4{\XINT_geq_pre {#2}{#1}}%
\def\XINT_geq_minusplus  #1#2#3#4{\XINT_geq_pre {#4#2}{#1}}%
\def\XINT_geq_plusminus  #1#2#3#4{\XINT_geq_pre {#2}{#3#1}}%
\def\XINT_geq_pre #1%
{%
  \expandafter\XINT_geq_pre_b\expandafter
  {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }%
}%
\def\XINT_geq_pre_b #1#2%
{%
    \expandafter\XINT_geq_A
    \expandafter1\expandafter{\expandafter}%
    \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z
        \W\X\Y\Z #1 \W\X\Y\Z
}%
%    \end{macrocode}
% \lverb|&
% PLUS GRAND OU ÉGAL$\ 
% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS
% POUR QUE LEURS LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS
% AUCUN NE SE TERMINE EN 0000$\ 
% routine appelée via$\ 
% \romannumeral0\XINT_geq_A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z$\ 
% ATTENTION RENVOIE 1 SI N1 < N2 ou N1 = N2 et 0 si N1 > N2|
%    \begin{macrocode}
\def\XINT_geq_A #1#2#3\W\X\Y\Z #4#5#6#7%
{%
    \xint_gob_til_W #4\xint_geq_az\W
    \XINT_geq_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z
}%
\def\XINT_geq_B #1#2#3#4#5#6#7%
{%
    \xint_gob_til_W #4\xint_geq_bz\W
    \XINT_geq_onestep #1#2{#7#6#5#4}{#3}%
}%
\def\XINT_geq_onestep #1#2#3#4#5#6%
{%
    \expandafter\XINT_geq_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.%
}%
\def\XINT_geq_backtoA #1#2#3.#4%
{%
    \XINT_geq_A #2{#3#4}%
}%
\def\xint_geq_bz\W\XINT_geq_onestep #1\W\X\Y\Z { 1}%
\def\xint_geq_az\W\XINT_geq_B #1#2#3#4#5#6#7%
{%
    \xint_gob_til_W #4\xint_geq_ez\W
    \XINT_geq_Eenter #1%
}%
\def\XINT_geq_Eenter #1\W\X\Y\Z { 0}%
\def\xint_geq_ez\W\XINT_geq_Eenter #1%
{%
    \xint_UDzerofork
      #1{ 0}             %     il y a une retenue
       0{ 1}             %     pas de retenue
    \krof
}%
%    \end{macrocode}
% \subsection{\csh{xintiMax}, \csh{xintiiMax}}
% \lverb|&
% The rationale is that it is more efficient than using \xintCmp.
% 1.03 makes the code a tiny bit slower but easier to re-use for fractions.
% Note: actually since 1.08a code for fractions does not all reduce to these
% entry points, so perhaps I should revert the changes made in 1.03. Release
% 1.09a has \xintnum added into \xintiMax.
%
% 1.1 adds the missing \xintiiMax. Using \xintMax and not \xintiMax in xint is
% deprecated.|
%    \begin{macrocode}
\def\xintiMax {\romannumeral0\xintimax }%
\def\xintimax #1%
{%
    \expandafter\xint_max\expandafter {\romannumeral0\xintnum{#1}}%
}%
\def\xint_max #1#2%
{%
    \expandafter\XINT_max_pre\expandafter {\romannumeral0\xintnum{#2}}{#1}%
}%
\def\xintiiMax {\romannumeral0\xintiimax }%
\def\xintiimax #1%
{%
    \expandafter\xint_iimax\expandafter {\romannumeral-`0#1}%
}%
\def\xint_iimax #1#2%
{%
    \expandafter\XINT_max_pre\expandafter {\romannumeral-`0#2}{#1}%
}%
\let\xintMax\xintiMax \let\xintmax\xintimax % deprecated, should be only with xintfrac
\def\XINT_max_pre #1#2{\XINT_max_fork #1\Z #2\Z {#2}{#1}}%
\def\XINT_Max #1#2{\romannumeral0\XINT_max_fork #2\Z #1\Z {#1}{#2}}%
%    \end{macrocode}
% \lverb|&
% #3#4 vient du *premier*,
% #1#2 vient du *second*|
%    \begin{macrocode}
\def\XINT_max_fork #1#2\Z #3#4\Z
{%
    \xint_UDsignsfork
          #1#3\XINT_max_minusminus  % A < 0, B < 0
           #1-\XINT_max_minusplus   % B < 0, A >= 0
           #3-\XINT_max_plusminus   % A < 0, B >= 0
            --{\xint_UDzerosfork
                      #1#3\XINT_max_zerozero % A = B = 0
                       #10\XINT_max_zeroplus % B = 0, A > 0
                       #30\XINT_max_pluszero % A = 0, B > 0
                        00\XINT_max_plusplus % A, B > 0
                      \krof }%
    \krof
    {#2}{#4}#1#3%
}%
%    \end{macrocode}
% \lverb|&
% A = #4#2, B = #3#1|
%    \begin{macrocode}
\def\XINT_max_zerozero  #1#2#3#4{\xint_firstoftwo_thenstop }%
\def\XINT_max_zeroplus  #1#2#3#4{\xint_firstoftwo_thenstop }%
\def\XINT_max_pluszero  #1#2#3#4{\xint_secondoftwo_thenstop }%
\def\XINT_max_minusplus #1#2#3#4{\xint_firstoftwo_thenstop }%
\def\XINT_max_plusminus #1#2#3#4{\xint_secondoftwo_thenstop }%
\def\XINT_max_plusplus  #1#2#3#4%
{%
    \ifodd\XINT_Geq {#4#2}{#3#1}
      \expandafter\xint_firstoftwo_thenstop
    \else
      \expandafter\xint_secondoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \lverb+#3=-, #4=-, #1 = |B| = -B, #2 = |A| = -A+
%    \begin{macrocode}
\def\XINT_max_minusminus #1#2#3#4%
{%
    \ifodd\XINT_Geq {#1}{#2}
      \expandafter\xint_firstoftwo_thenstop
    \else
      \expandafter\xint_secondoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintMaxof}}
% \lverb|New with 1.09a.|
%    \begin{macrocode}
\def\xintiMaxof      {\romannumeral0\xintimaxof }%
\def\xintimaxof    #1{\expandafter\XINT_imaxof_a\romannumeral-`0#1\relax }%
\def\XINT_imaxof_a #1{\expandafter\XINT_imaxof_b\romannumeral0\xintnum{#1}\Z }%
\def\XINT_imaxof_b #1\Z #2%
           {\expandafter\XINT_imaxof_c\romannumeral-`0#2\Z {#1}\Z}%
\def\XINT_imaxof_c #1%
           {\xint_gob_til_relax #1\XINT_imaxof_e\relax\XINT_imaxof_d #1}%
\def\XINT_imaxof_d #1\Z
           {\expandafter\XINT_imaxof_b\romannumeral0\xintimax {#1}}%
\def\XINT_imaxof_e #1\Z #2\Z { #2}%
\let\xintMaxof\xintiMaxof \let\xintmaxof\xintimaxof
%    \end{macrocode}
% \subsection{\csh{xintiMin}, \csh{xintiiMin}}
% \lverb|\xintnum added New with 1.09a. I add \xintiiMin in 1.1 and mark as
% deprecated \xintMin, renamed \xintiMin.|
%    \begin{macrocode}
\def\xintiMin {\romannumeral0\xintimin }%
\def\xintimin #1%
{%
    \expandafter\xint_min\expandafter {\romannumeral0\xintnum{#1}}%
}%
\def\xint_min #1#2%
{%
    \expandafter\XINT_min_pre\expandafter {\romannumeral0\xintnum{#2}}{#1}%
}%
\def\xintiiMin {\romannumeral0\xintiimin }%
\def\xintiimin #1%
{%
    \expandafter\xint_iimin\expandafter {\romannumeral-`0#1}%
}%
\def\xint_iimin #1#2%
{%
    \expandafter\XINT_min_pre\expandafter {\romannumeral-`0#2}{#1}%
}%
\let\xintMin\xintiMin \let\xintmin\xintimin % deprecated
\def\XINT_min_pre #1#2{\XINT_min_fork #1\Z #2\Z {#2}{#1}}%
\def\XINT_Min #1#2{\romannumeral0\XINT_min_fork #2\Z #1\Z {#1}{#2}}%
%    \end{macrocode}
% \lverb|&
% #3#4 vient du *premier*,
% #1#2 vient du *second*|
%    \begin{macrocode}
\def\XINT_min_fork #1#2\Z #3#4\Z
{%
    \xint_UDsignsfork
          #1#3\XINT_min_minusminus  % A < 0, B < 0
           #1-\XINT_min_minusplus   % B < 0, A >= 0
           #3-\XINT_min_plusminus   % A < 0, B >= 0
            --{\xint_UDzerosfork
                      #1#3\XINT_min_zerozero % A = B = 0
                       #10\XINT_min_zeroplus % B = 0, A > 0
                       #30\XINT_min_pluszero % A = 0, B > 0
                        00\XINT_min_plusplus % A, B > 0
                      \krof }%
    \krof
    {#2}{#4}#1#3%
}%
%    \end{macrocode}
% \lverb|&
% A = #4#2, B = #3#1|
%    \begin{macrocode}
\def\XINT_min_zerozero  #1#2#3#4{\xint_firstoftwo_thenstop }%
\def\XINT_min_zeroplus  #1#2#3#4{\xint_secondoftwo_thenstop }%
\def\XINT_min_pluszero  #1#2#3#4{\xint_firstoftwo_thenstop }%
\def\XINT_min_minusplus #1#2#3#4{\xint_secondoftwo_thenstop }%
\def\XINT_min_plusminus #1#2#3#4{\xint_firstoftwo_thenstop }%
\def\XINT_min_plusplus  #1#2#3#4%
{%
    \ifodd\XINT_Geq {#4#2}{#3#1}
      \expandafter\xint_secondoftwo_thenstop
    \else
      \expandafter\xint_firstoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \lverb+#3=-, #4=-, #1 = |B| = -B, #2 = |A| = -A+
%    \begin{macrocode}
\def\XINT_min_minusminus #1#2#3#4%
{%
    \ifodd\XINT_Geq {#1}{#2}
      \expandafter\xint_secondoftwo_thenstop
    \else
      \expandafter\xint_firstoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintiMinof}}
% \lverb|1.09a|
%    \begin{macrocode}
\def\xintiMinof      {\romannumeral0\xintiminof }%
\def\xintiminof    #1{\expandafter\XINT_iminof_a\romannumeral-`0#1\relax }%
\def\XINT_iminof_a #1{\expandafter\XINT_iminof_b\romannumeral0\xintnum{#1}\Z }%
\def\XINT_iminof_b #1\Z #2%
           {\expandafter\XINT_iminof_c\romannumeral-`0#2\Z {#1}\Z}%
\def\XINT_iminof_c #1%
           {\xint_gob_til_relax #1\XINT_iminof_e\relax\XINT_iminof_d #1}%
\def\XINT_iminof_d #1\Z
           {\expandafter\XINT_iminof_b\romannumeral0\xintimin {#1}}%
\def\XINT_iminof_e #1\Z #2\Z { #2}%
\let\xintMinof\xintiMinof \let\xintminof\xintiminof
%    \end{macrocode}
% \subsection{\csh{xintiiSum}}
% \lverb|&
% \xintSum {{a}{b}...{z}}$\ 
% \xintSumExpr {a}{b}...{z}\relax$\ 
% 1.03 (drastically) simplifies and makes the routines more efficient (for big
% computations). Also the way \xintSum and \xintSumExpr ...\relax are related.
% has been modified. Now \xintSumExpr \z \relax is accepted input when
% \z expands to a list of braced terms (prior only \xintSum {\z} or \xintSum \z
% was possible).
%
% 1.09a does NOT add the \xintnum overhead. 1.09h renames \xintiSum to
% \xintiiSum to correctly reflect this.|
%    \begin{macrocode}
\def\xintiiSum {\romannumeral0\xintiisum }%
\def\xintiisum #1{\xintiisumexpr #1\relax }%
\def\xintiiSumExpr {\romannumeral0\xintiisumexpr }%
\def\xintiisumexpr {\expandafter\XINT_sumexpr\romannumeral-`0}%
\let\xintSum\xintiiSum \let\xintsum\xintiisum
\let\xintSumExpr\xintiiSumExpr \let\xintsumexpr\xintiisumexpr
\def\XINT_sumexpr {\XINT_sum_loop {0000}{0000}}%
\def\XINT_sum_loop #1#2#3%
{%
    \expandafter\XINT_sum_checksign\romannumeral-`0#3\Z {#1}{#2}%
}%
\def\XINT_sum_checksign #1%
{%
    \xint_gob_til_relax #1\XINT_sum_finished\relax
    \xint_gob_til_zero #1\XINT_sum_skipzeroinput0%
    \xint_UDsignfork
      #1\XINT_sum_N
       -{\XINT_sum_P #1}%
    \krof
}%
\def\XINT_sum_finished #1\Z #2#3%
{%
    \XINT_sub_A 1{}#3\W\X\Y\Z #2\W\X\Y\Z
}%
\def\XINT_sum_skipzeroinput #1\krof #2\Z {\XINT_sum_loop }%
\def\XINT_sum_P #1\Z #2%
{%
    \expandafter\XINT_sum_loop\expandafter
    {\romannumeral0\expandafter
     \XINT_addr_A\expandafter0\expandafter{\expandafter}%
    \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z
    \W\X\Y\Z #2\W\X\Y\Z }%
}%
\def\XINT_sum_N #1\Z #2#3%
{%
    \expandafter\XINT_sum_NN\expandafter
    {\romannumeral0\expandafter
     \XINT_addr_A\expandafter0\expandafter{\expandafter}%
    \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z
    \W\X\Y\Z #3\W\X\Y\Z }{#2}%
}%
\def\XINT_sum_NN #1#2{\XINT_sum_loop {#2}{#1}}%
%    \end{macrocode}
% \subsection{\csh{xintiiPrd}}
% \lverb|&
% \xintPrd {{a}...{z}}$\ 
% \xintPrdExpr {a}...{z}\relax$\ 
% Release 1.02 modified the product routine.  The earlier version was faster in
% situations where each new term is bigger than the product of all previous
% terms, a situation which arises in the algorithm for computing powers. The
% 1.02 version was changed to be more efficient on big products, where the new
% term is small compared to what has been computed so far (the power algorithm
% now has its own product routine).
%
% Finally, the 1.03 version just simplifies everything as the multiplication now
% decides what is best, with the price of a little overhead. So the code has
% been dramatically reduced here.
%
% In 1.03 I also modify the way \xintPrd and \xintPrdExpr ...\relax are
% related. Now \xintPrdExpr \z \relax is accepted input when \z expands
% to a list of braced terms (prior only \xintPrd {\z} or \xintPrd \z was
% possible).
%
% In 1.06a I suddenly decide that \xintProductExpr was a silly name, and as the
% package is new and certainly not used, I decide I may just switch to
% \xintPrdExpr which I should have used from the beginning.
%
%  1.09a does NOT add the \xintnum overhead. 1.09h renames \xintiPrd to
%  \xintiiPrd to correctly reflect this.|
%    \begin{macrocode}
\def\xintiiPrd {\romannumeral0\xintiiprd }%
\def\xintiiprd #1{\xintiiprdexpr #1\relax }%
\let\xintPrd\xintiiPrd
\let\xintprd\xintiiprd
\def\xintiiPrdExpr {\romannumeral0\xintiiprdexpr }%
\def\xintiiprdexpr {\expandafter\XINT_prdexpr\romannumeral-`0}%
\let\xintPrdExpr\xintiiPrdExpr
\let\xintprdexpr\xintiiprdexpr
\def\XINT_prdexpr {\XINT_prod_loop_a 1\Z }%
\def\XINT_prod_loop_a #1\Z #2%
    {\expandafter\XINT_prod_loop_b \romannumeral-`0#2\Z #1\Z \Z}%
\def\XINT_prod_loop_b #1%
    {\xint_gob_til_relax #1\XINT_prod_finished\relax\XINT_prod_loop_c #1}%
\def\XINT_prod_loop_c
    {\expandafter\XINT_prod_loop_a\romannumeral0\XINT_mul_fork }%
\def\XINT_prod_finished #1\Z #2\Z \Z { #2}%
%    \end{macrocode}
% \subsection{\csh{xintFac}}
% \lverb|&
% Modified with 1.02 and again in 1.03 for greater efficiency. I am
% tempted,
% here and elsewhere, to use \ifcase\XINT_Geq  {#1}{1000000000}  rather than
% \ifnum\xintLength {#1}>9 but for the time being I leave things as they stand.
% With release 1.05, rather than using \xintLength I opt finally for direct use
% of \numexpr (which will throw a suitable number too big message), and to raise
% the \xintError:$\ FactorialOfTooBigNumber for argument larger than 1000000
% (rather than 1000000000). With 1.09a, \xintFac uses \xintnum.
%
% 1.09j for no special reason, I lower the maximal number from 999999 to 100000.
% Any how this computation would need more memory than TL2013 standard allows to
% TeX. And I don't even mention time... |
%    \begin{macrocode}
\def\xintiFac {\romannumeral0\xintifac }%
\def\xintifac #1%
{%
    \expandafter\XINT_fac_fork\expandafter{\the\numexpr #1}%
}%
\let\xintFac\xintiFac \let\xintfac\xintifac
\def\XINT_fac_fork #1%
{%
    \ifcase\XINT_cntSgn #1\Z
       \xint_afterfi{\expandafter\space\expandafter 1\xint_gobble_i }%
    \or
       \expandafter\XINT_fac_checklength
    \else
       \xint_afterfi{\expandafter\xintError:FactorialOfNegativeNumber
                \expandafter\space\expandafter 1\xint_gobble_i }%
    \fi
    {#1}%
}%
\def\XINT_fac_checklength #1%
{%
    \ifnum #1>100000
         \xint_afterfi{\expandafter\xintError:FactorialOfTooBigNumber
                       \expandafter\space\expandafter 1\xint_gobble_i }%
    \else
         \xint_afterfi{\ifnum #1>\xint_c_ixixixix
                          \expandafter\XINT_fac_big_loop
                       \else
                          \expandafter\XINT_fac_loop
                       \fi }%
    \fi
    {#1}%
}%
\def\XINT_fac_big_loop #1{\XINT_fac_big_loop_main {10000}{#1}{}}%
\def\XINT_fac_big_loop_main #1#2#3%
{%
    \ifnum #1<#2
        \expandafter
            \XINT_fac_big_loop_main
        \expandafter
           {\the\numexpr #1+1\expandafter }%
    \else
        \expandafter\XINT_fac_big_docomputation
    \fi
    {#2}{#3{#1}}%
}%
\def\XINT_fac_big_docomputation #1#2%
{%
    \expandafter \XINT_fac_bigcompute_loop \expandafter
    {\romannumeral0\XINT_fac_loop {9999}}#2\relax
}%
\def\XINT_fac_bigcompute_loop #1#2%
{%
    \xint_gob_til_relax #2\XINT_fac_bigcompute_end\relax
    \expandafter\XINT_fac_bigcompute_loop\expandafter
    {\expandafter\XINT_mul_enter
     \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z
     \Z\Z\Z\Z #1\W\W\W\W }%
}%
\def\XINT_fac_bigcompute_end #1#2#3#4#5%
{%
    \XINT_fac_bigcompute_end_ #5%
}%
\def\XINT_fac_bigcompute_end_ #1\R #2\Z \W\X\Y\Z #3\W\X\Y\Z { #3}%
\def\XINT_fac_loop #1{\XINT_fac_loop_main 1{1000}{#1}}%
\def\XINT_fac_loop_main #1#2#3%
{%
    \ifnum #3>#1
    \else
        \expandafter\XINT_fac_loop_exit
    \fi
    \expandafter\XINT_fac_loop_main\expandafter
    {\the\numexpr #1+1\expandafter }\expandafter
    {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z }%
    {#3}%
}%
\def\XINT_fac_loop_exit #1#2#3#4#5#6#7%
{%
    \XINT_fac_loop_exit_ #6%
}%
\def\XINT_fac_loop_exit_ #1#2#3%
{%
    \XINT_mul_M
}%
%    \end{macrocode}
% \lverb|&
% &
% -----------------------------------------------------------------$\ 
% -----------------------------------------------------------------$\ 
% DECIMAL OPERATIONS: FIRST DIGIT, LASTDIGIT, (<- moved to xintcore 
% because xintiiLDg need by division macros)
% ODDNESS,
% MULTIPLICATION BY TEN, QUOTIENT BY TEN, QUOTIENT OR
% MULTIPLICATION BY POWER OF TEN, SPLIT OPERATION.|
% \subsection{\csh{xintMON}, \csh{xintMMON}, \csh{xintiiMON}, \csh{xintiiMMON}}
% \lverb|&
% MINUS ONE TO THE POWER N and (-1)^{N-1}|
%    \begin{macrocode}
\def\xintiiMON {\romannumeral0\xintiimon }%
\def\xintiimon #1%
{%
    \ifodd\xintiiLDg {#1}
        \xint_afterfi{ -1}%
    \else
        \xint_afterfi{ 1}%
    \fi
}%
\def\xintiiMMON {\romannumeral0\xintiimmon }%
\def\xintiimmon #1%
{%
    \ifodd\xintiiLDg {#1}
        \xint_afterfi{ 1}%
    \else
        \xint_afterfi{ -1}%
    \fi
}%
\def\xintMON {\romannumeral0\xintmon }%
\def\xintmon #1%
{%
    \ifodd\xintLDg {#1}
        \xint_afterfi{ -1}%
    \else
        \xint_afterfi{ 1}%
    \fi
}%
\def\xintMMON {\romannumeral0\xintmmon }%
\def\xintmmon #1%
{%
    \ifodd\xintLDg {#1}
        \xint_afterfi{ 1}%
    \else
        \xint_afterfi{ -1}%
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintOdd}, \csh{xintiiOdd}, \csh{xintEven}, \csh{xintiiEven}}
% \lverb|1.05 has \xintiOdd, whereas \xintOdd parses through \xintNum.
% Inadvertently, 1.09a redefined \xintiLDg hence \xintiOdd also parsed through
% \xintNum. Anyway, having a \xintOdd and a \xintiOdd was silly. Removed in
% 1.09f, now only \xintOdd and \xintiiOdd. 1.1: \xintEven and \xintiiEven
% added for \xintiiexpr.|
%    \begin{macrocode}
\def\xintiiOdd {\romannumeral0\xintiiodd }%
\def\xintiiodd #1%
{%
    \ifodd\xintiiLDg{#1}
        \xint_afterfi{ 1}%
    \else
        \xint_afterfi{ 0}%
    \fi
}%
\def\xintiiEven {\romannumeral0\xintiieven }%
\def\xintiieven #1%
{%
    \ifodd\xintiiLDg{#1}
        \xint_afterfi{ 0}%
    \else
        \xint_afterfi{ 1}%
    \fi
}%
\def\xintOdd {\romannumeral0\xintodd }%
\def\xintodd #1%
{%
    \ifodd\xintLDg{#1}
        \xint_afterfi{ 1}%
    \else
        \xint_afterfi{ 0}%
    \fi
}%
\def\xintEven {\romannumeral0\xinteven }%
\def\xinteven #1%
{%
    \ifodd\xintLDg{#1}
        \xint_afterfi{ 0}%
    \else
        \xint_afterfi{ 1}%
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintDSL}}
% \lverb|&
% DECIMAL SHIFT LEFT (=MULTIPLICATION PAR 10)|
%    \begin{macrocode}
\def\xintDSL {\romannumeral0\xintdsl }%
\def\xintdsl #1%
{%
    \expandafter\XINT_dsl \romannumeral-`0#1\Z
}%
\def\XINT_DSL #1{\romannumeral0\XINT_dsl #1\Z }%
\def\XINT_dsl #1%
{%
    \xint_gob_til_zero #1\xint_dsl_zero 0\XINT_dsl_ #1%
}%
\def\xint_dsl_zero 0\XINT_dsl_ 0#1\Z { 0}%
\def\XINT_dsl_ #1\Z { #10}%
%    \end{macrocode}
% \subsection{\csh{xintDSR}}
% \lverb|&
% DECIMAL SHIFT RIGHT (=DIVISION PAR 10). Release 1.06b which replaced all @'s
% by
% underscores left undefined the \xint_minus used in \XINT_dsr_b, and this bug
% was fixed only later in release 1.09b|
%    \begin{macrocode}
\def\xintDSR {\romannumeral0\xintdsr }%
\def\xintdsr #1%
{%
    \expandafter\XINT_dsr_a\expandafter {\romannumeral-`0#1}\W\Z
}%
\def\XINT_DSR #1{\romannumeral0\XINT_dsr_a {#1}\W\Z }%
\def\XINT_dsr_a
{%
    \expandafter\XINT_dsr_b\romannumeral0\xintreverseorder
}%
\def\XINT_dsr_b #1#2#3\Z
{%
    \xint_gob_til_W #2\xint_dsr_onedigit\W
    \xint_gob_til_minus #2\xint_dsr_onedigit-%
    \expandafter\XINT_dsr_removew
    \romannumeral0\xintreverseorder {#2#3}%
}%
\def\xint_dsr_onedigit #1\xintreverseorder #2{ 0}%
\def\XINT_dsr_removew #1\W { }%
%    \end{macrocode}
% \subsection{\csh{xintDSH}, \csh{xintDSHr}}
% \lverb+DECIMAL SHIFTS \xintDSH {x}{A}$\ 
% si x <= 0, fait A -> A.10^(|x|). v1.03 corrige l'oversight pour A=0.$\ 
% si x >  0, et A >=0, fait A -> quo(A,10^(x))$\ 
% si x >  0, et A < 0, fait A -> -quo(-A,10^(x))$\ 
% (donc pour x > 0 c'est comme DSR itéré x fois)$\ 
% \xintDSHr donne le `reste' (si x<=0 donne zéro).
%
% Release 1.06 now feeds x to a \numexpr first. I will have to revise this code
% at some point.+
%    \begin{macrocode}
\def\xintDSHr {\romannumeral0\xintdshr }%
\def\xintdshr #1%
{%
    \expandafter\XINT_dshr_checkxpositive \the\numexpr #1\relax\Z
}%
\def\XINT_dshr_checkxpositive #1%
{%
    \xint_UDzerominusfork
      0#1\XINT_dshr_xzeroorneg
      #1-\XINT_dshr_xzeroorneg
       0-\XINT_dshr_xpositive
    \krof #1%
}%
\def\XINT_dshr_xzeroorneg #1\Z #2{ 0}%
\def\XINT_dshr_xpositive #1\Z
{%
    \expandafter\xint_secondoftwo_thenstop\romannumeral0\xintdsx {#1}%
}%
\def\xintDSH {\romannumeral0\xintdsh }%
\def\xintdsh #1#2%
{%
    \expandafter\xint_dsh\expandafter {\romannumeral-`0#2}{#1}%
}%
\def\xint_dsh #1#2%
{%
    \expandafter\XINT_dsh_checksignx \the\numexpr #2\relax\Z {#1}%
}%
\def\XINT_dsh_checksignx #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_dsh_xiszero
      0#1\XINT_dsx_xisNeg_checkA     % on passe direct dans DSx
       0-{\XINT_dsh_xisPos #1}%
    \krof
}%
\def\XINT_dsh_xiszero #1\Z #2{ #2}%
\def\XINT_dsh_xisPos #1\Z #2%
{%
    \expandafter\xint_firstoftwo_thenstop
    \romannumeral0\XINT_dsx_checksignA #2\Z {#1}% via DSx
}%
%    \end{macrocode}
% \subsection{\csh{xintDSx}}
% \lverb+Je fais cette routine pour la version 1.01, après modification de
% \xintDecSplit. Dorénavant \xintDSx fera appel à \xintDecSplit et de même
% \xintDSH fera appel à \xintDSx. J'ai donc supprimé entièrement l'ancien code
% de \xintDSH et re-écrit entièrement celui de \xintDecSplit pour x positif.
%
% --> Attention le cas x=0 est traité dans la même catégorie que x > 0 <--$\ 
% si x < 0, fait A -> A.10^(|x|)$\ 
% si x >=  0, et A >=0, fait A -> {quo(A,10^(x))}{rem(A,10^(x))}$\ 
% si x >=  0, et A < 0, d'abord on calcule {quo(-A,10^(x))}{rem(-A,10^(x))}$\ 
%    puis, si le premier n'est pas nul on lui donne le signe -$\ 
%          si le premier est nul on donne le signe - au second.
%
% On peut donc toujours reconstituer l'original A par 10^x Q \pm R
% où il faut prendre le signe plus si Q est positif ou nul et le signe moins si
% Q est strictement négatif.
%
% Release 1.06 has a faster and more compactly coded \XINT_dsx_zeroloop.
% Also, x is now given to a \numexpr. The earlier code should be then
% simplified, but I leave as is for the time being.
%
% Release 1.07 modified the coding of \XINT_dsx_zeroloop, to avoid impacting the
% input stack. Indeed the truncating, rounding, and conversion to float routines
% all use internally \XINT_dsx_zeroloop (via \XINT_dsx_addzerosnofuss), and they
% were thus roughly limited to generating N = 8 times the input save stack size
% digits. On TL2012 and TL2013, this means 40000 = 8x5000 digits. Although
% generating more than 40000 digits is more like a one shot thing, I wanted to
% open the possibility of outputting tens of thousands of digits to faile, thus
% I re-organized \XINT_dsx_zeroloop.
%
% January 5, 2014: but it is only with the new division implementation of 1.09j
% and also with its special \xintXTrunc routine that the possibility mentioned
% in the last paragraph has become a concrete one in terms of computation time.+
%    \begin{macrocode}
\def\xintDSx {\romannumeral0\xintdsx }%
\def\xintdsx #1#2%
{%
    \expandafter\xint_dsx\expandafter {\romannumeral-`0#2}{#1}%
}%
\def\xint_dsx #1#2%
{%
    \expandafter\XINT_dsx_checksignx \the\numexpr #2\relax\Z {#1}%
}%
\def\XINT_DSx #1#2{\romannumeral0\XINT_dsx_checksignx #1\Z {#2}}%
\def\XINT_dsx #1#2{\XINT_dsx_checksignx #1\Z {#2}}%
\def\XINT_dsx_checksignx #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_dsx_xisZero
      0#1\XINT_dsx_xisNeg_checkA
       0-{\XINT_dsx_xisPos #1}%
    \krof
}%
\def\XINT_dsx_xisZero #1\Z #2{ {#2}{0}}% attention comme x > 0
\def\XINT_dsx_xisNeg_checkA #1\Z #2%
{%
    \XINT_dsx_xisNeg_checkA_ #2\Z {#1}%
}%
\def\XINT_dsx_xisNeg_checkA_ #1#2\Z #3%
{%
    \xint_gob_til_zero #1\XINT_dsx_xisNeg_Azero 0%
    \XINT_dsx_xisNeg_checkx {#3}{#3}{}\Z {#1#2}%
}%
\def\XINT_dsx_xisNeg_Azero #1\Z #2{ 0}%
\def\XINT_dsx_xisNeg_checkx #1%
{%
    \ifnum #1>1000000
       \xint_afterfi
       {\xintError:TooBigDecimalShift
        \expandafter\space\expandafter 0\xint_gobble_iv }%
    \else
       \expandafter \XINT_dsx_zeroloop
    \fi
}%
\def\XINT_dsx_addzerosnofuss #1{\XINT_dsx_zeroloop {#1}{}\Z }%
\def\XINT_dsx_zeroloop #1#2%
{%
    \ifnum #1<\xint_c_ix \XINT_dsx_exita\fi
    \expandafter\XINT_dsx_zeroloop\expandafter
        {\the\numexpr #1-\xint_c_viii}{#200000000}%
}%
\def\XINT_dsx_exita\fi\expandafter\XINT_dsx_zeroloop
{%
    \fi\expandafter\XINT_dsx_exitb
}%
\def\XINT_dsx_exitb #1#2%
{%
    \expandafter\expandafter\expandafter
    \XINT_dsx_addzeros\csname xint_gobble_\romannumeral -#1\endcsname #2%
}%
\def\XINT_dsx_addzeros #1\Z #2{ #2#1}%
\def\XINT_dsx_xisPos #1\Z #2%
{%
    \XINT_dsx_checksignA #2\Z {#1}%
}%
\def\XINT_dsx_checksignA #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_dsx_AisZero
      0#1\XINT_dsx_AisNeg
       0-{\XINT_dsx_AisPos #1}%
    \krof
}%
\def\XINT_dsx_AisZero #1\Z #2{ {0}{0}}%
\def\XINT_dsx_AisNeg #1\Z #2%
{%
    \expandafter\XINT_dsx_AisNeg_dosplit_andcheckfirst
    \romannumeral0\XINT_split_checksizex {#2}{#1}%
}%
\def\XINT_dsx_AisNeg_dosplit_andcheckfirst #1%
{%
    \XINT_dsx_AisNeg_checkiffirstempty #1\Z
}%
\def\XINT_dsx_AisNeg_checkiffirstempty #1%
{%
    \xint_gob_til_Z #1\XINT_dsx_AisNeg_finish_zero\Z
    \XINT_dsx_AisNeg_finish_notzero #1%
}%
\def\XINT_dsx_AisNeg_finish_zero\Z
    \XINT_dsx_AisNeg_finish_notzero\Z #1%
{%
    \expandafter\XINT_dsx_end
    \expandafter {\romannumeral0\XINT_num {-#1}}{0}%
}%
\def\XINT_dsx_AisNeg_finish_notzero #1\Z #2%
{%
    \expandafter\XINT_dsx_end
    \expandafter {\romannumeral0\XINT_num {#2}}{-#1}%
}%
\def\XINT_dsx_AisPos #1\Z #2%
{%
    \expandafter\XINT_dsx_AisPos_finish
    \romannumeral0\XINT_split_checksizex {#2}{#1}%
}%
\def\XINT_dsx_AisPos_finish #1#2%
{%
    \expandafter\XINT_dsx_end
    \expandafter {\romannumeral0\XINT_num {#2}}%
                 {\romannumeral0\XINT_num {#1}}%
}%
\edef\XINT_dsx_end #1#2%
{%
    \noexpand\expandafter\space\noexpand\expandafter{#2}{#1}%
}%
%    \end{macrocode}
% \subsection{\csh{xintDecSplit}, \csh{xintDecSplitL}, \csh{xintDecSplitR}}
% \lverb!DECIMAL SPLIT
%
% The macro \xintDecSplit {x}{A} first replaces A with |A| (*)
% This macro cuts the number into two pieces L and R. The concatenation LR
% always reproduces |A|, and R may be empty or have leading zeros. The
% position of the cut is specified by the first argument x. If x is zero or
% positive the cut location is x slots to the left of the right end of the
% number. If x becomes equal to or larger than the length of the number then L
% becomes empty. If x is negative the location of the cut is |x| slots to the
% right of the left end of the number.
%
% (*) warning: this may change in a future version. Only the behavior
% for A non-negative is guaranteed to remain the same.
%
% v1.05a: \XINT_split_checksizex does not compute the length anymore, rather the
% error will be from a \numexpr; but the limit of 999999999 does not make much
% sense.
%
% v1.06: Improvements in \XINT_split_fromleft_loop, \XINT_split_fromright_loop
% and related macros. More readable coding, speed gains.
% Also, I now feed immediately a \numexpr with x. Some simplifications should
% probably be made to the code, which is kept as is for the time being.
%
% 1.09e pays attention to the use of xintiabs which acquired in 1.09a the
% xintnum overhead. So xintiiabs rather without that overhead.
% !
%    \begin{macrocode}
\def\xintDecSplitL {\romannumeral0\xintdecsplitl }%
\def\xintDecSplitR {\romannumeral0\xintdecsplitr }%
\def\xintdecsplitl
{%
    \expandafter\xint_firstoftwo_thenstop
    \romannumeral0\xintdecsplit
}%
\def\xintdecsplitr
{%
    \expandafter\xint_secondoftwo_thenstop
    \romannumeral0\xintdecsplit
}%
\def\xintDecSplit {\romannumeral0\xintdecsplit }%
\def\xintdecsplit #1#2%
{%
    \expandafter \xint_split \expandafter
    {\romannumeral0\xintiiabs {#2}}{#1}%  fait expansion de A
}%
\def\xint_split #1#2%
{%
    \expandafter\XINT_split_checksizex\expandafter{\the\numexpr #2}{#1}%
}%
\def\XINT_split_checksizex #1% 999999999 is anyhow very big, could be reduced
{%
    \ifnum\numexpr\XINT_Abs{#1}>999999999
       \xint_afterfi {\xintError:TooBigDecimalSplit\XINT_split_bigx }%
    \else
       \expandafter\XINT_split_xfork
    \fi
    #1\Z
}%
\def\XINT_split_bigx  #1\Z #2%
{%
    \ifcase\XINT_cntSgn #1\Z
    \or \xint_afterfi { {}{#2}}% positive big x
    \else
        \xint_afterfi { {#2}{}}% negative big x
    \fi
}%
\def\XINT_split_xfork #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_split_zerosplit
      0#1\XINT_split_fromleft
       0-{\XINT_split_fromright #1}%
    \krof
}%
\def\XINT_split_zerosplit #1\Z #2{ {#2}{}}%
\def\XINT_split_fromleft  #1\Z #2%
{%
    \XINT_split_fromleft_loop {#1}{}#2\W\W\W\W\W\W\W\W\Z
}%
\def\XINT_split_fromleft_loop #1%
{%
    \ifnum #1<\xint_c_viii\XINT_split_fromleft_exita\fi
    \expandafter\XINT_split_fromleft_loop_perhaps\expandafter
    {\the\numexpr #1-\xint_c_viii\expandafter}\XINT_split_fromleft_eight
}%
\def\XINT_split_fromleft_eight #1#2#3#4#5#6#7#8#9{#9{#1#2#3#4#5#6#7#8#9}}%
\def\XINT_split_fromleft_loop_perhaps #1#2%
{%
    \xint_gob_til_W #2\XINT_split_fromleft_toofar\W
    \XINT_split_fromleft_loop {#1}%
}%
\def\XINT_split_fromleft_toofar\W\XINT_split_fromleft_loop #1#2#3\Z
{%
    \XINT_split_fromleft_toofar_b #2\Z
}%
\def\XINT_split_fromleft_toofar_b #1\W #2\Z { {#1}{}}%
\def\XINT_split_fromleft_exita\fi
    \expandafter\XINT_split_fromleft_loop_perhaps\expandafter #1#2%
   {\fi \XINT_split_fromleft_exitb #1}%
\def\XINT_split_fromleft_exitb\the\numexpr #1-\xint_c_viii\expandafter
{%
    \csname XINT_split_fromleft_endsplit_\romannumeral #1\endcsname
}%
\def\XINT_split_fromleft_endsplit_ #1#2\W #3\Z { {#1}{#2}}%
\def\XINT_split_fromleft_endsplit_i #1#2%
                {\XINT_split_fromleft_checkiftoofar #2{#1#2}}%
\def\XINT_split_fromleft_endsplit_ii #1#2#3%
                {\XINT_split_fromleft_checkiftoofar #3{#1#2#3}}%
\def\XINT_split_fromleft_endsplit_iii #1#2#3#4%
                {\XINT_split_fromleft_checkiftoofar #4{#1#2#3#4}}%
\def\XINT_split_fromleft_endsplit_iv #1#2#3#4#5%
                {\XINT_split_fromleft_checkiftoofar #5{#1#2#3#4#5}}%
\def\XINT_split_fromleft_endsplit_v #1#2#3#4#5#6%
                {\XINT_split_fromleft_checkiftoofar #6{#1#2#3#4#5#6}}%
\def\XINT_split_fromleft_endsplit_vi #1#2#3#4#5#6#7%
                {\XINT_split_fromleft_checkiftoofar #7{#1#2#3#4#5#6#7}}%
\def\XINT_split_fromleft_endsplit_vii #1#2#3#4#5#6#7#8%
                {\XINT_split_fromleft_checkiftoofar #8{#1#2#3#4#5#6#7#8}}%
\def\XINT_split_fromleft_checkiftoofar #1#2#3\W #4\Z
{%
    \xint_gob_til_W #1\XINT_split_fromleft_wenttoofar\W
    \space {#2}{#3}%
}%
\def\XINT_split_fromleft_wenttoofar\W\space #1%
{%
    \XINT_split_fromleft_wenttoofar_b #1\Z
}%
\def\XINT_split_fromleft_wenttoofar_b #1\W #2\Z { {#1}}%
\def\XINT_split_fromright #1\Z #2%
{%
    \expandafter \XINT_split_fromright_a \expandafter
    {\romannumeral0\xintreverseorder {#2}}{#1}{#2}%
}%
\def\XINT_split_fromright_a #1#2%
{%
    \XINT_split_fromright_loop {#2}{}#1\W\W\W\W\W\W\W\W\Z
}%
\def\XINT_split_fromright_loop #1%
{%
    \ifnum #1<\xint_c_viii\XINT_split_fromright_exita\fi
    \expandafter\XINT_split_fromright_loop_perhaps\expandafter
    {\the\numexpr #1-\xint_c_viii\expandafter }\XINT_split_fromright_eight
}%
\def\XINT_split_fromright_eight #1#2#3#4#5#6#7#8#9{#9{#9#8#7#6#5#4#3#2#1}}%
\def\XINT_split_fromright_loop_perhaps #1#2%
{%
    \xint_gob_til_W #2\XINT_split_fromright_toofar\W
    \XINT_split_fromright_loop {#1}%
}%
\def\XINT_split_fromright_toofar\W\XINT_split_fromright_loop #1#2#3\Z { {}}%
\def\XINT_split_fromright_exita\fi
    \expandafter\XINT_split_fromright_loop_perhaps\expandafter #1#2%
    {\fi \XINT_split_fromright_exitb #1}%
\def\XINT_split_fromright_exitb\the\numexpr #1-\xint_c_viii\expandafter
{%
    \csname XINT_split_fromright_endsplit_\romannumeral #1\endcsname
}%
\edef\XINT_split_fromright_endsplit_ #1#2\W #3\Z #4%
{%
    \noexpand\expandafter\space\noexpand\expandafter
    {\noexpand\romannumeral0\noexpand\xintreverseorder {#2}}{#1}%
}%
\def\XINT_split_fromright_endsplit_i   #1#2%
            {\XINT_split_fromright_checkiftoofar #2{#2#1}}%
\def\XINT_split_fromright_endsplit_ii  #1#2#3%
            {\XINT_split_fromright_checkiftoofar #3{#3#2#1}}%
\def\XINT_split_fromright_endsplit_iii #1#2#3#4%
            {\XINT_split_fromright_checkiftoofar #4{#4#3#2#1}}%
\def\XINT_split_fromright_endsplit_iv  #1#2#3#4#5%
            {\XINT_split_fromright_checkiftoofar #5{#5#4#3#2#1}}%
\def\XINT_split_fromright_endsplit_v   #1#2#3#4#5#6%
            {\XINT_split_fromright_checkiftoofar #6{#6#5#4#3#2#1}}%
\def\XINT_split_fromright_endsplit_vi  #1#2#3#4#5#6#7%
            {\XINT_split_fromright_checkiftoofar #7{#7#6#5#4#3#2#1}}%
\def\XINT_split_fromright_endsplit_vii #1#2#3#4#5#6#7#8%
            {\XINT_split_fromright_checkiftoofar #8{#8#7#6#5#4#3#2#1}}%
\def\XINT_split_fromright_checkiftoofar #1%
{%
    \xint_gob_til_W #1\XINT_split_fromright_wenttoofar\W
    \XINT_split_fromright_endsplit_
}%
\def\XINT_split_fromright_wenttoofar\W\XINT_split_fromright_endsplit_ #1\Z #2%
    { {}{#2}}%
%    \end{macrocode}
% \subsection{\csh{xintiiSqrt}, \csh{xintiiSqrtR}, \csh{xintiiSquareRoot}}
% \lverb|v1.08. 1.09a uses \xintnum.
%
% Some overhead was added inadvertently in 1.09a to inner routines when
% \xintiquo and \xintidivision were also promoted to use \xintnum; release 1.09f
% thus uses \xintiiquo and \xintiidivision which avoid this \xintnum overhead.
%
% 1.09j replaced the previous long \ifcase from \XINT_sqrt_c by some nested
% \ifnum's.
% 
% 1.1 Ajout de \xintiiSqrt et \xintiiSquareRoot.
%
% 1.1a ajoute \xintiiSqrtR, which provides the rounded, not truncated square
% root.|
%    \begin{macrocode}
\def\xintiiSqrt  {\romannumeral0\xintiisqrt  }%
\def\xintiiSqrtR {\romannumeral0\xintiisqrtr }%
\def\xintiiSquareRoot {\romannumeral0\xintiisquareroot }%
\def\xintiSqrt        {\romannumeral0\xintisqrt        }%
\def\xintiSquareRoot  {\romannumeral0\xintisquareroot  }%
\def\xintisqrt   {\expandafter\XINT_sqrt_post\romannumeral0\xintisquareroot   }%
\def\xintiisqrt  {\expandafter\XINT_sqrt_post\romannumeral0\xintiisquareroot  }%
\def\xintiisqrtr {\expandafter\XINT_sqrtr_post\romannumeral0\xintiisquareroot }%
\def\XINT_sqrt_post #1#2{\XINT_dec_pos #1\R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W }%
%    \end{macrocode}
% \lverb|N = (#1)^2 - #2 avec #1 le plus petit possible et #2>0 (hence #2<2*#1).
% (#1-.5)^2=#1^2-#1+.25=N+#2-#1+.25. Si 0<#2<#1, <= N-0.75<N, donc rounded->#1
% si #2>=#1, (#1-.5)^2>=N+.25>N, donc rounded->#1-1.|
%    \begin{macrocode}
\def\XINT_sqrtr_post #1#2{\xintiiifLt {#2}{#1}% Lt <-> a<b
                          { #1}{\XINT_dec_pos #1\R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W }}%
\def\xintisquareroot  #1{\expandafter\XINT_sqrt_checkin\romannumeral0\xintnum{#1}\Z }%
\def\xintiisquareroot #1{\expandafter\XINT_sqrt_checkin\romannumeral-`0#1\Z }%
\def\XINT_sqrt_checkin #1%
{%
    \xint_UDzerominusfork
     #1-\XINT_sqrt_iszero
     0#1\XINT_sqrt_isneg
      0-{\XINT_sqrt #1}%
    \krof
}%
\def\XINT_sqrt_iszero #1\Z { 11}%
\edef\XINT_sqrt_isneg #1\Z {\noexpand\xintError:RootOfNegative\space 11}%
\def\XINT_sqrt #1\Z
{%
    \expandafter\XINT_sqrt_start\expandafter {\romannumeral0\xintlength {#1}}{#1}%
}%
\def\XINT_sqrt_start #1%
{%
    \ifnum #1<\xint_c_x
       \expandafter\XINT_sqrt_small_a
    \else
       \expandafter\XINT_sqrt_big_a
    \fi
    {#1}%
}%
\def\XINT_sqrt_small_a #1{\XINT_sqrt_a {#1}\XINT_sqrt_small_d }%
\def\XINT_sqrt_big_a   #1{\XINT_sqrt_a {#1}\XINT_sqrt_big_d   }%
\def\XINT_sqrt_a #1%
{%
   \ifodd #1
     \expandafter\XINT_sqrt_bB
   \else
     \expandafter\XINT_sqrt_bA
   \fi
   {#1}%
}%
\def\XINT_sqrt_bA #1#2#3%
{%
    \XINT_sqrt_bA_b #3\Z #2{#1}{#3}%
}%
\def\XINT_sqrt_bA_b #1#2#3\Z
{%
    \XINT_sqrt_c {#1#2}%
}%
\def\XINT_sqrt_bB #1#2#3%
{%
    \XINT_sqrt_bB_b #3\Z #2{#1}{#3}%
}%
\def\XINT_sqrt_bB_b #1#2\Z
{%
    \XINT_sqrt_c #1%
}%
\def\XINT_sqrt_c #1#2%
{%
    \expandafter #2\expandafter
    {\the\numexpr\ifnum #1>\xint_c_iii
                 \ifnum #1>\xint_c_viii
                 \ifnum #1>15 \ifnum #1>24 \ifnum #1>35
                 \ifnum #1>48 \ifnum #1>63 \ifnum #1>80
      10\else 9\fi \else 8\fi \else 7\fi \else 6\fi
        \else 5\fi \else 4\fi \else 3\fi \else 2\fi \relax }%
}%
\def\XINT_sqrt_small_d #1#2%
{%
   \expandafter\XINT_sqrt_small_e\expandafter
   {\the\numexpr #1\ifcase \numexpr #2/\xint_c_ii-\xint_c_i\relax
                   \or 0\or 00\or 000\or 0000\fi }%
}%
\def\XINT_sqrt_small_e #1#2%
{%
   \expandafter\XINT_sqrt_small_f\expandafter {\the\numexpr #1*#1-#2}{#1}%
}%
\def\XINT_sqrt_small_f #1#2%
{%
   \expandafter\XINT_sqrt_small_g\expandafter
   {\the\numexpr ((#1+#2)/(\xint_c_ii*#2))-\xint_c_i}{#1}{#2}%
}%
\def\XINT_sqrt_small_g #1%
{%
    \ifnum #1>\xint_c_
       \expandafter\XINT_sqrt_small_h
    \else
       \expandafter\XINT_sqrt_small_end
    \fi
    {#1}%
}%
\def\XINT_sqrt_small_h #1#2#3%
{%
    \expandafter\XINT_sqrt_small_f\expandafter
    {\the\numexpr #2-\xint_c_ii*#1*#3+#1*#1\expandafter}\expandafter
    {\the\numexpr #3-#1}%
}%
\def\XINT_sqrt_small_end #1#2#3{ {#3}{#2}}%
\def\XINT_sqrt_big_d #1#2%
{%
   \ifodd #2
     \expandafter\expandafter\expandafter\XINT_sqrt_big_eB
   \else
     \expandafter\expandafter\expandafter\XINT_sqrt_big_eA
   \fi
   \expandafter {\the\numexpr #2/\xint_c_ii }{#1}%
}%
\def\XINT_sqrt_big_eA  #1#2#3%
{%
    \XINT_sqrt_big_eA_a #3\Z {#2}{#1}{#3}%
}%
\def\XINT_sqrt_big_eA_a #1#2#3#4#5#6#7#8#9\Z
{%
    \XINT_sqrt_big_eA_b {#1#2#3#4#5#6#7#8}%
}%
\def\XINT_sqrt_big_eA_b #1#2%
{%
    \expandafter\XINT_sqrt_big_f
    \romannumeral0\XINT_sqrt_small_e {#2000}{#1}{#1}%
}%
\def\XINT_sqrt_big_eB #1#2#3%
{%
    \XINT_sqrt_big_eB_a #3\Z {#2}{#1}{#3}%
}%
\def\XINT_sqrt_big_eB_a #1#2#3#4#5#6#7#8#9%
{%
    \XINT_sqrt_big_eB_b {#1#2#3#4#5#6#7#8#9}%
}%
\def\XINT_sqrt_big_eB_b #1#2\Z #3%
{%
    \expandafter\XINT_sqrt_big_f
    \romannumeral0\XINT_sqrt_small_e {#30000}{#1}{#1}%
}%
\def\XINT_sqrt_big_f #1#2#3#4%
{%
   \expandafter\XINT_sqrt_big_f_a\expandafter
   {\the\numexpr #2+#3\expandafter}\expandafter
   {\romannumeral0\XINT_dsx_addzerosnofuss
                    {\numexpr #4-\xint_c_iv\relax}{#1}}{#4}%
}%
\def\XINT_sqrt_big_f_a #1#2#3#4%
{%
   \expandafter\XINT_sqrt_big_g\expandafter
   {\romannumeral0\xintiisub
       {\XINT_dsx_addzerosnofuss
        {\numexpr \xint_c_ii*#3-\xint_c_viii\relax}{#1}}{#4}}%
   {#2}{#3}%
}%
\def\XINT_sqrt_big_g #1#2%
{%
    \expandafter\XINT_sqrt_big_j
    \romannumeral0\xintiidivision{#1}%
        {\romannumeral0\XINT_dbl_pos #2\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W }{#2}%
}%
\def\XINT_sqrt_big_j #1%
{%
    \if0\XINT_Sgn #1\Z
          \expandafter \XINT_sqrt_big_end
    \else \expandafter \XINT_sqrt_big_k
    \fi {#1}%
}%
\def\XINT_sqrt_big_k #1#2#3%
{%
    \expandafter\XINT_sqrt_big_l\expandafter
    {\romannumeral0\xintiisub {#3}{#1}}%
    {\romannumeral0\xintiiadd {#2}{\xintiiSqr {#1}}}%
}%
\def\XINT_sqrt_big_l #1#2%
{%
   \expandafter\XINT_sqrt_big_g\expandafter
   {#2}{#1}%
}%
\def\XINT_sqrt_big_end #1#2#3#4{ {#3}{#2}}%
%    \end{macrocode}
% \subsection{\csh{xintiiE}}
% \lverb|Originally was used in \xintiiexpr. Transferred from xintfrac for 1.1.|
%    \begin{macrocode}
\def\xintiiE {\romannumeral0\xintiie }% used in \xintMod.
\def\xintiie #1#2%
   {\expandafter\XINT_iie\the\numexpr #2\expandafter.\expandafter{\romannumeral-`0#1}}%
\def\XINT_iie #1.#2{\ifnum#1>\xint_c_ \xint_dothis{\xint_dsh {#2}{-#1}}\fi
                    \xint_orthat{ #2}}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xint>\relax
%\def<*xintbinhex>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xint>
%<*xintbinhex>
%
% \StoreCodelineNo {xint}
%
% \section{Package \xintbinhexnameimp implementation}
% \label{sec:binheximp}
%
% \localtableofcontents
%
% The commenting is currently (\xintdocdate) very sparse.
%
% \subsection{Catcodes, \protect\eTeX{} and reload detection}
%
% The code for reload detection was initially copied from \textsc{Heiko
% Oberdiek}'s packages, then modified.
%
% The method for catcodes was also initially directly inspired by these
% packages.
%
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode64=11   % @
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \let\z\endgroup
  \expandafter\let\expandafter\x\csname ver@xintbinhex.sty\endcsname
  \expandafter\let\expandafter\w\csname ver@xintcore.sty\endcsname
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
  \ifx\csname numexpr\endcsname\relax
     \y{xintbinhex}{\numexpr not available, aborting input}%
     \aftergroup\endinput
  \else
    \ifx\x\relax   % plain-TeX, first loading of xintbinhex.sty
      \ifx\w\relax % but xintcore.sty not yet loaded.
         \def\z{\endgroup\input xintcore.sty\relax}%
      \fi
    \else
      \def\empty {}%
      \ifx\x\empty % LaTeX, first loading,
      % variable is initialized, but \ProvidesPackage not yet seen
          \ifx\w\relax % xintcore.sty not yet loaded.
            \def\z{\endgroup\RequirePackage{xintcore}}%
          \fi
      \else
        \aftergroup\endinput % xintbinhex already loaded.
      \fi
    \fi
  \fi
\z%
\XINTsetupcatcodes% defined in xintkernel.sty
%    \end{macrocode}
% \subsection{Package identification}
%    \begin{macrocode}
\XINT_providespackage
\ProvidesPackage{xintbinhex}%
  [2015/08/31 v1.1b Expandable binary and hexadecimal conversions (jfB)]%
%    \end{macrocode}
% \subsection{Constants, etc...}
% \lverb!v1.08!
%    \begin{macrocode}
\chardef\xint_c_xvi           16
% \chardef\xint_c_ii^v          32 % already in xint.sty
% \chardef\xint_c_ii^vi         64 % already in xint.sty
\chardef\xint_c_ii^vii       128
\mathchardef\xint_c_ii^viii  256
\mathchardef\xint_c_ii^xii  4096
\newcount\xint_c_ii^xv  \xint_c_ii^xv  32768
\newcount\xint_c_ii^xvi \xint_c_ii^xvi 65536
\newcount\xint_c_x^v    \xint_c_x^v    100000
\newcount\xint_c_x^ix   \xint_c_x^ix   1000000000
\def\XINT_tmpa #1{\ifx\relax#1\else
  \expandafter\edef\csname XINT_sdth_#1\endcsname
  {\ifcase #1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or
              8\or 9\or A\or B\or C\or D\or E\or F\fi}%
  \expandafter\XINT_tmpa\fi }%
\XINT_tmpa {0}{1}{2}{3}{4}{5}{6}{7}{8}{9}{10}{11}{12}{13}{14}{15}\relax
\def\XINT_tmpa #1{\ifx\relax#1\else
  \expandafter\edef\csname XINT_sdtb_#1\endcsname
  {\ifcase #1
   0000\or 0001\or 0010\or 0011\or 0100\or 0101\or 0110\or 0111\or
   1000\or 1001\or 1010\or 1011\or 1100\or 1101\or 1110\or 1111\fi}%
  \expandafter\XINT_tmpa\fi }%
\XINT_tmpa {0}{1}{2}{3}{4}{5}{6}{7}{8}{9}{10}{11}{12}{13}{14}{15}\relax
\let\XINT_tmpa\relax
\expandafter\def\csname XINT_sbtd_0000\endcsname {0}%
\expandafter\def\csname XINT_sbtd_0001\endcsname {1}%
\expandafter\def\csname XINT_sbtd_0010\endcsname {2}%
\expandafter\def\csname XINT_sbtd_0011\endcsname {3}%
\expandafter\def\csname XINT_sbtd_0100\endcsname {4}%
\expandafter\def\csname XINT_sbtd_0101\endcsname {5}%
\expandafter\def\csname XINT_sbtd_0110\endcsname {6}%
\expandafter\def\csname XINT_sbtd_0111\endcsname {7}%
\expandafter\def\csname XINT_sbtd_1000\endcsname {8}%
\expandafter\def\csname XINT_sbtd_1001\endcsname {9}%
\expandafter\def\csname XINT_sbtd_1010\endcsname {10}%
\expandafter\def\csname XINT_sbtd_1011\endcsname {11}%
\expandafter\def\csname XINT_sbtd_1100\endcsname {12}%
\expandafter\def\csname XINT_sbtd_1101\endcsname {13}%
\expandafter\def\csname XINT_sbtd_1110\endcsname {14}%
\expandafter\def\csname XINT_sbtd_1111\endcsname {15}%
\expandafter\let\csname XINT_sbth_0000\expandafter\endcsname
                \csname XINT_sbtd_0000\endcsname
\expandafter\let\csname XINT_sbth_0001\expandafter\endcsname
                \csname XINT_sbtd_0001\endcsname
\expandafter\let\csname XINT_sbth_0010\expandafter\endcsname
                \csname XINT_sbtd_0010\endcsname
\expandafter\let\csname XINT_sbth_0011\expandafter\endcsname
                \csname XINT_sbtd_0011\endcsname
\expandafter\let\csname XINT_sbth_0100\expandafter\endcsname
                \csname XINT_sbtd_0100\endcsname
\expandafter\let\csname XINT_sbth_0101\expandafter\endcsname
                \csname XINT_sbtd_0101\endcsname
\expandafter\let\csname XINT_sbth_0110\expandafter\endcsname
                \csname XINT_sbtd_0110\endcsname
\expandafter\let\csname XINT_sbth_0111\expandafter\endcsname
                \csname XINT_sbtd_0111\endcsname
\expandafter\let\csname XINT_sbth_1000\expandafter\endcsname
                \csname XINT_sbtd_1000\endcsname
\expandafter\let\csname XINT_sbth_1001\expandafter\endcsname
                \csname XINT_sbtd_1001\endcsname
\expandafter\def\csname XINT_sbth_1010\endcsname {A}%
\expandafter\def\csname XINT_sbth_1011\endcsname {B}%
\expandafter\def\csname XINT_sbth_1100\endcsname {C}%
\expandafter\def\csname XINT_sbth_1101\endcsname {D}%
\expandafter\def\csname XINT_sbth_1110\endcsname {E}%
\expandafter\def\csname XINT_sbth_1111\endcsname {F}%
\expandafter\def\csname XINT_shtb_0\endcsname {0000}%
\expandafter\def\csname XINT_shtb_1\endcsname {0001}%
\expandafter\def\csname XINT_shtb_2\endcsname {0010}%
\expandafter\def\csname XINT_shtb_3\endcsname {0011}%
\expandafter\def\csname XINT_shtb_4\endcsname {0100}%
\expandafter\def\csname XINT_shtb_5\endcsname {0101}%
\expandafter\def\csname XINT_shtb_6\endcsname {0110}%
\expandafter\def\csname XINT_shtb_7\endcsname {0111}%
\expandafter\def\csname XINT_shtb_8\endcsname {1000}%
\expandafter\def\csname XINT_shtb_9\endcsname {1001}%
\def\XINT_shtb_A {1010}%
\def\XINT_shtb_B {1011}%
\def\XINT_shtb_C {1100}%
\def\XINT_shtb_D {1101}%
\def\XINT_shtb_E {1110}%
\def\XINT_shtb_F {1111}%
\def\XINT_shtb_G {}%
\def\XINT_smallhex #1%
{%
    \expandafter\XINT_smallhex_a\expandafter
    {\the\numexpr (#1+\xint_c_viii)/\xint_c_xvi-\xint_c_i}{#1}%
}%
\def\XINT_smallhex_a #1#2%
{%
    \csname XINT_sdth_#1\expandafter\expandafter\expandafter\endcsname
    \csname XINT_sdth_\the\numexpr #2-\xint_c_xvi*#1\endcsname
}%
\def\XINT_smallbin #1%
{%
    \expandafter\XINT_smallbin_a\expandafter
    {\the\numexpr (#1+\xint_c_viii)/\xint_c_xvi-\xint_c_i}{#1}%
}%
\def\XINT_smallbin_a #1#2%
{%
    \csname XINT_sdtb_#1\expandafter\expandafter\expandafter\endcsname
    \csname XINT_sdtb_\the\numexpr #2-\xint_c_xvi*#1\endcsname
}%
%    \end{macrocode}
% \subsection{\csh{xintDecToHex}, \csh{xintDecToBin}}
% \lverb!v1.08!
%    \begin{macrocode}
\def\xintDecToHex {\romannumeral0\xintdectohex }%
\def\xintdectohex #1%
        {\expandafter\XINT_dth_checkin\romannumeral-`0#1\W\W\W\W \T}%
\def\XINT_dth_checkin #1%
{%
    \xint_UDsignfork
       #1\XINT_dth_N
        -{\XINT_dth_P #1}%
     \krof
}%
\def\XINT_dth_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_dth_P }%
\def\XINT_dth_P {\expandafter\XINT_dth_III\romannumeral-`0\XINT_dtbh_I {0.}}%
\def\xintDecToBin {\romannumeral0\xintdectobin }%
\def\xintdectobin #1%
        {\expandafter\XINT_dtb_checkin\romannumeral-`0#1\W\W\W\W \T }%
\def\XINT_dtb_checkin #1%
{%
    \xint_UDsignfork
       #1\XINT_dtb_N
        -{\XINT_dtb_P #1}%
     \krof
}%
\def\XINT_dtb_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_dtb_P }%
\def\XINT_dtb_P {\expandafter\XINT_dtb_III\romannumeral-`0\XINT_dtbh_I {0.}}%
\def\XINT_dtbh_I #1#2#3#4#5%
{%
    \xint_gob_til_W #5\XINT_dtbh_II_a\W\XINT_dtbh_I_a  {}{#2#3#4#5}#1\Z.%
}%
\def\XINT_dtbh_II_a\W\XINT_dtbh_I_a #1#2{\XINT_dtbh_II_b #2}%
\def\XINT_dtbh_II_b #1#2#3#4%
{%
    \xint_gob_til_W
      #1\XINT_dtbh_II_c
      #2\XINT_dtbh_II_ci
      #3\XINT_dtbh_II_cii
      \W\XINT_dtbh_II_ciii #1#2#3#4%
}%
\def\XINT_dtbh_II_c \W\XINT_dtbh_II_ci
                    \W\XINT_dtbh_II_cii
                    \W\XINT_dtbh_II_ciii \W\W\W\W {{}}%
\def\XINT_dtbh_II_ci #1\XINT_dtbh_II_ciii #2\W\W\W
   {\XINT_dtbh_II_d {}{#2}{0}}%
\def\XINT_dtbh_II_cii\W\XINT_dtbh_II_ciii #1#2\W\W
   {\XINT_dtbh_II_d {}{#1#2}{00}}%
\def\XINT_dtbh_II_ciii #1#2#3\W
   {\XINT_dtbh_II_d {}{#1#2#3}{000}}%
\def\XINT_dtbh_I_a #1#2#3.%
{%
    \xint_gob_til_Z #3\XINT_dtbh_I_z\Z
    \expandafter\XINT_dtbh_I_b\the\numexpr #2+#30000.{#1}%
}%
\def\XINT_dtbh_I_b #1.%
{%
    \expandafter\XINT_dtbh_I_c\the\numexpr
    (#1+\xint_c_ii^xv)/\xint_c_ii^xvi-\xint_c_i.#1.%
}%
\def\XINT_dtbh_I_c #1.#2.%
{%
    \expandafter\XINT_dtbh_I_d\expandafter
    {\the\numexpr #2-\xint_c_ii^xvi*#1}{#1}%
}%
\def\XINT_dtbh_I_d #1#2#3{\XINT_dtbh_I_a {#3#1.}{#2}}%
\def\XINT_dtbh_I_z\Z\expandafter\XINT_dtbh_I_b\the\numexpr #1+#2.%
{%
    \ifnum #1=\xint_c_ \expandafter\XINT_dtbh_I_end_zb\fi
    \XINT_dtbh_I_end_za {#1}%
}%
\def\XINT_dtbh_I_end_za #1#2{\XINT_dtbh_I {#2#1.}}%
\def\XINT_dtbh_I_end_zb\XINT_dtbh_I_end_za #1#2{\XINT_dtbh_I {#2}}%
\def\XINT_dtbh_II_d #1#2#3#4.%
{%
    \xint_gob_til_Z #4\XINT_dtbh_II_z\Z
    \expandafter\XINT_dtbh_II_e\the\numexpr #2+#4#3.{#1}{#3}%
}%
\def\XINT_dtbh_II_e #1.%
{%
    \expandafter\XINT_dtbh_II_f\the\numexpr
        (#1+\xint_c_ii^xv)/\xint_c_ii^xvi-\xint_c_i.#1.%
}%
\def\XINT_dtbh_II_f #1.#2.%
{%
    \expandafter\XINT_dtbh_II_g\expandafter
    {\the\numexpr #2-\xint_c_ii^xvi*#1}{#1}%
}%
\def\XINT_dtbh_II_g #1#2#3{\XINT_dtbh_II_d {#3#1.}{#2}}%
\def\XINT_dtbh_II_z\Z\expandafter\XINT_dtbh_II_e\the\numexpr #1+#2.%
{%
    \ifnum #1=\xint_c_ \expandafter\XINT_dtbh_II_end_zb\fi
    \XINT_dtbh_II_end_za {#1}%
}%
\def\XINT_dtbh_II_end_za #1#2#3{{}#2#1.\Z.}%
\def\XINT_dtbh_II_end_zb\XINT_dtbh_II_end_za #1#2#3{{}#2\Z.}%
\def\XINT_dth_III #1#2.%
{%
    \xint_gob_til_Z #2\XINT_dth_end\Z
    \expandafter\XINT_dth_III\expandafter
    {\romannumeral-`0\XINT_dth_small #2.#1}%
}%
\def\XINT_dth_small #1.%
{%
    \expandafter\XINT_smallhex\expandafter
    {\the\numexpr (#1+\xint_c_ii^vii)/\xint_c_ii^viii-\xint_c_i\expandafter}%
    \romannumeral-`0\expandafter\XINT_smallhex\expandafter
    {\the\numexpr
    #1-((#1+\xint_c_ii^vii)/\xint_c_ii^viii-\xint_c_i)*\xint_c_ii^viii}%
}%
\def\XINT_dth_end\Z\expandafter\XINT_dth_III\expandafter #1#2\T
{%
    \XINT_dth_end_b #1%
}%
\def\XINT_dth_end_b #1.{\XINT_dth_end_c }%
\def\XINT_dth_end_c #1{\xint_gob_til_zero #1\XINT_dth_end_d 0\space #1}%
\def\XINT_dth_end_d 0\space 0#1%
{%
    \xint_gob_til_zero #1\XINT_dth_end_e 0\space #1%
}%
\def\XINT_dth_end_e 0\space 0#1%
{%
    \xint_gob_til_zero #1\XINT_dth_end_f 0\space #1%
}%
\def\XINT_dth_end_f 0\space 0{ }%
\def\XINT_dtb_III #1#2.%
{%
    \xint_gob_til_Z #2\XINT_dtb_end\Z
    \expandafter\XINT_dtb_III\expandafter
    {\romannumeral-`0\XINT_dtb_small #2.#1}%
}%
\def\XINT_dtb_small #1.%
{%
    \expandafter\XINT_smallbin\expandafter
    {\the\numexpr (#1+\xint_c_ii^vii)/\xint_c_ii^viii-\xint_c_i\expandafter}%
    \romannumeral-`0\expandafter\XINT_smallbin\expandafter
    {\the\numexpr
    #1-((#1+\xint_c_ii^vii)/\xint_c_ii^viii-\xint_c_i)*\xint_c_ii^viii}%
}%
\def\XINT_dtb_end\Z\expandafter\XINT_dtb_III\expandafter #1#2\T
{%
    \XINT_dtb_end_b #1%
}%
\def\XINT_dtb_end_b #1.{\XINT_dtb_end_c }%
\def\XINT_dtb_end_c #1#2#3#4#5#6#7#8%
{%
    \expandafter\XINT_dtb_end_d\the\numexpr #1#2#3#4#5#6#7#8\relax
}%
\edef\XINT_dtb_end_d #1#2#3#4#5#6#7#8#9%
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8#9\relax
}%
%    \end{macrocode}
% \subsection{\csh{xintHexToDec}}
% \lverb!v1.08!
%    \begin{macrocode}
\def\xintHexToDec {\romannumeral0\xinthextodec }%
\def\xinthextodec #1%
        {\expandafter\XINT_htd_checkin\romannumeral-`0#1\W\W\W\W \T }%
\def\XINT_htd_checkin #1%
{%
    \xint_UDsignfork
       #1\XINT_htd_neg
        -{\XINT_htd_I {0000}#1}%
     \krof
}%
\def\XINT_htd_neg {\expandafter\xint_minus_thenstop
                   \romannumeral0\XINT_htd_I {0000}}%
\def\XINT_htd_I #1#2#3#4#5%
{%
    \xint_gob_til_W #5\XINT_htd_II_a\W
    \XINT_htd_I_a  {}{"#2#3#4#5}#1\Z\Z\Z\Z
}%
\def\XINT_htd_II_a \W\XINT_htd_I_a #1#2{\XINT_htd_II_b #2}%
\def\XINT_htd_II_b "#1#2#3#4%
{%
    \xint_gob_til_W
      #1\XINT_htd_II_c
      #2\XINT_htd_II_ci
      #3\XINT_htd_II_cii
      \W\XINT_htd_II_ciii #1#2#3#4%
}%
\def\XINT_htd_II_c \W\XINT_htd_II_ci
                   \W\XINT_htd_II_cii
                   \W\XINT_htd_II_ciii \W\W\W\W #1\Z\Z\Z\Z\T
{%
    \expandafter\xint_cleanupzeros_andstop
    \romannumeral0\XINT_rord_main {}#1%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
}%
\def\XINT_htd_II_ci #1\XINT_htd_II_ciii
                      #2\W\W\W {\XINT_htd_II_d {}{"#2}{\xint_c_xvi}}%
\def\XINT_htd_II_cii\W\XINT_htd_II_ciii
                      #1#2\W\W {\XINT_htd_II_d {}{"#1#2}{\xint_c_ii^viii}}%
\def\XINT_htd_II_ciii #1#2#3\W {\XINT_htd_II_d {}{"#1#2#3}{\xint_c_ii^xii}}%
\def\XINT_htd_I_a #1#2#3#4#5#6%
{%
    \xint_gob_til_Z #3\XINT_htd_I_end_a\Z
    \expandafter\XINT_htd_I_b\the\numexpr
    #2+\xint_c_ii^xvi*#6#5#4#3+\xint_c_x^ix\relax {#1}%
}%
\def\XINT_htd_I_b 1#1#2#3#4#5#6#7#8#9{\XINT_htd_I_c {#1#2#3#4#5}{#9#8#7#6}}%
\def\XINT_htd_I_c #1#2#3{\XINT_htd_I_a {#3#2}{#1}}%
\def\XINT_htd_I_end_a\Z\expandafter\XINT_htd_I_b\the\numexpr #1+#2\relax
{%
    \expandafter\XINT_htd_I_end_b\the\numexpr \xint_c_x^v+#1\relax
}%
\def\XINT_htd_I_end_b 1#1#2#3#4#5%
{%
    \xint_gob_til_zero #1\XINT_htd_I_end_bz0%
    \XINT_htd_I_end_c #1#2#3#4#5%
}%
\def\XINT_htd_I_end_c #1#2#3#4#5#6{\XINT_htd_I {#6#5#4#3#2#1000}}%
\def\XINT_htd_I_end_bz0\XINT_htd_I_end_c 0#1#2#3#4%
{%
    \xint_gob_til_zeros_iv #1#2#3#4\XINT_htd_I_end_bzz 0000%
    \XINT_htd_I_end_D {#4#3#2#1}%
}%
\def\XINT_htd_I_end_D  #1#2{\XINT_htd_I {#2#1}}%
\def\XINT_htd_I_end_bzz 0000\XINT_htd_I_end_D #1{\XINT_htd_I }%
\def\XINT_htd_II_d #1#2#3#4#5#6#7%
{%
    \xint_gob_til_Z #4\XINT_htd_II_end_a\Z
    \expandafter\XINT_htd_II_e\the\numexpr
    #2+#3*#7#6#5#4+\xint_c_x^viii\relax {#1}{#3}%
}%
\def\XINT_htd_II_e 1#1#2#3#4#5#6#7#8{\XINT_htd_II_f {#1#2#3#4}{#5#6#7#8}}%
\def\XINT_htd_II_f #1#2#3{\XINT_htd_II_d {#2#3}{#1}}%
\def\XINT_htd_II_end_a\Z\expandafter\XINT_htd_II_e
    \the\numexpr #1+#2\relax #3#4\T
{%
    \XINT_htd_II_end_b #1#3%
}%
\edef\XINT_htd_II_end_b #1#2#3#4#5#6#7#8%
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax
}%
%    \end{macrocode}
% \subsection{\csh{xintBinToDec}}
% \lverb!v1.08!
%    \begin{macrocode}
\def\xintBinToDec {\romannumeral0\xintbintodec }%
\def\xintbintodec #1{\expandafter\XINT_btd_checkin
                     \romannumeral-`0#1\W\W\W\W\W\W\W\W \T }%
\def\XINT_btd_checkin #1%
{%
    \xint_UDsignfork
       #1\XINT_btd_neg
        -{\XINT_btd_I {000000}#1}%
     \krof
}%
\def\XINT_btd_neg {\expandafter\xint_minus_thenstop
                               \romannumeral0\XINT_btd_I {000000}}%
\def\XINT_btd_I #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_W #9\XINT_btd_II_a {#2#3#4#5#6#7#8#9}\W
    \XINT_btd_I_a {}{\csname XINT_sbtd_#2#3#4#5\endcsname*\xint_c_xvi+%
                     \csname XINT_sbtd_#6#7#8#9\endcsname}%
    #1\Z\Z\Z\Z\Z\Z
}%
\def\XINT_btd_II_a #1\W\XINT_btd_I_a #2#3{\XINT_btd_II_b #1}%
\def\XINT_btd_II_b #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_W
      #1\XINT_btd_II_c
      #2\XINT_btd_II_ci
      #3\XINT_btd_II_cii
      #4\XINT_btd_II_ciii
      #5\XINT_btd_II_civ
      #6\XINT_btd_II_cv
      #7\XINT_btd_II_cvi
      \W\XINT_btd_II_cvii #1#2#3#4#5#6#7#8%
}%
\def\XINT_btd_II_c #1\XINT_btd_II_cvii \W\W\W\W\W\W\W\W #2\Z\Z\Z\Z\Z\Z\T
{%
    \expandafter\XINT_btd_II_c_end
    \romannumeral0\XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
}%
\edef\XINT_btd_II_c_end #1#2#3#4#5#6%
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6\relax
}%
\def\XINT_btd_II_ci  #1\XINT_btd_II_cvii #2\W\W\W\W\W\W\W
   {\XINT_btd_II_d {}{#2}{\xint_c_ii }}%
\def\XINT_btd_II_cii #1\XINT_btd_II_cvii #2\W\W\W\W\W\W
   {\XINT_btd_II_d {}{\csname XINT_sbtd_00#2\endcsname }{\xint_c_iv }}%
\def\XINT_btd_II_ciii #1\XINT_btd_II_cvii #2\W\W\W\W\W
   {\XINT_btd_II_d {}{\csname XINT_sbtd_0#2\endcsname }{\xint_c_viii }}%
\def\XINT_btd_II_civ #1\XINT_btd_II_cvii #2\W\W\W\W
   {\XINT_btd_II_d {}{\csname XINT_sbtd_#2\endcsname}{\xint_c_xvi }}%
\def\XINT_btd_II_cv #1\XINT_btd_II_cvii #2#3#4#5#6\W\W\W
{%
    \XINT_btd_II_d {}{\csname XINT_sbtd_#2#3#4#5\endcsname*\xint_c_ii+%
                          #6}{\xint_c_ii^v }%
}%
\def\XINT_btd_II_cvi #1\XINT_btd_II_cvii #2#3#4#5#6#7\W\W
{%
    \XINT_btd_II_d {}{\csname XINT_sbtd_#2#3#4#5\endcsname*\xint_c_iv+%
                      \csname XINT_sbtd_00#6#7\endcsname}{\xint_c_ii^vi }%
}%
\def\XINT_btd_II_cvii #1#2#3#4#5#6#7\W
{%
    \XINT_btd_II_d {}{\csname XINT_sbtd_#1#2#3#4\endcsname*\xint_c_viii+%
                      \csname XINT_sbtd_0#5#6#7\endcsname}{\xint_c_ii^vii }%
}%
\def\XINT_btd_II_d #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_Z #4\XINT_btd_II_end_a\Z
    \expandafter\XINT_btd_II_e\the\numexpr
    #2+(\xint_c_x^ix+#3*#9#8#7#6#5#4)\relax {#1}{#3}%
}%
\def\XINT_btd_II_e 1#1#2#3#4#5#6#7#8#9{\XINT_btd_II_f {#1#2#3}{#4#5#6#7#8#9}}%
\def\XINT_btd_II_f #1#2#3{\XINT_btd_II_d {#2#3}{#1}}%
\def\XINT_btd_II_end_a\Z\expandafter\XINT_btd_II_e
    \the\numexpr #1+(#2\relax #3#4\T
{%
    \XINT_btd_II_end_b #1#3%
}%
\edef\XINT_btd_II_end_b #1#2#3#4#5#6#7#8#9%
{%
    \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8#9\relax
}%
\def\XINT_btd_I_a #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_Z #3\XINT_btd_I_end_a\Z
    \expandafter\XINT_btd_I_b\the\numexpr
    #2+\xint_c_ii^viii*#8#7#6#5#4#3+\xint_c_x^ix\relax {#1}%
}%
\def\XINT_btd_I_b 1#1#2#3#4#5#6#7#8#9{\XINT_btd_I_c {#1#2#3}{#9#8#7#6#5#4}}%
\def\XINT_btd_I_c #1#2#3{\XINT_btd_I_a {#3#2}{#1}}%
\def\XINT_btd_I_end_a\Z\expandafter\XINT_btd_I_b
    \the\numexpr #1+\xint_c_ii^viii #2\relax
{%
    \expandafter\XINT_btd_I_end_b\the\numexpr 1000+#1\relax
}%
\def\XINT_btd_I_end_b 1#1#2#3%
{%
    \xint_gob_til_zeros_iii #1#2#3\XINT_btd_I_end_bz 000%
    \XINT_btd_I_end_c #1#2#3%
}%
\def\XINT_btd_I_end_c #1#2#3#4{\XINT_btd_I {#4#3#2#1000}}%
\def\XINT_btd_I_end_bz 000\XINT_btd_I_end_c 000{\XINT_btd_I }%
%    \end{macrocode}
% \subsection{\csh{xintBinToHex}}
% \lverb!v1.08!
%    \begin{macrocode}
\def\xintBinToHex {\romannumeral0\xintbintohex }%
\def\xintbintohex #1%
{%
    \expandafter\XINT_bth_checkin
                     \romannumeral0\expandafter\XINT_num_loop
                     \romannumeral-`0#1\xint_relax\xint_relax
                                       \xint_relax\xint_relax
                     \xint_relax\xint_relax\xint_relax\xint_relax\Z
    \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W
}%
\def\XINT_bth_checkin #1%
{%
    \xint_UDsignfork
       #1\XINT_bth_N
        -{\XINT_bth_P #1}%
     \krof
}%
\def\XINT_bth_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_bth_P }%
\def\XINT_bth_P {\expandafter\XINT_bth_I\expandafter{\expandafter}%
                 \romannumeral0\XINT_OQ {}}%
\def\XINT_bth_I #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_W #9\XINT_bth_end_a\W
    \expandafter\expandafter\expandafter
    \XINT_bth_I
    \expandafter\expandafter\expandafter
    {\csname XINT_sbth_#9#8#7#6\expandafter\expandafter\expandafter\endcsname
     \csname XINT_sbth_#5#4#3#2\endcsname #1}%
}%
\def\XINT_bth_end_a\W \expandafter\expandafter\expandafter
    \XINT_bth_I       \expandafter\expandafter\expandafter #1%
{%
    \XINT_bth_end_b #1%
}%
\def\XINT_bth_end_b  #1\endcsname #2\endcsname #3%
{%
    \xint_gob_til_zero #3\XINT_bth_end_z 0\space #3%
}%
\def\XINT_bth_end_z0\space 0{ }%
%    \end{macrocode}
% \subsection{\csh{xintHexToBin}}
% \lverb!v1.08!
%    \begin{macrocode}
\def\xintHexToBin {\romannumeral0\xinthextobin }%
\def\xinthextobin #1%
{%
    \expandafter\XINT_htb_checkin\romannumeral-`0#1GGGGGGGG\T
}%
\def\XINT_htb_checkin #1%
{%
    \xint_UDsignfork
       #1\XINT_htb_N
        -{\XINT_htb_P #1}%
     \krof
}%
\def\XINT_htb_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_htb_P }%
\def\XINT_htb_P {\XINT_htb_I_a {}}%
\def\XINT_htb_I_a #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_G #9\XINT_htb_II_a G%
    \expandafter\expandafter\expandafter
    \XINT_htb_I_b
    \expandafter\expandafter\expandafter
    {\csname XINT_shtb_#2\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#3\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#4\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#5\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#6\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#7\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#8\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#9\endcsname }{#1}%
}%
\def\XINT_htb_I_b #1#2{\XINT_htb_I_a {#2#1}}%
\def\XINT_htb_II_a G\expandafter\expandafter\expandafter\XINT_htb_I_b
{%
    \expandafter\expandafter\expandafter \XINT_htb_II_b
}%
\def\XINT_htb_II_b #1#2#3\T
{%
    \XINT_num_loop #2#1%
    \xint_relax\xint_relax\xint_relax\xint_relax
    \xint_relax\xint_relax\xint_relax\xint_relax\Z
}%
%    \end{macrocode}
% \subsection{\csh{xintCHexToBin}}
% \lverb!v1.08!
%    \begin{macrocode}
\def\xintCHexToBin {\romannumeral0\xintchextobin }%
\def\xintchextobin #1%
{%
    \expandafter\XINT_chtb_checkin\romannumeral-`0#1%
    \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W
}%
\def\XINT_chtb_checkin #1%
{%
    \xint_UDsignfork
       #1\XINT_chtb_N
        -{\XINT_chtb_P #1}%
     \krof
}%
\def\XINT_chtb_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_chtb_P }%
\def\XINT_chtb_P {\expandafter\XINT_chtb_I\expandafter{\expandafter}%
                  \romannumeral0\XINT_OQ {}}%
\def\XINT_chtb_I #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_W #9\XINT_chtb_end_a\W
    \expandafter\expandafter\expandafter
    \XINT_chtb_I
    \expandafter\expandafter\expandafter
    {\csname XINT_shtb_#9\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#8\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#7\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#6\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#5\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#4\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#3\expandafter\expandafter\expandafter\endcsname
     \csname XINT_shtb_#2\endcsname
     #1}%
}%
\def\XINT_chtb_end_a\W\expandafter\expandafter\expandafter
    \XINT_chtb_I\expandafter\expandafter\expandafter #1%
{%
    \XINT_chtb_end_b #1%
    \xint_relax\xint_relax\xint_relax\xint_relax
    \xint_relax\xint_relax\xint_relax\xint_relax\Z
}%
\def\XINT_chtb_end_b #1\W#2\W#3\W#4\W#5\W#6\W#7\W#8\W\endcsname
{%
    \XINT_num_loop
}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xintbinhex>\relax
%\def<*xintgcd>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xintbinhex>
%<*xintgcd>
%
% \StoreCodelineNo {xintbinhex}
%
% \section{Package \xintgcdnameimp implementation}
% \label{sec:gcdimp}
%
% \localtableofcontents
%
% The commenting is currently (\xintdocdate) very sparse. Release |1.09h| has
% modified a bit the |\xintTypesetEuclideAlgorithm| and
% |\xintTypesetBezoutAlgorithm| layout with respect to line indentation in
% particular. And they use the \xinttoolsnameimp |\xintloop| rather than the
% Plain \TeX{} or \LaTeX{}'s |\loop|.
%
% Since |1.1| the package only loads \xintcorenameimp, not \xintnameimp. And
% for the |\xintTypesetEuclideAlgorithm| and |\xintTypesetBezoutAlgorithm|
% macros to be functional the package \xinttoolsnameimp needs to be loaded
% explicitely by the user.
%
% \subsection{Catcodes, \protect\eTeX{} and reload detection}
%
% The code for reload detection was initially copied from \textsc{Heiko
% Oberdiek}'s packages, then modified.
%
% The method for catcodes was also initially directly inspired by these
% packages.
%
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode64=11   % @
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \let\z\endgroup
  \expandafter\let\expandafter\x\csname ver@xintgcd.sty\endcsname
  \expandafter\let\expandafter\w\csname ver@xintcore.sty\endcsname
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
  \ifx\csname numexpr\endcsname\relax
     \y{xintgcd}{\numexpr not available, aborting input}%
     \aftergroup\endinput
  \else
    \ifx\x\relax   % plain-TeX, first loading of xintgcd.sty
      \ifx\w\relax % but xintcore.sty not yet loaded.
         \def\z{\endgroup\input xintcore.sty\relax}%
      \fi
    \else
      \def\empty {}%
      \ifx\x\empty % LaTeX, first loading,
      % variable is initialized, but \ProvidesPackage not yet seen
          \ifx\w\relax % xintcore.sty not yet loaded.
            \def\z{\endgroup\RequirePackage{xintcore}}%
          \fi
      \else
        \aftergroup\endinput % xintgcd already loaded.
      \fi
    \fi
  \fi
\z%
\XINTsetupcatcodes% defined in xintkernel.sty
%    \end{macrocode}
% \subsection{Package identification}
%    \begin{macrocode}
\XINT_providespackage
\ProvidesPackage{xintgcd}%
  [2015/08/31 v1.1b Euclide algorithm with xint package (jfB)]%
%    \end{macrocode}
% \subsection{\csh{xintGCD}, \csh{xintiiGCD}}
% \lverb|The macros of 1.09a benefits from the \xintnum which has been inserted
% inside \xintiabs in \xintnameimp;
% this is a little overhead but is more convenient for the
% user and also makes it easier to use into \xintexpr-essions. 1.1a adds
% \xintiiGCD mainly for \xintiiexpr benefit. Perhaps one should always have
% had ONLY ii versions from the beginning. And perhaps for sake of
% consistency, \xintGCD should be named \xintiGCD? too late.|
%    \begin{macrocode}
\def\xintGCD {\romannumeral0\xintgcd }%
\def\xintgcd #1%
{%
    \expandafter\XINT_gcd\expandafter{\romannumeral0\xintiabs {#1}}%
}%
\def\XINT_gcd #1#2%
{%
    \expandafter\XINT_gcd_fork\romannumeral0\xintiabs {#2}\Z #1\Z
}%
\def\xintiiGCD {\romannumeral0\xintiigcd }%
\def\xintiigcd #1%
{%
    \expandafter\XINT_iigcd\expandafter{\romannumeral0\xintiiabs {#1}}%
}%
\def\XINT_iigcd #1#2%
{%
    \expandafter\XINT_gcd_fork\romannumeral0\xintiiabs {#2}\Z #1\Z
}%
%    \end{macrocode}
% \lverb|&
% Ici #3#4=A, #1#2=B|
%    \begin{macrocode}
\def\XINT_gcd_fork #1#2\Z #3#4\Z
{%
    \xint_UDzerofork
      #1\XINT_gcd_BisZero
      #3\XINT_gcd_AisZero
       0\XINT_gcd_loop
    \krof
    {#1#2}{#3#4}%
}%
\def\XINT_gcd_AisZero #1#2{ #1}%
\def\XINT_gcd_BisZero #1#2{ #2}%
\def\XINT_gcd_CheckRem #1#2\Z
{%
    \xint_gob_til_zero #1\xint_gcd_end0\XINT_gcd_loop {#1#2}%
}%
\def\xint_gcd_end0\XINT_gcd_loop #1#2{ #2}%
%    \end{macrocode}
% \lverb|#1=B, #2=A|
%    \begin{macrocode}
\def\XINT_gcd_loop #1#2%
{%
    \expandafter\expandafter\expandafter
        \XINT_gcd_CheckRem
    \expandafter\xint_secondoftwo
    \romannumeral0\XINT_div_prepare {#1}{#2}\Z
    {#1}%
}%
%    \end{macrocode}
% \subsection{\csh{xintLCM}, \csh{xintiiLCM}}
% \lverb|New with 1.09a. Inadvertent use of \xintiQuo which was promoted at the
% same time to add the \xintnum overhead. So with 1.09f \xintiiQuo without the
% overhead. However \xintiabs has the \xintnum thing. The advantage is that we
% can thus use lcm in \xintexpr. The disadvantage is that this has overhead in
% \xintiiexpr. Thus 1.1a has \xintiiLCM.|
%    \begin{macrocode}
\def\xintLCM {\romannumeral0\xintlcm}%
\def\xintlcm #1%
{%
    \expandafter\XINT_lcm\expandafter{\romannumeral0\xintiabs {#1}}%
}%
\def\XINT_lcm #1#2%
{%
    \expandafter\XINT_lcm_fork\romannumeral0\xintiabs {#2}\Z #1\Z
}%
\def\xintiiLCM {\romannumeral0\xintiilcm}%
\def\xintiilcm #1%
{%
    \expandafter\XINT_iilcm\expandafter{\romannumeral0\xintiiabs {#1}}%
}%
\def\XINT_iilcm #1#2%
{%
    \expandafter\XINT_lcm_fork\romannumeral0\xintiiabs {#2}\Z #1\Z
}%
\def\XINT_lcm_fork #1#2\Z #3#4\Z
{%
    \xint_UDzerofork
      #1\XINT_lcm_BisZero
      #3\XINT_lcm_AisZero
       0\expandafter
    \krof
    \XINT_lcm_notzero\expandafter{\romannumeral0\XINT_gcd_loop {#1#2}{#3#4}}%
    {#1#2}{#3#4}%
}%
\def\XINT_lcm_AisZero #1#2#3#4#5{ 0}%
\def\XINT_lcm_BisZero #1#2#3#4#5{ 0}%
\def\XINT_lcm_notzero #1#2#3{\xintiimul {#2}{\xintiiQuo{#3}{#1}}}%
%    \end{macrocode}
% \subsection{\csh{xintBezout}}
% \lverb|1.09a inserts use of \xintnum|
%    \begin{macrocode}
\def\xintBezout {\romannumeral0\xintbezout }%
\def\xintbezout #1%
{%
    \expandafter\xint_bezout\expandafter {\romannumeral0\xintnum{#1}}%
}%
\def\xint_bezout #1#2%
{%
    \expandafter\XINT_bezout_fork \romannumeral0\xintnum{#2}\Z #1\Z
}%
%    \end{macrocode}
% \lverb|#3#4 = A, #1#2=B|
%    \begin{macrocode}
\def\XINT_bezout_fork #1#2\Z #3#4\Z
{%
    \xint_UDzerosfork
     #1#3\XINT_bezout_botharezero
      #10\XINT_bezout_secondiszero
      #30\XINT_bezout_firstiszero
       00{\xint_UDsignsfork
          #1#3\XINT_bezout_minusminus % A < 0, B < 0
           #1-\XINT_bezout_minusplus  % A > 0, B < 0
           #3-\XINT_bezout_plusminus  % A < 0, B > 0
            --\XINT_bezout_plusplus   % A > 0, B > 0
         \krof }%
    \krof
    {#2}{#4}#1#3{#3#4}{#1#2}% #1#2=B, #3#4=A
}%
\edef\XINT_bezout_botharezero #1#2#3#4#5#6%
{%
    \noexpand\xintError:NoBezoutForZeros\space {0}{0}{0}{0}{0}%
}%
%    \end{macrocode}
% \lverb|&
% attention première entrée doit être ici (-1)^n donc 1$\ 
% #4#2 = 0 = A, B = #3#1|
%    \begin{macrocode}
\def\XINT_bezout_firstiszero #1#2#3#4#5#6%
{%
    \xint_UDsignfork
      #3{ {0}{#3#1}{0}{1}{#1}}%
       -{ {0}{#3#1}{0}{-1}{#1}}%
    \krof
}%
%    \end{macrocode}
% \lverb|#4#2 = A, B = #3#1 = 0|
%    \begin{macrocode}
\def\XINT_bezout_secondiszero #1#2#3#4#5#6%
{%
    \xint_UDsignfork
       #4{ {#4#2}{0}{-1}{0}{#2}}%
        -{ {#4#2}{0}{1}{0}{#2}}%
    \krof
}%
%    \end{macrocode}
% \lverb|#4#2= A < 0, #3#1 = B < 0|
%    \begin{macrocode}
\def\XINT_bezout_minusminus #1#2#3#4%
{%
    \expandafter\XINT_bezout_mm_post
    \romannumeral0\XINT_bezout_loop_a 1{#1}{#2}1001%
}%
\def\XINT_bezout_mm_post #1#2%
{%
    \expandafter\XINT_bezout_mm_postb\expandafter
    {\romannumeral0\xintiiopp{#2}}{\romannumeral0\xintiiopp{#1}}%
}%
\def\XINT_bezout_mm_postb #1#2%
{%
    \expandafter\XINT_bezout_mm_postc\expandafter {#2}{#1}%
}%
\edef\XINT_bezout_mm_postc #1#2#3#4#5%
{%
    \space {#4}{#5}{#1}{#2}{#3}%
}%
%    \end{macrocode}
% \lverb|minusplus  #4#2= A > 0, B < 0|
%    \begin{macrocode}
\def\XINT_bezout_minusplus #1#2#3#4%
{%
    \expandafter\XINT_bezout_mp_post
    \romannumeral0\XINT_bezout_loop_a 1{#1}{#4#2}1001%
}%
\def\XINT_bezout_mp_post #1#2%
{%
    \expandafter\XINT_bezout_mp_postb\expandafter
      {\romannumeral0\xintiiopp {#2}}{#1}%
}%
\edef\XINT_bezout_mp_postb #1#2#3#4#5%
{%
    \space {#4}{#5}{#2}{#1}{#3}%
}%
%    \end{macrocode}
% \lverb|plusminus  A < 0, B > 0|
%    \begin{macrocode}
\def\XINT_bezout_plusminus #1#2#3#4%
{%
    \expandafter\XINT_bezout_pm_post
    \romannumeral0\XINT_bezout_loop_a 1{#3#1}{#2}1001%
}%
\def\XINT_bezout_pm_post #1%
{%
    \expandafter \XINT_bezout_pm_postb \expandafter
        {\romannumeral0\xintiiopp{#1}}%
}%
\edef\XINT_bezout_pm_postb #1#2#3#4#5%
{%
    \space {#4}{#5}{#1}{#2}{#3}%
}%
%    \end{macrocode}
% \lverb|plusplus|
%    \begin{macrocode}
\def\XINT_bezout_plusplus #1#2#3#4%
{%
    \expandafter\XINT_bezout_pp_post
    \romannumeral0\XINT_bezout_loop_a 1{#3#1}{#4#2}1001%
}%
%    \end{macrocode}
% \lverb|la parité (-1)^N est en #1, et on la jette ici.|
%    \begin{macrocode}
\edef\XINT_bezout_pp_post #1#2#3#4#5%
{%
    \space {#4}{#5}{#1}{#2}{#3}%
}%
%    \end{macrocode}
% \lverb|&
% n = 0: 1BAalpha(0)beta(0)alpha(-1)beta(-1)$\ 
% n général:
% {(-1)^n}{r(n-1)}{r(n-2)}{alpha(n-1)}{beta(n-1)}{alpha(n-2)}{beta(n-2)}$\ 
% #2 = B, #3 = A|
%    \begin{macrocode}
\def\XINT_bezout_loop_a #1#2#3%
{%
    \expandafter\XINT_bezout_loop_b
    \expandafter{\the\numexpr -#1\expandafter }%
    \romannumeral0\XINT_div_prepare {#2}{#3}{#2}%
}%
%    \end{macrocode}
% \lverb|&
% Le q(n) a ici une existence éphémère, dans le version Bezout Algorithm
% il faudra le conserver. On voudra à la fin
% {{q(n)}{r(n)}{alpha(n)}{beta(n)}}.
% De plus ce n'est plus (-1)^n que l'on veut mais n. (ou dans un autre ordre)$\ 
% {-(-1)^n}{q(n)}{r(n)}{r(n-1)}{alpha(n-1)}{beta(n-1)}{alpha(n-2)}{beta(n-2)}|
%    \begin{macrocode}
\def\XINT_bezout_loop_b #1#2#3#4#5#6#7#8%
{%
    \expandafter \XINT_bezout_loop_c \expandafter
        {\romannumeral0\xintiiadd{\XINT_Mul{#5}{#2}}{#7}}%
        {\romannumeral0\xintiiadd{\XINT_Mul{#6}{#2}}{#8}}%
    {#1}{#3}{#4}{#5}{#6}%
}%
%    \end{macrocode}
% \lverb|{alpha(n)}{->beta(n)}{-(-1)^n}{r(n)}{r(n-1)}{alpha(n-1)}{beta(n-1)}|
%    \begin{macrocode}
\def\XINT_bezout_loop_c #1#2%
{%
    \expandafter \XINT_bezout_loop_d \expandafter
        {#2}{#1}%
}%
%    \end{macrocode}
% \lverb|{beta(n)}{alpha(n)}{(-1)^(n+1)}{r(n)}{r(n-1)}{alpha(n-1)}{beta(n-1)}|
%    \begin{macrocode}
\def\XINT_bezout_loop_d #1#2#3#4#5%
{%
    \XINT_bezout_loop_e #4\Z {#3}{#5}{#2}{#1}%
}%
%    \end{macrocode}
% \lverb|r(n)\Z {(-1)^(n+1)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}|
%    \begin{macrocode}
\def\XINT_bezout_loop_e #1#2\Z
{%
    \xint_gob_til_zero #1\xint_bezout_loop_exit0\XINT_bezout_loop_f
    {#1#2}%
}%
%    \end{macrocode}
% \lverb|{r(n)}{(-1)^(n+1)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}|
%    \begin{macrocode}
\def\XINT_bezout_loop_f #1#2%
{%
    \XINT_bezout_loop_a {#2}{#1}%
}%
%    \end{macrocode}
% \lverb|{(-1)^(n+1)}{r(n)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}
% et itération|
%    \begin{macrocode}
\def\xint_bezout_loop_exit0\XINT_bezout_loop_f #1#2%
{%
    \ifcase #2
    \or  \expandafter\XINT_bezout_exiteven
    \else\expandafter\XINT_bezout_exitodd
    \fi
}%
\edef\XINT_bezout_exiteven #1#2#3#4#5%
{%
    \space {#5}{#4}{#1}%
}%
\edef\XINT_bezout_exitodd #1#2#3#4#5%
{%
    \space {-#5}{-#4}{#1}%
}%
%    \end{macrocode}
% \subsection{\csh{xintEuclideAlgorithm}}
% \lverb|&
% Pour Euclide:
% {N}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0}$\ 
% u<2n> = u<2n+3>u<2n+2> + u<2n+4> à la n ième étape|
%    \begin{macrocode}
\def\xintEuclideAlgorithm {\romannumeral0\xinteuclidealgorithm }%
\def\xinteuclidealgorithm #1%
{%
    \expandafter \XINT_euc \expandafter{\romannumeral0\xintiabs {#1}}%
}%
\def\XINT_euc #1#2%
{%
    \expandafter\XINT_euc_fork \romannumeral0\xintiabs {#2}\Z #1\Z
}%
%    \end{macrocode}
% \lverb|Ici #3#4=A, #1#2=B|
%    \begin{macrocode}
\def\XINT_euc_fork #1#2\Z #3#4\Z
{%
    \xint_UDzerofork
      #1\XINT_euc_BisZero
      #3\XINT_euc_AisZero
       0\XINT_euc_a
    \krof
    {0}{#1#2}{#3#4}{{#3#4}{#1#2}}{}\Z
}%
%    \end{macrocode}
% \lverb|&
% Le {} pour protéger {{A}{B}} si on s'arrête après une étape (B divise
% A).
% On va renvoyer:$\ 
% {N}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0}|
%    \begin{macrocode}
\def\XINT_euc_AisZero #1#2#3#4#5#6{ {1}{0}{#2}{#2}{0}{0}}%
\def\XINT_euc_BisZero #1#2#3#4#5#6{ {1}{0}{#3}{#3}{0}{0}}%
%    \end{macrocode}
% \lverb|&
% {n}{rn}{an}{{qn}{rn}}...{{A}{B}}{}\Z$\ 
%  a(n) = r(n-1). Pour n=0 on a juste {0}{B}{A}{{A}{B}}{}\Z$\ 
% \XINT_div_prepare {u}{v} divise v par u|
%    \begin{macrocode}
\def\XINT_euc_a #1#2#3%
{%
    \expandafter\XINT_euc_b
    \expandafter {\the\numexpr #1+1\expandafter }%
    \romannumeral0\XINT_div_prepare {#2}{#3}{#2}%
}%
%    \end{macrocode}
% \lverb|{n+1}{q(n+1)}{r(n+1)}{rn}{{qn}{rn}}...|
%    \begin{macrocode}
\def\XINT_euc_b #1#2#3#4%
{%
    \XINT_euc_c #3\Z {#1}{#3}{#4}{{#2}{#3}}%
}%
%    \end{macrocode}
% \lverb|r(n+1)\Z {n+1}{r(n+1)}{r(n)}{{q(n+1)}{r(n+1)}}{{qn}{rn}}...$\ 
% Test si r(n+1) est nul.|
%    \begin{macrocode}
\def\XINT_euc_c #1#2\Z
{%
    \xint_gob_til_zero #1\xint_euc_end0\XINT_euc_a
}%
%    \end{macrocode}
% \lverb|&
% {n+1}{r(n+1)}{r(n)}{{q(n+1)}{r(n+1)}}...{}\Z
% Ici r(n+1) = 0. On arrête on se prépare à inverser
% {n+1}{0}{r(n)}{{q(n+1)}{r(n+1)}}.....{{q1}{r1}}{{A}{B}}{}\Z$\ 
% On veut renvoyer: {N=n+1}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0}|
%    \begin{macrocode}
\def\xint_euc_end0\XINT_euc_a #1#2#3#4\Z%
{%
    \expandafter\xint_euc_end_
    \romannumeral0%
    \XINT_rord_main {}#4{{#1}{#3}}%
    \xint_relax
      \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_bye\xint_bye\xint_bye\xint_bye
    \xint_relax
}%
\edef\xint_euc_end_ #1#2#3%
{%
    \space {#1}{#3}{#2}%
}%
%    \end{macrocode}
% \subsection{\csh{xintBezoutAlgorithm}}
% \lverb|&
% Pour Bezout: objectif, renvoyer$\ 
% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1}$\ 
%       {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}$\ 
% alpha0=1, beta0=0, alpha(-1)=0, beta(-1)=1|
%    \begin{macrocode}
\def\xintBezoutAlgorithm {\romannumeral0\xintbezoutalgorithm }%
\def\xintbezoutalgorithm #1%
{%
    \expandafter \XINT_bezalg \expandafter{\romannumeral0\xintiabs {#1}}%
}%
\def\XINT_bezalg #1#2%
{%
    \expandafter\XINT_bezalg_fork \romannumeral0\xintiabs {#2}\Z #1\Z
}%
%    \end{macrocode}
% \lverb|Ici #3#4=A, #1#2=B|
%    \begin{macrocode}
\def\XINT_bezalg_fork #1#2\Z #3#4\Z
{%
    \xint_UDzerofork
      #1\XINT_bezalg_BisZero
      #3\XINT_bezalg_AisZero
       0\XINT_bezalg_a
    \krof
    0{#1#2}{#3#4}1001{{#3#4}{#1#2}}{}\Z
}%
\def\XINT_bezalg_AisZero #1#2#3\Z{ {1}{0}{0}{1}{#2}{#2}{1}{0}{0}{0}{0}{1}}%
\def\XINT_bezalg_BisZero #1#2#3#4\Z{ {1}{0}{0}{1}{#3}{#3}{1}{0}{0}{0}{0}{1}}%
%    \end{macrocode}
% \lverb|&
% pour préparer l'étape n+1 il faut
% {n}{r(n)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}&
%                    {{q(n)}{r(n)}{alpha(n)}{beta(n)}}...
% division de #3 par #2|
%    \begin{macrocode}
\def\XINT_bezalg_a #1#2#3%
{%
    \expandafter\XINT_bezalg_b
    \expandafter {\the\numexpr #1+1\expandafter }%
    \romannumeral0\XINT_div_prepare {#2}{#3}{#2}%
}%
%    \end{macrocode}
% \lverb|&
% {n+1}{q(n+1)}{r(n+1)}{r(n)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}...|
%    \begin{macrocode}
\def\XINT_bezalg_b #1#2#3#4#5#6#7#8%
{%
    \expandafter\XINT_bezalg_c\expandafter
     {\romannumeral0\xintiiadd {\xintiiMul {#6}{#2}}{#8}}%
     {\romannumeral0\xintiiadd {\xintiiMul {#5}{#2}}{#7}}%
     {#1}{#2}{#3}{#4}{#5}{#6}%
}%
%    \end{macrocode}
% \lverb|&
% {beta(n+1)}{alpha(n+1)}{n+1}{q(n+1)}{r(n+1)}{r(n)}{alpha(n)}{beta(n}}|
%    \begin{macrocode}
\def\XINT_bezalg_c #1#2#3#4#5#6%
{%
    \expandafter\XINT_bezalg_d\expandafter {#2}{#3}{#4}{#5}{#6}{#1}%
}%
%    \end{macrocode}
% \lverb|{alpha(n+1)}{n+1}{q(n+1)}{r(n+1)}{r(n)}{beta(n+1)}|
%    \begin{macrocode}
\def\XINT_bezalg_d #1#2#3#4#5#6#7#8%
{%
    \XINT_bezalg_e #4\Z {#2}{#4}{#5}{#1}{#6}{#7}{#8}{{#3}{#4}{#1}{#6}}%
}%
%    \end{macrocode}
% \lverb|r(n+1)\Z {n+1}{r(n+1)}{r(n)}{alpha(n+1)}{beta(n+1)}$\ 
%                              {alpha(n)}{beta(n)}{q,r,alpha,beta(n+1)}$\ 
% Test si r(n+1) est nul.|
%    \begin{macrocode}
\def\XINT_bezalg_e #1#2\Z
{%
    \xint_gob_til_zero #1\xint_bezalg_end0\XINT_bezalg_a
}%
%    \end{macrocode}
% \lverb|&
% Ici r(n+1) = 0. On arrête on se prépare à inverser.$\ 
% {n+1}{r(n+1)}{r(n)}{alpha(n+1)}{beta(n+1)}{alpha(n)}{beta(n)}$\ 
%                     {q,r,alpha,beta(n+1)}...{{A}{B}}{}\Z$\ 
% On veut renvoyer$\ 
% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1}$\ 
%       {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}|
%    \begin{macrocode}
\def\xint_bezalg_end0\XINT_bezalg_a #1#2#3#4#5#6#7#8\Z
{%
    \expandafter\xint_bezalg_end_
    \romannumeral0%
    \XINT_rord_main {}#8{{#1}{#3}}%
    \xint_relax
      \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_bye\xint_bye\xint_bye\xint_bye
    \xint_relax
}%
%    \end{macrocode}
% \lverb|&
% {N}{D}{A}{B}{q1}{r1}{alpha1=q1}{beta1=1}{q2}{r2}{alpha2}{beta2}$\ 
%      ....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}$\ 
% On veut renvoyer$\ 
% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1}$\ 
%        {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}|
%    \begin{macrocode}
\edef\xint_bezalg_end_ #1#2#3#4%
{%
    \space {#1}{#3}{0}{1}{#2}{#4}{1}{0}%
}%
%    \end{macrocode}
% \subsection{\csh{xintGCDof}}
% \lverb|New with 1.09a. I also tried an optimization (not working two by two)
% which I thought was clever but
% it seemed to be less efficient ...|
%    \begin{macrocode}
\def\xintGCDof      {\romannumeral0\xintgcdof }%
\def\xintgcdof    #1{\expandafter\XINT_gcdof_a\romannumeral-`0#1\relax }%
\def\XINT_gcdof_a #1{\expandafter\XINT_gcdof_b\romannumeral-`0#1\Z }%
\def\XINT_gcdof_b #1\Z #2{\expandafter\XINT_gcdof_c\romannumeral-`0#2\Z {#1}\Z}%
\def\XINT_gcdof_c #1{\xint_gob_til_relax #1\XINT_gcdof_e\relax\XINT_gcdof_d #1}%
\def\XINT_gcdof_d #1\Z {\expandafter\XINT_gcdof_b\romannumeral0\xintgcd {#1}}%
\def\XINT_gcdof_e #1\Z #2\Z { #2}%
%    \end{macrocode}
% \subsection{\csh{xintLCMof}}
% \lverb|New with 1.09a|
%    \begin{macrocode}
\def\xintLCMof      {\romannumeral0\xintlcmof }%
\def\xintlcmof    #1{\expandafter\XINT_lcmof_a\romannumeral-`0#1\relax }%
\def\XINT_lcmof_a #1{\expandafter\XINT_lcmof_b\romannumeral-`0#1\Z }%
\def\XINT_lcmof_b #1\Z #2{\expandafter\XINT_lcmof_c\romannumeral-`0#2\Z {#1}\Z}%
\def\XINT_lcmof_c #1{\xint_gob_til_relax #1\XINT_lcmof_e\relax\XINT_lcmof_d #1}%
\def\XINT_lcmof_d #1\Z {\expandafter\XINT_lcmof_b\romannumeral0\xintlcm {#1}}%
\def\XINT_lcmof_e #1\Z #2\Z { #2}%
%    \end{macrocode}
% \subsection{\csh{xintTypesetEuclideAlgorithm}}
% \lverb|&
% TYPESETTING
%
% Organisation:
%
% {N}{A}{D}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0}$\ 
% \U1 = N = nombre d'étapes, \U3 = PGCD, \U2 = A, \U4=B
% q1 = \U5, q2 = \U7 --> qn = \U<2n+3>, rn = \U<2n+4>
% bn = rn. B = r0. A=r(-1)
%
% r(n-2) = q(n)r(n-1)+r(n) (n e étape)
%
% \U{2n} = \U{2n+3} \times \U{2n+2} + \U{2n+4}, n e étape.
% (avec n entre 1 et N)
%
% 1.09h uses \xintloop, and \par rather than \endgraf; and \par rather than
% \hfill\break|
%    \begin{macrocode}
\def\xintTypesetEuclideAlgorithm {%
    \unless\ifdefined\xintAssignArray
       \errmessage
        {xintgcd: package xinttools is required for \string\xintTypesetEuclideAlgorithm}%
       \expandafter\xint_gobble_iii
    \fi
    \XINT_TypesetEuclideAlgorithm
}%
\def\XINT_TypesetEuclideAlgorithm #1#2%
{% l'algo remplace #1 et #2 par |#1| et |#2|
  \par
  \begingroup
    \xintAssignArray\xintEuclideAlgorithm {#1}{#2}\to\U
    \edef\A{\U2}\edef\B{\U4}\edef\N{\U1}%
    \setbox 0 \vbox{\halign {$##$\cr \A\cr \B \cr}}%
    \count 255 1
    \xintloop
      \indent\hbox to \wd 0 {\hfil$\U{\numexpr 2*\count255\relax}$}%
      ${} =  \U{\numexpr 2*\count255 + 3\relax}
      \times \U{\numexpr 2*\count255 + 2\relax}
          +  \U{\numexpr 2*\count255 + 4\relax}$%
    \ifnum \count255 < \N
      \par
      \advance \count255 1
    \repeat
  \endgroup
}%
%    \end{macrocode}
% \subsection{\csh{xintTypesetBezoutAlgorithm}}
% \lverb|&
% Pour Bezout on a:
% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1}$\ 
%             {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}%
% Donc 4N+8 termes:
% U1 = N, U2= A, U5=D, U6=B, q1 = U9, qn = U{4n+5}, n au moins 1$\ 
% rn = U{4n+6}, n au moins -1$\ 
% alpha(n) = U{4n+7}, n au moins -1$\ 
% beta(n)  = U{4n+8}, n au moins -1
%
% 1.09h uses \xintloop, and \par rather than \endgraf; and no more \parindent0pt
% |
%    \begin{macrocode}
\def\xintTypesetBezoutAlgorithm {%
    \unless\ifdefined\xintAssignArray
       \errmessage
        {xintgcd: package xinttools is required for \string\xintTypesetBezoutAlgorithm}%
       \expandafter\xint_gobble_iii
    \fi
    \XINT_TypesetBezoutAlgorithm
}%
\def\XINT_TypesetBezoutAlgorithm #1#2%
{%
  \par
  \begingroup
    \xintAssignArray\xintBezoutAlgorithm {#1}{#2}\to\BEZ
    \edef\A{\BEZ2}\edef\B{\BEZ6}\edef\N{\BEZ1}% A = |#1|, B = |#2|
    \setbox 0 \vbox{\halign {$##$\cr \A\cr \B \cr}}%
    \count255 1
    \xintloop
      \indent\hbox to \wd 0 {\hfil$\BEZ{4*\count255 - 2}$}%
      ${} =  \BEZ{4*\count255 + 5}
      \times \BEZ{4*\count255 + 2}
          +  \BEZ{4*\count255 + 6}$\hfill\break
      \hbox to \wd 0 {\hfil$\BEZ{4*\count255 +7}$}%
      ${} = \BEZ{4*\count255 + 5}
      \times \BEZ{4*\count255 + 3}
          +  \BEZ{4*\count255 - 1}$\hfill\break
      \hbox to \wd 0 {\hfil$\BEZ{4*\count255 +8}$}%
      ${} =  \BEZ{4*\count255 + 5}
      \times \BEZ{4*\count255 + 4}
          +  \BEZ{4*\count255 }$
      \par
    \ifnum \count255 < \N
    \advance \count255 1
  \repeat
    \edef\U{\BEZ{4*\N + 4}}%
    \edef\V{\BEZ{4*\N + 3}}%
    \edef\D{\BEZ5}%
    \ifodd\N
       $\U\times\A  - \V\times \B = -\D$%
    \else
       $\U\times\A - \V\times\B = \D$%
    \fi
    \par
  \endgroup
}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xintgcd>\relax
%\def<*xintfrac>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xintgcd>
%<*xintfrac>
%
% \StoreCodelineNo {xintgcd}
%
% \section{Package \xintfracnameimp implementation}
% \label{sec:fracimp}
%
% \localtableofcontents
%
% The commenting is currently (\xintdocdate) very sparse.
%
% \subsection{Catcodes, \protect\eTeX{} and reload detection}
%
% The code for reload detection was initially copied from \textsc{Heiko
% Oberdiek}'s packages, then modified.
%
% The method for catcodes was also initially directly inspired by these
% packages.
%
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode64=11   % @
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \let\z\endgroup
  \expandafter\let\expandafter\x\csname ver@xintfrac.sty\endcsname
  \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
  \ifx\csname numexpr\endcsname\relax
     \y{xintfrac}{\numexpr not available, aborting input}%
     \aftergroup\endinput
  \else
    \ifx\x\relax   % plain-TeX, first loading of xintfrac.sty
      \ifx\w\relax % but xint.sty not yet loaded.
         \def\z{\endgroup\input xint.sty\relax}%
      \fi
    \else
      \def\empty {}%
      \ifx\x\empty % LaTeX, first loading,
      % variable is initialized, but \ProvidesPackage not yet seen
          \ifx\w\relax % xint.sty not yet loaded.
            \def\z{\endgroup\RequirePackage{xint}}%
          \fi
      \else
        \aftergroup\endinput % xintfrac already loaded.
      \fi
    \fi
  \fi
\z%
\XINTsetupcatcodes% defined in xintkernel.sty
%    \end{macrocode}
% \subsection{Package identification}
%    \begin{macrocode}
\XINT_providespackage
\ProvidesPackage{xintfrac}%
  [2015/08/31 v1.1b Expandable operations on fractions (jfB)]%
\chardef\xint_c_xviii 18
%    \end{macrocode}
% \subsection{\csh{XINT_cntSgnFork}}
% \lverb|1.09i. Used internally, #1 must expand to \m@ne, \z@, or \@ne or
% equivalent. Does not insert a space token to stop a romannumeral0 expansion.|
%    \begin{macrocode}
\def\XINT_cntSgnFork #1%
{%
    \ifcase #1\expandafter\xint_secondofthree
            \or\expandafter\xint_thirdofthree
          \else\expandafter\xint_firstofthree
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintLen}}
%    \begin{macrocode}
\def\xintLen {\romannumeral0\xintlen }%
\def\xintlen #1%
{%
    \expandafter\XINT_flen\romannumeral0\XINT_infrac {#1}%
}%
\def\XINT_flen #1#2#3%
{%
    \expandafter\space
    \the\numexpr -1+\XINT_Abs {#1}+\XINT_Len {#2}+\XINT_Len {#3}\relax
}%
%    \end{macrocode}
% \subsection{\csh{XINT_lenrord_loop}}
%    \begin{macrocode}
\def\XINT_lenrord_loop #1#2#3#4#5#6#7#8#9%
{%  faire \romannumeral-`0\XINT_lenrord_loop 0{}#1\Z\W\W\W\W\W\W\W\Z
    \xint_gob_til_W #9\XINT_lenrord_W\W
    \expandafter\XINT_lenrord_loop\expandafter
    {\the\numexpr #1+7}{#9#8#7#6#5#4#3#2}%
}%
\def\XINT_lenrord_W\W\expandafter\XINT_lenrord_loop\expandafter #1#2#3\Z
{%
    \expandafter\XINT_lenrord_X\expandafter {#1}#2\Z
}%
\def\XINT_lenrord_X #1#2\Z
{%
    \XINT_lenrord_Y #2\R\R\R\R\R\R\T {#1}%
}%
\def\XINT_lenrord_Y #1#2#3#4#5#6#7#8\T
{%
    \xint_gob_til_W
            #7\XINT_lenrord_Z \xint_c_viii
            #6\XINT_lenrord_Z \xint_c_vii
            #5\XINT_lenrord_Z \xint_c_vi
            #4\XINT_lenrord_Z \xint_c_v
            #3\XINT_lenrord_Z \xint_c_iv
            #2\XINT_lenrord_Z \xint_c_iii
            \W\XINT_lenrord_Z \xint_c_ii   \Z
}%
\def\XINT_lenrord_Z #1#2\Z #3% retourne: {longueur}renverse\Z
{%
    \expandafter{\the\numexpr #3-#1\relax}%
}%
%    \end{macrocode}
% \subsection{\csh{XINT_outfrac}}
% \lverb|&
% 1.06a version now outputs 0/1[0] and not 0[0] in case of zero. More generally
% all macros have been checked in xintfrac, xintseries, xintcfrac, to make sure
% the output format for fractions was always A/B[n]. (except \xintIrr,
% \xintJrr, \xintRawWithZeros)
%
% The problem with statements like those in the previous paragraph is that it is
% hard to maintain consistencies across relases. 
%
% Months later (2014/10/22): perhaps I should document what this macro does
% before I forget?  from {e}{N}{D} it outputs N/D[e], checking in passing if
% D=0 or if N=0. It also makes sure D is not < 0. I am not sure but I don't
% think there is any place in the code which could call \XINT_outfrac with a D
% < 0, but I should check.|
%    \begin{macrocode}
\def\XINT_outfrac #1#2#3%
{%
    \ifcase\XINT_cntSgn #3\Z
        \expandafter \XINT_outfrac_divisionbyzero
    \or
        \expandafter \XINT_outfrac_P
    \else
        \expandafter \XINT_outfrac_N
    \fi
    {#2}{#3}[#1]%
}%
\def\XINT_outfrac_divisionbyzero #1#2{\xintError:DivisionByZero\space #1/0}%
\edef\XINT_outfrac_P #1#2%
{%
    \noexpand\if0\noexpand\XINT_Sgn #1\noexpand\Z
        \noexpand\expandafter\noexpand\XINT_outfrac_Zero
    \noexpand\fi
    \space #1/#2%
}%
\def\XINT_outfrac_Zero #1[#2]{ 0/1[0]}%
\def\XINT_outfrac_N #1#2%
{%
    \expandafter\XINT_outfrac_N_a\expandafter
    {\romannumeral0\XINT_opp #2}{\romannumeral0\XINT_opp #1}%
}%
\def\XINT_outfrac_N_a #1#2%
{%
    \expandafter\XINT_outfrac_P\expandafter {#2}{#1}%
}%
%    \end{macrocode}
% \subsection{\csh{XINT_inFrac}}
% \lverb|Extended in 1.07 to accept scientific notation on input. With lowercase
% e only. The \xintexpr parser does accept uppercase E also. Ah, by the way,
% perhaps I should at least say what this macro does? (belated addition
% 2014/10/22...), before I forget! It prepares the fraction in the internal
% format {exponent}{Numerator}{Denominator} where Denominator is at least 1.|
%    \begin{macrocode}
\def\XINT_inFrac {\romannumeral0\XINT_infrac }%
\def\XINT_infrac #1%
{%
    \expandafter\XINT_infrac_ \romannumeral-`0#1[\W]\Z\T
}%
\def\XINT_infrac_ #1[#2#3]#4\Z
{%
    \xint_UDwfork
      #2\XINT_infrac_A
      \W\XINT_infrac_B
    \krof
    #1[#2#3]#4%
}%
\def\XINT_infrac_A #1[\W]\T
{%
    \XINT_frac #1/\W\Z
}%
\def\XINT_infrac_B #1%
{%
    \xint_gob_til_zero #1\XINT_infrac_Zero0\XINT_infrac_BB #1%
}%
\def\XINT_infrac_BB #1[\W]\T {\XINT_infrac_BC #1/\W\Z }%
\def\XINT_infrac_BC #1/#2#3\Z
{%
    \xint_UDwfork
     #2\XINT_infrac_BCa
     \W{\expandafter\XINT_infrac_BCb \romannumeral-`0#2}%
    \krof
    #3\Z #1\Z
}%
\def\XINT_infrac_BCa \Z #1[#2]#3\Z { {#2}{#1}{1}}%
\def\XINT_infrac_BCb #1[#2]/\W\Z #3\Z { {#2}{#3}{#1}}%
\def\XINT_infrac_Zero #1\T { {0}{0}{1}}%
%    \end{macrocode}
% \subsection{\csh{XINT_frac}}
% \lverb|Extended in 1.07 to recognize and accept scientific notation both at
% the numerator and (possible) denominator. Only a lowercase e will do here, but
% uppercase E is possible within an \xintexpr..\relax |
%    \begin{macrocode}
\def\XINT_frac #1/#2#3\Z
{%
    \xint_UDwfork
     #2\XINT_frac_A
     \W{\expandafter\XINT_frac_U \romannumeral-`0#2}%
    \krof
    #3e\W\Z #1e\W\Z
}%
\def\XINT_frac_U #1e#2#3\Z
{%
    \xint_UDwfork
      #2\XINT_frac_Ua
      \W{\XINT_frac_Ub #2}%
    \krof
    #3\Z #1\Z
}%
\def\XINT_frac_Ua      \Z #1/\W\Z  {\XINT_frac_B #1.\W\Z {0}}%
\def\XINT_frac_Ub #1/\W e\W\Z #2\Z {\XINT_frac_B #2.\W\Z {#1}}%
\def\XINT_frac_B #1.#2#3\Z
{%
    \xint_UDwfork
      #2\XINT_frac_Ba
      \W{\XINT_frac_Bb #2}%
    \krof
    #3\Z #1\Z
}%
\def\XINT_frac_Ba \Z #1\Z {\XINT_frac_T {0}{#1}}%
\def\XINT_frac_Bb #1.\W\Z #2\Z
{%
    \expandafter \XINT_frac_T \expandafter
    {\romannumeral0\xintlength {#1}}{#2#1}%
}%
\def\XINT_frac_A e\W\Z {\XINT_frac_T {0}{1}{0}}%
\def\XINT_frac_T #1#2#3#4e#5#6\Z
{%
    \xint_UDwfork
      #5\XINT_frac_Ta
      \W{\XINT_frac_Tb #5}%
    \krof
    #6\Z #4\Z {#1}{#2}{#3}%
}%
\def\XINT_frac_Ta \Z #1\Z      {\XINT_frac_C #1.\W\Z {0}}%
\def\XINT_frac_Tb #1e\W\Z #2\Z {\XINT_frac_C #2.\W\Z {#1}}%
\def\XINT_frac_C #1.#2#3\Z
{%
    \xint_UDwfork
      #2\XINT_frac_Ca
      \W{\XINT_frac_Cb #2}%
    \krof
    #3\Z #1\Z
}%
\def\XINT_frac_Ca \Z #1\Z {\XINT_frac_D {0}{#1}}%
\def\XINT_frac_Cb #1.\W\Z #2\Z
{%
    \expandafter\XINT_frac_D\expandafter
    {\romannumeral0\xintlength {#1}}{#2#1}%
}%
\def\XINT_frac_D #1#2#3#4#5#6%
{%
    \expandafter \XINT_frac_E \expandafter
    {\the\numexpr -#1+#3+#4-#6\expandafter}\expandafter
    {\romannumeral0\XINT_num_loop #2%
     \xint_relax\xint_relax\xint_relax\xint_relax
     \xint_relax\xint_relax\xint_relax\xint_relax\Z }%
    {\romannumeral0\XINT_num_loop #5%
     \xint_relax\xint_relax\xint_relax\xint_relax
     \xint_relax\xint_relax\xint_relax\xint_relax\Z }%
}%
\def\XINT_frac_E #1#2#3%
{%
   \expandafter \XINT_frac_F  #3\Z {#2}{#1}%
}%
\def\XINT_frac_F #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_frac_Gdivisionbyzero
      0#1\XINT_frac_Gneg
       0-{\XINT_frac_Gpos #1}%
    \krof
}%
\edef\XINT_frac_Gdivisionbyzero #1\Z #2#3%
{%
   \noexpand\xintError:DivisionByZero\space {0}{#2}{0}%
}%
\def\XINT_frac_Gneg #1\Z #2#3%
{%
    \expandafter\XINT_frac_H \expandafter{\romannumeral0\XINT_opp #2}{#3}{#1}%
}%
\def\XINT_frac_H #1#2{ {#2}{#1}}%
\def\XINT_frac_Gpos #1\Z #2#3{ {#3}{#2}{#1}}%
%    \end{macrocode}
% \subsection{\csh{XINT_factortens}, \csh{XINT_cuz_cnt}}
%    \begin{macrocode}
\def\XINT_factortens #1%
{%
    \expandafter\XINT_cuz_cnt_loop\expandafter
    {\expandafter}\romannumeral0\XINT_rord_main {}#1%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
    \R\R\R\R\R\R\R\R\Z
}%
\def\XINT_cuz_cnt #1%
{%
    \XINT_cuz_cnt_loop {}#1\R\R\R\R\R\R\R\R\Z
}%
\def\XINT_cuz_cnt_loop #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_R #9\XINT_cuz_cnt_toofara \R
    \expandafter\XINT_cuz_cnt_checka\expandafter
    {\the\numexpr #1+8\relax}{#2#3#4#5#6#7#8#9}%
}%
\def\XINT_cuz_cnt_toofara\R
    \expandafter\XINT_cuz_cnt_checka\expandafter #1#2%
{%
    \XINT_cuz_cnt_toofarb {#1}#2%
}%
\def\XINT_cuz_cnt_toofarb #1#2\Z {\XINT_cuz_cnt_toofarc #2\Z {#1}}%
\def\XINT_cuz_cnt_toofarc #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_R #2\XINT_cuz_cnt_toofard 7%
            #3\XINT_cuz_cnt_toofard 6%
            #4\XINT_cuz_cnt_toofard 5%
            #5\XINT_cuz_cnt_toofard 4%
            #6\XINT_cuz_cnt_toofard 3%
            #7\XINT_cuz_cnt_toofard 2%
            #8\XINT_cuz_cnt_toofard 1%
            \Z #1#2#3#4#5#6#7#8%
}%
\def\XINT_cuz_cnt_toofard #1#2\Z #3\R #4\Z #5%
{%
    \expandafter\XINT_cuz_cnt_toofare
    \the\numexpr #3\relax \R\R\R\R\R\R\R\R\Z
    {\the\numexpr #5-#1\relax}\R\Z
}%
\def\XINT_cuz_cnt_toofare #1#2#3#4#5#6#7#8%
{%
    \xint_gob_til_R #2\XINT_cuz_cnt_stopc 1%
            #3\XINT_cuz_cnt_stopc 2%
            #4\XINT_cuz_cnt_stopc 3%
            #5\XINT_cuz_cnt_stopc 4%
            #6\XINT_cuz_cnt_stopc 5%
            #7\XINT_cuz_cnt_stopc 6%
            #8\XINT_cuz_cnt_stopc 7%
            \Z #1#2#3#4#5#6#7#8%
}%
\def\XINT_cuz_cnt_checka #1#2%
{%
    \expandafter\XINT_cuz_cnt_checkb\the\numexpr #2\relax \Z {#1}%
}%
\def\XINT_cuz_cnt_checkb #1%
{%
    \xint_gob_til_zero #1\expandafter\XINT_cuz_cnt_loop\xint_gob_til_Z
    0\XINT_cuz_cnt_stopa #1%
}%
\def\XINT_cuz_cnt_stopa #1\Z
{%
    \XINT_cuz_cnt_stopb #1\R\R\R\R\R\R\R\R\Z %
}%
\def\XINT_cuz_cnt_stopb #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_R #2\XINT_cuz_cnt_stopc 1%
            #3\XINT_cuz_cnt_stopc 2%
            #4\XINT_cuz_cnt_stopc 3%
            #5\XINT_cuz_cnt_stopc 4%
            #6\XINT_cuz_cnt_stopc 5%
            #7\XINT_cuz_cnt_stopc 6%
            #8\XINT_cuz_cnt_stopc 7%
            #9\XINT_cuz_cnt_stopc 8%
            \Z #1#2#3#4#5#6#7#8#9%
}%
\def\XINT_cuz_cnt_stopc #1#2\Z #3\R #4\Z #5%
{%
    \expandafter\XINT_cuz_cnt_stopd\expandafter
    {\the\numexpr #5-#1}#3%
}%
\def\XINT_cuz_cnt_stopd #1#2\R #3\Z
{%
    \expandafter\space\expandafter
     {\romannumeral0\XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax }{#1}%
}%
%    \end{macrocode}
% \subsection{\csh{xintRaw}}
% \lverb|&
% 1.07: this macro simply prints in a user readable form the fraction after its
% initial scanning. Useful when put inside braces in an \xintexpr, when the
% input is not yet in the A/B[n] form.|
%    \begin{macrocode}
\def\xintRaw {\romannumeral0\xintraw }%
\def\xintraw
{%
    \expandafter\XINT_raw\romannumeral0\XINT_infrac
}%
\def\XINT_raw #1#2#3{ #2/#3[#1]}%
%    \end{macrocode}
% \subsection{\csh{xintPRaw}}
% \lverb|1.09b|
%    \begin{macrocode}
\def\xintPRaw {\romannumeral0\xintpraw }%
\def\xintpraw
{%
    \expandafter\XINT_praw\romannumeral0\XINT_infrac
}%
\def\XINT_praw #1%
{%
    \ifnum #1=\xint_c_ \expandafter\XINT_praw_a\fi \XINT_praw_A {#1}%
}%
\def\XINT_praw_A #1#2#3%
{%
    \if\XINT_isOne{#3}1\expandafter\xint_firstoftwo
                  \else\expandafter\xint_secondoftwo
    \fi { #2[#1]}{ #2/#3[#1]}%
}%
\def\XINT_praw_a\XINT_praw_A #1#2#3%
{%
    \if\XINT_isOne{#3}1\expandafter\xint_firstoftwo
                  \else\expandafter\xint_secondoftwo
    \fi { #2}{ #2/#3}%
}%
%    \end{macrocode}
% \subsection{\csh{xintRawWithZeros}}
% \lverb|&
% This was called \xintRaw in versions earlier than 1.07|
%    \begin{macrocode}
\def\xintRawWithZeros {\romannumeral0\xintrawwithzeros }%
\def\xintrawwithzeros
{%
    \expandafter\XINT_rawz\romannumeral0\XINT_infrac
}%
\def\XINT_rawz #1%
{%
    \ifcase\XINT_cntSgn #1\Z
      \expandafter\XINT_rawz_Ba
    \or
      \expandafter\XINT_rawz_A
    \else
      \expandafter\XINT_rawz_Ba
    \fi
    {#1}%
}%
\def\XINT_rawz_A  #1#2#3{\xint_dsh {#2}{-#1}/#3}%
\def\XINT_rawz_Ba #1#2#3{\expandafter\XINT_rawz_Bb
                        \expandafter{\romannumeral0\xint_dsh {#3}{#1}}{#2}}%
\def\XINT_rawz_Bb #1#2{ #2/#1}%
%    \end{macrocode}
% \subsection{\csh{xintFloor}, \csh{xintiFloor}}
% \lverb|1.09a, 1.1 for \xintiFloor/\xintFloor. Not efficient if big negative
% decimal exponent. Also sub-efficient if big positive decimal exponent.|
%    \begin{macrocode}
\def\xintFloor {\romannumeral0\xintfloor }%
\def\xintfloor #1% devrais-je faire \xintREZ?
    {\expandafter\XINT_ifloor \romannumeral0\xintrawwithzeros {#1}./1[0]}%
\def\xintiFloor {\romannumeral0\xintifloor }%
\def\xintifloor #1%
    {\expandafter\XINT_ifloor \romannumeral0\xintrawwithzeros {#1}.}%
\def\XINT_ifloor #1/#2.{\xintiiquo {#1}{#2}}%
%    \end{macrocode}
% \subsection{\csh{xintCeil}, \csh{xintiCeil}}
% \lverb|1.09a|
%    \begin{macrocode}
\def\xintCeil {\romannumeral0\xintceil }%
\def\xintceil #1{\xintiiopp {\xintFloor {\xintOpp{#1}}}}%
\def\xintiCeil {\romannumeral0\xinticeil }%
\def\xinticeil #1{\xintiiopp {\xintiFloor {\xintOpp{#1}}}}%
%    \end{macrocode}
% \subsection{\csh{xintNumerator}}
%    \begin{macrocode}
\def\xintNumerator {\romannumeral0\xintnumerator }%
\def\xintnumerator
{%
    \expandafter\XINT_numer\romannumeral0\XINT_infrac
}%
\def\XINT_numer #1%
{%
    \ifcase\XINT_cntSgn #1\Z
      \expandafter\XINT_numer_B
    \or
      \expandafter\XINT_numer_A
    \else
      \expandafter\XINT_numer_B
    \fi
    {#1}%
}%
\def\XINT_numer_A #1#2#3{\xint_dsh {#2}{-#1}}%
\def\XINT_numer_B #1#2#3{ #2}%
%    \end{macrocode}
% \subsection{\csh{xintDenominator}}
%    \begin{macrocode}
\def\xintDenominator {\romannumeral0\xintdenominator }%
\def\xintdenominator
{%
    \expandafter\XINT_denom\romannumeral0\XINT_infrac
}%
\def\XINT_denom #1%
{%
    \ifcase\XINT_cntSgn #1\Z
      \expandafter\XINT_denom_B
    \or
      \expandafter\XINT_denom_A
    \else
      \expandafter\XINT_denom_B
    \fi
    {#1}%
}%
\def\XINT_denom_A #1#2#3{ #3}%
\def\XINT_denom_B #1#2#3{\xint_dsh {#3}{#1}}%
%    \end{macrocode}
% \subsection{\csh{xintFrac}}
%    \begin{macrocode}
\def\xintFrac {\romannumeral0\xintfrac }%
\def\xintfrac #1%
{%
    \expandafter\XINT_fracfrac_A\romannumeral0\XINT_infrac {#1}%
}%
\def\XINT_fracfrac_A #1{\XINT_fracfrac_B #1\Z }%
\catcode`^=7
\def\XINT_fracfrac_B #1#2\Z
{%
    \xint_gob_til_zero #1\XINT_fracfrac_C 0\XINT_fracfrac_D {10^{#1#2}}%
}%
\def\XINT_fracfrac_C 0\XINT_fracfrac_D #1#2#3%
{%
    \if1\XINT_isOne {#3}%
        \xint_afterfi {\expandafter\xint_firstoftwo_thenstop\xint_gobble_ii }%
    \fi
    \space
    \frac {#2}{#3}%
}%
\def\XINT_fracfrac_D #1#2#3%
{%
    \if1\XINT_isOne {#3}\XINT_fracfrac_E\fi
    \space
    \frac {#2}{#3}#1%
}%
\def\XINT_fracfrac_E \fi\space\frac #1#2{\fi \space #1\cdot }%
%    \end{macrocode}
% \subsection{\csh{xintSignedFrac}}
%    \begin{macrocode}
\def\xintSignedFrac {\romannumeral0\xintsignedfrac }%
\def\xintsignedfrac #1%
{%
    \expandafter\XINT_sgnfrac_a\romannumeral0\XINT_infrac {#1}%
}%
\def\XINT_sgnfrac_a #1#2%
{%
    \XINT_sgnfrac_b #2\Z {#1}%
}%
\def\XINT_sgnfrac_b #1%
{%
    \xint_UDsignfork
      #1\XINT_sgnfrac_N
       -{\XINT_sgnfrac_P #1}%
    \krof
}%
\def\XINT_sgnfrac_P #1\Z #2%
{%
    \XINT_fracfrac_A {#2}{#1}%
}%
\def\XINT_sgnfrac_N
{%
    \expandafter\xint_minus_thenstop\romannumeral0\XINT_sgnfrac_P
}%
%    \end{macrocode}
% \subsection{\csh{xintFwOver}}
%    \begin{macrocode}
\def\xintFwOver {\romannumeral0\xintfwover }%
\def\xintfwover #1%
{%
    \expandafter\XINT_fwover_A\romannumeral0\XINT_infrac {#1}%
}%
\def\XINT_fwover_A #1{\XINT_fwover_B #1\Z }%
\def\XINT_fwover_B #1#2\Z
{%
    \xint_gob_til_zero #1\XINT_fwover_C 0\XINT_fwover_D {10^{#1#2}}%
}%
\catcode`^=11
\def\XINT_fwover_C #1#2#3#4#5%
{%
    \if0\XINT_isOne {#5}\xint_afterfi { {#4\over #5}}%
                   \else\xint_afterfi { #4}%
    \fi
}%
\def\XINT_fwover_D #1#2#3%
{%
    \if0\XINT_isOne {#3}\xint_afterfi { {#2\over #3}}%
                   \else\xint_afterfi { #2\cdot }%
    \fi
    #1%
}%
%    \end{macrocode}
% \subsection{\csh{xintSignedFwOver}}
%    \begin{macrocode}
\def\xintSignedFwOver {\romannumeral0\xintsignedfwover }%
\def\xintsignedfwover #1%
{%
    \expandafter\XINT_sgnfwover_a\romannumeral0\XINT_infrac {#1}%
}%
\def\XINT_sgnfwover_a #1#2%
{%
    \XINT_sgnfwover_b #2\Z {#1}%
}%
\def\XINT_sgnfwover_b #1%
{%
    \xint_UDsignfork
      #1\XINT_sgnfwover_N
       -{\XINT_sgnfwover_P #1}%
    \krof
}%
\def\XINT_sgnfwover_P #1\Z #2%
{%
    \XINT_fwover_A {#2}{#1}%
}%
\def\XINT_sgnfwover_N
{%
    \expandafter\xint_minus_thenstop\romannumeral0\XINT_sgnfwover_P
}%
%    \end{macrocode}
% \subsection{\csh{xintREZ}}
%    \begin{macrocode}
\def\xintREZ {\romannumeral0\xintrez }%
\def\xintrez
{%
    \expandafter\XINT_rez_A\romannumeral0\XINT_infrac
}%
\def\XINT_rez_A #1#2%
{%
    \XINT_rez_AB #2\Z {#1}%
}%
\def\XINT_rez_AB #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_rez_zero
      0#1\XINT_rez_neg
       0-{\XINT_rez_B #1}%
    \krof
}%
\def\XINT_rez_zero #1\Z #2#3{ 0/1[0]}%
\def\XINT_rez_neg {\expandafter\xint_minus_thenstop\romannumeral0\XINT_rez_B }%
\def\XINT_rez_B #1\Z
{%
    \expandafter\XINT_rez_C\romannumeral0\XINT_factortens {#1}%
}%
\def\XINT_rez_C #1#2#3#4%
{%
    \expandafter\XINT_rez_D\romannumeral0\XINT_factortens {#4}{#3}{#2}{#1}%
}%
\def\XINT_rez_D #1#2#3#4#5%
{%
    \expandafter\XINT_rez_E\expandafter
    {\the\numexpr #3+#4-#2}{#1}{#5}%
}%
\def\XINT_rez_E #1#2#3{ #3/#2[#1]}%
%    \end{macrocode}
% \subsection{\csh{xintE}, \csh{xintFloatE}, \csh{XINTinFloatE}}
% \lverb|1.07: The fraction is the first argument contrarily to \xintTrunc and
% \xintRound.
%
% \xintfE (1.07) and \xintiE (1.09i) are for \xintexpr and cousins. It is quite
% annoying that \numexpr does not know how to deal correctly with a minus sign -
% as prefix: \numexpr -(1)\relax is illegal! (one can do \numexpr 0-(1)\relax).
%
% the 1.07 \xintE puts directly its second argument in a \numexpr. The \xintfE
% first uses \xintNum on it, this is necessary for use in \xintexpr. (but
% one cannot use directly infix notation in the second argument of \xintfE)
%
% 1.09i also adds \xintFloatE and modifies \XINTinFloatfE, although currently
% the latter is only used from \xintfloatexpr hence always with \XINTdigits, it
% comes equipped with its first argument within brackets as the other
% \XINTinFloat... macros.
%
% 1.09m ceases here and elsewhere, also in \xintcfracname, to use \Z as
% delimiter in the code for the optional argument, as this is unsafe (it
% makes impossible to the user to employ \Z as argument to the macro).
% Replaced by \xint_relax. 1.09e had already done that in \xintSeq, but
% this should have been systematic. 
%
% 1.1 modifies and moves \xintiiE to xint.sty, and cleans up some unneeded
% stuff, now that expressions implement scientific notation directly at the
% number parsing level.|
%    \begin{macrocode}
\def\xintE {\romannumeral0\xinte }%
\def\xinte #1%
{%
    \expandafter\XINT_e \romannumeral0\XINT_infrac {#1}%
}%
\def\XINT_e #1#2#3#4%
{%
    \expandafter\XINT_e_end\expandafter{\the\numexpr #1+#4}{#2}{#3}%
}%
\def\XINT_e_end #1#2#3{ #2/#3[#1]}%
\def\xintFloatE   {\romannumeral0\xintfloate }%
\def\xintfloate #1{\XINT_floate_chkopt #1\xint_relax }%
\def\XINT_floate_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_floate_opt
       \else\expandafter\XINT_floate_noopt
    \fi  #1%
}%
\def\XINT_floate_noopt #1\xint_relax
{%
    \expandafter\XINT_floate_a\expandafter\XINTdigits
                \romannumeral0\XINT_infrac {#1}%
}%
\def\XINT_floate_opt [\xint_relax #1]#2%
{%
    \expandafter\XINT_floate_a\expandafter
    {\the\numexpr #1\expandafter}\romannumeral0\XINT_infrac {#2}%
}%
\def\XINT_floate_a #1#2#3#4#5%
{%
    \expandafter\expandafter\expandafter\XINT_float_a
    \expandafter\xint_exchangetwo_keepbraces\expandafter
            {\the\numexpr #2+#5}{#1}{#3}{#4}\XINT_float_Q
}%
\def\XINTinFloatE {\romannumeral0\XINTinfloate }%
\def\XINTinfloate {\expandafter\XINT_infloate\romannumeral0\XINTinfloat [\XINTdigits]}%
\def\XINT_infloate #1[#2]#3%
   {\expandafter\XINT_infloate_end\expandafter {\the\numexpr #3+#2}{#1}}%
\def\XINT_infloate_end #1#2{ #2[#1]}%
%    \end{macrocode}
% \subsection{\csh{xintIrr}}
% \lverb|&
% 1.04 fixes a buggy \xintIrr {0}.
% 1.05 modifies the initial parsing and post-processing to use \xintrawwithzeros
% and to
% more quickly deal with an input denominator equal to 1. 1.08 version does
% not remove a /1 denominator.|
%    \begin{macrocode}
\def\xintIrr {\romannumeral0\xintirr }%
\def\xintirr #1%
{%
    \expandafter\XINT_irr_start\romannumeral0\xintrawwithzeros {#1}\Z
}%
\def\XINT_irr_start #1#2/#3\Z
{%
    \if0\XINT_isOne {#3}%
      \xint_afterfi
          {\xint_UDsignfork
               #1\XINT_irr_negative
                -{\XINT_irr_nonneg #1}%
           \krof}%
    \else
      \xint_afterfi{\XINT_irr_denomisone #1}%
    \fi
    #2\Z {#3}%
}%
\def\XINT_irr_denomisone #1\Z #2{ #1/1}% changed in 1.08
\def\XINT_irr_negative   #1\Z #2{\XINT_irr_D #1\Z #2\Z \xint_minus_thenstop}%
\def\XINT_irr_nonneg     #1\Z #2{\XINT_irr_D #1\Z #2\Z \space}%
\def\XINT_irr_D #1#2\Z #3#4\Z
{%
    \xint_UDzerosfork
       #3#1\XINT_irr_indeterminate
       #30\XINT_irr_divisionbyzero
       #10\XINT_irr_zero
        00\XINT_irr_loop_a
    \krof
    {#3#4}{#1#2}{#3#4}{#1#2}%
}%
\def\XINT_irr_indeterminate #1#2#3#4#5{\xintError:NaN\space 0/0}%
\def\XINT_irr_divisionbyzero #1#2#3#4#5{\xintError:DivisionByZero #5#2/0}%
\def\XINT_irr_zero #1#2#3#4#5{ 0/1}% changed in 1.08
\def\XINT_irr_loop_a #1#2%
{%
    \expandafter\XINT_irr_loop_d
    \romannumeral0\XINT_div_prepare {#1}{#2}{#1}%
}%
\def\XINT_irr_loop_d #1#2%
{%
    \XINT_irr_loop_e #2\Z
}%
\def\XINT_irr_loop_e #1#2\Z
{%
    \xint_gob_til_zero #1\xint_irr_loop_exit0\XINT_irr_loop_a {#1#2}%
}%
\def\xint_irr_loop_exit0\XINT_irr_loop_a #1#2#3#4%
{%
    \expandafter\XINT_irr_loop_exitb\expandafter
    {\romannumeral0\xintiiquo {#3}{#2}}%
    {\romannumeral0\xintiiquo {#4}{#2}}%
}%
\def\XINT_irr_loop_exitb #1#2%
{%
   \expandafter\XINT_irr_finish\expandafter {#2}{#1}%
}%
\def\XINT_irr_finish #1#2#3{#3#1/#2}% changed in 1.08
%    \end{macrocode}
% \subsection{\csh{xintifInt}}
% \lverb|1.09e. xintfrac.sty only. Fixed in 1.1 to not use \xintIrr anymore
% as it was really stupid overhead.|
%    \begin{macrocode}
\def\xintifInt   {\romannumeral0\xintifint }%
\def\xintifint #1{\expandafter\XINT_ifint\romannumeral0\xintrawwithzeros {#1}.}%
\def\XINT_ifint #1/#2.%
{%
    \if 0\xintiiRem {#1}{#2}%
     \expandafter\xint_firstoftwo_thenstop
    \else
     \expandafter\xint_secondoftwo_thenstop
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintJrr}}
% \lverb|&
% Modified similarly as \xintIrr in release 1.05. 1.08 version does
% not remove a /1 denominator.|
%    \begin{macrocode}
\def\xintJrr {\romannumeral0\xintjrr }%
\def\xintjrr #1%
{%
    \expandafter\XINT_jrr_start\romannumeral0\xintrawwithzeros {#1}\Z
}%
\def\XINT_jrr_start #1#2/#3\Z
{%
    \if0\XINT_isOne {#3}\xint_afterfi
          {\xint_UDsignfork
               #1\XINT_jrr_negative
                -{\XINT_jrr_nonneg #1}%
           \krof}%
    \else
      \xint_afterfi{\XINT_jrr_denomisone #1}%
    \fi
    #2\Z {#3}%
}%
\def\XINT_jrr_denomisone #1\Z #2{ #1/1}% changed in 1.08
\def\XINT_jrr_negative   #1\Z #2{\XINT_jrr_D #1\Z #2\Z \xint_minus_thenstop }%
\def\XINT_jrr_nonneg     #1\Z #2{\XINT_jrr_D #1\Z #2\Z \space}%
\def\XINT_jrr_D #1#2\Z #3#4\Z
{%
    \xint_UDzerosfork
       #3#1\XINT_jrr_indeterminate
       #30\XINT_jrr_divisionbyzero
       #10\XINT_jrr_zero
        00\XINT_jrr_loop_a
    \krof
    {#3#4}{#1#2}1001%
}%
\def\XINT_jrr_indeterminate #1#2#3#4#5#6#7{\xintError:NaN\space 0/0}%
\def\XINT_jrr_divisionbyzero #1#2#3#4#5#6#7{\xintError:DivisionByZero #7#2/0}%
\def\XINT_jrr_zero #1#2#3#4#5#6#7{ 0/1}% changed in 1.08
\def\XINT_jrr_loop_a #1#2%
{%
    \expandafter\XINT_jrr_loop_b
    \romannumeral0\XINT_div_prepare {#1}{#2}{#1}%
}%
\def\XINT_jrr_loop_b #1#2#3#4#5#6#7%
{%
    \expandafter \XINT_jrr_loop_c \expandafter
        {\romannumeral0\xintiiadd{\XINT_Mul{#4}{#1}}{#6}}%
        {\romannumeral0\xintiiadd{\XINT_Mul{#5}{#1}}{#7}}%
    {#2}{#3}{#4}{#5}%
}%
\def\XINT_jrr_loop_c #1#2%
{%
    \expandafter \XINT_jrr_loop_d \expandafter{#2}{#1}%
}%
\def\XINT_jrr_loop_d #1#2#3#4%
{%
    \XINT_jrr_loop_e #3\Z {#4}{#2}{#1}%
}%
\def\XINT_jrr_loop_e #1#2\Z
{%
    \xint_gob_til_zero #1\xint_jrr_loop_exit0\XINT_jrr_loop_a {#1#2}%
}%
\def\xint_jrr_loop_exit0\XINT_jrr_loop_a #1#2#3#4#5#6%
{%
    \XINT_irr_finish {#3}{#4}%
}%
%    \end{macrocode}
% \subsection{\csh{xintTFrac}}
% \lverb|1.09i, for frac in \xintexpr. And \xintFrac is already assigned. T for
% truncation. However, potentially not very efficient with numbers in scientific
% notations, with big exponents. Will have to think it again some day. I
% hesitated how to call the macro. Same convention as in maple, but some people
% reserve fractional part to x - floor(x). Also, not clear if I had to make it
% negative (or zero) if x < 0, or rather always positive. There should be in
% fact such a thing for each rounding function, trunc, round, floor, ceil. |
%    \begin{macrocode}
\def\xintTFrac {\romannumeral0\xinttfrac }%
\def\xinttfrac #1{\expandafter\XINT_tfrac_fork\romannumeral0\xintrawwithzeros {#1}\Z }%
\def\XINT_tfrac_fork #1%
{%
    \xint_UDzerominusfork
        #1-\XINT_tfrac_zero
        0#1{\xintiiopp\XINT_tfrac_P }%
         0-{\XINT_tfrac_P #1}%
    \krof
}%
\def\XINT_tfrac_zero #1\Z { 0/1[0]}%
\def\XINT_tfrac_P #1/#2\Z {\expandafter\XINT_rez_AB
                           \romannumeral0\xintiirem{#1}{#2}\Z {0}{#2}}%
%    \end{macrocode}
% \subsection{\csh{XINTinFloatFracdigits}}
% \lverb|1.09i, for frac in \xintfloatexpr. This version computes
% exactly from the input the fractional part and then only converts it
% into a float with the asked-for number of digits. I will have to think
% it again some day, certainly.
%
% 1.1 removes optional argument for which there was anyhow no interface, for
% technical reasons having to do with \xintNewExpr.
%
% 1.1a renames the macro as \XINTinFloatFracdigits (from \XINTinFloatFrac) to
% be synchronous with the \XINTinFloatSqrt and \XINTinFloat habits related to
% \xintNewExpr problems.
%
% Note to myself: I still have to rethink the whole thing about what is the best
% to do, the initial way of going through \xinttfrac was just a first
% implementation.|
%    \begin{macrocode}
\def\XINTinFloatFracdigits {\romannumeral0\XINTinfloatfracdigits }%
\def\XINTinfloatfracdigits #1%
{%
    \expandafter\XINT_infloatfracdg_a\expandafter {\romannumeral0\xinttfrac{#1}}%
}%
\def\XINT_infloatfracdg_a {\XINTinfloat [\XINTdigits]}%
%    \end{macrocode}
% \subsection{\csh{xintTrunc}, \csh{xintiTrunc}}
% \lverb|&
% Modified in 1.06 to give the first argument to a \numexpr.
%
% 1.09f fixes the overhead added in 1.09a to some inner routines when \xintiquo
% was redefined to use \xintnum. Now uses \xintiiquo, rather.
%
% 1.09j: minor improvements, \XINT_trunc_E was very strange and defined two
% never occuring branches; also, optimizes the call to the division routine, and
% the zero loops.
%
% 1.1 adds \xintTTrunc as a shortcut to what \xintiTrunc 0 does, and maps \xintNum to it.|
%    \begin{macrocode}
\def\xintTrunc  {\romannumeral0\xinttrunc }%
\def\xintiTrunc {\romannumeral0\xintitrunc }%
\def\xinttrunc #1%
{%
    \expandafter\XINT_trunc\expandafter {\the\numexpr #1}%
}%
\def\XINT_trunc #1#2%
{%
    \expandafter\XINT_trunc_G
    \romannumeral0\expandafter\XINT_trunc_A
    \romannumeral0\XINT_infrac {#2}{#1}{#1}%
}%
\def\xintitrunc #1%
{%
    \expandafter\XINT_itrunc\expandafter {\the\numexpr #1}%
}%
\def\XINT_itrunc #1#2%
{%
    \expandafter\XINT_itrunc_G
    \romannumeral0\expandafter\XINT_trunc_A
    \romannumeral0\XINT_infrac {#2}{#1}{#1}%
}%
\def\XINT_trunc_A #1#2#3#4%
{%
    \expandafter\XINT_trunc_checkifzero
    \expandafter{\the\numexpr #1+#4}#2\Z {#3}%
}%
\def\XINT_trunc_checkifzero #1#2#3\Z
{%
    \xint_gob_til_zero #2\XINT_trunc_iszero0\XINT_trunc_B {#1}{#2#3}%
}%
\def\XINT_trunc_iszero0\XINT_trunc_B #1#2#3{ 0\Z 0}%
\def\XINT_trunc_B #1%
{%
    \ifcase\XINT_cntSgn #1\Z
      \expandafter\XINT_trunc_D
    \or
      \expandafter\XINT_trunc_D
    \else
      \expandafter\XINT_trunc_C
    \fi
    {#1}%
}%
\def\XINT_trunc_C #1#2#3%
{%
    \expandafter\XINT_trunc_CE\expandafter
    {\romannumeral0\XINT_dsx_zeroloop {-#1}{}\Z {#3}}{#2}%
}%
\def\XINT_trunc_CE #1#2{\XINT_trunc_E #2.{#1}}%
\def\XINT_trunc_D #1#2%
{%
    \expandafter\XINT_trunc_E
    \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z {#2}.%
}%
\def\XINT_trunc_E #1%
{%
    \xint_UDsignfork
       #1\XINT_trunc_Fneg
        -{\XINT_trunc_Fpos #1}%
    \krof
}%
\def\XINT_trunc_Fneg #1.#2{\expandafter\xint_firstoftwo_thenstop
           \romannumeral0\XINT_div_prepare {#2}{#1}\Z \xint_minus_thenstop}%
\def\XINT_trunc_Fpos #1.#2{\expandafter\xint_firstoftwo_thenstop
           \romannumeral0\XINT_div_prepare {#2}{#1}\Z \space }%
\def\XINT_itrunc_G #1#2\Z #3#4%
{%
    \xint_gob_til_zero #1\XINT_trunc_zero 0#3#1#2%
}%
\def\XINT_trunc_zero 0#1#20{ 0}%
\def\XINT_trunc_G #1\Z #2#3%
{%
    \xint_gob_til_zero #2\XINT_trunc_zero 0%
    \expandafter\XINT_trunc_H\expandafter
    {\the\numexpr\romannumeral0\xintlength {#1}-#3}{#3}{#1}#2%
}%
\def\XINT_trunc_H #1#2%
{%
    \ifnum #1 > \xint_c_
        \xint_afterfi {\XINT_trunc_Ha {#2}}%
    \else
        \xint_afterfi {\XINT_trunc_Hb {-#1}}% -0,--1,--2, ....
    \fi
}%
\def\XINT_trunc_Ha
{%
  \expandafter\XINT_trunc_Haa\romannumeral0\xintdecsplit
}%
\def\XINT_trunc_Haa #1#2#3%
{%
    #3#1.#2%
}%
\def\XINT_trunc_Hb #1#2#3%
{%
    \expandafter #3\expandafter0\expandafter.%
    \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z {}#2% #1=-0 autoris\'e !
}%
%    \end{macrocode}
% \subsection{\csh{xintTTrunc}}
% \lverb|1.1, a tiny bit more efficient than doing \xintiTrunc0. I map \xintNum
% to it, and I use it in \xintexpr for various things. Faster I guess than the \xintiFloor.|
%    \begin{macrocode}
\def\xintTTrunc {\romannumeral0\xintttrunc }%
\def\xintttrunc #1%
{%
    \expandafter\XINT_itrunc_G
    \romannumeral0\expandafter\XINT_ttrunc_A
    \romannumeral0\XINT_infrac {#1}0% this last 0 to let \XINT_itrunc_G be happy
}%
\def\XINT_ttrunc_A #1#2#3{\XINT_trunc_checkifzero {#1}#2\Z {#3}}%
%    \end{macrocode}
% \subsection{\csh{xintNum}}
% \lverb|This extension of the xint original xintNum is added in 1.05, as a
% synonym to \xintIrr, but raising an error when the input does not evaluate to
% an integer. Usable with not too much overhead on integer input as \xintIrr
% checks quickly for a denominator equal to 1 (which will be put there by the
% \XINT_infrac called by \xintrawwithzeros). This way, macros such as \xintQuo
% can be modified with minimal overhead to accept fractional input as long as
% it evaluates to an integer.
%
% 22 june 2014 (dev 1.1) I just don't understand what was the point of going through
% \xintIrr if to raise an arror afterwards...  and raising errors is silly, so
% let's do it sanely at last. In between I added \xintiFloor, thus, let's just
% let it to it.
%
% 24 october 2014 (final 1.1) (I left it taking dust since
% June...), I did \xintTTrunc, and will thus map \xintNum to it|
%    \begin{macrocode}
\let\xintNum \xintTTrunc
\let\xintnum \xintttrunc
%    \end{macrocode}
% \subsection{\csh{xintRound}, \csh{xintiRound}}
% \lverb|Modified in 1.06 to give the first argument to a \numexpr.|
%    \begin{macrocode}
\def\xintRound {\romannumeral0\xintround }%
\def\xintiRound {\romannumeral0\xintiround }%
\def\xintround #1%
{%
    \expandafter\XINT_round\expandafter {\the\numexpr #1}%
}%
\def\XINT_round
{%
    \expandafter\XINT_trunc_G\romannumeral0\XINT_round_A
}%
\def\xintiround #1%
{%
    \expandafter\XINT_iround\expandafter {\the\numexpr #1}%
}%
\def\XINT_iround
{%
    \expandafter\XINT_itrunc_G\romannumeral0\XINT_round_A
}%
\def\XINT_round_A #1#2%
{%
    \expandafter\XINT_round_B
    \romannumeral0\expandafter\XINT_trunc_A
    \romannumeral0\XINT_infrac {#2}{\the\numexpr #1+1\relax}{#1}%
}%
\def\XINT_round_B #1\Z
{%
    \expandafter\XINT_round_C
    \romannumeral0\XINT_rord_main {}#1%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax
    \Z
}%
\def\XINT_round_C #1%
{%
    \ifnum #1<5
        \expandafter\XINT_round_Daa
    \else
        \expandafter\XINT_round_Dba
    \fi
}%
\def\XINT_round_Daa #1%
{%
    \xint_gob_til_Z #1\XINT_round_Daz\Z \XINT_round_Da #1%
}%
\def\XINT_round_Daz\Z \XINT_round_Da \Z { 0\Z }%
\def\XINT_round_Da #1\Z
{%
    \XINT_rord_main {}#1%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax  \Z
}%
\def\XINT_round_Dba #1%
{%
    \xint_gob_til_Z #1\XINT_round_Dbz\Z \XINT_round_Db #1%
}%
\def\XINT_round_Dbz\Z \XINT_round_Db \Z { 1\Z }%
\def\XINT_round_Db #1\Z
{%
    \XINT_addm_A 0{}1000\W\X\Y\Z #1000\W\X\Y\Z \Z
}%
%    \end{macrocode}
% \subsection{\csh{xintXTrunc}}
% \lverb@1.09j [2014/01/06] This is completely expandable but not f-expandable.
% Designed be used inside an \edef or a \write, if one is interested in getting
% tens of thousands of digits from the decimal expansion of some fraction... it
% is not worth using it rather than \xintTrunc if for less than *hundreds* of
% digits. For efficiency it clones part of the preparatory division macros, as
% the same denominator will be used again and again. The D parameter which says
% how many digits to keep after decimal mark must be at least 1 (and it is
% forcefully set to such a value if found negative or zero, to avoid an eternal
% loop).
%
% For reasons of efficiency I try to use the shortest possible denominator, so
% if the fraction is A/B[N], I want to use B. For N at least zero, just
% immediately replace A by A.10^N. The first division then may be a little
% longish but the next ones will be fast (if B is not too big). For N<0, this is
% a bit more complicated. I thought somewhat about this, and I would need a
% rather complicated approach going through a long division algorithm, forcing
% me to essentially clone the actual division with some differences; a side
% thing is that as this would use blocks of four digits I would have a hard time
% allowing a non-multiple of four number of post decimal mark digits.
%
% Thus, for N<0, another method is followed. First the euclidean division
% A/B=Q+R/B is done. The number of digits of Q is M. If |N|\leq D, we launch
% inside a \csname the routine for obtaining D-|N| next digits (this may impact
% TeX's memory if D is very big), call them T. We then need to position the
% decimal mark D slots from the right of QT, which has length M+D-|N|, hence |N|
% slots from the right of Q. We thus avoid having to work will the T, as D may
% be very very big (\xintXTrunc's only goal is to make it possible to learn by
% hearts decimal expansions with thousands of digits). We can use the
% \xintDecSplit for that on Q . Computing the length M of Q was a more or less
% unavoidable step. If |N|>D, the \csname step is skipped we need to remove the
% D-|N| last digits from Q, etc.. we compare D-|N| with the length M of Q etc...
% (well in this last, very uncommon, branch, I stopped trying to optimize things
% and I even do an \xintnum to ensure a 0 if something comes out empty from
% \xintDecSplit).@
%    \begin{macrocode}
\def\xintXTrunc #1#2%
{%
    \expandafter\XINT_xtrunc_a\expandafter
    {\the\numexpr #1\expandafter}\romannumeral0\xintraw {#2}%
}%
\def\XINT_xtrunc_a #1%
{%
    \expandafter\XINT_xtrunc_b\expandafter
    {\the\numexpr\ifnum#1<\xint_c_i \xint_c_i-\fi #1}%
}%
\def\XINT_xtrunc_b #1%
{%
    \expandafter\XINT_xtrunc_c\expandafter
    {\the\numexpr (#1+\xint_c_ii^v)/\xint_c_ii^vi-\xint_c_i}{#1}%
}%
\def\XINT_xtrunc_c #1#2%
{%
    \expandafter\XINT_xtrunc_d\expandafter
    {\the\numexpr #2-\xint_c_ii^vi*#1}{#1}{#2}%
}%
\def\XINT_xtrunc_d #1#2#3#4/#5[#6]%
{%
    \XINT_xtrunc_e #4.{#6}{#5}{#3}{#2}{#1}%
}%
% #1=numerator.#2=N,#3=B,#4=D,#5=Blocs,#6=extra
\def\XINT_xtrunc_e #1%
{%
    \xint_UDzerominusfork
        #1-\XINT_xtrunc_zero
        0#1\XINT_xtrunc_N
        0-{\XINT_xtrunc_P #1}%
    \krof
}%
\def\XINT_xtrunc_zero .#1#2#3#4#5%
{%
    0.\romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                                  {\the\numexpr #5}{}\Z {}%
    \xintiloop [#4+-1]
    \ifnum \xintiloopindex>\xint_c_
    0000000000000000000000000000000000000000000000000000000000000000%
    \repeat
}%
\def\XINT_xtrunc_N {-\XINT_xtrunc_P }%
\def\XINT_xtrunc_P #1.#2%
{%
    \ifnum #2<\xint_c_
        \expandafter\XINT_xtrunc_negN_Q
    \else
        \expandafter\XINT_xtrunc_Q
    \fi  {#2}{#1}.%
}%
\def\XINT_xtrunc_negN_Q #1#2.#3#4#5#6%
{%
    \expandafter\XINT_xtrunc_negN_R
    \romannumeral0\XINT_div_prepare {#3}{#2}{#3}{#1}{#4}%
}%
% #1=Q, #2=R, #3=B, #4=N<0, #5=D
\def\XINT_xtrunc_negN_R #1#2#3#4#5%
{%
    \expandafter\XINT_xtrunc_negN_S\expandafter
    {\the\numexpr -#4}{#5}{#2}{#3}{#1}%
}%
\def\XINT_xtrunc_negN_S #1#2%
{%
    \expandafter\XINT_xtrunc_negN_T\expandafter
    {\the\numexpr #2-#1}{#1}{#2}%
}%
\def\XINT_xtrunc_negN_T #1%
{%
    \ifnum \xint_c_<#1
      \expandafter\XINT_xtrunc_negNA
    \else
      \expandafter\XINT_xtrunc_negNW
    \fi {#1}%
}%
% #1=D-|N|>0, #2=|N|, #3=D, #4=R, #5=B, #6=Q
\def\XINT_xtrunc_unlock #10.{ }%
\def\XINT_xtrunc_negNA #1#2#3#4#5#6%
{%
   \expandafter\XINT_xtrunc_negNB\expandafter
   {\romannumeral0\expandafter\expandafter\expandafter
    \XINT_xtrunc_unlock\expandafter\string
    \csname\XINT_xtrunc_b {#1}#4/#5[0]\expandafter\endcsname
    \expandafter}\expandafter
    {\the\numexpr\xintLength{#6}-#2}{#6}%
}%
\def\XINT_xtrunc_negNB #1#2#3{\XINT_xtrunc_negNC {#2}{#3}#1}%
\def\XINT_xtrunc_negNC #1%
{%
    \ifnum \xint_c_ < #1
      \expandafter\XINT_xtrunc_negNDa
    \else
      \expandafter\XINT_xtrunc_negNE
    \fi {#1}%
}%
\def\XINT_xtrunc_negNDa #1#2%
{%
    \expandafter\XINT_xtrunc_negNDb%
    \romannumeral0\XINT_split_fromleft_loop {#1}{}#2\W\W\W\W\W\W\W\W\Z
}%
\def\XINT_xtrunc_negNDb #1#2{#1.#2}%
\def\XINT_xtrunc_negNE #1#2%
{%
    0.\romannumeral0\XINT_dsx_zeroloop {-#1}{}\Z {}#2%
}%
% #1=D-|N|<=0, #2=|N|, #3=D, #4=R, #5=B, #6=Q
\def\XINT_xtrunc_negNW #1#2#3#4#5#6%
{%
    \expandafter\XINT_xtrunc_negNX\expandafter
    {\romannumeral0\xintnum{\xintDecSplitL {-#1}{#6}}}{#3}%
}%
\def\XINT_xtrunc_negNX #1#2%
{%
    \expandafter\XINT_xtrunc_negNC\expandafter
    {\the\numexpr\xintLength {#1}-#2}{#1}%
}%
\def\XINT_xtrunc_Q #1%
{%
    \expandafter\XINT_xtrunc_prepare_I
    \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z
}%
\def\XINT_xtrunc_prepare_I #1.#2#3%
{%
    \expandafter\XINT_xtrunc_prepareB_aa\expandafter
    {\romannumeral0\xintlength {#2}}{#2}{#1}%
}%
\def\XINT_xtrunc_prepareB_aa #1%
{%
    \ifnum #1=\xint_c_i
      \expandafter\XINT_xtrunc_prepareB_onedigit
    \else
      \expandafter\XINT_xtrunc_prepareB_PaBa
    \fi
    {#1}%
}%
\def\XINT_xtrunc_prepareB_onedigit #1#2%
{%
    \ifcase#2
    \or\expandafter\XINT_xtrunc_BisOne
    \or\expandafter\XINT_xtrunc_BisTwo
    \else\expandafter\XINT_xtrunc_prepareB_PaBe
    \fi {000}{0}{4}{#2}%
}%
\def\XINT_xtrunc_BisOne #1#2#3#4#5#6#7%
{%
    #5.\romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                                  {\the\numexpr #7}{}\Z {}%
    \xintiloop [#6+-1]
    \ifnum \xintiloopindex>\xint_c_
    0000000000000000000000000000000000000000000000000000000000000000%
    \repeat
}%
\def\XINT_xtrunc_BisTwo #1#2#3#4#5#6#7%
{%
    \xintHalf {#5}.\ifodd\xintiiLDg{#5} 5\else 0\fi
    \romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                                  {\the\numexpr #7-\xint_c_i}{}\Z {}%
    \xintiloop [#6+-1]
    \ifnum \xintiloopindex>\xint_c_
    0000000000000000000000000000000000000000000000000000000000000000%
    \repeat
}%
\def\XINT_xtrunc_prepareB_PaBa #1#2%
{%
    \expandafter\XINT_xtrunc_Pa\expandafter
    {\romannumeral0\XINT_xtrunc_prepareB_a {#1}{#2}}%
}%
\def\XINT_xtrunc_prepareB_a #1%
{%
  \expandafter\XINT_xtrunc_prepareB_c\expandafter
  {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}%
}%
\def\XINT_xtrunc_prepareB_c #1#2%
{%
    \csname XINT_xtrunc_prepareB_d\romannumeral\numexpr#1-#2\endcsname
    {#1}%
}%
\def\XINT_xtrunc_prepareB_d    {\XINT_xtrunc_prepareB_e {}{0000}}%
\def\XINT_xtrunc_prepareB_di   {\XINT_xtrunc_prepareB_e {0}{000}}%
\def\XINT_xtrunc_prepareB_dii  {\XINT_xtrunc_prepareB_e {00}{00}}%
\def\XINT_xtrunc_prepareB_diii {\XINT_xtrunc_prepareB_e {000}{0}}%
\def\XINT_xtrunc_prepareB_PaBe #1#2#3#4%
{%
    \expandafter\XINT_xtrunc_Pa\expandafter
    {\romannumeral0\XINT_xtrunc_prepareB_e {#1}{#2}{#3}{#4}}%
}%
\def\XINT_xtrunc_prepareB_e #1#2#3#4%
{%
    \ifnum#3=\xint_c_iv\expandafter\XINT_xtrunc_prepareLittleB_f
                   \else\expandafter\XINT_xtrunc_prepareB_f
    \fi
    #4#1{#3}{#2}{#1}%
}%
\def\XINT_xtrunc_prepareB_f #1#2#3#4#5#{%
    \expandafter\space
    \expandafter\XINT_div_prepareB_g
     \the\numexpr  #1#2#3#4+\xint_c_i\expandafter
    .\the\numexpr (#1#2#3#4+\xint_c_i)/\xint_c_ii\expandafter
    .\romannumeral0\xintreverseorder {#1#2#3#4#5}.{#1#2#3#4}%
}%
\def\XINT_xtrunc_prepareLittleB_f #1#{%
    \expandafter\space\expandafter
    \XINT_div_prepareB_g \the\numexpr  #1/\xint_c_ii.{}.{}.{#1}%
}%
\def\XINT_xtrunc_Pa #1#2%
{%
    \expandafter\XINT_xtrunc_Pb\romannumeral0#1{#2}{#1}%
}%
\def\XINT_xtrunc_Pb #1#2#3#4{#1.\XINT_xtrunc_A {#4}{#2}{#3}}%
\def\XINT_xtrunc_A #1%
{%
    \unless\ifnum #1>\xint_c_ \XINT_xtrunc_transition\fi
    \expandafter\XINT_xtrunc_B\expandafter{\the\numexpr #1-\xint_c_i}%
}%
\def\XINT_xtrunc_B #1#2#3%
{%
    \expandafter\XINT_xtrunc_D\romannumeral0#3%
    {#20000000000000000000000000000000000000000000000000000000000000000}%
    {#1}{#3}%
}%
\def\XINT_xtrunc_D #1#2#3%
{%
    \romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                  {\the\numexpr \xint_c_ii^vi-\xintLength{#1}}{}\Z {}#1%
    \XINT_xtrunc_A {#3}{#2}%
}%
\def\XINT_xtrunc_transition\fi
    \expandafter\XINT_xtrunc_B\expandafter #1#2#3#4%
{%
    \fi
    \ifnum #4=\xint_c_ \XINT_xtrunc_abort\fi
    \expandafter\XINT_xtrunc_x\expandafter
    {\romannumeral0\XINT_dsx_zeroloop {#4}{}\Z {#2}}{#3}{#4}%
}%
\def\XINT_xtrunc_x #1#2%
{%
    \expandafter\XINT_xtrunc_y\romannumeral0#2{#1}%
}%
\def\XINT_xtrunc_y #1#2#3%
{%
    \romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                  {\the\numexpr #3-\xintLength{#1}}{}\Z {}#1%
}%
\def\XINT_xtrunc_abort\fi\expandafter\XINT_xtrunc_x\expandafter #1#2#3{\fi}%
%    \end{macrocode}
% \subsection{\csh{xintDigits}}
% \lverb|The mathchardef used to be called \XINT_digits, but for reasons originating in
% \xintNewExpr (and now obsolete), release 1.09a uses \XINTdigits without underscore.|
%    \begin{macrocode}
\mathchardef\XINTdigits 16
\def\xintDigits #1#2%
   {\afterassignment \xint_gobble_i \mathchardef\XINTdigits=}%
\def\xinttheDigits {\number\XINTdigits }%
%    \end{macrocode}
% \subsection{\csh{xintFloat}}
% \lverb|1.07. Completely re-written in 1.08a, with spectacular speed
% gains. The earlier version was seriously silly when dealing with
% inputs having a big power of ten. Again some modifications in 1.08b
% for a better treatment of cases with long explicit numerators or
% denominators.
%
% Here again some inner macros used the \xintiquo with extra \xintnum overhead
% in 1.09a, 1.09f reinstalled use of \xintiiquo without this overhead.|
%    \begin{macrocode}
\def\xintFloat   {\romannumeral0\xintfloat }%
\def\xintfloat #1{\XINT_float_chkopt #1\xint_relax }%
\def\XINT_float_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_float_opt
       \else\expandafter\XINT_float_noopt
    \fi  #1%
}%
\def\XINT_float_noopt #1\xint_relax
{%
    \expandafter\XINT_float_a\expandafter\XINTdigits
    \romannumeral0\XINT_infrac {#1}\XINT_float_Q
}%
\def\XINT_float_opt [\xint_relax #1]#2%
{%
    \expandafter\XINT_float_a\expandafter
    {\the\numexpr #1\expandafter}%
    \romannumeral0\XINT_infrac {#2}\XINT_float_Q
}%
\def\XINT_float_a #1#2#3% #1=P, #2=n, #3=A, #4=B
{%
    \XINT_float_fork #3\Z {#1}{#2}% #1 = precision, #2=n
}%
\def\XINT_float_fork #1%
{%
    \xint_UDzerominusfork
     #1-\XINT_float_zero
     0#1\XINT_float_J
      0-{\XINT_float_K #1}%
    \krof
}%
\def\XINT_float_zero #1\Z #2#3#4#5{ 0.e0}%
\def\XINT_float_J {\expandafter\xint_minus_thenstop\romannumeral0\XINT_float_K }%
\def\XINT_float_K #1\Z #2% #1=A, #2=P, #3=n, #4=B
{%
    \expandafter\XINT_float_L\expandafter
    {\the\numexpr\xintLength{#1}\expandafter}\expandafter
    {\the\numexpr #2+\xint_c_ii}{#1}{#2}%
}%
\def\XINT_float_L #1#2%
{%
    \ifnum #1>#2
      \expandafter\XINT_float_Ma
    \else
      \expandafter\XINT_float_Mc
    \fi {#1}{#2}%
}%
\def\XINT_float_Ma #1#2#3%
{%
    \expandafter\XINT_float_Mb\expandafter
    {\the\numexpr #1-#2\expandafter\expandafter\expandafter}%
    \expandafter\expandafter\expandafter
    {\expandafter\xint_firstoftwo
     \romannumeral0\XINT_split_fromleft_loop {#2}{}#3\W\W\W\W\W\W\W\W\Z
     }{#2}%
}%
\def\XINT_float_Mb #1#2#3#4#5#6% #2=A', #3=P+2, #4=P, #5=n, #6=B
{%
   \expandafter\XINT_float_N\expandafter
   {\the\numexpr\xintLength{#6}\expandafter}\expandafter
   {\the\numexpr #3\expandafter}\expandafter
   {\the\numexpr #1+#5}%
   {#6}{#3}{#2}{#4}%
}% long de B, P+2, n', B, |A'|=P+2, A', P
\def\XINT_float_Mc #1#2#3#4#5#6%
{%
   \expandafter\XINT_float_N\expandafter
   {\romannumeral0\xintlength{#6}}{#2}{#5}{#6}{#1}{#3}{#4}%
}% long de B, P+2, n, B, |A|, A, P
\def\XINT_float_N #1#2%
{%
    \ifnum #1>#2
      \expandafter\XINT_float_O
    \else
      \expandafter\XINT_float_P
    \fi {#1}{#2}%
}%
\def\XINT_float_O #1#2#3#4%
{%
    \expandafter\XINT_float_P\expandafter
    {\the\numexpr #2\expandafter}\expandafter
    {\the\numexpr #2\expandafter}\expandafter
    {\the\numexpr #3-#1+#2\expandafter\expandafter\expandafter}%
    \expandafter\expandafter\expandafter
    {\expandafter\xint_firstoftwo
     \romannumeral0\XINT_split_fromleft_loop {#2}{}#4\W\W\W\W\W\W\W\W\Z
     }%
}% |B|,P+2,n,B,|A|,A,P
\def\XINT_float_P #1#2#3#4#5#6#7#8%
{%
    \expandafter #8\expandafter {\the\numexpr #1-#5+#2-\xint_c_i}%
    {#6}{#4}{#7}{#3}%
}% |B|-|A|+P+1,A,B,P,n
\def\XINT_float_Q #1%
{%
    \ifnum #1<\xint_c_
      \expandafter\XINT_float_Ri
    \else
      \expandafter\XINT_float_Rii
    \fi {#1}%
}%
\def\XINT_float_Ri #1#2#3%
{%
    \expandafter\XINT_float_Sa
    \romannumeral0\xintiiquo {#2}%
         {\XINT_dsx_addzerosnofuss {-#1}{#3}}\Z {#1}%
}%
\def\XINT_float_Rii #1#2#3%
{%
    \expandafter\XINT_float_Sa
    \romannumeral0\xintiiquo
         {\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}\Z {#1}%
}%
\def\XINT_float_Sa #1%
{%
    \if #19%
        \xint_afterfi {\XINT_float_Sb\XINT_float_Wb }%
    \else
        \xint_afterfi {\XINT_float_Sb\XINT_float_Wa }%
    \fi #1%
}%
\def\XINT_float_Sb #1#2\Z #3#4%
{%
    \expandafter\XINT_float_T\expandafter
    {\the\numexpr #4+\xint_c_i\expandafter}%
    \romannumeral-`0\XINT_lenrord_loop 0{}#2\Z\W\W\W\W\W\W\W\Z #1{#3}{#4}%
}%
\def\XINT_float_T #1#2#3%
{%
    \ifnum #2>#1
      \xint_afterfi{\XINT_float_U\XINT_float_Xb}%
    \else
      \xint_afterfi{\XINT_float_U\XINT_float_Xa #3}%
    \fi
}%
\def\XINT_float_U #1#2%
{%
    \ifnum #2<\xint_c_v
      \expandafter\XINT_float_Va
    \else
      \expandafter\XINT_float_Vb
    \fi #1%
}%
\def\XINT_float_Va #1#2\Z #3%
{%
    \expandafter#1%
    \romannumeral0\expandafter\XINT_float_Wa
    \romannumeral0\XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax \Z
}%
\def\XINT_float_Vb #1#2\Z #3%
{%
    \expandafter #1%
    \romannumeral0\expandafter #3%
    \romannumeral0\XINT_addm_A 0{}1000\W\X\Y\Z #2000\W\X\Y\Z \Z
}%
\def\XINT_float_Wa #1{ #1.}%
\def\XINT_float_Wb #1#2%
    {\if #11\xint_afterfi{ 10.}\else\xint_afterfi{ #1.#2}\fi }%
\def\XINT_float_Xa #1\Z #2#3#4%
{%
    \expandafter\XINT_float_Y\expandafter
    {\the\numexpr #3+#4-#2}{#1}%
}%
\def\XINT_float_Xb #1\Z #2#3#4%
{%
    \expandafter\XINT_float_Y\expandafter
    {\the\numexpr #3+#4+\xint_c_i-#2}{#1}%
}%
\def\XINT_float_Y #1#2{ #2e#1}%
%    \end{macrocode}
% \subsection{\csh{xintPFloat}}
% \lverb|1.1|
%    \begin{macrocode}
\def\xintPFloat   {\romannumeral0\xintpfloat }%
\def\xintpfloat #1{\XINT_pfloat_chkopt #1\xint_relax }%
\def\XINT_pfloat_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_pfloat_opt
       \else\expandafter\XINT_pfloat_noopt
    \fi  #1%
}%
\def\XINT_pfloat_noopt #1\xint_relax
{%
    \expandafter\XINT_pfloat_a\expandafter\XINTdigits
    \romannumeral0\XINTinfloat [\XINTdigits]{#1}%
}%
\def\XINT_pfloat_opt [\xint_relax #1]%#2%
{%
    \expandafter\XINT_pfloat_a\expandafter {\the\numexpr #1\expandafter}%
    \romannumeral0\XINTinfloat [\numexpr #1\relax]%{#2}%
}%
\def\XINT_pfloat_a #1#2%
{%
    \xint_UDzerominusfork
                           #2-\XINT_pfloat_zero
                           0#2\XINT_pfloat_neg
                            0-{\XINT_pfloat_pos #2}%
    \krof {#1}%
}%
\def\XINT_pfloat_zero #1[#2]{ 0}%
\def\XINT_pfloat_neg
   {\expandafter\xint_minus_thenstop\romannumeral0\XINT_pfloat_pos {}}%
\def\XINT_pfloat_pos #1#2#3[#4]%
{%
    \ifnum#4>0 \xint_dothis\XINT_pfloat_no\fi
    \ifnum#4>\numexpr-#2\relax \xint_dothis\XINT_pfloat_b\fi
    \ifnum#4>\numexpr-#2-\xint_c_v\relax \xint_dothis\XINT_pfloat_B\fi
    \xint_orthat\XINT_pfloat_no {#2}{#4}{#1#3}%
}%
\def\XINT_pfloat_no #1#2%
{%
 \expandafter\XINT_pfloat_no_b\expandafter{\the\numexpr #2+#1-\xint_c_i\relax}%
}%
\def\XINT_pfloat_no_b #1#2{\XINT_pfloat_no_c #2e#1}%
\def\XINT_pfloat_no_c #1{ #1.}%
\def\XINT_pfloat_b #1#2#3%
   {\expandafter\XINT_pfloat_c
    \romannumeral0\expandafter\XINT_split_fromleft_loop
    \expandafter {\the\numexpr #1+#2-\xint_c_i}#3\W\W\W\W\W\W\W\W\Z }%
\def\XINT_pfloat_c #1#2{ #1.#2}% #2 peut \^etre vide
\def\XINT_pfloat_B #1#2#3%
   {\expandafter\XINT_pfloat_C
    \romannumeral0\XINT_dsx_zeroloop {\numexpr -#1-#2}{}\Z {}#3}%
\def\XINT_pfloat_C { 0.}%
%    \end{macrocode}
% \subsection{\csh{XINTinFloat}}
% \lverb|1.07. Completely rewritten in 1.08a for immensely greater efficiency
% when the power of ten is big: previous version had some very serious
% bottlenecks arising from the creation of long strings of zeros, which made
% things such as 2^999999 completely impossible, but now even 2^999999999 with
% 24 significant digits is no problem! Again (slightly) improved in 1.08b.
%
% I decide in 1.09a not to use anymore \romannumeral`-0 mais \romannumeral0 also
% in the float routines, for consistency of style.
%
% Here again some inner macros used the \xintiquo with extra \xintnum overhead
% in 1.09a, 1.09f fixed that to use \xintiiquo for example.
%
% 1.09i added a stupid bug to \XINT_infloat_zero when it changed 0[0] to a silly
% 0/1[0], breaking in particular \xintFloatAdd when one of the argument is zero
%          :(((
%
% 1.09j fixes this. Besides, for notational coherence \XINT_inFloat and
% \XINT_infloat have been renamed respectively \XINTinFloat and \XINTinfloat in
% release 1.09j.|
%    \begin{macrocode}
\def\XINTinFloat {\romannumeral0\XINTinfloat }%
\def\XINTinfloat [#1]#2%
{%
    \expandafter\XINT_infloat_a\expandafter
    {\the\numexpr #1\expandafter}%
    \romannumeral0\XINT_infrac {#2}\XINT_infloat_Q
}%
\def\XINT_infloat_a #1#2#3% #1=P, #2=n, #3=A, #4=B
{%
    \XINT_infloat_fork #3\Z {#1}{#2}% #1 = precision, #2=n
}%
\def\XINT_infloat_fork #1%
{%
    \xint_UDzerominusfork
     #1-\XINT_infloat_zero
     0#1\XINT_infloat_J
      0-{\XINT_float_K #1}%
    \krof
}%
\def\XINT_infloat_zero #1\Z #2#3#4#5{ 0[0]}%
% the 0[0] was stupidly changed to 0/1[0] in 1.09i, with the result that the
% Float addition would crash when an operand was zero
\def\XINT_infloat_J {\expandafter-\romannumeral0\XINT_float_K }%
\def\XINT_infloat_Q #1%
{%
    \ifnum #1<\xint_c_
      \expandafter\XINT_infloat_Ri
    \else
      \expandafter\XINT_infloat_Rii
    \fi {#1}%
}%
\def\XINT_infloat_Ri #1#2#3%
{%
    \expandafter\XINT_infloat_S\expandafter
    {\romannumeral0\xintiiquo {#2}%
         {\XINT_dsx_addzerosnofuss {-#1}{#3}}}{#1}%
}%
\def\XINT_infloat_Rii #1#2#3%
{%
    \expandafter\XINT_infloat_S\expandafter
    {\romannumeral0\xintiiquo
         {\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}}{#1}%
}%
\def\XINT_infloat_S #1#2#3%
{%
    \expandafter\XINT_infloat_T\expandafter
    {\the\numexpr #3+\xint_c_i\expandafter}%
    \romannumeral-`0\XINT_lenrord_loop 0{}#1\Z\W\W\W\W\W\W\W\Z
    {#2}%
}%
\def\XINT_infloat_T #1#2#3%
{%
    \ifnum #2>#1
      \xint_afterfi{\XINT_infloat_U\XINT_infloat_Wb}%
    \else
      \xint_afterfi{\XINT_infloat_U\XINT_infloat_Wa #3}%
    \fi
}%
\def\XINT_infloat_U #1#2%
{%
    \ifnum #2<\xint_c_v
      \expandafter\XINT_infloat_Va
    \else
      \expandafter\XINT_infloat_Vb
    \fi #1%
}%
\def\XINT_infloat_Va #1#2\Z
{%
    \expandafter#1%
    \romannumeral0\XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax \Z
}%
\def\XINT_infloat_Vb #1#2\Z
{%
    \expandafter #1%
    \romannumeral0\XINT_addm_A 0{}1000\W\X\Y\Z #2000\W\X\Y\Z \Z
}%
\def\XINT_infloat_Wa #1\Z #2#3%
{%
    \expandafter\XINT_infloat_X\expandafter
    {\the\numexpr #3+\xint_c_i-#2}{#1}%
}%
\def\XINT_infloat_Wb #1\Z #2#3%
{%
    \expandafter\XINT_infloat_X\expandafter
    {\the\numexpr #3+\xint_c_ii-#2}{#1}%
}%
\def\XINT_infloat_X #1#2{ #2[#1]}%
%    \end{macrocode}
% \subsection{\csh{xintAdd}}
% \lverb|modified in v1.1. Et aussi 25 juin pour intercepter summand nul.|
%    \begin{macrocode}
\def\xintAdd {\romannumeral0\xintadd }%
\def\xintadd #1{\expandafter\xint_fadd\romannumeral0\xintraw {#1}}%
\def\xint_fadd #1{\xint_gob_til_zero #1\XINT_fadd_Azero 0\XINT_fadd_a #1}%
\def\XINT_fadd_Azero #1]{\xintraw }%
\def\XINT_fadd_a #1/#2[#3]#4%
   {\expandafter\XINT_fadd_b\romannumeral0\xintraw {#4}{#3}{#1}{#2}}%
\def\XINT_fadd_b #1{\xint_gob_til_zero #1\XINT_fadd_Bzero 0\XINT_fadd_c #1}%
\def\XINT_fadd_Bzero #1]#2#3#4{ #3/#4[#2]}%
\def\XINT_fadd_c #1/#2[#3]#4%
{%
    \expandafter\XINT_fadd_Aa\expandafter{\the\numexpr #4-#3}{#3}{#4}{#1}{#2}%
}%
\def\XINT_fadd_Aa #1%
{%
    \ifcase\XINT_cntSgn #1\Z
       \expandafter\XINT_fadd_B
    \or
       \expandafter \XINT_fadd_Ba
    \else
       \expandafter \XINT_fadd_Bb
    \fi {#1}%
}%
\def\XINT_fadd_B   #1#2#3#4#5#6#7{\XINT_fadd_C {#4}{#5}{#7}{#6}[#3]}%
\def\XINT_fadd_Ba  #1#2#3#4#5#6#7%
{%
    \expandafter\XINT_fadd_C\expandafter
        {\romannumeral0\XINT_dsx_zeroloop {#1}{}\Z {#6}}%
    {#7}{#5}{#4}[#2]%
}%
\def\XINT_fadd_Bb #1#2#3#4#5#6#7%
{%
    \expandafter\XINT_fadd_C\expandafter
        {\romannumeral0\XINT_dsx_zeroloop {-#1}{}\Z {#4}}%
    {#5}{#7}{#6}[#3]%
}%
\def\XINT_fadd_C #1#2#3%
{%
   \ifcase\romannumeral0\XINT_cmp_pre {#2}{#3} %<- intentional space here.
      \expandafter\XINT_fadd_eq
   \or\expandafter\XINT_fadd_D
   \else\expandafter\XINT_fadd_Da
   \fi {#2}{#3}{#1}%
}%
\def\XINT_fadd_eq #1#2#3#4%#5%
{%
   \expandafter\XINT_fadd_G
   \romannumeral0\xintiiadd {#3}{#4}/#1%[#5]%
}%
\def\XINT_fadd_D #1#2%
{%
   \expandafter\XINT_fadd_E\romannumeral0\XINT_div_prepare {#2}{#1}{#1}{#2}%
}%
\def\XINT_fadd_E #1#2%
{%
   \if0\XINT_Sgn #2\Z
        \expandafter\XINT_fadd_F
   \else\expandafter\XINT_fadd_K
   \fi {#1}%
}%
\def\XINT_fadd_F #1#2#3#4#5%#6%
{%
   \expandafter\XINT_fadd_G
   \romannumeral0\xintiiadd {\xintiiMul {#5}{#1}}{#4}/#2%[#6]%
}%
\def\XINT_fadd_Da #1#2%
{%
   \expandafter\XINT_fadd_Ea\romannumeral0\XINT_div_prepare {#1}{#2}{#1}{#2}%
}%
\def\XINT_fadd_Ea #1#2%
{%
   \if0\XINT_Sgn #2\Z
        \expandafter\XINT_fadd_Fa
   \else\expandafter\XINT_fadd_K
   \fi {#1}%
}%
\def\XINT_fadd_Fa #1#2#3#4#5%#6%
{%
   \expandafter\XINT_fadd_G
   \romannumeral0\xintiiadd {\xintiiMul {#4}{#1}}{#5}/#3%[#6]%
}%
\def\XINT_fadd_G #1{\if0#1\XINT_fadd_iszero\fi\space #1}%
\def\XINT_fadd_K #1#2#3#4#5%
{%
    \expandafter\XINT_fadd_L
    \romannumeral0\xintiiadd {\xintiiMul {#2}{#5}}{\xintiiMul {#3}{#4}}.%
    {{#2}{#3}}%
}%
\def\XINT_fadd_L #1{\if0#1\XINT_fadd_iszero\fi \XINT_fadd_M #1}%
\def\XINT_fadd_M #1.#2{\expandafter\XINT_fadd_N \expandafter
                       {\romannumeral0\xintiimul #2}{#1}}%
\def\XINT_fadd_N #1#2{ #2/#1}%
\edef\XINT_fadd_iszero\fi #1[#2]{\noexpand\fi\space 0/1[0]}% ou [#2] originel?
%    \end{macrocode}
% \subsection{\csh{xintSub}}
% \lverb|refait dans 1.1 pour vérifier si summands nuls.|
%    \begin{macrocode}
\def\xintSub   {\romannumeral0\xintsub }%
\def\xintsub #1{\expandafter\xint_fsub\romannumeral0\xintraw {#1}}%
\def\xint_fsub #1{\xint_gob_til_zero #1\XINT_fsub_Azero 0\XINT_fsub_a #1}%
\def\XINT_fsub_Azero #1]{\xintopp }%
\def\XINT_fsub_a #1/#2[#3]#4%
   {\expandafter\XINT_fsub_b\romannumeral0\xintraw {#4}{#3}{#1}{#2}}%
\def\XINT_fsub_b #1{\xint_UDzerominusfork
                      #1-\XINT_fadd_Bzero
                       0#1\XINT_fadd_c
                       0-{\XINT_fadd_c -#1}%
                     \krof }%
%    \end{macrocode}
% \subsection{\csh{xintSum}}
%    \begin{macrocode}
\def\xintSum {\romannumeral0\xintsum }%
\def\xintsum #1{\xintsumexpr #1\relax }%
\def\xintSumExpr {\romannumeral0\xintsumexpr }%
\def\xintsumexpr {\expandafter\XINT_fsumexpr\romannumeral-`0}%
\def\XINT_fsumexpr {\XINT_fsum_loop_a {0/1[0]}}%
\def\XINT_fsum_loop_a #1#2%
{%
    \expandafter\XINT_fsum_loop_b \romannumeral-`0#2\Z {#1}%
}%
\def\XINT_fsum_loop_b #1%
{%
    \xint_gob_til_relax #1\XINT_fsum_finished\relax
    \XINT_fsum_loop_c #1%
}%
\def\XINT_fsum_loop_c #1\Z #2%
{%
    \expandafter\XINT_fsum_loop_a\expandafter{\romannumeral0\xintadd {#2}{#1}}%
}%
\def\XINT_fsum_finished #1\Z #2{ #2}%
%    \end{macrocode}
% \subsection{\csh{xintMul}}
% \lverb|modif 1.1 25-juin-14 pour vérifier plus tôt si nul|
%    \begin{macrocode}
\def\xintMul {\romannumeral0\xintmul }%
\def\xintmul #1{\expandafter\xint_fmul\romannumeral0\xintraw {#1}.}%
\def\xint_fmul #1{\xint_gob_til_zero #1\XINT_fmul_zero 0\XINT_fmul_a #1}%
\def\XINT_fmul_a #1[#2].#3%
   {\expandafter\XINT_fmul_b\romannumeral0\xintraw {#3}#1[#2.]}%
\def\XINT_fmul_b #1{\xint_gob_til_zero #1\XINT_fmul_zero 0\XINT_fmul_c #1}%
\def\XINT_fmul_c #1/#2[#3]#4/#5[#6.]%
{%
    \expandafter\XINT_fmul_d
    \expandafter{\the\numexpr #3+#6\expandafter}%
    \expandafter{\romannumeral0\xintiimul {#5}{#2}}%
    {\romannumeral0\xintiimul {#4}{#1}}%
}%
\def\XINT_fmul_d #1#2#3%
{%
    \expandafter \XINT_fmul_e \expandafter{#3}{#1}{#2}%
}%
\def\XINT_fmul_e #1#2{\XINT_outfrac {#2}{#1}}%
\def\XINT_fmul_zero #1.#2{ 0/1[0]}%
%    \end{macrocode}
% \subsection{\csh{xintSqr}}
% \lverb|1.1 modifs comme xintMul|
%    \begin{macrocode}
\def\xintSqr {\romannumeral0\xintsqr }%
\def\xintsqr #1{\expandafter\xint_fsqr\romannumeral0\xintraw {#1}}%
\def\xint_fsqr #1{\xint_gob_til_zero #1\XINT_fsqr_zero 0\XINT_fsqr_a #1}%
\def\xint_fsqr_a #1/#2[#3]%
{%
    \expandafter\XINT_fsqr_b
    \expandafter{\the\numexpr #3+#3\expandafter}%
    \expandafter{\romannumeral0\xintiisqr {#2}}%
    {\romannumeral0\xintiisqr {#1}}%
}%
\def\XINT_fsqr_b #1#2#3{\expandafter \XINT_fmul_e \expandafter{#3}{#1}{#2}}%
\def\XINT_fsqr_zero #1]{ 0/1[0]}%
%    \end{macrocode}
% \subsection{\csh{xintPow}}
% \lverb|&
% Modified in 1.06 to give the exponent to a \numexpr.
%
% With 1.07 and for use within the \xintexpr parser, we must allow
% fractions (which are integers in disguise) as input to the exponent, so we
% must have a variant which uses \xintNum and not only \numexpr
% for normalizing the input. Hence the \xintfPow here.
%
% 1.08b: well actually I
% think that with xintfrac.sty loaded the exponent should always be allowed to
% be a fraction giving an integer. So I do as for \xintFac, and remove here the
% duplicated. Then \xintexpr can use the \xintPow as defined here.|
%    \begin{macrocode}
\def\xintPow {\romannumeral0\xintpow }%
\def\xintpow #1%
{%
    \expandafter\xint_fpow\expandafter {\romannumeral0\XINT_infrac {#1}}%
}%
\def\xint_fpow #1#2%
{%
    \expandafter\XINT_fpow_fork\the\numexpr \xintNum{#2}\relax\Z #1%
}%
\def\XINT_fpow_fork #1#2\Z
{%
    \xint_UDzerominusfork
      #1-\XINT_fpow_zero
      0#1\XINT_fpow_neg
       0-{\XINT_fpow_pos #1}%
    \krof
    {#2}%
}%
\def\XINT_fpow_zero #1#2#3#4{ 1/1[0]}%
\def\XINT_fpow_pos #1#2#3#4#5%
{%
    \expandafter\XINT_fpow_pos_A\expandafter
    {\the\numexpr #1#2*#3\expandafter}\expandafter
    {\romannumeral0\xintiipow {#5}{#1#2}}%
    {\romannumeral0\xintiipow {#4}{#1#2}}%
}%
\def\XINT_fpow_neg #1#2#3#4%
{%
    \expandafter\XINT_fpow_pos_A\expandafter
    {\the\numexpr -#1*#2\expandafter}\expandafter
    {\romannumeral0\xintiipow {#3}{#1}}%
    {\romannumeral0\xintiipow {#4}{#1}}%
}%
\def\XINT_fpow_pos_A #1#2#3%
{%
    \expandafter\XINT_fpow_pos_B\expandafter {#3}{#1}{#2}%
}%
\def\XINT_fpow_pos_B #1#2{\XINT_outfrac {#2}{#1}}%
%    \end{macrocode}
% \subsection{\csh{xintFac}}
% \lverb|1.07: to be used by the \xintexpr scanner which needs to be able to
% apply \xintFac
% to a fraction which is an integer in disguise; so we use \xintNum and not only
% \numexpr. Je modifie cela dans 1.08b, au lieu d'avoir un \xintfFac
% spécialement pour \xintexpr, tout simplement j'étends \xintFac comme les
% autres macros, pour qu'elle utilise \xintNum. |
%    \begin{macrocode}
\def\xintFac {\romannumeral0\xintfac }%
\def\xintfac #1%
{%
    \expandafter\XINT_fac_fork\expandafter{\the\numexpr \xintNum{#1}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintPrd}}
%    \begin{macrocode}
\def\xintPrd {\romannumeral0\xintprd }%
\def\xintprd #1{\xintprdexpr #1\relax }%
\def\xintPrdExpr {\romannumeral0\xintprdexpr }%
\def\xintprdexpr {\expandafter\XINT_fprdexpr \romannumeral-`0}%
\def\XINT_fprdexpr {\XINT_fprod_loop_a {1/1[0]}}%
\def\XINT_fprod_loop_a #1#2%
{%
    \expandafter\XINT_fprod_loop_b \romannumeral-`0#2\Z {#1}%
}%
\def\XINT_fprod_loop_b #1%
{%
    \xint_gob_til_relax #1\XINT_fprod_finished\relax
    \XINT_fprod_loop_c #1%
}%
\def\XINT_fprod_loop_c #1\Z #2%
{%
  \expandafter\XINT_fprod_loop_a\expandafter{\romannumeral0\xintmul {#1}{#2}}%
}%
\def\XINT_fprod_finished #1\Z #2{ #2}%
%    \end{macrocode}
% \subsection{\csh{xintDiv}}
%    \begin{macrocode}
\def\xintDiv {\romannumeral0\xintdiv }%
\def\xintdiv #1%
{%
    \expandafter\xint_fdiv\expandafter {\romannumeral0\XINT_infrac {#1}}%
}%
\def\xint_fdiv #1#2%
   {\expandafter\XINT_fdiv_A\romannumeral0\XINT_infrac {#2}#1}%
\def\XINT_fdiv_A #1#2#3#4#5#6%
{%
    \expandafter\XINT_fdiv_B
    \expandafter{\the\numexpr #4-#1\expandafter}%
    \expandafter{\romannumeral0\xintiimul {#2}{#6}}%
    {\romannumeral0\xintiimul {#3}{#5}}%
}%
\def\XINT_fdiv_B #1#2#3%
{%
    \expandafter\XINT_fdiv_C
    \expandafter{#3}{#1}{#2}%
}%
\def\XINT_fdiv_C #1#2{\XINT_outfrac {#2}{#1}}%
%    \end{macrocode}
% \subsection{\csh{xintDivFloor}}
% \lverb|1.1|
%    \begin{macrocode}
\def\xintDivFloor     {\romannumeral0\xintdivfloor }%
\def\xintdivfloor #1#2{\xintfloor{\xintDiv {#1}{#2}}}%
%    \end{macrocode}
% \subsection{\csh{xintDivTrunc}}
% \lverb|1.1. \xintttrunc rather than \xintitrunc0 in 1.1a|
%    \begin{macrocode}
\def\xintDivTrunc     {\romannumeral0\xintdivtrunc }%
\def\xintdivtrunc #1#2{\xintttrunc {\xintDiv {#1}{#2}}}%
%    \end{macrocode}
% \subsection{\csh{xintDivRound}}
% \lverb|1.1|
%    \begin{macrocode}
\def\xintDivRound     {\romannumeral0\xintdivround }%
\def\xintdivround #1#2{\xintiround 0{\xintDiv {#1}{#2}}}%
%    \end{macrocode}
% \subsection{\csh{xintMod}}
% \lverb|1.1. \xintMod {q1}{q2} computes q2*t(q1/q2) with t(q1/q2) equal to
% the truncated division of two arbitrary fractions q1 and q2. We put some
% efforts into minimizing the amount of computations.|
%    \begin{macrocode}
\def\xintMod {\romannumeral0\xintmod }%
\def\xintmod #1{\expandafter\XINT_mod_a\romannumeral0\xintraw{#1}.}%
\def\XINT_mod_a #1#2.#3%
   {\expandafter\XINT_mod_b\expandafter #1\romannumeral0\xintraw{#3}#2.}%
\def\XINT_mod_b #1#2% #1 de A, #2 de B.
{%
    \if0#2\xint_dothis\XINT_mod_divbyzero\fi
    \if0#1\xint_dothis\XINT_mod_aiszero\fi
    \if-#2\xint_dothis{\XINT_mod_bneg #1}\fi
          \xint_orthat{\XINT_mod_bpos #1#2}%
}%
\def\XINT_mod_bpos #1%
{%
    \xint_UDsignfork
            #1{\xintiiopp\XINT_mod_pos {}}%
             -{\XINT_mod_pos #1}%
    \krof
}%
\def\XINT_mod_bneg #1%
{%
    \xint_UDsignfork
            #1{\xintiiopp\XINT_mod_pos {}}%
             -{\XINT_mod_pos #1}%
    \krof
}%
\def\XINT_mod_divbyzero #1.{\xintError:DivisionByZero\space 0/1[0]}%
\def\XINT_mod_aiszero #1.{ 0/1[0]}%
\def\XINT_mod_pos #1#2/#3[#4]#5/#6[#7].%
{%
    \expandafter\XINT_mod_pos_a
    \the\numexpr\ifnum#7>#4 #4\else #7\fi\expandafter.\expandafter
    {\romannumeral0\xintiimul {#6}{#3}}%       n fois u
    {\xintiiE{\xintiiMul {#1#5}{#3}}{#7-#4}}%  m fois u
    {\xintiiE{\xintiiMul {#2}{#6}}{#4-#7}}%    t fois n
}%
\def\XINT_mod_pos_a #1.#2#3#4{\xintiirem {#3}{#4}/#2[#1]}%
%    \end{macrocode}
% \subsection{\csh{XINTinFloatMod}}
% \lverb|Pour emploi dans xintexpr 1.1|
%    \begin{macrocode}
\def\XINTinFloatMod {\romannumeral0\XINTinfloatmod [\XINTdigits]}%
\def\XINTinfloatmod [#1]#2#3{\expandafter\XINT_infloatmod\expandafter
                           {\romannumeral0\XINTinfloat[#1]{#2}}%
                           {\romannumeral0\XINTinfloat[#1]{#3}}{#1}}%
\def\XINT_infloatmod #1#2{\expandafter\XINT_infloatmod_a\expandafter {#2}{#1}}%
\def\XINT_infloatmod_a #1#2#3{\XINTinfloat [#3]{\xintMod {#2}{#1}}}%
%    \end{macrocode}
% \subsection{\csh{xintIsOne}}
% \lverb|&
% New with 1.09a. Could be more efficient. For fractions with big powers of
% tens, it is better to use \xintCmp{f}{1}. Restyled in 1.09i.|
%    \begin{macrocode}
\def\xintIsOne   {\romannumeral0\xintisone }%
\def\xintisone #1{\expandafter\XINT_fracisone
                  \romannumeral0\xintrawwithzeros{#1}\Z }%
\def\XINT_fracisone #1/#2\Z
    {\if0\XINT_Cmp {#1}{#2}\xint_afterfi{ 1}\else\xint_afterfi{ 0}\fi}%
%    \end{macrocode}
% \subsection{\csh{xintGeq}}
% \lverb|&
% Rewritten completely in 1.08a to be less dumb when comparing fractions having
% big powers of tens.|
%    \begin{macrocode}
\def\xintGeq {\romannumeral0\xintgeq }%
\def\xintgeq #1%
{%
    \expandafter\xint_fgeq\expandafter {\romannumeral0\xintabs {#1}}%
}%
\def\xint_fgeq #1#2%
{%
    \expandafter\XINT_fgeq_A \romannumeral0\xintabs {#2}#1%
}%
\def\XINT_fgeq_A #1%
{%
    \xint_gob_til_zero #1\XINT_fgeq_Zii 0%
    \XINT_fgeq_B #1%
}%
\def\XINT_fgeq_Zii 0\XINT_fgeq_B #1[#2]#3[#4]{ 1}%
\def\XINT_fgeq_B #1/#2[#3]#4#5/#6[#7]%
{%
    \xint_gob_til_zero #4\XINT_fgeq_Zi 0%
    \expandafter\XINT_fgeq_C\expandafter
    {\the\numexpr #7-#3\expandafter}\expandafter
    {\romannumeral0\xintiimul {#4#5}{#2}}%
    {\romannumeral0\xintiimul {#6}{#1}}%
}%
\def\XINT_fgeq_Zi 0#1#2#3#4#5#6#7{ 0}%
\def\XINT_fgeq_C #1#2#3%
{%
    \expandafter\XINT_fgeq_D\expandafter
    {#3}{#1}{#2}%
}%
\def\XINT_fgeq_D #1#2#3%
{%
    \expandafter\XINT_cntSgnFork\romannumeral-`0\expandafter\XINT_cntSgn
     \the\numexpr #2+\xintLength{#3}-\xintLength{#1}\relax\Z
    { 0}{\XINT_fgeq_E #2\Z {#3}{#1}}{ 1}%
}%
\def\XINT_fgeq_E #1%
{%
    \xint_UDsignfork
        #1\XINT_fgeq_Fd
         -{\XINT_fgeq_Fn #1}%
    \krof
}%
\def\XINT_fgeq_Fd #1\Z #2#3%
{%
    \expandafter\XINT_fgeq_Fe\expandafter
    {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#3}}{#2}%
}%
\def\XINT_fgeq_Fe #1#2{\XINT_geq_pre {#2}{#1}}%
\def\XINT_fgeq_Fn #1\Z #2#3%
{%
    \expandafter\XINT_geq_pre\expandafter
    {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}%
}%
%    \end{macrocode}
% \subsection{\csh{xintMax}}
% \lverb|&
% Rewritten completely in 1.08a.|
%    \begin{macrocode}
\def\xintMax {\romannumeral0\xintmax }%
\def\xintmax #1%
{%
    \expandafter\xint_fmax\expandafter {\romannumeral0\xintraw {#1}}%
}%
\def\xint_fmax #1#2%
{%
    \expandafter\XINT_fmax_A\romannumeral0\xintraw {#2}#1%
}%
\def\XINT_fmax_A #1#2/#3[#4]#5#6/#7[#8]%
{%
    \xint_UDsignsfork
      #1#5\XINT_fmax_minusminus
       -#5\XINT_fmax_firstneg
       #1-\XINT_fmax_secondneg
        --\XINT_fmax_nonneg_a
    \krof
    #1#5{#2/#3[#4]}{#6/#7[#8]}%
}%
\def\XINT_fmax_minusminus --%
   {\expandafter\xint_minus_thenstop\romannumeral0\XINT_fmin_nonneg_b }%
\def\XINT_fmax_firstneg #1-#2#3{ #1#2}%
\def\XINT_fmax_secondneg -#1#2#3{ #1#3}%
\def\XINT_fmax_nonneg_a #1#2#3#4%
{%
    \XINT_fmax_nonneg_b {#1#3}{#2#4}%
}%
\def\XINT_fmax_nonneg_b #1#2%
{%
    \if0\romannumeral0\XINT_fgeq_A #1#2%
          \xint_afterfi{ #1}%
    \else \xint_afterfi{ #2}%
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintMaxof}}
%    \begin{macrocode}
\def\xintMaxof      {\romannumeral0\xintmaxof }%
\def\xintmaxof    #1{\expandafter\XINT_maxof_a\romannumeral-`0#1\relax }%
\def\XINT_maxof_a #1{\expandafter\XINT_maxof_b\romannumeral0\xintraw{#1}\Z }%
\def\XINT_maxof_b #1\Z #2%
           {\expandafter\XINT_maxof_c\romannumeral-`0#2\Z {#1}\Z}%
\def\XINT_maxof_c #1%
           {\xint_gob_til_relax #1\XINT_maxof_e\relax\XINT_maxof_d #1}%
\def\XINT_maxof_d #1\Z
           {\expandafter\XINT_maxof_b\romannumeral0\xintmax {#1}}%
\def\XINT_maxof_e #1\Z #2\Z { #2}%
%    \end{macrocode}
% \subsection{\csh{xintMin}}
% \lverb|&
% Rewritten completely in 1.08a.|
%    \begin{macrocode}
\def\xintMin {\romannumeral0\xintmin }%
\def\xintmin #1%
{%
    \expandafter\xint_fmin\expandafter {\romannumeral0\xintraw {#1}}%
}%
\def\xint_fmin #1#2%
{%
    \expandafter\XINT_fmin_A\romannumeral0\xintraw {#2}#1%
}%
\def\XINT_fmin_A #1#2/#3[#4]#5#6/#7[#8]%
{%
    \xint_UDsignsfork
      #1#5\XINT_fmin_minusminus
       -#5\XINT_fmin_firstneg
       #1-\XINT_fmin_secondneg
        --\XINT_fmin_nonneg_a
    \krof
    #1#5{#2/#3[#4]}{#6/#7[#8]}%
}%
\def\XINT_fmin_minusminus --%
   {\expandafter\xint_minus_thenstop\romannumeral0\XINT_fmax_nonneg_b }%
\def\XINT_fmin_firstneg #1-#2#3{ -#3}%
\def\XINT_fmin_secondneg -#1#2#3{ -#2}%
\def\XINT_fmin_nonneg_a #1#2#3#4%
{%
    \XINT_fmin_nonneg_b {#1#3}{#2#4}%
}%
\def\XINT_fmin_nonneg_b #1#2%
{%
    \if0\romannumeral0\XINT_fgeq_A #1#2%
          \xint_afterfi{ #2}%
    \else \xint_afterfi{ #1}%
    \fi
}%
%    \end{macrocode}
% \subsection{\csh{xintMinof}}
%    \begin{macrocode}
\def\xintMinof      {\romannumeral0\xintminof }%
\def\xintminof    #1{\expandafter\XINT_minof_a\romannumeral-`0#1\relax }%
\def\XINT_minof_a #1{\expandafter\XINT_minof_b\romannumeral0\xintraw{#1}\Z }%
\def\XINT_minof_b #1\Z #2%
           {\expandafter\XINT_minof_c\romannumeral-`0#2\Z {#1}\Z}%
\def\XINT_minof_c #1%
           {\xint_gob_til_relax #1\XINT_minof_e\relax\XINT_minof_d #1}%
\def\XINT_minof_d #1\Z
           {\expandafter\XINT_minof_b\romannumeral0\xintmin {#1}}%
\def\XINT_minof_e #1\Z #2\Z { #2}%
%    \end{macrocode}
% \subsection{\csh{xintCmp}}
% \lverb|Rewritten completely in 1.08a to be less dumb when comparing fractions having
% big powers of tens.|
%    \begin{macrocode}
%\def\xintCmp {\romannumeral0\xintcmp }%
\def\xintcmp #1%
{%
    \expandafter\xint_fcmp\expandafter {\romannumeral0\xintraw {#1}}%
}%
\def\xint_fcmp #1#2%
{%
    \expandafter\XINT_fcmp_A\romannumeral0\xintraw {#2}#1%
}%
\def\XINT_fcmp_A #1#2/#3[#4]#5#6/#7[#8]%
{%
    \xint_UDsignsfork
      #1#5\XINT_fcmp_minusminus
       -#5\XINT_fcmp_firstneg
       #1-\XINT_fcmp_secondneg
        --\XINT_fcmp_nonneg_a
    \krof
    #1#5{#2/#3[#4]}{#6/#7[#8]}%
}%
\def\XINT_fcmp_minusminus --#1#2{\XINT_fcmp_B #2#1}%
\def\XINT_fcmp_firstneg #1-#2#3{ -1}%
\def\XINT_fcmp_secondneg -#1#2#3{ 1}%
\def\XINT_fcmp_nonneg_a #1#2%
{%
    \xint_UDzerosfork
      #1#2\XINT_fcmp_zerozero
       0#2\XINT_fcmp_firstzero
       #10\XINT_fcmp_secondzero
        00\XINT_fcmp_pos
    \krof
    #1#2%
}%
\def\XINT_fcmp_zerozero   #1#2#3#4{ 0}%  1.08b had some [ and ] here!!!
\def\XINT_fcmp_firstzero  #1#2#3#4{ -1}% incredibly I never saw that until
\def\XINT_fcmp_secondzero #1#2#3#4{ 1}%  preparing 1.09a.
\def\XINT_fcmp_pos #1#2#3#4%
{%
    \XINT_fcmp_B #1#3#2#4%
}%
\def\XINT_fcmp_B #1/#2[#3]#4/#5[#6]%
{%
    \expandafter\XINT_fcmp_C\expandafter
    {\the\numexpr #6-#3\expandafter}\expandafter
    {\romannumeral0\xintiimul {#4}{#2}}%
    {\romannumeral0\xintiimul {#5}{#1}}%
}%
\def\XINT_fcmp_C #1#2#3%
{%
    \expandafter\XINT_fcmp_D\expandafter
    {#3}{#1}{#2}%
}%
\def\XINT_fcmp_D #1#2#3%
{%
    \expandafter\XINT_cntSgnFork\romannumeral-`0\expandafter\XINT_cntSgn
    \the\numexpr #2+\xintLength{#3}-\xintLength{#1}\relax\Z
    { -1}{\XINT_fcmp_E #2\Z {#3}{#1}}{ 1}%
}%
\def\XINT_fcmp_E #1%
{%
    \xint_UDsignfork
        #1\XINT_fcmp_Fd
         -{\XINT_fcmp_Fn #1}%
    \krof
}%
\def\XINT_fcmp_Fd #1\Z #2#3%
{%
    \expandafter\XINT_fcmp_Fe\expandafter
    {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#3}}{#2}%
}%
\def\XINT_fcmp_Fe #1#2{\XINT_cmp_pre {#2}{#1}}%
\def\XINT_fcmp_Fn #1\Z #2#3%
{%
    \expandafter\XINT_cmp_pre\expandafter
    {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}%
}%
%    \end{macrocode}
% \subsection{\csh{xintAbs}}
% \lverb|Simplified in 1.09i. (original macro had been written before \xintRaw)|
%    \begin{macrocode}
\def\xintAbs   {\romannumeral0\xintabs }%
\def\xintabs #1{\expandafter\XINT_abs\romannumeral0\xintraw {#1}}%
%    \end{macrocode}
% \subsection{\csh{xintOpp}}
% \lverb|caution that -#1 would not be ok if #1 has [n]
% stuff. Simplified in 1.09i. (original macro had been written before \xintRaw)|
%    \begin{macrocode}
\def\xintOpp   {\romannumeral0\xintopp }%
\def\xintopp #1{\expandafter\XINT_opp\romannumeral0\xintraw {#1}}%
%    \end{macrocode}
% \subsection{\csh{xintSgn}}
% \lverb|Simplified in 1.09i. (original macro had been written before \xintRaw)|
%    \begin{macrocode}
\def\xintSgn   {\romannumeral0\xintsgn }%
\def\xintsgn #1{\expandafter\XINT_sgn\romannumeral0\xintraw {#1}\Z }%
%    \end{macrocode}
% \subsection{\csh{xintFloatAdd}, \csh{XINTinFloatAdd}}
% \lverb|1.07; 1.09ka improves a bit the efficieny of the coding of
% \XINT_FL_Add_d.|
%    \begin{macrocode}
\def\xintFloatAdd      {\romannumeral0\xintfloatadd }%
\def\xintfloatadd    #1{\XINT_fladd_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatAdd    {\romannumeral0\XINTinfloatadd }%
\def\XINTinfloatadd  #1{\XINT_fladd_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_fladd_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_fladd_opt
       \else\expandafter\XINT_fladd_noopt
    \fi  #1#2%
}%
\def\XINT_fladd_noopt #1#2\xint_relax #3%
{%
    #1[\XINTdigits]{\XINT_FL_Add {\XINTdigits+\xint_c_ii}{#2}{#3}}%
}%
\def\XINT_fladd_opt #1[\xint_relax #2]#3#4%
{%
    #1[#2]{\XINT_FL_Add {#2+\xint_c_ii}{#3}{#4}}%
}%
\def\XINT_FL_Add #1#2%
{%
    \expandafter\XINT_FL_Add_a\expandafter{\the\numexpr #1\expandafter}%
    \expandafter{\romannumeral0\XINTinfloat [#1]{#2}}%
}%
\def\XINT_FL_Add_a #1#2#3%
{%
    \expandafter\XINT_FL_Add_b\romannumeral0\XINTinfloat [#1]{#3}#2{#1}%
}%
\def\XINT_FL_Add_b #1%
{%
    \xint_gob_til_zero #1\XINT_FL_Add_zero 0\XINT_FL_Add_c #1%
}%
\def\XINT_FL_Add_c #1[#2]#3%
{%
    \xint_gob_til_zero #3\XINT_FL_Add_zerobis 0\XINT_FL_Add_d #1[#2]#3%
}%
\def\XINT_FL_Add_d #1[#2]#3[#4]#5%
{%
    \ifnum \numexpr #2-#4-#5>\xint_c_i
       \expandafter \xint_secondofthree_thenstop
    \else
       \ifnum \numexpr #4-#2-#5>\xint_c_i
              \expandafter\expandafter\expandafter\xint_thirdofthree_thenstop
       \fi
    \fi
    \xintadd {#1[#2]}{#3[#4]}%
}%
\def\XINT_FL_Add_zero 0\XINT_FL_Add_c 0[0]#1[#2]#3{#1[#2]}%
\def\XINT_FL_Add_zerobis 0\XINT_FL_Add_d #1[#2]0[0]#3{#1[#2]}%
%    \end{macrocode}
% \subsection{\csh{xintFloatSub}, \csh{XINTinFloatSub}}
% \lverb|1.07|
%    \begin{macrocode}
\def\xintFloatSub {\romannumeral0\xintfloatsub }%
\def\xintfloatsub    #1{\XINT_flsub_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatSub {\romannumeral0\XINTinfloatsub }%
\def\XINTinfloatsub #1{\XINT_flsub_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flsub_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flsub_opt
       \else\expandafter\XINT_flsub_noopt
    \fi  #1#2%
}%
\def\XINT_flsub_noopt #1#2\xint_relax #3%
{%
    #1[\XINTdigits]{\XINT_FL_Add {\XINTdigits+\xint_c_ii}{#2}{\xintOpp{#3}}}%
}%
\def\XINT_flsub_opt #1[\xint_relax #2]#3#4%
{%
    #1[#2]{\XINT_FL_Add {#2+\xint_c_ii}{#3}{\xintOpp{#4}}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFloatMul}, \csh{XINTinFloatMul}}
% \lverb|1.07|
%    \begin{macrocode}
\def\xintFloatMul    {\romannumeral0\xintfloatmul}%
\def\xintfloatmul    #1{\XINT_flmul_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatMul {\romannumeral0\XINTinfloatmul }%
\def\XINTinfloatmul #1{\XINT_flmul_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flmul_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flmul_opt
       \else\expandafter\XINT_flmul_noopt
    \fi  #1#2%
}%
\def\XINT_flmul_noopt #1#2\xint_relax #3%
{%
    #1[\XINTdigits]{\XINT_FL_Mul {\XINTdigits+\xint_c_ii}{#2}{#3}}%
}%
\def\XINT_flmul_opt #1[\xint_relax #2]#3#4%
{%
    #1[#2]{\XINT_FL_Mul {#2+\xint_c_ii}{#3}{#4}}%
}%
\def\XINT_FL_Mul #1#2%
{%
    \expandafter\XINT_FL_Mul_a\expandafter{\the\numexpr #1\expandafter}%
    \expandafter{\romannumeral0\XINTinfloat [#1]{#2}}%
}%
\def\XINT_FL_Mul_a #1#2#3%
{%
    \expandafter\XINT_FL_Mul_b\romannumeral0\XINTinfloat [#1]{#3}#2%
}%
\def\XINT_FL_Mul_b #1[#2]#3[#4]{\xintE{\xintiiMul {#1}{#3}}{#2+#4}}%
%    \end{macrocode}
% \subsection{\csh{xintFloatDiv}, \csh{XINTinFloatDiv}}
% \lverb|1.07|
%    \begin{macrocode}
\def\xintFloatDiv    {\romannumeral0\xintfloatdiv}%
\def\xintfloatdiv    #1{\XINT_fldiv_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatDiv  {\romannumeral0\XINTinfloatdiv }%
\def\XINTinfloatdiv  #1{\XINT_fldiv_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_fldiv_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_fldiv_opt
       \else\expandafter\XINT_fldiv_noopt
    \fi  #1#2%
}%
\def\XINT_fldiv_noopt #1#2\xint_relax #3%
{%
    #1[\XINTdigits]{\XINT_FL_Div {\XINTdigits+\xint_c_ii}{#2}{#3}}%
}%
\def\XINT_fldiv_opt #1[\xint_relax #2]#3#4%
{%
    #1[#2]{\XINT_FL_Div {#2+\xint_c_ii}{#3}{#4}}%
}%
\def\XINT_FL_Div #1#2%
{%
    \expandafter\XINT_FL_Div_a\expandafter{\the\numexpr #1\expandafter}%
    \expandafter{\romannumeral0\XINTinfloat [#1]{#2}}%
}%
\def\XINT_FL_Div_a #1#2#3%
{%
    \expandafter\XINT_FL_Div_b\romannumeral0\XINTinfloat [#1]{#3}#2%
}%
\def\XINT_FL_Div_b #1[#2]#3[#4]{\xintE{#3/#1}{#4-#2}}%
%    \end{macrocode}
% \subsection{\csh{xintFloatPow}, \csh{XINTinFloatPow}}
% \lverb|1.07. Release 1.09j has re-organized the core loop, and
% \XINT_flpow_prd sub-routine has been removed.|
%    \begin{macrocode}
\def\xintFloatPow {\romannumeral0\xintfloatpow}%
\def\xintfloatpow #1{\XINT_flpow_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatPow {\romannumeral0\XINTinfloatpow }%
\def\XINTinfloatpow #1{\XINT_flpow_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flpow_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flpow_opt
       \else\expandafter\XINT_flpow_noopt
    \fi
     #1#2%
}%
\def\XINT_flpow_noopt  #1#2\xint_relax #3%
{%
   \expandafter\XINT_flpow_checkB_start\expandafter
                {\the\numexpr #3\expandafter}\expandafter
                {\the\numexpr \XINTdigits}{#2}{#1[\XINTdigits]}%
}%
\def\XINT_flpow_opt #1[\xint_relax #2]#3#4%
{%
   \expandafter\XINT_flpow_checkB_start\expandafter
               {\the\numexpr #4\expandafter}\expandafter
               {\the\numexpr #2}{#3}{#1[#2]}%
}%
\def\XINT_flpow_checkB_start #1{\XINT_flpow_checkB_a #1\Z }%
\def\XINT_flpow_checkB_a #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_flpow_BisZero
      0#1{\XINT_flpow_checkB_b 1}%
       0-{\XINT_flpow_checkB_b 0#1}%
    \krof
}%
\def\XINT_flpow_BisZero \Z #1#2#3{#3{1/1[0]}}%
\def\XINT_flpow_checkB_b #1#2\Z #3%
{%
    \expandafter\XINT_flpow_checkB_c \expandafter
    {\romannumeral0\xintlength{#2}}{#3}{#2}#1%
}%
\def\XINT_flpow_checkB_c #1#2%
{%
    \expandafter\XINT_flpow_checkB_d \expandafter
    {\the\numexpr \expandafter\xintLength\expandafter
                  {\the\numexpr #1*20/\xint_c_iii }+#1+#2+\xint_c_i }%
}%
\def\XINT_flpow_checkB_d #1#2#3#4%
{%
    \expandafter \XINT_flpow_a
    \romannumeral0\XINTinfloat [#1]{#4}{#1}{#2}#3%
}%
\def\XINT_flpow_a #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_flpow_zero
      0#1{\XINT_flpow_b 1}%
       0-{\XINT_flpow_b 0#1}%
    \krof
}%
\def\XINT_flpow_b #1#2[#3]#4#5%
{%
    \XINT_flpow_loopI {#5}{#2[#3]}{\romannumeral0\XINTinfloatmul [#4]}%
    {#1*\ifodd #5 1\else 0\fi}%
}%
\def\XINT_flpow_zero [#1]#2#3#4#5%
% xint is not equipped to signal infinity, the 2^31 will provoke
% deliberately a number too big and arithmetic overflow in \XINT_float_Xb
{%
    \if #41\xint_afterfi {\xintError:DivisionByZero #5{1[2147483648]}}%
    \else \xint_afterfi {#5{0[0]}}\fi
}%
\def\XINT_flpow_loopI #1%
{%
    \ifnum #1=\xint_c_i\XINT_flpow_ItoIII\fi
    \ifodd #1
       \expandafter\XINT_flpow_loopI_odd
    \else
       \expandafter\XINT_flpow_loopI_even
    \fi
    {#1}%
}%
\def\XINT_flpow_ItoIII\fi #1\fi #2#3#4#5%
{%
    \fi\expandafter\XINT_flpow_III\the\numexpr #5\relax #3%
}%
\def\XINT_flpow_loopI_even #1#2#3%
{%
    \expandafter\XINT_flpow_loopI\expandafter
    {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter
    {#3{#2}{#2}}{#3}%
}%
\def\XINT_flpow_loopI_odd #1#2#3%
{%
    \expandafter\XINT_flpow_loopII\expandafter
    {\the\numexpr #1/\xint_c_ii-\xint_c_i\expandafter}\expandafter
    {#3{#2}{#2}}{#3}{#2}%
}%
\def\XINT_flpow_loopII #1%
{%
    \ifnum #1 = \xint_c_i\XINT_flpow_IItoIII\fi
    \ifodd #1
       \expandafter\XINT_flpow_loopII_odd
    \else
       \expandafter\XINT_flpow_loopII_even
    \fi
    {#1}%
}%
\def\XINT_flpow_loopII_even #1#2#3%
{%
    \expandafter\XINT_flpow_loopII\expandafter
    {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter
    {#3{#2}{#2}}{#3}%
}%
\def\XINT_flpow_loopII_odd #1#2#3#4%
{%
    \expandafter\XINT_flpow_loopII_odda\expandafter
    {#3{#2}{#4}}{#1}{#2}{#3}%
}%
\def\XINT_flpow_loopII_odda #1#2#3#4%
{%
    \expandafter\XINT_flpow_loopII\expandafter
    {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter
    {#4{#3}{#3}}{#4}{#1}%
}%
\def\XINT_flpow_IItoIII\fi #1\fi #2#3#4#5#6%
{%
    \fi\expandafter\XINT_flpow_III\the\numexpr #6\expandafter\relax
    #4{#3}{#5}%
}%
\def\XINT_flpow_III #1#2[#3]#4%
{%
    \expandafter\XINT_flpow_IIIend\expandafter
    {\the\numexpr\if #41-\fi#3\expandafter}%
    \xint_UDzerofork
        #4{{#2}}%
         0{{1/#2}}%
    \krof #1%
}%
\def\XINT_flpow_IIIend #1#2#3#4%
{%
    \xint_UDzerofork
    #3{#4{#2[#1]}}%
     0{#4{-#2[#1]}}%
    \krof
}%
%    \end{macrocode}
% \subsection{\csh{xintFloatPower}, \csh{XINTinFloatPower}}
% \lverb|1.07. The core loop has been re-organized in 1.09j for some slight
% efficiency gain. |
%    \begin{macrocode}
\def\xintFloatPower {\romannumeral0\xintfloatpower}%
\def\xintfloatpower #1{\XINT_flpower_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatPower {\romannumeral0\XINTinfloatpower}%
\def\XINTinfloatpower #1{\XINT_flpower_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flpower_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flpower_opt
       \else\expandafter\XINT_flpower_noopt
    \fi
     #1#2%
}%
\def\XINT_flpower_noopt  #1#2\xint_relax #3%
{%
   \expandafter\XINT_flpower_checkB_start\expandafter
                {\the\numexpr \XINTdigits\expandafter}\expandafter
                {\romannumeral0\xintnum{#3}}{#2}{#1[\XINTdigits]}%
}%
\def\XINT_flpower_opt #1[\xint_relax #2]#3#4%
{%
   \expandafter\XINT_flpower_checkB_start\expandafter
               {\the\numexpr #2\expandafter}\expandafter
               {\romannumeral0\xintnum{#4}}{#3}{#1[#2]}%
}%
\def\XINT_flpower_checkB_start #1#2{\XINT_flpower_checkB_a #2\Z {#1}}%
\def\XINT_flpower_checkB_a #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_flpower_BisZero
      0#1{\XINT_flpower_checkB_b 1}%
       0-{\XINT_flpower_checkB_b 0#1}%
    \krof
}%
\def\XINT_flpower_BisZero \Z #1#2#3{#3{1/1[0]}}%
\def\XINT_flpower_checkB_b #1#2\Z #3%
{%
    \expandafter\XINT_flpower_checkB_c \expandafter
    {\romannumeral0\xintlength{#2}}{#3}{#2}#1%
}%
\def\XINT_flpower_checkB_c #1#2%
{%
    \expandafter\XINT_flpower_checkB_d \expandafter
    {\the\numexpr \expandafter\xintLength\expandafter
                  {\the\numexpr #1*20/\xint_c_iii }+#1+#2+\xint_c_i }%
}%
\def\XINT_flpower_checkB_d #1#2#3#4%
{%
    \expandafter \XINT_flpower_a
    \romannumeral0\XINTinfloat [#1]{#4}{#1}{#2}#3%
}%
\def\XINT_flpower_a #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_flpow_zero
      0#1{\XINT_flpower_b 1}%
       0-{\XINT_flpower_b 0#1}%
    \krof
}%
\def\XINT_flpower_b #1#2[#3]#4#5%
{%
    \XINT_flpower_loopI {#5}{#2[#3]}{\romannumeral0\XINTinfloatmul [#4]}%
    {#1*\xintiiOdd {#5}}%
}%
\def\XINT_flpower_loopI #1%
{%
    \if1\XINT_isOne {#1}\XINT_flpower_ItoIII\fi
    \if1\xintiiOdd{#1}%
       \expandafter\expandafter\expandafter\XINT_flpower_loopI_odd
    \else
       \expandafter\expandafter\expandafter\XINT_flpower_loopI_even
    \fi
    \expandafter {\romannumeral0\xinthalf{#1}}%
}%
\def\XINT_flpower_ItoIII\fi #1\fi\expandafter #2#3#4#5%
{%
    \fi\expandafter\XINT_flpow_III \the\numexpr #5\relax #3%
}%
\def\XINT_flpower_loopI_even #1#2#3%
{%
    \expandafter\XINT_flpower_toI\expandafter {#3{#2}{#2}}{#1}{#3}%
}%
\def\XINT_flpower_loopI_odd #1#2#3%
{%
    \expandafter\XINT_flpower_toII\expandafter {#3{#2}{#2}}{#1}{#3}{#2}%
}%
\def\XINT_flpower_toI  #1#2{\XINT_flpower_loopI  {#2}{#1}}%
\def\XINT_flpower_toII #1#2{\XINT_flpower_loopII {#2}{#1}}%
\def\XINT_flpower_loopII #1%
{%
    \if1\XINT_isOne {#1}\XINT_flpower_IItoIII\fi
    \if1\xintiiOdd{#1}%
       \expandafter\expandafter\expandafter\XINT_flpower_loopII_odd
    \else
       \expandafter\expandafter\expandafter\XINT_flpower_loopII_even
    \fi
    \expandafter {\romannumeral0\xinthalf{#1}}%
}%
\def\XINT_flpower_loopII_even #1#2#3%
{%
    \expandafter\XINT_flpower_toII\expandafter
    {#3{#2}{#2}}{#1}{#3}%
}%
\def\XINT_flpower_loopII_odd #1#2#3#4%
{%
    \expandafter\XINT_flpower_loopII_odda\expandafter
    {#3{#2}{#4}}{#2}{#3}{#1}%
}%
\def\XINT_flpower_loopII_odda #1#2#3#4%
{%
    \expandafter\XINT_flpower_toII\expandafter
    {#3{#2}{#2}}{#4}{#3}{#1}%
}%
\def\XINT_flpower_IItoIII\fi #1\fi\expandafter #2#3#4#5#6%
{%
    \fi\expandafter\XINT_flpow_III\the\numexpr #6\expandafter\relax
    #4{#3}{#5}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFloatSqrt}, \csh{XINTinFloatSqrt}}
% \lverb|1.08|
%    \begin{macrocode}
\def\xintFloatSqrt     {\romannumeral0\xintfloatsqrt }%
\def\xintfloatsqrt   #1{\XINT_flsqrt_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatSqrt   {\romannumeral0\XINTinfloatsqrt }%
\def\XINTinfloatsqrt #1{\XINT_flsqrt_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flsqrt_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flsqrt_opt
       \else\expandafter\XINT_flsqrt_noopt
    \fi  #1#2%
}%
\def\XINT_flsqrt_noopt #1#2\xint_relax
{%
    #1[\XINTdigits]{\XINT_FL_sqrt \XINTdigits {#2}}%
}%
\def\XINT_flsqrt_opt #1[\xint_relax #2]#3%
{%
    #1[#2]{\XINT_FL_sqrt {#2}{#3}}%
}%
\def\XINT_FL_sqrt #1%
{%
    \ifnum\numexpr #1<\xint_c_xviii
        \xint_afterfi {\XINT_FL_sqrt_a\xint_c_xviii}%
    \else
        \xint_afterfi {\XINT_FL_sqrt_a {#1+\xint_c_i}}%
    \fi
}%
\def\XINT_FL_sqrt_a #1#2%
{%
    \expandafter\XINT_FL_sqrt_checkifzeroorneg
    \romannumeral0\XINTinfloat [#1]{#2}%
}%
\def\XINT_FL_sqrt_checkifzeroorneg #1%
{%
    \xint_UDzerominusfork
     #1-\XINT_FL_sqrt_iszero
     0#1\XINT_FL_sqrt_isneg
      0-{\XINT_FL_sqrt_b #1}%
    \krof
}%
\def\XINT_FL_sqrt_iszero #1[#2]{0[0]}%
\def\XINT_FL_sqrt_isneg  #1[#2]{\xintError:RootOfNegative 0[0]}%
\def\XINT_FL_sqrt_b #1[#2]%
{%
    \ifodd #2
        \xint_afterfi{\XINT_FL_sqrt_c 01}%
    \else
        \xint_afterfi{\XINT_FL_sqrt_c {}0}%
    \fi
    {#1}{#2}%
}%
\def\XINT_FL_sqrt_c #1#2#3#4%
{%
    \expandafter\XINT_flsqrt\expandafter {\the\numexpr #4-#2}{#3#1}%
}%
\def\XINT_flsqrt #1#2%
{%
    \expandafter\XINT_sqrt_a
    \expandafter{\romannumeral0\xintlength {#2}}\XINT_flsqrt_big_d {#2}{#1}%
}%
\def\XINT_flsqrt_big_d #1#2%
{%
   \ifodd #2
     \expandafter\expandafter\expandafter\XINT_flsqrt_big_eB
   \else
     \expandafter\expandafter\expandafter\XINT_flsqrt_big_eA
   \fi
   \expandafter {\the\numexpr (#2-\xint_c_i)/\xint_c_ii }{#1}%
}%
\def\XINT_flsqrt_big_eA  #1#2#3%
{%
    \XINT_flsqrt_big_eA_a #3\Z {#2}{#1}{#3}%
}%
\def\XINT_flsqrt_big_eA_a #1#2#3#4#5#6#7#8#9\Z
{%
    \XINT_flsqrt_big_eA_b {#1#2#3#4#5#6#7#8}%
}%
\def\XINT_flsqrt_big_eA_b #1#2%
{%
    \expandafter\XINT_flsqrt_big_f
    \romannumeral0\XINT_flsqrt_small_e {#2001}{#1}%
}%
\def\XINT_flsqrt_big_eB #1#2#3%
{%
    \XINT_flsqrt_big_eB_a #3\Z {#2}{#1}{#3}%
}%
\def\XINT_flsqrt_big_eB_a #1#2#3#4#5#6#7#8#9%
{%
    \XINT_flsqrt_big_eB_b {#1#2#3#4#5#6#7#8#9}%
}%
\def\XINT_flsqrt_big_eB_b #1#2\Z #3%
{%
    \expandafter\XINT_flsqrt_big_f
    \romannumeral0\XINT_flsqrt_small_e {#30001}{#1}%
}%
\def\XINT_flsqrt_small_e #1#2%
{%
   \expandafter\XINT_flsqrt_small_f\expandafter
   {\the\numexpr #1*#1-#2-\xint_c_i}{#1}%
}%
\def\XINT_flsqrt_small_f #1#2%
{%
   \expandafter\XINT_flsqrt_small_g\expandafter
   {\the\numexpr (#1+#2)/(2*#2)-\xint_c_i }{#1}{#2}%
}%
\def\XINT_flsqrt_small_g #1%
{%
    \ifnum #1>\xint_c_
       \expandafter\XINT_flsqrt_small_h
    \else
       \expandafter\XINT_flsqrt_small_end
    \fi
    {#1}%
}%
\def\XINT_flsqrt_small_h #1#2#3%
{%
    \expandafter\XINT_flsqrt_small_f\expandafter
    {\the\numexpr #2-\xint_c_ii*#1*#3+#1*#1\expandafter}\expandafter
    {\the\numexpr #3-#1}%
}%
\def\XINT_flsqrt_small_end #1#2#3%
{%
    \expandafter\space\expandafter
    {\the\numexpr \xint_c_i+#3*\xint_c_x^iv-
                      (#2*\xint_c_x^iv+#3)/(\xint_c_ii*#3)}%
}%
\def\XINT_flsqrt_big_f #1%
{%
    \expandafter\XINT_flsqrt_big_fa\expandafter
    {\romannumeral0\xintiisqr {#1}}{#1}%
}%
\def\XINT_flsqrt_big_fa #1#2#3#4%
{%
    \expandafter\XINT_flsqrt_big_fb\expandafter
    {\romannumeral0\XINT_dsx_addzerosnofuss
                     {\numexpr #3-\xint_c_viii\relax}{#2}}%
    {\romannumeral0\xintiisub
      {\XINT_dsx_addzerosnofuss
           {\numexpr \xint_c_ii*(#3-\xint_c_viii)\relax}{#1}}{#4}}%
    {#3}%
}%
\def\XINT_flsqrt_big_fb #1#2%
{%
    \expandafter\XINT_flsqrt_big_g\expandafter {#2}{#1}%
}%
\def\XINT_flsqrt_big_g #1#2%
{%
    \expandafter\XINT_flsqrt_big_j
    \romannumeral0\xintiidivision
    {#1}{\romannumeral0\XINT_dbl_pos #2\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W }{#2}%
}%
\def\XINT_flsqrt_big_j #1%
{%
    \if0\XINT_Sgn #1\Z
        \expandafter \XINT_flsqrt_big_end_a
    \else \expandafter \XINT_flsqrt_big_k
    \fi {#1}%
}%
\def\XINT_flsqrt_big_k #1#2#3%
{%
    \expandafter\XINT_flsqrt_big_l\expandafter
    {\romannumeral0\XINT_sub_pre {#3}{#1}}%
    {\romannumeral0\xintiiadd {#2}{\romannumeral0\XINT_sqr {#1}}}%
}%
\def\XINT_flsqrt_big_l #1#2%
{%
   \expandafter\XINT_flsqrt_big_g\expandafter
   {#2}{#1}%
}%
\def\XINT_flsqrt_big_end_a #1#2#3#4#5%
{%
   \expandafter\XINT_flsqrt_big_end_b\expandafter
   {\the\numexpr -#4+#5/\xint_c_ii\expandafter}\expandafter
   {\romannumeral0\xintiisub
    {\XINT_dsx_addzerosnofuss {#4}{#3}}%
    {\xintHalf{\xintiiQuo{\XINT_dsx_addzerosnofuss {#4}{#2}}{#3}}}}%
}%
\def\XINT_flsqrt_big_end_b #1#2{#2[#1]}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xintfrac>\relax
%\def<*xintseries>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xintfrac>
%<*xintseries>
%
% \StoreCodelineNo {xintfrac}
%
% \section{Package \xintseriesnameimp implementation}
% \label{sec:seriesimp}
%
% \localtableofcontents
%
% The commenting is currently (\xintdocdate) very sparse.
%
% \subsection{Catcodes, \protect\eTeX{} and reload detection}
%
% The code for reload detection was initially copied from \textsc{Heiko
% Oberdiek}'s packages, then modified.
%
% The method for catcodes was also initially directly inspired by these
% packages.
%
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode64=11   % @
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \let\z\endgroup
  \expandafter\let\expandafter\x\csname ver@xintseries.sty\endcsname
  \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
  \ifx\csname numexpr\endcsname\relax
     \y{xintseries}{\numexpr not available, aborting input}%
     \aftergroup\endinput
  \else
    \ifx\x\relax   % plain-TeX, first loading of xintseries.sty
      \ifx\w\relax % but xintfrac.sty not yet loaded.
         \def\z{\endgroup\input xintfrac.sty\relax}%
      \fi
    \else
      \def\empty {}%
      \ifx\x\empty % LaTeX, first loading,
      % variable is initialized, but \ProvidesPackage not yet seen
          \ifx\w\relax % xintfrac.sty not yet loaded.
            \def\z{\endgroup\RequirePackage{xintfrac}}%
          \fi
      \else
        \aftergroup\endinput % xintseries already loaded.
      \fi
    \fi
  \fi
\z%
\XINTsetupcatcodes% defined in xintkernel.sty
%    \end{macrocode}
% \subsection{Package identification}
%    \begin{macrocode}
\XINT_providespackage
\ProvidesPackage{xintseries}%
  [2015/08/31 v1.1b Expandable partial sums with xint package (jfB)]%
%    \end{macrocode}
% \subsection{\csh{xintSeries}}
% \lverb|&
% Modified in 1.06 to give the indices first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
% 1.08a adds the forgotten optimization following that previous change.|
%    \begin{macrocode}
\def\xintSeries {\romannumeral0\xintseries }%
\def\xintseries #1#2%
{%
    \expandafter\XINT_series\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_series #1#2#3%
{%
   \ifnum #2<#1
      \xint_afterfi { 0/1[0]}%
   \else
      \xint_afterfi {\XINT_series_loop {#1}{0}{#2}{#3}}%
   \fi
}%
\def\XINT_series_loop #1#2#3#4%
{%
    \ifnum #3>#1 \else \XINT_series_exit \fi
    \expandafter\XINT_series_loop\expandafter
    {\the\numexpr #1+1\expandafter }\expandafter
    {\romannumeral0\xintadd {#2}{#4{#1}}}%
    {#3}{#4}%
}%
\def\XINT_series_exit \fi #1#2#3#4#5#6#7#8%
{%
    \fi\xint_gobble_ii #6%
}%
%    \end{macrocode}
% \subsection{\csh{xintiSeries}}
% \lverb|&
% Modified in 1.06 to give the indices first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
% 1.08a adds the forgotten optimization following that previous change.|
%    \begin{macrocode}
\def\xintiSeries {\romannumeral0\xintiseries }%
\def\xintiseries #1#2%
{%
    \expandafter\XINT_iseries\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_iseries #1#2#3%
{%
   \ifnum #2<#1
      \xint_afterfi { 0}%
   \else
      \xint_afterfi {\XINT_iseries_loop {#1}{0}{#2}{#3}}%
   \fi
}%
\def\XINT_iseries_loop #1#2#3#4%
{%
    \ifnum #3>#1 \else \XINT_iseries_exit \fi
    \expandafter\XINT_iseries_loop\expandafter
    {\the\numexpr #1+1\expandafter }\expandafter
    {\romannumeral0\xintiiadd {#2}{#4{#1}}}%
    {#3}{#4}%
}%
\def\XINT_iseries_exit \fi #1#2#3#4#5#6#7#8%
{%
    \fi\xint_gobble_ii #6%
}%
%    \end{macrocode}
% \subsection{\csh{xintPowerSeries}}
% \lverb|&
% The 1.03 version was very lame and created a build-up of denominators.
% (this was at a time \xintAdd always multiplied denominators, by the way)
% The Horner scheme for polynomial evaluation is used in 1.04, this
% cures the denominator problem and drastically improves the efficiency
% of the macro.
% Modified in 1.06 to give the indices first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
% 1.08a adds the forgotten optimization following that previous change.|
%    \begin{macrocode}
\def\xintPowerSeries {\romannumeral0\xintpowerseries }%
\def\xintpowerseries #1#2%
{%
    \expandafter\XINT_powseries\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_powseries #1#2#3#4%
{%
   \ifnum #2<#1
      \xint_afterfi { 0/1[0]}%
   \else
      \xint_afterfi
      {\XINT_powseries_loop_i {#3{#2}}{#1}{#2}{#3}{#4}}%
   \fi
}%
\def\XINT_powseries_loop_i #1#2#3#4#5%
{%
    \ifnum #3>#2 \else\XINT_powseries_exit_i\fi
    \expandafter\XINT_powseries_loop_ii\expandafter
    {\the\numexpr #3-1\expandafter}\expandafter
    {\romannumeral0\xintmul {#1}{#5}}{#2}{#4}{#5}%
}%
\def\XINT_powseries_loop_ii #1#2#3#4%
{%
   \expandafter\XINT_powseries_loop_i\expandafter
   {\romannumeral0\xintadd {#4{#1}}{#2}}{#3}{#1}{#4}%
}%
\def\XINT_powseries_exit_i\fi #1#2#3#4#5#6#7#8#9%
{%
    \fi \XINT_powseries_exit_ii  #6{#7}%
}%
\def\XINT_powseries_exit_ii #1#2#3#4#5#6%
{%
    \xintmul{\xintPow {#5}{#6}}{#4}%
}%
%    \end{macrocode}
% \subsection{\csh{xintPowerSeriesX}}
% \lverb|&
% Same as \xintPowerSeries except for the initial expansion of the x parameter.
% Modified in 1.06 to give the indices first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
% 1.08a adds the forgotten optimization following that previous change.|
%    \begin{macrocode}
\def\xintPowerSeriesX {\romannumeral0\xintpowerseriesx }%
\def\xintpowerseriesx #1#2%
{%
    \expandafter\XINT_powseriesx\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_powseriesx #1#2#3#4%
{%
   \ifnum #2<#1
      \xint_afterfi { 0/1[0]}%
   \else
      \xint_afterfi
      {\expandafter\XINT_powseriesx_pre\expandafter
                  {\romannumeral-`0#4}{#1}{#2}{#3}%
      }%
   \fi
}%
\def\XINT_powseriesx_pre #1#2#3#4%
{%
    \XINT_powseries_loop_i {#4{#3}}{#2}{#3}{#4}{#1}%
}%
%    \end{macrocode}
% \subsection{\csh{xintRationalSeries}}
% \lverb|&
% This computes F(a)+...+F(b) on the basis of the value of F(a) and the
% ratios F(n)/F(n-1). As in \xintPowerSeries we use an iterative scheme which
% has the great advantage to avoid denominator build-up. This makes exact
% computations possible with exponential type series, which would be completely
% inaccessible to \xintSeries.
% #1=a, #2=b, #3=F(a), #4=ratio function
% Modified in 1.06 to give the indices first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
% 1.08a adds the forgotten optimization following that previous change.|
%    \begin{macrocode}
\def\xintRationalSeries {\romannumeral0\xintratseries }%
\def\xintratseries #1#2%
{%
    \expandafter\XINT_ratseries\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_ratseries #1#2#3#4%
{%
   \ifnum #2<#1
      \xint_afterfi { 0/1[0]}%
   \else
      \xint_afterfi
      {\XINT_ratseries_loop {#2}{1}{#1}{#4}{#3}}%
   \fi
}%
\def\XINT_ratseries_loop #1#2#3#4%
{%
    \ifnum #1>#3 \else\XINT_ratseries_exit_i\fi
    \expandafter\XINT_ratseries_loop\expandafter
    {\the\numexpr #1-1\expandafter}\expandafter
    {\romannumeral0\xintadd {1}{\xintMul {#2}{#4{#1}}}}{#3}{#4}%
}%
\def\XINT_ratseries_exit_i\fi #1#2#3#4#5#6#7#8%
{%
    \fi \XINT_ratseries_exit_ii  #6%
}%
\def\XINT_ratseries_exit_ii #1#2#3#4#5%
{%
    \XINT_ratseries_exit_iii #5%
}%
\def\XINT_ratseries_exit_iii #1#2#3#4%
{%
    \xintmul{#2}{#4}%
}%
%    \end{macrocode}
% \subsection{\csh{xintRationalSeriesX}}
% \lverb|&
% a,b,initial,ratiofunction,x$\ 
% This computes F(a,x)+...+F(b,x) on the basis of the value of F(a,x) and the
% ratios F(n,x)/F(n-1,x). The argument x is first expanded and it is the value
% resulting from this which is used then throughout. The initial term F(a,x)
% must be defined as one-parameter macro which will be given x.
% Modified in 1.06 to give the indices first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
% 1.08a adds the forgotten optimization following that previous change.|
%    \begin{macrocode}
\def\xintRationalSeriesX {\romannumeral0\xintratseriesx }%
\def\xintratseriesx #1#2%
{%
    \expandafter\XINT_ratseriesx\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_ratseriesx #1#2#3#4#5%
{%
   \ifnum #2<#1
      \xint_afterfi { 0/1[0]}%
   \else
      \xint_afterfi
      {\expandafter\XINT_ratseriesx_pre\expandafter
                   {\romannumeral-`0#5}{#2}{#1}{#4}{#3}%
      }%
   \fi
}%
\def\XINT_ratseriesx_pre #1#2#3#4#5%
{%
    \XINT_ratseries_loop {#2}{1}{#3}{#4{#1}}{#5{#1}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFxPtPowerSeries}}
% \lverb|&
% I am not two happy with this piece of code. Will make it more economical
% another day.
% Modified in 1.06 to give the indices first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
% 1.08a: forgot last time some optimization from the change to \numexpr.|
%    \begin{macrocode}
\def\xintFxPtPowerSeries {\romannumeral0\xintfxptpowerseries }%
\def\xintfxptpowerseries #1#2%
{%
    \expandafter\XINT_fppowseries\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_fppowseries #1#2#3#4#5%
{%
   \ifnum #2<#1
      \xint_afterfi { 0}%
   \else
      \xint_afterfi
        {\expandafter\XINT_fppowseries_loop_pre\expandafter
           {\romannumeral0\xinttrunc {#5}{\xintPow {#4}{#1}}}%
          {#1}{#4}{#2}{#3}{#5}%
        }%
   \fi
}%
\def\XINT_fppowseries_loop_pre #1#2#3#4#5#6%
{%
    \ifnum #4>#2 \else\XINT_fppowseries_dont_i \fi
    \expandafter\XINT_fppowseries_loop_i\expandafter
    {\the\numexpr #2+\xint_c_i\expandafter}\expandafter
    {\romannumeral0\xintitrunc {#6}{\xintMul {#5{#2}}{#1}}}%
    {#1}{#3}{#4}{#5}{#6}%
}%
\def\XINT_fppowseries_dont_i \fi\expandafter\XINT_fppowseries_loop_i
    {\fi \expandafter\XINT_fppowseries_dont_ii }%
\def\XINT_fppowseries_dont_ii #1#2#3#4#5#6#7{\xinttrunc {#7}{#2[-#7]}}%
\def\XINT_fppowseries_loop_i #1#2#3#4#5#6#7%
{%
    \ifnum #5>#1 \else \XINT_fppowseries_exit_i \fi
    \expandafter\XINT_fppowseries_loop_ii\expandafter
    {\romannumeral0\xinttrunc {#7}{\xintMul {#3}{#4}}}%
    {#1}{#4}{#2}{#5}{#6}{#7}%
}%
\def\XINT_fppowseries_loop_ii #1#2#3#4#5#6#7%
{%
    \expandafter\XINT_fppowseries_loop_i\expandafter
    {\the\numexpr #2+\xint_c_i\expandafter}\expandafter
    {\romannumeral0\xintiiadd {#4}{\xintiTrunc {#7}{\xintMul {#6{#2}}{#1}}}}%
    {#1}{#3}{#5}{#6}{#7}%
}%
\def\XINT_fppowseries_exit_i\fi\expandafter\XINT_fppowseries_loop_ii
    {\fi \expandafter\XINT_fppowseries_exit_ii }%
\def\XINT_fppowseries_exit_ii #1#2#3#4#5#6#7%
{%
    \xinttrunc {#7}
    {\xintiiadd {#4}{\xintiTrunc {#7}{\xintMul {#6{#2}}{#1}}}[-#7]}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFxPtPowerSeriesX}}
% \lverb|&
% a,b,coeff,x,D$\ 
% Modified in 1.06 to give the indices first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
% 1.08a adds the forgotten optimization following that previous change.|
%    \begin{macrocode}
\def\xintFxPtPowerSeriesX {\romannumeral0\xintfxptpowerseriesx }%
\def\xintfxptpowerseriesx #1#2%
{%
    \expandafter\XINT_fppowseriesx\expandafter
    {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_fppowseriesx #1#2#3#4#5%
{%
   \ifnum #2<#1
      \xint_afterfi { 0}%
   \else
      \xint_afterfi
        {\expandafter \XINT_fppowseriesx_pre \expandafter
         {\romannumeral-`0#4}{#1}{#2}{#3}{#5}%
        }%
   \fi
}%
\def\XINT_fppowseriesx_pre #1#2#3#4#5%
{%
    \expandafter\XINT_fppowseries_loop_pre\expandafter
       {\romannumeral0\xinttrunc {#5}{\xintPow {#1}{#2}}}%
       {#2}{#1}{#3}{#4}{#5}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFloatPowerSeries}}
% \lverb|1.08a. I still have to re-visit \xintFxPtPowerSeries; temporarily I
% just adapted the code to the case of floats.|
%    \begin{macrocode}
\def\xintFloatPowerSeries {\romannumeral0\xintfloatpowerseries }%
\def\xintfloatpowerseries #1{\XINT_flpowseries_chkopt #1\xint_relax }%
\def\XINT_flpowseries_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_flpowseries_opt
       \else\expandafter\XINT_flpowseries_noopt
    \fi
    #1%
}%
\def\XINT_flpowseries_noopt  #1\xint_relax #2%
{%
    \expandafter\XINT_flpowseries\expandafter
    {\the\numexpr #1\expandafter}\expandafter
    {\the\numexpr #2}\XINTdigits
}%
\def\XINT_flpowseries_opt [\xint_relax #1]#2#3%
{%
    \expandafter\XINT_flpowseries\expandafter
    {\the\numexpr #2\expandafter}\expandafter
    {\the\numexpr #3\expandafter}{\the\numexpr #1}%
}%
\def\XINT_flpowseries #1#2#3#4#5%
{%
   \ifnum #2<#1
      \xint_afterfi { 0.e0}%
   \else
      \xint_afterfi
        {\expandafter\XINT_flpowseries_loop_pre\expandafter
           {\romannumeral0\XINTinfloatpow [#3]{#5}{#1}}%
          {#1}{#5}{#2}{#4}{#3}%
        }%
   \fi
}%
\def\XINT_flpowseries_loop_pre #1#2#3#4#5#6%
{%
    \ifnum #4>#2 \else\XINT_flpowseries_dont_i \fi
    \expandafter\XINT_flpowseries_loop_i\expandafter
    {\the\numexpr #2+\xint_c_i\expandafter}\expandafter
    {\romannumeral0\XINTinfloatmul [#6]{#5{#2}}{#1}}%
    {#1}{#3}{#4}{#5}{#6}%
}%
\def\XINT_flpowseries_dont_i \fi\expandafter\XINT_flpowseries_loop_i
    {\fi \expandafter\XINT_flpowseries_dont_ii }%
\def\XINT_flpowseries_dont_ii #1#2#3#4#5#6#7{\xintfloat [#7]{#2}}%
\def\XINT_flpowseries_loop_i #1#2#3#4#5#6#7%
{%
    \ifnum #5>#1 \else \XINT_flpowseries_exit_i \fi
    \expandafter\XINT_flpowseries_loop_ii\expandafter
    {\romannumeral0\XINTinfloatmul [#7]{#3}{#4}}%
    {#1}{#4}{#2}{#5}{#6}{#7}%
}%
\def\XINT_flpowseries_loop_ii #1#2#3#4#5#6#7%
{%
    \expandafter\XINT_flpowseries_loop_i\expandafter
    {\the\numexpr #2+\xint_c_i\expandafter}\expandafter
    {\romannumeral0\XINTinfloatadd [#7]{#4}%
                        {\XINTinfloatmul [#7]{#6{#2}}{#1}}}%
    {#1}{#3}{#5}{#6}{#7}%
}%
\def\XINT_flpowseries_exit_i\fi\expandafter\XINT_flpowseries_loop_ii
    {\fi \expandafter\XINT_flpowseries_exit_ii }%
\def\XINT_flpowseries_exit_ii #1#2#3#4#5#6#7%
{%
    \xintfloatadd [#7]{#4}{\XINTinfloatmul [#7]{#6{#2}}{#1}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFloatPowerSeriesX}}
% \lverb|1.08a|
%    \begin{macrocode}
\def\xintFloatPowerSeriesX {\romannumeral0\xintfloatpowerseriesx }%
\def\xintfloatpowerseriesx #1{\XINT_flpowseriesx_chkopt #1\xint_relax }%
\def\XINT_flpowseriesx_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_flpowseriesx_opt
       \else\expandafter\XINT_flpowseriesx_noopt
    \fi
    #1%
}%
\def\XINT_flpowseriesx_noopt  #1\xint_relax #2%
{%
    \expandafter\XINT_flpowseriesx\expandafter
    {\the\numexpr #1\expandafter}\expandafter
    {\the\numexpr #2}\XINTdigits
}%
\def\XINT_flpowseriesx_opt [\xint_relax #1]#2#3%
{%
    \expandafter\XINT_flpowseriesx\expandafter
    {\the\numexpr #2\expandafter}\expandafter
    {\the\numexpr #3\expandafter}{\the\numexpr #1}%
}%
\def\XINT_flpowseriesx #1#2#3#4#5%
{%
   \ifnum #2<#1
      \xint_afterfi { 0.e0}%
   \else
      \xint_afterfi
        {\expandafter \XINT_flpowseriesx_pre \expandafter
         {\romannumeral-`0#5}{#1}{#2}{#4}{#3}%
        }%
   \fi
}%
\def\XINT_flpowseriesx_pre #1#2#3#4#5%
{%
    \expandafter\XINT_flpowseries_loop_pre\expandafter
       {\romannumeral0\XINTinfloatpow [#5]{#1}{#2}}%
       {#2}{#1}{#3}{#4}{#5}%
}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xintseries>\relax
%\def<*xintcfrac>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xintseries>
%<*xintcfrac>
%
% \StoreCodelineNo {xintseries}
%
% \section{Package \xintcfracnameimp implementation}
% \label{sec:cfracimp}
%
% \localtableofcontents
%
% The commenting is currently (\xintdocdate) very sparse. Release |1.09m|
% (|2014/02/26|) has modified a few things: |\xintFtoCs| and
% |\xintCntoCs| insert spaces after the commas, |\xintCstoF| and
% |\xintCstoCv| authorize spaces in the input also before the commas,
% |\xintCntoCs| does not brace the produced coefficients, new macros
% |\xintFtoC|, |\xintCtoF|, |\xintCtoCv|, |\xintFGtoC|, and
% |\xintGGCFrac|.
%
% \subsection{Catcodes, \protect\eTeX{} and reload detection}
%
% The code for reload detection was initially copied from \textsc{Heiko
% Oberdiek}'s packages, then modified.
%
% The method for catcodes was also initially directly inspired by these
% packages.
%
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode64=11   % @
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \let\z\endgroup
  \expandafter\let\expandafter\x\csname ver@xintcfrac.sty\endcsname
  \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
  \ifx\csname numexpr\endcsname\relax
     \y{xintcfrac}{\numexpr not available, aborting input}%
     \aftergroup\endinput
  \else
    \ifx\x\relax   % plain-TeX, first loading of xintcfrac.sty
      \ifx\w\relax % but xintfrac.sty not yet loaded.
         \def\z{\endgroup\input xintfrac.sty\relax}%
      \fi
    \else
      \def\empty {}%
      \ifx\x\empty % LaTeX, first loading,
      % variable is initialized, but \ProvidesPackage not yet seen
          \ifx\w\relax % xintfrac.sty not yet loaded.
            \def\z{\endgroup\RequirePackage{xintfrac}}%
          \fi
      \else
        \aftergroup\endinput % xintcfrac already loaded.
      \fi
    \fi
  \fi
\z%
\XINTsetupcatcodes% defined in xintkernel.sty
%    \end{macrocode}
% \subsection{Package identification}
%    \begin{macrocode}
\XINT_providespackage
\ProvidesPackage{xintcfrac}%
  [2015/08/31 v1.1b Expandable continued fractions with xint package (jfB)]%
%    \end{macrocode}
% \subsection{\csh{xintCFrac}}
%    \begin{macrocode}
\def\xintCFrac {\romannumeral0\xintcfrac }%
\def\xintcfrac #1%
{%
    \XINT_cfrac_opt_a #1\xint_relax
}%
\def\XINT_cfrac_opt_a #1%
{%
    \ifx[#1\XINT_cfrac_opt_b\fi \XINT_cfrac_noopt #1%
}%
\def\XINT_cfrac_noopt #1\xint_relax
{%
    \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z
    \relax\relax
}%
\def\XINT_cfrac_opt_b\fi\XINT_cfrac_noopt [\xint_relax #1]%
{%
    \fi\csname XINT_cfrac_opt#1\endcsname
}%
\def\XINT_cfrac_optl #1%
{%
    \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z
    \relax\hfill
}%
\def\XINT_cfrac_optc #1%
{%
    \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z
    \relax\relax
}%
\def\XINT_cfrac_optr #1%
{%
    \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z
    \hfill\relax
}%
\def\XINT_cfrac_A #1/#2\Z
{%
    \expandafter\XINT_cfrac_B\romannumeral0\xintiidivision {#1}{#2}{#2}%
}%
\def\XINT_cfrac_B #1#2%
{%
    \XINT_cfrac_C #2\Z {#1}%
}%
\def\XINT_cfrac_C #1%
{%
    \xint_gob_til_zero #1\XINT_cfrac_integer 0\XINT_cfrac_D #1%
}%
\def\XINT_cfrac_integer 0\XINT_cfrac_D 0#1\Z #2#3#4#5{ #2}%
\def\XINT_cfrac_D #1\Z #2#3{\XINT_cfrac_loop_a {#1}{#3}{#1}{{#2}}}%
\def\XINT_cfrac_loop_a
{%
    \expandafter\XINT_cfrac_loop_d\romannumeral0\XINT_div_prepare
}%
\def\XINT_cfrac_loop_d #1#2%
{%
    \XINT_cfrac_loop_e #2.{#1}%
}%
\def\XINT_cfrac_loop_e #1%
{%
    \xint_gob_til_zero #1\xint_cfrac_loop_exit0\XINT_cfrac_loop_f #1%
}%
\def\XINT_cfrac_loop_f #1.#2#3#4%
{%
    \XINT_cfrac_loop_a {#1}{#3}{#1}{{#2}#4}%
}%
\def\xint_cfrac_loop_exit0\XINT_cfrac_loop_f #1.#2#3#4#5#6%
   {\XINT_cfrac_T #5#6{#2}#4\Z }%
\def\XINT_cfrac_T #1#2#3#4%
{%
  \xint_gob_til_Z #4\XINT_cfrac_end\Z\XINT_cfrac_T #1#2{#4+\cfrac{#11#2}{#3}}%
}%
\def\XINT_cfrac_end\Z\XINT_cfrac_T #1#2#3%
{%
    \XINT_cfrac_end_b #3%
}%
\def\XINT_cfrac_end_b \Z+\cfrac#1#2{ #2}%
%    \end{macrocode}
% \subsection{\csh{xintGCFrac}}
%    \begin{macrocode}
\def\xintGCFrac {\romannumeral0\xintgcfrac }%
\def\xintgcfrac #1{\XINT_gcfrac_opt_a #1\xint_relax }%
\def\XINT_gcfrac_opt_a #1%
{%
    \ifx[#1\XINT_gcfrac_opt_b\fi \XINT_gcfrac_noopt #1%
}%
\def\XINT_gcfrac_noopt #1\xint_relax
{%
    \XINT_gcfrac #1+\xint_relax/\relax\relax
}%
\def\XINT_gcfrac_opt_b\fi\XINT_gcfrac_noopt [\xint_relax #1]%
{%
    \fi\csname XINT_gcfrac_opt#1\endcsname
}%
\def\XINT_gcfrac_optl #1%
{%
    \XINT_gcfrac #1+\xint_relax/\relax\hfill
}%
\def\XINT_gcfrac_optc #1%
{%
    \XINT_gcfrac #1+\xint_relax/\relax\relax
}%
\def\XINT_gcfrac_optr #1%
{%
    \XINT_gcfrac #1+\xint_relax/\hfill\relax
}%
\def\XINT_gcfrac
{%
    \expandafter\XINT_gcfrac_enter\romannumeral-`0%
}%
\def\XINT_gcfrac_enter {\XINT_gcfrac_loop {}}%
\def\XINT_gcfrac_loop #1#2+#3/%
{%
    \xint_gob_til_xint_relax #3\XINT_gcfrac_endloop\xint_relax
    \XINT_gcfrac_loop {{#3}{#2}#1}%
}%
\def\XINT_gcfrac_endloop\xint_relax\XINT_gcfrac_loop #1#2#3%
{%
    \XINT_gcfrac_T #2#3#1\xint_relax\xint_relax
}%
\def\XINT_gcfrac_T #1#2#3#4{\XINT_gcfrac_U #1#2{\xintFrac{#4}}}%
\def\XINT_gcfrac_U #1#2#3#4#5%
{%
    \xint_gob_til_xint_relax #5\XINT_gcfrac_end\xint_relax\XINT_gcfrac_U
              #1#2{\xintFrac{#5}%
               \ifcase\xintSgn{#4}
               +\or+\else-\fi
               \cfrac{#1\xintFrac{\xintAbs{#4}}#2}{#3}}%
}%
\def\XINT_gcfrac_end\xint_relax\XINT_gcfrac_U #1#2#3%
{%
    \XINT_gcfrac_end_b #3%
}%
\def\XINT_gcfrac_end_b #1\cfrac#2#3{ #3}%
%    \end{macrocode}
% \subsection{\csh{xintGGCFrac}}
% \lverb|New with 1.09m|
%    \begin{macrocode}
\def\xintGGCFrac {\romannumeral0\xintggcfrac }%
\def\xintggcfrac #1{\XINT_ggcfrac_opt_a #1\xint_relax }%
\def\XINT_ggcfrac_opt_a #1%
{%
    \ifx[#1\XINT_ggcfrac_opt_b\fi \XINT_ggcfrac_noopt #1%
}%
\def\XINT_ggcfrac_noopt #1\xint_relax
{%
    \XINT_ggcfrac #1+\xint_relax/\relax\relax
}%
\def\XINT_ggcfrac_opt_b\fi\XINT_ggcfrac_noopt [\xint_relax #1]%
{%
    \fi\csname XINT_ggcfrac_opt#1\endcsname
}%
\def\XINT_ggcfrac_optl #1%
{%
    \XINT_ggcfrac #1+\xint_relax/\relax\hfill
}%
\def\XINT_ggcfrac_optc #1%
{%
    \XINT_ggcfrac #1+\xint_relax/\relax\relax
}%
\def\XINT_ggcfrac_optr #1%
{%
    \XINT_ggcfrac #1+\xint_relax/\hfill\relax
}%
\def\XINT_ggcfrac
{%
    \expandafter\XINT_ggcfrac_enter\romannumeral-`0%
}%
\def\XINT_ggcfrac_enter {\XINT_ggcfrac_loop {}}%
\def\XINT_ggcfrac_loop #1#2+#3/%
{%
    \xint_gob_til_xint_relax #3\XINT_ggcfrac_endloop\xint_relax
    \XINT_ggcfrac_loop {{#3}{#2}#1}%
}%
\def\XINT_ggcfrac_endloop\xint_relax\XINT_ggcfrac_loop #1#2#3%
{%
    \XINT_ggcfrac_T #2#3#1\xint_relax\xint_relax
}%
\def\XINT_ggcfrac_T #1#2#3#4{\XINT_ggcfrac_U #1#2{#4}}%
\def\XINT_ggcfrac_U #1#2#3#4#5%
{%
    \xint_gob_til_xint_relax #5\XINT_ggcfrac_end\xint_relax\XINT_ggcfrac_U
              #1#2{#5+\cfrac{#1#4#2}{#3}}%
}%
\def\XINT_ggcfrac_end\xint_relax\XINT_ggcfrac_U #1#2#3%
{%
    \XINT_ggcfrac_end_b #3%
}%
\def\XINT_ggcfrac_end_b #1\cfrac#2#3{ #3}%
%    \end{macrocode}
% \subsection{\csh{xintGCtoGCx}}
%    \begin{macrocode}
\def\xintGCtoGCx {\romannumeral0\xintgctogcx }%
\def\xintgctogcx #1#2#3%
{%
    \expandafter\XINT_gctgcx_start\expandafter {\romannumeral-`0#3}{#1}{#2}%
}%
\def\XINT_gctgcx_start #1#2#3{\XINT_gctgcx_loop_a {}{#2}{#3}#1+\xint_relax/}%
\def\XINT_gctgcx_loop_a #1#2#3#4+#5/%
{%
    \xint_gob_til_xint_relax #5\XINT_gctgcx_end\xint_relax
    \XINT_gctgcx_loop_b {#1{#4}}{#2{#5}#3}{#2}{#3}%
}%
\def\XINT_gctgcx_loop_b #1#2%
{%
    \XINT_gctgcx_loop_a {#1#2}%
}%
\def\XINT_gctgcx_end\xint_relax\XINT_gctgcx_loop_b #1#2#3#4{ #1}%
%    \end{macrocode}
% \subsection{\csh{xintFtoCs}}
% \lverb|Modified in 1.09m: a space is added after the inserted commas.|
%    \begin{macrocode}
\def\xintFtoCs {\romannumeral0\xintftocs }%
\def\xintftocs #1%
{%
    \expandafter\XINT_ftc_A\romannumeral0\xintrawwithzeros {#1}\Z
}%
\def\XINT_ftc_A #1/#2\Z
{%
    \expandafter\XINT_ftc_B\romannumeral0\xintiidivision {#1}{#2}{#2}%
}%
\def\XINT_ftc_B #1#2%
{%
    \XINT_ftc_C #2.{#1}%
}%
\def\XINT_ftc_C #1%
{%
    \xint_gob_til_zero #1\XINT_ftc_integer 0\XINT_ftc_D #1%
}%
\def\XINT_ftc_integer 0\XINT_ftc_D 0#1.#2#3{ #2}%
\def\XINT_ftc_D #1.#2#3{\XINT_ftc_loop_a {#1}{#3}{#1}{#2, }}% 1.09m adds a space
\def\XINT_ftc_loop_a
{%
    \expandafter\XINT_ftc_loop_d\romannumeral0\XINT_div_prepare
}%
\def\XINT_ftc_loop_d #1#2%
{%
    \XINT_ftc_loop_e #2.{#1}%
}%
\def\XINT_ftc_loop_e #1%
{%
    \xint_gob_til_zero #1\xint_ftc_loop_exit0\XINT_ftc_loop_f #1%
}%
\def\XINT_ftc_loop_f #1.#2#3#4%
{%
    \XINT_ftc_loop_a {#1}{#3}{#1}{#4#2, }% 1.09m has an added space here
}%
\def\xint_ftc_loop_exit0\XINT_ftc_loop_f #1.#2#3#4{ #4#2}%
%    \end{macrocode}
% \subsection{\csh{xintFtoCx}}
%    \begin{macrocode}
\def\xintFtoCx {\romannumeral0\xintftocx }%
\def\xintftocx #1#2%
{%
    \expandafter\XINT_ftcx_A\romannumeral0\xintrawwithzeros {#2}\Z {#1}%
}%
\def\XINT_ftcx_A #1/#2\Z
{%
    \expandafter\XINT_ftcx_B\romannumeral0\xintiidivision {#1}{#2}{#2}%
}%
\def\XINT_ftcx_B #1#2%
{%
    \XINT_ftcx_C #2.{#1}%
}%
\def\XINT_ftcx_C #1%
{%
    \xint_gob_til_zero #1\XINT_ftcx_integer 0\XINT_ftcx_D #1%
}%
\def\XINT_ftcx_integer 0\XINT_ftcx_D 0#1.#2#3#4{ #2}%
\def\XINT_ftcx_D #1.#2#3#4{\XINT_ftcx_loop_a {#1}{#3}{#1}{{#2}#4}{#4}}%
\def\XINT_ftcx_loop_a
{%
    \expandafter\XINT_ftcx_loop_d\romannumeral0\XINT_div_prepare
}%
\def\XINT_ftcx_loop_d #1#2%
{%
    \XINT_ftcx_loop_e #2.{#1}%
}%
\def\XINT_ftcx_loop_e #1%
{%
    \xint_gob_til_zero #1\xint_ftcx_loop_exit0\XINT_ftcx_loop_f #1%
}%
\def\XINT_ftcx_loop_f #1.#2#3#4#5%
{%
    \XINT_ftcx_loop_a {#1}{#3}{#1}{#4{#2}#5}{#5}%
}%
\def\xint_ftcx_loop_exit0\XINT_ftcx_loop_f #1.#2#3#4#5{ #4{#2}}%
%    \end{macrocode}
% \subsection{\csh{xintFtoC}}
% \lverb|New in 1.09m: this is the same as \xintFtoCx with empty separator. I
% had temporarily during preparation of 1.09m removed braces from \xintFtoCx,
% but I recalled later why that was useful (see doc), thus let's just here do
% \xintFtoCx {}|
%    \begin{macrocode}
\def\xintFtoC {\romannumeral0\xintftoc }%
\def\xintftoc {\xintftocx {}}%
%    \end{macrocode}
% \subsection{\csh{xintFtoGC}}
%    \begin{macrocode}
\def\xintFtoGC {\romannumeral0\xintftogc }%
\def\xintftogc {\xintftocx {+1/}}%
%    \end{macrocode}
% \subsection{\csh{xintFGtoC}}
% \lverb|New with 1.09m of 2014/02/26. Computes the common initial coefficients
% for the two fractions f and g, and outputs them as a sequence of braced
% items.|
%    \begin{macrocode}
\def\xintFGtoC {\romannumeral0\xintfgtoc}%
\def\xintfgtoc#1%
{%
    \expandafter\XINT_fgtc_a\romannumeral0\xintrawwithzeros {#1}\Z
}%
\def\XINT_fgtc_a #1/#2\Z #3%
{%
    \expandafter\XINT_fgtc_b\romannumeral0\xintrawwithzeros {#3}\Z #1/#2\Z { }%
}%
\def\XINT_fgtc_b #1/#2\Z
{%
    \expandafter\XINT_fgtc_c\romannumeral0\xintiidivision {#1}{#2}{#2}%
}%
\def\XINT_fgtc_c #1#2#3#4/#5\Z
{%
    \expandafter\XINT_fgtc_d\romannumeral0\xintiidivision
                                         {#4}{#5}{#5}{#1}{#2}{#3}%
}%
\def\XINT_fgtc_d #1#2#3#4%#5#6#7%
{%
    \xintifEq {#1}{#4}{\XINT_fgtc_da {#1}{#2}{#3}{#4}}%
                      {\xint_thirdofthree}%
}%
\def\XINT_fgtc_da #1#2#3#4#5#6#7%
{%
     \XINT_fgtc_e {#2}{#5}{#3}{#6}{#7{#1}}%
}%
\def\XINT_fgtc_e #1%
{%
    \xintifZero {#1}{\expandafter\xint_firstofone\xint_gobble_iii}%
                    {\XINT_fgtc_f {#1}}%
}%
\def\XINT_fgtc_f #1#2%
{%
   \xintifZero {#2}{\xint_thirdofthree}{\XINT_fgtc_g {#1}{#2}}%
}%
\def\XINT_fgtc_g #1#2#3%
{%
    \expandafter\XINT_fgtc_h\romannumeral0\XINT_div_prepare {#1}{#3}{#1}{#2}%
}%
\def\XINT_fgtc_h #1#2#3#4#5%
{%
    \expandafter\XINT_fgtc_d\romannumeral0\XINT_div_prepare
                       {#4}{#5}{#4}{#1}{#2}{#3}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFtoCC}}
%    \begin{macrocode}
\def\xintFtoCC {\romannumeral0\xintftocc }%
\def\xintftocc #1%
{%
    \expandafter\XINT_ftcc_A\expandafter {\romannumeral0\xintrawwithzeros {#1}}%
}%
\def\XINT_ftcc_A #1%
{%
    \expandafter\XINT_ftcc_B
    \romannumeral0\xintrawwithzeros {\xintAdd {1/2[0]}{#1[0]}}\Z {#1[0]}%
}%
\def\XINT_ftcc_B #1/#2\Z
{%
    \expandafter\XINT_ftcc_C\expandafter {\romannumeral0\xintiiquo {#1}{#2}}%
}%
\def\XINT_ftcc_C #1#2%
{%
    \expandafter\XINT_ftcc_D\romannumeral0\xintsub {#2}{#1}\Z {#1}%
}%
\def\XINT_ftcc_D #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_ftcc_integer
      0#1\XINT_ftcc_En
       0-{\XINT_ftcc_Ep #1}%
    \krof
}%
\def\XINT_ftcc_Ep #1\Z #2%
{%
    \expandafter\XINT_ftcc_loop_a\expandafter
    {\romannumeral0\xintdiv {1[0]}{#1}}{#2+1/}%
}%
\def\XINT_ftcc_En #1\Z #2%
{%
    \expandafter\XINT_ftcc_loop_a\expandafter
    {\romannumeral0\xintdiv {1[0]}{#1}}{#2+-1/}%
}%
\def\XINT_ftcc_integer #1\Z #2{ #2}%
\def\XINT_ftcc_loop_a #1%
{%
    \expandafter\XINT_ftcc_loop_b
    \romannumeral0\xintrawwithzeros {\xintAdd {1/2[0]}{#1}}\Z {#1}%
}%
\def\XINT_ftcc_loop_b #1/#2\Z
{%
    \expandafter\XINT_ftcc_loop_c\expandafter
    {\romannumeral0\xintiiquo {#1}{#2}}%
}%
\def\XINT_ftcc_loop_c #1#2%
{%
    \expandafter\XINT_ftcc_loop_d
    \romannumeral0\xintsub {#2}{#1[0]}\Z {#1}%
}%
\def\XINT_ftcc_loop_d #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_ftcc_end
      0#1\XINT_ftcc_loop_N
       0-{\XINT_ftcc_loop_P #1}%
    \krof
}%
\def\XINT_ftcc_end #1\Z #2#3{ #3#2}%
\def\XINT_ftcc_loop_P #1\Z #2#3%
{%
    \expandafter\XINT_ftcc_loop_a\expandafter
    {\romannumeral0\xintdiv {1[0]}{#1}}{#3#2+1/}%
}%
\def\XINT_ftcc_loop_N #1\Z #2#3%
{%
    \expandafter\XINT_ftcc_loop_a\expandafter
    {\romannumeral0\xintdiv {1[0]}{#1}}{#3#2+-1/}%
}%
%    \end{macrocode}
% \subsection{\csh{xintCtoF}, \csh{xintCstoF}}
% \lverb|1.09m uses \xintCSVtoList on the argument of \xintCstoF to allow
% spaces also before the commas. And the original \xintCstoF code became the
% one of the new \xintCtoF dealing with a braced rather than comma separated
% list.|
%    \begin{macrocode}
\def\xintCstoF {\romannumeral0\xintcstof }%
\def\xintcstof #1%
{%
    \expandafter\XINT_ctf_prep \romannumeral0\xintcsvtolist{#1}\xint_relax
}%
\def\xintCtoF {\romannumeral0\xintctof }%
\def\xintctof #1%
{%
    \expandafter\XINT_ctf_prep \romannumeral-`0#1\xint_relax
}%
\def\XINT_ctf_prep
{%
    \XINT_ctf_loop_a 1001%
}%
\def\XINT_ctf_loop_a #1#2#3#4#5%
{%
    \xint_gob_til_xint_relax #5\XINT_ctf_end\xint_relax
    \expandafter\XINT_ctf_loop_b
    \romannumeral0\xintrawwithzeros {#5}.{#1}{#2}{#3}{#4}%
}%
\def\XINT_ctf_loop_b #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_ctf_loop_c\expandafter
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}%
}%
\def\XINT_ctf_loop_c #1#2%
{%
    \expandafter\XINT_ctf_loop_d\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_ctf_loop_d #1#2%
{%
    \expandafter\XINT_ctf_loop_e\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_ctf_loop_e #1#2%
{%
    \expandafter\XINT_ctf_loop_a\expandafter{#2}#1%
}%
\def\XINT_ctf_end #1.#2#3#4#5{\xintrawwithzeros {#2/#3}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintiCstoF}}
%    \begin{macrocode}
\def\xintiCstoF {\romannumeral0\xinticstof }%
\def\xinticstof #1%
{%
    \expandafter\XINT_icstf_prep \romannumeral-`0#1,\xint_relax,%
}%
\def\XINT_icstf_prep
{%
    \XINT_icstf_loop_a 1001%
}%
\def\XINT_icstf_loop_a #1#2#3#4#5,%
{%
    \xint_gob_til_xint_relax #5\XINT_icstf_end\xint_relax
    \expandafter
    \XINT_icstf_loop_b \romannumeral-`0#5.{#1}{#2}{#3}{#4}%
}%
\def\XINT_icstf_loop_b #1.#2#3#4#5%
{%
    \expandafter\XINT_icstf_loop_c\expandafter
    {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}%
    {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}%
    {#2}{#3}%
}%
\def\XINT_icstf_loop_c #1#2%
{%
    \expandafter\XINT_icstf_loop_a\expandafter {#2}{#1}%
}%
\def\XINT_icstf_end#1.#2#3#4#5{\xintrawwithzeros {#2/#3}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintGCtoF}}
%    \begin{macrocode}
\def\xintGCtoF {\romannumeral0\xintgctof }%
\def\xintgctof #1%
{%
    \expandafter\XINT_gctf_prep \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_gctf_prep
{%
    \XINT_gctf_loop_a 1001%
}%
\def\XINT_gctf_loop_a #1#2#3#4#5+%
{%
    \expandafter\XINT_gctf_loop_b
    \romannumeral0\xintrawwithzeros {#5}.{#1}{#2}{#3}{#4}%
}%
\def\XINT_gctf_loop_b #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_gctf_loop_c\expandafter
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}%
}%
\def\XINT_gctf_loop_c #1#2%
{%
    \expandafter\XINT_gctf_loop_d\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctf_loop_d #1#2%
{%
    \expandafter\XINT_gctf_loop_e\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctf_loop_e #1#2%
{%
    \expandafter\XINT_gctf_loop_f\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctf_loop_f #1#2/%
{%
    \xint_gob_til_xint_relax #2\XINT_gctf_end\xint_relax
    \expandafter\XINT_gctf_loop_g
    \romannumeral0\xintrawwithzeros {#2}.#1%
}%
\def\XINT_gctf_loop_g #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_gctf_loop_h\expandafter
    {\romannumeral0\XINT_mul_fork #1\Z #6\Z }%
    {\romannumeral0\XINT_mul_fork #1\Z #5\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
}%
\def\XINT_gctf_loop_h #1#2%
{%
    \expandafter\XINT_gctf_loop_i\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctf_loop_i #1#2%
{%
    \expandafter\XINT_gctf_loop_j\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctf_loop_j #1#2%
{%
    \expandafter\XINT_gctf_loop_a\expandafter {#2}#1%
}%
\def\XINT_gctf_end #1.#2#3#4#5{\xintrawwithzeros {#2/#3}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintiGCtoF}}
%    \begin{macrocode}
\def\xintiGCtoF {\romannumeral0\xintigctof }%
\def\xintigctof #1%
{%
    \expandafter\XINT_igctf_prep \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_igctf_prep
{%
    \XINT_igctf_loop_a 1001%
}%
\def\XINT_igctf_loop_a #1#2#3#4#5+%
{%
    \expandafter\XINT_igctf_loop_b
    \romannumeral-`0#5.{#1}{#2}{#3}{#4}%
}%
\def\XINT_igctf_loop_b #1.#2#3#4#5%
{%
    \expandafter\XINT_igctf_loop_c\expandafter
    {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}%
    {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}%
    {#2}{#3}%
}%
\def\XINT_igctf_loop_c #1#2%
{%
    \expandafter\XINT_igctf_loop_f\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_igctf_loop_f #1#2#3#4/%
{%
    \xint_gob_til_xint_relax #4\XINT_igctf_end\xint_relax
    \expandafter\XINT_igctf_loop_g
    \romannumeral-`0#4.{#2}{#3}#1%
}%
\def\XINT_igctf_loop_g #1.#2#3%
{%
    \expandafter\XINT_igctf_loop_h\expandafter
    {\romannumeral0\XINT_mul_fork #1\Z #3\Z }%
    {\romannumeral0\XINT_mul_fork #1\Z #2\Z }%
}%
\def\XINT_igctf_loop_h #1#2%
{%
    \expandafter\XINT_igctf_loop_i\expandafter {#2}{#1}%
}%
\def\XINT_igctf_loop_i #1#2#3#4%
{%
    \XINT_igctf_loop_a {#3}{#4}{#1}{#2}%
}%
\def\XINT_igctf_end #1.#2#3#4#5{\xintrawwithzeros {#4/#5}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintCtoCv}, \csh{xintCstoCv}}
% \lverb|1.09m uses \xintCSVtoList on the argument of \xintCstoCv to allow
% spaces also before the commas. The original \xintCstoCv code became the
% one of the new \xintCtoF dealing with a braced rather than comma separated
% list.|
%    \begin{macrocode}
\def\xintCstoCv {\romannumeral0\xintcstocv }%
\def\xintcstocv #1%
{%
    \expandafter\XINT_ctcv_prep\romannumeral0\xintcsvtolist{#1}\xint_relax
}%
\def\xintCtoCv {\romannumeral0\xintctocv }%
\def\xintctocv #1%
{%
    \expandafter\XINT_ctcv_prep\romannumeral-`0#1\xint_relax
}%
\def\XINT_ctcv_prep
{%
    \XINT_ctcv_loop_a {}1001%
}%
\def\XINT_ctcv_loop_a #1#2#3#4#5#6%
{%
    \xint_gob_til_xint_relax #6\XINT_ctcv_end\xint_relax
    \expandafter\XINT_ctcv_loop_b
    \romannumeral0\xintrawwithzeros {#6}.{#2}{#3}{#4}{#5}{#1}%
}%
\def\XINT_ctcv_loop_b #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_ctcv_loop_c\expandafter
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}%
}%
\def\XINT_ctcv_loop_c #1#2%
{%
    \expandafter\XINT_ctcv_loop_d\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_ctcv_loop_d #1#2%
{%
    \expandafter\XINT_ctcv_loop_e\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_ctcv_loop_e #1#2%
{%
    \expandafter\XINT_ctcv_loop_f\expandafter{#2}#1%
}%
\def\XINT_ctcv_loop_f #1#2#3#4#5%
{%
    \expandafter\XINT_ctcv_loop_g\expandafter
    {\romannumeral0\xintrawwithzeros {#1/#2}}{#5}{#1}{#2}{#3}{#4}%
}%
\def\XINT_ctcv_loop_g #1#2{\XINT_ctcv_loop_a {#2{#1}}}% 1.09b removes [0]
\def\XINT_ctcv_end #1.#2#3#4#5#6{ #6}%
%    \end{macrocode}
% \subsection{\csh{xintiCstoCv}}
%    \begin{macrocode}
\def\xintiCstoCv {\romannumeral0\xinticstocv }%
\def\xinticstocv #1%
{%
    \expandafter\XINT_icstcv_prep \romannumeral-`0#1,\xint_relax,%
}%
\def\XINT_icstcv_prep
{%
    \XINT_icstcv_loop_a {}1001%
}%
\def\XINT_icstcv_loop_a #1#2#3#4#5#6,%
{%
    \xint_gob_til_xint_relax #6\XINT_icstcv_end\xint_relax
    \expandafter
    \XINT_icstcv_loop_b \romannumeral-`0#6.{#2}{#3}{#4}{#5}{#1}%
}%
\def\XINT_icstcv_loop_b #1.#2#3#4#5%
{%
    \expandafter\XINT_icstcv_loop_c\expandafter
    {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}%
    {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}%
    {{#2}{#3}}%
}%
\def\XINT_icstcv_loop_c #1#2%
{%
    \expandafter\XINT_icstcv_loop_d\expandafter {#2}{#1}%
}%
\def\XINT_icstcv_loop_d #1#2%
{%
    \expandafter\XINT_icstcv_loop_e\expandafter
    {\romannumeral0\xintrawwithzeros {#1/#2}}{{#1}{#2}}%
}%
\def\XINT_icstcv_loop_e #1#2#3#4{\XINT_icstcv_loop_a {#4{#1}}#2#3}%
\def\XINT_icstcv_end #1.#2#3#4#5#6{ #6}%  1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintGCtoCv}}
%    \begin{macrocode}
\def\xintGCtoCv {\romannumeral0\xintgctocv }%
\def\xintgctocv #1%
{%
    \expandafter\XINT_gctcv_prep \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_gctcv_prep
{%
    \XINT_gctcv_loop_a {}1001%
}%
\def\XINT_gctcv_loop_a #1#2#3#4#5#6+%
{%
    \expandafter\XINT_gctcv_loop_b
    \romannumeral0\xintrawwithzeros {#6}.{#2}{#3}{#4}{#5}{#1}%
}%
\def\XINT_gctcv_loop_b #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_gctcv_loop_c\expandafter
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}%
}%
\def\XINT_gctcv_loop_c #1#2%
{%
    \expandafter\XINT_gctcv_loop_d\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctcv_loop_d #1#2%
{%
    \expandafter\XINT_gctcv_loop_e\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctcv_loop_e #1#2%
{%
    \expandafter\XINT_gctcv_loop_f\expandafter {#2}#1%
}%
\def\XINT_gctcv_loop_f #1#2%
{%
    \expandafter\XINT_gctcv_loop_g\expandafter
    {\romannumeral0\xintrawwithzeros {#1/#2}}{{#1}{#2}}%
}%
\def\XINT_gctcv_loop_g #1#2#3#4%
{%
    \XINT_gctcv_loop_h {#4{#1}}{#2#3}% 1.09b removes [0]
}%
\def\XINT_gctcv_loop_h #1#2#3/%
{%
    \xint_gob_til_xint_relax #3\XINT_gctcv_end\xint_relax
    \expandafter\XINT_gctcv_loop_i
    \romannumeral0\xintrawwithzeros {#3}.#2{#1}%
}%
\def\XINT_gctcv_loop_i #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_gctcv_loop_j\expandafter
    {\romannumeral0\XINT_mul_fork #1\Z #6\Z }%
    {\romannumeral0\XINT_mul_fork #1\Z #5\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
}%
\def\XINT_gctcv_loop_j #1#2%
{%
    \expandafter\XINT_gctcv_loop_k\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctcv_loop_k #1#2%
{%
    \expandafter\XINT_gctcv_loop_l\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctcv_loop_l #1#2%
{%
    \expandafter\XINT_gctcv_loop_m\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctcv_loop_m #1#2{\XINT_gctcv_loop_a {#2}#1}%
\def\XINT_gctcv_end #1.#2#3#4#5#6{ #6}%
%    \end{macrocode}
% \subsection{\csh{xintiGCtoCv}}
%    \begin{macrocode}
\def\xintiGCtoCv {\romannumeral0\xintigctocv }%
\def\xintigctocv #1%
{%
    \expandafter\XINT_igctcv_prep \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_igctcv_prep
{%
    \XINT_igctcv_loop_a {}1001%
}%
\def\XINT_igctcv_loop_a #1#2#3#4#5#6+%
{%
    \expandafter\XINT_igctcv_loop_b
    \romannumeral-`0#6.{#2}{#3}{#4}{#5}{#1}%
}%
\def\XINT_igctcv_loop_b #1.#2#3#4#5%
{%
    \expandafter\XINT_igctcv_loop_c\expandafter
    {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}%
    {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}%
    {{#2}{#3}}%
}%
\def\XINT_igctcv_loop_c #1#2%
{%
    \expandafter\XINT_igctcv_loop_f\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_igctcv_loop_f #1#2#3#4/%
{%
    \xint_gob_til_xint_relax #4\XINT_igctcv_end_a\xint_relax
    \expandafter\XINT_igctcv_loop_g
    \romannumeral-`0#4.#1#2{#3}%
}%
\def\XINT_igctcv_loop_g #1.#2#3#4#5%
{%
    \expandafter\XINT_igctcv_loop_h\expandafter
    {\romannumeral0\XINT_mul_fork #1\Z #5\Z }%
    {\romannumeral0\XINT_mul_fork #1\Z #4\Z }%
    {{#2}{#3}}%
}%
\def\XINT_igctcv_loop_h #1#2%
{%
    \expandafter\XINT_igctcv_loop_i\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_igctcv_loop_i #1#2{\XINT_igctcv_loop_k #2{#2#1}}%
\def\XINT_igctcv_loop_k #1#2%
{%
    \expandafter\XINT_igctcv_loop_l\expandafter
    {\romannumeral0\xintrawwithzeros {#1/#2}}%
}%
\def\XINT_igctcv_loop_l #1#2#3{\XINT_igctcv_loop_a {#3{#1}}#2}%1.09i removes [0]
\def\XINT_igctcv_end_a #1.#2#3#4#5%
{%
    \expandafter\XINT_igctcv_end_b\expandafter
    {\romannumeral0\xintrawwithzeros {#2/#3}}%
}%
\def\XINT_igctcv_end_b #1#2{ #2{#1}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintFtoCv}}
% \lverb|Still uses \xinticstocv \xintFtoCs rather than \xintctocv \xintFtoC.|
%    \begin{macrocode}
\def\xintFtoCv {\romannumeral0\xintftocv }%
\def\xintftocv #1%
{%
    \xinticstocv {\xintFtoCs {#1}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFtoCCv}}
%    \begin{macrocode}
\def\xintFtoCCv {\romannumeral0\xintftoccv }%
\def\xintftoccv #1%
{%
    \xintigctocv {\xintFtoCC {#1}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintCntoF}}
% \lverb|&
% Modified in 1.06 to give the N first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.|
%    \begin{macrocode}
\def\xintCntoF {\romannumeral0\xintcntof }%
\def\xintcntof #1%
{%
    \expandafter\XINT_cntf\expandafter {\the\numexpr #1}%
}%
\def\XINT_cntf #1#2%
{%
   \ifnum #1>\xint_c_
      \xint_afterfi {\expandafter\XINT_cntf_loop\expandafter
                     {\the\numexpr #1-1\expandafter}\expandafter
                     {\romannumeral-`0#2{#1}}{#2}}%
   \else
      \xint_afterfi
         {\ifnum #1=\xint_c_
              \xint_afterfi {\expandafter\space \romannumeral-`0#2{0}}%
          \else \xint_afterfi { }% 1.09m now returns nothing.
          \fi}%
   \fi
}%
\def\XINT_cntf_loop #1#2#3%
{%
    \ifnum #1>\xint_c_ \else \XINT_cntf_exit \fi
    \expandafter\XINT_cntf_loop\expandafter
    {\the\numexpr #1-1\expandafter }\expandafter
    {\romannumeral0\xintadd {\xintDiv {1[0]}{#2}}{#3{#1}}}%
    {#3}%
}%
\def\XINT_cntf_exit \fi
    \expandafter\XINT_cntf_loop\expandafter
    #1\expandafter #2#3%
{%
    \fi\xint_gobble_ii #2%
}%
%    \end{macrocode}
% \subsection{\csh{xintGCntoF}}
% \lverb|Modified in 1.06 to give the N argument first to a \numexpr rather
% than expanding twice. I just use \the\numexpr and maintain the previous code
% after that.|
%    \begin{macrocode}
\def\xintGCntoF {\romannumeral0\xintgcntof }%
\def\xintgcntof #1%
{%
    \expandafter\XINT_gcntf\expandafter {\the\numexpr #1}%
}%
\def\XINT_gcntf #1#2#3%
{%
   \ifnum #1>\xint_c_
      \xint_afterfi {\expandafter\XINT_gcntf_loop\expandafter
                     {\the\numexpr #1-1\expandafter}\expandafter
                     {\romannumeral-`0#2{#1}}{#2}{#3}}%
   \else
      \xint_afterfi
         {\ifnum #1=\xint_c_
              \xint_afterfi {\expandafter\space\romannumeral-`0#2{0}}%
          \else \xint_afterfi { }% 1.09m now returns nothing rather than 0/1[0]
          \fi}%
   \fi
}%
\def\XINT_gcntf_loop #1#2#3#4%
{%
    \ifnum #1>\xint_c_ \else \XINT_gcntf_exit \fi
    \expandafter\XINT_gcntf_loop\expandafter
    {\the\numexpr #1-1\expandafter }\expandafter
    {\romannumeral0\xintadd {\xintDiv {#4{#1}}{#2}}{#3{#1}}}%
    {#3}{#4}%
}%
\def\XINT_gcntf_exit \fi
    \expandafter\XINT_gcntf_loop\expandafter
    #1\expandafter #2#3#4%
{%
    \fi\xint_gobble_ii #2%
}%
%    \end{macrocode}
% \subsection{\csh{xintCntoCs}}
% \lverb|Modified in 1.09m: added spaces after the commas in the produced list.
% Moreover the coefficients are not braced anymore. A slight induced limitation
% is that the macro argument should not contain some explicit comma (cf.
% \XINT_cntcs_exit_b), hence \xintCntoCs {\macro,} with \def\macro,#1{<stuff>}
% would crash. Not a very serious limitation, I believe. |
%    \begin{macrocode}
\def\xintCntoCs {\romannumeral0\xintcntocs }%
\def\xintcntocs #1%
{%
    \expandafter\XINT_cntcs\expandafter {\the\numexpr #1}%
}%
\def\XINT_cntcs #1#2%
{%
   \ifnum #1<0
      \xint_afterfi { }% 1.09i: a 0/1[0] was here, now the macro returns nothing
   \else
      \xint_afterfi {\expandafter\XINT_cntcs_loop\expandafter
                     {\the\numexpr #1-\xint_c_i\expandafter}\expandafter
                     {\romannumeral-`0#2{#1}}{#2}}% produced coeff not braced
   \fi
}%
\def\XINT_cntcs_loop #1#2#3%
{%
    \ifnum #1>-\xint_c_i \else \XINT_cntcs_exit \fi
    \expandafter\XINT_cntcs_loop\expandafter
    {\the\numexpr #1-\xint_c_i\expandafter}\expandafter
    {\romannumeral-`0#3{#1}, #2}{#3}% space added, 1.09m
}%
\def\XINT_cntcs_exit \fi
    \expandafter\XINT_cntcs_loop\expandafter
    #1\expandafter #2#3%
{%
    \fi\XINT_cntcs_exit_b #2%
}%
\def\XINT_cntcs_exit_b #1,{}% romannumeral stopping space already there
%    \end{macrocode}
% \subsection{\csh{xintCntoGC}}
% \lverb|&
% Modified in 1.06 to give the N first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
%
% 1.09m maintains the braces, as the coeff are allowed to be fraction and the
% slash can not be naked in the GC format, contrarily to what happens in
% \xintCntoCs. Also the separators given to \xintGCtoGCx may then fetch the
% coefficients as argument, as they are braced.|
%    \begin{macrocode}
\def\xintCntoGC {\romannumeral0\xintcntogc }%
\def\xintcntogc #1%
{%
    \expandafter\XINT_cntgc\expandafter {\the\numexpr #1}%
}%
\def\XINT_cntgc #1#2%
{%
   \ifnum #1<0
      \xint_afterfi { }% 1.09i there was as strange 0/1[0] here, removed
   \else
      \xint_afterfi {\expandafter\XINT_cntgc_loop\expandafter
                     {\the\numexpr #1-\xint_c_i\expandafter}\expandafter
                     {\expandafter{\romannumeral-`0#2{#1}}}{#2}}%
   \fi
}%
\def\XINT_cntgc_loop #1#2#3%
{%
    \ifnum #1>-\xint_c_i \else \XINT_cntgc_exit \fi
    \expandafter\XINT_cntgc_loop\expandafter
    {\the\numexpr #1-\xint_c_i\expandafter }\expandafter
    {\expandafter{\romannumeral-`0#3{#1}}+1/#2}{#3}%
}%
\def\XINT_cntgc_exit \fi
    \expandafter\XINT_cntgc_loop\expandafter
    #1\expandafter #2#3%
{%
    \fi\XINT_cntgc_exit_b #2%
}%
\def\XINT_cntgc_exit_b #1+1/{ }%
%    \end{macrocode}
% \subsection{\csh{xintGCntoGC}}
% \lverb|&
% Modified in 1.06 to give the N first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.|
%    \begin{macrocode}
\def\xintGCntoGC {\romannumeral0\xintgcntogc }%
\def\xintgcntogc #1%
{%
    \expandafter\XINT_gcntgc\expandafter {\the\numexpr #1}%
}%
\def\XINT_gcntgc #1#2#3%
{%
   \ifnum #1<0
      \xint_afterfi { }% 1.09i now returns nothing
   \else
      \xint_afterfi {\expandafter\XINT_gcntgc_loop\expandafter
                     {\the\numexpr #1-\xint_c_i\expandafter}\expandafter
                     {\expandafter{\romannumeral-`0#2{#1}}}{#2}{#3}}%
   \fi
}%
\def\XINT_gcntgc_loop #1#2#3#4%
{%
    \ifnum #1>-\xint_c_i \else \XINT_gcntgc_exit \fi
    \expandafter\XINT_gcntgc_loop_b\expandafter
    {\expandafter{\romannumeral-`0#4{#1}}/#2}{#3{#1}}{#1}{#3}{#4}%
}%
\def\XINT_gcntgc_loop_b #1#2#3%
{%
    \expandafter\XINT_gcntgc_loop\expandafter
    {\the\numexpr #3-\xint_c_i \expandafter}\expandafter
    {\expandafter{\romannumeral-`0#2}+#1}%
}%
\def\XINT_gcntgc_exit \fi
    \expandafter\XINT_gcntgc_loop_b\expandafter #1#2#3#4#5%
{%
    \fi\XINT_gcntgc_exit_b #1%
}%
\def\XINT_gcntgc_exit_b #1/{ }%
%    \end{macrocode}
% \subsection{\csh{xintCstoGC}}
%    \begin{macrocode}
\def\xintCstoGC {\romannumeral0\xintcstogc }%
\def\xintcstogc #1%
{%
    \expandafter\XINT_cstc_prep \romannumeral-`0#1,\xint_relax,%
}%
\def\XINT_cstc_prep #1,{\XINT_cstc_loop_a {{#1}}}%
\def\XINT_cstc_loop_a #1#2,%
{%
    \xint_gob_til_xint_relax #2\XINT_cstc_end\xint_relax
    \XINT_cstc_loop_b {#1}{#2}%
}%
\def\XINT_cstc_loop_b #1#2{\XINT_cstc_loop_a {#1+1/{#2}}}%
\def\XINT_cstc_end\xint_relax\XINT_cstc_loop_b #1#2{ #1}%
%    \end{macrocode}
% \subsection{\csh{xintGCtoGC}}
%    \begin{macrocode}
\def\xintGCtoGC {\romannumeral0\xintgctogc }%
\def\xintgctogc #1%
{%
    \expandafter\XINT_gctgc_start \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_gctgc_start {\XINT_gctgc_loop_a {}}%
\def\XINT_gctgc_loop_a #1#2+#3/%
{%
    \xint_gob_til_xint_relax #3\XINT_gctgc_end\xint_relax
    \expandafter\XINT_gctgc_loop_b\expandafter
    {\romannumeral-`0#2}{#3}{#1}%
}%
\def\XINT_gctgc_loop_b #1#2%
{%
    \expandafter\XINT_gctgc_loop_c\expandafter
    {\romannumeral-`0#2}{#1}%
}%
\def\XINT_gctgc_loop_c #1#2#3%
{%
    \XINT_gctgc_loop_a {#3{#2}+{#1}/}%
}%
\def\XINT_gctgc_end\xint_relax\expandafter\XINT_gctgc_loop_b
{%
    \expandafter\XINT_gctgc_end_b
}%
\def\XINT_gctgc_end_b #1#2#3{ #3{#1}}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xintcfrac>\relax
%\def<*xintexpr>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xintcfrac>
%<*xintexpr>
%
% \StoreCodelineNo {xintcfrac}
%
% \section{Package \xintexprnameimp implementation}
% \label{sec:exprimp}
%
% \etocstandarddisplaystyle
% \etocstandardlines
% \localtableofcontents
% \etocmarkbothnouc {Package \xintexprnameimp implementation}
%
% The first version was released in June 2013. I was greatly helped in this task
% of writing an expandable parser of infix operations by the comments provided
% in |l3fp-parse.dtx| (in its version as available in April-May 2013). One will
% recognize in particular the idea of the `until' macros; I have not looked into
% the actual |l3fp| code beyond the very useful comments provided in its
% documentation.
%
% A main worry was that my data has no a priori bound on its size; to keep the
% code reasonably efficient, I experimented with a technique of storing and
% retrieving data expandably as \emph{names} of control sequences. Intermediate
% computation results are stored as control sequences |\.=a/b[n]|.
%
% Release |1.1| |[2014/10/28]| has made many extensions, some bug fixes, and
% some breaking changes:
% \begin{description}
% \item[bug fixes]  \begin{itemize}
%   \item |\xintiiexpr| did not strip leading zeroes,
%   \item |\xinttheexpr \xintiexpr 1.23\relax\relax| should have produced |1|,
%     but it produced |1.23|
%   \item the catcode of |;| was not set at package launching time.
%   \end{itemize}
% \item[breaking changes] \begin{itemize}
%   \item in |\xintiiexpr|, |/| does \emph{rounded} division, rather than the
%     Euclidean division (for positive arguments, this is truncated division).
%     The new |//| operator does truncated division,
%   \item the |:| operator for three-way branching is gone, replaced with |??|,
%   \item |1e(3+5)| is now illegal. The number parser identifies |e| and |E|
%     in the same way it does for the decimal mark, earlier versions treated
%     |e| as |E| rather as postfix operators,
%   \item the |add| and |mul| have a new syntax, old syntax is with |`+`| and
%     |`*`| (quotes mandatory), |sum| and |prd| are gone,
%   \item no more special treatment for encountered brace pairs |{..}| by the
%     number scanner, |a/b[N]| notation can be used without use of braces (the
%     |N| will end up as is in a |\numexpr|, it is not parsed by the
%     |\xintexpr|-ession scanner).
%   \item although |&| and \verb+|+ are still available as Boolean
%     operators the use of |&&| and \verb+||+ is strongly recommended.
%     The single letter operators might be assigned some other meaning
%     in later releases (bitwise operations, perhaps). Do not use them.
%   \item place holders for |\xintNewExpr| could be denoted |#1|, |#2|,
%     ... or also, for special purposes |$1|,
%     |$2|, ... Only the first form is now accepted and the special cases
%     previously treated via the second form are now managed via a
%     |protect(...)| function.
% \end{itemize}
% \item[novelties] They are quite a few. \begin{itemize}
%    \item |\xintiexpr|, |\xinttheiexpr| admit an optional argument within brackets
%     |[d]|, they round the computation result (or results, if comma separated)
%     to |d| digits after decimal mark, (the whole computation is done exactly,
%     as in |xintexpr|),
%
%    \item |\xintfloatexpr|, |\xintthefloatexpr| similarly admit an optional
%     argument which serves to keep only |d| digits of precision, getting rid
%     of cumulated uncertainties in the last digits (the whole computation is
%     done according to the precision set via |\xintDigits|),
%
%    \item |\xinttheexpr| and |\xintthefloatexpr| ''pretty-print'' if possible,
%     the former removing unit denominator or |[0]| brackets, the latter
%     avoiding scientific notation if decimal notation is practical,
%
%    \item the |//| does truncated division and |/:| is the associated modulo,
%
%    \item multi-character operators |&&|, \verb+||+, |==|, |<=|, |>=|, |!=|,
%     |**|,
%
%    \item multi-letter infix binary words |'and'|, |'or'|, |'xor'|, |'mod'|
%     (quotes mandatory),
%
%    \item functions |even|, |odd|, 
%
%    \item |\xintdefvar A3:=3.1415;| for variable definitions (non expandable,
%     naturally), usable in subsequent expressions; variable names may contain
%     letters, digits, underscores. They should not start with a digit, the
%     |@| is reserved, and single lowercase and uppercase Latin letters are
%     predefined to work as dummy variables (see next),
%
%    \item generation of comma separated lists |a..b|, |a..[d]..b|,
%
%    \item Python syntax-like list extractors |[list][n:]|, |[list][:n]|,
%     |[list][a:b]| allowing negative indices, but no optional step argument,
%     and |[list][n]| (|n=0| for the number of items in the list),
%
%    \item functions |first|, |last|, |reversed|,
%
%    \item itemwise operations on comma separated lists |a*[list]|, etc.., possible
%     on both sides |a*[list]^b|, an obeying the same precedence rules as with
%     numbers,
%
%    \item |add| and |mul| must use a dummy variable: |add(x(x+1)(x-1), x=-10..10)|,
%
%    \item variable substitutions with |subs|: |subs(subs(add(x^2+y^2,x=1..y),y=t),t=20)|,
%
%    \item sequence generation using |seq| with a dummy variable: |seq(x^3, x=-10..10)|,
%
%    \item simple recursive lists with |rseq|, with |@| given the last value,
%      |rseq(1;2@+1,i=1..10)|,
%
%    \item higher recursion with |rrseq|, |@1|, |@2|, |@3|, |@4|, and |@@(n)|
%      for earlier values, up to |n=K| where |K| is the number of terms of the
%      initial stretch |rrseq(0,1;@1+@2,i=2..100)|,
%
%    \item iteration with |iter| which is like |rrseq| but outputs only the
%      last |K| terms, where |K| was the number of initial terms,
%
%    \item inside |seq|, |rseq|, |rrseq|, |iter|, possibility to use |omit|,
%      |abort| and |break| to control termination,
%
%    \item |n++| potentially infinite index generation for |seq|, |rseq|,
%      |rrseq|, and |iter|, it is advised to use |abort| or |break(..)| at
%      some point,
%
%    \item the |add|, |mul|, |seq|, ... are nestable,
%
%    \item |\xintthecoords| converts a comma separated list of an even number
%      of items to the format as expected by the |TikZ| |coordinates| syntax,
%
%    \item completely rewritten |\xintNewExpr|, |protect| function to handle
%      external macros. However not all constructs are compatible with
%      |\xintNewExpr|.
%
% \end{itemize}
% \end{description}
%
% Comments dating back to earlier releases:
% 
% Roughly speaking, the parser mechanism is as follows: at any given time the
% last found ``operator'' has its associated |until| macro awaiting some news
% from the token flow; first |getnext| expands forward in the hope to construct
% some number, which may come from a parenthesized sub-expression, from some
% braced material, or from a digit by digit scan. After this number has been
% formed the next operator is looked for by the |getop| macro. Once |getop| has
% finished its job, |until| is presented with three tokens: the first one is the
% precedence level of the new found operator (which may be an end of expression
% marker), the second is the operator character token (earlier versions had here
% already some macro name, but in order to keep as much common code to expr and
% floatexpr common as possible, this was modified) of the new found operator, and
% the third one is the newly found number (which was encountered just before the
% new operator).
%
% The |until| macro of the earlier operator examines the precedence level of the
% new found one, and either executes the earlier operator (in the case of a
% binary operation, with the found number and a previously stored one) or it
% delays execution, giving the hand to the |until| macro of the operator having
% been found of higher precedence.
%
% A minus sign acting as prefix gets converted into a (unary) operator
% inheriting the precedence level of the previous operator.
%
% Once the end of the expression is found (it has to be marked by a |\relax|)
% the final result is output as four tokens (five tokens since |1.09j|) the
% first one a catcode 11 exclamation mark, the second one an error generating
% macro, the third one is a protection mechanism, the fourth one a printing
% macro and the fifth is |\.=a/b[n]|. The prefix |\xintthe| makes the output
% printable by killing the first three tokens.
%
% \begin{description}
% \item[{|1.08b [2013/06/14]|}] corrected a problem originating in the attempt
% to attribute a special rôle to braces: expansion could be stopped by space
% tokens, as various macros tried to expand without grabbing what came next.
% They now have a doubled |\romannumeral-`0|.
%
% \item[{|1.09a| |[2013/09/24]|}] has a better mechanism regarding |\xintthe|,
% more commenting and better organization of the code, and most importantly it
% implements functions, comparison operators, logic operators, conditionals. The
% code was reorganized and expansion proceeds a bit differently in order to have
% the |_getnext| and |_getop| codes entirely shared by |\xintexpr| and
% |\xintfloatexpr|. |\xintNewExpr| was rewritten in order to work with the
% standard macro parameter character |#|, to be catcode protected and to also
% allow comma separated expressions.
%
% \item[{|1.09c| |[2013/10/09]|}] added the |bool| and |togl| operators,
% |\xintboolexpr|, and |\xintNewNumExpr|, |\xintNewBoolExpr|. The code for
% |\xintNewExpr| is shared with |float|, |num|, and |bool|-expressions. Also the
% precedence level of the postfix operators |!|, |?| and |:| has been made lower
% than the one of functions. 
%
% \item[{|1.09i| |[2013/12/18]|}] unpacks count and dimen registers and control
% squences, with tacit multiplication. It has also made small improvements.
% (speed gains in macro expansions in quite a few places.)
%
% Also, |1.09i| implements |\xintiiexpr|, |\xinttheiiexpr|. New function |frac|.
% And encapsulation in |\csname..\endcsname| is done with |.=| as first tokens,
% so unpacking with |\string| can be done in a completely escape char agnostic
% way.
%
% \item[{|1.09j| |[2014/01/09]|}] extends the tacit multiplication to the case of
% a sub |\xintexpr|-essions. Also, it now |\xint_protect|s the result of the
% |\xintexpr| full expansions, thus, an |\xintexpr| without |\xintthe| prefix
% can be used not only as the first item within an ``|\fdef|'' as previously but
% also now anywhere within an |\edef|. Five tokens are used to pack the
% computation result rather than the possibly hundreds or thousands of digits of
% an |\xintthe| unlocked result. I deliberately omit a second |\xint_protect|
% which, however would be necessary if some macro |\.=digits/digits[digits]| had
% acquired some expandable meaning elsewhere. But this seems not that probable,
% and adding the protection would mean impacting everything only to allow some
% crazy user which has loaded something else than xint to do an |\edef|... the
% |\xintexpr| computations are otherwise in no way affected if such control
% sequences have a meaning.
%
% \item[{|1.09k| |[2014/01/21]|}] does tacit multiplication also for an opening
% parenthesis encountered during the scanning of a number, or at a time when the
% parser expects an infix operator.
%
% And it adds to the syntax recognition of hexadecimal numbers starting with a
% |"|, and having possibly a fractional part (except in |\xintiiexpr|,
% naturally).
%
% \item[{|1.09kb| |[2014/02/13]|}] fixes the bug introduced in |\xintNewExpr|
% in |1.09i| of December 2013: an |\endlinechar -1| was removed, but without
% it there is a spurious trailing space token in the outputs of the created
% macros, and nesting is then impossible.
%
% \end{description}
%
% This is release \expandafter|\xintbndlversion| of
% \expandafter|\expandafter[\xintbndldate]|.
%
% \subsection{Catcodes, \protect\eTeX{} and reload detection}
%
% The code for reload detection was initially copied from \textsc{Heiko
% Oberdiek}'s packages, then modified.
%
% The method for catcodes was also initially directly inspired by these
% packages.
%
%    \begin{macrocode}
\begingroup\catcode61\catcode48\catcode32=10\relax%
  \catcode13=5    % ^^M
  \endlinechar=13 %
  \catcode123=1   % {
  \catcode125=2   % }
  \catcode64=11   % @
  \catcode35=6    % #
  \catcode44=12   % ,
  \catcode45=12   % -
  \catcode46=12   % .
  \catcode58=12   % :
  \def\z {\endgroup}%
  \expandafter\let\expandafter\x\csname ver@xintexpr.sty\endcsname
  \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname
  \expandafter\let\expandafter\t\csname ver@xinttools.sty\endcsname
  \expandafter
    \ifx\csname PackageInfo\endcsname\relax
      \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
    \else
      \def\y#1#2{\PackageInfo{#1}{#2}}%
    \fi
  \expandafter
  \ifx\csname numexpr\endcsname\relax
     \y{xintexpr}{\numexpr not available, aborting input}%
     \aftergroup\endinput
  \else
    \ifx\x\relax   % plain-TeX, first loading of xintexpr.sty
      \ifx\w\relax % but xintfrac.sty not yet loaded.
         \expandafter\def\expandafter\z\expandafter
                    {\z\input xintfrac.sty\relax}%
      \fi
      \ifx\t\relax % but xinttools.sty not yet loaded.
         \expandafter\def\expandafter\z\expandafter
                    {\z\input xinttools.sty\relax}%
      \fi
    \else
      \def\empty {}%
      \ifx\x\empty % LaTeX, first loading,
      % variable is initialized, but \ProvidesPackage not yet seen
          \ifx\w\relax % xintfrac.sty not yet loaded.
            \expandafter\def\expandafter\z\expandafter
                           {\z\RequirePackage{xintfrac}}%
          \fi
          \ifx\t\relax % xinttools.sty not yet loaded.
            \expandafter\def\expandafter\z\expandafter
                           {\z\RequirePackage{xinttools}}%
          \fi
      \else
        \aftergroup\endinput % xintexpr already loaded.
      \fi
    \fi
  \fi
\z%
\XINTsetupcatcodes%
%    \end{macrocode}
% \subsection{Package identification}
%    \begin{macrocode}
\XINT_providespackage
\ProvidesPackage{xintexpr}%
  [2015/08/31 v1.1b Expandable expression parser (jfB)]%
%    \end{macrocode}
% \subsection{Locking and unlocking}
% je dois réfléchir si je dois bloquer expansion après |unlock_a|, à
% cause de nil.
%    \begin{macrocode}
\def\xint_gob_til_! #1!{}% this ! has catcode 11
\edef\XINT_expr_lockscan#1!{\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }%
\edef\XINT_expr_lockit   #1{\noexpand\expandafter\space\noexpand\csname .=#1\endcsname }%
\def\XINT_expr_inintpart  #1!{\XINT_num{#1}}%
\def\XINT_expr_infracpart #1e#2!{#1![\the\numexpr#2-\xintLength{#1}]!}%
\def\XINT_expr_inexppart  e#1!{![\the\numexpr #1]!}%
\def\XINT_expr_unlock  {\expandafter\XINT_expr_unlock_a\string }%
\def\XINT_expr_unlock_a #1.={}%
\def\XINT_expr_unexpectedtoken {\xintError:ignored }%
\let\XINT_expr_done\space
%    \end{macrocode}
% \subsection{\csh{XINT_expr_wrap}, \csh{XINT_iiexpr_wrap}}
%    \begin{macrocode}
\def\XINT_expr_wrap   { !\XINT_expr_usethe\XINT_protectii\XINT_expr_print }%
\def\XINT_iiexpr_wrap { !\XINT_expr_usethe\XINT_protectii\XINT_iiexpr_print }%
%    \end{macrocode}
% \subsection{\csh{XINT_protectii}, \csh{XINT_expr_usethe}}
%    \begin{macrocode}
\def\XINT_protectii #1{\noexpand\XINT_protectii\noexpand #1\noexpand }%
\protected\def\XINT_expr_usethe\XINT_protectii {\xintError:missing_xintthe!}%
%    \end{macrocode}
% \subsection{\csh{XINT_expr_print}, \csh{XINT_iiexpr_print}, \csh{XINT_boolexpr_print}}
% \lverb|See also the \XINT_flexpr_print which is special, below.|
%    \begin{macrocode}
\def\XINT_expr_print     #1{\xintSPRaw::csv  {\XINT_expr_unlock #1}}%
\def\XINT_iiexpr_print   #1{\xintCSV::csv    {\XINT_expr_unlock #1}}%
\def\XINT_boolexpr_print #1{\xintIsTrue::csv {\XINT_expr_unlock #1}}%
%    \end{macrocode}
% \subsection{\csh{xintexpr}, \csh{xintiexpr}, \csh{xintfloatexpr},
% \csh{xintiiexpr}, \csh{xinttheexpr}, etc\dots}
%    \begin{macrocode}
\def\xintexpr       {\romannumeral0\xinteval      }%
\def\xintiexpr      {\romannumeral0\xintieval     }%
\def\xintfloatexpr  {\romannumeral0\xintfloateval }%
\def\xintiiexpr     {\romannumeral0\xintiieval    }%
\def\xinttheexpr    
   {\romannumeral-`0\expandafter\XINT_expr_print\romannumeral0\xintbareeval  }%
\def\xinttheiexpr     {\romannumeral-`0\xintthe\xintiexpr }%
\def\xintthefloatexpr {\romannumeral-`0\xintthe\xintfloatexpr }%
\def\xinttheiiexpr 
   {\romannumeral-`0\expandafter\XINT_iiexpr_print\romannumeral0\xintbareiieval }%
% \let\xintnumexpr   \xintiexpr    % was deprecated, now obsolete with 1.1
% \let\xintthenumexpr\xinttheiexpr % was deprecated, now obsolete with 1.1
%    \end{macrocode}
% \subsection{\csh{xintthe}}
%    \begin{macrocode}
\def\xintthe #1{\romannumeral-`0\expandafter\xint_gobble_iii\romannumeral-`0#1}%
%    \end{macrocode}
% \subsection{\csh{xintthecoords}}
% \lverb|1.1 Wraps up an even number of comma separated items into pairs of
% TikZ coordinates; for use in the following way: 
%
% coordinates {\xintthecoords\xintfloatexpr ... \relax}
%
% The crazyness with the \csname and unlock is due to TikZ somewhat STRANGE
% control of the TOTAL number of expansions which should not exceed the very low
% value of 100 !! As we implemented \XINT_thecoords_b in an "inline" style for
% efficiency, we need to hide its expansions.
%
% Not to be used as \xintthecoords\xintthefloatexpr, only as
% \xintthecoords\xintfloatexpr (or \xintiexpr etc...). Perhaps \xintthecoords
% could make an extra check, but one should not accustome users to too loose
% requirements!|
%    \begin{macrocode}
\def\xintthecoords  #1{\romannumeral-`0\expandafter\expandafter\expandafter
                     \XINT_thecoords_a
                     \expandafter\xint_gobble_iii\romannumeral0#1}%
\def\XINT_thecoords_a #1#2% #1=print macro, indispensible for scientific notation
   {\expandafter\XINT_expr_unlock\csname.=\expandafter\XINT_thecoords_b
                         \romannumeral-`0#1#2,!,!,^\endcsname }%
\def\XINT_thecoords_b #1#2,#3#4,%
   {\xint_gob_til_! #3\XINT_thecoords_c ! (#1#2, #3#4)\XINT_thecoords_b }%
\def\XINT_thecoords_c #1^{}%
%    \end{macrocode}
% \subsection{\csh{xintbareeval}, \csh{xintbarefloateval}, \csh{xintbareiieval}}
%    \begin{macrocode}
\def\xintbareeval      
   {\expandafter\XINT_expr_until_end_a\romannumeral-`0\XINT_expr_getnext }%
\def\xintbarefloateval 
   {\expandafter\XINT_flexpr_until_end_a\romannumeral-`0\XINT_expr_getnext }%
\def\xintbareiieval    
   {\expandafter\XINT_iiexpr_until_end_a\romannumeral-`0\XINT_expr_getnext }%
%    \end{macrocode}
% \subsection{\csh{xinteval}, \csh{xintiieval}}
%    \begin{macrocode}
\def\xinteval   {\expandafter\XINT_expr_wrap\romannumeral0\xintbareeval }%
\def\xintiieval {\expandafter\XINT_iiexpr_wrap\romannumeral0\xintbareiieval }%
%    \end{macrocode}
% \subsection{\csh{xintieval}, \csh{XINT_iexpr_wrap}}
% \lverb|Optional argument since 1.1|
%    \begin{macrocode}
\def\xintieval #1%
   {\ifx [#1\expandafter\XINT_iexpr_withopt\else\expandafter\XINT_iexpr_noopt \fi #1}%
\def\XINT_iexpr_noopt 
   {\expandafter\XINT_iexpr_wrap \expandafter 0\romannumeral0\xintbareeval }%
\def\XINT_iexpr_withopt [#1]%
{%
    \expandafter\XINT_iexpr_wrap\expandafter
    {\the\numexpr \xint_zapspaces #1 \xint_gobble_i\expandafter}%
    \romannumeral0\xintbareeval 
}%
\def\XINT_iexpr_wrap #1#2%
{%
    \expandafter\XINT_expr_wrap
    \csname .=\xintRound::csv {#1}{\XINT_expr_unlock #2}\endcsname 
}%
%    \end{macrocode}
% \subsection{\csh{xintfloateval}, \csh{XINT_flexpr_wrap}, \csh{XINT_flexpr_print}}
% \lverb|Optional argument since 1.1|
%    \begin{macrocode}
\def\xintfloateval #1%
{%
    \ifx [#1\expandafter\XINT_flexpr_withopt_a\else\expandafter\XINT_flexpr_noopt
    \fi #1%
}%
\def\XINT_flexpr_noopt
{%
   \expandafter\XINT_flexpr_withopt_b\expandafter\xinttheDigits
   \romannumeral0\xintbarefloateval 
}%
\def\XINT_flexpr_withopt_a [#1]%
{%
   \expandafter\XINT_flexpr_withopt_b\expandafter
    {\the\numexpr\xint_zapspaces #1 \xint_gobble_i\expandafter}%
    \romannumeral0\xintbarefloateval 
}%
\def\XINT_flexpr_withopt_b #1#2%
{%
    \expandafter\XINT_flexpr_wrap\csname .;#1.=% ; and not : as before b'cause NewExpr
    \XINTinFloat::csv {#1}{\XINT_expr_unlock #2}\endcsname 
}%
\def\XINT_flexpr_wrap { !\XINT_expr_usethe\XINT_protectii\XINT_flexpr_print }%
\def\XINT_flexpr_print #1%
{%
    \expandafter\xintPFloat::csv
    \romannumeral-`0\expandafter\XINT_expr_unlock_sp\string #1!%
}%
\catcode`: 12
    \def\XINT_expr_unlock_sp #1.;#2.=#3!{{#2}{#3}}%
\catcode`: 11
%    \end{macrocode}
% \subsection{\csh{xintboolexpr}, \csh{xinttheboolexpr}}
%    \begin{macrocode}
\def\xintboolexpr      {\romannumeral0\expandafter\expandafter\expandafter
    \XINT_boolexpr_done \expandafter\xint_gobble_iv\romannumeral0\xinteval }%
\def\xinttheboolexpr   {\romannumeral-`0\expandafter\expandafter\expandafter
    \XINT_boolexpr_print\expandafter\xint_gobble_iv\romannumeral0\xinteval }%
\def\XINT_boolexpr_done { !\XINT_expr_usethe\XINT_protectii\XINT_boolexpr_print }%
%    \end{macrocode}
% \subsection{\csh{xintifboolexpr}, \csh{xintifboolfloatexpr}, \csh{xintifbooliiexpr}}
% \lverb|Do not work with comma separated expressions.|
%    \begin{macrocode}
\def\xintifboolexpr      #1{\romannumeral0\xintifnotzero {\xinttheexpr #1\relax}}%
\def\xintifboolfloatexpr #1{\romannumeral0\xintifnotzero {\xintthefloatexpr #1\relax}}%
\def\xintifbooliiexpr    #1{\romannumeral0\xintifnotzero {\xinttheiiexpr #1\relax}}%
%    \end{macrocode}
% \subsection{Macros handling csv lists on output (for \csh{XINT_expr_print} et
% al. routines)}
% \lverb|Changed completely for 1.1, which adds the optional arguments to
% \xintiexpr and \xintfloatexpr.|
% \subsubsection{\csh{XINT_::_end}}
% \lverb|Le mécanisme est le suivant, #2 est dans des accolades et commence par
% ,<sp>. Donc le gobble se débarrasse du, et le <sp> après brace stripping
% arrête un \romannumeral0 ou \romannumeral-`0|
%    \begin{macrocode}
\def\XINT_::_end #1,#2{\xint_gobble_i #2}%
%    \end{macrocode}
% \subsubsection{\csh{xintCSV::csv}}
% \lverb|pour \xinttheiiexpr. 1.1a adds the \romannumeral-`0 for each item,
% which have no use for \xintiiexpr etc..., but are necessary for \xintNewExpr
% to be able to handle comma separated inputs. I am not sure but I think I had
% them just prior to releasing 1.1 but removed them foolishsly.|
%    \begin{macrocode}
\def\xintCSV::csv #1{\expandafter\XINT_csv::_a\romannumeral-`0#1,^,}%
\def\XINT_csv::_a {\XINT_csv::_b {}}%
\def\XINT_csv::_b #1#2,{\expandafter\XINT_csv::_c \romannumeral-`0#2,{#1}}%
\def\XINT_csv::_c #1{\if ^#1\expandafter\XINT_::_end\fi\XINT_csv::_d #1}%
\def\XINT_csv::_d #1,#2{\XINT_csv::_b {#2, #1}}% possibly, item #1 is empty.
%    \end{macrocode}
% \subsubsection{\csh{xintSPRaw}, \csh{xintSPRaw::csv}}
% \lverb|Pour \xinttheexpr.
% J'avais voulu optimiser en testant si présence ou non de [N],
% cependant reduce() produit résultat sans, et du coup, le /1 peut ne pas
% être retiré. Bon je rajoute un [0] dans reduce. 14/10/25 au moment de
% boucler.
%
% Same added \romannumeral-`0 in 1.1a for \xintNewExpr purposes.|
%    \begin{macrocode}
\def\xintSPRaw    {\romannumeral0\xintspraw }%
\def\xintspraw  #1{\expandafter\XINT_spraw\romannumeral-`0#1[\W]}%
\def\XINT_spraw #1[#2#3]{\xint_gob_til_W #2\XINT_spraw_a\W\XINT_spraw_p #1[#2#3]}%
\def\XINT_spraw_a\W\XINT_spraw_p #1[\W]{ #1}%
\def\XINT_spraw_p #1[\W]{\xintpraw {#1}}%
\def\xintSPRaw::csv #1{\romannumeral0\expandafter\XINT_spraw::_a\romannumeral-`0#1,^,}%
\def\XINT_spraw::_a {\XINT_spraw::_b {}}%
\def\XINT_spraw::_b #1#2,{\expandafter\XINT_spraw::_c \romannumeral-`0#2,{#1}}%
\def\XINT_spraw::_c #1{\if ,#1\xint_dothis\XINT_spraw::_e\fi
                       \if ^#1\xint_dothis\XINT_::_end\fi
                       \xint_orthat\XINT_spraw::_d #1}%
\def\XINT_spraw::_d #1,{\expandafter\XINT_spraw::_e\romannumeral0\XINT_spraw #1[\W],}%
\def\XINT_spraw::_e #1,#2{\XINT_spraw::_b {#2, #1}}%
%    \end{macrocode}
% \subsubsection{\csh{xintIsTrue::csv}}
%    \begin{macrocode}
\def\xintIsTrue::csv #1{\romannumeral0\expandafter\XINT_istrue::_a\romannumeral-`0#1,^,}%
\def\XINT_istrue::_a {\XINT_istrue::_b {}}%
\def\XINT_istrue::_b #1#2,{\expandafter\XINT_istrue::_c \romannumeral-`0#2,{#1}}%
\def\XINT_istrue::_c #1{\if ,#1\xint_dothis\XINT_istrue::_e\fi
                        \if ^#1\xint_dothis\XINT_::_end\fi
                        \xint_orthat\XINT_istrue::_d #1}%
\def\XINT_istrue::_d #1,{\expandafter\XINT_istrue::_e\romannumeral0\xintisnotzero {#1},}%
\def\XINT_istrue::_e #1,#2{\XINT_istrue::_b {#2, #1}}%
%    \end{macrocode}
% \subsubsection{\csh{xintRound::csv}}
% \lverb|Pour \xintiexpr avec argument optionnel (finalement, malgré un
% certain overhead lors de l'exécution, pour économiser du code je ne
% distingue plus les deux cas). Reason for annoying expansion bridge is
% related to \xintNewExpr. Attention utilise \XINT_:::_end.|
%    \begin{macrocode}
\def\XINT_:::_end #1,#2#3{\xint_gobble_i #3}%
\def\xintRound::csv #1#2{\romannumeral0\expandafter\XINT_round::_b\expandafter
    {\the\numexpr#1\expandafter}\expandafter{\expandafter}\romannumeral-`0#2,^,}%
\def\XINT_round::_b #1#2#3,{\expandafter\XINT_round::_c \romannumeral-`0#3,{#1}{#2}}%
\def\XINT_round::_c #1{\if ,#1\xint_dothis\XINT_round::_e\fi
                       \if ^#1\xint_dothis\XINT_:::_end\fi
                       \xint_orthat\XINT_round::_d #1}%
\def\XINT_round::_d #1,#2{%
      \expandafter\XINT_round::_e\romannumeral0\ifnum#2>\xint_c_
      \expandafter\xintround\else\expandafter\xintiround\fi {#2}{#1},{#2}}%
\def\XINT_round::_e #1,#2#3{\XINT_round::_b {#2}{#3, #1}}%
%    \end{macrocode}
% \subsubsection{\csh{XINTinFloat::csv}}
% \lverb|Pour \xintfloatexpr. Attention, prépare sous la forme digits[N] pour
% traitement par les macros. Pas utilisé en sortie. Utilise \XINT_:::_end.
%
% 1.1a I believe this is not needed for \xintNewExpr, as it is removed by
% re-defined by \XINT_flexpr_wrap code, hence no need to add the extra
% \romannumeral-`0. Sub-expressions in \xintNewExpr are not supported.
%
% I didn't start and don't want now to think about it at all.
%|
%    \begin{macrocode}
\def\XINTinFloat::csv #1#2{\romannumeral0\expandafter\XINT_infloat::_b\expandafter
   {\the\numexpr #1\expandafter}\expandafter{\expandafter}\romannumeral-`0#2,^,}%
\def\XINT_infloat::_b #1#2#3,{\XINT_infloat::_c #3,{#1}{#2}}%
\def\XINT_infloat::_c #1{\if ,#1\xint_dothis\XINT_infloat::_e\fi
                       \if ^#1\xint_dothis\XINT_:::_end\fi
                       \xint_orthat\XINT_infloat::_d #1}%
\def\XINT_infloat::_d #1,#2%
        {\expandafter\XINT_infloat::_e\romannumeral0\XINTinfloat [#2]{#1},{#2}}%
\def\XINT_infloat::_e #1,#2#3{\XINT_infloat::_b {#2}{#3, #1}}%
%    \end{macrocode}
% \subsubsection{\csh{xintPFloat::csv}}
% \lverb|Expansion à cause de \xintNewExpr. Attention à l'ordre, pas le même que pour
% \XINTinFloat::csv. Donc c'est cette routine qui imprime. Utilise \XINT_:::_end|
%    \begin{macrocode}
\def\xintPFloat::csv #1#2{\romannumeral0\expandafter\XINT_pfloat::_b\expandafter
   {\the\numexpr #1\expandafter}\expandafter{\expandafter}\romannumeral-`0#2,^,}%
\def\XINT_pfloat::_b #1#2#3,{\expandafter\XINT_pfloat::_c \romannumeral-`0#3,{#1}{#2}}%
\def\XINT_pfloat::_c #1{\if ,#1\xint_dothis\XINT_pfloat::_e\fi
                       \if ^#1\xint_dothis\XINT_:::_end\fi
                       \xint_orthat\XINT_pfloat::_d #1}%
\def\XINT_pfloat::_d #1,#2%
 {\expandafter\XINT_pfloat::_e\romannumeral0\XINT_pfloat_opt [\xint_relax #2]{#1},{#2}}%
\def\XINT_pfloat::_e #1,#2#3{\XINT_pfloat::_b {#2}{#3, #1}}%
%    \end{macrocode}
% \subsection{\csh{XINT_expr_getnext}: fetching some number then an operator}
% \lverb|Big change in 1.1, no attempt to detect braced stuff anymore as the
% [N] notation is implemented otherwise. Now, braces should not be used at
% all; one level removed, then \romannumeral-`0 expansion.|
%    \begin{macrocode}
\def\XINT_expr_getnext #1%
{%
    \expandafter\XINT_expr_getnext_a\romannumeral-`0#1%
}%
\def\XINT_expr_getnext_a #1%
{% screens out sub-expressions and \count or \dimen registers/variables
    \xint_gob_til_! #1\XINT_expr_subexpr !% recall this ! has catcode 11
    \ifcat\relax#1% \count or \numexpr etc... token or count, dimen, skip cs
       \expandafter\XINT_expr_countetc
    \else
       \expandafter\expandafter\expandafter\XINT_expr_getnextfork\expandafter\string
    \fi
    #1%
}%
\def\XINT_expr_subexpr !#1\fi !{\expandafter\XINT_expr_getop\xint_gobble_iii }%
\def\XINT_expr_countetc #1%
{%
    \ifx\count#1\else\ifx#1\dimen\else\ifx#1\numexpr\else\ifx#1\dimexpr\else
    \ifx\skip#1\else\ifx\glueexpr#1\else\ifx\fontdimen#1\else
      \XINT_expr_unpackvar
    \fi\fi\fi\fi\fi\fi\fi
    \expandafter\XINT_expr_getnext\number #1%
}%
\def\XINT_expr_unpackvar\fi\fi\fi\fi\fi\fi\fi\expandafter\XINT_expr_getnext\number #1%
    {\fi\fi\fi\fi\fi\fi\fi\expandafter\XINT_expr_getop\csname .=\number#1\endcsname }%
\begingroup
\lccode`*=`#
\lowercase{\endgroup
\def\XINT_expr_getnextfork #1{%
    \if#1*\xint_dothis {\XINT_expr_scan_macropar *}\fi
    \if#1[\xint_dothis {\xint_c_xviii ({}}\fi
    \if#1+\xint_dothis \XINT_expr_getnext \fi
    \if#1.\xint_dothis {\XINT_expr_scandec_II\XINT_expr_infracpart}\fi
    \if#1-\xint_dothis -\fi
    \if#1(\xint_dothis {\xint_c_xviii ({}}\fi
    \xint_orthat {\XINT_expr_scan_nbr_or_func #1}%
}}%
\def\XINT_expr_scan_macropar #1#2{\expandafter\XINT_expr_getop\csname .=#1#2\endcsname }%
%    \end{macrocode}
% \subsection{The  integer or decimal number or hexa-decimal number or
% function name or variable name or special hacky things big parser}
%    \begin{macrocode}
\catcode96 11 % `
\def\XINT_expr_scan_nbr_or_func #1% this #1 has necessarily here catcode 12
{%
    \if "#1\xint_dothis \XINT_expr_scanhex_I\fi
    \if `#1\xint_dothis {\XINT_expr_onlitteral_`}\fi
    \ifnum \xint_c_ix<1#1 \xint_dothis \XINT_expr_scandec_I\fi
    \xint_orthat \XINT_expr_scanfunc #1%
}%
\catcode96 12 % `
\def\XINT_expr_scandec_I
{%
    \expandafter\XINT_expr_getop\romannumeral-`0\expandafter
    \XINT_expr_lockscan\romannumeral0\expandafter\XINT_expr_inintpart
    \romannumeral-`0\XINT_expr_scanintpart_b
}%
\def\XINT_expr_scandec_II
{%
    \expandafter\XINT_expr_getop\romannumeral-`0\expandafter
    \XINT_expr_lockscan\romannumeral0\expandafter\XINT_expr_inintpart
    \romannumeral-`0\XINT_expr_scanfracpart_b
}%
%    \end{macrocode}
% \subsubsection{Integral part}
%    \begin{macrocode}
\def\XINT_expr_scanintpart_a #1%
{% careful that ! has catcode letter here
    \ifcat \relax #1\xint_dothis{!!#1}\fi % stops the scan
    \if    e#1\xint_dothis{\expandafter\XINT_expr_inexppart
                  \romannumeral-`0\XINT_expr_scanexppart_a e}\fi
    \if    E#1\xint_dothis{\expandafter\XINT_expr_inexppart
                  \romannumeral-`0\XINT_expr_scanexppart_a e}\fi
    % \if    @#1\xint_dothis{!*#1}\fi % tacit multiplication later
    % \if    _#1\xint_dothis{!*#1}\fi % tacit multiplication for variables
    \ifcat a#1\xint_dothis{!!*#1}\fi % includes subexpressions (#1=! letter)
    \xint_orthat {\expandafter\XINT_expr_scanintpart_aa\string #1}%
}%
\def\XINT_expr_scanintpart_aa #1%
{%
    \if .#1\xint_dothis\XINT_expr_scandec_transition\fi
    \ifnum \xint_c_ix<1#1 \xint_dothis\XINT_expr_scanintpart_b\fi
    \xint_orthat {!!}#1%
}%
\def\XINT_expr_scanintpart_b #1#2%
{%
    \expandafter #1\romannumeral-`0\expandafter
    \XINT_expr_scanintpart_a\romannumeral-`0#2%
}%
\def\XINT_expr_scandec_transition .#1%
{%
    \expandafter\XINT_expr_scandec_trans_a\romannumeral-`0#1%
}%
\def\XINT_expr_scandec_trans_a #1%
{%
    \if .#1\xint_dothis{!!..}\fi
    \xint_orthat {\expandafter\XINT_expr_infracpart
                  \romannumeral-`0\XINT_expr_scanfracpart_a #1}%
}%
%    \end{macrocode}
% \subsubsection{Fractional part}
%    \begin{macrocode}
\def\XINT_expr_scanfracpart_a #1%
{%
    \ifcat \relax #1\xint_dothis{e!#1}\fi % stops the scan
    \if    e#1\xint_dothis{\XINT_expr_scanexppart_a e}\fi
    \if    E#1\xint_dothis{\XINT_expr_scanexppart_a e}\fi
    \ifcat a#1\xint_dothis{e!*#1}\fi % and also the case of subexpressions (!)
    \xint_orthat {\expandafter\XINT_expr_scanfracpart_aa\string #1}%
}%
\def\XINT_expr_scanfracpart_aa #1%
{%
    \ifnum \xint_c_ix<1#1
       \expandafter\XINT_expr_scanfracpart_b
    \else
       \xint_afterfi {e!}%
    \fi
    #1%
}%
\def\XINT_expr_scanfracpart_b #1#2%
{%
    \expandafter #1\romannumeral-`0\expandafter
    \XINT_expr_scanfracpart_a\romannumeral-`0#2%
}%
%    \end{macrocode}
% \subsubsection{Scientific notation}
%    \begin{macrocode}
\def\XINT_expr_scanexppart_a #1#2%
{%
    \expandafter #1\romannumeral-`0\expandafter
    \XINT_expr_scanexppart_b\romannumeral-`0#2%
}%
\def\XINT_expr_scanexppart_b #1%
{%
    \ifcat \relax #1\xint_dothis{0!#1}\fi % stops the scan (incorrect syntax)
    \ifcat a#1\xint_dothis{0!*#1}\fi % idem
    \if    +#1\xint_dothis {\XINT_expr_scanexppart_a +}\fi
    \if    -#1\xint_dothis {\XINT_expr_scanexppart_a -}\fi
    \xint_orthat {\expandafter\XINT_expr_scanexppart_c\string #1}%
}%
\def\XINT_expr_scanexppart_c #1%
{%
    \ifnum \xint_c_ix<1#1
       \expandafter\XINT_expr_scanexppart_d
    \else
       \expandafter !%
    \fi
    #1%
}%
\def\XINT_expr_scanexppart_d #1#2%
{%
    \expandafter #1\romannumeral-`0\expandafter
    \XINT_expr_scanexppart_e\romannumeral-`0#2%
}%
\def\XINT_expr_scanexppart_e #1%
{%
    \ifcat \relax #1\xint_dothis{!#1}\fi % stops the scan
    \ifcat a#1\xint_dothis{!*#1}\fi % idem
    \xint_orthat {\expandafter\XINT_expr_scanexppart_f\string #1}%
}%
\def\XINT_expr_scanexppart_f #1%
{%
    \ifnum \xint_c_ix<1#1
       \expandafter\XINT_expr_scanexppart_d
    \else
       \expandafter !%
    \fi
    #1%
}%
%    \end{macrocode}
% \subsubsection{Hexadecimal numbers}
%    \begin{macrocode}
\def\XINT_expr_scanhex_I #1%
{%
    \expandafter\XINT_expr_getop\romannumeral-`0\expandafter
    \XINT_expr_lockscan\expandafter\XINT_expr_inhex
    \romannumeral-`0\XINT_expr_scanhexI_a
}%
\def\XINT_expr_inhex #1.#2#3;% expanded inside \csname..\endcsname
{%
    \if#2I\xintHexToDec{#1}%
    \else
      \xintiiMul{\xintiiPow{625}{\xintLength{#3}}}{\xintHexToDec{#1#3}}%
      [\the\numexpr-4*\xintLength{#3}]%
    \fi
}%
\def\XINT_expr_scanhexI_a #1%
{%
    \ifcat #1\relax\xint_dothis{.I;!#1}\fi
    \ifx        !#1\xint_dothis{.I;!*!}\fi % tacit multiplication
    \xint_orthat {\expandafter\XINT_expr_scanhexI_aa\string #1}%
}%
\def\XINT_expr_scanhexI_aa #1%
{%
    \if\ifnum`#1>`/
       \ifnum`#1>`9
       \ifnum`#1>`@
       \ifnum`#1>`F
       0\else1\fi\else0\fi\else1\fi\else0\fi 1%
       \expandafter\XINT_expr_scanhexI_b
    \else
       \if .#1%
           \expandafter\xint_firstoftwo
       \else % gather what we got so far, leave catcode 12 #1 in stream
           \expandafter\xint_secondoftwo
       \fi
       {\expandafter\XINT_expr_scanhex_transition}%
       {\xint_afterfi {.I;!}}%
    \fi
    #1%
}%
\def\XINT_expr_scanhexI_b #1#2%
{%
    \expandafter #1\romannumeral-`0\expandafter
    \XINT_expr_scanhexI_a\romannumeral-`0#2%
}%
\def\XINT_expr_scanhex_transition .#1%
{%
    \expandafter.\expandafter.\romannumeral-`0\expandafter
    \XINT_expr_scanhexII_a\romannumeral-`0#1%
}%
\def\XINT_expr_scanhexII_a #1%
{%
    \ifcat #1\relax\xint_dothis{;!#1}\fi
    \ifx        !#1\xint_dothis{;!*!}\fi % tacit multiplication
    \xint_orthat {\expandafter\XINT_expr_scanhexII_aa\string #1}%
}%
\def\XINT_expr_scanhexII_aa #1%
{%
    \if\ifnum`#1>`/
       \ifnum`#1>`9
       \ifnum`#1>`@
       \ifnum`#1>`F
       0\else1\fi\else0\fi\else1\fi\else0\fi 1%
       \expandafter\XINT_expr_scanhexII_b
    \else
       \xint_afterfi {;!}%
    \fi
    #1%
}%
\def\XINT_expr_scanhexII_b #1#2%
{%
    \expandafter #1\romannumeral-`0\expandafter
    \XINT_expr_scanhexII_a\romannumeral-`0#2%
}%
%    \end{macrocode}
% \subsubsection{Function and variable names}
%    \begin{macrocode}
\def\XINT_expr_scanfunc
{%
    \expandafter\XINT_expr_func\romannumeral-`0\XINT_expr_scanfunc_a
}%
\def\XINT_expr_scanfunc_a #1#2%
{%
    \expandafter #1\romannumeral-`0\expandafter\XINT_expr_scanfunc_b\romannumeral-`0#2%
}%
\def\XINT_expr_scanfunc_b #1%
{%
  \ifx !#1\xint_dothis{\xint_firstoftwo{(_*!}}\fi
  \ifcat \relax#1\xint_dothis{(_}\fi 
  \if (#1\xint_dothis{\xint_firstoftwo{(`}}\fi
  \if _#1\xint_dothis \XINT_expr_scanfunc_a \fi
  \if @#1\xint_dothis \XINT_expr_scanfunc_a \fi
  \ifnum \xint_c_ix<1\string#1 \xint_dothis \XINT_expr_scanfunc_a \fi
  \ifcat a#1\xint_dothis \XINT_expr_scanfunc_a \fi
  \xint_orthat {(_}%
    #1%
}%
\def\XINT_expr_func #1(#2%
{% #2=` pour une fonction, #2=_ pour une variable
   \if #2`\ifcsname XINT_expr_var_#1\endcsname
          \expandafter\expandafter\expandafter\xint_thirdofthree
   \fi\fi
   \xint_firstoftwo {\xint_c_xviii #2{#1}}{\xint_c_xviii _{#1}*(}%
}%
%    \end{macrocode}
% \subsection{\csh{XINT_expr_getop}: finding the next operator or closing
% parenthesis or end of expression}
% \lverb|Release 1.1 implements multi-character operators.|
%    \begin{macrocode}
\def\XINT_expr_getop #1#2% this #1 is the current locked computed value
{%
    \expandafter\XINT_expr_getop_a\expandafter #1\romannumeral-`0#2%
}%
\catcode`* 11
\def\XINT_expr_getop_a #1#2%
{%
    \ifx   \relax #2\xint_dothis\xint_firstofthree\fi
    \ifcat \relax #2\xint_dothis\xint_secondofthree\fi
    \if    _#2\xint_dothis      \xint_secondofthree\fi
    \if    @#2\xint_dothis      \xint_secondofthree\fi
    \if    (#2\xint_dothis      \xint_secondofthree\fi
    \ifx   !#2\xint_dothis      \xint_secondofthree\fi
    \xint_orthat \xint_thirdofthree
    {\XINT_expr_foundend #1}%
    {\XINT_expr_precedence_* *#1#2}% tacit multiplication
    {\XINT_expr_getop_b #2#1}%
}%
\catcode`* 12
\def\XINT_expr_foundend {\xint_c_ \relax }% \relax is a place holder here.
\def\XINT_expr_getop_b #1%
{% ? and : a special syntax in \xintexpr as they are
 % followed by braced arguments, and thus we must intercept them here.
 % I wanted to change this but now I don't have time to think about it.
 % 1.1 removes : as logic operator. Replaced by ??.
     \if '#1\xint_dothis{\XINT_expr_binopwrd }\fi
     \if ?#1\xint_dothis{\XINT_expr_precedence_? ?}\fi
     \xint_orthat       {\XINT_expr_scanop_a #1}%
}%
\def\XINT_expr_binopwrd #1#2'{\expandafter\XINT_expr_foundop_a
    \csname XINT_expr_itself_\xint_zapspaces #2 \xint_gobble_i\endcsname #1}%
\def\XINT_expr_scanop_a #1#2#3%
    {\expandafter\XINT_expr_scanop_b\expandafter #1\expandafter #2\romannumeral-`0#3}%
\def\XINT_expr_scanop_b #1#2#3%
{%
  \ifcat#3\relax\xint_dothis{\XINT_expr_foundop_a #1#2#3}\fi
  \ifcsname XINT_expr_itself_#1#3\endcsname
  \xint_dothis
        {\expandafter\XINT_expr_scanop_c\csname XINT_expr_itself_#1#3\endcsname #2}\fi
  \xint_orthat {\XINT_expr_foundop_a #1#2#3}%
}%
\def\XINT_expr_scanop_c #1#2#3%
{%
  \expandafter\XINT_expr_scanop_d\expandafter #1\expandafter #2\romannumeral-`0#3%
}%
\def\XINT_expr_scanop_d #1#2#3%
{%
  \ifcat#3\relax \xint_dothis{\XINT_expr_foundop #1#2#3}\fi
  \ifcsname XINT_expr_itself_#1#3\endcsname
  \xint_dothis
        {\expandafter\XINT_expr_scanop_c\csname XINT_expr_itself_#1#3\endcsname #2}\fi
  \xint_orthat {\csname XINT_expr_precedence_#1\endcsname #1#2#3}%
}%
\def\XINT_expr_foundop_a #1%
{%
    \ifcsname XINT_expr_precedence_#1\endcsname
        \csname XINT_expr_precedence_#1\expandafter\endcsname
        \expandafter #1%
    \else
        \xint_afterfi{\XINT_expr_unknown_operator {#1}\XINT_expr_getop}%
    \fi
}%
\def\XINT_expr_unknown_operator #1{\xintError:removed \xint_gobble_i {#1}}%
\def\XINT_expr_foundop #1{\csname XINT_expr_precedence_#1\endcsname #1}%
%    \end{macrocode}
% \subsection{Opening and closing parentheses, square brackets for lists, the
% \textasciicircum C for omit and abort within seq or rseq}
%    \begin{macrocode}
\catcode`) 11
\def\XINT_tmpa #1#2#3#4% (avant #4#5)
{%
    \def#1##1%
    {%
        \xint_UDsignfork
                     ##1{\expandafter#1\romannumeral-`0#3}%
                       -{#2##1}%
        \krof
    }%
    \def#2##1##2%
    {%
        \ifcase ##1\xint_afterfi
        {\ifx\XINT_expr_itself_^C ##2\xint_dothis
        {\expandafter#1\romannumeral-`0\expandafter\XINT_expr_getnext\xint_gobble_i}\fi
         \xint_orthat \XINT_expr_done }%
        \or\xint_afterfi{\XINT_expr_extra_)
                          \expandafter #1\romannumeral-`0\XINT_expr_getop }%
        \else
        \xint_afterfi{\expandafter#1\romannumeral-`0\csname XINT_#4_op_##2\endcsname }%
        \fi
    }%
}%
\def\XINT_expr_extra_) {\xintError:removed }%
\xintFor #1 in {expr,flexpr,iiexpr} \do {%
    \expandafter\XINT_tmpa
    \csname XINT_#1_until_end_a\expandafter\endcsname
    \csname XINT_#1_until_end_b\expandafter\endcsname
    \csname XINT_#1_op_-vi\endcsname
    {#1}%
}%
\def\XINT_tmpa #1#2#3#4#5#6%
{%
    \def #1##1{\expandafter #3\romannumeral-`0\XINT_expr_getnext }%
    \def #2{\expandafter #3\romannumeral-`0\XINT_expr_getnext }%
    \def #3##1{\xint_UDsignfork
                ##1{\expandafter #3\romannumeral-`0#5}%
                  -{#4##1}%
               \krof }%
    \def #4##1##2{\ifcase ##1%
      \xint_afterfi{\ifx\XINT_expr_itself_^C ##2\xint_dothis{\xint_c_ ##2}\fi
                    \xint_orthat\XINT_expr_missing_) }%
      \or   \csname XINT_#6_op_##2\expandafter\endcsname
      \else 
      \xint_afterfi{\expandafter #3\romannumeral-`0\csname XINT_#6_op_##2\endcsname }%
      \fi
    }%
}%
\def\XINT_expr_missing_) {\xintError:inserted \xint_c_ \XINT_expr_done }%
\catcode`) 12
\xintFor #1 in {expr,flexpr,iiexpr} \do {%
    \expandafter\XINT_tmpa
    \csname XINT_#1_op_(\expandafter\endcsname
    \csname XINT_#1_oparen\expandafter\endcsname
    \csname XINT_#1_until_)_a\expandafter\endcsname
    \csname XINT_#1_until_)_b\expandafter\endcsname
    \csname XINT_#1_op_-vi\endcsname
    {#1}%
}%
\expandafter\let\csname XINT_expr_precedence_)\endcsname\xint_c_i
\expandafter\let\csname XINT_expr_precedence_]\endcsname\xint_c_i
\expandafter\let\csname XINT_expr_precedence_;\endcsname\xint_c_i
\let\XINT_expr_precedence_a \xint_c_xviii
\expandafter\let\csname XINT_expr_precedence_^C\endcsname \xint_c_
\expandafter\let\csname XINT_expr_precedence_++)\endcsname \xint_c_i
\catcode`. 11 \catcode`= 11 \catcode`+ 11
\xintFor #1 in {expr,flexpr,iiexpr} \do {%
    \expandafter\let\csname XINT_#1_op_)\endcsname \XINT_expr_getop
    \expandafter\let\csname XINT_#1_op_;\endcsname \space
    \expandafter\def\csname XINT_#1_op_]\endcsname ##1{\XINT_expr_getop ##1a}%
    \expandafter\let\csname XINT_#1_op_a\endcsname \XINT_expr_getop
    \expandafter\def\csname XINT_#1_op_++)\endcsname ##1##2\relax
  {\expandafter\XINT_expr_foundend \expandafter 
                                  {\expandafter\.=+\xintiCeil{\XINT_expr_unlock ##1}}}%
}%
\catcode`. 12 \catcode`= 12 \catcode`+ 12
%    \end{macrocode}
% \subsection{\textbar, \textbar\textbar, \&,
% \&\&, <, >, =, ==, <=, >=, !=, +, \textendash, \texorpdfstring{\protect\lowast}{*}, /, \textasciicircum, \texorpdfstring{\protect\lowast\protect\lowast}{**}, //, /:, .., ..[, ].., ][, ][:, :], \textasciicircum C, and ++ operators}
%    \begin{macrocode}
\xintFor* #1 in {{==}{<=}{>=}{!=}{&&}{||}{**}{//}{/:}{..}{..[}{].}{]..}%
                 {+[}{-[}{*[}{/[}{**[}{^[}{a+}{a-}{a*}{a/}{a**}{a^}%
                 {][}{][:}{:]}{^C}{++}{++)}}
    \do {\expandafter\def\csname XINT_expr_itself_#1\endcsname {#1}}%
%    \end{macrocode}
% \subsubsection{The \textbar,
% \&, xor, <, >, =, <=, >=, !=, //, /:, .., ..[, and ].. operators}
%    \begin{macrocode}
\def\XINT_tmpc #1#2#3#4#5#6#7#8%
{%
  \def #1##1% \XINT_expr_op_<op> ou flexpr ou iiexpr
  {% keep value, get next number and operator, then do until
    \expandafter #2\expandafter ##1%
    \romannumeral-`0\expandafter\XINT_expr_getnext }%
  \def #2##1##2% \XINT_expr_until_<op>_a ou flexpr ou iiexpr
  {\xint_UDsignfork ##2{\expandafter #2\expandafter ##1\romannumeral-`0#4}%
    -{#3##1##2}%
    \krof }%
  \def #3##1##2##3##4% \XINT_expr_until_<op>_b ou flexpr ou iiexpr
  {% either execute next operation now, or first do next (possibly unary)
    \ifnum ##2>#5%
    \xint_afterfi {\expandafter #2\expandafter ##1\romannumeral-`0%
      \csname XINT_#8_op_##3\endcsname {##4}}%
    \else \xint_afterfi {\expandafter ##2\expandafter ##3%
      \csname .=#6{\XINT_expr_unlock ##1}{\XINT_expr_unlock ##4}\endcsname }%
    \fi }%
  \let #7#5%
}%
\def\XINT_tmpb #1#2#3#4#5#6%
{%
  \expandafter\XINT_tmpc
  \csname XINT_#1_op_#3\expandafter\endcsname
  \csname XINT_#1_until_#3_a\expandafter\endcsname
  \csname XINT_#1_until_#3_b\expandafter\endcsname
  \csname XINT_#1_op_-#5\expandafter\endcsname
  \csname xint_c_#4\expandafter\endcsname
  \csname #2#6\expandafter\endcsname
  \csname XINT_expr_precedence_#3\endcsname {#1}%
}%
\xintFor #1 in {expr, flexpr} \do {%
  \def\XINT_tmpa ##1{\XINT_tmpb {#1}{xint}##1}%
  \xintApplyInline {\XINT_tmpa }{%
    {|{iii}{vi}{OR}}%
    {&{iv}{vi}{AND}}%
    {{xor}{iii}{vi}{XOR}}%
    {<{v}{vi}{Lt}}%
    {>{v}{vi}{Gt}}%
    {={v}{vi}{Eq}}%
    {{<=}{v}{vi}{LtorEq}}%
    {{>=}{v}{vi}{GtorEq}}%
    {{!=}{v}{vi}{Neq}}%
    {{..}{iii}{vi}{Seq::csv}}%
    {{//}{vii}{vii}{DivTrunc}}%
    {{/:}{vii}{vii}{Mod}}%
  }%
}%
\def\XINT_tmpa #1{\XINT_tmpb {expr}{xint}#1}%
\xintApplyInline {\XINT_tmpa }{%
  {+{vi}{vi}{Add}}%
  {-{vi}{vi}{Sub}}%
  {*{vii}{vii}{Mul}}%
  {/{vii}{vii}{Div}}%
  {^{viii}{viii}{Pow}}%
  {{..[}{iii}{vi}{SeqA::csv}}%
  {{]..}{iii}{vi}{SeqB::csv}}%
}%
\def\XINT_tmpa #1{\XINT_tmpb {flexpr}{XINTinFloat}#1}%
\xintApplyInline {\XINT_tmpa }{%
  {+{vi}{vi}{Add}}%
  {-{vi}{vi}{Sub}}%
  {*{vii}{vii}{Mul}}%
  {/{vii}{vii}{Div}}%
  {^{viii}{viii}{Power}}%
  {{..[}{iii}{vi}{SeqA::csv}}%
  {{]..}{iii}{vi}{SeqB::csv}}%
}%
\def\XINT_tmpa #1{\XINT_tmpb {iiexpr}{xint}#1}%
\xintApplyInline {\XINT_tmpa }{%
    {|{iii}{vi}{OR}}%
    {&{iv}{vi}{AND}}%
    {{xor}{iii}{vi}{XOR}}%
    {<{v}{vi}{iiLt}}%
    {>{v}{vi}{iiGt}}%
    {={v}{vi}{iiEq}}%
    {{<=}{v}{vi}{iiLtorEq}}%
    {{>=}{v}{vi}{iiGtorEq}}%
    {{!=}{v}{vi}{iiNeq}}%
  {+{vi}{vi}{iiAdd}}%
  {-{vi}{vi}{iiSub}}%
  {*{vii}{vii}{iiMul}}%
  {/{vii}{vii}{iiDivRound}}% CHANGED IN 1.1! PREVIOUSLY DID EUCLIDEAN QUOTIENT
  {^{viii}{viii}{iiPow}}%
  {{..[}{iii}{vi}{iiSeqA::csv}}%
  {{]..}{iii}{vi}{iiSeqB::csv}}%
  {{..}{iii}{vi}{iiSeq::csv}}%
  {{//}{vii}{vii}{iiDivTrunc}}%
  {{/:}{vii}{vii}{iiMod}}%
}%
%    \end{macrocode}
% \subsubsection{The ]+, ]\textendash, ]\texorpdfstring{\protect\lowast}{*}, ]/, ]\textasciicircum, +[, \textendash[, \texorpdfstring{\protect\lowast}{*}[, /[, and \textasciicircum[ list
% operators}
% \paragraph{\csh{XINT_expr_binop_inline_b}}\par
%    \begin{macrocode}
\def\XINT_expr_binop_inline_a
   {\expandafter\xint_gobble_i\romannumeral-`0\XINT_expr_binop_inline_b }%
\def\XINT_expr_binop_inline_b #1#2,{\XINT_expr_binop_inline_c #2,{#1}}%
\def\XINT_expr_binop_inline_c #1{%
   \if ,#1\xint_dothis\XINT_expr_binop_inline_e\fi
   \if ^#1\xint_dothis\XINT_expr_binop_inline_end\fi
   \xint_orthat\XINT_expr_binop_inline_d #1}%
\def\XINT_expr_binop_inline_d #1,#2{,#2{#1}\XINT_expr_binop_inline_b {#2}}%
\def\XINT_expr_binop_inline_e #1,#2{,\XINT_expr_binop_inline_b {#2}}%
\def\XINT_expr_binop_inline_end #1,#2{}%
\def\XINT_tmpc #1#2#3#4#5#6#7#8%
{%
  \def #1##1% \XINT_expr_op_<op> ou flexpr ou iiexpr
  {% keep value, get next number and operator, then do until
    \expandafter #2\expandafter ##1%
    \romannumeral-`0\expandafter\XINT_expr_getnext }%
  \def #2##1##2% \XINT_expr_until_<op>_a ou flexpr ou iiexpr
  {\xint_UDsignfork ##2{\expandafter #2\expandafter ##1\romannumeral-`0#4}%
    -{#3##1##2}%
    \krof }%
  \def #3##1##2##3##4% \XINT_expr_until_<op>_b ou flexpr ou iiexpr
  {% either execute next operation now, or first do next (possibly unary)
    \ifnum ##2>#5%
    \xint_afterfi {\expandafter #2\expandafter ##1\romannumeral-`0%
      \csname XINT_#8_op_##3\endcsname {##4}}%
    \else \xint_afterfi {\expandafter ##2\expandafter ##3%
      \csname .=\expandafter\XINT_expr_binop_inline_a\expandafter
      {\expandafter\expandafter\expandafter#6\expandafter
       \xint_exchangetwo_keepbraces\expandafter
      {\expandafter\XINT_expr_unlock\expandafter ##4\expandafter}\expandafter}%
         \romannumeral-`0\XINT_expr_unlock ##1,^,\endcsname }%
    \fi }%
  \let #7#5%
}%
\def\XINT_tmpb #1#2#3#4%
{%
  \expandafter\XINT_tmpc
  \csname XINT_#1_op_#2\expandafter\endcsname
  \csname XINT_#1_until_#2_a\expandafter\endcsname
  \csname XINT_#1_until_#2_b\expandafter\endcsname
  \csname XINT_#1_op_-#3\expandafter\endcsname
  \csname xint_c_#3\expandafter\endcsname
  \csname #4\expandafter\endcsname
  \csname XINT_expr_precedence_#2\endcsname {#1}%
}%
\xintApplyInline {\expandafter\XINT_tmpb \xint_firstofone}{%
  {{expr}{a+}{vi}{xintAdd}}%
  {{expr}{a-}{vi}{xintSub}}%
  {{expr}{a*}{vii}{xintMul}}%
  {{expr}{a/}{vii}{xintDiv}}%
  {{expr}{a^}{viii}{xintPow}}%
  {{iiexpr}{a+}{vi}{xintiiAdd}}%
  {{iiexpr}{a-}{vi}{xintiiSub}}%
  {{iiexpr}{a*}{vii}{xintiiMul}}%
  {{iiexpr}{a/}{vii}{xintiiDivRound}}%
  {{iiexpr}{a^}{viii}{xintiiPow}}%
  {{flexpr}{a+}{vi}{XINTinFloatAdd}}%
  {{flexpr}{a-}{vi}{XINTinFloatSub}}%
  {{flexpr}{a*}{vii}{XINTinFloatMul}}%
  {{flexpr}{a/}{vii}{XINTinFloatDiv}}%
  {{flexpr}{a^}{viii}{XINTinFloatPower}}%
}%
\def\XINT_tmpc #1#2#3#4#5#6#7%
{%
  \def #1##1{\expandafter#2\expandafter##1\romannumeral-`0%
             \expandafter #3\romannumeral-`0\XINT_expr_getnext }%
  \def #2##1##2##3##4%
  {% either execute next operation now, or first do next (possibly unary)
    \ifnum ##2>#4%
    \xint_afterfi {\expandafter #2\expandafter ##1\romannumeral-`0%
      \csname XINT_#7_op_##3\endcsname {##4}}%
    \else \xint_afterfi {\expandafter ##2\expandafter ##3%
      \csname .=\expandafter\XINT_expr_binop_inline_a\expandafter
      {\expandafter#5\expandafter
      {\expandafter\XINT_expr_unlock\expandafter ##1\expandafter}\expandafter}%
         \romannumeral-`0\XINT_expr_unlock ##4,^,\endcsname }%
    \fi }%
  \let #6#4%
}%
\def\XINT_tmpb #1#2#3#4%
{%
  \expandafter\XINT_tmpc
  \csname XINT_#1_op_#2\expandafter\endcsname
  \csname XINT_#1_until_#2\expandafter\endcsname
  \csname XINT_#1_until_)_a\expandafter\endcsname
  \csname xint_c_#3\expandafter\endcsname
  \csname #4\expandafter\endcsname
  \csname XINT_expr_precedence_#2\endcsname {#1}%
}%
\xintApplyInline {\expandafter\XINT_tmpb\xint_firstofone }{%
  {{expr}{+[}{vi}{xintAdd}}%
  {{expr}{-[}{vi}{xintSub}}%
  {{expr}{*[}{vii}{xintMul}}%
  {{expr}{/[}{vii}{xintDiv}}%
  {{expr}{^[}{viii}{xintPow}}%
  {{iiexpr}{+[}{vi}{xintiiAdd}}%
  {{iiexpr}{-[}{vi}{xintiiSub}}%
  {{iiexpr}{*[}{vii}{xintiiMul}}%
  {{iiexpr}{/[}{vii}{xintiiDivRound}}%
  {{iiexpr}{^[}{viii}{xintiiPow}}%
  {{flexpr}{+[}{vi}{XINTinFloatAdd}}%
  {{flexpr}{-[}{vi}{XINTinFloatSub}}%
  {{flexpr}{*[}{vii}{XINTinFloatMul}}%
  {{flexpr}{/[}{vii}{XINTinFloatDiv}}%
  {{flexpr}{^[}{viii}{XINTinFloatPower}}%
}%
%    \end{macrocode}
% \subsubsection{The \textquotesingle and\textquotesingle, \textquotesingle
% or\textquotesingle, \textquotesingle xor\textquotesingle, and
% \textquotesingle mod\textquotesingle\ as infix operator words}
%    \begin{macrocode}
\xintFor #1 in {and,or,xor,mod} \do {%
   \expandafter\def\csname XINT_expr_itself_#1\endcsname {#1}}%
\expandafter\let\csname XINT_expr_precedence_and\expandafter\endcsname
                \csname XINT_expr_precedence_&\endcsname
\expandafter\let\csname XINT_expr_precedence_or\expandafter\endcsname
                \csname XINT_expr_precedence_|\endcsname
\expandafter\let\csname XINT_expr_precedence_mod\expandafter\endcsname
                \csname XINT_expr_precedence_/:\endcsname
\xintFor #1 in {expr, flexpr, iiexpr} \do {%
   \expandafter\let\csname XINT_#1_op_and\expandafter\endcsname
                   \csname XINT_#1_op_&\endcsname
   \expandafter\let\csname XINT_#1_op_or\expandafter\endcsname
                   \csname XINT_#1_op_|\endcsname
   \expandafter\let\csname XINT_#1_op_mod\expandafter\endcsname
                   \csname XINT_#1_op_/:\endcsname
}%
%    \end{macrocode}
% \subsubsection{The \textbar\textbar,
% \&\&, \texorpdfstring{\protect\lowast\protect\lowast,
% \protect\lowast\protect\lowast[, ]\protect\lowast\protect\lowast}{**, **[, ]**}{} operators as synonyms}
%    \begin{macrocode}
\expandafter\let\csname XINT_expr_precedence_==\expandafter\endcsname
                \csname XINT_expr_precedence_=\endcsname
\expandafter\let\csname XINT_expr_precedence_&&\expandafter\endcsname
                \csname XINT_expr_precedence_&\endcsname
\expandafter\let\csname XINT_expr_precedence_||\expandafter\endcsname
                \csname XINT_expr_precedence_|\endcsname
\expandafter\let\csname XINT_expr_precedence_**\expandafter\endcsname
                \csname XINT_expr_precedence_^\endcsname
\expandafter\let\csname XINT_expr_precedence_a**\expandafter\endcsname
                \csname XINT_expr_precedence_a^\endcsname
\expandafter\let\csname XINT_expr_precedence_**[\expandafter\endcsname
                \csname XINT_expr_precedence_^[\endcsname
\xintFor #1 in {expr, flexpr, iiexpr} \do {%
   \expandafter\let\csname XINT_#1_op_==\expandafter\endcsname
                   \csname XINT_#1_op_=\endcsname
   \expandafter\let\csname XINT_#1_op_&&\expandafter\endcsname
                   \csname XINT_#1_op_&\endcsname
   \expandafter\let\csname XINT_#1_op_||\expandafter\endcsname
                   \csname XINT_#1_op_|\endcsname
   \expandafter\let\csname XINT_#1_op_**\expandafter\endcsname
                   \csname XINT_#1_op_^\endcsname
   \expandafter\let\csname XINT_#1_op_a**\expandafter\endcsname
                   \csname XINT_#1_op_a^\endcsname
   \expandafter\let\csname XINT_#1_op_**[\expandafter\endcsname
                   \csname XINT_#1_op_^[\endcsname
}%
%    \end{macrocode}
% \subsubsection{List selectors: [list][N], [list][:b], [list][a:], [list][a:b]}
% \lverb|1.1 (27 octobre 2014) I implement Python syntax, see
% http://stackoverflow.com/a/13005464/4184837. Do not implement third
% argument giving the step. Also, I gather that [5:2] selector returns empty
% and not, as I could have been tempted to do, (list[5], list[4], list[3]).
% Anyway, it is simpler not to go that way. For reversing I could implement
% [::-1] but this would get confusing, better to do function "reversed".
%
% This gets the job done, but I would definitely need \xintTrim::csv, \xintKeep::csv,
% \xintNthElt::csv for better efficiency. Not for 1.1.|
%    \begin{macrocode}
\def\XINT_tmpa #1#2#3#4#5#6%
{%
    \def #1##1% \XINT_expr_op_][
    {%
        \expandafter #2\expandafter ##1\romannumeral-`0\XINT_expr_getnext
    }%
    \def #2##1##2% \XINT_expr_until_][_a
    {\xint_UDsignfork
        ##2{\expandafter #2\expandafter ##1\romannumeral-`0#4}%
          -{#3##1##2}%
     \krof }%
    \def #3##1##2##3##4% \XINT_expr_until_][_b
    {%
      \ifnum ##2>\xint_c_ii
        \xint_afterfi {\expandafter #2\expandafter ##1\romannumeral-`0%
                       \csname XINT_#6_op_##3\endcsname {##4}}%
      \else
        \xint_afterfi
        {\expandafter ##2\expandafter ##3\csname 
              .=\expandafter\xintListSel:csv \romannumeral-`0\XINT_expr_unlock ##4;%
                \XINT_expr_unlock ##1;\endcsname % unlock for \xintNewExpr
        }%
      \fi
    }%
    \let #5\xint_c_ii
}%
\xintFor #1 in {expr,flexpr,iiexpr} \do {%
\expandafter\XINT_tmpa
    \csname XINT_#1_op_][\expandafter\endcsname
    \csname XINT_#1_until_][_a\expandafter\endcsname
    \csname XINT_#1_until_][_b\expandafter\endcsname
    \csname XINT_#1_op_-vi\expandafter\endcsname
    \csname XINT_expr_precedence_][\endcsname {#1}%
}%
\def\XINT_tmpa #1#2#3#4#5#6%
{%
    \def #1##1% \XINT_expr_op_:
    {%
        \expandafter #2\expandafter ##1\romannumeral-`0\XINT_expr_getnext
    }%
    \def #2##1##2% \XINT_expr_until_:_a
    {\xint_UDsignfork
        ##2{\expandafter #2\expandafter ##1\romannumeral-`0#4}%
          -{#3##1##2}%
     \krof }%
    \def #3##1##2##3##4% \XINT_expr_until_:_b
    {%
      \ifnum ##2>\xint_c_iii
        \xint_afterfi {\expandafter #2\expandafter ##1\romannumeral-`0%
                       \csname XINT_#6_op_##3\endcsname {##4}}%
      \else
        \xint_afterfi
        {\expandafter ##2\expandafter ##3\csname 
              .=:\xintiiifSgn{\XINT_expr_unlock ##1}NPP.% : and dots for expansion
                \xintiiifSgn{\XINT_expr_unlock ##4}NPP.%  in \xintNewExpr context
% reason for \xintNum is a/1[x]  format, but -0.5 will not work, seen <0, but 0 after
                \xintNum{\XINT_expr_unlock ##1};\xintNum{\XINT_expr_unlock ##4}\endcsname 
        }%
      \fi
    }%
    \let #5\xint_c_iii
}%
\xintFor #1 in {expr,flexpr,iiexpr} \do {%
\expandafter\XINT_tmpa
    \csname XINT_#1_op_:\expandafter\endcsname
    \csname XINT_#1_until_:_a\expandafter\endcsname
    \csname XINT_#1_until_:_b\expandafter\endcsname
    \csname XINT_#1_op_-vi\expandafter\endcsname
    \csname XINT_expr_precedence_:\endcsname {#1}%
}%
\catcode`[ 11 \catcode`] 11
\let\XINT_expr_precedence_:] \xint_c_iii
\def\XINT_expr_op_:] #1{\expandafter\xint_c_i\expandafter )%
    \csname .=]\xintiiifSgn{\XINT_expr_unlock #1}npp\XINT_expr_unlock #1\endcsname }%
\let\XINT_flexpr_op_:] \XINT_expr_op_:]
\let\XINT_iiexpr_op_:] \XINT_expr_op_:]
\let\XINT_expr_precedence_][: \xint_c_iii
\edef\XINT_expr_op_][: #1{\xint_c_ii \expandafter\noexpand
                          \csname XINT_expr_itself_][\endcsname #10\string :}% 
% : must be catcode 12, else will be mistaken for start of variable by expression parser
\let\XINT_flexpr_op_][: \XINT_expr_op_][:
\let\XINT_iiexpr_op_][: \XINT_expr_op_][:
\catcode`[ 12 \catcode`] 12
\def\xintListSel:csv #1{% these complications are due to \xintNewExpr matters
    \if ]\noexpand#1\xint_dothis{\expandafter\XINT_listsel:_s\romannumeral-`0}\fi
    \if :\noexpand#1\xint_dothis{\XINT_listsel:_:}\fi
    \xint_orthat {\XINT_listsel:_nth #1}%
}%
\def\XINT_listsel:_s #1{\if p#1\expandafter\XINT_listsel:_trim\else
                               \expandafter\XINT_listsel:_keep\fi }%
\def\XINT_listsel:_: #1.#2.{\csname XINT_listsel:_#1#2\endcsname }%
\def\XINT_listsel:_trim #1;#2;%
   {\xintListWithSep,{\xintTrim {\xintNum{#1}}{\xintCSVtoListNonStripped{#2}}}}%
\def\XINT_listsel:_keep #1;#2;%
   {\xintListWithSep,{\xintKeep {\xintNum{#1}}{\xintCSVtoListNonStripped{#2}}}}%
\def\XINT_listsel:_nth#1;#2;%
   {\xintNthElt {\xintNum{#1}}{\xintCSVtoListNonStripped{#2}}}%
\def\XINT_listsel:_PP #1;#2;#3;%
   {\xintListWithSep,%
    {\xintTrim {\xintNum{#1}}%
               {\xintKeep {\xintNum{#2}}%
                          {\xintCSVtoListNonStripped{#3}}%
               }%
    }%
   }%
\def\XINT_listsel:_NN #1;#2;#3;%
   {\xintListWithSep,%
    {\xintTrim {\xintNum{#2}}%
               {\xintKeep {\xintNum{#1}}%
                          {\xintCSVtoListNonStripped{#3}}%
               }%
    }%
   }%
\def\XINT_listsel:_NP #1;#2;#3;%
   {\expandafter\XINT_listsel:_NP_a \the\numexpr #1+%
             \xintNthElt{0}{\xintCSVtoListNonStripped{#3}};#2;#3;}%
\def\XINT_listsel:_NP_a #1#2;{\if -#1\expandafter\XINT_listsel:_OP\fi
                              \XINT_listsel:_PP #1#2;}%
\def\XINT_listsel:_OP\XINT_listsel:_PP #1;{\XINT_listsel:_PP 0;}%
\def\XINT_listsel:_PN #1;#2;#3;%
   {\expandafter\XINT_listsel:_PN_a \the\numexpr #2+%
             \xintNthElt{0}{\xintCSVtoListNonStripped{#3}};#1;#3;}%
\def\XINT_listsel:_PN_a #1#2;#3;{\if -#1\expandafter\XINT_listsel:_PO\fi
                              \XINT_listsel:_PP #3;#1#2;}%
\def\XINT_listsel:_PO\XINT_listsel:_PP #1;#2;{\XINT_listsel:_PP #1;0;}%
%    \end{macrocode}
%\subsection{Macros for a..b list generation}
%\lverb|Attention, ne produit que des listes de petits entiers!|
%\subsubsection{\csh{xintSeq::csv}}
%\lverb|Commence par remplacer a par ceil(a) et b par floor(b) et renvoie
% ensuite les entiers entre les deux, possiblement en décroissant, et
% extrémités comprises. Si a=b est non entier en obtient donc ceil(a) et
% floor(a). Ne renvoie jamais une liste vide.|
%    \begin{macrocode}
\def\xintSeq::csv {\romannumeral0\xintseq::csv }%
\def\xintseq::csv #1#2%
{%
    \expandafter\XINT_seq::csv\expandafter
       {\the\numexpr \xintiCeil{#1}\expandafter}\expandafter
       {\the\numexpr \xintiFloor{#2}}%
}%
\def\XINT_seq::csv #1#2%
{%
   \ifcase\ifnum #1=#2 0\else\ifnum #2>#1 1\else -1\fi\fi\space
      \expandafter\XINT_seq::csv_z
   \or
      \expandafter\XINT_seq::csv_p
   \else
      \expandafter\XINT_seq::csv_n
   \fi
   {#2}{#1}%
}%
\def\XINT_seq::csv_z #1#2{ #1/1[0]}%
\def\XINT_seq::csv_p #1#2%
{%
    \ifnum #1>#2
      \expandafter\expandafter\expandafter\XINT_seq::csv_p
    \else
      \expandafter\XINT_seq::csv_e
    \fi
    \expandafter{\the\numexpr #1-\xint_c_i}{#2},#1/1[0]%
}%
\def\XINT_seq::csv_n #1#2%
{%
    \ifnum #1<#2
      \expandafter\expandafter\expandafter\XINT_seq::csv_n
    \else
      \expandafter\XINT_seq::csv_e
    \fi
     \expandafter{\the\numexpr #1+\xint_c_i}{#2},#1/1[0]%
}%
\def\XINT_seq::csv_e #1,{ }%
%    \end{macrocode}
%\subsubsection{\csh{xintiiSeq::csv}}
%    \begin{macrocode}
\def\xintiiSeq::csv {\romannumeral0\xintiiseq::csv }%
\def\xintiiseq::csv #1#2%
{%
    \expandafter\XINT_iiseq::csv\expandafter
       {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}%
}%
\def\XINT_iiseq::csv #1#2%
{%
   \ifcase\ifnum #1=#2 0\else\ifnum #2>#1 1\else -1\fi\fi\space
      \expandafter\XINT_iiseq::csv_z
   \or
      \expandafter\XINT_iiseq::csv_p
   \else
      \expandafter\XINT_iiseq::csv_n
   \fi
   {#2}{#1}%
}%
\def\XINT_iiseq::csv_z #1#2{ #1}%
\def\XINT_iiseq::csv_p #1#2%
{%
    \ifnum #1>#2
      \expandafter\expandafter\expandafter\XINT_iiseq::csv_p
    \else
      \expandafter\XINT_seq::csv_e
    \fi
    \expandafter{\the\numexpr #1-\xint_c_i}{#2},#1%
}%
\def\XINT_iiseq::csv_n #1#2%
{%
    \ifnum #1<#2
      \expandafter\expandafter\expandafter\XINT_iiseq::csv_n
    \else
      \expandafter\XINT_seq::csv_e
    \fi
     \expandafter{\the\numexpr #1+\xint_c_i}{#2},#1%
}%
\def\XINT_seq::csv_e #1,{ }%
%    \end{macrocode}
%\subsection{Macros for a..[d]..b list generation}
%\lverb|Contrarily to a..b which is limited to small integers, this works with a, b, and d
%(big) fractions. It will produce a «nil» list, if a>b and d<0 or a<b and d>0.|
%\subsubsection{\csh{xintSeqA::csv}, \csh{xintiiSeqA::csv}, \csh{XINTinFloatSeqA::csv}}
%    \begin{macrocode}
\def\xintSeqA::csv #1%
   {\expandafter\XINT_seqa::csv\expandafter{\romannumeral0\xintraw {#1}}}%
\def\XINT_seqa::csv #1#2{\expandafter\XINT_seqa::csv_a \romannumeral0\xintraw {#2};#1;}%
\def\xintiiSeqA::csv #1#2{\XINT_iiseqa::csv #1#2}%
\def\XINT_iiseqa::csv #1#2#3#4{\expandafter\XINT_seqa::csv_a
    \romannumeral-`0\expandafter \XINT_expr_unlock\expandafter#4%
    \expandafter;\romannumeral-`0\XINT_expr_unlock #2;}%
\def\XINTinFloatSeqA::csv #1{\expandafter\XINT_flseqa::csv\expandafter
   {\romannumeral0\XINTinfloat [\XINTdigits]{#1}}}%
\def\XINT_flseqa::csv #1#2%
   {\expandafter\XINT_seqa::csv_a\romannumeral0\XINTinfloat [\XINTdigits]{#2};#1;}%
\def\XINT_seqa::csv_a #1{\xint_UDzerominusfork
                                   #1-{z}%
                                   0#1{n}%
                                   0-{p}%
                        \krof #1}%
%    \end{macrocode}
%\subsubsection{\csh{xintSeqB::csv}}
%    \begin{macrocode}
\def\xintSeqB::csv #1#2%
   {\expandafter\XINT_seqb::csv \expandafter{\romannumeral0\xintraw{#2}}{#1}}%
\def\XINT_seqb::csv #1#2{\expandafter\XINT_seqb::csv_a\romannumeral-`0#2#1!}%
\def\XINT_seqb::csv_a #1#2;#3;#4!{\expandafter\XINT_expr_seq_empty?
      \romannumeral0\csname XINT_seqb::csv_#1\endcsname {#3}{#4}{#2}}%
\def\XINT_seqb::csv_p #1#2#3%
{%
   \xintifCmp {#1}{#2}{,#1\expandafter\XINT_seqb::csv_p\expandafter}%
   {,#1\xint_gobble_iii}{\xint_gobble_iii}%
% \romannumeral0 stopped by \endcsname, XINT_expr_seq_empty? constructs "nil".
   {\romannumeral0\xintadd {#3}{#1}}{#2}{#3}%
}%
\def\XINT_seqb::csv_n #1#2#3%
{%
    \xintifCmp {#1}{#2}{\xint_gobble_iii}{,#1\xint_gobble_iii}%
    {,#1\expandafter\XINT_seqb::csv_n\expandafter}%
    {\romannumeral0\xintadd {#3}{#1}}{#2}{#3}%
}%
\def\XINT_seqb::csv_z #1#2#3{,#1}%
%    \end{macrocode}
%\subsubsection{\csh{xintiiSeqB::csv}}
%    \begin{macrocode}
\def\xintiiSeqB::csv #1#2{\XINT_iiseqb::csv #1#2}%
\def\XINT_iiseqb::csv #1#2#3#4%
   {\expandafter\XINT_iiseqb::csv_a
    \romannumeral-`0\expandafter \XINT_expr_unlock\expandafter#2%
    \romannumeral-`0\XINT_expr_unlock #4!}%
\def\XINT_iiseqb::csv_a #1#2;#3;#4!{\expandafter\XINT_expr_seq_empty?
      \romannumeral-`0\csname XINT_iiseqb::csv_#1\endcsname {#3}{#4}{#2}}%
\def\XINT_iiseqb::csv_p #1#2#3%
{%
  \xintSgnFork{\XINT_Cmp {#1}{#2}}{,#1\expandafter\XINT_iiseqb::csv_p\expandafter}%
  {,#1\xint_gobble_iii}{\xint_gobble_iii}%
  {\romannumeral0\xintiiadd {#3}{#1}}{#2}{#3}%
}%
\def\XINT_iiseqb::csv_n #1#2#3%
{%
  \xintSgnFork{\XINT_Cmp {#1}{#2}}{\xint_gobble_iii}{,#1\xint_gobble_iii}%
  {,#1\expandafter\XINT_iiseqb::csv_n\expandafter}%
  {\romannumeral0\xintiiadd {#3}{#1}}{#2}{#3}%
}%
\def\XINT_iiseqb::csv_z #1#2#3{,#1}%
%    \end{macrocode}
%\subsubsection{\csh{XINTinFloatSeqB::csv}}
%    \begin{macrocode}
\def\XINTinFloatSeqB::csv #1#2{\expandafter\XINT_flseqb::csv \expandafter
    {\romannumeral0\XINTinfloat [\XINTdigits]{#2}}{#1}}%
\def\XINT_flseqb::csv #1#2{\expandafter\XINT_flseqb::csv_a\romannumeral-`0#2#1!}%
\def\XINT_flseqb::csv_a #1#2;#3;#4!{\expandafter\XINT_expr_seq_empty?
      \romannumeral-`0\csname XINT_flseqb::csv_#1\endcsname {#3}{#4}{#2}}%
\def\XINT_flseqb::csv_p #1#2#3%
{%
  \xintifCmp {#1}{#2}{,#1\expandafter\XINT_flseqb::csv_p\expandafter}%
  {,#1\xint_gobble_iii}{\xint_gobble_iii}%
  {\romannumeral0\XINTinfloatadd {#3}{#1}}{#2}{#3}%
}%
\def\XINT_flseqb::csv_n #1#2#3%
{%
  \xintifCmp {#1}{#2}{\xint_gobble_iii}{,#1\xint_gobble_iii}%
  {,#1\expandafter\XINT_flseqb::csv_n\expandafter}%
  {\romannumeral0\XINTinfloatadd {#3}{#1}}{#2}{#3}%
}%
\def\XINT_flseqb::csv_z #1#2#3{,#1}%
%    \end{macrocode}
% \subsection{The comma as binary operator}
% \lverb|New with 1.09a.|
%    \begin{macrocode}
\def\XINT_tmpa #1#2#3#4#5#6%
{%
    \def #1##1% \XINT_expr_op_,
    {%
        \expandafter #2\expandafter ##1\romannumeral-`0\XINT_expr_getnext
    }%
    \def #2##1##2% \XINT_expr_until_,_a
    {\xint_UDsignfork
        ##2{\expandafter #2\expandafter ##1\romannumeral-`0#4}%
          -{#3##1##2}%
     \krof }%
    \def #3##1##2##3##4% \XINT_expr_until_,_b
    {%
      \ifnum ##2>\xint_c_ii
        \xint_afterfi {\expandafter #2\expandafter ##1\romannumeral-`0%
                       \csname XINT_#6_op_##3\endcsname {##4}}%
      \else
        \xint_afterfi
        {\expandafter ##2\expandafter ##3%
         \csname .=\XINT_expr_unlock ##1,\XINT_expr_unlock ##4\endcsname }%
      \fi
    }%
    \let #5\xint_c_ii
}%
\xintFor #1 in {expr,flexpr,iiexpr} \do {%
\expandafter\XINT_tmpa
    \csname XINT_#1_op_,\expandafter\endcsname
    \csname XINT_#1_until_,_a\expandafter\endcsname
    \csname XINT_#1_until_,_b\expandafter\endcsname
    \csname XINT_#1_op_-vi\expandafter\endcsname
    \csname XINT_expr_precedence_,\endcsname {#1}%
}%
%    \end{macrocode}
% \subsection{The minus as prefix operator of variable precedence level}
%    \begin{macrocode}
\def\XINT_tmpa #1#2#3%
{%
    \expandafter\XINT_tmpb
    \csname XINT_#1_op_-#3\expandafter\endcsname
    \csname XINT_#1_until_-#3_a\expandafter\endcsname
    \csname XINT_#1_until_-#3_b\expandafter\endcsname
    \csname xint_c_#3\endcsname {#1}#2%
}%
\def\XINT_tmpb #1#2#3#4#5#6%
{%
    \def #1% \XINT_expr_op_-<level>
    {%  get next number+operator then switch to _until macro
        \expandafter #2\romannumeral-`0\XINT_expr_getnext
    }%
    \def #2##1% \XINT_expr_until_-<l>_a
    {\xint_UDsignfork
        ##1{\expandafter #2\romannumeral-`0#1}%
          -{#3##1}%
     \krof }%
    \def #3##1##2##3% \XINT_expr_until_-<l>_b
    {%  _until tests precedence level with next op, executes now or postpones
        \ifnum ##1>#4%
         \xint_afterfi {\expandafter #2\romannumeral-`0%
                        \csname XINT_#5_op_##2\endcsname {##3}}%
        \else
         \xint_afterfi {\expandafter ##1\expandafter ##2%
                        \csname .=#6{\XINT_expr_unlock ##3}\endcsname }%
        \fi
    }%
}%
\xintApplyInline{\XINT_tmpa {expr}\xintOpp}{{vi}{vii}{viii}{ix}}%
\xintApplyInline{\XINT_tmpa {flexpr}\xintOpp}{{vi}{vii}{viii}{ix}}%
\xintApplyInline{\XINT_tmpa {iiexpr}\xintiiOpp}{{vi}{vii}{viii}{ix}}%
%    \end{macrocode}
% \subsection{? as two-way and ?? as three-way conditionals with braced branches}
% \lverb|In 1.1, I overload ? with ??, as : will be used for list extraction,
% problem with (stuff)?{?(1)}{0} for example, one should put a space (stuff)?{
% ?(1)}{0} will work. Small idiosyncrasy. ?{yes}{no} and ??{<0}{=0}{>0}|
%    \begin{macrocode}
\let\XINT_expr_precedence_? \xint_c_x
\def\XINT_expr_op_? #1#2{\if ?#2\expandafter \XINT_expr_op_??\fi
                         \XINT_expr_op_?a #1{#2}}% 
\def\XINT_expr_op_?a #1#2#3%
{%
    \xintiiifNotZero{\XINT_expr_unlock  #1}{\XINT_expr_getnext #2}{\XINT_expr_getnext #3}%
}%
\let\XINT_flexpr_op_?\XINT_expr_op_?
\let\XINT_iiexpr_op_?\XINT_expr_op_?
\def\XINT_expr_op_?? #1#2#3#4#5#6%
{%
     \xintiiifSgn {\XINT_expr_unlock  #2}{\XINT_expr_getnext #4}{\XINT_expr_getnext #5}%
                  {\XINT_expr_getnext #6}%
}%
%    \end{macrocode}
% \subsection{! as postfix factorial operator}
% \lverb|As of 2014/11/07, not yet a float version of factorial. I must do it!|
%    \begin{macrocode}
\let\XINT_expr_precedence_! \xint_c_x
\def\XINT_expr_op_! #1{\expandafter\XINT_expr_getop
                                    \csname .=\xintFac{\XINT_expr_unlock #1}\endcsname }%
\let\XINT_flexpr_op_!\XINT_expr_op_!
\def\XINT_iiexpr_op_! #1{\expandafter\XINT_expr_getop
                                   \csname .=\xintiFac{\XINT_expr_unlock #1}\endcsname }%
%    \end{macrocode}
% \subsection{The A/B[N] mechanism}
% \lverb|Releases earlier than 1.1 required the use of braces around A/B[N]
% input. The [N] is now implemented directly. *BUT* uses a delimited macro!
% thus N is not allowed to be itself an expression (I could add it...).
% \xintE, \xintiiE, and \XINTinFloatE all put #2 in a \numexpr. BUT ATTENTION
% TO CRAZYNESS OF NUMEXPR: \the\numexpr 3 + 7 9 \relax !! Hence we have to do
% the job ourselves.|
%    \begin{macrocode}
\catcode`[ 11
\catcode`* 11
\let\XINT_expr_precedence_[ \xint_c_vii
\def\XINT_expr_op_[ #1#2]{\expandafter\XINT_expr_getop
                \csname .=\xintE{\XINT_expr_unlock #1}%
                {\xint_zapspaces #2 \xint_gobble_i}\endcsname}%
\def\XINT_iiexpr_op_[ #1#2]{\expandafter\XINT_expr_getop
                \csname .=\xintiiE{\XINT_expr_unlock #1}%
                {\xint_zapspaces #2 \xint_gobble_i}\endcsname}%
\def\XINT_flexpr_op_[ #1#2]{\expandafter\XINT_expr_getop
                \csname .=\XINTinFloatE{\XINT_expr_unlock #1}%
                {\xint_zapspaces #2 \xint_gobble_i}\endcsname}%
\catcode`[ 12
\catcode`* 12
%    \end{macrocode}
% \subsection{For variables}
%    \begin{macrocode}
\def\XINT_expr_op__  #1% op__ with two _'s
     {%
         \ifcsname XINT_expr_var_#1\endcsname
           \expandafter\xint_firstoftwo
         \else
           \expandafter\xint_secondoftwo
         \fi
         {\expandafter\expandafter\expandafter\expandafter
          \expandafter\expandafter\expandafter
          \XINT_expr_getop\csname XINT_expr_var_#1\endcsname}%
         {\XINT_expr_unknown_variable {#1}%
          \expandafter\XINT_expr_getop\csname .=0\endcsname}%
     }%
\def\XINT_expr_unknown_variable #1{\xintError:removed \xint_gobble_i {#1}}%
\let\XINT_flexpr_op__ \XINT_expr_op__
\let\XINT_iiexpr_op__ \XINT_expr_op__
%    \end{macrocode}
% \subsubsection{Defining variables}
% \lverb|1.1 An active : character will be a pain and I almot decided not to
% use := but rather = as affectation operator, but this is the same problem
% inside expressions with the modulo operator /:, or with babel+frenchb with
% all high punctuation ?, !, :, ;.
%
% It is not recommended to overwrite single Latin letters which are
% pre-defined to serve as dummy variables. Variable names may contains
% letters, digits, underscores, and must not start with a digit.|
%    \begin{macrocode}
\catcode`: 12
\def\xintdefvar #1:=#2;{\expandafter\odef
   \csname XINT_expr_var_\xint_zapspaces #1 \xint_gobble_i\endcsname
                           {\expandafter\empty\romannumeral0\xintbareeval #2\relax }}%
\def\xintdefiivar #1:=#2;{\expandafter\odef
   \csname XINT_expr_var_\xint_zapspaces #1 \xint_gobble_i\endcsname
  {\expandafter\empty\romannumeral0\xintbareiieval #2\relax }%
}%
\def\xintdeffloatvar #1:=#2;{\expandafter\odef
   \csname XINT_expr_var_\xint_zapspaces #1 \xint_gobble_i\endcsname
  {\expandafter\empty\romannumeral0\xintbarefloateval #2\relax }%
}%
\catcode`: 11
%    \end{macrocode}
% \subsubsection{Letters as dummy variables; the nil list}
%    \begin{macrocode}
\def\XINT_tmpa #1%
{%
   \expandafter\def\csname XINT_expr_var_#1\endcsname ##1\relax !#1##2%
   {\romannumeral0\XINT_expr_lockscan ##2!##1\relax !#1{##2}}%
}%
\xintApplyUnbraced \XINT_tmpa {abcdefghijklmnopqrstuvwxyz}%
\xintApplyUnbraced \XINT_tmpa {ABCDEFGHIJKLMNOPQRSTUVWXYZ}%
\expandafter\def\expandafter\XINT_expr_var_nil\expandafter 
               {\expandafter\empty\csname .= \endcsname}%
%    \end{macrocode}
% \subsubsection{The omit and abort constructs}
%    \begin{macrocode}
\catcode`. 11 \catcode`= 11
\def\XINT_expr_var_omit  #1\relax !{1^C!{}{}{}\.=!\relax !}% 24 juin
\def\XINT_expr_var_abort #1\relax !{1^C!{}{}{}\.=^\relax !}% 25 juin
\catcode`. 12 \catcode`= 12
%    \end{macrocode}
% \subsubsection{The @, @1, @2, @3, @4, @@, @@(1), \dots, @@@, @@@(1), \dots
% for recursion} 
% \lverb|I had completely forgotten what the @@@ etc... stuff were supposed to
% do: this is for nesting recursions! (I was mad back in June 2014). @@(N)
% gives the Nth back, @@@(N) gives the Nth back of the higher recursion!|
%    \begin{macrocode}
\catcode`? 3
\def\XINT_expr_var_@ #1~#2{ #2#1~#2}%
\expandafter\let\csname XINT_expr_var_@1\endcsname \XINT_expr_var_@
\expandafter\def\csname XINT_expr_var_@2\endcsname #1~#2#3{ #3#1~#2#3}%
\expandafter\def\csname XINT_expr_var_@3\endcsname #1~#2#3#4{ #4#1~#2#3#4}%
\expandafter\def\csname XINT_expr_var_@4\endcsname #1~#2#3#4#5{ #5#1~#2#3#4#5}%
\def\XINT_expr_func_@@ #1#2#3#4~#5?%
{%
   \expandafter#1\expandafter#2\romannumeral0\xintntheltnoexpand 
                             {\xintNum{\XINT_expr_unlock#3}}{#5}#4~#5?%
}%
\def\XINT_expr_func_@@@ #1#2#3#4~#5~#6?%
{%
   \expandafter#1\expandafter#2\romannumeral0\xintntheltnoexpand 
   {\xintNum{\XINT_expr_unlock#3}}{#6}#4~#5~#6?%
}%
\def\XINT_expr_func_@@@@ #1#2#3#4~#5~#6~#7?%
{%
   \expandafter#1\expandafter#2\romannumeral0\xintntheltnoexpand 
   {\xintNum{\XINT_expr_unlock#3}}{#7}#4~#5~#6~#7?%
}%
\let\XINT_flexpr_func_@@\XINT_expr_func_@@
\let\XINT_flexpr_func_@@@\XINT_expr_func_@@@
\let\XINT_flexpr_func_@@@@\XINT_expr_func_@@@@
\def\XINT_iiexpr_func_@@ #1#2#3#4~#5?%
{%
    \expandafter#1\expandafter#2\romannumeral0\xintntheltnoexpand 
    {\XINT_expr_unlock#3}{#5}#4~#5?%
}%
\def\XINT_iiexpr_func_@@@ #1#2#3#4~#5~#6?%
{%
    \expandafter#1\expandafter#2\romannumeral0\xintntheltnoexpand 
    {\XINT_expr_unlock#3}{#6}#4~#5~#6?%
}%
\def\XINT_iiexpr_func_@@@@ #1#2#3#4~#5~#6~#7?%
{%
    \expandafter#1\expandafter#2\romannumeral0\xintntheltnoexpand 
    {\XINT_expr_unlock#3}{#7}#4~#5~#6~#7?%
}%
\catcode`? 11
%    \end{macrocode}
% \subsection{For functions}
%    \begin{macrocode}
\def\XINT_tmpa #1#2#3{%
    \def #1##1% \XINT_expr_op_`, #2=\XINT_expr_oparen
    {%
        \ifcsname XINT_expr_onlitteral_##1\endcsname
          \xint_dothis{\csname XINT_expr_onlitteral_##1\endcsname}\fi
        \ifcsname XINT_#3_func_##1\endcsname
          \xint_dothis{\expandafter\expandafter
                     \csname XINT_#3_func_##1\endcsname\romannumeral-`0#2}\fi
        \xint_orthat{\XINT_expr_unknown_function {##1}%
           \expandafter\XINT_expr_func_unknown\romannumeral-`0#2}%
   }%
}%
\def\XINT_expr_unknown_function #1{\xintError:removed \xint_gobble_i {#1}}%
\xintFor #1 in {expr,flexpr,iiexpr} \do {%
     \expandafter\XINT_tmpa
                 \csname XINT_#1_op_`\expandafter\endcsname
                 \csname XINT_#1_oparen\endcsname
                 {#1}%
}%
\expandafter\def\csname XINT_expr_onlitteral_`\endcsname #1#2#3({\xint_c_xviii `{#2}}%
%    \end{macrocode}
% \subsection{The bool, togl, protect, unknown, and break "functions"}
% \lverb|bool, togl and protect use delimited macros. Only unknown and break
% are true functions with a more flexible parsing of the opening and closing
% parentheses, which may possibly arise from expansion itself.|
%    \begin{macrocode}
\def\XINT_expr_onlitteral_bool #1)%
        {\expandafter\XINT_expr_getop\csname .=\xintBool{#1}\endcsname }%
\def\XINT_expr_onlitteral_togl #1)%
        {\expandafter\XINT_expr_getop\csname .=\xintToggle{#1}\endcsname }%
\def\XINT_expr_onlitteral_protect #1)%
        {\expandafter\XINT_expr_getop\csname .=\detokenize{#1}\endcsname }%
\def\XINT_expr_func_unknown #1#2#3{\expandafter #1\expandafter #2\csname .=0\endcsname }%
\def\XINT_expr_func_break #1#2#3%
{\expandafter #1\expandafter #2\csname.=?\romannumeral-`0\XINT_expr_unlock #3\endcsname }%
\let\XINT_flexpr_func_break \XINT_expr_func_break
\let\XINT_iiexpr_func_break \XINT_expr_func_break
%    \end{macrocode}
% \subsection{seq and the implementation of dummy variables}
% \lverb|All of seq, add, mul, rseq, etc... (actually all of the extensive
% changes from xintexpr 1.09n to 1.1) was done around June 15-25th 2014, but the
% problem is that I did not document the code enough, and I had a hard time
% understanding in October what I had done in June. Despite the lesson, again
% being short on time, I do not document enough my current understanding of the
% innards of the beast...
%
% I added subs, and iter in October (also the [:n], [n:] list extractors),
% proving I did at least understand a bit (or rather could imitate) my earlier
% code (but don't ask me to explain \xintNewExpr !)
% 
% The \XINT_expr_onlitteral_seq_a parses: "expression, variable=list)" (when it is called
% the opening ( has been swallowed, and it looks for the ending one.) Both expression and
% list may themselves contain parentheses and commas, we allow nesting. For example
% "x^2,x=1..10)", at the end of seq_a we have {variable{expression}}{list}, in this
% example {x{x^2}}{1..10}, or more complicated "seq(add(y,y=1..x),x=1..10)" will work
% too. The variable is a single lowercase Latin letter.
%
% The complications with \xint_c_xviii in seq_f is for the recurrent thing that we don't
% know in what type of expressions we are, hence we must move back up, with some loss of
% efficiency (superfluous check for minus sign, etc...). But the code manages
% simultaneously expr, flexpr and iiexpr.|
%
% \subsubsection{\csh{XINT_expr_onlitteral_seq}}
%    \begin{macrocode}
\def\XINT_expr_onlitteral_seq 
 {\expandafter\XINT_expr_onlitteral_seq_f\romannumeral-`0\XINT_expr_onlitteral_seq_a {}}%
\def\XINT_expr_onlitteral_seq_f #1#2{\xint_c_xviii `{seqx}#2)\relax #1}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_onlitteral_seq_a}}
%    \begin{macrocode}
\def\XINT_expr_onlitteral_seq_a #1#2,%
{% checks balancing of parentheses
    \ifcase\XINT_isbalanced_a \relax #1#2(\xint_bye)\xint_bye
           \expandafter\XINT_expr_onlitteral_seq_c
        \or\expandafter\XINT_expr_onlitteral_seq_b
      \else\expandafter\xintError:we_are_doomed
    \fi {#1#2},%
}%
\def\XINT_expr_onlitteral_seq_b #1,{\XINT_expr_onlitteral_seq_a {#1,}}%
\def\XINT_expr_onlitteral_seq_c #1,#2#3% #3 pour absorber le =
{%
    \XINT_expr_onlitteral_seq_d {#2{#1}}{}%
}%
\def\XINT_expr_onlitteral_seq_d #1#2#3)%
{%
    \ifcase\XINT_isbalanced_a \relax #2#3(\xint_bye)\xint_bye
        \or\expandafter\XINT_expr_onlitteral_seq_e
       \else\expandafter\xintError:we_are_doomed
    \fi
    {#1}{#2#3}%
}%
\def\XINT_expr_onlitteral_seq_e #1#2{\XINT_expr_onlitteral_seq_d {#1}{#2)}}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_isbalanced_a}  for \csh{XINT_expr_onlitteral_seq_a}}
%\lverb|Expands to \m@ne in case a closing ) had no opening ( matching it, to
% \@ne if opening ) had no closing ) matching it, to \z@ if expression was
% balanced.|
%    \begin{macrocode}
% use as \XINT_isbalanced_a \relax #1(\xint_bye)\xint_bye
\def\XINT_isbalanced_a #1({\XINT_isbalanced_b #1)\xint_bye }%
\def\XINT_isbalanced_b #1)#2%
   {\xint_bye #2\XINT_isbalanced_c\xint_bye\XINT_isbalanced_error }%
%    \end{macrocode}
% \lverb|if #2 is not \xint_bye, a ) was found, but there was no (. Hence error -> -1|
%    \begin{macrocode}
\def\XINT_isbalanced_error #1)\xint_bye {\m@ne}%
%    \end{macrocode}
% \lverb|#2 was \xint_bye, was there a ) in original #1?|
%    \begin{macrocode}
\def\XINT_isbalanced_c\xint_bye\XINT_isbalanced_error #1%
    {\xint_bye #1\XINT_isbalanced_yes\xint_bye\XINT_isbalanced_d #1}%
%    \end{macrocode}
% \lverb|#1 is \xint_bye, there was never ( nor ) in original #1, hence OK.|
%    \begin{macrocode}
\def\XINT_isbalanced_yes\xint_bye\XINT_isbalanced_d\xint_bye )\xint_bye {\xint_c_ }%
%    \end{macrocode}
% \lverb|#1 is not \xint_bye, there was indeed a ( in original #1. We check if
% we see a ). If we do, we then loop until no ( nor ) is to be found.|
%    \begin{macrocode}
\def\XINT_isbalanced_d #1)#2%
   {\xint_bye #2\XINT_isbalanced_no\xint_bye\XINT_isbalanced_a #1#2}%
%    \end{macrocode}
% \lverb|#2 was \xint_bye, we did not find a closing ) in original #1. Error.|
%    \begin{macrocode}
\def\XINT_isbalanced_no\xint_bye #1\xint_bye\xint_bye {\xint_c_i }%
%    \end{macrocode}
% \subsubsection{\csh{XINT_allexpr_func_seqx}, \csh{XINT_allexpr_func_subx}}
%    \begin{macrocode}
\def\XINT_expr_func_seqx   #1#2{\XINT_allexpr_seqx \xintbareeval }%
\def\XINT_flexpr_func_seqx #1#2{\XINT_allexpr_seqx \xintbarefloateval}%
\def\XINT_iiexpr_func_seqx #1#2{\XINT_allexpr_seqx \xintbareiieval }%
\def\XINT_allexpr_seqx #1#2#3#4% #2 is the index list, fully evaluated and encapsulated
{% #3 is the Latin letter serving as dummy variable, #4 is the expression to evaluate
    \expandafter \XINT_expr_getop
    \csname .=\expandafter\XINT_expr_seq:_aa
           \romannumeral-`0\XINT_expr_unlock #2!{#1#4\relax !#3},^,\endcsname
}%
\def\XINT_expr_seq:_aa #1{\if +#1\expandafter\XINT_expr_seq:_A\else
                                 \expandafter\XINT_expr_seq:_a\fi #1}%
%    \end{macrocode}
% \subsubsection{break, abort, omit within seq}
% \lverb|when evaluation is done in seq:_d, after the ! we find: the Latin
% letter, the braced evaluated value to which it will be assigned, a saved copy of the
% the \xintexpr stuff, the braced accumulated comma separated list of previous
% computations, and the rest of the list of comma separated values to assign to
% the dummy letter and at the very end there is ^ and the final comma.|
%    \begin{macrocode}
\def\XINT_expr_seq:_a #1!#2{\expandafter\XINT_expr_seq_empty?
                             \romannumeral0\XINT_expr_seq:_b {#2}#1}%
\def\XINT_expr_seq:_b #1#2,{\XINT_expr_seq:_c #2,{#1}}%
\def\XINT_expr_seq:_c #1{\if ,#1\xint_dothis\XINT_expr_seq:_noop\fi
                         \if ^#1\xint_dothis\XINT_expr_seq:_end\fi
                         \xint_orthat\XINT_expr_seq:_d #1}%
\def\XINT_expr_seq:_d #1,#2{\expandafter\XINT_expr_seq:_e 
    \romannumeral-`0\expandafter\XINT_expr_unlock\romannumeral0#2{#1}{#2}}%
\def\XINT_expr_seq:_e #1{\if #1^\xint_dothis\XINT_expr_seq:_abort\fi
                         \if #1?\xint_dothis\XINT_expr_seq:_break\fi
                         \if #1!\xint_dothis\XINT_expr_seq:_omit\fi
                         \xint_orthat{\XINT_expr_seq:_goon #1}}%
\def\XINT_expr_seq:_goon  #1!#2#3#4{,#1\XINT_expr_seq:_b {#4}}%
\def\XINT_expr_seq:_omit  #1!#2#3#4{\XINT_expr_seq:_b {#4}}%
\def\XINT_expr_seq:_abort #1!#2#3#4#5^,{}%
\def\XINT_expr_seq:_break #1!#2#3#4#5^,{,#1}%
\def\XINT_expr_seq:_noop ,#1{\XINT_expr_seq:_b {#1}}%
\def\XINT_expr_seq:_end ^,#1{}% if all is omit, _empty? constructs "nil"
\def\XINT_expr_seq_empty? #1{%
\def\XINT_expr_seq_empty? ##1{\if ,##1\expandafter\xint_gobble_i\fi #1\endcsname }}%
\XINT_expr_seq_empty? { }%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_seq:_A}}
% \lverb|This is for index lists generated by ++. The starting point will have
% been replaced by its ceil. For efficiency I use \numexpr rather than
% \xintInc, hence the indexing is limited to small integers.|
%    \begin{macrocode}
\def\XINT_expr_seq:_A +#1!#2,^,%
   {\expandafter\XINT_expr_seq_empty?\romannumeral0\XINT_expr_seq:_D {#1}{#2}}%
\def\XINT_expr_seq:_D #1#2{\expandafter\XINT_expr_seq:_E
    \romannumeral-`0\expandafter\XINT_expr_unlock\romannumeral0#2{#1}{#2}}%
\def\XINT_expr_seq:_E #1{\if #1^\xint_dothis\XINT_expr_seq:_Abort\fi
                         \if #1?\xint_dothis\XINT_expr_seq:_Break\fi
                         \if #1!\xint_dothis\XINT_expr_seq:_Omit\fi
                         \xint_orthat{\XINT_expr_seq:_Goon #1}}%
\def\XINT_expr_seq:_Goon  #1!#2#3#4%
    {,#1\expandafter\XINT_expr_seq:_D\expandafter{\the\numexpr #3+\xint_c_i}{#4}}%
\def\XINT_expr_seq:_Omit  #1!#2#3#4%
    {\expandafter\XINT_expr_seq:_D\expandafter{\the\numexpr #3+\xint_c_i}{#4}}%
\def\XINT_expr_seq:_Abort #1!#2#3#4{}%
\def\XINT_expr_seq:_Break #1!#2#3#4{,#1}%
%    \end{macrocode}
% \subsubsection{add and mul, \csh{XINT_expr_onlitteral_add},
% \csh{XINT_expr_onlitteral_mul}}
%    \begin{macrocode}
\def\XINT_expr_onlitteral_add 
 {\expandafter\XINT_expr_onlitteral_add_f\romannumeral-`0\XINT_expr_onlitteral_seq_a {}}%
\def\XINT_expr_onlitteral_add_f #1#2{\xint_c_xviii `{opx}#2)\relax #1+}%
\def\XINT_expr_onlitteral_mul 
 {\expandafter\XINT_expr_onlitteral_mul_f\romannumeral-`0\XINT_expr_onlitteral_seq_a {}}%
\def\XINT_expr_onlitteral_mul_f #1#2{\xint_c_xviii `{opx}#2)\relax #1*}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_func_opx}, \csh{XINT_flexpr_func_opx},
% \csh{XINT_iiexpr_func_opx}}
%    \begin{macrocode}
\expandafter\edef\csname XINT_expr_op:_+\endcsname 
     {\noexpand\xint_gobble_v {}{}{}\expandafter\noexpand\csname .=0\endcsname}%
\expandafter\edef\csname XINT_expr_op:_*\endcsname 
     {\noexpand\xint_gobble_v {}{}{}\expandafter\noexpand\csname .=1\endcsname}%
\def\XINT_expr_func_opx   #1#2{\XINT_allexpr_opx \xintexpr }%
\def\XINT_flexpr_func_opx #1#2{\XINT_allexpr_opx \xintfloatexpr }%
\def\XINT_iiexpr_func_opx #1#2{\XINT_allexpr_opx \xintiiexpr }%
\def\XINT_allexpr_opx #1#2#3#4#5%
{% au d\'epart on avait op(#4,#3=#2 (\'evalu\'e ici)) #3=la variable, #4=expression, #5=+ ou*.
    \expandafter\XINT_expr_getop\romannumeral0\expandafter\XINT_expr_op:_a
    \csname XINT_expr_op:_#5\expandafter\endcsname
    \romannumeral-`0\XINT_expr_unlock #2!#5#1#3{#4}%
}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_op:_a}, \dots}
%    \begin{macrocode}
\def\XINT_expr_op:_a #1#2!#3#4#5#6{\XINT_expr_op:_b {#1#4#3{#6\relax\relax !#5}}#2,^,}%
% #1=op_+ ou op_*, #2=liste, #3=+ou*,#4=\xintexpr, etc, #5=la var,#6=expression
\def\XINT_expr_op:_b #1#2,{\XINT_expr_op:_c #2,#1}%
\def\XINT_expr_op:_c #1{\if ,#1\xint_dothis\XINT_expr_op:_noop\fi
                        \if ^#1\xint_dothis\XINT_expr_op:_end\fi
                        \xint_orthat\XINT_expr_op:_d #1}%
\def\XINT_expr_op:_noop #1,#2#3#4#5{\XINT_expr_op:_b {{#2}#3#4{#5}}}%
\def\XINT_expr_op:_d #1,#2#3#4#5%
% #1=valeur, #2=partiel, #3=\xintexpr #4=+ ou *, #5 = expression
  {\expandafter\expandafter\expandafter\XINT_expr_op:_e #3#2#4#3#5{#1}{#3#4{#5}}}%
% #2=nom de la variable, #3=ancienne valeur variable,
\def\XINT_expr_op:_e   !#1!#2#3#4{\XINT_expr_op:_b {{!#1}#4}}%
\def\XINT_expr_op:_end ^,#1#2#3#4{\expandafter\expandafter\expandafter\space
                                  \expandafter\xint_gobble_iv #1}%
%    \end{macrocode}
% \subsubsection{subs, \csh{XINT_expr_onlitteral_subs}}
%    \begin{macrocode}
\def\XINT_expr_onlitteral_subs 
 {\expandafter\XINT_expr_onlitteral_subs_f\romannumeral-`0\XINT_expr_onlitteral_seq_a {}}%
\def\XINT_expr_onlitteral_subs_f #1#2{\xint_c_xviii `{subx}#2)\relax #1}%
\def\XINT_expr_func_subx   #1#2{\XINT_allexpr_subx \xintbareeval }%
\def\XINT_flexpr_func_subx #1#2{\XINT_allexpr_subx \xintbarefloateval}%
\def\XINT_iiexpr_func_subx #1#2{\XINT_allexpr_subx \xintbareiieval }%
\def\XINT_allexpr_subx #1#2#3#4% #2 is the value to assign to the dummy variable
{% #3 is the dummy variable, #4 is the expression to evaluate
    \expandafter \XINT_expr_getop
    \csname .=\expandafter\XINT_expr_subx:_a
           \romannumeral-`0\XINT_expr_unlock #2!{#1#4\relax !#3}\endcsname
}%
\def\XINT_expr_subx:_a #1!#2%  10/25 that was a quick addition!
   {\expandafter\XINT_expr_subx:_end \romannumeral0#2{#1}}%
% attention, if one day I add a space in unlock, will need \romannumeral-`0
\def\XINT_expr_subx:_end #1!#2#3{\XINT_expr_unlock #1}%
%    \end{macrocode}
% \subsection{rseq}
% \lverb|When func_rseq has its turn, initial segment has been scanned by oparen, the ;
% mimicking the rôle of a closing parenthesis, and stopping further expansion.|
%    \begin{macrocode}
\def\XINT_expr_func_rseq   {\XINT_allexpr_rseq \xintbareeval      }%
\def\XINT_flexpr_func_rseq {\XINT_allexpr_rseq \xintbarefloateval }%
\def\XINT_iiexpr_func_rseq {\XINT_allexpr_rseq \xintbareiieval    }%
\def\XINT_allexpr_rseq #1#2%
{%
    \expandafter\XINT_expr_rseqx\expandafter #1\expandafter
    #2\romannumeral-`0\XINT_expr_onlitteral_seq_a {}%
}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_rseqx}}
% \lverb|The (#4) is for ++ mechanism which must have its closing parenthesis.|
%    \begin{macrocode}
\def\XINT_expr_rseqx #1#2#3#4%
{%
    \expandafter\XINT_expr_rseqy\romannumeral0#1(#4)\relax
    #2#3#1%
}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_rseqy}}
%    \begin{macrocode}
\def\XINT_expr_rseqy #1#2#3#4#5% #1=valeurs pour variable (locked),
              % #2=toutes les valeurs initiales  (csv,locked),
              % #3=variable, #4=expr,
              % #5=\xintbareeval ou \xintbarefloateval ou \xintbareiieval
{%
    \expandafter \XINT_expr_getop
    \csname .=\XINT_expr_unlock #2%
    \expandafter\XINT_expr_rseq:_aa
                \romannumeral-`0\XINT_expr_unlock #1!{#5#4\relax !#3}#2,^,\endcsname
}%
\def\XINT_expr_rseq:_aa #1{\if +#1\expandafter\XINT_expr_rseq:_A\else
                                  \expandafter\XINT_expr_rseq:_a\fi #1}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_rseq:_a} etc\dots}
%    \begin{macrocode}
\def\XINT_expr_rseq:_a #1!#2#3{\XINT_expr_rseq:_b #3{#2}#1}%
\def\XINT_expr_rseq:_b #1#2#3,{\XINT_expr_rseq:_c #3,~#1{#2}}%
\def\XINT_expr_rseq:_c #1{\if ,#1\xint_dothis\XINT_expr_rseq:_noop\fi
                          \if ^#1\xint_dothis\XINT_expr_rseq:_end\fi
                          \xint_orthat\XINT_expr_rseq:_d #1}%
\def\XINT_expr_rseq:_d #1,~#2#3{\expandafter\XINT_expr_rseq:_e
       \romannumeral-`0\expandafter\XINT_expr_unlock\romannumeral0#3{#1}~#2{#3}}%
\def\XINT_expr_rseq:_e #1{%
    \if ^#1\xint_dothis\XINT_expr_rseq:_abort\fi
    \if ?#1\xint_dothis\XINT_expr_rseq:_break\fi
    \if !#1\xint_dothis\XINT_expr_rseq:_omit\fi
    \xint_orthat{\XINT_expr_rseq:_goon #1}}%
\def\XINT_expr_rseq:_goon  #1!#2#3~#4#5{,#1\expandafter\XINT_expr_rseq:_b 
        \romannumeral0\XINT_expr_lockit {#1}{#5}}%
\def\XINT_expr_rseq:_omit  #1!#2#3~{\XINT_expr_rseq:_b }%
\def\XINT_expr_rseq:_abort #1!#2#3~#4#5#6^,{}%
\def\XINT_expr_rseq:_break #1!#2#3~#4#5#6^,{,#1}%
\def\XINT_expr_rseq:_noop ,~#1#2{\XINT_expr_rseq:_b #1{#2}}%
\def\XINT_expr_rseq:_end ^,~#1#2{}% no nil for rseq
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_rseq:_A} etc\dots}
% \lverb |n++ for rseq|
%    \begin{macrocode}
\def\XINT_expr_rseq:_A +#1!#2#3,^,{\XINT_expr_rseq:_D {#1}#3{#2}}%
\def\XINT_expr_rseq:_D #1#2#3{\expandafter\XINT_expr_rseq:_E
    \romannumeral-`0\expandafter\XINT_expr_unlock\romannumeral0#3{#1}~#2{#3}}%
\def\XINT_expr_rseq:_E #1{\if #1^\xint_dothis\XINT_expr_rseq:_Abort\fi
                         \if #1?\xint_dothis\XINT_expr_rseq:_Break\fi
                         \if #1!\xint_dothis\XINT_expr_rseq:_Omit\fi
                         \xint_orthat{\XINT_expr_rseq:_Goon #1}}%
\def\XINT_expr_rseq:_Goon  #1!#2#3~#4#5%
    {,#1\expandafter\XINT_expr_rseq:_D\expandafter{\the\numexpr #3+\xint_c_i\expandafter}%
    \romannumeral0\XINT_expr_lockit{#1}{#5}}%
\def\XINT_expr_rseq:_Omit  #1!#2#3~%#4#5%
    {\expandafter\XINT_expr_rseq:_D\expandafter{\the\numexpr #3+\xint_c_i}}%
\def\XINT_expr_rseq:_Abort #1!#2#3~#4#5{}%
\def\XINT_expr_rseq:_Break #1!#2#3~#4#5{,#1}%
%    \end{macrocode}
% \subsection{rrseq}
% \lverb|When func_rrseq has its turn, initial segment has been scanned by oparen, the ;
% mimicking the rôle of a closing parenthesis, and stopping further expansion.|
%    \begin{macrocode}
\def\XINT_expr_func_rrseq   {\XINT_allexpr_rrseq \xintbareeval      }%
\def\XINT_flexpr_func_rrseq {\XINT_allexpr_rrseq \xintbarefloateval }%
\def\XINT_iiexpr_func_rrseq {\XINT_allexpr_rrseq \xintbareiieval    }%
\def\XINT_allexpr_rrseq #1#2%
{%
    \expandafter\XINT_expr_rrseqx\expandafter #1\expandafter
    #2\romannumeral-`0\XINT_expr_onlitteral_seq_a {}%
}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_rrseqx}}
% \lverb|The (#4) is for ++ mechanism which must have its closing parenthesis.|
%    \begin{macrocode}
\def\XINT_expr_rrseqx #1#2#3#4%
{%
    \expandafter\XINT_expr_rrseqy\romannumeral0#1(#4)\expandafter\relax
    \expandafter{\romannumeral0\xintapply \XINT_expr_lockit
       {\xintRevWithBraces{\xintCSVtoListNonStripped{\XINT_expr_unlock #2}}}}%
    #2#3#1%
}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_rrseqy}}
%    \begin{macrocode}
\def\XINT_expr_rrseqy #1#2#3#4#5#6% #1=valeurs pour variable (locked),
              % #2=initial values (reversed, one (braced) token each)
              % #3=toutes les valeurs initiales  (csv,locked),
              % #4=variable, #5=expr,
              % #6=\xintbareeval ou \xintbarefloateval ou \xintbareiieval
{%
    \expandafter \XINT_expr_getop
    \csname .=\XINT_expr_unlock #3%
    \expandafter\XINT_expr_rrseq:_aa
                \romannumeral-`0\XINT_expr_unlock #1!{#6#5\relax !#4}{#2},^,\endcsname
}%
\def\XINT_expr_rrseq:_aa #1{\if +#1\expandafter\XINT_expr_rrseq:_A\else
                                  \expandafter\XINT_expr_rrseq:_a\fi #1}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_rrseq:_a} etc\dots}
%    \begin{macrocode}
\catcode`? 3
\def\XINT_expr_rrseq:_a #1!#2#3{\XINT_expr_rrseq:_b {#3}{#2}#1}%
\def\XINT_expr_rrseq:_b #1#2#3,{\XINT_expr_rrseq:_c #3,~#1?{#2}}%
\def\XINT_expr_rrseq:_c #1{\if ,#1\xint_dothis\XINT_expr_rrseq:_noop\fi
                          \if ^#1\xint_dothis\XINT_expr_rrseq:_end\fi
                          \xint_orthat\XINT_expr_rrseq:_d #1}%
\def\XINT_expr_rrseq:_d #1,~#2?#3{\expandafter\XINT_expr_rrseq:_e
       \romannumeral-`0\expandafter\XINT_expr_unlock\romannumeral0#3{#1}~#2?{#3}}%
\def\XINT_expr_rrseq:_goon  #1!#2#3~#4?#5{,#1\expandafter\XINT_expr_rrseq:_b\expandafter
        {\romannumeral0\xinttrim{-1}{\XINT_expr_lockit{#1}#4}}{#5}}%
\def\XINT_expr_rrseq:_omit  #1!#2#3~{\XINT_expr_rrseq:_b }%
\def\XINT_expr_rrseq:_abort #1!#2#3~#4?#5#6^,{}%
\def\XINT_expr_rrseq:_break #1!#2#3~#4?#5#6^,{,#1}%
\def\XINT_expr_rrseq:_noop ,~#1?#2{\XINT_expr_rrseq:_b {#1}{#2}}%
\def\XINT_expr_rrseq:_end ^,~#1?#2{}% No nil for rrseq.
\catcode`? 11
\def\XINT_expr_rrseq:_e #1{%
    \if ^#1\xint_dothis\XINT_expr_rrseq:_abort\fi
    \if ?#1\xint_dothis\XINT_expr_rrseq:_break\fi
    \if !#1\xint_dothis\XINT_expr_rrseq:_omit\fi
    \xint_orthat{\XINT_expr_rrseq:_goon #1}%
}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_rrseq:_A} etc\dots}
% \lverb |n++ for rrseq|
%    \begin{macrocode}
\catcode`? 3
\def\XINT_expr_rrseq:_A +#1!#2#3,^,{\XINT_expr_rrseq:_D {#1}{#3}{#2}}%
\def\XINT_expr_rrseq:_D #1#2#3{\expandafter\XINT_expr_rrseq:_E
    \romannumeral-`0\expandafter\XINT_expr_unlock\romannumeral0#3{#1}~#2?{#3}}%
\def\XINT_expr_rrseq:_Goon  #1!#2#3~#4?#5%
   {,#1\expandafter\XINT_expr_rrseq:_D\expandafter{\the\numexpr #3+\xint_c_i\expandafter}%
    \expandafter{\romannumeral0\xinttrim{-1}{\XINT_expr_lockit{#1}#4}}{#5}}%
\def\XINT_expr_rrseq:_Omit  #1!#2#3~%#4?#5%
    {\expandafter\XINT_expr_rrseq:_D\expandafter{\the\numexpr #3+\xint_c_i}}%
\def\XINT_expr_rrseq:_Abort #1!#2#3~#4?#5{}%
\def\XINT_expr_rrseq:_Break #1!#2#3~#4?#5{,#1}%
\catcode`? 11
\def\XINT_expr_rrseq:_E #1{\if #1^\xint_dothis\XINT_expr_rrseq:_Abort\fi
                         \if #1?\xint_dothis\XINT_expr_rrseq:_Break\fi
                         \if #1!\xint_dothis\XINT_expr_rrseq:_Omit\fi
                         \xint_orthat{\XINT_expr_rrseq:_Goon #1}}%
%    \end{macrocode}
% \subsection{iter}
%    \begin{macrocode}
\def\XINT_expr_func_iter   {\XINT_allexpr_iter \xintbareeval      }%
\def\XINT_flexpr_func_iter {\XINT_allexpr_iter \xintbarefloateval }%
\def\XINT_iiexpr_func_iter {\XINT_allexpr_iter \xintbareiieval    }%
\def\XINT_allexpr_iter #1#2%
{%
    \expandafter\XINT_expr_iterx\expandafter #1\expandafter
    #2\romannumeral-`0\XINT_expr_onlitteral_seq_a {}%
}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_iterx}}
% \lverb|The (#4) is for ++ mechanism which must have its closing parenthesis.|
%    \begin{macrocode}
\def\XINT_expr_iterx #1#2#3#4%
{%
    \expandafter\XINT_expr_itery\romannumeral0#1(#4)\expandafter\relax
    \expandafter{\romannumeral0\xintapply \XINT_expr_lockit
       {\xintRevWithBraces{\xintCSVtoListNonStripped{\XINT_expr_unlock #2}}}}%
    #2#3#1%
}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_itery}}
%    \begin{macrocode}
\def\XINT_expr_itery #1#2#3#4#5#6% #1=valeurs pour variable (locked),
              % #2=initial values (reversed, one (braced) token each)
              % #3=toutes les valeurs initiales  (csv,locked),
              % #4=variable, #5=expr,
              % #6=\xintbareeval ou \xintbarefloateval ou \xintbareiieval
{%
    \expandafter \XINT_expr_getop
    \csname .=%
    \expandafter\XINT_expr_iter:_aa
                \romannumeral-`0\XINT_expr_unlock #1!{#6#5\relax !#4}{#2},^,\endcsname
}%
\def\XINT_expr_iter:_aa #1{\if +#1\expandafter\XINT_expr_iter:_A\else
                                  \expandafter\XINT_expr_iter:_a\fi #1}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_iter:_a} etc\dots}
%    \begin{macrocode}
\catcode`? 3
\def\XINT_expr_iter:_a #1!#2#3{\XINT_expr_iter:_b {#3}{#2}#1}%
\def\XINT_expr_iter:_b #1#2#3,{\XINT_expr_iter:_c #3,~#1?{#2}}%
\def\XINT_expr_iter:_c #1{\if ,#1\xint_dothis\XINT_expr_iter:_noop\fi
                          \if ^#1\xint_dothis\XINT_expr_iter:_end\fi
                          \xint_orthat\XINT_expr_iter:_d #1}%
\def\XINT_expr_iter:_d #1,~#2?#3{\expandafter\XINT_expr_iter:_e
       \romannumeral-`0\expandafter\XINT_expr_unlock\romannumeral0#3{#1}~#2?{#3}}%
\def\XINT_expr_iter:_goon  #1!#2#3~#4?#5{\expandafter\XINT_expr_iter:_b\expandafter
        {\romannumeral0\xinttrim{-1}{\XINT_expr_lockit{#1}#4}}{#5}}%
\def\XINT_expr_iter:_omit  #1!#2#3~{\XINT_expr_iter:_b }%
\def\XINT_expr_iter:_abort #1!#2#3~#4?#5#6^,%
   {\expandafter\xint_gobble_i\romannumeral0\xintapplyunbraced
          {,\XINT_expr:_unlock}{\xintReverseOrder{#4\space}}}%
\def\XINT_expr_iter:_break #1!#2#3~#4?#5#6^,%
   {\expandafter\xint_gobble_iv\romannumeral0\xintapplyunbraced
          {,\XINT_expr:_unlock}{\xintReverseOrder{#4\space}},#1}%
\def\XINT_expr_iter:_noop ,~#1?#2{\XINT_expr_iter:_b {#1}{#2}}%
\def\XINT_expr_iter:_end ^,~#1?#2%
   {\expandafter\xint_gobble_i\romannumeral0\xintapplyunbraced
          {,\XINT_expr:_unlock}{\xintReverseOrder{#1\space}}}%
\catcode`? 11
\def\XINT_expr_iter:_e #1{%
    \if ^#1\xint_dothis\XINT_expr_iter:_abort\fi
    \if ?#1\xint_dothis\XINT_expr_iter:_break\fi
    \if !#1\xint_dothis\XINT_expr_iter:_omit\fi
    \xint_orthat{\XINT_expr_iter:_goon #1}%
}%
\def\XINT_expr:_unlock #1{\XINT_expr_unlock #1}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_iter:_A} etc\dots}
% \lverb |n++ for iter|
%    \begin{macrocode}
\catcode`? 3
\def\XINT_expr_iter:_A +#1!#2#3,^,{\XINT_expr_iter:_D {#1}{#3}{#2}}%
\def\XINT_expr_iter:_D #1#2#3{\expandafter\XINT_expr_iter:_E
    \romannumeral-`0\expandafter\XINT_expr_unlock\romannumeral0#3{#1}~#2?{#3}}%
\def\XINT_expr_iter:_Goon  #1!#2#3~#4?#5%
   {\expandafter\XINT_expr_iter:_D\expandafter{\the\numexpr #3+\xint_c_i\expandafter}%
    \expandafter{\romannumeral0\xinttrim{-1}{\XINT_expr_lockit{#1}#4}}{#5}}%
\def\XINT_expr_iter:_Omit  #1!#2#3~%#4?#5%
    {\expandafter\XINT_expr_iter:_D\expandafter{\the\numexpr #3+\xint_c_i}}%
\def\XINT_expr_iter:_Abort #1!#2#3~#4?#5%
   {\expandafter\xint_gobble_i\romannumeral0\xintapplyunbraced
          {,\XINT_expr:_unlock}{\xintReverseOrder{#4\space}}}%
\def\XINT_expr_iter:_Break #1!#2#3~#4?#5%
   {\expandafter\xint_gobble_iv\romannumeral0\xintapplyunbraced
          {,\XINT_expr:_unlock}{\xintReverseOrder{#4\space}},#1}%
\catcode`? 11
\def\XINT_expr_iter:_E #1{\if #1^\xint_dothis\XINT_expr_iter:_Abort\fi
                         \if #1?\xint_dothis\XINT_expr_iter:_Break\fi
                         \if #1!\xint_dothis\XINT_expr_iter:_Omit\fi
                         \xint_orthat{\XINT_expr_iter:_Goon #1}}%
%    \end{macrocode}
% \subsection{Macros handling csv lists for functions with multiple comma
% separated arguments in expressions}
% \lverb|These 17 macros are used inside \csname...\endcsname. These things
% are not initiated by a \romannumeral in general, but in some cases they are,
% especially when involved in an \xintNewExpr. They will then be protected
% against expansion and expand only later in contexts governed by an
% initial \romannumeral-`0. There each new item may need to be expanded, which
% would not be the case in the use for the _func_ things.|
%    \begin{macrocode}
%    \end{macrocode}
% \subsubsection{\csh{xintANDof:csv}}
% \lverb|1.09a. For use by \xintexpr inside \csname. 1.1, je remplace
% ifTrueAelseB par iiNotZero pour des raisons d'optimisations.|
%    \begin{macrocode}
\def\xintANDof:csv #1{\expandafter\XINT_andof:_a\romannumeral-`0#1,,^}%
\def\XINT_andof:_a #1{\if ,#1\expandafter\XINT_andof:_e
                      \else\expandafter\XINT_andof:_c\fi #1}%
\def\XINT_andof:_c #1,{\xintiiifNotZero {#1}{\XINT_andof:_a}{\XINT_andof:_no}}%
\def\XINT_andof:_no #1^{0}%
\def\XINT_andof:_e  #1^{1}% works with empty list
%    \end{macrocode}
% \subsubsection{\csh{xintORof:csv}}
% \lverb|1.09a. For use by \xintexpr.|
%    \begin{macrocode}
\def\xintORof:csv #1{\expandafter\XINT_orof:_a\romannumeral-`0#1,,^}%
\def\XINT_orof:_a #1{\if ,#1\expandafter\XINT_orof:_e
                      \else\expandafter\XINT_orof:_c\fi #1}%
\def\XINT_orof:_c #1,{\xintiiifNotZero{#1}{\XINT_orof:_yes}{\XINT_orof:_a}}%
\def\XINT_orof:_yes #1^{1}%
\def\XINT_orof:_e   #1^{0}% works with empty list
%    \end{macrocode}
% \subsubsection{\csh{xintXORof:csv}}
% \lverb|1.09a. For use by \xintexpr (inside a \csname..\endcsname).|
%    \begin{macrocode}
\def\xintXORof:csv #1{\expandafter\XINT_xorof:_a\expandafter 0\romannumeral-`0#1,,^}%
\def\XINT_xorof:_a #1#2,{\XINT_xorof:_b #2,#1}%
\def\XINT_xorof:_b #1{\if ,#1\expandafter\XINT_xorof:_e
                      \else\expandafter\XINT_xorof:_c\fi #1}%
\def\XINT_xorof:_c #1,#2%
           {\xintiiifNotZero {#1}{\if #20\xint_afterfi{\XINT_xorof:_a 1}%
                                   \else\xint_afterfi{\XINT_xorof:_a 0}\fi}%
                                  {\XINT_xorof:_a #2}%
           }%
\def\XINT_xorof:_e ,#1#2^{#1}% allows empty list (then returns 0)
%    \end{macrocode}
% \subsubsection{Generic csv routine}
% \lverb|1.1. generic routine. up to the loss of some efficiency, especially
% for Sum:csv and Prod:csv, where \XINTinFloat will be done twice for each
% argument.|
%    \begin{macrocode}
\def\XINT_oncsv:_empty  #1,^,#2{#2}%
\def\XINT_oncsv:_end    ^,#1#2#3#4{#1}%
\def\XINT_oncsv:_a #1#2#3%
   {\if ,#3\expandafter\XINT_oncsv:_empty\else\expandafter\XINT_oncsv:_b\fi #1#2#3}%
\def\XINT_oncsv:_b #1#2#3,%
   {\expandafter\XINT_oncsv:_c \expandafter{\romannumeral-`0#2{#3}}#1#2}%
\def\XINT_oncsv:_c #1#2#3#4,{\expandafter\XINT_oncsv:_d \romannumeral-`0#4,{#1}#2#3}%
\def\XINT_oncsv:_d #1%
   {\if ^#1\expandafter\XINT_oncsv:_end\else\expandafter\XINT_oncsv:_e\fi #1}%
\def\XINT_oncsv:_e #1,#2#3#4%
   {\expandafter\XINT_oncsv:_c\expandafter {\romannumeral-`0#3{#4{#1}}{#2}}#3#4}%
%    \end{macrocode}
% \subsubsection{\csh{xintMaxof:csv}, \csh{xintiiMaxof:csv}}
% \lverb|1.09i. Rewritten for 1.1. Compatible avec liste vide donnant valeur par
% défaut. Pas compatible avec items manquants.
% ah je m'aperçois au dernier moment que je n'ai pas en effet de \xintiiMax.
% Je devrais le rajouter. En tout cas ici c'est uniquement pour xintiiexpr,
% dans il faut bien sûr ne pas faire de xintNum, donc il faut un iimax.|
%    \begin{macrocode}
\def\xintMaxof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintmax
                      \expandafter\xint_firstofone\romannumeral-`0#1,^,{0/1[0]}}%
\def\xintiiMaxof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintiimax
                       \expandafter\xint_firstofone\romannumeral-`0#1,^,0}%
%    \end{macrocode}
% \subsubsection{\csh{xintMinof:csv}, \csh{xintiiMinof:csv}}
% \lverb|1.09i. Rewritten for 1.1. For use by \xintiiexpr.|
%    \begin{macrocode}
\def\xintMinof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintmin
                       \expandafter\xint_firstofone\romannumeral-`0#1,^,{0/1[0]}}%
\def\xintiiMinof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintiimin
                       \expandafter\xint_firstofone\romannumeral-`0#1,^,0}%
%    \end{macrocode}
% \subsubsection{\csh{xintSum:csv}, \csh{xintiiSum:csv}}
% \lverb|1.09a. Rewritten for 1.1. For use by \xintexpr.|
%    \begin{macrocode}
\def\xintSum:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintadd
                    \expandafter\xint_firstofone\romannumeral-`0#1,^,{0/1[0]}}%
\def\xintiiSum:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintiiadd
                      \expandafter\xint_firstofone\romannumeral-`0#1,^,0}%
%    \end{macrocode}
% \subsubsection{\csh{xintPrd:csv}, \csh{xintiiPrd:csv}}
% \lverb|1.09a. Rewritten for 1.1. For use by \xintexpr.|
%    \begin{macrocode}
\def\xintPrd:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintmul
                    \expandafter\xint_firstofone\romannumeral-`0#1,^,{1/1[0]}}%
\def\xintiiPrd:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintiimul
                      \expandafter\xint_firstofone\romannumeral-`0#1,^,1}%
%    \end{macrocode}
% \subsubsection{\csh{xintGCDof:csv}, \csh{xintLCMof:csv}}
% \lverb|1.09a. Rewritten for 1.1. For use by \xintexpr. Expansion réinstaurée
% pour besoins de xintNewExpr de version 1.1|
%    \begin{macrocode}
\def\xintGCDof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintgcd
                      \expandafter\xint_firstofone\romannumeral-`0#1,^,1}%
\def\xintLCMof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintlcm
                      \expandafter\xint_firstofone\romannumeral-`0#1,^,0}%
%    \end{macrocode}
% \subsubsection{\csh{xintiiGCDof:csv}, \csh{xintiiLCMof:csv}}
% \lverb|1.1a pour \xintiiexpr. Ces histoires de ii sont pénibles à la fin.|
%    \begin{macrocode}
\def\xintiiGCDof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintiigcd
                      \expandafter\xint_firstofone\romannumeral-`0#1,^,1}%
\def\xintiiLCMof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintiilcm
                      \expandafter\xint_firstofone\romannumeral-`0#1,^,0}%
%    \end{macrocode}
% \subsubsection{\csh{XINTinFloatdigits}, \csh{XINTinFloatSqrtdigits}}
% \lverb|for \xintNewExpr matters, mainly.|
%    \begin{macrocode}
\def\XINTinFloatdigits     {\XINTinFloat [\XINTdigits]}%
\def\XINTinFloatSqrtdigits {\XINTinFloatSqrt [\XINTdigits]}%
%    \end{macrocode}
% \subsubsection{\csh{XINTinFloatMaxof:csv}, \csh{XINTinFloatMinof:csv}}
% \lverb|1.09a. Rewritten for 1.1. For use by \xintfloatexpr. Name changed in 1.09h|
%    \begin{macrocode}
\def\XINTinFloatMaxof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintmax
                         \expandafter\XINTinFloatdigits\romannumeral-`0#1,^,{0[0]}}%
\def\XINTinFloatMinof:csv #1{\expandafter\XINT_oncsv:_a\expandafter\xintmin
                         \expandafter\XINTinFloatdigits\romannumeral-`0#1,^,{0[0]}}%
%    \end{macrocode}
% \subsubsection{\csh{XINTinFloatSum:csv}, \csh{XINTinFloatPrd:csv}}
% \lverb|1.09a. Rewritten for 1.1. For use by \xintfloatexpr.|
%    \begin{macrocode}
\def\XINTinFloatSum:csv #1{\expandafter\XINT_oncsv:_a\expandafter\XINTinfloatadd
                        \expandafter\XINTinFloatdigits\romannumeral-`0#1,^,{0[0]}}%
\def\XINTinFloatPrd:csv #1{\expandafter\XINT_oncsv:_a\expandafter\XINTinfloatmul
                        \expandafter\XINTinFloatdigits\romannumeral-`0#1,^,{1[0]}}%
%    \end{macrocode}
% \subsection{The num, reduce, abs, sgn, frac, floor, ceil, sqr, sqrt, sqrtr, float,
% round, trunc, mod, quo, rem, gcd, lcm, max, min, \textasciigrave
% +\textasciigrave, \textasciigrave
% \texorpdfstring{\protect\lowast}{*}\textasciigrave, ?, !, not, all, any,
% xor, if, ifsgn, first, last, even, odd, and reversed functions}
%    \begin{macrocode}
\def\XINT_expr_twoargs #1,#2,{{#1}{#2}}%
\def\XINT_expr_argandopt #1,#2,#3.#4#5%
{%
    \if\relax#3\relax\expandafter\xint_firstoftwo\else
                     \expandafter\xint_secondoftwo\fi
    {#4}{#5[\xintNum {#2}]}{#1}%
}%
\def\XINT_expr_oneortwo #1#2#3,#4,#5.%
{%
    \if\relax#5\relax\expandafter\xint_firstoftwo\else
                     \expandafter\xint_secondoftwo\fi
    {#1{0}}{#2{\xintNum {#4}}}{#3}%
}%
\def\XINT_iiexpr_oneortwo #1#2,#3,#4.%
{%
    \if\relax#4\relax\expandafter\xint_firstoftwo\else
                     \expandafter\xint_secondoftwo\fi
    {#1{0}}{#1{#3}}{#2}%
}%
\def\XINT_expr_func_num #1#2#3%
    {\expandafter #1\expandafter #2\csname.=\xintNum {\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_num\XINT_expr_func_num
\let\XINT_iiexpr_func_num\XINT_expr_func_num
% [0] added Oct 25. For interaction with SPRaw::csv
\def\XINT_expr_func_reduce #1#2#3%
  {\expandafter #1\expandafter #2\csname.=\xintIrr {\XINT_expr_unlock #3}[0]\endcsname }%
\let\XINT_flexpr_func_reduce\XINT_expr_func_reduce
% no \XINT_iiexpr_func_reduce
\def\XINT_expr_func_abs #1#2#3%
  {\expandafter #1\expandafter #2\csname.=\xintAbs {\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_abs\XINT_expr_func_abs
\def\XINT_iiexpr_func_abs #1#2#3%
  {\expandafter #1\expandafter #2\csname.=\xintiiAbs {\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_sgn #1#2#3%
  {\expandafter #1\expandafter #2\csname.=\xintSgn {\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_sgn\XINT_expr_func_sgn
\def\XINT_iiexpr_func_sgn #1#2#3%
  {\expandafter #1\expandafter #2\csname.=\xintiiSgn {\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_frac #1#2#3%
  {\expandafter #1\expandafter #2\csname.=\xintTFrac {\XINT_expr_unlock #3}\endcsname }%
\def\XINT_flexpr_func_frac #1#2#3{\expandafter #1\expandafter #2\csname 
    .=\XINTinFloatFracdigits {\XINT_expr_unlock #3}\endcsname }%
% no \XINT_iiexpr_func_frac
\def\XINT_expr_func_floor #1#2#3%
  {\expandafter #1\expandafter #2\csname .=\xintFloor {\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_floor\XINT_expr_func_floor
\def\XINT_iiexpr_func_floor #1#2#3%
{% mais absurde si on ne peut pas avoir quotient comme input
  \expandafter #1\expandafter #2\csname.=\xintiFloor {\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_ceil #1#2#3%
  {\expandafter #1\expandafter #2\csname .=\xintCeil {\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_ceil\XINT_expr_func_ceil
\def\XINT_iiexpr_func_ceil #1#2#3%
{% mais absurde si on ne peut pas avoir quotient comme input
  \expandafter #1\expandafter #2\csname.=\xintiCeil {\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_sqr #1#2#3%
    {\expandafter #1\expandafter #2\csname.=\xintSqr {\XINT_expr_unlock #3}\endcsname }%
\def\XINT_flexpr_func_sqr #1#2#3%
{%
    \expandafter #1\expandafter #2\csname
    .=\XINTinFloatMul % [\XINTdigits]% pour simplifier mes affaires avec \xintNewExpr
            {\XINT_expr_unlock #3}{\XINT_expr_unlock #3}\endcsname
}%
\def\XINT_iiexpr_func_sqr #1#2#3%
  {\expandafter #1\expandafter #2\csname.=\xintiiSqr {\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_sqrt #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\XINT_expr_argandopt
    \romannumeral-`0\XINT_expr_unlock#3,,.\XINTinFloatSqrtdigits\XINTinFloatSqrt
    \endcsname
}%
\let\XINT_flexpr_func_sqrt\XINT_expr_func_sqrt
\def\XINT_iiexpr_func_sqrt #1#2#3%
 {\expandafter #1\expandafter #2\csname.=\xintiiSqrt {\XINT_expr_unlock #3}\endcsname }%
\def\XINT_iiexpr_func_sqrtr #1#2#3%
 {\expandafter #1\expandafter #2\csname.=\xintiiSqrtR {\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_round #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\XINT_expr_oneortwo
    \expandafter\xintiRound\expandafter\xintRound
    \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname
}%
\let\XINT_flexpr_func_round\XINT_expr_func_round
\def\XINT_iiexpr_func_round #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\XINT_iiexpr_oneortwo\expandafter\xintiRound
    \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname
}%
\def\XINT_expr_func_trunc #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\XINT_expr_oneortwo
    \expandafter\xintiTrunc\expandafter\xintTrunc
    \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname
}%
\let\XINT_flexpr_func_trunc\XINT_expr_func_trunc
\def\XINT_iiexpr_func_trunc #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\XINT_iiexpr_oneortwo\expandafter\xintiTrunc
    \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname
}%
\def\XINT_expr_func_float #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\XINT_expr_argandopt
    \romannumeral-`0\XINT_expr_unlock #3,,.\XINTinFloatdigits\XINTinFloat
    \endcsname
}%
\let\XINT_flexpr_func_float\XINT_expr_func_float
% \XINT_iiexpr_func_float not defined
\def\XINT_expr_func_mod #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\expandafter\expandafter\xintMod
    \expandafter\XINT_expr_twoargs
    \romannumeral-`0\XINT_expr_unlock #3,\endcsname
}%
\def\XINT_flexpr_func_mod #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\XINTinFloatMod
    \romannumeral-`0\expandafter\XINT_expr_twoargs
    \romannumeral-`0\XINT_expr_unlock #3,\endcsname
}%
\def\XINT_iiexpr_func_mod #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\expandafter\expandafter\xintiiMod
    \expandafter\XINT_expr_twoargs
    \romannumeral-`0\XINT_expr_unlock #3,\endcsname
}%
\def\XINT_expr_func_quo #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\expandafter\expandafter\xintiQuo
    \expandafter\XINT_expr_twoargs
    \romannumeral-`0\XINT_expr_unlock #3,\endcsname
}%
\let\XINT_flexpr_func_quo\XINT_expr_func_quo
\def\XINT_iiexpr_func_quo #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\expandafter\expandafter\xintiiQuo
    \expandafter\XINT_expr_twoargs
    \romannumeral-`0\XINT_expr_unlock #3,\endcsname
}%
\def\XINT_expr_func_rem #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\expandafter\expandafter\xintiRem
    \expandafter\XINT_expr_twoargs
    \romannumeral-`0\XINT_expr_unlock #3,\endcsname
}%
\let\XINT_flexpr_func_rem\XINT_expr_func_rem
\def\XINT_iiexpr_func_rem #1#2#3%
{%
    \expandafter #1\expandafter #2\csname .=%
    \expandafter\expandafter\expandafter\xintiiRem
    \expandafter\XINT_expr_twoargs
    \romannumeral-`0\XINT_expr_unlock #3,\endcsname
}%
\def\XINT_expr_func_gcd #1#2#3%
   {\expandafter #1\expandafter #2\csname
                                      .=\xintGCDof:csv{\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_gcd\XINT_expr_func_gcd
\def\XINT_iiexpr_func_gcd #1#2#3%
   {\expandafter #1\expandafter #2\csname
                                      .=\xintiiGCDof:csv{\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_lcm #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                      .=\xintLCMof:csv{\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_lcm\XINT_expr_func_lcm
\def\XINT_iiexpr_func_lcm #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                      .=\xintiiLCMof:csv{\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_max #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                      .=\xintMaxof:csv{\XINT_expr_unlock #3}\endcsname }%
\def\XINT_iiexpr_func_max #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                     .=\xintiiMaxof:csv{\XINT_expr_unlock #3}\endcsname }%
\def\XINT_flexpr_func_max #1#2#3%
    {\expandafter #1\expandafter #2\csname
                               .=\XINTinFloatMaxof:csv{\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_min #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                      .=\xintMinof:csv{\XINT_expr_unlock #3}\endcsname }%
\def\XINT_iiexpr_func_min #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                     .=\xintiiMinof:csv{\XINT_expr_unlock #3}\endcsname }%
\def\XINT_flexpr_func_min #1#2#3%
    {\expandafter #1\expandafter #2\csname
                               .=\XINTinFloatMinof:csv{\XINT_expr_unlock #3}\endcsname }%
\expandafter\def\csname XINT_expr_func_+\endcsname #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                        .=\xintSum:csv{\XINT_expr_unlock #3}\endcsname }%
\expandafter\def\csname XINT_flexpr_func_+\endcsname #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                 .=\XINTinFloatSum:csv{\XINT_expr_unlock #3}\endcsname }%
\expandafter\def\csname XINT_iiexpr_func_+\endcsname #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                      .=\xintiiSum:csv{\XINT_expr_unlock #3}\endcsname }%
\expandafter\def\csname XINT_expr_func_*\endcsname #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                        .=\xintPrd:csv{\XINT_expr_unlock #3}\endcsname }%
\expandafter\def\csname XINT_flexpr_func_*\endcsname #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                 .=\XINTinFloatPrd:csv{\XINT_expr_unlock #3}\endcsname }%
\expandafter\def\csname XINT_iiexpr_func_*\endcsname #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                      .=\xintiiPrd:csv{\XINT_expr_unlock #3}\endcsname }%
\def\XINT_expr_func_? #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                   .=\xintiiIsNotZero {\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_? \XINT_expr_func_?
\let\XINT_iiexpr_func_? \XINT_expr_func_?
\def\XINT_expr_func_! #1#2#3%
 {\expandafter #1\expandafter #2\csname.=\xintiiIsZero {\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_! \XINT_expr_func_!
\let\XINT_iiexpr_func_! \XINT_expr_func_!
\def\XINT_expr_func_not #1#2#3%
 {\expandafter #1\expandafter #2\csname.=\xintiiIsZero {\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_not \XINT_expr_func_not
\let\XINT_iiexpr_func_not \XINT_expr_func_not
\def\XINT_expr_func_all #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                      .=\xintANDof:csv{\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_all\XINT_expr_func_all
\let\XINT_iiexpr_func_all\XINT_expr_func_all
\def\XINT_expr_func_any #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                       .=\xintORof:csv{\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_any\XINT_expr_func_any
\let\XINT_iiexpr_func_any\XINT_expr_func_any
\def\XINT_expr_func_xor #1#2#3%
    {\expandafter #1\expandafter #2\csname
                                      .=\xintXORof:csv{\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_xor\XINT_expr_func_xor
\let\XINT_iiexpr_func_xor\XINT_expr_func_xor
\def\xintifNotZero: #1,#2,#3,{\xintiiifNotZero{#1}{#2}{#3}}%
\def\XINT_expr_func_if #1#2#3%
    {\expandafter #1\expandafter #2\csname
         .=\expandafter\xintifNotZero:\romannumeral-`0\XINT_expr_unlock #3,\endcsname }%
\let\XINT_flexpr_func_if\XINT_expr_func_if
\let\XINT_iiexpr_func_if\XINT_expr_func_if
\def\xintifSgn: #1,#2,#3,#4,{\xintiiifSgn{#1}{#2}{#3}{#4}}%
\def\XINT_expr_func_ifsgn #1#2#3%
{%
    \expandafter #1\expandafter #2\csname
         .=\expandafter\xintifSgn:\romannumeral-`0\XINT_expr_unlock #3,\endcsname
}%
\let\XINT_flexpr_func_ifsgn\XINT_expr_func_ifsgn
\let\XINT_iiexpr_func_ifsgn\XINT_expr_func_ifsgn
\def\XINT_expr_func_first #1#2#3%
    {\expandafter #1\expandafter #2\csname.=\expandafter\XINT_expr_func_firsta 
     \romannumeral-`0\XINT_expr_unlock #3,^\endcsname }%
\def\XINT_expr_func_firsta #1,#2^{#1}%
\let\XINT_flexpr_func_first\XINT_expr_func_first
\let\XINT_iiexpr_func_first\XINT_expr_func_first
\def\XINT_expr_func_last #1#2#3% will not work in \xintNewExpr if macro param involved
    {\expandafter #1\expandafter #2\csname.=\expandafter\XINT_expr_func_lasta 
     \romannumeral-`0\XINT_expr_unlock #3,^\endcsname }%
\def\XINT_expr_func_lasta #1,#2%
   {\if ^#2 #1\expandafter\xint_gobble_ii\fi \XINT_expr_func_lasta #2}%
\let\XINT_flexpr_func_last\XINT_expr_func_last
\let\XINT_iiexpr_func_last\XINT_expr_func_last
\def\XINT_expr_func_odd #1#2#3%
    {\expandafter #1\expandafter #2\csname.=\xintOdd{\XINT_expr_unlock #3}\endcsname}% 
\let\XINT_flexpr_func_odd\XINT_expr_func_odd
\def\XINT_iiexpr_func_odd #1#2#3%
    {\expandafter #1\expandafter #2\csname.=\xintiiOdd{\XINT_expr_unlock #3}\endcsname}% 
\def\XINT_expr_func_even #1#2#3%
    {\expandafter #1\expandafter #2\csname.=\xintEven{\XINT_expr_unlock #3}\endcsname}% 
\let\XINT_flexpr_func_even\XINT_expr_func_even
\def\XINT_iiexpr_func_even #1#2#3%
    {\expandafter #1\expandafter #2\csname.=\xintiiEven{\XINT_expr_unlock #3}\endcsname}% 
\def\XINT_expr_func_nuple #1#2#3%
   {\expandafter #1\expandafter #2\csname .=\XINT_expr_unlock #3\endcsname }%
\let\XINT_flexpr_func_nuple\XINT_expr_func_nuple
\let\XINT_iiexpr_func_nuple\XINT_expr_func_nuple
\def\XINT_expr_func_reversed #1#2#3%
   {\expandafter #1\expandafter #2\csname .=\xintReversed::csv 
                                                      {\XINT_expr_unlock #3}\endcsname }%
\let\XINT_flexpr_func_reversed\XINT_expr_func_reversed
\let\XINT_iiexpr_func_reversed\XINT_expr_func_reversed
\def\xintReversed::csv #1% should be done directly, of course
   {\xintListWithSep,{\xintRevWithBraces {\xintCSVtoListNonStripped{#1}}}}%
%    \end{macrocode}
%
% \subsection{f-expandable versions of the SeqB::csv routines, for
% \csh{xintNewExpr}}
% 
% \subsubsection{\csh{xintSeqB:f:csv}}
% \lverb|Produces in f-expandable way. If the step is zero, gives empty result
% except if start and end coincide.|
%    \begin{macrocode}
\def\xintSeqB:f:csv #1#2%
   {\expandafter\XINT_seqb:f:csv \expandafter{\romannumeral0\xintraw{#2}}{#1}}%
\def\XINT_seqb:f:csv #1#2{\expandafter\XINT_seqb:f:csv_a\romannumeral-`0#2#1!}%
\def\XINT_seqb:f:csv_a #1#2;#3;#4!{%
   \expandafter\xint_gobble_i\romannumeral-`0%
   \xintifCmp {#3}{#4}\XINT_seqb:f:csv_bl\XINT_seqb:f:csv_be\XINT_seqb:f:csv_bg
   #1{#3}{#4}{}{#2}}%
\def\XINT_seqb:f:csv_be #1#2#3#4#5{,#2}%
\def\XINT_seqb:f:csv_bl #1{\if #1p\expandafter\XINT_seqb:f:csv_pa\else
                                  \xint_afterfi{\expandafter,\xint_gobble_iv}\fi }%
\def\XINT_seqb:f:csv_pa #1#2#3#4{\expandafter\XINT_seqb:f:csv_p\expandafter
                                 {\romannumeral0\xintadd{#4}{#1}}{#2}{#3,#1}{#4}}%
\def\XINT_seqb:f:csv_p #1#2%
{%
   \xintifCmp {#1}{#2}\XINT_seqb:f:csv_pa\XINT_seqb:f:csv_pb\XINT_seqb:f:csv_pc
   {#1}{#2}%
}%
\def\XINT_seqb:f:csv_pb #1#2#3#4{#3,#1}%
\def\XINT_seqb:f:csv_pc #1#2#3#4{#3}%
\def\XINT_seqb:f:csv_bg #1{\if #1n\expandafter\XINT_seqb:f:csv_na\else
                                  \xint_afterfi{\expandafter,\xint_gobble_iv}\fi }%
\def\XINT_seqb:f:csv_na #1#2#3#4{\expandafter\XINT_seqb:f:csv_n\expandafter
                                 {\romannumeral0\xintadd{#4}{#1}}{#2}{#3,#1}{#4}}%
\def\XINT_seqb:f:csv_n #1#2%
{%
   \xintifCmp {#1}{#2}\XINT_seqb:f:csv_nc\XINT_seqb:f:csv_nb\XINT_seqb:f:csv_na
   {#1}{#2}%
}%
\def\XINT_seqb:f:csv_nb #1#2#3#4{#3,#1}%
\def\XINT_seqb:f:csv_nc #1#2#3#4{#3}%
%    \end{macrocode}
%\subsubsection{\csh{xintiiSeqB:f:csv}}
% \lverb|Produces in f-expandable way. If the step is zero, gives empty result
% except if start and end coincide.|
%    \begin{macrocode}
\def\xintiiSeqb:f:csv #1#2%
   {\expandafter\XINT_iiseqb:f:csv \expandafter{\romannumeral-`0#2}{#1}}%
\def\XINT_iiseqb:f:csv #1#2{\expandafter\XINT_iiseqb:f:csv_a\romannumeral-`0#2#1!}%
\def\XINT_iiseqb:f:csv_a #1#2;#3;#4!{%
   \expandafter\xint_gobble_i\romannumeral-`0%
   \xintSgnFork{\XINT_Cmp {#3}{#4}}%
                 \XINT_iiseqb:f:csv_bl\XINT_seqb:f:csv_be\XINT_iiseqb:f:csv_bg
   #1{#3}{#4}{}{#2}}%
\def\XINT_iiseqb:f:csv_bl #1{\if #1p\expandafter\XINT_iiseqb:f:csv_pa\else
                                  \xint_afterfi{\expandafter,\xint_gobble_iv}\fi }%
\def\XINT_iiseqb:f:csv_pa #1#2#3#4{\expandafter\XINT_iiseqb:f:csv_p\expandafter
                               {\romannumeral0\xintiiadd{#4}{#1}}{#2}{#3,#1}{#4}}%
\def\XINT_iiseqb:f:csv_p #1#2%
{%
    \xintSgnFork{\XINT_Cmp {#1}{#2}}%
    \XINT_iiseqb:f:csv_pa\XINT_iiseqb:f:csv_pb\XINT_iiseqb:f:csv_pc {#1}{#2}%
}%
\def\XINT_iiseqb:f:csv_pb #1#2#3#4{#3,#1}%
\def\XINT_iiseqb:f:csv_pc #1#2#3#4{#3}%
\def\XINT_iiseqb:f:csv_bg #1{\if #1n\expandafter\XINT_iiseqb:f:csv_na\else
                                  \xint_afterfi{\expandafter,\xint_gobble_iv}\fi }%
\def\XINT_iiseqb:f:csv_na #1#2#3#4{\expandafter\XINT_iiseqb:f:csv_n\expandafter
                               {\romannumeral0\xintiiadd{#4}{#1}}{#2}{#3,#1}{#4}}%
\def\XINT_iiseqb:f:csv_n #1#2%
{%
    \xintSgnFork{\XINT_Cmp {#1}{#2}}%
    \XINT_seqb:f:csv_nc\XINT_seqb:f:csv_nb\XINT_iiseqb:f:csv_na {#1}{#2}%
}%
%    \end{macrocode}
%\subsubsection{\csh{XINTinFloatSeqB:f:csv}}
% \lverb|Produces in f-expandable way. If the step is zero, gives empty result
% except if start and end coincide. This is all for \xintNewExpr.|
%    \begin{macrocode}
\def\XINTinFloatSeqB:f:csv #1#2{\expandafter\XINT_flseqb:f:csv \expandafter
   {\romannumeral0\XINTinfloat [\XINTdigits]{#2}}{#1}}%
\def\XINT_flseqb:f:csv #1#2{\expandafter\XINT_flseqb:f:csv_a\romannumeral-`0#2#1!}%
\def\XINT_flseqb:f:csv_a #1#2;#3;#4!{%
   \expandafter\xint_gobble_i\romannumeral-`0%
   \xintifCmp {#3}{#4}\XINT_flseqb:f:csv_bl\XINT_seqb:f:csv_be\XINT_flseqb:f:csv_bg
   #1{#3}{#4}{}{#2}}%
\def\XINT_flseqb:f:csv_bl #1{\if #1p\expandafter\XINT_flseqb:f:csv_pa\else
                                  \xint_afterfi{\expandafter,\xint_gobble_iv}\fi }%
\def\XINT_flseqb:f:csv_pa #1#2#3#4{\expandafter\XINT_flseqb:f:csv_p\expandafter
                          {\romannumeral0\XINTinfloatadd{#4}{#1}}{#2}{#3,#1}{#4}}%
\def\XINT_flseqb:f:csv_p #1#2%
{%
    \xintifCmp {#1}{#2}%
    \XINT_flseqb:f:csv_pa\XINT_flseqb:f:csv_pb\XINT_flseqb:f:csv_pc {#1}{#2}%
}%
\def\XINT_flseqb:f:csv_pb #1#2#3#4{#3,#1}%
\def\XINT_flseqb:f:csv_pc #1#2#3#4{#3}%
\def\XINT_flseqb:f:csv_bg #1{\if #1n\expandafter\XINT_flseqb:f:csv_na\else
                                  \xint_afterfi{\expandafter,\xint_gobble_iv}\fi }%
\def\XINT_flseqb:f:csv_na #1#2#3#4{\expandafter\XINT_flseqb:f:csv_n\expandafter
                          {\romannumeral0\XINTinfloatadd{#4}{#1}}{#2}{#3,#1}{#4}}%
\def\XINT_flseqb:f:csv_n #1#2%
{%
    \xintifCmp {#1}{#2}%
    \XINT_seqb:f:csv_nc\XINT_seqb:f:csv_nb\XINT_flseqb:f:csv_na {#1}{#2}%
}%
%    \end{macrocode}
% \subsection{\csh{xintNewExpr}, \csh{xintNewIExpr}, \csh{xintNewFloatExpr},
% \csh{xintNewIIExpr}}
% \subsubsection{\csh{xintApply::csv}}
% \lverb|Don't ask me what this if for. I wrote it in June, and we are now
% late October.|
%    \begin{macrocode}
\def\xintApply::csv #1#2%
   {\expandafter\XINT_applyon::_a\expandafter {\romannumeral-`0#2}{#1}}%
\def\XINT_applyon::_a #1#2{\XINT_applyon::_b {#2}{}#1,,}%
\def\XINT_applyon::_b #1#2#3,{\expandafter\XINT_applyon::_c \romannumeral-`0#3,{#1}{#2}}%
\def\XINT_applyon::_c #1{\if #1,\expandafter\XINT_applyon::_end
                                \else\expandafter\XINT_applyon::_d\fi #1}%
\def\XINT_applyon::_d #1,#2{\expandafter\XINT_applyon::_e\romannumeral-`0#2{#1},{#2}}%
\def\XINT_applyon::_e #1,#2#3{\XINT_applyon::_b {#2}{#3, #1}}%
\def\XINT_applyon::_end #1,#2#3{\xint_secondoftwo #3}%
%    \end{macrocode}
% \subsubsection{\csh{xintApply:::csv}}
%    \begin{macrocode}
\def\xintApply:::csv #1#2#3%
   {\expandafter\XINT_applyon:::_a\expandafter{\romannumeral-`0#2}{#1}{#3}}%
\def\XINT_applyon:::_a #1#2#3{\XINT_applyon:::_b {#2}{#3}{}#1,,}%
\def\XINT_applyon:::_b #1#2#3#4,%
   {\expandafter\XINT_applyon:::_c \romannumeral-`0#4,{#1}{#2}{#3}}%
\def\XINT_applyon:::_c #1{\if #1,\expandafter\XINT_applyon:::_end
                     \else\expandafter\XINT_applyon:::_d\fi #1}%
\def\XINT_applyon:::_d #1,#2#3%
   {\expandafter\XINT_applyon:::_e\expandafter
    {\romannumeral-`0\xintApply::csv {#2{#1}}{#3}},{#2}{#3}}%
\def\XINT_applyon:::_e #1,#2#3#4{\XINT_applyon:::_b {#2}{#3}{#4, #1}}%
\def\XINT_applyon:::_end #1,#2#3#4{\xint_secondoftwo #4}%
%    \end{macrocode}
% \subsubsection{\csh{XINT_expr_RApply::csv}, \csh{XINT_expr_LApply::csv},
% \csh{XINT_expr_RLApply:::csv}}
% \lverb|The #1 in _Rapply will start with a ~. No risk of glueing to previous
% ~expandafter during the \scantokens.|
%    \begin{macrocode}
\def\XINT_expr_RApply::csv #1#2#3#4%
   {~xintApply::csv{~expandafter#1~xint_exchangetwo_keepbraces{#4}}{#3}}%
\def\XINT_expr_LApply::csv #1#2#3#4{~xintApply::csv{#1{#3}}{#4}}%
\def\XINT_expr_RLApply:::csv #1#2{~xintApply:::csv{#1}}%
%    \end{macrocode}
% \subsubsection{Mysterious stuff}
% \lverb|actually I dimly remember that the whole point is to allow maximal
% evaluation as long as macro parameters not encountered. Else it would be
% easier. \xintNewIExpr \f [2]{[12] #1+#2+3*6*1} will correctly compute the 18.
%
% 1.1a re-establishes the trick with \toks0\expandafter{\the\toks0\expandafter
% etc...} with *no* space after 0. I don't know why it was removed at some
% point before releasing 1.1, but \XINT_expr_redefinemacros is even bigger
% without the trick.|
%    \begin{macrocode}
\catcode`~ 12 % by the way, catcode is set to 3 in \XINTsetupcatcodes
\catcode`$ 12 % $
\def\XINT_xptwo_getab_b #1#2!#3%
   {\expandafter\XINT_xptwo_getab_c\romannumeral-`0#3!#1{#1#2}}%
\def\XINT_xptwo_getab_c #1#2!#3#4#5#6{#1#3{#5}{#6}{#1#2}{#4}}%
\def\xint_ddfork #1$$#2#3\krof {#2}% $$
\def\XINT_NEfork #1#2{\xint_ddfork
                          #1#2\XINT_expr_RLApply:::csv
                           #1$\XINT_expr_RApply::csv% $
                           $#2\XINT_expr_LApply::csv% $
                            $${\XINT_NEfork_nn #1#2}% $$
                      \krof }%
\def\XINT_NEfork_nn #1#2#3#4{%
        \if #1##\xint_dothis{#3}\fi
        \if  #1~\xint_dothis{#3}\fi
        \if #2##\xint_dothis{#3}\fi
        \if  #2~\xint_dothis{#3}\fi
        \xint_orthat {\csname #4NE\endcsname }%
        }%
\def\XINT_NEfork_one #1#2!#3#4#5#6{%
    \if ###1\xint_dothis {#3}\fi
    \if  ~#1\xint_dothis {#3}\fi
    \if  $#1\xint_dothis {~xintApply::csv{#3#5}}\fi %$
    \xint_orthat {\csname #4NE\endcsname #6}{#1#2}%
}%
\toks0 {}%
\xintFor #1 in {DivTrunc,iiDivTrunc,iiDivRound,Mod,iiMod,iRound,Round,iTrunc,Trunc,%
        Lt,Gt,Eq,LtorEq,GtorEq,Neq,AND,OR,XOR,iQuo,iRem,Add,Sub,Mul,Div,Pow,E,%
        iiAdd,iiSub,iiMul,iiPow,iiQuo,iiRem,iiE,SeqA::csv,iiSeqA::csv}\do
 {\toks0
  \expandafter{\the\toks0% no space! (makes shorter macro in the end)
  \expandafter\let\csname xint#1NE\expandafter\endcsname\csname xint#1\expandafter
  \endcsname\expandafter\def\csname xint#1\endcsname ####1####2{%
        \expandafter\XINT_NEfork
        \romannumeral-`0\expandafter\XINT_xptwo_getab_b
        \romannumeral-`0####2!{####1}{~xint#1}{xint#1}}%
  }%
}%
\xintFor #1 in {Num,Irr,Abs,iiAbs,Sgn,iiSgn,TFrac,Floor,iFloor,Ceil,iCeil,%
    Sqr,iiSqr,iiSqrt,iiSqrtR,iiIsZero,iiIsNotZero,iiifNotZero,iiifSgn,%
    Odd,Even,iiOdd,iiEven,Opp,iiOpp,iiifZero,Fac,iFac,Bool,Toggle}\do
{\toks0
  \expandafter{\the\toks0%
  \expandafter\let\csname xint#1NE\expandafter\endcsname\csname xint#1\expandafter
  \endcsname\expandafter\def\csname xint#1\endcsname ####1{%
      \expandafter\XINT_NEfork_one\romannumeral-`0####1!{~xint#1}{xint#1}{}{}}%
  }%
}%
\xintFor #1 in {Add,Sub,Mul,Div,Power,E,Mod,SeqA::csv}\do
{\toks0
  \expandafter{\the\toks0%
  \expandafter\let\csname XINTinFloat#1NE\expandafter\endcsname
                                \csname XINTinFloat#1\expandafter\endcsname
  \expandafter\def\csname XINTinFloat#1\endcsname ####1####2{%
        \expandafter\XINT_NEfork
        \romannumeral-`0\expandafter\XINT_xptwo_getab_b
        \romannumeral-`0####2!{####1}{~XINTinFloat#1}{XINTinFloat#1}}%
  }%
}%
\xintFor #1 in {XINTinFloatdigits,XINTinFloatFracdigits,XINTinFloatSqrtdigits}\do
{\toks0
  \expandafter{\the\toks0%
  \expandafter\let\csname #1NE\expandafter\endcsname\csname #1\expandafter
  \endcsname\expandafter\def\csname #1\endcsname ####1{\expandafter
       \XINT_NEfork_one\romannumeral-`0####1!{~#1}{#1}{}{}}%
  }%
}%
\xintFor #1 in {xintSeq::csv,xintiiSeq::csv,XINTinFloatSeq::csv}\do
 {\toks0
  \expandafter{\the\toks0% no space
  \expandafter\let\csname #1NE\expandafter\endcsname\csname #1\expandafter
  \endcsname\expandafter\def\csname #1\endcsname ####1####2{%
        \expandafter\XINT_NEfork
        \romannumeral-`0\expandafter\XINT_xptwo_getab_b
        \romannumeral-`0####2!{####1}{$noexpand$#1}{#1}}%
  }%
}%
\xintFor #1 in {xintSeqB,xintiiSeqB,XINTinFloatSeqB}\do
 {\toks0
  \expandafter{\the\toks0% no space
  \expandafter\let\csname #1::csvNE\expandafter\endcsname\csname #1::csv\expandafter
  \endcsname\expandafter\def\csname #1::csv\endcsname ####1####2{%
        \expandafter\XINT_NEfork
        \romannumeral-`0\expandafter\XINT_xptwo_getab_b
        \romannumeral-`0####2!{####1}{$noexpand$#1:f:csv}{#1::csv}}%
  }%
}%
\toks0
  \expandafter{\the\toks0
  \let\XINTinFloatNE\XINTinFloat
  \def\XINTinFloat [##1]##2{% not ultimately general, but got tired
        \expandafter\XINT_NEfork_one
        \romannumeral-`0##2!{~XINTinFloat[##1]}{XINTinFloat}{}{[##1]}}%
  \let\XINTinFloatSqrtNE\XINTinFloatSqrt
  \def\XINTinFloatSqrt [##1]##2{%
        \expandafter\XINT_NEfork_one
        \romannumeral-`0##2!{~XINTinFloatSqrt[##1]}{XINTinFloatSqrt}{}{[##1]}}%
}%
\xintFor #1 in {ANDof,ORof,XORof,iiMaxof,iiMinof,iiSum,iiPrd,
                GCDof,LCMof,Sum,Prd,Maxof,Minof}\do
{\toks0
  \expandafter{\the\toks0\expandafter\def\csname xint#1:csv\endcsname {~xint#1:csv}}%
}%
\xintFor #1 in {XINTinFloatMaxof,XINTinFloatMinof,XINTinFloatSum,XINTinFloatPrd}\do
{\toks0
  \expandafter{\the\toks0\expandafter\def\csname #1:csv\endcsname {~#1:csv}}%
}%
\expandafter\def\expandafter\XINT_expr_redefinemacros\expandafter
  {\the\toks0
   \def\XINT_flexpr_noopt {\expandafter\XINT_flexpr_withopt_b\expandafter-%
                           \romannumeral0\xintbarefloateval }%
   \def\XINT_flexpr_withopt_b ##1##2%
     {\expandafter\XINT_flexpr_wrap\csname .;##1.=\XINT_expr_unlock  ##2\endcsname }%
   \def\XINT_expr_unlock_sp ##1.;##2##3.=##4!{\if -##2\expandafter\xint_firstoftwo
         \else\expandafter\xint_secondoftwo\fi \XINTdigits{{##2##3}}{##4}}%
   \def\XINT_expr_print     ##1{\expandafter\xintSPRaw::csv\expandafter  
                             {\romannumeral-`0\XINT_expr_unlock ##1}}%
   \def\XINT_iiexpr_print   ##1{\expandafter\xintCSV::csv\expandafter  
                             {\romannumeral-`0\XINT_expr_unlock ##1}}%
   \def\XINT_boolexpr_print ##1{\expandafter\xintIsTrue::csv\expandafter
                               {\romannumeral-`0\XINT_expr_unlock ##1}}%
   \def\xintCSV::csv     {~xintCSV::csv }% spaces to separate from possible catcode 11 
   \def\xintSPRaw::csv   {~xintSPRaw::csv }% stuff after
   \def\xintPFloat::csv  {~xintPFloat::csv }%
   \def\xintIsTrue::csv  {~xintIsTrue::csv }%
   \def\xintRound::csv   {~xintRound::csv }%
%   \def\XINTinFloat::csv {~XINTinFloat::csv }% should not be needed.
   \def\xintReversed::csv {~xintReversed::csv }%
   \def\xintListSel:csv   {~xintListSel:csv }%
}%
\toks0 {}%
\def\xintNewExpr      {\xint_NewExpr\xinttheexpr      }%
\def\xintNewFloatExpr {\xint_NewExpr\xintthefloatexpr }%
\def\xintNewIExpr     {\xint_NewExpr\xinttheiexpr     }%
% \let\xintNewNumExpr\xintNewIExpr % made obsolete for 1.1 release
\def\xintNewIIExpr    {\xint_NewExpr\xinttheiiexpr    }%
\def\xintNewBoolExpr  {\xint_NewExpr\xinttheboolexpr  }%
%    \end{macrocode}
%    \begin{macrocode}
\def\XINT_newexpr_finish #1>{\noexpand\romannumeral-`0}%
\def\xint_NewExpr #1#2[#3]%
{%
 \begingroup
    \ifcase #3\relax
        \toks0 {\xdef #2}%
    \or \toks0 {\xdef #2##1}%
    \or \toks0 {\xdef #2##1##2}%
    \or \toks0 {\xdef #2##1##2##3}%
    \or \toks0 {\xdef #2##1##2##3##4}%
    \or \toks0 {\xdef #2##1##2##3##4##5}%
    \or \toks0 {\xdef #2##1##2##3##4##5##6}%
    \or \toks0 {\xdef #2##1##2##3##4##5##6##7}%
    \or \toks0 {\xdef #2##1##2##3##4##5##6##7##8}%
    \or \toks0 {\xdef #2##1##2##3##4##5##6##7##8##9}%
    \fi
    \xintexprSafeCatcodes
    \XINT_NewExpr #1%
}%
\catcode`~ 13 \catcode`@ 14 \catcode`\% 6 \catcode`# 12 \catcode`$ 11 @ $
\def\XINT_NewExpr %1%2@
{@
    \def\XINT_tmpa %%1%%2%%3%%4%%5%%6%%7%%8%%9{%2}@
    \XINT_expr_redefinemacros
    \def~{$noexpand$}@
    \catcode`: 11 \catcode`_ 11 
    \catcode`# 12 \catcode`~ 13 \escapechar 126
    \endlinechar -1 \everyeof {\noexpand }@
    \edef\XINT_tmpb
    {\scantokens\expandafter
     {\romannumeral-`0\expandafter%1\XINT_tmpa {#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}{#9}\relax}@
    }@
    \escapechar 92 \catcode`# 6 \catcode`$ 0 @ $
    \the\toks0
    {\scantokens\expandafter{\expandafter\XINT_newexpr_finish\meaning\XINT_tmpb}}@
  \endgroup
}@
\catcode`% 14
\let\xintexprRestoreCatcodes\empty
\def\xintexprSafeCatcodes
{%
    \edef\xintexprRestoreCatcodes  {%
        \catcode59=\the\catcode59   % ;
        \catcode34=\the\catcode34   % "
        \catcode63=\the\catcode63   % ?
        \catcode124=\the\catcode124 % |
        \catcode38=\the\catcode38   % &
        \catcode33=\the\catcode33   % !
        \catcode93=\the\catcode93   % ]
        \catcode91=\the\catcode91   % [
        \catcode94=\the\catcode94   % ^
        \catcode95=\the\catcode95   % _
        \catcode47=\the\catcode47   % /
        \catcode41=\the\catcode41   % )
        \catcode40=\the\catcode40   % (
        \catcode42=\the\catcode42   % *
        \catcode43=\the\catcode43   % +
        \catcode62=\the\catcode62   % >
        \catcode60=\the\catcode60   % <
        \catcode58=\the\catcode58   % :
        \catcode46=\the\catcode46   % .
        \catcode45=\the\catcode45   % -
        \catcode44=\the\catcode44   % ,
        \catcode61=\the\catcode61   % =
        \catcode32=\the\catcode32\relax % space
    }%
        \catcode59=12  % ;
        \catcode34=12  % "
        \catcode63=12  % ?
        \catcode124=12 % |
        \catcode38=4   % &
        \catcode33=12  % !
        \catcode93=12  % ]
        \catcode91=12  % [
        \catcode94=7   % ^
        \catcode95=8   % _
        \catcode47=12  % /
        \catcode41=12  % )
        \catcode40=12  % (
        \catcode42=12  % *
        \catcode43=12  % +
        \catcode62=12  % >
        \catcode60=12  % <
        \catcode58=12  % :
        \catcode46=12  % .
        \catcode45=12  % -
        \catcode44=12  % ,
        \catcode61=12  % =
        \catcode32=10  % space
}%
\let\XINT_tmpa\relax \let\XINT_tmpb\relax \let\XINT_tmpc\relax
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
% \DeleteShortVerb{\|}
% \MakePercentComment
%</xintexpr>
%<*dtx>
\StoreCodelineNo {xintexpr}

\def\mymacro #1{\mymacroaux #1}
\def\mymacroaux #1#2{\strut \csname #1nameimp\endcsname:& \dtt{ #2.}\tabularnewline }
\indent
\begin{tabular}[t]{r@{}r}
\xintApplyInline\mymacro\storedlinecounts
\end{tabular}
\def\mymacroaux #1#2{#2}%
%
\parbox[t]{10cm}{Total number of code lines:
  \dtt{\the\numexpr
  \xintListWithSep+{\xintApply\mymacro\storedlinecounts}\relax }. Each
  package starts
  with circa \dtt{50} lines dealing with catcodes, package identification
  and reloading management, also for Plain \TeX\strut. Version
  {\xintbndlversion} of {\xintbndldate}.\par}

\CharacterTable
 {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
  Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
  Digits        \0\1\2\3\4\5\6\7\8\9
  Exclamation   \!     Double quote  \"     Hash (number) \#
  Dollar        \$     Percent       \%     Ampersand     \&
  Acute accent  \'     Left paren    \(     Right paren   \)
  Asterisk      \*     Plus          \+     Comma         \,
  Minus         \-     Point         \.     Solidus       \/
  Colon         \:     Semicolon     \;     Less than     \<
  Equals        \=     Greater than  \>     Question mark \?
  Commercial at \@     Left bracket  \[     Backslash     \\
  Right bracket \]     Circumflex    \^     Underscore    \_
  Grave accent  \`     Left brace    \{     Vertical bar  \|
  Right brace   \}     Tilde         \~}
\CheckSum {25574}
\makeatletter\check@checksum\makeatother
\Finale
%% End of file xint.dtx
