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authorKarl Berry <karl@freefriends.org>2014-02-14 23:26:40 +0000
committerKarl Berry <karl@freefriends.org>2014-02-14 23:26:40 +0000
commit85276df4f1ff4a7de8716186b240501de96ff6df (patch)
tree9e0688450c17dca1cbad5fe98433abc281a0bcf6 /Master/texmf-dist/doc/generic
parentbdd296aedf9d020d8458f3ab47ef242cc69029e3 (diff)
xint (14feb14)
git-svn-id: svn://tug.org/texlive/trunk@32969 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/generic')
-rw-r--r--Master/texmf-dist/doc/generic/xint/README161
-rw-r--r--Master/texmf-dist/doc/generic/xint/xint.dtx24759
-rw-r--r--Master/texmf-dist/doc/generic/xint/xint.pdfbin998916 -> 1000698 bytes
3 files changed, 161 insertions, 24759 deletions
diff --git a/Master/texmf-dist/doc/generic/xint/README b/Master/texmf-dist/doc/generic/xint/README
new file mode 100644
index 00000000000..f34411fd82e
--- /dev/null
+++ b/Master/texmf-dist/doc/generic/xint/README
@@ -0,0 +1,161 @@
+The xint bundle
+Release 1.09kb (2014/02/13). Documentation date: 2014/02/13
+
+Copyright (C) 2013-2014 by Jean-Francois Burnol
+License: LaTeX Project Public License 1.3c or later.
+
+Contents: Abstract, Installation, License.
+
+Abstract
+========
+
+xinttools is loaded by xint (hence by all other packages of the
+bundle, too): it provides utilities of independent interest such as
+expandable and non-expandable loops.
+
+xint implements with expandable TEX macros additions, subtractions,
+multiplications, divisions and powers with arbitrarily long numbers.
+
+xintfrac extends the scope of xint to decimal numbers, to numbers in
+scientific notation and also to fractions with arbitrarily long such
+numerators and denominators separated by a forward slash.
+
+xintexpr extends xintfrac with an expandable parser
+ \xintexpr . . . \relax
+of expressions involving arithmetic operations in infix notation on
+decimal numbers, fractions, numbers in scientific notation, with
+parentheses, factorial symbol, function names, comparison operators,
+logic operators, twofold and threefold way conditionals,
+sub-expressions, macros expanding to the previous items.
+
+Further modules:
+
+xintbinhex is for conversions to and from binary and hexadecimal
+bases.
+
+xintseries provides some basic functionality for computing in an
+expandable manner partial sums of series and power series with
+fractional coefficients.
+
+xintgcd implements the Euclidean algorithm and its typesetting.
+
+xintcfrac deals with the computation of continued fractions.
+
+Most macros, and all of those doing computations, work purely by
+expansion without assignments, and may thus be used almost everywhere
+in TeX. The packages may be used with any flavor of TeX supporting the
+e-TeX extensions. LaTeX users will use \usepackage and others \input
+to load the package components.
+
+Installation
+============
+
+A. Installation using xint.tds.zip:
+-----------------------------------
+
+obtain xint.tds.zip from CTAN:
+ http://mirror.ctan.org/install/macros/generic/xint.tds.zip
+
+cd to the download repertory and issue
+ unzip xint.tds.zip -d <TEXMF>
+for example: (assuming standard access rights, so sudo needed)
+ sudo unzip xint.tds.zip -d /usr/local/texlive/texmf-local
+ sudo mktexlsr
+
+On Mac OS X, installation into user home folder:
+ unzip xint.tds.zip -d ~/Library/texmf
+
+B. Installation after file extractions:
+---------------------------------------
+
+obtain xint.dtx, xint.ins and the README from CTAN:
+ http://www.ctan.org/pkg/xint
+
+- "tex xint.ins" generates the style files
+(pre-existing files in the same repertory will be overwritten).
+
+- without xint.ins: "tex or latex or pdflatex or xelatex xint.dtx"
+will also generate the style files (and xint.ins).
+
+xint.tex is also extracted, use it for the documentation:
+
+- with latex+dvipdfmx: latex xint.tex thrice then dvipdfmx xint.dvi
+Ignore dvipdfmx warnings, but if the pdf file has problems with fonts
+(possibly from an old dvipdfmx), use then rather pdflatex or xelatex.
+
+- with pdflatex or xelatex: run it directly thrice on xint.dtx, or run
+it on xint.tex after having edited the suitable toggle therein.
+
+When compiling xint.tex, the documentation is by default produced
+with the source code included. See instructions in the file for
+changing this default.
+
+When compiling directly xint.dtx, the documentation is produced
+without the source code (latex+dvips or pdflatex or xelatex).
+
+Finishing the installation: (on first installation the destination
+repertories may need to be created)
+
+ xinttools.sty |
+ xint.sty |
+ xintfrac.sty |
+ xintexpr.sty | --> TDS:tex/generic/xint/
+ xintbinhex.sty |
+ xintgcd.sty |
+ xintseries.sty |
+ xintcfrac.sty |
+
+ xint.dtx --> TDS:source/generic/xint/
+ xint.ins --> TDS:source/generic/xint/
+ xint.tex --> TDS:source/generic/xint/
+
+ xint.pdf --> TDS:doc/generic/xint/
+ README --> TDS:doc/generic/xint/
+
+Depending on the TDS destination and the TeX installation, it may be
+necessary to refresh the TeX installation filename database (mktexlsr)
+
+C. Usage:
+---------
+
+Usage with LaTeX: \usepackage{xinttools}
+ \usepackage{xint} % (loads xinttools)
+ \usepackage{xintfrac} % (loads xint)
+ \usepackage{xintexpr} % (loads xintfrac)
+
+ \usepackage{xintbinhex} % (loads xint)
+ \usepackage{xintgcd} % (loads xint)
+ \usepackage{xintseries} % (loads xintfrac)
+ \usepackage{xintcfrac} % (loads xintfrac)
+
+Usage with TeX: \input xinttools.sty\relax
+ \input xint.sty\relax % (loads xinttools)
+ \input xintfrac.sty\relax % (loads xint)
+ \input xintexpr.sty\relax % (loads xintfrac)
+
+ \input xintbinhex.sty\relax % (loads xint)
+ \input xintgcd.sty\relax % (loads xint)
+ \input xintseries.sty\relax % (loads xintfrac)
+ \input xintcfrac.sty\relax % (loads xintfrac)
+
+License
+=======
+
+ 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 consists of the source file xint.dtx and of its derived files:
+xinttools.sty xint.sty, xintfrac.sty, xintexpr.sty, xintbinhex.sty,
+xintgcd.sty, xintseries.sty, xintcfrac.sty, as well as xint.ins, xint.tex
+and the documentation xint.pdf (or xint.dvi).
+
+The author of this work is Jean-Francois Burnol <jfbu at free dot fr>.
+This work has the LPPL maintenance status `author-maintained'.
+
diff --git a/Master/texmf-dist/doc/generic/xint/xint.dtx b/Master/texmf-dist/doc/generic/xint/xint.dtx
deleted file mode 100644
index 6f868a9a451..00000000000
--- a/Master/texmf-dist/doc/generic/xint/xint.dtx
+++ /dev/null
@@ -1,24759 +0,0 @@
-% -*- coding: iso-latin-1; time-stamp-format: "%02d-%02m-%:y %02H:%02M:%02S %Z" -*-
-% File: xint.dtx, package: 1.09ka (2014/02/05), documentation: 2014/02/05
-% License: LaTeX Project Public License 1.3c or later.
-% Copyright (C) 2013-2014 by Jean-Francois Burnol <jfbu at free dot fr>
-%<*dtx>
-\def\lasttimestamp{Time-stamp: <05-02-2014 21:53:23 CET>}
-%</dtx>
-%<*drv>
-\def\xintdate {2014/02/05}
-\def\xintversion {1.09ka}
-%</drv>
-%%----------------------------------------------------------------
-%% The xint bundle (version 1.09ka of February 5, 2014)
-%<xinttools>%% xinttools: Expandable and non-expandable utilities
-%<xint>%% xint: Expandable operations on long numbers
-%<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
-%% Copyright (C) 2013-2014 by Jean-Francois Burnol
-%%----------------------------------------------------------------
-% Installation
-% ============
-%
-% A. Installation using xint.tds.zip:
-% -----------------------------------
-%
-% obtain xint.tds.zip from CTAN:
-% http://mirror.ctan.org/install/macros/generic/xint.tds.zip
-%
-% cd to the download repertory and issue
-% unzip xint.tds.zip -d <TEXMF>
-% for example: (assuming standard access rights, so sudo needed)
-% sudo unzip xint.tds.zip -d /usr/local/texlive/texmf-local
-% sudo mktexlsr
-%
-% On Mac OS X, installation into user home folder:
-% unzip xint.tds.zip -d ~/Library/texmf
-%
-% B. Installation after file extractions:
-% ---------------------------------------
-%
-% obtain xint.dtx, xint.ins and the README from CTAN:
-% http://www.ctan.org/pkg/xint
-%
-% - "tex xint.ins" generates the style files
-% (pre-existing files in the same repertory will be overwritten).
-%
-% - without xint.ins: "tex or latex or pdflatex or xelatex xint.dtx"
-% will also generate the style files (and xint.ins).
-%
-% xint.tex is also extracted, use it for the documentation:
-%
-% - with latex+dvipdfmx: latex xint.tex thrice then dvipdfmx xint.dvi
-% Ignore dvipdfmx warnings, but if the pdf file has problems with fonts
-% (possibly from an old dvipdfmx), use then rather pdflatex or xelatex.
-%
-% - with pdflatex or xelatex: run it directly thrice on xint.dtx, or run
-% it on xint.tex after having edited the suitable toggle therein.
-%
-% When compiling xint.tex, the documentation is by default produced
-% with the source code included. See instructions in the file for
-% changing this default.
-%
-% When compiling directly xint.dtx, the documentation is produced
-% without the source code (latex+dvips or pdflatex or xelatex).
-%
-% Finishing the installation: (on first installation the destination
-% repertories may need to be created)
-%
-% xinttools.sty |
-% xint.sty |
-% xintfrac.sty |
-% xintexpr.sty | --> TDS:tex/generic/xint/
-% xintbinhex.sty |
-% xintgcd.sty |
-% xintseries.sty |
-% xintcfrac.sty |
-%
-% xint.dtx --> TDS:source/generic/xint/
-% xint.ins --> TDS:source/generic/xint/
-% xint.tex --> TDS:source/generic/xint/
-%
-% xint.pdf --> TDS:doc/generic/xint/
-% README --> TDS:doc/generic/xint/
-%
-% Depending on the TDS destination and the TeX installation, it may be
-% necessary to refresh the TeX installation filename database (mktexlsr)
-%
-% C. Usage:
-% ---------
-%
-% Usage with LaTeX: \usepackage{xinttools}
-% \usepackage{xint} % (loads xinttools)
-% \usepackage{xintfrac} % (loads xint)
-% \usepackage{xintexpr} % (loads xintfrac)
-%
-% \usepackage{xintbinhex} % (loads xint)
-% \usepackage{xintgcd} % (loads xint)
-% \usepackage{xintseries} % (loads xintfrac)
-% \usepackage{xintcfrac} % (loads xintfrac)
-%
-% Usage with TeX: \input xinttools.sty\relax
-% \input xint.sty\relax % (loads xinttools)
-% \input xintfrac.sty\relax % (loads xint)
-% \input xintexpr.sty\relax % (loads xintfrac)
-%
-% \input xintbinhex.sty\relax % (loads xint)
-% \input xintgcd.sty\relax % (loads xint)
-% \input xintseries.sty\relax % (loads xintfrac)
-% \input xintcfrac.sty\relax % (loads xintfrac)
-%
-% License
-% =======
-%
-% 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 consists of the source file xint.dtx and of its derived files:
-% xinttools.sty xint.sty, xintfrac.sty, xintexpr.sty, xintbinhex.sty,
-% xintgcd.sty, xintseries.sty, xintcfrac.sty, as well as xint.ins, xint.tex
-% and the documentation xint.pdf (or xint.dvi).
-%
-% The author of this work is Jean-Francois Burnol <jfbu at free dot fr>.
-% This work has the LPPL maintenance status `author-maintained'.
-%
-%<*dtx>
-\iffalse
-%</dtx>
-%<*drv>----------------------------------------------------------------------
-%% This is a generated file. Run latex thrice on this file xint.tex then
-%% run dvipdfmx on xint.dvi to produce the documentation xint.pdf, with
-%% source code included. (ignore the dvipdfmx warnings)
-%%
-%% Customize as desired the class options and the two toggles below.
-%%
-%% See xint.dtx for the copyright and the conditions for distribution
-%% and/or modification of this work.
-%%
-\NeedsTeXFormat{LaTeX2e}
-\ProvidesFile{xint.tex}%
-[\xintdate\space v\xintversion\space driver file for xint documentation (jfB)]%
-\PassOptionsToClass{a4paper,fontsize=11pt}{scrdoc}
-\chardef\Withdvipdfmx 1 % replace 1<space> by 0<space> for using latex/pdflatex
-\chardef\NoSourceCode 0 % replace 0<space> by 1<space> for no source code
-\input xint.dtx
-%%% Local Variables:
-%%% mode: latex
-%%% End:
-%</drv>----------------------------------------------------------------------
-%<*ins>-------------------------------------------------------------------------
-%% This is a generated file.
-%% "tex xint.ins" extracts from xint.dtx:
-%% xinttools.sty, xint.sty, xintfrac.sty, xintexpr.sty, xintbinhex.sty,
-%% xintgcd.sty, xintseries.sty and xintcfrac.sty as well as xint.tex
-%% (for typesetting the documentation).
-%%
-%% See xint.dtx for the copyright and the conditions for distribution
-%% and/or modification of this work.
-%%
-\input docstrip.tex
-\askforoverwritefalse
-\generate{\nopreamble
-\file{xint.tex}{\from{xint.dtx}{drv}}
-\usepreamble\defaultpreamble
-\file{xinttools.sty}{\from{xint.dtx}{xinttools}}
-\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{* xinttools.sty}
-\Msg{* xint.sty}
-\Msg{* xintbinhex.sty}
-\Msg{* xintgcd.sty}
-\Msg{* xintfrac.sty}
-\Msg{* xintseries.sty}
-\Msg{* xintcfrac.sty}
-\Msg{* xintexpr.sty}
-\Msg{*}
-\Msg{* To produce the documentation run latex thrice on file xint.tex}
-\Msg{* and then run dvipdfmx on file xint.dvi (ignore dvipdfmx warnings)}
-\Msg{*}
-\Msg{* Happy TeXing!}
-\Msg{*}
-\Msg{************************************************************************}
-\endbatchfile
-%</ins>-------------------------------------------------------------------------
-%<*dtx>
-\fi % end of \iffalse block
-\def\striptimestamp #1 <#2 #3 #4>{#2 at #3 #4}
-\def\getdocdate #1 <#2-#3-#4 #5>{#4/#3/#2}
-\edef\docdate{\expandafter\getdocdate\lasttimestamp}
-\edef\dtxtimestamp{\expandafter\striptimestamp\lasttimestamp}
-\chardef\noetex 0
-\expandafter\ifx\csname numexpr\endcsname\relax \chardef\noetex 1 \fi
-\ifnum\noetex=1 \chardef\extractfiles 0 % extract files, then stop
-\else
- \expandafter\ifx\csname ProvidesFile\endcsname\relax
- \chardef\extractfiles 0 % etex etc.. on xint.dtx
- \else % latex/pdflatex on xint.tex or on xint.dtx
- \expandafter\ifx\csname Withdvipdfmx\endcsname\relax
- % latex run is on etoc.dtx, we will extract all files
- \chardef\extractfiles 1 % 1 = extract all and typeset doc
- \chardef\Withdvipdfmx 0 % 0 = pdflatex or latex+dvips
- \chardef\NoSourceCode 1 %
- \NeedsTeXFormat{LaTeX2e}%
- \PassOptionsToClass{a4paper,11pt}{scrdoc}%
- \else % latex run is on etoc.tex,
- \chardef\extractfiles 2 % no extractions
- \fi
- \ProvidesFile{xint.dtx}%
- [bundle source (\xintversion, \xintdate) and documentation (\docdate)]%
- \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 xinttools.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{*^^J}%
-\Msg{* To produce the documentation with source code included run latex^^J}%
-\Msg{* thrice on file xint.tex and then dvipdfmx on xint.dvi^^J}%
-\Msg{* \space\space\space\space(ignore the dvipdfmx warnings)^^J}%
-\Msg{*^^J}%
-\Msg{* Happy TeXing!^^J}%
-\Msg{*^^J}%
-\Msg{********************************************************************^^J}%
-}%
-\begingroup
- \input docstrip.tex
- \askforoverwritefalse
- \generate{\nopreamble
- \file{xint.ins}{\from{xint.dtx}{ins}}
- \file{xint.tex}{\from{xint.dtx}{drv}}
- \usepreamble\defaultpreamble
- \file{xinttools.sty}{\from{xint.dtx}{xinttools}}
- \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
-\ifnum\extractfiles=0
-% direct tex/etex/xetex/etc on xint.dtx, files now extracted, stop
- \MessageDeFin\expandafter\end
-\fi
-% no use of docstrip to extract files if latex compilation was on etoc.tex
-\ifdefined\MessageDeFin\AtEndDocument{\MessageDeFin}\fi
-%-------------------------------------------------------------------------------
-\documentclass {scrdoc}
-\ifnum\NoSourceCode=1 \OnlyDescription\fi
-\makeatletter
-\ifnum\Withdvipdfmx=1
- \@for\@tempa:=hyperref,bookmark,graphicx,xcolor\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
-% January 4, 2014
-% took me a while to pinpoint yesterday evening the origin of the problem, if
-% only I had visited
-% http://www.komascript.de/release3.12 immediately!
-%
-% as I subscribe to c.t.tex and d.c.t.tex I thought a problem with KOMA scrartcl
-% would have been mentioned there, if as crippling as is this one, so I
-% initially thought something related to TOCs had changed in KOMA and that etoc
-% was now incompatible, and thus I started examining this, until finally
-% understanding this had nothing to do with the TOC but originated in a
-% buggy \sectionmark, revealed with pagestyle headings.
-%
-% This morning I see this is fixed in the experimental archive
-% http://www.komascript.de/~mkohm/texlive-KOMA/archive/ and appears in the
-% CHANGELOG as r1584. It is a bit hard for me to understand why such a typo with
-% big consequences is not yet fixed in the CTAN distributed version. I did waste
-% 90 minutes on that, at a time I was concentrating on xint things. Bugs are
-% unavoidable, especially typos like this originating from modifying earlier
-% code, but this tiny typo is severely annoying to users (*) and in my humble
-% opinion a CTAN update should have been done sooner. Ok, this was a
-% turn-of-year time...
-%
-% (*) compiling old documents is broken, and one sometimes does not want to
-% modify the source files.
-%
-\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}
-
-%---- GEOMETRY WILL BE CHANGED FOR SOURCE CODE SECTIONS
-\usepackage[hscale=0.66,vscale=0.75]{geometry}
-
-\usepackage{xintexpr}
-
-\usepackage{xintbinhex}
-\usepackage{xintgcd}
-\usepackage{xintseries}
-\usepackage{xintcfrac}
-
-\usepackage{amsmath} % for \cfrac in the documentation
-\usepackage{varioref}
-
-\usepackage{etoolbox}
-
-\usepackage{etoc}[2013/10/16] % I need \etocdepthtag.toc
-
-%---- USE OF ETOC FOR THE TABLES OF CONTENTS
-
-\def\gobbletodot #1.{}
-\makeatletter
-\let\savedsectionline\l@section
-\makeatother
-
-\def\sectioncouleur{{cyan}}
-
-% attention à ce 22 hard codé. 23 maintenant,... 24; et 31 non 32...
-% et ça continue de changer
-
-\etocsetstyle{section}{}
- {}
- {\ifnum\etocthenumber=23 \gdef\sectioncouleur{{joli}}\fi
- \ifnum\etocthenumber=31 \gdef\sectioncouleur{[named]{RoyalPurple}}\fi
- \savedsectionline{\numberline{\expandafter\textcolor\sectioncouleur
- {\etocnumber}}\etocname}
- {{\mdseries\etocpage}}%
- }% cf l@section en classe scrartcl
- {}%
-
-\def\MARGEPAGENO {1.5em}
-\etocsetstyle{subsection}
- {\begingroup
- \setlength{\premulticols}{0pt}
- \setlength{\multicolsep}{0pt}
- \setlength{\columnsep}{1em}
- \setlength{\columnseprule}{.4pt}
- \raggedcolumns % only added for 1.08a, I should have done it long time ago!
- \begin{multicols}{2}
- \leftskip 2.3em
- \rightskip \MARGEPAGENO plus 2em minus 1em % 18 octobre 2013
- \parfillskip -\MARGEPAGENO\relax
- }
- {}
- {\noindent
- \llap{\makebox[2.3em][l]
- {\ttfamily\bfseries\etoclink
- {.\expandafter\gobbletodot\etocthenumber}}}%
- \strut
- \etocname\nobreak\leaders\etoctoclineleaders\hfill\nobreak
- \strut\makebox[1.5em][r]{\normalfont\small\etocpage}\endgraf }
- {\end{multicols}\endgroup }%
-
-\makeatother
-
-\addtocontents{toc}{\protect\hypersetup{hidelinks}}
-% je rends le @ actif... après begin document... (donc ok pour aux)
-\addtocontents{toc}{\protect\makeatother}
-
-%--- TXFONTS: TXTT WILL BE MADE SMALLER AND WILL ALLOW HYPHENATION
-\usepackage{txfonts}
-\usepackage{pifont}
-
-% malheureusement, comme j'utilise des diacritiques dans mes
-% parties commentées, imprimées verbatim, je ne pourrai pas
-% utiliser dvipdfmx qui a un problème avec txtt
-
-\DeclareFontFamily{T1}{txtt}{}
-\DeclareFontShape{T1}{txtt}{m}{n}{ %medium
- <->s*[.96] t1xtt%
-}{}
-\DeclareFontShape{T1}{txtt}{m}{sc}{ %cap & small cap
- <->s*[.96] t1xttsc%
-}{}
-\DeclareFontShape{T1}{txtt}{m}{sl}{ %slanted
- <->s*[.96] t1xttsl%
-}{}
-\DeclareFontShape{T1}{txtt}{m}{it}{ %italic
- <->ssub * txtt/m/sl%
-}{}
-\DeclareFontShape{T1}{txtt}{m}{ui}{ %unslanted italic
- <->ssub * txtt/m/sl%
-}{}
-\DeclareFontShape{T1}{txtt}{bx}{n}{ %bold extended
- <->t1xbtt%
-}{}
-\DeclareFontShape{T1}{txtt}{bx}{sc}{ %bold extended cap & small cap
- <->t1xbttsc%
-}{}
-\DeclareFontShape{T1}{txtt}{bx}{sl}{ %bold extended slanted
- <->t1xbttsl%
-}{}
-\DeclareFontShape{T1}{txtt}{bx}{it}{ %bold extended italic
- <->ssub * txtt/bx/sl%
-}{}
-\DeclareFontShape{T1}{txtt}{bx}{ui}{ %bold extended unslanted italic
- <->ssub * txtt/bx/sl%
-}{}
-\DeclareFontShape{T1}{txtt}{b}{n}{ %bold
- <->ssub * txtt/bx/n%
-}{}
-\DeclareFontShape{T1}{txtt}{b}{sc}{ %bold cap & small cap
- <->ssub * txtt/bx/sc%
-}{}
-\DeclareFontShape{T1}{txtt}{b}{sl}{ %bold slanted
- <->ssub * txtt/bx/sl%
-}{}
-\DeclareFontShape{T1}{txtt}{b}{it}{ %bold italic
- <->ssub * txtt/bx/it%
-}{}
-\DeclareFontShape{T1}{txtt}{b}{ui}{ %bold unslanted italic
- <->ssub * txtt/bx/ui%
-}{}
-
-\def\digitstt {\bgroup\fontfamily {lmtt}\selectfont\let\next=}
-
-\usepackage{xspace}
-%\usepackage[dvipsnames]{color}
-\usepackage[dvipsnames]{xcolor}
-\usepackage{framed}
-
-\definecolor{joli}{RGB}{225,95,0}
-\definecolor{JOLI}{RGB}{225,95,0}
-\definecolor{BLUE}{RGB}{0,0,255}
-\definecolor{niceone}{RGB}{38,128,192}
-
-% for the quick sort algorithm illustration
-\definecolor{LEFT}{RGB}{216,195,88}
-\definecolor{RIGHT}{RGB}{208,231,153}
-\definecolor{INERT}{RGB}{199,200,194}
-\definecolor{PIVOT}{RGB}{109,8,57}
-
-\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}
-\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 1.5cm\rightskip.5cm minus.5cm
- \hss\vtop{\noindent #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 {IMPORTANT!}}
-% 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}}}}
-\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
-% use therein \verb or other commands changing catcodes.
-% A proposito, the \LaTeX \centerline uses \hsize and not \linewidth !
-% (which in my humble opinion is bad)
-
-% \ignorespaces ajouté le 9 juin.
-
-\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@ }
-\makeatother
-
-%---- MODIFIED \verb, and verbatim like `environments' FITS BETTER OUR USE OF IT
-% le \verb de doc.sty est très chiant car il a retiré \verbatim@font pour mettre
-% un \ttfamily hard-coded à la place. [en fin de compte j'utilise dorénavant le
-% vocable \MicroFont plutôt que \verbatim@font]
-%
-% à propos \do@noligs:
-% macro:#1->\catcode `#1\active \begingroup \lccode `\~=`#1\relax \lowercase
-% {\endgroup \def ~{\leavevmode \kern \z@ \char `#1}}
-% ne manque-t-il pas un espace après le \char `#1? En effet! ça me pose des
-% problèmes lorsque l'espace a catcode 10!! Ils ont voulu optimiser et gagner
-% un token mais du coup ça en limite l'employabilité.
-%
-\def\MicroFont {\ttfamily\hyphenchar\font45 }
-\def\MacroFont {\ttfamily\baselineskip12pt\relax}
-\makeatletter
-
-% \makestarlowast ajouté le 8 juin 2013
-
-% 18 octobre 2013, hyphénation dans les blocs verbatim
-\def\dobackslash
-{%
- \catcode92 \active
- \begingroup \lccode `\~=92\relax
- \lowercase {\endgroup \def ~{\hskip \z@\@plus.1pt\@minus.1pt \char 92 }}%
-}%
-\def\dobraces
-{%
- \catcode123 \active
- \begingroup \lccode `\~=123\relax
- \lowercase {\endgroup \def ~{\hskip \z@\@plus.1pt\@minus.1pt
- \char 123 }}%
- \catcode125 \active
- \begingroup \lccode `\~=125\relax
- \lowercase {\endgroup \def ~{\char 125 \hskip \z@\@plus.1pt\@minus.1pt }}%
-}%
-% modif de \do@noligs: \char`#1} --> \char`#1 }
-\def\do@noligs #1%
-{%
- \catcode `#1\active
- \begingroup \lccode `\~=`#1\relax
- \lowercase {\endgroup \def ~{\leavevmode \kern \z@ \char `#1 }}%
-}%
-% *** \verb utilise \MicroFont
-\def\verb
-{%
- \relax \ifmmode\hbox\else\leavevmode\null\fi
- \bgroup \MicroFont
- \let\do\do@noligs \verbatim@nolig@list
- \let\do\@makeother \dospecials \catcode32 10
- \dobackslash
- \dobraces
- \makestarlowast \@jfverb
-}%
-%
-\long\def\lverb % pour utilisation dans la partie implémentation
-% *** \lverb utilise \MacroFont (comme \verbatim)
-{%
- \relax\par\smallskip\noindent\null
- \begingroup
- \let\par\@@par\hbadness 100 \hfuzz 100pt\relax
- \hsize .85\hsize
- \MacroFont
- \bgroup
- \aftergroup\@@par \aftergroup\endgroup \aftergroup\medskip
- \let\do\do@noligs \verbatim@nolig@list
- \let\do\@makeother \dospecials
- \catcode32 10 \catcode`\% 9 \catcode`\& 14 \catcode`\$ 0
- \@jfverb
-}
-% et voilà. Comme quoi, on peut aussi faire sans \trivlist si on veut.
-% Voir aussi la re-définition de \MacroFont au moment du \StopEventually
-%
-% *** \dverb utilise \MacroFont (comme \verbatim)
-%
-% J'ai parfois besoin d'un caractère de contrôle, j'avais dans les premières
-% versions de cette doc utilisé & ou $ mais ceci est devenu très peu commode
-% lorsque j'ai commencé à insérer des tabular. Finalement j'ai fait sans, mais
-% je prends aujourd'hui " qui par miracle est compatible aux emplois de \dverb
-% dans la doc, et va me permettre par exemple d'en colorier des parties, via
-% méthode sioux pour disposer des { et } temporairement.
-%
-\long\def\dverb % pour utilisation dans le manuel de l'utilisateur
-{%
- \relax\par\smallskip
- \bgroup
- \parindent0pt
- \def\par{\@@par\leavevmode\null}%
- \let\do\do@noligs \verbatim@nolig@list
- \let\do\@makeother \dospecials
- \def\"{\begingroup\catcode123 1 \catcode 125 2 \dverbescape}%
- \catcode`\@ 14 \catcode`\" 0 \makestarlowast
- \MacroFont \obeylines \@vobeyspaces
- \@jfverb
-}
-\def\dverbescape #1;!{#1\endgroup }
-
-\def\@jfverb #1{\catcode`#1\active
- \lccode`\~`#1\lowercase{\let~\egroup}}%
-\makeatother
-
-\catcode`\_=11
-
-\def\csa_aux #1{\ttfamily\hyphenchar\font45 \char`\\%
- \scantokens{#1}\endgroup }
-\def\csb_aux #1{\hyperref[\detokenize{xint#1}]{\ttfamily
- \hyphenchar\font45 \char`\\\mbox{xint}\-%
- \scantokens{#1}}\endgroup }
-
-\DeclareRobustCommand\csa {\begingroup\catcode`\_=11
- \everyeof{\noexpand}\endlinechar -1
- \makeatother
- \makestarlowast
- \csa_aux }
-\DeclareRobustCommand\csbnolk {\begingroup\catcode`\_=11
- \everyeof{\noexpand}\endlinechar -1
- \makestarlowast
- \makeatother
- \color{blue}%
- \csa_aux }
-\DeclareRobustCommand\csbxint {\begingroup\catcode`\_=11
- \everyeof{\noexpand}\endlinechar -1
- \makestarlowast
- \makeatother
- \csb_aux }
-\catcode`\_=8
-
-\newcommand\csh[1]{\texorpdfstring{\csa{#1}}{\textbackslash #1}}
-\newcommand\csbh[1]{\texorpdfstring{\csbnolk{#1}}{\textbackslash #1}}
-
-% emploi de \xintFor à partir de 1.09c
-% There were some color leaks in 1.09i from dvipdfmx (not pdflatex) compilation,
-% due to missing braces around use of \color, I have now added them.
-\xintForpair #1#2 in
-{(xinttools,tools),(xint,xint),(xintbinhex,binhex),(xintgcd,gcd),%
- (xintfrac,frac),(xintseries,series),(xintcfrac,cfrac),(xintexpr,expr)}
-\do
-{%
- \expandafter\def\csname #1name\endcsname
- {\texorpdfstring
- {\hyperref[sec:#2]%
- {{\color{joli}\bfseries\ttfamily\hyphenchar\font45 #1}}}
- {#1}%
- \xspace }%
- \expandafter\def\csname #1nameimp\endcsname
- {\texorpdfstring
- {\hyperref[sec:#2imp]%
- {{\color[named]{RoyalPurple}%
- \bfseries\ttfamily\hyphenchar\font45 #1}}}
- {#1}%
- \xspace }%
-}%
-
-\frenchspacing
-\renewcommand\familydefault\sfdefault
-
-%---- QUICK WAY TO PRINT LONG THINGS, IN PARTICULAR, BUT NOT EXCLUSIVELY, LONG
-% NUMBERS
-\def\allowsplits #1%
-{%
- \ifx #1\relax \else #1\hskip 0pt plus 1pt\relax
- \expandafter\allowsplits\fi
-}%
-\def\printnumber #1% first ``fully'' expands its argument.
-{\expandafter\allowsplits \romannumeral-`0#1\relax }%
-
-
-%--- counts used in particular in the samples from the documentation of the
-% xintseries.sty package
-\newcount\cnta
-\newcount\cntb
-\newcount\cntc
-
-%--- printing (systematically) * in a lowered position in the various verbatim
-% blocks using txtt.
-
-\def\lowast{\raisebox{-.25\height}{*}}
-\begingroup
- \catcode`* 13
- \gdef\makestarlowast {\let*\lowast\catcode`\*\active}%
-\endgroup
-
-% 22 octobre 2013
-\newcommand\fexpan {\textit{f}-expan}
-
-% 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}}
-
-\begin{document}\thispagestyle{empty}\rmfamily
-\pdfbookmark[1]{Title page}{TOP}
-\makeatletter
-
-\begingroup\lccode`\~=`@
-\lowercase{\endgroup\def~}{\begingroup\fontfamily{lmtt}\selectfont
- \let\do\@makeother\dospecials
- \catcode`\@ \active
- \jfendshrtverb }
-\catcode`\@ \active
-\def\jfendshrtverb #1@{#1\endgroup }
-
-% 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}%
- \digitstt{\specialprintnumber{F(1250)=}%
- \specialprintnumber{\Fibonacci{1250}}}\par}}%
- }
- }%
-}
-
-% Octobre 2013: le & est devenu assez pénible puisque j'en ai besoin dans mes
-% exemples de \xintApplyInline/\xintFor avec des tabulars. Donc je décide
-% (après avoir temporairement fait des choses un peu lourdes avec \lverb) de
-% le remplacer par @ car il n'y en a quasi pas dans la partie user manual;
-% idem pour \dverb. Cependant je dois faire attention avec un @ actif par
-% exemple dans les tables de matières. Bon on va voir.
-
-{\normalfont\Large\parindent0pt \parfillskip 0pt\relax
- \leftskip 2cm plus 1fil \rightskip 2cm plus 1fil
- The \xintname bundle\par}%
-{\centering
- \textsc{Jean-François Burnol}\par
- \footnotesize \ttfamily
- jfbu (at) free (dot) fr\par
- Package version: \xintversion\ (\xintdate)%
- \let\thefootnote\empty
- \footnote{Documentation generated from the
- source file with timestamp ``\dtxtimestamp''.}\par
-}
-\setcounter{footnote}{0}
-
-\bigskip
-
-% comme \dverb ne fait pas un \par à la fin, il y a un problème avec le
-% \baselineskip si on ne le spécifie pas en plus; il faudra que je voie si
-% vraiment j'utilise \dverb sans terminer un paragraphe, il doit y avoir au plus
-% quelque cas.
-\begingroup\footnotesize\def\MacroFont {\ttfamily\baselineskip10pt\relax}
-\baselineskip 10pt
-\dverb|@
-\input xintexpr.sty
-% December 7, 2013. Expandably computing a big Fibonacci number
-% using TeX+\numexpr+\xintexpr, (c) Jean-François Burnol
-% January 17, 2014: algorithm modified to be more economical in computations.
-
-\catcode`_ 11
-
-\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
-
-% This \Fibonacci macro is designed to compute *one* Fibonacci number, not a
-% whole sequence of them. Let's reap the fruits of our work:
-
-\message{F(1250)=\Fibonacci {1250}}
-\bye |\ttfamily\% see \autoref{ssec:fibonacci} for some explanations and
-more.\par
-\endgroup
-
-\clearpage
-
-% \pagebreak[3]
-
-\pdfbookmark[1]{Abstract}{ABSTRACT}
-
-\begin{addmargin}{1cm}\footnotesize
- \begin{center} \bfseries\large Description of the packages\par\smallskip
- \end{center}\medskip
-\makeatletter
-\renewenvironment{description}
- {\list{}{\topsep\z@ \parsep\z@ \labelwidth\z@ \itemindent-\leftmargin
- \let\makelabel\descriptionlabel}}
- {\endlist}
-\makeatother
-\begin{description}
-\item[\xinttoolsname] is loaded by \xintname (hence by all other packages of the
- bundle, too): it provides utilities of independent interest such as expandable
- and non-expandable loops.
-
-\item[\xintname] implements with expandable \TeX{} macros additions,
- subtractions, multiplications, divisions and powers with arbitrarily long
- numbers.
-
-\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 an expandable parser |\xintexpr
- . . . \relax| of expressions involving arithmetic operations in infix notation
- on decimal numbers, fractions, numbers in scientific notation, with
- parentheses, factorial symbol, function names, comparison operators, logic
- operators, twofold and threefold way conditionals, sub-expressions, macros
- expanding to the previous items.
-\end{description}
-
-\noindent 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}
-
- Most macros, and all of those doing computations, work purely by expansion
- without assignments, and may thus be used almost everywhere in \TeX{}.
-
- The packages may be used with any flavor of \TeX{} supporting the \eTeX{}
- extensions. \LaTeX{} users will use |\usepackage| and others |\input| to
- load the package components.
-
-\end{addmargin}
-
-\bigskip
-
-% \clearpage
-% 18 octobre 2013, je remets la TOC ici.
-
-% je ne veux pas non plus que la main toc se liste elle-même donc je passe pour
-% elle aussi à \section*
-
-\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 [2]{%
- \phantomsection\section* {Contents}
- \etoctoccontentsline*{toctobookmark}{Contents}{1}%
-}
-
- \etocsettagdepth {description}{section}
- \etocsettagdepth {commandsA} {none}
- \etocsettagdepth {xintexpr} {none}
- \etocsettagdepth {commandsB} {none}
- \etocsettagdepth {implementation}{none}
-\tableofcontents
-\etocmulticolstyle [2]{\raggedcolumns}{}
- \etocsettagdepth {description}{none}
- \etocsettagdepth {commandsA} {section}
- \etocsettagdepth {xintexpr} {section}
- \etocsettagdepth {commandsB} {section}
- \etocsettagdepth {implementation}{section}
-\tableofcontents
-\medskip
-
-% pour la suite:
-\etocignoredepthtags
-\etocmulticolstyle [1]{%
- \phantomsection\section* {Contents}
- \etoctoccontentsline*{toctobookmark}{Contents}{2}%
-}
-
-\etocdepthtag.toc {description}
-
-% \pdfbookmark[1]{Snapshot}{SNAPSHOT}
-
-\section{Read me first}\label{sec:quickintro}
-
-This section provides recommended reading on first discovering the package;
-complete details are given later in the manual.
-
-{\etocdefaultlines\etocsettocstyle{}{}\localtableofcontents}
-
-\subsection{Presentation of the package}
-
-The components of the \xintname bundle provide macros dedicated to
-\emph{expandable} computations on numbers exceeding the \TeX{} (and \eTeX{})
-limit of \digitstt{\number"7FFFFFFF}.
-
-The \eTeX{} extensions must be enabled; this is the case in modern
-distributions by default, except if \TeX{} is invoked under the name
-|tex| in command line (|etex| should be used then, or |pdftex| in |DVI|
-output mode). All components may be used as regular \LaTeX{} packages
-or, with any other format based on \TeX{}, loaded directly via
-\string\input{} (e.g. |\input
-xint.sty\relax|).
-%
-% {\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.
-
-The \xintname bundle consists of the three principal components \xintname,
-\xintfracname (which loads \xintname), and \xintexprname (which loads
-\xintfracname), and four additional modules. The macros of the \xintname bundle
-not dealing directly with the manipulation of big numbers belong to a package
-\xinttoolsname (automatically loaded by all others), which is of independent
-interest.
-
-\subsection{User interface}
-
-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 |^| 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.
-
-In the latter case 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. In the former (macro) case 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 \csbxint{expr}|..\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 its 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 (1.)~\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.} (2.)~\emph{apply this \fexpan sion to each one of their
- arguments.} 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}.
-
-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:
-%
-\centeredline{|\def\x{17} \def\y{35} \edef\z{\xintMul {\x}{\y}}|}
-%
-As a faster alternative to |\edef| (when hundreds of digits are involved), the
-package provides |\oodef| which only expands twice its argument.
-
-The \xintexprname package has a private internal
-representation for the evaluated computation result. With
-%
-\centeredline{|\oodef\z {\xintexpr 3.141^17\relax}|}
-%
-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
-%
-\centeredline{|\xintexpr \z+1/\z\relax|}
-%
-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 example above produces a somewhat large output:
-\digitstt{\oodef\z {\xintexpr 3.141^17\relax}%
- \printnumber {\xinttheexpr \z+1/\z\relax }}
-
- \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.
- \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
-%
-\centeredline{|-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
-%
-\centeredline{\digitstt{\xintRaw{-123.45602e78/+765.987e-123}}}
-%
-The optional forward slash |/| introducing a denominator is not an operation,
-but a denomination for a fractional input. Reduction to the irreducible form
-must be asked for explicitely via the \csbxint{Irr} macro or the |reduce|
-function within |\xintexpr..\relax|. Elementary operations on fractions work
-blindly (addition does not even check for equality of the denominators and
-multiply them automatically) and do none of the simplifications which
-could be obvious to (some) human beings.
-
-
-\subsection{Space and time, floating point macros}
-
-The size of the manipulated numbers is limited by two
-factors:\footnote{there is an intrinsic limit of
- \digitstt{\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 \digitstt{32767} may be passed
- as optional argument to the used macro.} The exponents may be as big as
-\digitstt{$\pm$\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 to close to the \TeX{} bound
- \digitstt{$\pm$\number"7FFFFFFF}.}
-
-Here is such a floating point computation: \centeredline{|\xintFloatPower [48]
- {1.1547}{\xintiiPow {2}{35}}|} which thus computes
-$(1.1547)^{2^{35}}=(1.1547)^{\xintiiPow {2}{35}}$ to be approximately
-\centeredline{\digitstt{\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 \centeredline{|\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:
-%
-\centeredline{|\xintthefloatexpr 1.1547^\xintiiexpr 2^35\relax\relax|}
-%
-The |\xintiiexpr| is a cousin of |\xintexpr| 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.
-
-
-\subsection{Printing big numbers on the page}
-
-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:)
-
-%
-\begingroup\baselineskip11pt\def\MacroFont{\small\ttfamily\baselineskip11pt\relax }%
-\dverb|@
-\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.|
-\par\endgroup
-%
-An alternative (\autoref{fn:np}) is to suitably configure the thousand separator
-with the \href{http://ctan.org/pkg/numprint}{numprint} package (does not work 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.\par
-
-\subsection{Expandable implementations of mathematical algorithms}
-
-Another use of the |\xintexpr|-essions is illustrated with the algorithm on the
-title page: it shows how one may chain expandable evaluations, almost as if one
-were using the |\numexpr| facilities.\footnote{The implementation uses the
- (already once-expanded) integer only variant \csa{xintiiexpr} as \csa{romannumeral0}\csa{xintiieval..}\csa{relax}.}
-Notice that the @47@th Fibonacci number is \digitstt{\Fibonacci {47}} thus
-already too big for \TeX{} and \eTeX{}, a difficulty which our front page showed
-how to overcome (see \autoref{ssec:fibonacci} for more). The |\Fibonacci| macro
-is completely expandable hence can be used for example within |\message| to
-write to the log and terminal.
-
-It is even \fexpan dable (although not in only two steps, this could be added
-but does not matter here), thus if we are interested in knowing how many digits
-@F(1250)@ has, suffices to use |\xintLen {\Fibonacci {1250}}| (which expands to
-\digitstt{\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{xintGCD} macro is provided by the \xintgcdname package.}
-\centeredline{|\xintGCD{\Fibonacci{1859}}{\Fibonacci{1573}}=\Fibonacci{\xintGCD{1859}{1573}}|}
-\centeredline{\digitstt{\printnumber{\xintGCD{\Fibonacci{1859}}{\Fibonacci{1573}}}=\printnumber{\Fibonacci{\xintGCD{1859}{1573}}}}}
-
-The |\Fibonacci| macro expanded its |\xintGCD{1859}{1573}| argument via the
-services of |\numexpr|: this step allows only things obeying the \TeX{} bound,
-naturally! (but \digitstt{F(\xintiiPow2{31}}) would be rather big anyhow...).
-
-
-\section{Recent changes}
-
-\footnotesize
-
-\noindent Release |1.09ka| (|[2014/02/05]|):
-\begin{itemize}
-\item bug fix (\xinttoolsname): \csbxint{BreakFor} and \csbxint{BreakForAndDo}
- were buggy when used in the last iteration of an |\xintFor| loop.
-\item bug fix (\xinttoolsname): \csbxint{Seq} from |1.09k| needed a |\chardef|
- which was missing from |xinttools.sty|, it was in |xint.sty|.
-\end{itemize}
-
-\noindent Release |1.09k| (|[2014/01/21]|):
-\begin{itemize}
-\item inside |\xintexpr..\relax| (and its variants) tacit multiplication
- is implied when a number or operand is followed directly with an
- opening parenthesis,
-\item the |"| for denoting (arbitrarily big) hexadecimal numbers is recognized
- by |\xintexpr| and its variants (package \xintbinhexname is required); a
- fractional hexadecimal part introduced by a
- dot |.| is allowed.
-\item re-organization of the first sections of the user manual.
-\item bug fix: forgotten loading time |"| catcode sanity check has been added.
-\end{itemize}
-
-For a more detailed change history, see \autoref{sec:releases}. Main recent
-additions: \smallskip
-
-\noindent Release |1.09j| (|[2014/01/09]|):
-\begin{itemize}
-\item the core division routines have been re-written for some (limited)
- efficiency gain, more pronounced for small divisors. As a result the
- \hyperlink{Machin1000}{computation of one thousand digits of $\pi$}
- is close to three times faster than with earlier releases.
-\item a new macro \csbxint{XTrunc} is designed to produce thousands or even tens
- of thousands of digits of the decimal expansion of a fraction.
-\item the tacit multiplication done in \csbxint{expr}|..\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.
-\item \csbxint{expr} can now be used in an |\edef| with no |\xintthe|
- prefix.
-\end{itemize}
-
-\noindent Release |1.09i| (|[2013/12/18]|):
-\begin{itemize}
-\item \csbxint{iiexpr} is a variant of \csbxint{expr} which is optimized to deal
- only with (long) integers, |/| does a euclidean quotient.
-\item |\xintnumexpr|, |\xintthenumexpr|, |\xintNewNumExpr| are renamed,
- respectively, \csbxint{iexpr}, \csbxint{theiexpr}, \csbxint{NewIExpr}. The
- earlier denominations are kept but to be removed at some point.
-\item 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.
-\item \xinttoolsname defines \hyperref[odef]{\ttfamily\char92odef},
- \hyperref[oodef]{\ttfamily\char92oodef},
- \hyperref[fdef]{\ttfamily\char92fdef}. These tools are provided for the case
- one uses the package macros in a non-expandable context, particularly
- \hyperref[oodef]{\ttfamily\char92oodef} which expands twice the macro
- replacement text and is thus a faster alternative to |\edef|. This can be
- significant when repeatedly making |\def|-initions expanding to hundreds of
- digits.
-\end{itemize}
-
-
-\noindent Release |1.09h| (|[2013/11/28]|):
-\begin{itemize}
-\item all macros of \xinttoolsname for which it makes sense are now
- declared |\long|.
-\end{itemize}
-
-\noindent Release |1.09g| (|[2013/11/22]|):
-\begin{itemize}
-\item package \xinttoolsname is detached from \xintname, to make tools such as
- \csbxint{For}, \csbxint{ApplyUnbraced}, and \csbxint{iloop} available without
- the \xintname overhead.
-\item new expandable nestable loops \csbxint{loop} and \csbxint{iloop}.
-\end{itemize}
-
-\noindent Release |1.09f| (|[2013/11/04]|):
-\begin{itemize}
-\item new \csbxint{ZapFirstSpaces}, \csbxint{ZapLastSpaces},
- \csbxint{ZapSpaces}, \csbxint{ZapSpacesB}, for expandably stripping away
- leading and/or ending spaces.
-\item \csbxint{CSVtoList} by default uses \csbxint{ZapSpacesB} to strip away
- spaces around commas (or at the start and end of the comma separated list).
-\item also the \csbxint{For} loop will strip out all spaces around commas and at
- the start and the end of its list argument; and similarly for
- \csbxint{Forpair}, \csbxint{Forthree}, \csbxint{Forfour}.
-\item \csbxint{For} \emph{et al.} accept all macro parameters
- from |#1| to |#9|.
-\end{itemize}
-
-
-\noindent Release |1.09e| (|[2013/10/29]|):
-\begin{itemize}
-\item new \csbxint{integers}, \csbxint{dimensions}, \csbxint{rationals} for
- infinite \csbxint{For} loops, interrupted with \csbxint{BreakFor} and
- \csbxint{BreakForAndDo}.
-\item new \csbxint{ifForFirst}, \csbxint{ifForLast} for the \csa{xintFor} and
- \csa{xintFor*} loops,
-\item the \csa{xintFor} and \csa{xintFor*} loops are now |\long|, the
- replacement text and the items may contain explicit |\par|'s.
-\item new conditionals \csbxint{ifCmp}, \csbxint{ifInt}, \csbxint{ifOdd}.
-\item the documentation has been enriched with various additional examples,
- such as the \hyperref[ssec:quicksort]{the quick sort algorithm illustrated} or
- the computation of prime numbers (\autoref{ssec:primesI},
- \autoref{ssec:primesII}, \autoref{ssec:primesIII}).
-\end{itemize}
-
-\noindent Release |1.09c| (|[2013/10/09]|):
-\begin{itemize}
-\item added \hyperlink{item:bool}{|bool|} and \hyperlink{item:bool}{|togl|} to
- the
- \csbxint{expr} syntax; also added \csbxint{boolexpr} and \csbxint{ifboolexpr}.
-\item \csbxint{For} 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,
-\item \csbxint{ApplyInline} has been enhanced in order to be usable for
- generating rows (partially or completely) in an alignment,
-\item new command \csbxint{Seq} to generate (expandably) arithmetic sequences of
- (short) integers,
-\end{itemize}
-
-
-\noindent Release |1.09a| (|[2013/09/24]|):
-\begin{itemize}
-\item \csbxint{expr}|..\relax| and
- \csbxint{floatexpr}|..\relax| admit functions in their
- syntax, with comma separated values as arguments, among them \texttt{reduce,
- sqr, sqrt, abs, sgn, floor, ceil, quo, rem, round, trunc, float, gcd, lcm,
- max, min, sum, prd, add, mul, not, all, any, xor}.
-\item comparison (|<|, |>|, |=|) and logical (\verb$|$, |&|) operators.
-\item \csbxint{NewExpr} now works with the standard macro parameter character
- |#|.
-\item both regular |\xintexpr|-essions and commands defined by |\xintNewExpr|
- will work with comma separated lists of expressions,
-\item new commands \csbxint{Floor}, \csbxint{Ceil}, \csbxint{Maxof},
- \csbxint{Minof} (package \xintfracname), \csbxint{GCDof}, \csbxint{LCM},
- \csbxint{LCMof} (package \xintgcdname), \csbxint{ifLt}, \csbxint{ifGt},
- \csbxint{ifSgn}, \csbxint{ANDof}, ...
-\item The arithmetic macros from package \xintname now filter their operands via
- \csbxint{Num} 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 but with \xintfracname loaded.
-\end{itemize}
-
-See \autoref{sec:releases} for more.
-
-\normalsize
-
-
-
-\section{Some examples}
-
-The main initial goal is to allow computations with integers and fractions of
-arbitrary sizes.
-
-Here are some examples. 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.
-
-{\color{magenta}@123456^99@: }\\
-{\color[named]{Purple}\csa{xintiPow}|{123456}{99}|}: \digitstt{\printnumber{\xintiPow {123456}{99}}}
-
-{\color{magenta}1234/56789 with 1500 digits after the decimal point: }\\
-{\color[named]{Purple}\csa{xintTrunc}|{1500}{1234/56789}\dots|}:
-\digitstt{\printnumber {\xintTrunc {1500}{1234/56789}}\dots }
-
-{\color{magenta}@0.99^{-100}@ with 200 digits after the decimal point:}\\
-{\color[named]{Purple}\csa{xinttheexpr trunc}|(.99^-100,200)\relax\dots|}:
-\digitstt{\printnumber{\xinttheexpr trunc(.99^-100,200)\relax}\dots }
-
-
-{\color{magenta}%
- Computation of a Bezout identity with |7^200-3^200| and |2^200-1|:}\par
-{\color[named]{Purple}
-\dverb|@
-\xintAssign \xintBezout {\xinttheiexpr 7^200-3^200\relax}
- {\xinttheiexpr 2^200-1\relax}\to\A\B\U\V\D|%
-\centeredline {|\U$\times$(7^200-3^200)+\xintiOpp\V$\times$(2^200-1)=\D|}}%
-%
-\xintAssign \xintBezout {\xinttheiexpr 7^200-3^200\relax}
- {\xinttheiexpr 2^200-1\relax}\to\A\B\U\V\D
-\digitstt
-{\printnumber\U$\times$(@7^200-3^200@)+%
- \printnumber{\xintiOpp\V}$\times$(@2^200-1@)=\printnumber\D}
-
-\textcolor{magenta}{The Euclide algorithm applied to
-\np{22206980239027589097} and \np{8169486210102119256}:}%
-\footnote{this example is computed tremendously faster than the
- other ones, but we had to limit the space taken by the output.}\par
-{\color[named]{Purple}
-\noindent|\xintTypesetEuclideAlgorithm
-{22206980239027589097}{8169486210102119256}|\endgraf}
-\xintTypesetEuclideAlgorithm
-{22206980239027589097}{8169486210102119256} \smallskip
-
-{\color{magenta}$\sum_{n=1}^{500} (4n^2 - 9)^{-2}$ with each term rounded to
- twelve digits, and the sum to nine digits:} {\color[named]{Purple}%
- |\def\coeff #1%|\\
- | {\xintiRound {12}{1/\xintiSqr{\the\numexpr 4*#1*#1-9\relax }[0]}}|\\
- |\xintRound {9}{\xintiSeries {1}{500}{\coeff}[-12]}|:} \def\coeff #1%
-{\xintiRound {12}{1/\xintiSqr{\the\numexpr 4*#1*#1-9\relax }[0]}}
-\digitstt{\xintRound {9}{\xintiSeries {1}{500}{\coeff}[-12]}}\endgraf
-
-The complete series, extended to
-infinity, has value
-$\frac{\pi^2}{144}-\frac1{162}={}$%
-\digitstt{\np{0.06236607994583659534684445}\dots}\,%
-\footnote{\label{fn:np}This number is typeset using the
- \href{http://www.ctan.org/pkg/numprint}{numprint} package, with
- \texttt{\detokenize{\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:
-{\color[named]{Purple}\dverb|@
-\def\coeff #1%
-{\xintiRound {22}{1/\xintiSqr{\xintiMul{\the\numexpr 2*#1-3\relax}
- {\the\numexpr 2*#1+3\relax}}[0]}}
-|%
-}%
-
-
-{\color{magenta}Computation of $2^{\np{999999999}}$ with |24| significant
- figures:}\\
-|\numprint{|{\color[named]{Purple}|\xintFloatPow [24] {2}{999999999}|}|}| expands to:
-\centeredline{\digitstt{\np{\xintFloatPow [24] {2}{999999999}}}} where the
-|\numprint| macro from the \hyperref[fn:np]{eponym package} was used.
-
-\edef\x{\xintQuo{\xintPow {2}{1000}}{\xintFac{100}}}
-\edef\y{\xintLen{\x}}
-
-As an example of chaining package macros, let us consider the following
-code snippet within a file with filename |myfile.tex|:
-\dverb|@
-\newwrite\outstream
-\immediate\openout\outstream \jobname-out\relax
-\immediate\write\outstream {\xintQuo{\xintPow{2}{1000}}{\xintFac{100}}}
-% \immediate\closeout\outstream
-|%
-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{\xintQuo{\xintPow{2}{1000}}{\xintFac{100}}}| which expands (in two
-steps) and tells us that @[2^{1000}/100!]@ has {\y} digits. This is not so many,
-let us print them here: \digitstt{\printnumber\x}.
-
-For the sake of typesetting this documentation and not have big numbers
-extend into the margin and go beyond the page physical limits, I use
-these commands (not provided by the package):
-\dverb|@
-\def\allowsplits #1{\ifx #1\relax \else #1\hskip 0pt plus 1pt \relax
- \expandafter\allowsplits\fi}%
-\def\printnumber #1% first ``fully'' expands its argument.
-{\expandafter\allowsplits \romannumeral-`0#1\relax }|
-
-The |\printnumber| macro is not part of the package and would need additional
-thinking for more general use.\footnote{as explained in \hyperref[fn:np]{a
- previous footnote}, the |numprint| package may also be used, in text mode
- only (as the thousand separator seemingly ends up typeset in a |\string\hbox|
- when in math mode).} It may be used like this:
-%
-\centeredline{|\printnumber {\xintQuo{\xintPow {2}{1000}}{\xintFac{100}}}|}
-or as |\printnumber\mynumber| or |\printnumber{\mynumber}| if
-|\mynumber| was previously defined via a |\newcommand|, or a |\def|:
-%
-\centeredline{%
- |\def\mynumber {\xintQuo {\xintPow {2}{1000}}{\xintFac{100}}}|}%
-
-
-Just to show off (again), let's print 300 digits (after the decimal point) of
-the decimal expansion of @0.7^{-25}@:\footnote{the |\string\np| typesetting
- macro is from the |numprint| package.}
-\centeredline{|\np {\xinttheexpr trunc(.7^-25,300)\relax}\dots|}
- \digitstt{\np {\xinttheexpr trunc(.7^-25,300)\relax}\dots }
-
-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}. The fraction |.7^-25| is first evaluated \emph{exactly}; for
-some more complex inputs, such as |.7123045678952^-243|, the exact evaluation
-before truncation would be expensive, and (assuming one needs twenty digits) one
-would rather use floating mode: \centeredline{|\xintDigits:=20;
- \np{\xintthefloatexpr .7123045678952^-243\relax}|}%
-\xintDigits:=20;%
-\centeredline{|.7123045678952^-243|${}\approx{}$%
- \digitstt{\np{\xintthefloatexpr .7123045678952^-243\relax }}} 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
-
-\xintDigits:=16;
-
-
-\section {Further illustrative examples within this document}
-\label{sec:awesome}
-
-
-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}).
-
-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}).
-
-The title page fun with Fibonacci numbers is continued in
-\autoref{ssec:fibonacci} with \csbxint{For*} joining the game.
-
-The computations of \hyperref[ssec:Machin]{ $\pi$ and $\log 2$}
-(\autoref{ssec:Machin}) using \xintname and the computation of the
-\hyperlink{e-convergents}{convergents of $e$} with the further help of
-the \xintcfracname package are among further examples.
-There is also an example of an \hyperref[xintXTrunc]{interactive
- session}, where results are output to the log or to a file.
-
-Almost all of the computational results interspersed through the
-documentation are not hard-coded in the source of the document but just written
-there using the package macros, and were selected to not impact too much the
-compilation time.
-
-
-\section{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': this means with less than
- |2^31-1|\digitstt{=\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 \digitstt{\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=\string\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
- $\pm$\digitstt{9999}.}
-
-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 do 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.
-
-
-\section{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}
-
-
-
-\section{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 \digitstt{\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*}\ntype{{\lowast f}} 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 in the margin. Some other macros
-such as \csbxint{Sum}\ntype{f{$\to$}{\lowast\Ff}} 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 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
- \centeredline{|\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
-\centeredline{|\xinttheiexpr\ifnum3>2 143\else 33\fi 0^2\relax|$\to$\digitstt{\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 \centeredline{%
- |\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
- \digitstt{\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\centeredline {|\def\AplusBC
- #1#2#3{|{\color{blue}|\romannumeral-`0\xintAdd |}|{#1}{\xintMul {#2}{#3}}}|}or use the \emph{lowercase} form of
- \csa{xintAdd}: \smallskip\centeredline {|\def\AplusBC
- #1#2#3{|{\color{blue}|\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: \centeredline{|\xintNewExpr\AplusBC[3]{#1+#2*#3}|}
-creates the |\AplusBC| macro doing the above and expanding in two expansion
-steps.
-
-
-\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 disposals the general tools of
-the \TeX{} language: |\def| or (in \LaTeX) |\newcommand|, and |\edef|.
-
-The \xinttoolsname package provides |\oodef| which expands twice the replacement
-text, 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:\centeredline{|\xintNum
- {+-+-+----++-++----00000000009876543210}|\digitstt{=\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.} \centeredline{|\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}:
- \centeredline{|\xintAdd{10.1e1}{101.010e3}|%
- \digitstt{=\xintAdd{10.1e1}{101.010e3}}}
- \centeredline{|\xintFloatAdd{10.1e1}{101.010e3}|%
- \digitstt{=\xintFloatAdd{10.1e1}{101.010e3}}}
- \centeredline{|\xintPow {2}{100}|%
- \digitstt{=\xintPow {2}{100}}}
- \centeredline{|\xintFloat{\xintPow {2}{100}}|%
- \digitstt{=\xintFloat{\xintPow {2}{100}}}}
- \centeredline{|\xintFloatPow {2}{100}|%
- \digitstt{=\xintFloatPow {2}{100}}}
-%
-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.
-%
-\centeredline{|\xintPRaw{\xintAdd{10.1e1}{101.010e3}}|%
- \digitstt{=\xintPRaw{\xintAdd{10.1e1}{101.010e3}}}}
-\centeredline{|\xintRaw{1.234e5/6.789e3}|%
- \digitstt{=\xintRaw{1.234e5/6.789e3}}}%
-\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 its devoted section
- (\autoref{sec:exprsummaryII}).
-\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 accept
-a fraction reducing to an integer. For example |\xintQuo {100/2}{12/3}| works,
-because its arguments are, after simplification, integers.
-%
-% In this
-% documentation, I often say ``numbers or fractions'', although at times the
-% vocable ``numbers'' by itself may also include ``fractions''; and ``decimal
-% numbers'' are counted among ``fractions''.
-
-With \xintfracname loaded, a number may be empty or start directly with a
-decimal point: \centeredline{|\xintRaw{}=\xintRaw{.}|\digitstt{=\xintRaw{}}}
-\centeredline{|\xintPow{-.3/.7}{11}|\digitstt{=\xintPow{-.3/+.7}{11}}}
-\centeredline{|\xinttheexpr (-.3/.7)^11\relax|%
- \digitstt{=\xinttheexpr (-.3/.7)^11\relax}} 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 |\string\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. \centeredline{|\xintRaw
- {+--+1253.2782e++--3/---0087.123e---5}|\digitstt{=\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}, \csbxint{Sum}, \csbxint{Prd} are modified to allow
-fractions on input,\footnote{the power function does not accept a fractional
- exponent. Or rather, does not expect, and errors will result if one is
- provided.}\,\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, if these fractions
- reduce to integers, and then the output format is the original \xintname's
- 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.} and 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|. The fraction |f| may be, and
-generally is, reducible, and |A| and |B| may well end up with zeroes (\emph{i.e.}
-|n| does not contain all powers of 10). Conversely, this format is accepted on
-input (and is parsed more quickly than fractions containing decimal points; the
-input may be a number without denominator).\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\string^\string{31\string}-9|.}
-
-Thus loading \xintfracname not only relaxes the format of the inputs; it
-also modifies the format of the outputs: except when a fraction is
-filtered on output by \csbxint{Irr} or \csbxint{RawWithZeros}, or
-\csbxint{PRaw}, or by the truncation or rounding macros, or is given as
-argument in math mode to \csbxint{Frac}, the output format is normally
-of the \fbox{|A/B[n]|} form (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.
-
-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).
-
-The macro \csbxint{Raw} prints the fraction
-directly from its internal representation in |A/B[n]| form. The macro
-\csbxint{PRaw} does the same but without printing the |[n]| if |n=0| and without
-printing |/1| if |B=1|.
-
-% To convert the trailing |[n]| into explicit zeroes either at the
-% numerator or the denominator, use \csbxint{RawWithZeros}. In both cases
-% the |B| is printed even if it has value |1|. Conversely (sort of), the
-% macro \csbxint{REZ} puts all powers of ten into the |[n]| (REZ stands
-% for remove zeroes). Here also, the |B| is printed even if it has value
-% |1|.
-
-The macro \csbxint{Irr} reduces the fraction to its irreducible form
-|C/D| (without a trailing |[0]|), and it prints the |D| even if |D=1|.
-
-The macro \csbxint{Num} from package \xintname is extended: it now does
-like \csbxint{Irr}, raises an error if the fraction did not reduce to an
-integer, and outputs the numerator. This macro should be used when one
-knows that necessarily the result of a computation is an integer, and
-one wants to get rid of its denominator |/1| which would be left by
-\csa{xintIrr} (or one can use \csbxint{PRaw} on top of \csbxint{Irr}).
-
-
-% The macro \csbxint{Trunc}|{N}{f}| prints\footnote{`prints' does not at all mean
-% that this macro is designed for typesetting; I am just using the verb here in
-% analogy to the effect of the functioning of a computing software in console
-% mode. The package does not provide any `printing' facility, besides its
-% rudimentary \csbxint{Frac} and \csbxint{FwOver} math-mode only macros. To deal
-% with really long numbers, some macros are necessary as \TeX{} by default will
-% print a long number on a single line extending beyond the page limits. The
-% \csa{printnumber} command used in this documentation is just one way to
-% address this problem, some other method should be used if it is important that
-% digits occupy the same width always.} the decimal expansion of |f| with |N|
-% digits after the decimal point.\footnote{the current release does not provide a
-% macro to get the period of the decimal expansion.} Currently, it does not
-% verify that |N| is non-negative and strange things could happen with a negative
-% |N|. A negative |f| is no problem, needless to say. When the original
-% fraction is negative and its truncation has only zeroes, it is printed as
-% |-0.0...0|, with |N| zeroes following the decimal point:
-% \centeredline{|\xintTrunc {5}{\xintPow {-13}{-9}}|\digitstt{=\xintTrunc
-% {5}{\xintPow {-13}{-9}}}}%
-% \centeredline{|\xintTrunc {20}{\xintPow {-13}{-9}}|\digitstt{=\xintTrunc
-% {20}{\xintPow {-13}{-9}}}} The output always contains a decimal point (even
-% for |N=0|) followed by |N| digits, except when the original fraction was zero.
-% In that case the output is |0|, with no decimal point. \centeredline{|\xintTrunc
-% {10}{\xintSum {{1/2}{1/3}{1/5}{-31/30}}}|%
-% \digitstt{=\xintTrunc {10}{\xintSum {{1/2}{1/3}{1/5}{-31/30}}}}}
-
-% \edef\z {\xintPow {1.01}{100}}
-
-% The macro \csbxint{iTrunc}|{N}{f}| is like \csa{xintTrunc}|{N}{f}|
-% followed by multiplication by |10^N|. Thus, it outputs an integer
-% in a format acceptable by the integer-only macros.
-% To get the integer part of the decimal expansion of |f|, use
-% |\xintiTrunc{0}{f}|: \centeredline{|\xintiTrunc {0}{\xintPow
-% {1.01}{100}}|\digitstt{=\xintiTrunc {0}\z}}%
-% \centeredline{|\xintiTrunc {0}{\xintPow{0.123}{-10}}|\digitstt{=\xintiTrunc
-% {0}{\xintPow{0.123}{-10}}}}
-
-See also the documentations of \csbxint{Trunc}, \csbxint{iTrunc},
-\csbxint{XTrunc}, \csbxint{Round}, \csbxint{iRound} and
-\csbxint{Float}.
-
-The \csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul}, \csbxint{iPow}, and
-some others accept fractions on input under
-the condition that they are (big) integers in disguise and then output a
-(possibly big) integer, without fraction slash nor trailing |[n]|.
-
-The \csbxint{iiAdd}, \csbxint{iiSub}, \csbxint{iiMul}, \csbxint{iiPow}, and
-some others with `\textcolor{blue}{ii}' in their names accept on input
-only integers in strict format (skipping the overhead of the
-\csbxint{Num} parsing) and output naturally a
-(possibly big) integer, without fraction slash nor trailing |[n]|.
-
-
-\subsection{Multiple outputs}\label{sec:multout}
-
-Some macros have an output consisting of more than one number or
-fraction, each one is then returned within braces. Examples of
-multiple-output macros are \csbxint{Division} 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 outputs is when using commas inside
-\csbxint{expr}|..\relax|:
-\centeredline{|\xinttheiexpr 10!,2^20,lcm(1000,725)\relax|%
- $\to$\digitstt{\xinttheiexpr 10!,2^20,lcm(1000,725)\relax}}
-
-
-\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 ensues 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).
-\centeredline{|\cnta 10 \cntb 35 \xintRaw
- {\cnta+\cntb{/}17/1+\cnta*\cntb}|\digitstt{->\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}
-\dverb|@
-\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
-\centeredline{\digitstt{\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
-\digitstt{72.27/10/2.54*65536=\xinttheexpr trunc(72.27/10/2.54*65536,3)\relax
- ...} and \TeX{} uses internally \digitstt{\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@:
-\centeredline{|\xintListWithSep{, }{\xintFtoCCv{70124/65536}}|}
-%
-\xintFor* #1 in {\xintFtoCCv{70124/65536}}\do {@#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):
-\centeredline{``@1157@\,|dd|''\digitstt{=\xinttheexpr trunc(1157\dimexpr
- 1dd\relax/\dimexpr 1pt\relax,12)\relax}\dots|pt|}
-\centeredline{``@1357@\,|dd|''\digitstt{=\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
-%
-\centeredline{|\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:
-\centeredline{@1157@\,|dd|=|\number\dimexpr 1157dd\relax/65536|%
- \digitstt{=\xintTrunc{12}{\number\dimexpr 1157dd\relax/65536}}\dots|pt|}
-\centeredline{@1357@\,|dd|=|\number\dimexpr 1357dd\relax/65536|%
- \digitstt{=\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: \centeredline{|\xinttheexpr
- trunc(1238/1157*25.4/72.27,12)\relax|%
- \digitstt{=\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 the \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
-%
-\centeredline
-{|\xinttheexpr trunc(\dimexpr 1dd\relax/65536/72.27*25.4,12)\relax|}
-\centeredline{%
-\digitstt{=\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
-\centeredline{|\xinttheexpr reduce(.376065/(25.4/72.27))\relax|%
- \digitstt{=\xinttheexpr reduce(.376065/(25.4/72.27))\relax}\,|pt|}
-and the centered convergents of this fraction are \xintFor* #1 in
-{\xintFtoCCv{543564351/508000000}}\do {@#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|%
-% \digitstt{=\number\numexpr\dimexpr1in\relax-72*\dimexpr1bp\relax\relax}\,|sp|}
-% \centeredline{|\number\dimexpr1in-72bp\relax|%
-% \digitstt{=\number\dimexpr1in-72bp\relax}\,|sp|}
-% on the other hand
-% \centeredline{|\xinttheexpr reduce(\dimexpr1in\relax-72.27*\dimexpr1pt\relax)\relax|}
-% \centeredline
-% \digitstt{=\xinttheexpr reduce(\dimexpr1in\relax-72.27*\dimexpr1pt\relax)\relax}\,|sp|=@-0.72@\,|sp|}
-% \centeredline
-% {\digitstt{=\number\dimexpr1in-72.27pt\relax}\,|sp|=@-0.72@\,|sp|}
-
-
-
-\section{\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.
-With |\def\A{1}|:
-\dverb|@
-\ifcase \xintSgn\A 0\or OK\else ERROR\fi ---> gives ERROR
-\ifcase \xintSgn\A\space 0\or OK\else ERROR\fi ---> gives OK
-\ifcase \xintSgn{\A} 0\or OK\else ERROR\fi ---> gives OK|
-% \def\A{1}
-% \ifcase \xintSgn\A 0\or OK\else ERROR\fi\
-% \ifcase \xintSgn{\A} 0\or OK\else ERROR\fi
-
-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:
-\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.
-
-
-
-\section{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 \csa{xintDivision} macro which computes both quotient and remainder at
-the same time:
-\centeredline{\csbxint{Assign}\csa{xintDivision}|{100}{3}|\csbnolk{to}|\A\B|}
-\centeredline{\csbxint{Assign}\csa{xintDivision}%
- |{\xintiPow {2}{1000}}{\xintFac{100}}|\csbnolk{to}|\A\B|} gives
-\xintAssign\xintDivision{\xintiPow {2}{1000}}{\xintFac{100}}\to\A\B
-|\meaning\A|\digitstt{: \expandafter\allowsplits\meaning\A\relax} and
-|\meaning\B|\digitstt{: \expandafter\allowsplits\meaning\B\relax}.
-
-%
-Another example (which uses \csbxint{Bezout} from the \xintgcdname package):
-\centeredline{\csbxint{Assign}\csa{xintBezout}|{357}{323}|%
-\csbnolk{to}|\A\B\U\V\D|} is equivalent to setting |\A| to \digitstt{\tmpA},
-|\B| to \digitstt{\tmpB}, |\U| to \digitstt{\tmpU}, |\V| to \digitstt{\tmpV},
-and |\D| to \digitstt{\tmpD}. And indeed
-\digitstt{(\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
-
-\centeredline{\csbxint{Assign}\csa{xintBezout}|{3570902836026}{200467139463}|%
- \csbnolk{to}|\A\B\U\V\D|}
-\noindent
-gives then |\U|\digitstt{:
- \expandafter\allowsplits\meaning\tmpU\relax},
- |\V|\digitstt{:
- \expandafter\allowsplits\meaning\tmpV\relax} and |\D|\digitstt{=\tmpD}.
-
-%
-In situations when one does not know in advance the number of items, one has
-\csbxint{AssignArray} or its synonym \csbxint{DigitsOf}:
-\centeredline{\csbxint{DigitsOf}\csa{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:
-%
-\dverb+@
-\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
-+
-
-\edef\z{\xintiPow {2}{100}}
-
-\begingroup
-\xintDigitsOf\z\to\DIGITS
-\cnta = 1
-\cntb = 0
-\loop
-\advance \cntb \xintiSqr{\DIGITS{\cnta}}
-\ifnum \cnta < \DIGITS{0}
-\advance\cnta 1
-\repeat
-
-@2^{100}@ (=\z) 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
-
-% We used a group in order to release the memory taken by the
-% \csa{DIGITS} array: indeed internally, besides \csa{DIGITS} itself,
-% additional macros are defined which are \csa{DIGITS0}, \csa{DIGITS00},
-% \csa{DIGITS1}, \csa{DIGITS2}, ..., \csa{DIGITSN}, where |N| is the size of
-% the array (which is the value returned by |\DIGITS{0}|; the digits
-% are parts of the names not arguments).
-
-% The command \csbxint{RelaxArray}\csa{DIGITS} sets all these macros to
-% \csa{relax}, but it was simpler to put everything withing a group.
-
-Warning: \csbxint{Assign}, \csbxint{AssignArray} and \csbxint{DigitsOf}
-\emph{do not do any check} on whether the macros they define are already
-defined.
-
-% In the example above, we deliberately broke all rules of complete expandability,
-% but had we wanted to compute the sum of the digits, not the sum of the squares,
-% we could just have written: \csbxint{iiSum}|{\xintiPow{2}{100}}|\digitstt{=%
-% \xintiiSum\z}. Indeed, \csa{xintiiSum} is usually used on braced items as in
-% \centeredline{%
-% \csbxint{iiSum}|{{123}{-345}{\xintFac{7}}{\xintiOpp{\xintRem{3347}{591}}}}|%
-% \digitstt{=%
-% \xintiiSum{{123}{-345}{\xintFac{7}}{\xintiOpp{\xintRem{3347}{591}}}}}} but
-% in the previous example each digit of @2^{100}@ was treated as one item due to
-% the rules of \TeX{} for parsing macro arguments.
-
-% Note: |{-\xintRem{3347}{591}}| would not be a valid input, because
-% the expansion will apply only to the minus sign and leave
-% unaffected the |\xintRem|. So we used \csa{xint}\-|iOpp| which replaces
-% a number with its opposite.
-
-
-% As a last example with \csa{xintAssignArray} here is one line
-% extracted from the source code of the \xintgcdname macro
-% \csbxint{TypesetEuclideAlgorithm}:
-% \centeredline{|\xintAssignArray\xintEuclideAlgorithm
-% {#1}{#2}\to\U|}
-% This is done inside a group. After this command |\U{1}| contains
-% the number |N| of steps of the algorithm (not to be confused with
-% |\U{0}=2N+4| which is the number of elements in the |\U| array),
-% and the GCD is to be found in |\U{3}|, a convenient location
-% between |\U{2}| and |\U{4}| which are (absolute values of the
-% expansion of) the
-% initial inputs. Then follow |N| quotients and remainders
-% from the first to the last step of the algorithm. The
-% \csa{xintTypesetEuclideAlgorithm} macro organizes this data
-% for typesetting: this is just an example of one way to do it.
-
-\section{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.}
-
-\edef\z{\xintiPow {2}{100}}
-
-As an example the following code uses only expandable operations:
-\dverb+@
-|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}}}.
-+
-|2^{100}| (=\z) has \xintLen{\z} digits and the sum of
-their squares is \xintiiSum{\xintApply\xintiSqr\z}. These digits are, from the
-least to the most significant: \xintListWithSep {, }{\xintRev\z}. The
-thirteenth most
-significant digit is \xintNthElt{13}{\z}. The seventh
-least significant one is \xintNthElt{7}{\xintRev\z}.
-
-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}.
-
-
-\section{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}).
-
-\section{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}).
-
-\section{Exceptions (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:
-\dverb|@
-\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:bigtroubleahead
-\xintError:unknownfunction|
-
-\section{Common input errors when using the package macros}
-
-\edef\x{\xintMul {3}{5}/\xintMul{7}{9}}
-
-Here is a list of common 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 |\string\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}|\digitstt{=\xintRaw{1.5/-3.5e-2}},
- |\xintRaw{-1.5e2/3.5}|\digitstt{=\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 \digitstt{\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| (but |/| therein is euclidean
- quotient, which on positive operands is like truncating to the integer part,
- not rounding).
-\end{itemize}
-
-
-\section{Package namespace}
-
-Inner macros of \xinttoolsname, \xintname, \xintfracname, \xintexprname,
-\xintbinhexname, \xintgcdname, \xintseriesname, and \xintcfracname{} all begin
-either with |\XINT_| or with |\xint_|.\footnote{starting with release |1.06b|
- the style files use for macro names a more modern underscore |\_| rather than
- the \texttt{\char`\@} sign. A handful of private macros starting with
- |\string\XINT| do not have the underscore for technical reasons:
- \csa{XINTsetupcatcodes}, \csa{XINTdigits} and macros with names starting with
- |XINTinFloat| or |XINTinfloat|.} The package public commands all start with
-|\xint|. Some other control sequences are used only as delimiters, and left
-undefined, they may have been defined elsewhere, their meaning doesn't matter
-and is not touched.
-
-\xinttoolsname defines \hyperref[odef]{\ttfamily\char92odef},
-\hyperref[oodef]{\ttfamily\char92oodef}, \hyperref[fdef]{\ttfamily\char92fdef},
-but only if macros with these names do not already exist (|\xintoodef| etc...
-are defined anyhow for use in \csbxint{Assign} and \csbxint{AssignArray}).
-
-{\makeatother The \xintname packages presuppose that the \csa{space},
-\csa{empty}, |\m@ne|, |\z@| and |\@ne| control sequences
-have their meanings as in Plain \TeX{} or \LaTeX2e.}
-
-
-\section{Loading and usage}
-
-\dverb|@
-Usage with LaTeX: \usepackage{xinttools}
- \usepackage{xint} % (loads xinttools)
- \usepackage{xintfrac} % (loads xint)
- \usepackage{xintexpr} % (loads xintfrac)
-
- \usepackage{xintbinhex} % (loads xint)
- \usepackage{xintgcd} % (loads xint)
- \usepackage{xintseries} % (loads xintfrac)
- \usepackage{xintcfrac} % (loads xintfrac)
-
-Usage with TeX: \input xinttools.sty\relax
- \input xint.sty\relax % (loads xinttools)
- \input xintfrac.sty\relax % (loads xint)
- \input xintexpr.sty\relax % (loads xintfrac)
-
- \input xintbinhex.sty\relax % (loads xint)
- \input xintgcd.sty\relax % (loads xint)
- \input xintseries.sty\relax % (loads xintfrac)
- \input xintcfrac.sty\relax % (loads xintfrac)
-|
-
-We have added, directly copied from packages by \textsc{Heiko Oberdiek}, a
-mechanism of re-load and \eTeX{} detection, especially for Plain \TeX{}. As
-\eTeX{} is required, the executable |tex| can not be used, |etex| or |pdftex|
-(version |1.40| or later) or ..., must be invoked. Each package refuses to be
-loaded twice and automatically loads the other components on which it has
-dependencies.\footnote{exception: \xintexprname needs the user to explicitely
- load \xintgcdname, resp. \xintbinhexname, if use is to be made in
- \csa{xintexpr} of the \texttt{lcm} and \texttt{gcd} functions, and, resp.,
- hexadecimal numbers.}
-
-Also initially inspired from the \textsc{Heiko Oberdiek} packages we have
-included a complete catcode protection mecanism. The packages may be loaded in
-any catcode configuration satisfying these 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, and the previous configuration is restored
-after the loading of each one of the packages.
-
-This is for the loading of the packages.
-
-For the input of numbers as macro arguments the minus sign must have its
-standard category code (``\emph{other}''). Similarly the slash used for
-fractions must have its standard category code. And the square brackets, if made
-use of in the input, also must be of category code \emph{other}. The `e' of the
-scientific notation must be of category code \emph{letter}.
-
-All these requirements (which are anyhow satisfied by default) are
-relaxed for the contents of an |\xintexpr|-ession: spaces are gobbled,
-catcodes mostly do not matter, the |e| of scientific notation may be |E|
-(on input) \dots{}
-
-
-\section{Installation}\label{sec:install}
-
-\begingroup
-\def\MacroFont {\ttfamily\small\baselineskip11pt\relax\catcode`\"=12 }
-\dverb!@
-A. Installation using xint.tds.zip:
------------------------------------
-
-obtain xint.tds.zip from CTAN:
- http://mirror.ctan.org/install/macros/generic/xint.tds.zip
-
-cd to the download repertory and issue
- unzip xint.tds.zip -d <TEXMF>
-for example: (assuming standard access rights, so sudo needed)
- sudo unzip xint.tds.zip -d /usr/local/texlive/texmf-local
- sudo mktexlsr
-
-On Mac OS X, installation into user home folder:
- unzip xint.tds.zip -d ~/Library/texmf
-
-B. Installation after file extractions:
----------------------------------------
-
-obtain xint.dtx, xint.ins and the README from CTAN:
- http://www.ctan.org/pkg/xint
-
-- "tex xint.ins" generates the style files
-(pre-existing files in the same repertory will be overwritten).
-
-- without xint.ins: "tex or latex or pdflatex or xelatex xint.dtx"
-will also generate the style files (and xint.ins).
-
-xint.tex is also extracted, use it for the documentation:
-
-- with latex+dvipdfmx: latex xint.tex thrice then dvipdfmx xint.dvi
-Ignore dvipdfmx warnings, but if the pdf file has problems with fonts
-(possibly from an old dvipdfmx), use then rather pdflatex or xelatex.
-
-- with pdflatex or xelatex: run it directly thrice on xint.dtx, or run
-it on xint.tex after having edited the suitable toggle therein.
-
-When compiling xint.tex, the documentation is by default produced
-with the source code included. See instructions in the file for
-changing this default.
-
-When compiling directly xint.dtx, the documentation is produced
-without the source code (latex+dvips or pdflatex or xelatex).
-
-Finishing the installation: (on first installation the destination
-repertories may need to be created)
-
- xinttools.sty |
- xint.sty |
- xintfrac.sty |
- xintexpr.sty | --> TDS:tex/generic/xint/
- xintbinhex.sty |
- xintgcd.sty |
- xintseries.sty |
- xintcfrac.sty |
-
- xint.dtx --> TDS:source/generic/xint/
- xint.ins --> TDS:source/generic/xint/
- xint.tex --> TDS:source/generic/xint/
-
- xint.pdf --> TDS:doc/generic/xint/
- README --> TDS:doc/generic/xint/
-
-Depending on the TDS destination and the TeX installation, it may be
-necessary to refresh the TeX installation filename database (mktexlsr)!
-\endgroup
-
-\section{The \csh{xintexpr} math parser (I)}
-\label{sec:exprsummary}
-
-% 27 octobre 2013 plus de problème avec &... il n'est plus actif (ouf)
-\xintexprSafeCatcodes
-\newcommand\formula[3]{\xinttheexpr round((#1 & (#2 | #3)) * (355/113*#3 -
- (#1 - #2/2)^2), 8)\relax }
-\xintexprRestoreCatcodes
-
-
-Here is some random formula, defining a \LaTeX{} command with three parameters,
-\centeredline{\verb$\newcommand\formula[3]$}
-\centeredline{\verb${\xinttheexpr round((#1 & (#2 | #3)) * (355/113*#3 - (#1 -
- #2/2)^2), 8) \relax}$}
-
-\smallskip
-
-Let |a=#1|, |b=#2|, |c=#3| be the parameters. The first term is the logical
-operation |a and (b or c)| where a number or fraction has truth value @1@ if it
-is non-zero, and @0@ otherwise. So here it means that |a| must be non-zero as
-well as |b| or |c|, for this first operand to be @1@, else the formula returns
-@0@. This multiplies a second term which is algebraic. Finally the result (where
-all intermediate computations are done \emph{exactly}) is rounded to a value
-with @8@ digits after the decimal mark, and printed.
-\centeredline{|\formula
- {771.3/9.1}{1.51e2}{37.73}| expands to
- \digitstt{\formula {771.3/9.1}{1.51e2}{37.73}}}
-Note that |#1|, |#2|, and |#3| are not protected by parentheses in the
-definition of |\formula|, this is something to keep in mind if for example we
-want to use |2+5| as third argument: it should be |(2+5)| then.
-
-
-\begingroup % 9 octobre pour une meilleure gestion de l'indentation
-\leftmargini 0pt
-\list\labelitemi{\def\makelabel#1{\hss\llap{#1}}\listparindent\parindent
- \labelwidth\parindent
- \itemindent\labelwidth}%
-%
-\item as everything gets expanded, the characters
- \verb$+,-,*,/,^,!,&,|,?,:,<,>,=,(,),"$ and the comma, which may appear
- in the |infix| syntax, should not (if actually used in the expression) be
- active (for example from serving as
- shorthands for some language in the |Babel| system).
- The command \csbxint{exprSafeCatcodes} resets these characters to their
- standard catcodes and \csbxint{exprRestoreCatcodes} restores the status
- prevailing at the time of the previous \csa{xintexprSafeCatcodes}.
-\item many expressions have equivalent macro formulations written without
- |\xinttheexpr|.\footnote{Not everything allows a straightforward reformulation
- because the package macros only \fexpan d their arguments while
- \csa{xintexpr} expands everything from left to right.} Here for |\formula|
- we could have used: \centeredline {|\xintRound {8}{\xintMul {\xintAND
- {#1}{\xintOR {#2}{#3}}}{\xintSub |} \centeredline {| {\xintMul
- {355/113}{#3}}{\xintPow {\xintSub {#1}{\xintDiv {#2}{2}}}{2}}}}|} with
- the inherent difficulty of keeping up with braces and everything...
-\item if such a formula is used thousands of times in a document (for plots?),
- this could impact some parts of the \TeX{} program memory (for technical
- reasons explained in \autoref{sec:expr}). So, a utility \csbxint{NewExpr}
- is provided to do the work of translating an |\xintexpr|-ession with
- parameters into a chain of macro evaluations.\footnote{As its makes some macro
- definitions, it is not an expandable command. It does not need protection
- against active characters as it does it itself.} With
- \centeredline{|\xintNewExpr\formula[3]|}
- \centeredline{\verb${ round((#1 & (#2 | #3)) * (355/113*#3 - (#1 - #2/2)^2),
- 8) }$}
- one gets a command |\formula| with three parameters and meaning:
-
-\xintNewExpr\formula[3]
-{ round((#1 & (#2 | #3)) * (355/113*#3 - (#1 - #2/2)^2),
- 8) }
-
-{\centering\ttfamily
-
-\meaning\formula
-
-}This does the same thing as the hand-written version from the previous item
-(but expands in only two steps).\footnote{But the hand-written version as well
- as the \csa{xintNewExpr} generated one differ from the original \csa{formula}
- command which allowed each of its argument to use all the operators and
- functions recognized by \csa{xintexpr}, and this aspect is lost. To recover it
- the arguments themselves should be passed as \csa{xinttheexpr..\char92relax}
- to the defined macro.} The use
-even thousands of times of such an |\xintNewExpr|-generated |\formula| has no
-memory impact.
-\item count registers and |\numexpr|-essions are accepted (LaTeX{}'s counters
- can be inserted using |\value|) without needing |\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 if 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 is looking
- for an infix operator, and: (1.)\inmarg{v1.09i}~\emph{encounters a register,
- variable or \eTeX{} expression (as described in the previous item)},
- (2.)\inmarg{v1.09j}~\emph{encounters a sub-\csa{xintexpr}-ession}, or
- (3.)\inmarg{\\ v1.09k}~\emph{encounters an opening parenthesis.}
-\item so far only |\xinttheexpr| was mentioned: there is also |\xintexpr| which,
- like a |\numexpr|, needs a prefix which is called \csbxint{the}. Thus
- \csbxint{theexpr} as was done in the definition of |\formula| is equivalent to
- \csbxint{the}|\xintexpr|.
-\item This latter form is convenient when one has defined for
- example:
-%
-\centeredline{|\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 use it in some other package macros. 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.
-\item in both cases (the `yet-to-be computed' and the
-`already computed') |\x| can then be inserted in other expressions, as
-for example
-%
-\centeredline {|\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 sometimes one needs an integer, not a fraction or decimal number. The
- |round| function rounds to the nearest integer, and |\xintexpr
- round(...)\relax| has an alternative and equivalent syntax as \csbxint{iexpr}|
- ... \relax|. There is also \csbxint{theiexpr}. The rounding is applied to the
- final result only, intermediate computations are not rounded.
-\item \csbxint{iiexpr}|..\relax| and \csbxint{theiiexpr}|..\relax| deal only
- with (long) integers and skip the overhead of the fraction internal format.
- The infix operator |/| does euclidean division, thus |2+5/3| will not be
- treated exactly but be like |2+1|.
-\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:exprsummary}. Here is an
- example:
-\begingroup
-\def\MacroFont {\ttfamily\parskip0pt \parindent 15pt \baselineskip 12pt \relax }
-\dverb!@
-\xintNewBoolExpr \AssertionA[3]{ #1 & (#2|#3) }
-\xintNewBoolExpr \AssertionB[3]{ #1 | (#2&#3) }
-\xintNewBoolExpr \AssertionC[3]{ xor(#1,#2,#3) }
-\xintFor #1 in {0,1} \do {%
- \xintFor #2 in {0,1} \do {%
- \xintFor #3 in {0,1} \do {%
- \centerline{#1 AND (#2 OR #3) is \AssertionA {#1}{#2}{#3}\hfil
- #1 OR (#2 AND #3) is \AssertionB {#1}{#2}{#3}\hfil
- #1 XOR #2 XOR #3 is \AssertionC {#1}{#2}{#3}}}}}
-!%
-\endgroup
-\xintNewBoolExpr \AssertionA[3]{ #1 & (#2|#3) }
-\xintNewBoolExpr \AssertionB[3]{ #1 | (#2&#3) }
-\xintNewBoolExpr \AssertionC[3]{ xor(#1,#2,#3) }
-\xintFor #1 in {0,1} \do {%
- \xintFor #2 in {0,1} \do {%
- \xintFor #3 in {0,1} \do {%
- \centeredline{#1 AND (#2 OR #3) is \AssertionA {#1}{#2}{#3}\hfil
- #1 OR (#2 AND #3) is \AssertionB {#1}{#2}{#3}\hfil
- #1 XOR #2 XOR #3 is \AssertionC {#1}{#2}{#3}}}}}
-
-%
-\item there is also \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.
- \centeredline{|\xintthefloatexpr 2^100000\relax:| \digitstt{\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: \centeredline{|\xinttheexpr sqrt(2,60)\relax|:}
- \centeredline{\digitstt{\xinttheexpr sqrt(2,60)\relax }} Notice the |a/b[n]|
- notation: usually the denominator |b| even if |1| gets printed; it does not
- show here because the square root is computed by a version of
- \csbxint{FloatSqrt} which for efficiency when used in such expressions outputs
- the result in a format |d_1 d_2 .... d_P [N]| equivalent to the usual float
- output format |d_1.d_2...d_P e<expon.>|. To get a float
- format, it is easier to use an |\xintfloatexpr|, but the precision must be set
- using the non expandable |\xintDigits:=60;| assignment, there is no optional
- parameter possible currently to |\xintfloatexpr|:
-%
-\centeredline{|\xintDigits:=60;\xintthefloatexpr sqrt(2)\relax|}
-\centeredline{\digitstt{\xintDigits:=60;\xintthefloatexpr sqrt(2)\relax}}
-%
-Or, without manipulating |\xintDigits|, another option to convert to float a
-computation done by an |\xintexpr|:
-\centeredline{|\xintFloat[60]{\xinttheexpr sqrt(2,60)\relax}|}
-\centeredline{\digitstt{\xintFloat[60]{\xinttheexpr sqrt(2,60)\relax}}}
-%
-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.
-%
-\centeredline{|\xintDigits:=48;
- \xintthefloatexpr 2^100000\relax|: }
-\centeredline{\xintDigits:=48;\digitstt{\xintthefloatexpr 2^100000\relax}}
-%
-\item hexadecimal \TeX{} number\inmarg{New with 1.09k!} denotations
- (\emph{i.e.}, with a |"| prefix) are recognized by the |\xintexpr| parser and
- its variants. Except in |\xintiiexpr|, a (possibly empty) fractional part
- with the dot |.| as ``hexadecimal'' mark is allowed.
-%
-\centeredline{|\xinttheexpr "FEDCBA9876543210\relax|$\to$\digitstt{\xinttheexpr
- "FEDCBA9876543210\relax}}
-\centeredline{|\xinttheiexpr 16^5-("F75DE.0A8B9+"8A21.F5746+16^-5)\relax|$\to$\digitstt{\xinttheiexpr 16^5-("F75DE.0A8B9+"8A21.F5746+16^-5)\relax}}
-%
-Letters must be uppercased, as with standard
- \TeX{} hexadecimal denotations. Loading the \xintbinhexname package is required
- for this functionality.
-\endlist
-\endgroup
-
-\section{The \csh{xintexpr} math parser (II)}
-\label{sec:exprsummaryII}
-
-An expression is built with infix operators (including comparison and boolean
-operators), parentheses, functions, and the two branching operators |?| and |:|.
-The parser expands everything from the left to the right and everything may thus
-be revealed step by step by expansion of macros. Spaces anywhere are allowed.
-
-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 take into
-account that:
-\begin{enumerate}
-\item the parser will not see the macro arguments, (but they may themselves be
- set-up as |\xinttheexpr...\relax|),
-\item the output format of most \xintfracname macros is |A/B[N]|, and square
- brackets are \emph{not understood by the parser}. One \emph{must} enclose the
- macro and its arguments inside a brace pair |{..}|, which will be recognized
- and treated specially,
-\item a macro outputting numbers in scientific notation |x.yEz| (either with a
- lowercase |e| or uppercase |E|), must \emph{not} be enclosed
- in a brace pair, this is the exact opposite of the |A/B[N]| case; scientific
- numbers, explicit or implicit, should just be inserted directly in the
- expression.
-\end{enumerate}
-
-One may insert a sub-|\xintexpr|-expression within a larger one. Each one of
-|\xintexpr|, |\xintiexpr|, |\xintfloatexpr|, |\xintboolexpr| may be inserted in
-another one. On the other hand the integer only |\xintiiexpr| will generally
-choke on a sub-|\xintexpr| as the latter (except if it did not do any operation
-or had an overall top level |round| or |trunc| or |?(..)| or\dots) produces (in
-internal format) an |A/B[N]| which the strictly integer only \csbxint{iiexpr}
-does not understand. See \autoref{xintiiexpr} for more information.
-
-Here is, listed from the highest priority to the lowest, the complete list of
-operators and functions. Functions are at the top level of priority. Next are
-the postfix operators: |!| for the factorial, |?| and |:| are two-fold way and
-three-fold way branching constructs. Also at the top level of priority the |e|
-and |E| of the scientific notation and the |"|\inmarg{\string" is new in 1.09k}
-for hexadecimal numbers, then power, multiplication/division,
-addition/subtraction, comparison, and logical operators. At the lowest level:
-commas then parentheses.
-
-
-The |\relax| at the end of an expression is \emph{mandatory}.
-
- % 1.09c ajoute bool et togl
- % 1.09a:
- % reduce,
- % sqr, sqrt, abs, sgn, floor, ceil, quo, rem, round, trunc, float, gcd, lcm,
- % max, min, sum, prd, add, mul, not, all, any, xor
- % ?, !, if, ifsgn, ?, :.
-
-\newcommand\ctexttt [1]{\begingroup\color[named]{DarkOrchid}\ttfamily\bfseries
- #1\endgroup}
-
-\begingroup % 9 octobre pour la gestion de l'indentation et couleurs
-\leftmargini 0pt
-\leftmarginii .5\parindent
-\list\labelitemi{\def\makelabel#1{\hss\llap{#1}}\listparindent\parindent
- \labelwidth\parindent
- \itemindent\labelwidth}%
-\item
- Functions are at the same top level of priority. All functions even
- |?| and |!| (as prefix) require parentheses around their argument
- (possibly a comma separated list).
- \begin{framed}
- \ctexttt{floor, ceil, frac, reduce, sqr, abs, sgn, ?, !, not, bool,
- togl, round, trunc, float, sqrt, quo, rem, if, ifsgn, all, any,
- xor, add (=sum), mul (=prd), max, min, gcd, lcm.}
-
- |quo| and |rem|
- operate only on integers; |gcd| and |lcm| also and require
- \xintgcdname loaded; |togl| requires the |etoolbox| package; |all|, |any|,
- |xor|, |add|, |mul|, |max| and |min| are functions with arbitrarily many
- comma separated arguments.
- \end{framed}
- \begin{description}
- \item[functions with one (numeric) argument] (numeric: any expression leading
- to an integer, decimal number, fraction, or floating number in scientific
- notation) \ctexttt{floor, ceil, frac, reduce, sqr, abs, sgn, ?, !, not}. The
- |?(x)| function returns the truth value, @1@ if |x<>0|, @0@ if |x=0|. The
- |!(x)| is the logical not. The |reduce| function puts the fraction in
- irreducible form. The |frac| function is fractional part,
- same sign as the number:\newline
- \null\quad\quad|\xinttheexpr
- frac(-3.57)\relax|$\to$\digitstt{\xinttheexpr frac(-3.57)\relax}\newline
- \null\quad\quad|\xinttheexpr
- trunc(frac(-3.57),2)\relax|$\to$\digitstt{\xinttheexpr
- trunc(frac(-3.57),2)\relax}\newline
- \null\quad\quad|\xintthefloatexpr
- frac(-3.57)\relax|$\to$\digitstt{\xintthefloatexpr
- frac(-3.57)\relax}.\newline
- Like
- the other functions |!| and |?| \emph{must} use parentheses.
-
- \item[functions with one (alphabetical) argument] \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:
- \centeredline{|\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 optional argument]
- \ctexttt{round, trunc,\\ float, sqrt}. For
- example |round(2^9/3^5,12)=|\digitstt{\xinttheexpr round(2^9/3^5,12)\relax.}
- The |sqrt| is available also in |\xintexpr|, not only in |\xintfloatexpr|.
- The second optional argument is the required float precision.
- \item[functions with two arguments]
- \ctexttt{quo, rem}. These functions are integer only, they give the quotient
- and remainder in Euclidean division (more generally one can use
- the |floor| function; related: the |frac| function).
- \item[the if conditional (twofold way)] \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)]
- \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] \ctexttt{all, any,
- xor, add (=sum), mul (=prd), max, min, gcd, lcm}: |gcd| and |lcm| are
- integer-only and require the \xintgcdname package. Currently, the |and| and
- |or| keywords are left undefined by the package, which uses rather |all|
- and |any|. They must have at least one argument.
- \end{description}
-\item The three postfix operators \ctexttt{!, ?, :}.
- \begin{description}
- \item[{\color[named]{DarkOrchid}!}] computes the factorial of an integer. |sqrt(36)!| evaluates to |6!|
- (\digitstt{=\np{\xinttheexpr sqrt(36)!\relax}}) and not to the square root of
- |36!| (\digitstt{$\approx$\np{\xintthefloatexpr sqrt(36!)\relax}}). 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
- \centeredline{|\xinttheiexpr (3>2)?{5+6}{7-1}2^3\relax|} is legal and
- computes |5+62^3=|\digitstt{\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 (it is not necessary to use |sgn(cond):{<}{=}{>}|) 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.
- \centeredline{|\def\x{0.33}\def\y{1/3}|} \centeredline{|\xinttheexpr
- (\x-\y):{sqrt}{0}{1/}(\y-\x)\relax|%
- \digitstt{=\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\bfseries}
- 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 {\def\MicroFont{\ttfamily}|\xinttheexpr
- .^2+2^.\relax|$\to$\digitstt{\xinttheexpr .^2+2^.\relax} (which is
- |0^2+2^0|)}.
-\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 {\def\MicroFont{\ttfamily}|\xintiiexpr..\relax|}. The |"|
- functionality requires that the user loaded \xintbinhexname (there is no
- warning, but an ``undefined control sequence'' error will naturally results if
- the package has not been loaded).
-\item
- %
- The |e| and |E| for scientific notation. They are treated as infix operators
- of highest priority: this means that they serve as an end marker (possibly
- arising from macro expansion) for the scanned number, and then will pre-empt
- the number coming next, either explicit, or arising from expansion, from
- parenthesized material, from a sub-expression etc..., to serve as exponent.
- \begingroup
- \def\MicroFont{\ttfamily}%
- From
- the rules above, inside |\xintexpr|, |1e3-1|
- is \digitstt{\xinttheexpr 1e3-1\relax}, |1e3^2| is \digitstt{\xinttheexpr
- 1e3^2\relax}, and |"Ae("A+"F)^"A|
- is \digitstt{\xinttheexpr "Ae("A+"F)^"A\relax}.\endgroup
-\item The power operator |^|. It is left associative:
-\begingroup\def\MicroFont{\ttfamily}%
-|\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 withing |\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
-(\digitstt{={\xintiiPow{12}{16}}}), hence the result is wrong:
-%
-\centeredline{$\np{\xintthefloatexpr (1+1e-8)^(12^16)\relax }$}
-%
-One should code
-%
-\centeredline{|\xintthe\xintfloatexpr (1+1e-8)^\xintiiexpr 12^20\relax \relax|}
-%
-to obtain the correct floating point evaluation
-%
-\centeredline{$\np{1.00000001}^{12^{16}}\approx\np{\xintthefloatexpr
- (1+1e-8)^\xintiiexpr 12^16\relax\relax }$}%
-%
-\endgroup
-\item Multiplication and division \raisebox{-.3\height}{|*|}, |/|. The division
- is left associative, too: \begingroup\def\MicroFont{\ttfamily}%
- |\xinttheiexpr 100/50/2\relax| evaluates to
- \xinttheiexpr 100/50/2\relax, not \xinttheiexpr 100/(50/2)\relax.\endgroup
-\item Addition and subtraction |+|, |-|. Again, |-| is left
- associative: \begingroup\def\MicroFont{\ttfamily}%
- |\xinttheiexpr 100-50-2\relax| evaluates to
- \xinttheiexpr 100-50-2\relax, not \xinttheiexpr 100-(50-2)\relax.\endgroup
-\item Comparison operators |<|, |>|, |=| (currently, no @<=@, @>=@,
- \dots ).
-\item Conjunction (logical and): |&|. (no @&&@)
-\item Inclusive disjunction (logical or): \verb$|$. (no @||@)
-\item The comma |,|. \def\MicroFont{\ttfamily}%
- With |\xinttheiexpr 2^3, 3^4, 5^6\relax| one obtains as output
- \xinttheiexpr 2^3,3^4,5^6\relax{} (no space after the commas on output).
-\item The parentheses.
-\endlist
-\endgroup
-
-See \autoref{ssec:countinexpr} for count and dimen registers and variables.
-
-
-\section{Change log for earlier releases}
-\label{sec:releases}
-
-% peut-être je devrais mettre ici le dernier aussi?
-
-\footnotesize
-
-\noindent Release |1.09j| (|[2014/01/09]|):
-\begin{itemize}
-\item the core division routines have been re-written for some (limited)
- efficiency gain, more pronounced for small divisors. As a result the
- \hyperlink{Machin1000}{computation of one thousand digits of $\pi$}
- is close to three times faster than with earlier releases.
-\item some various other small improvements, particularly in the power routines.
-\item a new macro \csbxint{XTrunc} 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.
-\item the tacit multiplication done in \csbxint{expr}|..\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.
-\item \csbxint{expr} 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.
-\item the default behavior of \csbxint {Assign} 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.
-\item bug-fix: |1.09i| did an unexplainable change to |\XINT_infloat_zero| which
- broke the floating point routines for vanishing operands =:(((
-\item dtx bug-fix: the |1.09i .ins| file produced a buggy |.tex| file.
-\end{itemize}
-
-\noindent Release |1.09i| (|[2013/12/18]|):
-\begin{itemize}
-\item \csbxint{iiexpr} is a variant of \csbxint{expr} which is optimized to deal
- only with (long) integers, |/| does a euclidean quotient.
-\item |\xintnumexpr|, |\xintthenumexpr|, |\xintNewNumExpr| are renamed,
- respectively, \csbxint{iexpr}, \csbxint{theiexpr}, \csbxint{NewIExpr}. The
- earlier denominations are kept but to be removed at some point.
-\item 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
- \autoref{sec:Dimensions}.
-\item new conditional \csbxint{ifOne}; |\xintifTrueFalse| renamed to
- \csbxint{ifTrueAelseB}; new macros \csbxint{TFrac} (`fractional part', mapped
- to function |frac| in |\xintexpr|-essions), \csbxint{FloatE}.
-\item \csbxint{Assign} 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
-\item related to the previous item, \xinttoolsname defines
- \hyperref[odef]{\ttfamily\char92odef},
- \hyperref[oodef]{\ttfamily\char92oodef},
- \hyperref[fdef]{\ttfamily\char92fdef} (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
- \hyperref[oodef]{\ttfamily\char92oodef} which expands twice the macro
- replacement text and is thus a faster alternative to |\edef| taking into
- account that the \xintname bundle macros expand already completely in only
- two steps. This can be significant when repeatedly making |\def|-initions
- expanding to hundreds of digits.
-\item 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.
-\item bug-fix: |\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 \verb+|+ in
- \csa{xintexpr-}essions, it affected only the macro form)
- |:-((|.
-\item bug-fix: \csbxint{FtoCCv} still ended fractions with the |[0]|'s which
- were supposed to have been removed since release |1.09b|.
-\end{itemize}
-
-\noindent Release |1.09h| (|[2013/11/28]|):
-\begin{itemize}
-\item parts of the documentation have been re-written or re-organized,
- particularly the discussion of expansion issues and of input and
- output formats.
-\item 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.
-\item a dependency of \xinttoolsname on \xintname (inside \csbxint{Seq}) has
- been removed.
-\item \csbxint{TypesetEuclideAlgorithm} and \csbxint{TypesetBezoutAlgorithm}
- have been slightly modified (regarding indentation).
-\item macros \csa{xintiSum} and \csa{xintiPrd} are renamed to \csbxint{iiSum}
- and \csbxint{iiPrd}.
-\item a count register used in |1.09g| in the \csbxint{For} loops for parsing
- purposes has been removed and replaced by use of a |\numexpr|.
-\item the few uses of |\loop| have been replaced by |\xintloop/\xintiloop|.
-\item all macros of \xinttoolsname for which it makes sense are now
- declared |\long|.
-\end{itemize}
-
-\noindent Release |1.09g| (|[2013/11/22]|):
-\begin{itemize}
-\item package \xinttoolsname is detached from \xintname, to make tools such as
- \csbxint{For}, \csbxint{ApplyUnbraced}, and \csbxint{iloop} available without
- the \xintname overhead.
-\item new expandable nestable loops \csbxint{loop} and \csbxint{iloop}.
-\item bugfix: \csbxint{For} and \csbxint{For*} do not modify anymore the value
- of |\count 255|.
-\end{itemize}
-
-\noindent Release |1.09f| (|[2013/11/04]|):
-\begin{itemize}
-\item new \csbxint{ZapFirstSpaces}, \csbxint{ZapLastSpaces},
- \csbxint{ZapSpaces}, \csbxint{ZapSpacesB}, for expandably stripping away
- leading and/or ending spaces.
-\item \csbxint{CSVtoList} by default uses \csbxint{ZapSpacesB} to strip away
- spaces around commas (or at the start and end of the comma separated list).
-\item also the \csbxint{For} loop will strip out all spaces around commas and at
- the start and the end of its list argument; and similarly for
- \csbxint{Forpair}, \csbxint{Forthree}, \csbxint{Forfour}.
-\item \csbxint{For} \emph{et al.} accept all macro parameters
- from
- |#1| to |#9|.
-\item for reasons of inner coherence some macros previously with one extra `|i|'
- in their names (e.g. \csa{xint\-iMON}) now have a doubled `|ii|'
- (\csbxint{iiMON}) to indicate that they skip the overhead of parsing their
- inputs via \csbxint{Num}. Macros with a \emph{single} `|i|' such as
- \csbxint{iAdd} are those which maintain the non-\xintfracname output format
- for big integers, but do parse their inputs via \csbxint{Num} (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.
-\end{itemize}
-
-
-\noindent Release |1.09e| (|[2013/10/29]|):
-\begin{itemize}
-\item new \csbxint{integers}, \csbxint{dimensions}, \csbxint{rationals} for
- infinite \csbxint{For} loops, interrupted with \csbxint{BreakFor} and
- \csbxint{BreakForAndDo}.
-\item new \csbxint{ifForFirst}, \csbxint{ifForLast} for the \csa{xintFor} and
- \csa{xintFor*} loops,
-\item the \csa{xintFor} and \csa{xintFor*} loops are now |\long|, the
- replacement text and the items may contain explicit |\par|'s.
-\item bug fix, the \csbxint{For} loop (not \csbxint{For*}) did not correctly
- detect an
- empty list.
-\item new conditionals \csbxint{ifCmp}, \csbxint{ifInt}, \csbxint{ifOdd}.
-\item bug fix, |\xintiSqrt {0}| crashed. |:-((|
-\item the documentation has been enriched with various additional examples,
- such as the \hyperref[ssec:quicksort]{the quick sort algorithm illustrated} or
- the computation of prime numbers (\autoref{ssec:primesI},
- \autoref{ssec:primesII}, \autoref{ssec:primesIII}).
-\item the documentation explains with more details various expansion related
- issues, particularly in relation to conditionals.
-\end{itemize}
-
-\noindent Release |1.09d| (|[2013/10/22]|):\nobreak
-\begin{itemize}
-\item \csbxint{For*} is modified to gracefully handle a space token (or
- more than one) located at the
- very end of its list argument (as in for example |\xintFor* #1 in
- {{a}{b}{c}<space>} \do {stuff}|;
- spaces at other locations were already harmless). Furthermore this new
-version \fexpan ds the un-braced list items. After
-|\def\x{{1}{2}}| and |\def\y{{a}\x {b}{c}\x }|, |\y| will appear to
-\csbxint{For*} exactly as if it had been defined as
-|\def\y{{a}{1}{2}{b}{c}{1}{2}}|.
-\item same bug fix in \csbxint{ApplyInline}.
-\end{itemize}
-
-\noindent Release |1.09c| (|[2013/10/09]|):
-\begin{itemize}
-\item added \hyperlink{item:bool}{|bool|} and \hyperlink{item:bool}{|togl|} to
- the
- \csbxint{expr} syntax; also added \csbxint{boolexpr} and \csbxint{ifboolexpr}.
-\item added |\xintNewNumExpr| (now \csbxint{NewIExpr} and \csbxint{NewBoolExpr},
-\item \csbxint{For} 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,
-\item \csbxint{Forpair}, \csbxint{Forthree}, \csbxint{Forfour} are experimental
- variants of \csbxint{For},
-\item \csbxint{ApplyInline} has been enhanced in order to be usable for
- generating rows (partially or completely) in an alignment,
-\item new command \csbxint{Seq} to generate (expandably) arithmetic sequences of
- (short) integers,
-\item the factorial |!| and branching |?|, |:|, operators (in
- \csbxint{expr}|...\relax|) have now less precedence than a function name
- located just before: |func(x)!| is the factorial of |func(x)|, not |func(x!)|,
-\item again various improvements and changes in the documentation.
-\end{itemize}
-
-\noindent Release |1.09b| (|[2013/10/03]|):
-\begin{itemize}
-\item various improvements in the documentation,
-\item more economical catcode management and re-loading handling,
-\item removal of all those |[0]|'s previously forcefully added at the end of
- fractions by various macros of \xintcfracname,
-\item \csbxint{NthElt} with a negative index returns from the tail of the list,
-\item new macro \csbxint{PRaw} to have something like what |\xintFrac| does in
- math
- mode; i.e. a |\xintRaw| which does not print the denominator if it is one.
-\end{itemize}
-
-\noindent Release |1.09a| (|[2013/09/24]|):
-\begin{itemize}
-\item \csbxint{expr}|..\relax| and
- \csbxint{floatexpr}|..\relax| admit functions in their
- syntax, with comma separated values as arguments, among them \texttt{reduce,
- sqr, sqrt, abs, sgn, floor, ceil, quo, rem, round, trunc, float, gcd, lcm,
- max, min, sum, prd, add, mul, not, all, any, xor}.
-\item comparison (|<|, |>|, |=|) and logical (\verb$|$, |&|) operators.
-\item 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 def'ined to be an |\xintexpr..\relax|:
-\centeredline{|\def\x{\xintexpr 3^57\relax}\def\y{\xintexpr \x^(-2)\relax}|}
-\centeredline{|\def\z{\xintexpr
- \y-3^-114\relax}|\hspace{1cm}|\xintthe\z=|\begingroup
-\def\x{\xintexpr 3^57\relax}\def\y{\xintexpr \x^(-2)\relax}%
-\def\z{\xintexpr \y-3^-114\relax}\digitstt{\xintthe\z}\endgroup}
-\item |\xintnumexpr .. \relax| (now renamed \csbxint{iexpr}) is |\xintexpr
- round( .. ) \relax|.
-\item \csbxint{NewExpr} now works with the standard macro parameter character
- |#|.
-\item both regular |\xintexpr|-essions and commands defined by |\xintNewExpr|
- will work with comma separated lists of expressions,
-\item new commands \csbxint{Floor}, \csbxint{Ceil}, \csbxint{Maxof},
- \csbxint{Minof} (package \xintfracname), \csbxint{GCDof}, \csbxint{LCM},
- \csbxint{LCMof} (package \xintgcdname), \csbxint{ifLt}, \csbxint{ifGt},
- \csbxint{ifSgn}, \csbxint{ANDof}, ...
-\item The arithmetic macros from package \xintname now filter their operands via
- \csbxint{Num} 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 but with \xintfracname loaded.
-\item a bug introduced in |1.08b| made \csbxint{Cmp} crash when one of its
- arguments was zero. |:-((|
-\end{itemize}
-
-
-\noindent Release |1.08b| (|[2013/06/14]|):
-\begin{itemize}
-\item Correction of a problem with spaces inside |\xintexpr|-essions.
-\item Additional improvements to the handling of floating point numbers.
-\item The macros of \xintfracname allow to use count registers in their
- arguments in ways which were not previously documented. See
- \hyperref[sec:useofcount]{Use of count registers}.
-\end{itemize}
-
-\noindent Release |1.08a| (|[2013/06/11]|):
-\begin{itemize}
-\item Improved efficiency of the basic conversion from exact
- fractions to floating point numbers,
- with ensuing speed gains especially for the power function macros
- \csbxint{FloatPow} and \csbxint{FloatPower},
-\item Better management by the \xintfracname macros \csbxint{Cmp},
- \csbxint{Max}, \csbxint{Min} and \csbxint{Geq} of inputs having big powers
- of ten in them.
-\item Macros for floating point numbers added to the \xintseriesname package.
-\end{itemize}
-
-\noindent Release |1.08| (|[2013/06/07]|):
-\begin{itemize}
-\item Extraction of square roots, for floating point numbers
- (\csbxint{FloatSqrt}), and also in
- a version adapted to integers (\csbxint{iSqrt}).
-\item New package \xintbinhexname providing \hyperref[sec:binhex]{conversion
- routines} to and from binary and hexadecimal bases.
-\end{itemize}
-
-\noindent Release |1.07| (|[2013/05/25)]|):
-\begin{itemize}
-\item The \xintfracname macros accept numbers written in scientific notation,
- the \csbxint{Float} 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 \csbxint{Float} or set with |\xintDigits := D;|. The default value
- is |16|.
-\item The \xintexprname package is a new core constituent (which loads
- automatically \xintfracname and \xintname) and implements the expandable
- expanding parsers \centeredline{\csbxint{expr}| . . . \relax|,
- and its variant
- \csbxint{floatexpr}| . . . \relax|} allowing on input formulas using the
- standard form with infix
- operators |+|, |-|, |*|, |/|, and |^|, and arbitrary levels of
- parenthesizing. Within a float expression the operations are executed
- according to the current value of \csbxint{Digits}. Within an
- |\xintexpr|-ession the binary operators are computed exactly.
-\item 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|.\footnote{but
- values higher than 100 or 200 will presumably give too slow evaluations.} The
-macro incarnations of the binary operations admit an optional argument which
-will replace pointwise |D|; this argument may exceed the |32767| bound.
-\item To write the |\xintexpr| parser I benefited from the commented source of
- the
-\LaTeX3 parser; the |\xintexpr| parser has its own features and peculiarities.
-See \hyperref[sec:expr]{its documentation}.
-\end{itemize}
-
-Initial release |1.0| was on |2013/03/28|.
-
-
-% \noindent Historians debate the early history of the \xintname bundle, whose
-% details will need patient reconstruction from the scattered archeological
-% remnants. It has been established that the initial release |1.0| was on
-% |2013/03/28|, although only closer scrutiny of the CTAN logs could help
-% completely exclude possibility of an earlier |0.9|.
-
-
-
-\normalsize
-
-
-\etocdepthtag.toc {commandsA}
-
-\section{Commands of the \xinttoolsname package}
-\label{sec:tools}
-
-\def\n{\string{N\string}}
-\def\m{\string{M\string}}
-\def\x{\string{x\string}}
-
-These utilities used to be provided within the \xintname package; since |1.09g|
-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.
-
-\clearpage
-
-\localtableofcontents
-
-
-\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
-reverted. Unprotected spaces (of any character code) are gobbled.
-\centeredline{|\xintReverseOrder{\xintDigitsOf\xintiPow {2}{100}\to\Stuff}|}
-\centeredline{gives:
- \ttfamily{\string\Stuff\string\to1002\string\xintiPow\string\xintDigitsOf}}
-
-\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, adding a pair of braces to each (thus, maintaining brace pairs
-already existing). Spaces (in-between external brace pairs) 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. \centeredline{|\edef\x{\xintRevWithBraces{12345}}|}
-\centeredline{|\meaning\x:|\ttfamily{\meaning\X}}
-\centeredline{|\edef\y{\xintRevWithBraces\x}|}%
-\centeredline{|\meaning\y:|\ttfamily{\meaning\y}} The examples above could be
-defined with |\edef|'s because the braced material did not contain macros.
-Alternatively: \centeredline{|\expandafter\def\expandafter\w\expandafter|}%
-\centeredline{|{\romannumeral0\xintrevwithbraces{{\A}{\B}{\C}{\D}{\E}}}|}
-\centeredline{|\meaning\w:|\ttfamily{\meaning\w}} The macro
-\csa{xintReverseWithBracesNoExpand}\etype{n} does the same job without the
-initial
-expansion of its argument.
-
-\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}| for a variant which first \fexpan ds
-its argument. \centeredline{|\xintLength {\xintiPow
- {2}{100}}|\digitstt{=\xintLength {\xintiPow{2}{100}}}}
-\centeredline{${}\neq{}$|\xintLen {\xintiPow {2}{100}}|\digitstt{=\xintLen
- {\xintiPow{2}{100}}}}
-
-\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 \csa{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 } }
-\centeredline{|\xintZapFirstSpaces { \a { \X } { \b \Y } }->|%
-\digitstt{\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 } }
-\centeredline{|\xintZapLastSpaces { \a { \X } { \b \Y } }->|%
-\digitstt{\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 } }
-\centeredline{|\xintZapSpaces { \a { \X } { \b \Y } }->|%
-\digitstt{\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 } }
-\centeredline{|\xintZapSpacesB { \a { \X } { \b \Y } }->|%
-\digitstt{\color{magenta}{}\expandafter\detokenize\expandafter
-{\romannumeral0\expandafter\xintzapspacesb\expandafter{\x}}}+++}
-\def\x { { \a { \X } { \b \Y } } }
-\centeredline{|\xintZapSpacesB { { \a { \X } { \b \Y } } }->|%
-\digitstt{\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} } }}
-
-\centeredline{|\xintCSVtoList { 1 ,{ 2 , 3 , 4 , 5 }, a , {b,T} U , { c , d } ,
- { {x , y} } }|}
-\centeredline{|->|%
-{\makeatletter\digitstt{\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
-\digitstt{\expandafter\detokenize\expandafter{\romannumeral0\xintcsvtolist { }}}
-(a list with one empty item),
-for ``|<opt. spaces>{}<opt.
-spaces>|'' the output is
-\digitstt{\expandafter\detokenize\expandafter
- {\romannumeral0\xintcsvtolist { {} }}}
-(again a list with one empty item, the braces were removed),
-for ``|{ }|'' the output is
-\digitstt{\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
-\digitstt{\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
-\digitstt{\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
-\digitstt{\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}}
-
-\centeredline{|\def\y{ \a,\b,\c,\d,\e} \xintCSVtoList\y->|%
- {\makeatletter\digitstt{\expandafter\strip@prefix\meaning\Y}}}
-\centeredline{|\def\t {{\if},\ifnum,\ifx,\ifdim,\ifcat,\ifmmode}|} \centeredline
-{|\xintCSVtoList\t->|\makeatletter\digitstt{\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:
-\centeredline{|\expandafter\xintCSVtoListNoExpand\expandafter {\y}|} Else, we
-may have direct use: \centeredline{|\xintCSVtoListNoExpand
- {\if,\ifnum,\ifx,\ifdim,\ifcat,\ifmmode}|}
-\centeredline{|->|\digitstt{\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. \centeredline{|\xintNthElt
- {3}{{agh}\u{zzz}\v{Z}}| is \texttt{\xintNthElt {3}{{agh}\u{zzz}\v{Z}}}}
-\centeredline{|\xintNthElt {3}{{agh}\u{{zzz}}\v{Z}}| is
- \texttt{\expandafter\expandafter\expandafter
- \detokenize\expandafter\expandafter\expandafter {\xintNthElt
- {3}{{agh}\u{{zzz}}\v{Z}}}}} \centeredline{|\xintNthElt
- {2}{{agh}\u{{zzz}}\v{Z}}| is \texttt{\expandafter\expandafter\expandafter
- \detokenize\expandafter\expandafter\expandafter {\xintNthElt
- {2}{{agh}\u{{zzz}}\v{Z}}}}} \centeredline{|\xintNthElt {37}{\xintFac
- {100}}|\digitstt{=\xintNthElt {37}{\xintFac {100}}} is the thirty-seventh
- digit of @100!@.} \centeredline{|\xintNthElt {10}{\xintFtoCv
- {566827/208524}}|\digitstt{=\xintNthElt {10}{\xintFtoCv {566827/208524}}}}
-is the tenth convergent of @566827/208524@ (uses \xintcfracname package).
-\centeredline{|\xintNthElt {7}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}|%
- \digitstt{=\xintNthElt {7}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}}}%
-\centeredline{|\xintNthElt {0}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}|%
- \digitstt{=\xintNthElt {0}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}}}
-\centeredline{|\xintNthElt {-3}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}|%
- \digitstt{=\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 \texttt{|x|}th element from the end of the list.
- \centeredline{|\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{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
-\digitstt{\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).
-\centeredline{|\xintListWithSep{:}{\xintFac
- {20}}|\digitstt{=\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. \centeredline{|\def\macro #1{\the\numexpr
- 9-#1\relax}|} \centeredline{|\xintApply\macro{\xintFac
- {20}}|\digitstt{=\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}
-
-\def\macro #1{\expandafter\def\csname myself#1\endcsname {#1}}
-\xintApplyUnbraced\macro{{elta}{eltb}{eltc}}
-
-\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
-\centeredline{|\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.
-%
-\dverb|@
- \def\macro #1{\expandafter\def\csname myself#1\endcsname {#1}}
- \xintApplyUnbraced\macro{{elta}{eltb}{eltc}}
- \meaning\myselfelta: "meaning"myselfelta
- \meaning\myselfeltb: "meaning"myselfeltb
- \meaning\myselfeltc: "meaning"myselfeltc|
-
-%
-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|.
-
-
-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.
-
-\centeredline{|\xintListWithSep{,\hskip2pt
- plus 1pt minus 1pt }{\xintSeq {12}{-25}}|}
-\noindent\digitstt{\xintListWithSep{,\hskip2pt plus 1pt minus 1pt }{\xintSeq
- {12}{-25}}}
-\centeredline{|\xintiiSum{\xintSeq [3]{1}{1000}}|\digitstt{=\xintiiSum{\xintSeq [3]{1}{1000}}}}
-
-\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 \digitstt{\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:
-\dverb|@
-\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{Rem} 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:
-\dverb|@
-\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, 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:
-%
-\dverb|@
-\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:
-%
-\dverb|@
-\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:
-%
-\dverb|@
-\newcommand{\IsNotDivisibleBy}[2]{\ifnumequal{#1-(#1/#2)*#2}{0}{0}{1}}
-|%
-Let us enhance our prime macro to work also on the small primes:
-\dverb|@
-\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.
-%
-\dverb?@
-\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.}
-\begingroup
-\makeatother
-\begin{verbatim}
-\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 % etc...
-\[\begin{pmatrix}
- \A11&\A12&\A13&\A14\\
- \A21&\A22&\A23&\A24\\
- \A31&\A32&\A33&\A34
- \end{pmatrix}
-\times
- \begin{pmatrix}
- \B11&\B12&\B13\\
- \B21&\B22&\B23\\
- \B31&\B32&\B33\\
- \B41&\B42&\B43
- \end{pmatrix}
-=
-\begin{pmatrix}
- \C11&\C12&\C13\\
- \C21&\C22&\C23\\
- \C31&\C32&\C33
-\end{pmatrix}\]
-\[\begin{pmatrix}
- \C11&\C12&\C13\\
- \C21&\C22&\C23\\
- \C31&\C32&\C33
-\end{pmatrix}^2 = \begin{pmatrix}
- \D11&\D12&\D13\\
- \D21&\D22&\D23\\
- \D31&\D32&\D33
-\end{pmatrix}\]
-\end{verbatim}
-\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
-\setlength{\unitlength}{1cm}%
-% le picture de LaTeX est tout de même assez génial!
-\begin{picture}(0,0)
-\put(5,11){\vtop{\hsize8cm
-\[\begin{pmatrix}
- \A11&\A12&\A13&\A14\\
- \A21&\A22&\A23&\A24\\
- \A31&\A32&\A33&\A34
- \end{pmatrix}
-\times
- \begin{pmatrix}
- \B11&\B12&\B13\\
- \B21&\B22&\B23\\
- \B31&\B32&\B33\\
- \B41&\B42&\B43
- \end{pmatrix}
-=
-\begin{pmatrix}
- \C11&\C12&\C13\\
- \C21&\C22&\C23\\
- \C31&\C32&\C33
-\end{pmatrix}\]
-\[\begin{pmatrix}
- \C11&\C12&\C13\\
- \C21&\C22&\C23\\
- \C31&\C32&\C33
-\end{pmatrix}^2 = \begin{pmatrix}
- \D11&\D12&\D13\\
- \D21&\D22&\D23\\
- \D31&\D32&\D33
-\end{pmatrix}\]\MatrixMultiplication\C\D\E
-\[\begin{pmatrix}
- \C11&\C12&\C13\\
- \C21&\C22&\C23\\
- \C31&\C32&\C33
-\end{pmatrix}^3 = \begin{pmatrix}
- \E11&\E12&\E13\\
- \E21&\E22&\E23\\
- \E31&\E32&\E33
-\end{pmatrix}\]\MatrixMultiplication\C\E\F
-\[\begin{pmatrix}
- \C11&\C12&\C13\\
- \C21&\C22&\C23\\
- \C31&\C32&\C33
-\end{pmatrix}^4 = \begin{pmatrix}
- \F11&\F12&\F13\\
- \F21&\F22&\F23\\
- \F31&\F32&\F33
-\end{pmatrix}\]}}
-\end{picture}\par
-\endgroup
-
-\kern-2\baselineskip
-
-\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:
-%
-\centeredline{|\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:
-%
-\centeredline
-{|\expandafter\xintbreakiloopanddo\expandafter\macro\xintiloopindex.%|}
-\centeredline{|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:
-\centeredline
-{|\xint_afterfi{\expandafter\xintbreakiloopanddo\expandafter\macro\xintiloopindex.}%|}
-\centeredline{|\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:
-\dverb|@
-\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|
-\begingroup%\ttfamily
-\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 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.
-\endgroup
-
-
-Let us create an alignment where each row will contain all divisors of its
-first entry.
-\dverb|@
-\tabskip1ex
-\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 % \space is CRUCIAL
- \repeat \cr }%
- \ifnum\xintiloopindex<30
- \repeat }|
-
-%
-\noindent 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 is:
-%
-\dverb|@ \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 }|
-
-\noindent
-Here is the output, thus obtained without any count register:
-\begingroup\catcode`_ 11
-\begin{multicols}2
-\tabskip1ex
-\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}
-\endgroup
-
-\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.
-%
-\dverb?@
-\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
- \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}
-?
-\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 }
-
-\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}.
-
-\begingroup
-\def\MacroFont{\ttfamily\baselineskip12pt\relax \catcode`\" 12 }
-\dverb|@
-\tabskip1ex
-\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}
-}|\par\smallskip
-\endgroup
-
-The \hyperref[floatfactorize]{table} has been made into a float which appears
-\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 \texttt{\char92\string@firstoftwo} and
-\texttt{\char92\string@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.
-
-\begingroup
-\def\MacroFont{\ttfamily\baselineskip12pt\relax \catcode`\" 12 }
-\dverb|@
-\catcode`_ 11
-\def\abortfactorize #1\xint_secondoftwo\fi #2#3{\fi}
-
-\def\factorize #1.{\ifnum#1=1 \abortfactorize\fi
- % avoid overflow if #1="7FFFFFFF
- \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
- % this will avoid overflow which could result from #2*#2
- \ifnum\numexpr #1-(#2-1)*#2<#2
- #1\abortfactorize % this #1 is prime
- \fi
- % again, avoiding overflow as \numexpr integer division
- % rounds rather than truncates.
- \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|
-\endgroup
-
-\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}
-\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*}
-
-
-\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.
-
-\dverb|@
-\def\Macro #1{\advance\cnta #1 , \the\cnta}
-\cnta 0
-0\xintApplyInline\Macro {3141592653}.|
-\def\Macro #1{\advance\cnta #1 , \the\cnta}
-\cnta 0
-Output: 0\xintApplyInline\Macro {3141592653}.
-
-
-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 for example:\par
-\smallskip
-\centeredline
- {\begin{tabular}{ccc}
- $N$ & $N^2$ & $N^3$ \\ \hline
- \def\Row #1{ #1 & \xintiSqr {#1} & \xintiPow {#1}{3} \\ \hline }%
- \xintApplyInline \Row {\xintCSVtoList{17,28,39,50,61}}
- \end{tabular}}
-\smallskip
-% 38 = &, 43 = +, 36=$, 45 = -
-was obtained from the following input:
-\dverb|@
-\begin{tabular}{ccc}
- $N$ & $N^2$ & $N^3$ \\ \hline
- \def\Row #1{ #1 & \xintiSqr {#1} & \xintiPow {#1}{3} \\ \hline }%
- \xintApplyInline \Row {\xintCSVtoList{17,28,39,50,61}}
-\end{tabular}|%
-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
-\dverb|@
-\def\Row #1{#1:\xintApplyInline {\Item {#1}}{0123456789}\\ }%
-\def\Item #1#2{&\xintiPow {#1}{#2}}%
-\begin{tabular}{ccccccccccc}
- &0&1&2&3&4&5&6&7&8&9\\ \hline
- \xintApplyInline \Row {0123456789}
-\end{tabular}|
-\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*}
-
-
-One could not move the definition of |\Item| inside the tabular,
-as it would get lost after the first |&|. But this
-works:
-\dverb|@
-\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
-\dverb|@
- \def\Row #1{#1:\def\Item ##1{&\xintiPow {#1}{##1}}%
- \xintApplyInline \Item {0123456789}\\ }%
- \xintApplyInline \Row {0123456789} % does not work
-|%
-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:
-%
-\dverb|@
-\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 {%
- $$#9\times#1\times#3\times#2=\xintiiPrd{{#1}{#2}{#3}{#9}}$$}}}}
-|%
-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). \centeredline{|\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:
-%
-\dverb|@
- \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 .. }|
-
-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):
-
-\begingroup
-\centeredline{\begin{tabular}{rccccc}
- \xintFor #7 in {A,B,C} \do {%
- #7:\xintFor* #3 in {abcde} \do {&($ #3 \to #7 $)}\\ }%
-\end{tabular}}
-\endgroup
-
-\dverb|@
-\begin{tabular}{rccccc}
- \xintFor #7 in {A,B,C} \do {%
- #7:\xintFor* #3 in {abcde} \do {&($ #3 \to #7 $)}\\ }%
-\end{tabular}|
-
-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:
-%
-\dverb|@
-\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 |%
-\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)} }%
- }}%
-\centeredline{\T\def\sep {\def\sep{, }}\z} 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*} intially 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.
-
-\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
-% \xintNewNumExpr \FA [2] {{_DimToNum {$2}}^3/{_DimToNum {$1}}^2} %$
-% % square root
-% \xintNewNumExpr \FB [2] {sqrt ({_DimToNum {$2}}*{_DimToNum {$1}})}
-% \xintNewExpr \Ratio [2] {trunc({_DimToNum {$2}}/{_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}{\xintQuo {\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{1cm}%
-\scriptsize\def\MacroFont {\ttfamily\baselineskip8pt\relax}
-\begin{minipage}{\dimexpr\linewidth-3cm-\parindent\relax}
-\dverb|@
-\def\DimToNum #1{\number\dimexpr #1\relax }
-\xintNewNumExpr \FA [2] {{_DimToNum {$2}}^3/{_DimToNum {$1}}^2} % cube
-\xintNewNumExpr \FB [2] {sqrt ({_DimToNum {$2}}*{_DimToNum {$1}})} % sqrt
-\xintNewExpr \Ratio [2] {trunc({_DimToNum {$2}}/{_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
-|\par
-\end{minipage}}
-\end{figure*}
-
-% attention, pour le \meaning dans cette note de base de page
-
-The\xintNewNumExpr \FA [2] {{_DimToNum {$2}}^3/{_DimToNum {$1}}^2} %$
-\hyperlink{graphic}{graphic}, with the code on its right\footnote{the somewhat
- peculiar use of |\_| and |\$| is explained in \autoref{xintNewExpr}; they are
- made necessary from the fact that the parameters are passed to a \emph{macro}
- (\csa{DimToNum}) and not only to \emph{functions}, as are known to
- \hyperref[sec:exprsummary]{\csa{xintexpr}}. But one can also define directly
- the desired function, for example the constructed \csa{FA} turns out to have
- meaning \texttt{\meaning\FA}, where the \csa{romannumeral} part is only to
- ensure it expands in only two steps, and could be removed. A handwritten macro
- would use here \csa{xintiPow} and not \csa{xintPow}, as we know it has to deal
- with integers only. See 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}{\xintQuo {\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).
-
-\begingroup\small
-\noindent\dverb|@
-\xintFor #1 in {\xintrationals [10/21+1/21]} \do
-{#1=\xintifInt {#1}
- {\textcolor{blue}{\xintTrunc{10}{#1}}}
- {\xintTrunc{10}{#1}}% in blue if an integer
- \xintifGt {#1}{1.123}{\xintBreakFor}{, }%
-}|
-
-\smallskip
-\centeredline{\parbox{\dimexpr\linewidth-3em}{\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 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.
-
-\begingroup\footnotesize \def\MacroFont {\ttfamily\relax}
-\noindent\dverb|@
-\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
-
-\centeredline{\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
-
-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
-\endgroup
-
-
-\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}}}%
-}%
-
-\dverb|@
-\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}
-\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*}
-
-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 appears
-\vpageref[above]{primes}):
-\dverb?@
-\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*}?
-
-\subsection{Some arithmetic with Fibonacci numbers}
-\label{ssec:fibonacci}
-
-Here is again the code employed on the title page to compute Fibonacci numbers:
-
-\begingroup\footnotesize\baselineskip10pt
-\def\MacroFont {\ttfamily}
-\dverb|@
-\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
-\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}
-\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 }%
-|\par\endgroup
-
-\catcode`_ 11
-\def\Fibonacci_end_i #1#2#3#4#5{{#4}{#5}}%
-\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.
-% \Fibonacci returns {F(N+1)}{F(N)} (both in \xintexpr encapsulation)
-% \FibonacciN returns F(N) (also in encapsulated format)
-\def\FibonacciN {\expandafter\xint_secondoftwo\romannumeral-`0\Fibonacci }%
-\catcode`_ 8
-
-% 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, as I now 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\let\MacroFont\ttfamily
-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}).
-%
-\dverb|@
-\def\Fibo #1.{\xintthe\FibonacciN {#1}}%
-\message{\xintiloop [0+1] \expandafter\Fibo\xintiloopindex.,
- \ifnum\xintiloopindex<49 \repeat \xintthe\FibonacciN{50}.}|\par
-\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.\par
- Technical note: I decided (somewhat hesitantly) for
- reasons of optimization purposes to skip in the private \csa{xintexpr}
- format a \csa{protect}-ion for the \csa{.=digits/digits[digits]}
- control sequences used internally. Thus in the improbable case that
- some macro package (such control sequence names are unavailable to the
- casual user) has given a meaning to one such control sequence, there
- is a possibility of a crash when embedding an \csa{xintexpr} without
- \csa{xintthe} prefix in an \csa{edef} (the computations by themselves
- do proceed perfectly correctly even if these control sequences have
- acquired some non \csa{relax} meaning).}
-
-Our |\Fibonacci| expands completely under \fexpan sion,
-so we can use \hyperref[fdef]{\ttfamily\char92fdef} rather than |\edef| in a
-situation such as \centeredline {|\fdef \X {\FibonacciN {100}}|} but for the
-reasons explained above, it is as efficient to employ |\edef|. And if we want
-\centeredline{|\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
-\def\MacroFont {\ttfamily}
-\dverb|@
-\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
-|\par\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 &
- \xintRem{\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 &
- \xintRem{\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 &
- \xintRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}%
-}}%
-%
-\centeredline{Some Fibonacci numbers together with their residues modulo
- |F(30)|\digitstt{=\xintthe\Fibxxx}}
-\end{figure*}
-
-\begingroup\footnotesize\baselineskip10pt
-\def\MacroFont {\ttfamily}
-\dverb|@
-\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 &
- \xintRem{\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 &
- \xintRem{\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 &
- \xintRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}%
-}%
-|\par\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.
-%
-\dverb|@
-\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}|%
-\centeredline{\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}}
-
-\smallskip 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 \xintiPow {7}{13}\to\SevenToThePowerThirteen
-\xintAssign \xintDivision{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}.
-\centeredline{|\xintAssign \xintDivision{1000000000000}{133333333}\to\Q\R|}
-\centeredline{|\meaning\Q: |\digitstt{\meaning\Q}, |\meaning\R:|
- \digitstt{\meaning\R}} \centeredline{|\xintAssign \xintiPow
- {7}{13}\to\SevenToThePowerThirteen|}
-\centeredline{|\SevenToThePowerThirteen|\digitstt{=\SevenToThePowerThirteen}}
-\centeredline{(same as |\edef\SevenToThePowerThirteen{\xintiPow {7}{13}}|)}
-
-
-\noindent\csa{xintAssign}\MyMarginNote{Changed!} 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|.
-
-% 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}|.
-\centeredline{|\xintAssignArray \xintBezout {1000}{113}\to\Bez|} will set
-|\Bez{0}| to \digitstt{\Bez0}, |\Bez{1}| to \digitstt{\Bez1}, |\Bez{2}| to
-\digitstt{\Bez2}, |\Bez{3}| to \digitstt{\Bez3}, |\Bez{4}| to
-\digitstt{\Bez4}, and |\Bez{5}| to \digitstt{\Bez5}:
-\digitstt{(\Bez3)${}\times{}$\Bez1${}-{}$(\Bez4)${}\times{}$\Bez2${}={}$\Bez5.}
-This macro is incompatible with expansion-only contexts.
-
-\csa{xintAssignArray}\MyMarginNote{Changed!} 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{\csbh{odef}, \csbh{oodef}, \csbh{fdef}}
-\label{odef}
-\label{oodef}
-\label{fdef}
-
-\csa{oodef}|\controlsequence {<stuff>}| does
-\dverb|@
- \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:
-\dverb|@
-\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 \xinttoolsname finds them currently undefined. They
-can be prefixed with |\global|.
-
-
-\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}.
-
-\begingroup\makeatletter\let\check@percent\relax
-\def\MacroFont{\small\baselineskip12pt \ttfamily }
-\begin{verbatim}
-% THE QUICK SORT ALGORITHM EXPANDABLY
-\input xintfrac.sty
-% HELPER COMPARISON MACROS
-\def\QSMore #1#2{\xintifGt {#2}{#1}{{#2}}{ }}
-% the spaces are there to stop the \romannumeral-`0 originating
-% in \xintapplyunbraced when it applies a macro to an item
-\def\QSEqual #1#2{\xintifEq {#2}{#1}{{#2}}{ }}
-\def\QSLess #1#2{\xintifLt {#2}{#1}{{#2}}{ }}
-%
-\makeatletter
-\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}
-% for simplicity of implementation, we pick up the first item as pivot
-\def\qsfull@c #1{\qsfull@ci #1\undef {#1}}
-\def\qsfull@ci #1#2\undef {\qsfull@d {#1}}% #3 is the list, #1 its first item
-\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}
-\makeatother
-% EXAMPLE
-\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}}}
-\tt\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
-\meaning\z
-\end{verbatim}
-
-% THE QUICK SORT ALGORITHM EXPANDABLY
-\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}
-\makeatother
-% EXAMPLE
-\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}}}
-\noindent Output:\par
-\texttt{\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
-\texttt{\printnumber{\meaning\z}}
-\endgroup
-
-
-
-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.
-
-\begingroup
-\makeatletter
-\let\check@percent\relax
-% il utilise MacroFont
-\def\MacroFont{\small\baselineskip 12pt \ttfamily }
-\begin{verbatim}
-\input xintfrac.sty % if Plain TeX
-%
-\definecolor{LEFT}{RGB}{216,195,88}
-\definecolor{RIGHT}{RGB}{208,231,153}
-\definecolor{INERT}{RGB}{199,200,194}
-\definecolor{PIVOT}{RGB}{109,8,57}
-%
-\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}}{ }}
-%
-\makeatletter
-\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}}%
-% Here \QSLr, \QSIr, \QSr have been let to \relax, so expansion stops.
-% #2= elements < pivot, #1 = elements = pivot, #3 = elements > pivot
-\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
-% \QS@list \par
-\par\centerline{\QS@list}
-}
-\def\QSoneStep {\let\QSLr\DecoLEFTwithPivot
- \let\QSIr\DecoINERT
- \let\QSRr\DecoRIGHTwithPivot
-% \QS@list
-\centerline{\QS@list}
-% \par
- \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
-% \QS@list
-\centerline{\QS@list}
-% \par
-}
-\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{verbatim}
-
-\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
-% \QS@list \par
-\par\centerline{\QS@list}
-}
-\def\QSoneStep {\let\QSLr\DecoLEFTwithPivot
- \let\QSIr\DecoINERT
- \let\QSRr\DecoRIGHTwithPivot
-% \QS@list
-\centerline{\QS@list}
-% \par
- \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
-% \QS@list
-\centerline{\QS@list}
-% \par
-}
-
-\phantomsection\label{quicksort}
-\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
-
-
-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{verbatim}
-\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
-% \QS@list \par
-\par\centerline{\QS@list}
-}
-\end{verbatim}
-\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
-% \QS@list \par
-\par\centerline{\QS@list}
-}
-\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
-
-\endgroup
-
-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 \xintname package}
-\label{sec:xint}
-
-
-In the description of the macros \texttt{\n} and \texttt{\m} stand for (long)
-numbers within braces or for a control sequence possibly within braces and
-\hyperref[sec:expansions]{\fexpan ding} to such a number (without the braces!),
-or for material within braces which \fexpan ds to such a number, as is
-acceptable on input by the \csbxint{Num} macro: a sequence of plus and minus
-signs, followed by some string of zeroes, followed by digits. The margin
-annotation for such an argument which is parsed by \csbxint{Num} is
-\textcolor[named]{PineGreen}{\Numf}. Sometimes however only a
-\textcolor[named]{PineGreen}{\emph{f}} symbol appears in the margin, signaling
-that the input will not be parsed via \csbxint{Num}.
-
-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 argument to the package macros, see the
-\hyperref[sec:useofcount]{Use of count} section.
-
-Some of these macros are extended by \xintfracname to accept fractions
-on input, and, generally, to output a fraction. But this means that
-additions, subtractions, multiplications output in fraction format; to
-guarantee the integer format on output when the inputs are integers, the
-original integer-only macros \csa{xintAdd}, \csa{xintSub},
-\csa{xintMul}, etc\dots are available under the names \csa{xintiAdd},
-\csa{xintiSub}, \csa{xintiMul}, \dots, also when \xintfracname is not
-loaded. Even these originally integer-only macros will accept fractions
-on input if \xintfracname is loaded as long as they are integers in
-disguise; they produce on output integers without any forward
-slash mark nor trailing |[n]|.
-
-But |\xintAdd| will output fractions |A/B[n]|, with |B| present even if its
-value is one. See the \xintfracname \hyperref[sec:frac]{documentation} for
-additional information.
-
-% on how macros of \xintname are modified after loading
-% \xintfracname (or \xintexprname).
-
-
-% \xintfracname will extend \csbxint{Num} for it to remove this unit
-% denominator and convert the |[n]| part into explicit zeros; see also
-% \csbxint{PRaw} which does not make the assumption that the fraction is an
-% integer in disguise.
-
-% This is mandatory when the computation result is fetched
-% into a context where \TeX{} expects a number (assuming it does not exceed
-% @2^31@). See the also the \xintfracname \hyperref[sec:frac]{documentation} for
-% more information on how macros of \xintname are modified after loading
-% \xintfracname (or \xintexprname).
-
-
-% Package \xintname also provides some general macro programming or token
-% manipulation utilities (expandable as well as non-expandable), which are
-% described in the next section (\autoref{sec:tools}).
-
-\localtableofcontents
-
-\subsection{\csbh{xintRev}} \label{xintRev}
-
-\csa{xintRev\n}\etype{f} will revert 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.
-\centeredline{|\xintRev{-123000}|\digitstt{=\xintRev{-123000}}}
-\centeredline{|\xintNum{\xintRev{-123000}}|%
- \digitstt{=\xintNum{\xintRev{-123000}}}}
-
-
-\subsection{\csbh{xintLen}}\label{xintiLen}
-
-\csa{xintLen\n}\etype{\Numf} returns the length of the number, not counting the
-sign. \centeredline{|\xintLen{-12345678901234567890123456789}|\digitstt
- {=\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). \centeredline{|\xintLen{-1e3/5.425}|\digitstt
- {=\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}|\digitstt{=\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
-\centeredline{|\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{xintNum}}\label{xintiNum}
-
-\csa{xintNum\n}\etype{f} removes chains of plus or minus signs, followed by
-zeroes. \centeredline{|\xintNum{+---++----+--000000000367941789479}|\digitstt
- {=\xintNum{+---++----+--000000000367941789479}}} Extended by \xintfracname to
-accept also a fraction on input, as long as it reduces to an integer after
-division of the numerator by the denominator.
-\centeredline{|\xintNum{123.48/-0.03}|\digitstt{=\xintNum{123.48/-0.03}}}
-
-
-\subsection{\csbh{xintSgn}}\label{xintiiSgn}
-
-\csa{xintSgn\n}\etype{\Numf} returns 1 if the number is positive, 0 if it is
-zero and -1 if it is negative. Extended by \xintfracname to fractions.
-\csbxint{iiSgn} skips the \csbxint{Num} overhead.\etype{f}
-
-\subsection{\csbh{xintOpp}}\label{xintiOpp}\label{xintiiOpp}
-
-\csa{xintOpp\n}\etype{\Numf} return the opposite |-N| of the number |N|.
-Extended by \xintfracname to fractions. \csa{xintiOpp} is a synonym not modified
-by \xintfracname\footnote{here, and in all similar instances, this means that
- the macro remains integer-only both on input and output, but it does accept on
- input a fraction which in disguise is a (big) integer.}, and
-\csa{xintiiOpp} skips the \csbxint{Num} overhead.\etype{f}
-
-
-\subsection{\csbh{xintAbs}}\label{xintiAbs}\label{xintiiAbs}
-
-\csa{xintAbs\n}\etype{\Numf} returns the absolute value of the number. Extended
-by \xintfracname to fractions. \csa{xintiAbs} is a synonym not modified
-by \xintfracname, and \csa{xintiiAbs} skips the \csbxint{Num} overhead.\etype{f}
-
-
-\subsection{\csbh{xintAdd}}\label{xintiAdd}\label{xintiiAdd}
-
-\csa{xintAdd\n\m}\etype{\Numf\Numf} returns the sum of the two numbers. Extended
-by \xintfracname to fractions. \csa{xintiAdd} is a synonym not modified by
-\xintfracname, and \csa{xintiiAdd} skips the \csbxint{Num} overhead.\etype{ff}
-
-
-
-\subsection{\csbh{xintSub}}\label{xintiSub}\label{xintiiSub}
-
-\csa{xintSub\n\m}\etype{\Numf\Numf} returns the difference |N-M|. Extended
-by \xintfracname to fractions. \csa{xintiSub} is a synonym not modified by
-\xintfracname, and \csa{xintiiSub} skips the \csbxint{Num} overhead.\etype{ff}
-
-
-\subsection{\csbh{xintCmp}}\label{xintiCmp}
-
-\csa{xintCmp\n\m}\etype{\Numf\Numf} returns 1 if |N>M|, 0 if |N=M|, and -1
-if |N<M|. Extended by \xintfracname to fractions.
-
-\subsection{\csbh{xintEq}}\label{xintEq}
-%{\small New with release |1.09a|.\par}
-
-\csa{xintEq\n\m}\etype{\Numf\Numf} returns 1 if |N=M|, 0 otherwise. Extended
-by \xintfracname to fractions.
-
-\subsection{\csbh{xintGt}}\label{xintGt}
-%{\small New with release |1.09a|.\par}
-
-\csa{xintGt\n\m}\etype{\Numf\Numf} returns 1 if |N|$>$|M|, 0 otherwise.
-Extended by \xintfracname to fractions.
-
-\subsection{\csbh{xintLt}}\label{xintLt}
-%{\small New with release |1.09a|.\par}
-
-\csa{xintLt\n\m}\etype{\Numf\Numf} returns 1 if |N|$<$|M|, 0 otherwise.
-Extended by \xintfracname to fractions.
-
-\subsection{\csbh{xintIsZero}}\label{xintIsZero}
-%{\small New with release |1.09a|.\par}
-
-\csa{xintIsZero\n}\etype{\Numf} returns 1 if |N=0|, 0 otherwise.
-Extended by \xintfracname to fractions.
-
-\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}}\label{xintIsNotZero}
-%{\small New with release |1.09a|.\par}
-
-\csa{xintIsNotZero\n}\etype{\Numf} returns 1 if |N<>0|, 0 otherwise.
-Extended by \xintfracname to fractions.
-
-\subsection{\csbh{xintIsOne}}\label{xintIsOne}
-%{\small New with release |1.09a|.\par}
-
-\csa{xintIsOne\n}\etype{\Numf} returns 1 if |N=1|, 0 otherwise.
-Extended by \xintfracname to fractions.
-
-\subsection{\csbh{xintAND}}\label{xintAND}
-%{\small New with release |1.09a|.\par}
-
-\csa{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}
-
-\csa{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}
-
-\csa{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}
-
-\csa{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|)
-Please note that the macro compares
-\emph{absolute values}.
-
-\subsection{\csbh{xintMax}}\label{xintiMax}
-
-\csa{xintMax\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}|\digitstt{=\xintiMax{-5}{-6}}. Extended by \xintfracname to
-fractions. \csa{xintiMax} is a synonym not modified by
-\xintfracname.
-
-\subsection{\csbh{xintMaxof}}\label{xintMaxof}
-%{\small New with release |1.09a|.\par}
-
-\csa{xintMaxof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns the maximum.
-The list argument may be a macro, it is \fexpan ded first. Extended by
-\xintfracname to fractions. \csa{xintiMaxof} is a
-synonym not modified by \xintfracname.
-
-
-\subsection{\csbh{xintMin}}\label{xintiMin}
-
-\csa{xintMin\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}|\digitstt{=\xintiMin{-5}{-6}}. Extended by \xintfracname to fractions.
-\csa{xintiMin} is a synonym not modified by
-\xintfracname.
-
-\subsection{\csbh{xintMinof}}\label{xintMinof}
-%{\small New with release |1.09a|.\par}
-
-\csa{xintMinof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns the minimum.
-The list argument may be a macro, it is \fexpan ded first. Extended by
-\xintfracname to fractions. \csa{xintiMinof} is a synonym not modified by
-\xintfracname.
-
-
-\subsection{\csbh{xintSum}}\label{xintiiSum}
-
-\csa{xintSum}\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}.
-
-\csa{xintSum} is
-extended by \xintfracname to fractions. The original, which accepts (after
-\fexpan sion) only (big) integers in the strict format and produces a (big)
-integer is available as \csa{xintiiSum}, also with \xintfracname loaded.
-
-\centeredline{%
- \csa{xintiiSum}|{{123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}}|%
- \digitstt{=\xintiiSum{{123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}}}}
-\centeredline{\csa{xintiiSum}|{1234567890}|\digitstt{=\xintiiSum{1234567890}}}
-An empty sum is no error and returns zero: |\xintiiSum
-{}|\digitstt{=\xintiiSum {}}. A sum with only one term returns that
-number: |\xintiiSum {{-1234}}|\digitstt{=\xintiiSum {{-1234}}}.
-Attention that |\xintiiSum {-1234}| is not legal input and will make the
-\TeX{} run fail. On the other hand |\xintiiSum
-{1234}|\digitstt{=\xintiiSum{1234}}. Extended by \xintfracname to
-fractions.
-
-% retiré de la doc le 22 octobre 2013
-
-% \subsection{\csbh{xintSumExpr}}\label{xintiiSumExpr}
-
-% \csa{xintSumExpr}\meta{braced things}\csa{relax} is to what \csa{xintSum}
-% expands. The argument is then expanded (with the usual meaning) and should give
-% a list of braced quantities or macros, each one will be expanded in turn.
-% \centeredline{%
-% \csa{xintiiSumExpr}| {123}{-98763450}|%
-% |{\xintFac{7}}{\xintiMul{3347}{591}}\relax|\digitstt{=%
-% \xintiiSumExpr {123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}\relax}}
-
-% Note: I am not so happy with the name which seems to suggest that the
-% |+| sign should be used instead of braces. Perhaps this will change
-% in the future.
-
-% Extended by \xintfracname to fractions.
-
-\subsection{\csbh{xintMul}}\label{xintiMul}\label{xintiiMul}
-%{\small Modified in release |1.03|.\par}
-
-\csa{xintMul\n\m}\etype{\Numf\Numf} returns the product of the two numbers.
-% Starting with release |1.03| of \xintname, the macro checks the lengths of the
-% two numbers and then activates its algorithm with the best (or at least,
-% hoped-so) choice of which one to put first. This makes the macro a bit slower
-% for numbers up to 50 digits, but may give substantial speed gain when one of the
-% number has 100 digits or more.
-Extended by \xintfracname to fractions.
-\csa{xintiMul} is a synonym not modified by \xintfracname, and
-\csa{xintiiMul} skips the \csbxint{Num} overhead.\etype{ff}
-
-\subsection{\csbh{xintSqr}}\label{xintiSqr}\label{xintiiSqr}
-
-\csa{xintSqr\n}\etype{\Numf} returns the square. Extended by \xintfracname to
-fractions. \csa{xintiSqr} is a synonym not modified by
-\xintfracname, and \csa{xintiiSqr} skips the \csbxint{Num} overhead.\etype{f}
-
-
-
-\subsection{\csbh{xintPrd}}\label{xintiiPrd}
-
-\csa{xintPrd}\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}.
-\centeredline{%
- \csa{xintiiPrd}|{{-9876}{\xintFac{7}}{\xintiMul{3347}{591}}}|%
- \digitstt{=%
- \xintiiPrd{{-9876}{\xintFac{7}}{\xintiMul{3347}{591}}}}}
-\centeredline{\csa{xintiiPrd}|{123456789123456789}|\digitstt{=%
- \xintiiPrd{123456789123456789}}} An empty product is no error and returns 1:
-|\xintiiPrd {}|\digitstt{=\xintiiPrd {}}. A product reduced to a single term
-returns this number: |\xintiiPrd {{-1234}}|\digitstt{=\xintiiPrd {{-1234}}}.
-Attention that |\xintiiPrd {-1234}| is not legal input and will make the \TeX{}
-compilation fail. On the other hand |\xintiiPrd {1234}|\digitstt{=\xintiiPrd
- {1234}}. \centeredline{$\displaystyle 2^{200}3^{100}7^{100}$}
-\centeredline{|=\xintiiPrd {{\xintiPow {2}{200}}{\xintiPow {3}{100}}{\xintiPow
- {7}{100}}}|}
-\digitstt{=\printnumber{\xintNum {\xinttheexpr
- 2^200*3^100*7^100\relax }}}
-
-With \xintexprname, the above could be coded simply as \centeredline
-{|\xinttheiiexpr 2^200*3^100*7^100\relax |}
-
-Extended by \xintfracname to fractions. The original, which accepts (after
-\fexpan sion) only (big) integers in the strict format and produces a (big)
-integer is available as \csbxint{iiPrd}, also with \xintfracname loaded.
-
-
-% I temporarily remove mention of \xintPrdExpr from the documentation; I
-% really dislike the name now.
-
-% \subsection{\csbh{xintPrdExpr}}\label{xintiiPrdExpr}
-
-% {\small Name change in |1.06a|! I apologize, but I suddenly decided that
-% \csa{xintProductExpr} was a bad choice; so I just replaced it by the current
-% name. \par}
-
-% \csa{xintPrdExpr}\marg{argument}\csa{relax} is to what \csa{xintPrd} expands
-% ; its argument is expanded (with the usual meaning) and should give a list of
-% braced numbers or macros. Each will be expanded when it is its turn.
-% \centeredline{\csa{xintiiPrdExpr}| 123456789123456789\relax|\digitstt{=%
-% \xintiiPrdExpr 123456789123456789\relax}}
-
-% Note: I am not so happy with the name which seems to suggest that the
-% |*| sign should be used instead of braces. Perhaps this will change
-% in the future.
-
-% Extended by \xintfracname to fractions.
-
-\subsection{\csbh{xintPow}}\label{xintiPow}\label{xintiiPow}
-
-\csa{xintPow\n\x}\etype{\Numf\numx} returns |N^x|. When |x| is zero, this is 1.
-If |N| is zero and |x<0|, if \verb+|N|>1+ and |x<0| negative, or if \verb+|N|>1+
-and |x>100000|,\MyMarginNote{Changed!} then an error is raised. Indeed |2^50000|
-already has \digitstt{\xintLen{\xintFloatPow [1]{2}{50000}}} digits; each exact
-multiplication of two one thousand digits numbers already takes a few seconds,
-and it would take hours for the expandable computation to conclude with two
-numbers with each circa @15000@ digits. Perhaps some completely expandable but
-not \fexpan dable variants could fare better?
-
-Extended by \xintfracname to fractions (\csbxint{Pow}) and to floats
-(\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 cause errors anymore. The float
-version is able to deal with things such as |2^999999999| without any problem.
-For example |\xintFloatPow[4]{2}{50000}|\digitstt{=\xintFloatPow[4]{2}{50000}}
-and |\xintFloatPow[4]{2}{999999999}|
-\digitstt{=\xintFloatPow[4]{2}{999999999}}.\footnote{On my laptop
- |\string\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 $\pm$\digitstt{9999}.}
-
-\csa{xintiPow} is a synonym not modified by \xintfracname, and \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.
-
-Within an \csbxint{iiexpr}|..\relax| the infix operator |^| is mapped to
-\csa{xintiiPow}; within an \csbxint{expr}-ession\MyMarginNote{corr. of the
- previous doc.} it is mapped to \csbxint{Pow}
-(as extended by \xintfracname); in \csbxint{floatexpr}, it is mapped to
-\csbxint{FloatPower}.
-
-
-
-
-\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| (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}}\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.
-
-\subsection{\csbh{xintifZero}}\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.
-
-\subsection{\csbh{xintifNotZero}}\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.
-
-\subsection{\csbh{xintifOne}}\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.
-
-
-\subsection{\csbh{xintifTrueAelseB}, \csbh{xint\-ifFalseAelseB}}
-\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}}\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.
-
-\subsection{\csbh{xintifEq}}\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.
-
-\subsection{\csbh{xintifGt}}\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.
-
-\subsection{\csbh{xintifLt}}\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.
-
-\begin{framed}
- The macros described next are all integer-only on input. With \xintfracname
- loaded their argument is first given to \csbxint{Num} and may thus be
- a fraction, as long as it is in fact an integer in disguise.
-\end{framed}
-
-\subsection{\csbh{xintifOdd}}\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.
-
-
-\subsection{\csbh{xintFac}}\label{xintiFac}
-
-\csa{xintFac\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.
-
-With \xintfracname loaded, the macro is modified to accept a fraction as
-argument, as long as this fraction turns out to be an integer: |\xintFac
-{66/3}|\digitstt{=\xintFac {66/3}}. \csa{xintiFac} is a synonym not modified by
-the loading of \xintfracname.
-
-% the construct |\xintFac{\xintAdd {2}{3}}| will fail,
-% use either |\xintFac{\xintiAdd {2}{3}}| or |\xintFac{\xintNum{\xintAdd
-% {2}{3}}}|.
-
-% 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{xintDivision}}\label{xintDivision}\label{xintiiDivision}
-
-\csa{xintDivision\n\m}\etype{\Numf\Numf} 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}|. The variant \csa{xintiiDivision}\etype{ff}
-skips the overhead of parsing via \csbxint{Num}.
-
-This macro is integer only (with \xintfracname loaded it accepts
-fractions on input, but they must be integers in disguise) and not to be
-confused with the \xintfracname macro \csbxint{Div} which divides one
-fraction by another.
-
-\subsection{\csbh{xintQuo}}\label{xintQuo}\label{xintiiQuo}
-
-\csa{xintQuo\n\m}\etype{\Numf\Numf} returns the quotient from the euclidean
-division. When both |N| and |M| are positive one has
-\csa{xintQuo\n\m}|=\xintiTrunc {0}{N/M}| (using package \xintfracname). With
-\xintfracname loaded it accepts fractions on input, but they must be integers in
-disguise. The variant \csa{xintiiQuo}\etype{ff}
-skips the overhead of parsing via \csbxint{Num}.
-
-\subsection{\csbh{xintRem}}\label{xintRem}\label{xintiiRem}
-
-\csa{xintRem\n\m}\etype{\Numf\Numf} returns the remainder from the euclidean
-division. With \xintfracname loaded it accepts fractions on input, but they must
-be integers in disguise. The variant \csa{xintiiRem}\etype{ff}
-skips the overhead of parsing via \csbxint{Num}.
-
-
-\subsection{\csbh{xintFDg}}\label{xintFDg}\label{xintiiFDg}
-
-\csa{xintFDg\n}\etype{\Numf} returns the first digit (most significant) of the
-decimal expansion. The variant \csa{xintiiFDg}\etype{f}
-skips the overhead of parsing via \csbxint{Num}.
-
-\subsection{\csbh{xintLDg}}\label{xintLDg}\label{xintiiLDg}
-
-\csa{xintLDg\n}\etype{\Numf} returns the least significant digit. When the
-number is positive, this is the same as the remainder in the euclidean division
-by ten. The variant \csa{xintiiLDg}\etype{f}
-skips the overhead of parsing via \csbxint{Num}.
-
-\subsection{\csbh{xintMON}, \csbh{xintMMON}}
-\label{xintMON}\label{xintMMON}\label{xintiiMON}\label{xintiiMMON}
-%{\small New in version |1.03|.\par}
-
-\csa{xintMON\n}\etype{\Numf} returns |(-1)^N| and \csa{xintMMON\n} returns
-|(-1)^{N-1}|. \centeredline{|\xintMON {-280914019374101929}|\digitstt{=\xintMON
- {280914019374101929}}, |\xintMMON {-280914019374101929}|\digitstt{=\xintMMON
- {280914019374101929}}}
-The variants \csa{xintiiMON}\etype{f} and \csa{xintiiMMON}
-skip the overhead of parsing via \csbxint{Num}.
-
-\subsection{\csbh{xintOdd}}\label{xintOdd}\label{xintiiOdd}
-
-\csa{xintOdd\n}\etype{\Numf} is 1 if the number is odd and 0 otherwise. The
-variant \csa{xintiiOdd} skip the overhead of parsing via \csbxint{Num}.\etype{f}
-
-
-
-\subsection{\csbh{xintiSqrt}, \csbh{xintiSquareRoot}}\label{xintiSqrt}
-\label{xintiSquareRoot}
-%{\small New with |1.08|.\par}
-
-\xintAssign\xintiSquareRoot {17000000000000000000000000}\to\A\B
-
-\noindent\csa{xintiSqrt\n}\etype{\Numf} returns the largest integer whose
-square is at most equal to |N|. \centeredline{|\xintiSqrt
- {2000000000000000000000000000000000000}=|%
- \digitstt{\xintiSqrt{2000000000000000000000000000000000000}}}
-\centeredline{|\xintiSqrt {3000000000000000000000000000000000000}=|%
- \digitstt{\xintiSqrt{3000000000000000000000000000000000000}}}
-\centeredline{|\xintiSqrt {\xintDSH {-120}{2}}=|}%
-\centeredline{\digitstt{\xintiSqrt {\xintDSH {-120}{2}}}}
-\csa{xintiSquareRoot\n}\etype{\Numf} returns |{M}{d}| with |d>0|, |M^2-d=N| and
-|M| smallest (hence |=1+|\csa{xint\-iSqrt}|{N}|).
-\centeredline{|\xintAssign\xintiSquareRoot
- {17000000000000000000000000}\to\A\B|}%
-\centeredline{|\xintiSub{\xintiSqr\A}\B=\A^2-\B|}%
-\centeredline{\digitstt{\xintiSub{\xintiSqr\A}\B=\A\string^2-\B}} 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}.
-\end{framed}
-
-\subsection{\csbh{xintInc}, \csbh{xintDec}}
-\label{xintInc}
-\label{xintDec}
-%{\small New with |1.08|.\par}
-
-\csa{xintInc\n}\etype{f} is |N+1| and \csa{xintDec\n} is |N-1|. These macros
-remain integer-only, even with \xintfracname loaded.
-
-\subsection{\csbh{xintDouble}, \csbh{xintHalf}}
-\label{xintDouble}
-\label{xintHalf}
-%{\small New with |1.08|.\par}
-
-\csa{xintDouble\n}\etype{f} returns |2N| and \csa{xintHalf\n} is |N/2| rounded
-towards zero. These macros remain integer-only, even with \xintfracname loaded.
-
-\subsection{\csbh{xintDSL}}\label{xintDSL}
-
-\csa{xintDSL\n}\etype{f} is decimal shift left, \emph{i.e.} multiplication by
-ten.
-
-\subsection{\csbh{xintDSR}}\label{xintDSR}
-
-\csa{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}
-
-\csa{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}
-
-\csa{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
-\csa{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{xintDivision}),
-\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|.
-
-
-\csa{xintDSx\x\n}\etype{\numx f} for |x| negative is exactly as
-\csa{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 \csa{xintDSH\x\n}, and |R| is \csa{xintDSHr\x\n}, but computed
-simultaneously.
-
-\begin{flushleft}
- \xintAssign\xintDSx {-1}{-123456789}\to\M
- \noindent{|\xintAssign\xintDSx {-1}{-123456789}\to\M|}\\
- |\meaning\M: |\digitstt{\meaning\M}.\\
- \xintAssign\xintDSx {-20}{1234567689}\to\M
- {|\xintAssign\xintDSx {-20}{123456789}\to\M|}\\
- |\meaning\M: |\digitstt{\meaning\M}.\\
- \xintAssign\xintDSx{0}{-123004321}\to\Q\R
- {|\xintAssign\xintDSx {0}{-123004321}\to\Q\R|}\\
- \noindent|\meaning\Q: |\digitstt{\meaning\Q}, |\meaning\R:
- |\digitstt{\meaning\R.}\\
- |\xintDSH {0}{-123004321}|\digitstt{=\xintDSH {0}{-123004321}},
- |\xintDSHr {0}{-123004321}|\digitstt{=\xintDSHr {0}{-123004321}}\\
- \xintAssign\xintDSx {6}{-123004321}\to\Q\R
- {|\xintAssign\xintDSx {6}{-123004321}\to\Q\R|}\\
- |\meaning\Q: |\digitstt{\meaning\Q},
- |\meaning\R: |\digitstt{\meaning\R.}\\
- |\xintDSH {6}{-123004321}|\digitstt{=\xintDSH {6}{-123004321}},
- |\xintDSHr {6}{-123004321}|\digitstt{=\xintDSHr {6}{-123004321}}\\
- \xintAssign\xintDSx {8}{-123004321}\to\Q\R
- {|\xintAssign\xintDSx {8}{-123004321}\to\Q\R|}\\
- |\meaning\Q: |\digitstt{\meaning\Q},
- |\meaning\R: |\digitstt{\meaning\R.} \\
- |\xintDSH {8}{-123004321}|\digitstt{=\xintDSH {8}{-123004321}},
- |\xintDSHr {8}{-123004321}|\digitstt{=\xintDSHr {8}{-123004321}}\\
- \xintAssign\xintDSx {9}{-123004321}\to\Q\R
- {|\xintAssign\xintDSx {9}{-123004321}\to\Q\R|}\\
- |\meaning\Q: |\digitstt{\meaning\Q},
- |\meaning\R: |\digitstt{\meaning\R.}\\
- |\xintDSH {9}{-123004321}|\digitstt{=\xintDSH {9}{-123004321}},
- |\xintDSHr {9}{-123004321}|\digitstt{=\xintDSHr {9}{-123004321}}\\
-\end{flushleft}
-
-\subsection{\csbh{xintDecSplit}}\label{xintDecSplit}
-
-%{\small This has been modified in release |1.01|.\par}
-
-\csa{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
-\centeredline{|\xintAssign\xintDecSplit {0}{-123004321}\to\L\R|}
-\noindent|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.}
-\xintAssign\xintDecSplit {5}{-123004321}\to\L\R
-\centeredline{|\xintAssign\xintDecSplit {5}{-123004321}\to\L\R|}
-|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.}
-\xintAssign\xintDecSplit {9}{-123004321}\to\L\R
-\centeredline{|\xintAssign\xintDecSplit {9}{-123004321}\to\L\R|}
-|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.}
-\xintAssign\xintDecSplit {10}{-123004321}\to\L\R
-\centeredline{|\xintAssign\xintDecSplit {10}{-123004321}\to\L\R|}
-|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.}
-\xintAssign\xintDecSplit {-5}{-12300004321}\to\L\R
-\centeredline{|\xintAssign\xintDecSplit {-5}{-12300004321}\to\L\R|}
-|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.}
-\xintAssign\xintDecSplit {-11}{-12300004321}\to\L\R
-\centeredline{|\xintAssign\xintDecSplit {-11}{-12300004321}\to\L\R|}
-|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.}
-\xintAssign\xintDecSplit {-15}{-12300004321}\to\L\R
-\centeredline{|\xintAssign\xintDecSplit {-15}{-12300004321}\to\L\R|}
-|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.}
-
-\subsection{\csbh{xintDecSplitL}}\label{xintDecSplitL}
-
-\csa{xintDecSplitL\x\n}\etype{\numx f} returns the first piece after the action
-of \csa{xintDecSplit}.
-
-\subsection{\csbh{xintDecSplitR}}\label{xintDecSplitR}
-
-\csa{xintDecSplitR\x\n}\etype{\numx f} returns the second piece after the action
-of \csa{xintDecSplit}.
-
-
-
-\section{Commands of the \xintfracname package}
-\label{sec:frac}
-
-\def\x{\string{x\string}}
-
-This package was first included in release |1.03| 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
-\digitstt{\xintPRaw {\xintAdd {2/5}{3/5}}}\footnote{yes, \csbxint{Add} blindly
- multiplies denominators... }, whereas |\xintPRaw {\xintIrr {\xintAdd
- {2/5}{3/5}}}| returns \digitstt{\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.
-
-
-\localtableofcontents
-
-\subsection{\csbh{xintNum}}\label{xintNum}
-
-The macro\etype{f} is extended to accept a fraction on input. But this fraction
-should reduce to an integer. If not an error will be raised. The original is
-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 add the necessary zeroes to get an explicit integer.
-
-\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
-|{}|. As will all other \xintname conditionals, 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.
-\centeredline {|\xintLen {201710/298219}|\digitstt{=\xintLen {201710/298219}},
-|\xintLen {1234/1}|\digitstt{=\xintLen {1234/1}}, |\xintLen {1234}|%
- \digitstt{=\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|.
-\centeredline{|\xintRaw{\the\numexpr 571*987\relax.123e-10/\the\numexpr
- -201+59\relax e-7}=|}%
-\centeredline{\digitstt{\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|. \centeredline{|\xintPRaw
- {123e10/321e10}=|\digitstt{\xintPRaw {123e10/321e10}}, |\xintPRaw
- {123e9/321e10}=|\digitstt{\xintPRaw {123e9/321e10}}} \centeredline{|\xintPRaw
- {\xintIrr{861/123}}=|\digitstt{\xintPRaw{\xintIrr{861/123}}} \ vz.\
- |\xintIrr{861/123}=|\digitstt{\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 \csbxint{Num} for that, but there will be an
-error message if the fraction was not an integer; so the combination
-|\xintPRaw{\xintIrr{f}}| is the way to go.
-
-\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: \centeredline{|\xintNumerator
- {178000/25600000[17]}|\digitstt{=\xintNumerator {178000/25600000[17]}}}
-\centeredline{|\xintNumerator {312.289001/20198.27}|%
- \digitstt{=\xintNumerator {312.289001/20198.27}}}
-\centeredline{|\xintNumerator {178000e-3/256e5}|\digitstt{=\xintNumerator
- {178000e-3/256e5}}} \centeredline{|\xintNumerator
- {178.000/25600000}|\digitstt{=\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.} \centeredline{|\xintDenominator
- {178000/25600000[17]}|\digitstt{=\xintDenominator {178000/25600000[17]}}}%
-\centeredline{|\xintDenominator {312.289001/20198.27}|%
- \digitstt{=\xintDenominator {312.289001/20198.27}}}
-\centeredline{|\xintDenominator {178000e-3/256e5}|\digitstt{=\xintDenominator
- {178000e-3/256e5}}} \centeredline{|\xintDenominator
- {178.000/25600000}|\digitstt{=\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}|.
-\centeredline{|\xintRawWithZeros{\the\numexpr 571*987\relax.123e-10/\the\numexpr
- -201+59\relax e-7}=|}%
-\centeredline{\digitstt{\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: \centeredline{|\xintREZ
- {178000/25600000[17]}|\digitstt{=\xintREZ {178000/25600000[17]}}}
-\centeredline{|\xintREZ {1780000000000e30/2560000000000e15}|\digitstt{=\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. \centeredline{|\[\xintFrac
- {-355/113}=\xintSignedFrac {-355/113}\]|}
-\[\xintFrac {-355/113}=\xintSignedFrac {-355/113}\]
-
-\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. \centeredline{|\[\xintFwOver
- {-355/113}=\xintSignedFwOver {-355/113}\]|}
-\[\xintFwOver {-355/113}=\xintSignedFwOver {-355/113}\]
-
-
-\subsection{\csbh{xintIrr}}\label{xintIrr}
-
-This puts the fraction\etype{\Ff} into its unique irreducible form:
-\centeredline{|\xintIrr {178.256/256.178}|%
- \digitstt{=\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:
-\centeredline{|\xintJrr {178.256/256.178}|%
- \digitstt{=\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 }|\digitstt{=%
- \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...@.
-\centeredline{|\xintTrunc
- {16}{-803.2028/20905.298}|\digitstt{=\xintTrunc {16}{-803.2028/20905.298}}}%
-\centeredline{|\xintTrunc {20}{-803.2028/20905.298}|\digitstt{=\xintTrunc
- {20}{-803.2028/20905.298}}}%
-\centeredline{|\xintTrunc {10}{\xintPow {-11}{-11}}|\digitstt{=\xintTrunc
- {10}{\xintPow {-11}{-11}}}}%
-\centeredline{|\xintTrunc {12}{\xintPow {-11}{-11}}|\digitstt{=\xintTrunc
- {12}{\xintPow {-11}{-11}}}}%
-\centeredline{|\xintTrunc {12}{\xintAdd {-1/3}{3/9}}|\digitstt{=\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.
-%
-\centeredline{|\xintiTrunc
- {16}{-803.2028/20905.298}|\digitstt{=\xintiTrunc {16}{-803.2028/20905.298}}}%
-\centeredline{|\xintiTrunc {10}{\xintPow {-11}{-11}}|\digitstt{=\xintiTrunc
- {10}{\xintPow {-11}{-11}}}}%
-\centeredline{|\xintiTrunc {12}{\xintPow {-11}{-11}}|\digitstt{=\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 ``\digitstt{\xintTrunc
- 0{-0.5}}'' whereas \csa{xintiTrunc}|{0}{-0.5}| simply returns
-``\digitstt{\xintiTrunc 0{-0.5}}''.
-
-\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
-\dverb|@
-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 |\char92 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
- |\char 92 xintXTrunc|, especially for the case of a negative |N|. Doing these
- things with \TeX{} is an effort. Besides an
- |\char 92 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 |\char 92
- xintexpr| context where things are expanded inside |\char92 csname..\char92
- 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. \centeredline{|\xintRound {16}{-803.2028/20905.298}|\digitstt{=\xintRound
- {16}{-803.2028/20905.298}}}%
-\centeredline{|\xintRound {20}{-803.2028/20905.298}|\digitstt{=\xintRound
- {20}{-803.2028/20905.298}}}%
-\centeredline{|\xintRound {10}{\xintPow {-11}{-11}}|\digitstt{=\xintRound
- {10}{\xintPow {-11}{-11}}}}%
-\centeredline{|\xintRound {12}{\xintPow {-11}{-11}}|\digitstt{=\xintRound
- {12}{\xintPow {-11}{-11}}}}%
-\centeredline{|\xintRound {12}{\xintAdd {-1/3}{3/9}}|\digitstt{=\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:
-\centeredline{\digitstt{\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. \centeredline{|\xintiRound
- {16}{-803.2028/20905.298}|\digitstt{=\xintiRound {16}{-803.2028/20905.298}}}%
-\centeredline{|\xintiRound {10}{\xintPow {-11}{-11}}|\digitstt{=\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}}\label{xintFloor}
-%{\small New with release |1.09a|.\par}
-
-|\xintFloor {f}|\etype{\Ff} returns the largest relative integer |N| with
-|N|${}\leq{}$|f|. \centeredline{|\xintFloor {-2.13}|\digitstt{=\xintFloor
- {-2.13}}, |\xintFloor {-2}|\digitstt{=\xintFloor {-2}}, |\xintFloor
- {2.13}|\digitstt{=\xintFloor {2.13}}%
-}
-
-\subsection{\csbh{xintCeil}}\label{xintCeil}
-%{\small New with release |1.09a|.\par}
-
-|\xintCeil {f}|\etype{\Ff} returns the smallest relative integer |N| with
-|N|${}>{}$|f|. \centeredline{|\xintCeil {-2.13}|\digitstt{=\xintCeil {-2.13}},
- |\xintCeil {-2}|\digitstt{=\xintCeil {-2}}, |\xintCeil
- {2.13}|\digitstt{=\xintCeil {2.13}}%
-}
-
-\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.
-\centeredline{|\xintTFrac {1235/97}|\digitstt{=\xintTFrac {1235/97}}\quad
- |\xintTFrac {-1235/97}|\digitstt{=\xintTFrac {-1235/97}}}
-\centeredline{|\xintTFrac {1235.973}|\digitstt{=\xintTFrac {1235.973}}\quad
- |\xintTFrac {-1235.973}|\digitstt{=\xintTFrac {-1235.973}}}
-\centeredline{|\xintTFrac {1.122435727e5}|%
- \digitstt{=\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:
-\centeredline{|\count 255 123456789 \xintE {10}{\count 255}|\digitstt{->\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:\centeredline{|\xintFloatAdd
- {1e1234567890}{1}|\digitstt{=\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|.
-\centeredline{|\xintFloatE {1.23e37}{53}|\digitstt{=\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).
-\centeredline{|\xintFloat[32]{1234567/7654321}|%
- \digitstt{=\xintFloat[32]{1234567/7654321}}}
-% \pdfresettimer
-\centeredline{|\xintFloat[32]{1/\xintFac{100}}|%
- \digitstt{=\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{xintAdd}}\label{xintAdd}
-
-The original\etype{\Ff\Ff} macro is extended to accept fractions on input. Its
-output will now always be in the form |A/B[n]|. The original is available as
-\csbxint{iAdd}.
-
-\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}
-
-The original\etype{\Ff\Ff} macro is extended to accept fractions on input. Its
-output will now always be in the form |A/B[n]|. The original is available as
-\csbxint{iSub}.
-
-\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}
-
-The original\etype{\Ff\Ff} macro is extended to accept fractions on input. Its
-output will now always be in the form |A/B[n]|. The original, only for big
-integers, and outputting a big integer, is available as \csbxint{iMul}.
-
-\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{xintSqr}}\label{xintSqr}
-
-The original\etype{\Ff} macro is extended to accept a fraction on input. Its
-output will now always be in the form |A/B[n]|. The original which outputs only
-big integers is available as \csbxint{iSqr}.
-
-\subsection{\csbh{xintDiv}}\label{xintDiv}
-
-\csa{xintDiv}|{f}{g}|\etype{\Ff\Ff} computes the fraction |f/g|. As with all
-other computation macros, no simplification is done on the output, which is in
-the form |A/B[n]|.
-
-\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 on input but this
-fraction |f| must simplify to a integer |n| (non negative and at most |999999|,
-but already |100000!| is prohibitively time-costly). On output |n!| (no trailing
-|/1[0]|). The original macro (which has less overhead) is still available as
-\csbxint{iFac}.
-
-\subsection{\csbh{xintPow}}\label{xintPow}
-
-\csa{xintPow}{|{f}{g}|}:\etype{\Ff\Numf} the original macro is extended to
-accept fractions on input. The output will now always be in the form |A/B[n]|
-(even when the exponent vanishes: |\xintPow
-{2/3}{0}|\digitstt{=\xintPow{2/3}{0}}). The original is available as
-\csbxint{iPow}.
-
-The exponent is allowed to be input as a fraction but it must simplify to an
-integer: |\xintPow {2/3}{10/2}|\digitstt{=\xintPow {2/3}{10/2}}. This integer
-will be checked to not exceed |100000|. Indeed |2^50000| already has
-\digitstt{\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).
-
-\centeredline{|\xintFloatPow [8]{3.1415}{1234567890}|%
- \digitstt{=\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.
-\centeredline{|\xintFloatPower [8]{1.000000000001}{1e12}|%
- \digitstt{=\xintFloatPower [8]{1.000000000001}{1e12}}}
-\centeredline{|\xintFloatPower [8]{3.1415}{3e9}|%
- \digitstt{=\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{}
-\digitstt{\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).
-\centeredline{|\xintFloatSqrt [50]{12.3456789e12}|}%
-\centeredline{${}\approx{}$\digitstt{\xintFloatSqrt [50]{12.3456789e12}}}%
-\centeredline{|\xintDigits:=50;\xintFloatSqrt {\xintFloatSqrt {2}}|}%
-\centeredline{%
- ${}\approx{}$\xintDigits:=50;\digitstt{\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}.
-
-The original\etype{f{$\to$}{\lowast\Ff}} command is extended to accept fractions
-on input and produce fractions on output. 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}.
-
-
-% \subsection{\csbh{xintPrd}, \csbh{xintPrdExpr}}\label{xintPrd}\label{xintPrdExpr}
-
-\subsection{\csbh{xintPrd}}\label{xintPrd}\label{xintPrdExpr}
-
-The original\etype{f{$\to$}{\lowast\Ff}} is extended to accept fractions on
-input and produce fractions on output. 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}.
-
-\subsection{\csbh{xintCmp}}\label{xintCmp}
-%{\small Rewritten in |1.08a|.\par}
-
-The macro\etype{\Ff\Ff} is extended to fractions. Its output is still either
-|-1|, |0|, or |1| with no forward slash nor trailing |[n]|.
-
-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}}
-See \csbxint{IsOne}\etype{\Ff} (\autoref{xintIsOne}).
-
-\subsection{\csbh{xintGeq}}\label{xintGeq}
-%{\small Rewritten in |1.08a|.\par}
-
-The macro\etype{\Ff\Ff} is extended to fractions. Beware that the comparison is
-on the \emph{absolute values} of the fractions. Can be used 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 macro is extended to fractions.\etype{\Ff\Ff} But now |\xintMax {2}{3}|
-returns \digitstt{\xintMax {2}{3}}. The original, for use with (possibly big)
-integers only, is available as \csbxint{iMax}: |\xintiMax
-{2}{3}=|\digitstt{\xintiMax {2}{3}}.
-
-\subsection{\csbh{xintMaxof}}
-See \csbxint{Maxof} (\autoref{xintMaxof}).\etype{f{$\to$}{\lowast\Ff}}
-
-\subsection{\csbh{xintMin}}\label{xintMin}
-%{\small Rewritten in |1.08a|.\par}
-
-The macro is extended to fractions.\etype{\Ff\Ff} The original, for (big)
-integers only, is available as \csbxint{iMin}.
-
-\subsection{\csbh{xintMinof}}
-See \csbxint{Minof} (\autoref{xintMinof}).\etype{f{$\to$}{\lowast\Ff}}
-
-\subsection{\csbh{xintAbs}}\label{xintAbs}
-
-The macro is extended to fractions.\etype{\Ff} The original, for (big) integers
-only, is available as \csbxint{iAbs}. Note that |\xintAbs
-{-2}|\digitstt{=\xintAbs {-2}} whereas |\xintiAbs {-2}|\digitstt{=\xintiAbs
- {-2}}.
-
-\subsection{\csbh{xintSgn}}\label{xintSgn}
-
-The macro is extended to fractions.\etype{\Ff} Naturally, its output is still
-either |-1|, |0|, or |1| with no forward slash nor trailing |[n]|.
-
-\subsection{\csbh{xintOpp}}\label{xintOpp}
-
-The macro is extended to fractions.\etype{\Ff} The original is available as
-\csbxint{iOpp}. Note that |\xintOpp {3}| now outputs \digitstt{\xintOpp {3}}
-whereas |\xintiOpp {3}| returns \digitstt{\xintiOpp {3}}.
-
-\subsection{\csbh{xintDivision}, \csbh{xint\-Quo}, \csbh{xint\-Rem},
- \csbh{xintFDg}, \csbh{xintLDg}, \csbh{xintMON}, \csbh{xintMMON}, \csbh{xintOdd}}
-
-These macros\etype{\Ff\Ff} accept a fraction on input if this fraction in fact
-reduces to an integer (if not an |\xintError:NotAnInteger| will be
-raised).\etype{{\textcolor{black}{\upshape or}}\Ff} There is no difference in
-the format of the outputs, which are still (possibly big) integers without
-fraction slash nor trailing |[n]|, the sole difference is in the extended range
-of accepted inputs.
-
-All have variants 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. These variants are already available in
-\xintname, there is no need for \xintfracname to be loaded.
-
-\centeredline{|\xintNum {1e80}|}
-\centeredline{\digitstt{\xintNum{1e80}}}
-
-
-\etocdepthtag.toc {xintexpr}
-
-\section{Expandable expressions with the \xintexprname package}%
-\label{sec:expr}
-
-The \xintexprname package was first released with version |1.07| of the
-\xintname bundle. It loads automatically \xintfracname, hence
-also \xintname and \xinttoolsname.
-
-% Release |1.09a| has extended the scope of |\xintexpr|-essions: infix
-% comparison operators (|<|, |>|, |=|), logical operators (|&|, \verb+|+),
-% functions (|round|, |sqrt|, |max|, |all|, etc...), conditional ``branching''
-% (|if| and |?|, |ifsgn| and |:|).
-
-The syntax is described in \autoref{sec:exprsummary} and
-\autoref{sec:exprsummaryII}.
-
-\localtableofcontents
-
-
-\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 \marg{expandable} may be serving as a
-macro parameter, or a branch of the |?| two-way and |:| three-way operators;
-else it is treated in a special manner:
-\begin{enumerate}
-\item it is allowed to occur only at the spots where numbers are legal,
-\item the \meta{expandable} contents is then completely expanded as if it were
- put in a |\csname..\endcsname|,\footnote{well, actually it \emph{is} put in
- such a \texttt{\char92csname..\char92endcsname}.} thus it escapes entirely
- the parser scope and infix notations will not be recognized except if the
- expanded macros know how to handle them by themselves,
-\item and this complete expansion \emph{must} produce a number or a fraction,
- possibly with decimal mark and trailing |[n]|, the scientific notation is
- \emph{not} authorized.
-\end{enumerate}
-Braces are the only way to input some number or fraction with
-a trailing |[n]|: square brackets are
-\emph{not} accepted in a |\xintexpr...\relax| if not within such braces.
-
-
-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 output format for the result can be coded inside the expression
-through the use of one of the functions |round|, |trunc|, |float|,
-|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
-\begingroup\raggedright\leftskip.5cm
-{|\xinttheexpr 1/5!-1/7!-1/9!\relax|%
- \digitstt{=\xinttheexpr 1/5!-1/7!-1/9!\relax}}\\
-{|\xinttheexpr round(1/5!-1/7!-1/9!,18)\relax|%
- \digitstt{=\xinttheexpr round(1/5!-1/7!-1/9!,18)\relax}}\\
-{|\xinttheexpr float(1/5!-1/7!-1/9!,18)\relax|%
- \digitstt{=\xinttheexpr float(1/5!-1/7!-1/9!,18)\relax}}\\
-{|\xinttheexpr reduce(1/5!-1/7!-1/9!)\relax|%
- \digitstt{=\xinttheexpr reduce(1/5!-1/7!-1/9!)\relax}}\\
-{|\xinttheexpr 1.99^-2 - 2.01^-2 \relax|%
- \digitstt{=\xinttheexpr 1.99^-2 - 2.01^-2 \relax}}\\
-{|\xinttheexpr round(1.99^-2 - 2.01^-2, 10)\relax|%
- \digitstt{=\xinttheexpr round(1.99^-2 - 2.01^-2, 10) \relax}}\par
-\endgroup
-
-\smallskip
-\begingroup % 18 octobre, je reprends la méthode déjà utilisée au début du
- % document le 9 octobre.
-\leftmargini 0pt
-\list\labelitemi{\def\makelabel#1{\hss\llap{#1}}\listparindent\parindent
- \labelwidth\parindent
- \itemindent\labelwidth}%
-\item the expression may contain arbitrarily many levels of nested parenthesized
- sub-expressions.
-\item sub-contents giving numbers of fractions should be either
- \begin{enumerate}
- \item parenthesized,
- \item a sub-expression |\xintexpr...\relax|,
- \item or within braces.
- \end{enumerate}
- When a sub-expression is hit against in the midst of absorbing the
- digits of a number, a |*| to force tacit multiplication is
- inserted.\inmarg{1.09j}. Similarly, if it is an opening parenthesis
- which is hit against.\inmarg{1.09k}
- \item an expression can not be given as argument to the other package macros,
- nor printed, for this one must use |\xinttheexpr...\relax| or
- |\xintthe\xintexpr...\relax|.
- \item one does not use |\xinttheexpr...\relax| as a sub-constituent of an
- |\xintexpr...\relax| but simply |\xintexpr...\relax|; this is mainly because
- most of the time |\xinttheexpr..\relax| will insert explicit square brackets
- which are not parsable, as already mentioned, in the surrounding expression.
-\item each \xintexprname{}ession is completely expandable and obtains
- its result in two expansion steps.
-\endlist
-\endgroup
-
-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 |\oodef\x {\xintexpr \a+\b\relax}| will do the
-computation on the spot.
-
-\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|.
-
-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{\digitstt{\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}
-
-\subsubsection{\csbh{xintexprSafeCatcodes}}
-\label{xintexprSafeCatcodes}
-%{\small New with release |1.09a|.\par}
-
-Active characters will interfere with |\xintexpr|-essions. One may prefix them
-with |\string| within |\xintexpr..\relax|, thus preserving expandability, or
-there is the non-expandable \csa{xintexprSafeCatcodes} which can be issued
-before the use of |\xintexpr|. This command sets (not globally) 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
-
-Unbraced 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 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.
-
-
-% In the case of the factorial, the macro
-% |\xintFac| may be used rather than the postfix |!|, preferably within braces as
-% this will avoid the subsequent slow scan digit by digit of its expansion (other
-% macros from the \xintfracname package generally \emph{must} be used within a
-% brace pair, as they expand to fractions |A/B[n]| with the trailing |[n]|; the
-% |\xintFac| produces an integer with no |[n]| and braces are only optional, but
-% preferable, as the scanner will get the job done faster.)
-
-% Sub-material within braces is treated technically in a different manner, and
-% depending on the macros used therein may be more sensitive to the catcode of the
-% five operations.
-
-Digits, slash, square brackets, minus sign, in the output from an |\xinttheexpr|
-are all of catcode 12. For |\xintthefloatexpr| the `e' in the output is of
-catcode 11.
-
-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 (or within braces used to protect
-square brackets).
-
-
-\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|.\MyMarginNote{New with 1.09j!} 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 exception is with brace pairs which, apart from their usual r\^ole
-for macro arguments, are also allowed in places where the scanner expects a
-numeric operand, the braced material should expand to some number (or fraction),
-but scientific notation is not allowed. Conversely fractions in |A/B[N]| format
-(either explicit or from macro expansion) must be enclosed in such a brace pair.
-
-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}.\MyMarginNote{New with 1.09j!}
-
-An expression can only be legally finished by a |\relax| token, which
-will be absorbed.
-
-
-\subsection{Memory considerations}
-
-The parser creates an undefined control sequence for each intermediate
-computation (this does not refer to the intermediate steps needed in
-the evaluations of the \csbxint{Add}, \csbxint{Mul}, etc... macros corresponding
-to the infix operators, but only to each conversion of such an infix operator
-into a computation). So, 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 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:\centeredline{|\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.
-\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. 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.
-
-\begin{framed}
- 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; hence one can not use infix notation inside the
- arguments, as in for example |\myformula {28^7-35^12}| which would have been
- allowed by
- \centeredline{|\def\myformula #1{\xinttheexpr (#1)^3\relax}|}
- One will have to do |\myformula {\xinttheexpr 28^7-35^12\relax}|, or redefine
- |\myformula| to have more parameters.
-\end{framed}
-
-% The formula may contain besides the infix operators and macro
-% parameters some arbitrary decimal numbers, fractions (within braces) and also
-% macros. If these macros do not involve the parameters, nothing special needs to
-% be done, they will be expanded once during the construction of the formula. But
-% if the parameters are to be used within the macros themselves, then the macro
-% should be code with an underscore |_| rather than a backslash |\|.
-
-\dverb|@
-@\xintNewExpr\myformA[4]{ #1 + #2 * #3^#4 }
-@\xintNewExpr\myformB[3]{ (#1 + 1.75)^#2 + #3*2.7 }
-@\xintNewExpr\myformC[3]{ #1*#1+#2*#2+#3*#3-(#1*#2+#2*#3+#3*#1) }
-@\xintNewExpr\myformD[2]{ (1+1.5*#1)^#2 - (1+1.5*#2)^#1 }
-@\xintNewExpr\myformE[2]{ -----((((((#1*10-5*#2)))))) }
-@\xintNewExpr\myformF[4]{ -#1^-#2*-#3-#4 }
-@\xintNewExpr\myformG[4]{ -#1*-#2^-#3-#4 }
-\xintNewExpr\DET[9]{ #1*#5*#9+#2*#6*#7+#3*#4*#8-#1*#6*#8-#2*#4*#9-#3*#5*#7 }|
-
-% \xintNewExpr\myformA[4]{ #1 + #2 * #3^#4 }
-% \xintNewExpr\myformB[3]{ (#1 + 1.75)^#2 + #3*2.7 }
-% \xintNewExpr\myformC[3]{ #1*#1+#2*#2+#3*#3-(#1*#2+#2*#3+#3*#1) }
-% \xintNewExpr\myformD[2]{ (1+1.5*#1)^#2 - (1+1.5*#2)^#1 }
-% \xintNewExpr\myformE[2]{ -----((((((#1*10-5*#2)))))) }
-% \xintNewExpr\myformF[4]{ -#1^-#2*-#3-#4 }
-% \xintNewExpr\myformG[4]{ -#1*-#2^-#3-#4 }
-\xintNewExpr\DET[9]{ #1*#5*#9+#2*#6*#7+#3*#4*#8-#1*#6*#8-#2*#4*#9-#3*#5*#7 }
-
-\ttfamily
-% |\meaning\myformA:|\printnumber{\meaning\myformA}\endgraf
-% |\meaning\myformB:|\printnumber{\meaning\myformB}\endgraf
-% |\meaning\myformC:|\printnumber{\meaning\myformC}\endgraf
-% |\meaning\myformD:|\printnumber{\meaning\myformD}\endgraf
-% |\meaning\myformE:|\printnumber{\meaning\myformE}\endgraf
-% |\meaning\myformF:|\printnumber{\meaning\myformF}\endgraf
-% |\meaning\myformG:|\printnumber{\meaning\myformG}\endgraf
-|\meaning\DET:|\printnumber{\meaning\DET}\endgraf
-
-
-\centeredline{|\xintNum{\DET {1}{1}{1}{10}{-10}{5}{11}{-9}{6}}|%
- \digitstt{=\xintNum{\DET {1}{1}{1}{10}{-10}{5}{11}{-9}{6}}}}%
-\centeredline{|\xintNum{\DET {1}{2}{3}{10}{0}{-10}{21}{2}{-17}}|%
- \digitstt{=\xintNum{\DET {1}{2}{3}{10}{0}{-10}{21}{2}{-17}}}}
-
-
-\rmfamily
-
-
-\emph{Remark:} |\meaning| has been used within the argument to a |\printnumber|
-command, to avoid going into the right margin, but this zaps all spaces
-originally in the output from |\meaning|. Here is as an illustration the raw
-output of
-|\meaning| on the previous example:
-
-\ttfamily
-\meaning\DET
-\rmfamily
-
-This is why |\printnumber| was used, to have breaks across lines.
-
-\subsubsection {Use of conditional operators}
-
-The |1.09a| conditional operators |?| and |:| 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|.
-
-\xintNewExpr\Formula [3]{ if((#1>#2) & (#2>#3),
- sqrt(#1-#2)*sqrt(#2-#3), #1^2+#3/#2) }
-
-\centeredline{|\xintNewExpr\Formula [3]|}
-\centeredline{|{ if((#1>#2) & (#2>#3),
- sqrt(#1-#2)*sqrt(#2-#3), #1^2+#3/#2) }|}
-
-\ttfamily
-\noindent|\meaning\Formula:|\printnumber{\meaning\Formula}\endgraf
-
-\rmfamily
-This formula (with |\xintifNotZero|) will gobble the false branch.
-
-Remark: this
-|\XINTinFloatSqrt| macro is a non-user package macro used internally within
-|\xintexpr|-essions, it produces the result in |A[n]| form rather
-than in scientific notation, and for reasons of the inner workings of
-|\xintexpr|-essions, this is necessary; a hand-made macro would
-have used instead the equivalent |\xintFloatSqrt|.
-
-Another example
-
-\xintNewExpr\myformula[3]{ ifsgn(#1,#2/#3,#2-#3,#2*#3) }
-\centeredline{|\xintNewExpr\myformula [3]|}
-\centeredline{|{ ifsgn(#1,#2/#3,#2-#3,#2*#3) }|}
-
-\ttfamily
-\noindent\printnumber{\meaning\myformula}\endgraf
-
-\rmfamily
-Again, this macro gobbles the false branches, as would have the operator |:|
-inside an |\xintexpr|-ession.
-
-
-
-\subsubsection{Use of macros}
-
-
-For macros to be inserted within such a created \xintname-formula command, there
-are two cases:
-\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 parameters as argument. Then:
- \begin{enumerate}
- \item the whole thing (macro + argument) should be braced (not necessary if it
- is already included into a braced group),
- \item the macro should be coded with an underscore |_| in place of the
- backslash |\|.
- \item the parameters should be coded with a dollar sign |$1|, |$2|, etc...
- \end{enumerate}
-\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):
-
-\dverb|@
-\xintNewExpr\myformI[2]{ {_xintRound{$1}{$2}} - {_xintTrunc{$1}{$2}} }
-\meaning\myformI:|
-
-\xintNewExpr\myformI[2]{ {_xintRound{$1}{$2}} - {_xintTrunc{$1}{$2}} }
-\ttfamily
-\centeredline{\meaning\myformI}
-
-\dverb|@
-\xintNewIIExpr\formula [3]{rem(#1,quo({_the_numexpr $2_relax},#3))}
-\meaning\formula:|%$
-
-\xintNewIIExpr\formula [3]{rem(#1,quo({_the_numexpr $2_relax},#3))}%$
-\noindent{\meaning\formula}\endgraf
-
-\rmfamily
-
-\subsection{\csbh{xintiexpr}, \csbh{xinttheiexpr}}
-\label{xintiexpr}\label{xinttheiexpr}
-% \label{xintnumexpr}\label{xintthenumexpr}
-
-Equivalent\etype{x} to doing |\xintexpr round(...)\relax|. Thus, only the final
-result is rounded to an integer. Half integers are rounded towards $+\infty$ for
-positive numbers and towards $-\infty$ for negative ones. Can be used on comma
-separated lists of expressions.
-
-Initially\MyMarginNote{|1.09i| warning} 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} maps |/| to the euclidean quotient and deals almost only
-with (long) integers. It uses the `ii' macros for addition, subtraction,
-multiplication, power, square, sums, products, euclidean quotient and remainder.
-The |round| and |trunc|, in the presence of the second optional argument, are
-mapped to \csbxint{iRound}, respectively \csbxint{iTrunc}, hence they always
-produce (long) integers.
-
-To input a fraction to |round|, |trunc|, |floor| or |ceil| one can
-use braces, else the |/| will do the euclidean quotient.
-The minus sign should be put together with the fraction: |round(-{30/18})| is
-illegal (even if the fraction had been an integer), use
-|round({-30/18})|\digitstt{=\xinttheiiexpr round({-30/18})\relax}.
-
-Decimal numbers are allowed only if postfixed immediately with |e| or |E|, the
-number will then be truncated to an integer after multiplication by the power of
-ten with exponent the number following |e| or |E|.
-\centeredline{|\xinttheiiexpr 13.4567e3+10000123e-3\relax|%
- \digitstt{=\xinttheiiexpr 13.4567e3+10000123e-3\relax}}
-%
-
-A fraction within braces should be followed immediately by an |e| (or inside a
-|round|, |trunc|, etc...) to convert it
-into an integer as expected by the main operations. The truncation is only done
-after the |e| action.
-
-The |reduce| function is not available and will raise un error. The |frac|
-function also. The |sqrt| function is mapped to \csbxint{iSqrt}.
-
-Numbers in float notation, obtained via a macro like \csbxint{FloatSqrt}, are a
-bit of a challenge: they can not be within braces (this has been mentioned
-already, |e| is not legal within braces) and if not braced they will be
-truncated when the parser meets the |e|. The way out of the dilemma is to use a
-sub-expression:
-\centeredline{|\xinttheiiexpr \xintFloatSqrt{2}\relax|%
- \digitstt{=\xinttheiiexpr \xintFloatSqrt{2}\relax}}
-\centeredline{|\xinttheiiexpr \xintexpr\xintFloatSqrt{2}\relax e10\relax|%
- \digitstt{=\xinttheiiexpr \xintexpr\xintFloatSqrt{2}\relax e10\relax}}
-\centeredline{|\xinttheiiexpr round(\xintexpr\xintFloatSqrt{2}\relax,10)\relax|%
- \digitstt{=\xinttheiiexpr round(\xintexpr\xintFloatSqrt{2}\relax,10)\relax}}
-(recall that |round| is mapped within |\xintiiexpr..\relax| to \csbxint{iRound}
-which always outputs an integer).
-
-The whole point of \csbxint{iiexpr} is to gain some speed in integer only
-algorithms, and the above explanations related to how to 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{xintthe\-float\-expr}}\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 is from the current setting of
-|\xintDigits| (it can not be given as an optional parameter).
-
-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. \centeredline{|\xintDigits:= 9; \xintthefloatexpr
- (1+1e-9)-1\relax|\digitstt{=\xintthefloatexpr (1+1e-9)-1\relax}}
-\centeredline{|\xintDigits:= 9; \xintthefloatexpr
- 1.000000001-1\relax|\digitstt{=\xintthefloatexpr 1.000000001-1\relax}}
-
-For the fun of it:\xintDigits:=20; |\xintDigits:=20;|%
-\centeredline{|\xintthefloatexpr (1+1e-7)^1e7\relax|%
- \digitstt{=\xintthefloatexpr (1+1e-7)^1e7\relax}}
-
-|\xintDigits:=36;|\xintDigits:=36;
-\centeredline{|\xintthefloatexpr
- ((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax|}
-\centeredline{\digitstt{\xintthefloatexpr
- ((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax}}
-\centeredline{|\xintFloat{\xinttheexpr
- ((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax}|}
-\centeredline{\digitstt{\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:
-%
-\centeredline{|\xintDigits:=12;\xintthefloatexpr 1.000000000000001^1e15\relax|}
-%
-\centeredline{\xintDigits:=12;%
- \digitstt{\xintthefloatexpr 1.000000000000001^1e15\relax}}
-%
-Contrarily to some professional computing sofware which are our concurrents on
-this market, the \digitstt{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 involove 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 numbers fetched as
-parameters will be the one locally given by |\xintDigits| at the time of use of
-the created formulas, not |\xintNewFloatExpr|. However, the numbers hard-wired
-in the original expression will have been evaluated with the then current
-setting for |\xintDigits|.
-
-\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{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 withing 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
-|\XINT_expr_usethe| which prints an error message in the document and in
-the log file if it is executed, then a |\xint_protect| token, a token
-doing the actual printing and finally a token |\.=A/B[n]|. Using
-|\xinttheexpr| means zapping the first three things, the fourth one will
-then unlock |A/B[n]| from the (presumably undefined, but it does not
-matter) control sequence |\.=A/B[n]|.
-
-Thanks to the release |1.09j| added |\xint_protect| token and the fact
-that |\XINT_expr_usethe| is |\protected|, one can now use |\xintexpr|
-inside an |\edef|, with no need of the |\xintthe| prefix.
-
-\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.
-%
-% \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.
-
-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 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}
-
-This package was first included in the |1.08| release of \xintname. It
-provides expandable conversions of arbitrarily long numbers
-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
-
-\localtableofcontents
-
-
-
-\subsection{\csbh{xintDecToHex}}\label{xintDecToHex}
-
-Converts from decimal to hexadecimal.\etype{f}
-
-\texttt{\string\xintDecToHex \string{\printnumber{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}\string}}\endgraf\noindent\digitstt{->\printnumber{\xintDecToHex{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}}}
-
-\subsection{\csbh{xintDecToBin}}\label{xintDecToBin}
-
-Converts from decimal to binary.\etype{f}
-
-\texttt{\string\xintDecToBin \string{\printnumber{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}\string}}\endgraf\noindent\digitstt{->\printnumber{\xintDecToBin{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}}}
-
-\subsection{\csbh{xintHexToDec}}\label{xintHexToDec}
-
-Converts from hexadecimal to decimal.\etype{f}
-
-\texttt{\string\xintHexToDec
- \string{\printnumber{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}\string}}\endgraf\noindent
-\digitstt{->\printnumber{\xintHexToDec{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}}}
-
-\subsection{\csbh{xintBinToDec}}\label{xintBinToDec}
-
-Converts from binary to decimal.\etype{f}
-
-\texttt{\string\xintBinToDec
- \string{\printnumber{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}\string}}\endgraf\noindent
-\digitstt{->\printnumber{\xintBinToDec{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}}}
-
-\subsection{\csbh{xintBinToHex}}\label{xintBinToHex}
-
-Converts from binary to hexadecimal.\etype{f}
-
-\texttt{\string\xintBinToHex
- \string{\printnumber{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}\string}}\endgraf\noindent
-\digitstt{->\printnumber{\xintBinToHex{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}}}
-
-\subsection{\csbh{xintHexToBin}}\label{xintHexToBin}
-
-Converts from hexadecimal to binary.\etype{f}
-
-\texttt{\string\xintHexToBin
- \string{\printnumber{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}\string}}\endgraf\noindent
-\digitstt{->\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
-\digitstt{->\printnumber{\xintCHexToBin{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}}}
-
-
-
-\section{Commands of the \xintgcdname package}
-\label{sec:gcd}
-
-
-This package was included in the original release |1.0| 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.
-
-%% \clearpage
-
-\localtableofcontents
-
-\subsection{\csbh{xintGCD}}\label{xintGCD}
-
-\csa{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.
-\centeredline{\csa{xintGCD}|{10000}{1113}|\digitstt{=\xintGCD{10000}{1113}}}
-\centeredline{|\xintGCD{123456789012345}{9876543210321}=|\digitstt
- {\xintGCD{123456789012345}{9876543210321}}}
-
-\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}}\label{xintLCM}
-%{\small New with release |1.09a|.\par}
-
-\csa{xintGCD\n\m}\etype{\Numf\Numf} computes the least common multiple. It is
-|0| if one of the two integers vanishes.
-
-\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
-
-\csa{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 \digitstt{UA - VB = D}. \centeredline{|\xintAssign
- {{\xintBezout {10000}{1113}}}\to\X|} \centeredline{|\meaning\X:
- |\digitstt{\meaning\X }.}
-\noindent{|\xintAssign {\xintBezout {10000}{1113}}\to\A\B\U\V\D|}\\
-|\A: |\digitstt{\A },
-|\B: |\digitstt{\B },
-|\U: |\digitstt{\U },
-|\V: |\digitstt{\V },
-|\D: |\digitstt{\D }.\\
-\xintAssign {\xintBezout {123456789012345}{9876543210321}}\to\A\B\U\V\D
-\noindent{|\xintAssign {\xintBezout {123456789012345}{9876543210321}}\to\A\B\U\V\D
-|}\\
-|\A: |\digitstt{\A },
-|\B: |\digitstt{\B },
-|\U: |\digitstt{\U },
-|\V: |\digitstt{\V },
-|\D: |\digitstt{\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
-
-\csa{xintEuclideAlgorithm\n\m}\etype{\Numf\Numf} applies the Euclide algorithm
-and keeps a copy of all quotients and remainders. \centeredline{|\xintAssign
- {{\xintEuclideAlgorithm {10000}{1113}}}\to\X|}
-
-|\meaning\X: |\digitstt{\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
-
-\csa{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. \centeredline{|\xintAssign
- {{\xintEuclideAlgorithm {10000}{1113}}}\to\X|}
-
-|\meaning\X: |\digitstt{\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}\texorpdfstring{\allowbreak\null\hspace*{.25cm}}{}}%
-\label{xintTypesetEuclideAlgorithm}
-
-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.
-\centeredline{|\xintTypesetEuclideAlgorithm {123456789012345}{9876543210321}|}
-\xintTypesetEuclideAlgorithm {123456789012345}{9876543210321}
-
-
-\subsection{\csbh{xintTypesetBezoutAlgorithm}}%
-\label{xintTypesetBezoutAlgorithm}
-
-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.
-\centeredline{|\xintTypesetBezoutAlgorithm {10000}{1113}|}
-\xintTypesetBezoutAlgorithm {10000}{1113}
-
-
-\section{Commands of the \xintseriesname package}
-\label{sec:series}
-
-Some arguments to the package commands are macros which are expanded only later,
-when given their parameters. The arguments serving as indices are systematically
-given to a |\numexpr| expressions (new with |1.06|!) , hence \fexpan ded,
-they may be count registers, etc...
-
-This package was first released with version |1.03| of the \xintname bundle.
-
-We use \Ff{} for the expansion type of various macro arguments, but if only
-\xintname and not \xintfracname is loaded this should be more appropriately
-\Numf. The macro \csbxint{iSeries} is special and expects summing big integers
-obeying the strict format, even if \xintfracname is loaded.
-
-%% \clearpage
-
-\localtableofcontents
-
-\subsection{\csbh{xintSeries}}\label{xintSeries}
-
-\def\coeff #1{\xintiiMON{#1}/#1.5} % (-1)^n/(n+1/2)
-\edef\w {\xintSeries {0}{50}{\coeff}}
-\edef\z {\xintJrr {\w}[0]}
-
-\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}
-%
-\dverb|@
-\def\coeff #1{\xintiiMON{#1}/#1.5} % (-1)^n/(n+1/2)
-\edef\w {\xintSeries {0}{50}{\coeff}} % we want to re-use it
-\edef\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.
-\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} = \xintFrac\z \]|
-\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} = \xintFrac\z \]
-For info,
-before action by |\xintJrr| the inner representation of the result has a
-denominator of |\xintLen {\xintDenominator\w}=|\xintLen
-{\xintDenominator\w} digits. This troubled me as @101!!@ has only 81
-digits: |\xintLen {\xintQuo {\xintFac {101}}{\xintiMul {\xintiPow
- {2}{50}}{\xintFac{50}}}}|\digitstt{=\xintLen {\xintQuo {\xintFac
- {101}}{\xintiMul {\xintiPow {2}{50}}{\xintFac{50}}}}}. The
-explanation lies in the too clever to be efficient |#1.5| trick. It
-leads to a silly extra @5^{51}@ (which has \xintLen {\xintPow {5}{51}}
-digits) in the denominator. See the explanations in the next section.
-
-\begin{framed}
- Note: as soon as the coefficients look like factorials, it is more
- efficient to use the \csbxint{RationalSeries} macro whose evaluation
- will avoid a denominator build-up; indeed the raw operations of
- addition and subtraction of fractions blindly multiply out
- denominators. So the raw evaluation of $\sum_{n=0}^{|N|}1/n!$ with
- \csa{xintSeries} will have a denominator equal to $\prod_{n=0}^{|N|}
- n!$. Needless to say this makes it more difficult to compute the exact
- value of this sum with |N=50|, for example, whereas with
- \csbxint{RationalSeries} the denominator does not
- get bigger than $50!$.
-
-\footnotesize
- For info: by the way $\prod_{n=0}^{50} n!$ is easily computed by \xintname
- and is a number with 1394 digits. And $\prod_{n=0}^{100} n!$ is also
- computable by \xintname (24 seconds on my laptop for the brute force
- iterated multiplication of all factorials, a
- specialized routine would do it faster) and has 6941 digits (this
- means more than two pages if printed...). Whereas $100!$ only has
- 158 digits.
-\end{framed}
-
-% \newcount\cntb
-% \cnta 2
-% \loop
-% \advance\cntb by \xintLen{\xintFac{\the\cnta}}%
-% \ifnum\cnta < 50
-% \advance\cnta 1
-% \repeat
-% \the\cntb
-
-% \cnta 2
-% \def\z{1}
-% \pdfresettimer
-% \loop
-% \edef\z {\xintiMul\z{\xintFac{\the\cnta}}}%
-% \ifnum\cnta < 100
-% \advance\cnta 1
-% \repeat
-% \edef\temps{\the\pdfelapsedtime}%
-
-% \temps: \xintQuo\temps{\xintiMul{60}{65536}} minutes,
-% \xintQuo{\xintRem\temps{\xintiMul{60}{65536}}}{65536} secondes et
-% \xintiTrunc {2}{\xintRem\temps{65536}/65536} centièmes de secondes
-% 1573518: 0 minutes, 24 secondes et 0 centièmes de secondes
-% nota bene, marrant c'était 0,99 centièmes en fait.
-
-% \xintLen\z
-
-% \printnumber\z
-
-\setlength{\columnsep}{0pt}
-\dverb|@
-\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\digitstt{\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.
-\dverb|@
-\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.
-\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx
- \xintTrunc {40}{\xintiSeries {0}{50}{\coeff}[-40]}\dots\]|
-
-The |#1.5| trick to define the |\coeff| macro was neat, but |1/3.5|, for
-example, turns internally into |10/35| whereas it would be more efficient to
-have |2/7|. The second way of coding the wanted coefficient avoids a superfluous
-factor of five and leads to a faster evaluation. The third way is faster, after
-all there is no need to use \csbxint{MON} (or rather
-\hyperref[fn:xintiiMON]{\csa{xintiiMON}} which has
-less parsing overhead) on integers
-obeying the \TeX{} bound. The denominator having no sign, we have added the
-|[0]| as this speeds up (infinitesimally) the parsing.
-\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx \xintTrunc
-{40}{\xintiSeries {0}{50}{\coeff}[-40]}\] 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:
-\dverb|@
-\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)|.
-
-\dverb|@
-\def\ratio #1{2/#1[0]}% 2/n, to compute exp(2)
-\cnta 0 % previously declared count
-\loop \edef\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{}\endgraf
-\ifnum\cnta<20 \advance\cnta 1 \repeat|
-
-\def\ratio #1{2/#1[0]}% 2/n, comes from the series of exp(2)
-\cnta 0
-\loop
-\edef\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{}\endgraf
-\ifnum\cnta<20 \advance\cnta 1 \repeat
-
-\medskip
-Such computations would become quickly completely inaccessible via the
-\csbxint{Series} macros, as the factorials in the denominators would get
-all multiplied together: the raw addition and subtraction on fractions
-just blindly multiplies denominators! Whereas \csa{xintRationalSeries}
-evaluate the partial sums via a less silly iterative scheme.
-\dverb|@
-\def\ratio #1{-1/#1[0]}% -1/n, comes from the series of exp(-1)
-\cnta 0 % previously declared count
-\loop
-\edef\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{}\endgraf
-\ifnum\cnta<20 \advance\cnta 1 \repeat|
-
-\def\ratio #1{-1/#1[0]}% -1/n, comes from the series of exp(-1)
-\cnta 0 % previously declared count
-
-\loop
-\edef\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{}\endgraf
-\ifnum\cnta<20 \advance\cnta 1 \repeat
-
-
- \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 \centeredline{|\xintRationalSeries {0}{b}{1}{\ratioexp{\x}}|}
-will compute $\sum_{n=0}^{n=b} x^n/n!$:\par
-\dverb|@
-\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|
-
-\cnta 0
-\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
-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:
-\dverb|@
-\cnta 1
-\loop \edef\z {\xintRationalSeries
- {\cnta}
- {2*\cnta-1}
- {\xintiPow {\the\cnta}{\cnta}/\xintFac{\cnta}}
- {\ratioexp{\the\cnta}}}%
-\edef\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|
-
-\cnta 1
-\begin{multicols}{2}
-\loop \edef\z {\xintRationalSeries
- {\cnta}
- {2*\cnta-1}
- {\xintiPow {\the\cnta}{\cnta}/\xintFac{\cnta}}
- {\ratioexp{\the\cnta}}}%
-\edef\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}
-
-\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\centeredline{|\xintRationalSeries {A}{B}{\first}{\ratio{\g}}|}
-\centeredline{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.
-\dverb|@
-\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.
-\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|
-
-\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/12)) for a=1,..., 12.
-\begin{multicols}{3}\raggedcolumns
- \cnta 1
- \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}
- % to see how they look like...
- % \loop
- % \noindent\printnumber{%
- % \xintRationalSeriesX {0}{9}{\firstterm}{\ratioexp}
- % {\xintPowerSeries{1}{10}{\coefflog}{\the\cnta[-2]}}}\dots
- % \endgraf
- % \ifnum\cnta < 60 \advance \cnta 1 \repeat
-
-These completely exact operations rapidly create numbers with many digits. Let
-us print in full the raw fractions created by the operation illustrated above:
-
-\edef\z{\xintRationalSeriesX {0}{9}{\firstterm}
-{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1[-1]}}}
-
-|E(L(1[-1]))=|\digitstt{\printnumber{\z}} (length of numerator:
-\xintLen {\xintNumerator \z})
-
-\edef\z{\xintRationalSeriesX {0}{9}{\firstterm}
-{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{12[-2]}}}
-
-|E(L(12[-2]))=|\digitstt{\printnumber{\z}} (length of numerator:
-\xintLen {\xintNumerator \z})
-
-\edef\z{\xintRationalSeriesX {0}{9}{\firstterm}
-{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{123[-3]}}}
-
-|E(L(123[-3]))=|\digitstt{\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:
-
-
-\edef\z{\xintRationalSeriesX {0}{9}{\firstterm}
-{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/7}}}
-
-|E(L(1/7))=|\digitstt{\printnumber{\z}} (length of numerator:
-\xintLen {\xintNumerator \z}; length of denominator:
-\xintLen {\xintDenominator \z})
-
-\edef\z{\xintRationalSeriesX {0}{9}{\firstterm}
-{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/71}}}
-
-|E(L(1/71))=|\digitstt{\printnumber{\z}} (length of numerator:
-\xintLen {\xintNumerator \z}; length of denominator:
-\xintLen {\xintDenominator \z})
-
-
-\edef\z{\xintRationalSeriesX {0}{9}{\firstterm}
-{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/712}}}
-
-|E(L(1/712))=|\digitstt{\printnumber{\z}} (length of numerator:
-\xintLen {\xintNumerator \z}; length of denominator:
-\xintLen {\xintDenominator \z})
-
-% \pdfresettimer
-% \edef\w{\xintDenominator{\xintIrr{\z}}}
-% \the\pdfelapsedtime
-
-For info the last fraction put into irreducible form still has 288 digits in its
-denominator.\footnote{putting this fraction in irreducible form takes more time
- than is typical of the other computations in this document; so exceptionally I
- have hard-coded the 288 in the document source.} 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!
-Briefly said, imagine that the rules of the game make the programmer like a
-security guard at an airport scanning machine: a never-ending flux of passengers
-keep on arriving and all you can do is re-shuffle the first nine of them,
-organize marriages among some, execute some, move children farther back among
-the first nine only. If a passenger comes along with many hand luggages, this
-will slow down the process even if you move him to ninth position, because
-sooner or later you will have to digest him, and the children will be big too.
-There is no way to move some guy out of the file and to a discrete interrogatory
-room for separate treatment or to give him/her some badge saying ``I left my
-stuff in storage box 357''.
-
-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, also has
-\csbxint{FxPtPowerSeries} for fixed-point computations.
-
-Update: release |1.08a| of \xintseriesname now includes 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). Starting with release |1.04| a Horner scheme for polynomial
-evaluation is used, which has the advantage to avoid a denominator build-up
-which was plaguing the |1.03| version. \footnote{with powers |f\string^k|, from
- |k=0| to |N|, a denominator |d| of |f| became
- |d\string^\string{1+2+\dots+N\string}|, which is bad. With the |1.04| method,
- the part of the denominator originating from |f| does not accumulate to more
- than |d\string^N|. }
-
-\begin{framed}
- Note: as soon as the coefficients look like factorials, it is more efficient
- to use the \csbxint{RationalSeries} macro whose evaluation, also based on a
- similar Horner scheme, will avoid a denominator build-up originating in the
- coefficients themselves.
-\end{framed}
-
-\dverb|@
-\def\geom #1{1[0]} % the geometric series
-\def\f {5/17[0]}
-\[ \sum_{n=0}^{n=20} \Bigl(\frac 5{17}\Bigr)^n
- =\xintFrac{\xintIrr{\xintPowerSeries {0}{20}{\geom}{\f}}}
- =\xintFrac{\xinttheexpr (17^21-5^21)/12/17^20\relax}\]|%
-\def\geom #1{1[0]} % the geometric series
-\def\f {5/17[0]} %
-\[ \sum_{n=0}^{n=20} \Bigl(\frac 5{17}\Bigr)^n
- =\xintFrac{\xintIrr{\xintPowerSeries {0}{20}{\geom}{\f}}}
- =\xintFrac{\xinttheexpr (17^21-5^21)/12/17^20\relax}\]
-
-\dverb|@
-\def\coefflog #1{1/#1[0]}% 1/n
-\def\f {1/2[0]}%
-\[ \log 2 \approx \sum_{n=1}^{20} \frac1{n\cdot 2^n}
- = \xintFrac {\xintIrr {\xintPowerSeries {1}{20}{\coefflog}{\f}}}\]
-\[ \log 2 \approx \sum_{n=1}^{50} \frac1{n\cdot 2^n}
- = \xintFrac {\xintIrr {\xintPowerSeries {1}{50}{\coefflog}{\f}}}\]|%
-\def\coefflog #1{1/#1[0]} % 1/n
-\def\f {1/2[0]}%
-\[ \log 2 \approx \sum_{n=1}^{20} \frac1{n\cdot 2^n}
- = \xintFrac {\xintIrr {\xintPowerSeries
- {1}{20}{\coefflog}{\f}}}\]
-\[ \log 2 \approx \sum_{n=1}^{50} \frac1{n\cdot 2^n}
- = \xintFrac {\xintIrr {\xintPowerSeries {1}{50}{\coefflog}{\f}}}\]
-\dverb|@
-\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|
-\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}
-\dverb|@
-%\def\coeffarctg #1{1/\the\numexpr\xintMON{#1}*(2*#1+1)\relax }%
-\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)
-% NOTE in passing this aspect of \numexpr:
-% **** \numexpr -(1)\relax does not work!!! ****
-\def\f {1/25[0]}% 1/5^2
-\[\mathrm{Arctg}(\frac15)\approx
- \frac15\sum_{n=0}^{15} \frac{(-1)^n}{(2n+1)25^n}
-= \xintFrac{\xintIrr {\xintDiv
- {\xintPowerSeries {0}{15}{\coeffarctg}{\f}}{5}}}\]|
-
-\def\coeffarctg #1{1/\the\numexpr\ifodd #1 -2*#1-1\else2*#1+1\fi\relax }%
-\def\f {1/25[0]}% 1/5^2
-\[\mathrm{Arctg}(\frac15)\approx
- \frac15\sum_{n=0}^{15} \frac{(-1)^n}{(2n+1)25^n}
-= \xintFrac{\xintIrr {\xintDiv
- {\xintPowerSeries {0}{15}{\coeffarctg}{\f}}{5}}}\]
-
-\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.
-%
-\dverb|@
-\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.
-\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|
-
-\cnta 0
-\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}
-
-
-\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}
-\dverb|@
-\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: |
-
-\centeredline{|\xintFxPtPowerSeries {0}{19}{\coeffexp}{\f}{25}=|
-\digitstt{\xintFxPtPowerSeries {0}{19}{\coeffexp}{\f}{25}}}
-\edef\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:
-\centeredline{|\xintPowerSeries {0}{19}{\coeffexp}{\f}| ${}=
- \displaystyle\xintFrac{\z}$%
- \vphantom{\vrule height 20pt depth 12pt}}%
-\centeredline{${}=\xintTrunc {30}{\z}\dots$} 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
-\centeredline{|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.
-\dverb|@
-\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
-\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|
-
-\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): }}%
-\digitstt{\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}
-
-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.
-\dverb|@
-\loop
-\noindent \hbox to 2.5cm {\hss\texttt{D(\the\cnta/100): }}%
-\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|
-
-\begin{multicols}2
-\loop
-\noindent \hbox to 2.5cm {\hss\texttt{D(\the\cnta/100): }}%
-\digitstt{\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}
-
-
-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\etype{\Ff\Ff}
-precision given by the optional parameter |P| or by the current setting of
-|\xintDigits|.
-
-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]}%
-
-\dverb+@
-\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}%
-\xintFloatPowerSeries [8]{1}{30}{\coefflog}{-1/2[0]}+%
-\centeredline{\digitstt{\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]}%
-
-\dverb+@
-\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]}}+%
-\centeredline{\digitstt{\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): \centeredline{\digitstt{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
-
-\dverb|@
-\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|
-
-\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) with #1 digits precision
-{% this #1 may be a count register, if desired
- \romannumeral0\expandafter\LogTwoDoIt \expandafter
- {\the\numexpr #1*150/143\expandafter}\expandafter % Nb Terms for 1/9
- {\the\numexpr #1*100/129\expandafter}\expandafter % Nb Terms for 13/256
- {\the\numexpr #1\relax }%
-}%
-\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
- \xinttrunc {#3}
- {\xintAdd
- {\xintMul {2}{\xintFxPtPowerSeries {1}{#2}{\coefflog}{\xa}{#3}}}
- {\xintMul {5}{\xintFxPtPowerSeries {1}{#1}{\coefflog}{\xb}{#3}}}%
- }%
-}%
-
-\noindent $\log 2 \approx {}$\digitstt{\LogTwo {60}\dots}\endgraf
-\noindent\phantom{$\log 2$}${}\approx{}$\digitstt{\printnumber{\LogTwo
- {65}}\dots}\endgraf
-\noindent\phantom{$\log 2$}${}\approx{}$\digitstt{\printnumber{\LogTwo
- {70}}\dots}\endgraf
-
-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}.
-\dverb|@
-\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}{}
-\dverb|@
-% 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]}%
-% the above computes (-1)^n/(2n+1).
-\def\xa {1/25[0]}% 1/5^2, the [0] for (infinitesimally) faster parsing
-\def\xb {1/57121[0]}% 1/239^2, the [0] for faster parsing
-\def\Machin #1{% \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%
-% #4: digits to keep after decimal point for final printing
-% #3=#4+3: digits for evaluation of the necessary number of terms
-% to be kept in the arctangent series, also used to truncate each
-% individual summand.
-{\xinttrunc {#4} % lowercase macro to match the initial \romannumeral0.
- {\xintSub
- {\xintMul {16/5}{\xintFxPtPowerSeries {0}{#1}{\coeffarctg}{\xa}{#3}}}
- {\xintMul {4/239}{\xintFxPtPowerSeries {0}{#2}{\coeffarctg}{\xb}{#3}}}%
- }}%
-\[ \pi = \Machin {60}\dots \]|
-
-\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}
-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.
-\dverb|@
-\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}}}%
-}}%|
-
-\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}}}%
-}}%
-
-Let us use this variant for a loop showing the build-up of digits:
-\dverb|@
- \cnta 0 % previously declared \count register
- \loop
- \MachinBis{\cnta} \endgraf % Plain's \loop does not accept \par
- \ifnum\cnta < 30 \advance\cnta 1 \repeat|
-
-\begin{multicols}{2}
- \cnta 0 % previously declared \count register
- \loop \noindent
- \centeredline{\digitstt{\MachinBis{\cnta}}}%
- \ifnum\cnta < 30
- \advance\cnta 1 \repeat
-\end{multicols}
-
-
-\hypertarget{Machin1000}{}
-%
-You want more digits and have some time? compile this copy of the
-\hyperlink{MachinCode}{|\char 92 Machin|} with |etex| (or |pdftex|):
-%
-\dverb|@
-% 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
- \digitstt{42} seconds; the \texttt{1.09j} division is much faster with small
- denominators as occurs here with \digitstt{\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}
-
-This package was first included in release |1.04| of the \xintname bundle.
-
-
-\localtableofcontents
-
-
-\subsection{Package overview}
-
-A \emph{simple} continued fraction has coefficients
-|[c0,c1,...,cN]| (usually called partial quotients, but I really
-dislike this entrenched terminology), where |c0| is a positive or
-negative integer and the others are positive integers. As we will
-see it is possible with \xintcfracname to specify the coefficient
-function |c:n->cn|. Note that the index then starts at zero as
-indicated. With the |amsmath| macro |\cfrac| one can display such a
-continued fraction as
-\[ c_0 + \cfrac{1}{c_1+\cfrac1{c_2+\cfrac1{c_3+\cfrac1{\ddots}}}}\]
-Here is a concrete example:
-\[ \xintFrac {208341/66317}=\xintCFrac {208341/66317}\] But the
-difference with |amsmath|'s |\cfrac| is that this was input as
-\centeredline{|\[ \xintFrac {208341/66317}=\xintCFrac
- {208341/66317} \]|} The command \csbxint{CFrac} produces in two
-expansion steps the whole thing with the many chained |\cfrac|'s and all
-necessary braces, ready to be printed, in math mode. This is \LaTeX{}
-only and with the |amsmath| package (we shall mention another method for
-Plain \TeX{} users of |amstex|).
-
-A \emph{generalized} continued fraction has the same structure but
-the numerators are not restricted to be ones, and numbers used in
-the continued fraction may be arbitrary, also fractions,
-irrationals, indeterminates. The \emph{centered} continued
-fraction associated to a rational number is an
-example:
-\[ \xintFrac {915286/188421}=\xintGCFrac {\xintFtoCC {915286/188421}}
-=\xintCFrac {915286/188421}\]
- \centeredline{|\[ \xintFrac {915286/188421}=\xintGCFrac {\xintFtoCC
- {915286/188421}} \]|}
-The command \csbxint{GCFrac}, contrarily to
-\csbxint{CFrac}, does not compute anything, it just typesets. Here, it is the
-command \csbxint{FtoCC} which did the computation of
-the centered continued fraction of |f|. Its output has the `inline format'
-described in the next paragraph. In the display, we also used \csa{xintCFrac}
-(code not shown), for comparison of the two types of continued fractions.
-
-A generalized continued fraction may be input `inline' as:
-\centeredline{|a0+b0/a1+b1/a2+b2/...../a(n-1)+b(n-1)/an|}%
-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). Or, they may
-be macros expanding (in two steps) to some number or fractional number.
-\centeredline{|\xintGCFrac {1+-1/57+\xintPow {-3}{7}/\xintQuo {132}{25}}|}
-\[ \xintFrac{\xintGCtoF {1+-1/57+\xintPow {-3}{7}/\xintQuo {132}{25}}}=
- \xintGCFrac {1+-1/57+\xintPow {-3}{7}/\xintQuo {132}{25}}\]
-The left hand side was obtained with the following code:
-\centeredline{|\xintFrac{\xintGCtoF {1+-1/57+\xintPow {-3}{7}/\xintQuo
- {132}{25}}}|}
-It uses the macro \csbxint{GCtoF} to convert a generalized fraction from the
-`inline format' to the fraction it evaluates to.
-
-A simple continued fraction is a special case of a generalized continued
-fraction and may be input as such to macros expecting the `inline format', for
-example |-7+1/6+1/19+1/1+1/33|. There is a simpler comma separated format:
-\centeredline
-{|\xintFrac{\xintCstoF{-7,6,19,1,33}}=\xintCFrac{\xintCstoF{-7,6,19,1,33}}|}
-\[
-\xintFrac{\xintCstoF{-7,6,19,1,33}}=\xintCFrac{\xintCstoF{-7,6,19,1,33}}\] This
-comma separated format may also be used with fractions among the coefficients:
-in that case, computing with \csbxint{FtoCs} from the resulting |f|
-its real coefficients will give a new comma separated list
-with only integers. This list has no spaces: the spaces in the display below
-arise from the math mode processing.
-\centeredline{|\xintFrac{1084483/398959}=[\xintFtoCs{1084483/398959}]|}
-\[\xintFrac{1084483/398959}=[\xintFtoCs{1084483/398959}]\]
-If one prefers other separators, one can use \csbxint{FtoCx} whose first
-argument will be the separator to be used.
-\centeredline{|\xintFrac{2721/1001}=\xintFtoCx {+1/(}{2721/1001})\cdots)|}
-\[\xintFrac{2721/1001}=\xintFtoCx {+1/(}{2721/1001})\cdots)\]
-People using Plain \TeX{} and |amstex| can achieve the same effect as
-|\xintCFrac| with:
-|$$\xintFwOver{2721/1001}=\xintFtoCx {+\cfrac1\\ }{2721/1001}\endcfrac$$|
-
-Using \csa{xintFtoCx} with first argument an empty pair of braces |{}| will
-return the list of the coefficients of the continued fraction of |f|, without
-separator, and each one enclosed in a pair of group braces. This can then be
-manipulated by the non-expandable macro \csbxint{AssignArray} or the expandable
-ones \csbxint{Apply} and \csbxint{ListWithSep}.
-
-As a shortcut to using \csa{xintFtoCx} with separator |1+/|, there is
-\csbxint{FtoGC}:
-\centeredline{|2721/1001=\xintFtoGC {2721/1001}|}%
-\centeredline{\digitstt{2721/1001=\xintFtoGC {2721/1001}}}
-Let us compare in that case with the output of \csbxint{FtoCC}:
-\centeredline{|2721/1001=\xintFtoCC {2721/1001}|}%
-\centeredline{\digitstt{2721/1001=\xintFtoCC {2721/1001}}}
-
-The `|\printnumber|' macro which we use to print long numbers can also
-be useful on long continued fractions.
-\centeredline{|\printnumber{\xintFtoCC {35037018906350720204351049/%|}%
-\centeredline{|244241737886197404558180}}|}%
-\digitstt{\printnumber{\xintFtoCC {35037018906350720204351049/244241737886197404558180}}}.
-If we apply \csbxint{GCtoF} to this generalized continued fraction, we
-discover that the original fraction was reducible:
-\centeredline{|\xintGCtoF
- {143+1/2+...+-1/9}|\digitstt{=\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{\fontfamily{lmtt}\selectfont #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:
-\centeredline{|\xintGCtoF {143+1/2+...+-1/6}|\digitstt{=\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:
-\[ \begin{vmatrix}
- ^2897319801297630107^ & ^328124887710626729^\\
- ^20197107104701740^ & ^2287346221788023^
- \end{vmatrix} = \mbox{\digitstt{\xintiSub {\xintiMul {2897319801297630107}{2287346221788023}}{\xintiMul{20197107104701740}{328124887710626729}}}}\]
-
-\endgroup
-More generally 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. This list can then be treated either
-with \csa{xint\-AssignArray}, or \csa{xintListWithSep}, or any other way
-(but then, some \TeX{} programming knowledge will be necessary). Here
-is an example:
-
-\noindent
-\centeredline{|$$\xintFrac{915286/188421}\to \xintListWithSep {,}%|}%
-\centeredline{|{\xintApply{\xintFrac}{\xintFtoCv{915286/188421}}}$$|}
-\[ \xintFrac{915286/188421}\to \xintListWithSep {,}
-{\xintApply\xintFrac{\xintFtoCv{915286/188421}}}\]
-\centeredline{|$$\xintFrac{915286/188421}\to \xintListWithSep {,}%|}%
-\centeredline{|{\xintApply{\xintFrac}{\xintFtoCCv{915286/188421}}}$$|}
-\[ \xintFrac{915286/188421}\to \xintListWithSep {,}
-{\xintApply\xintFrac{\xintFtoCCv{915286/188421}}}\] 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:\centeredline{|\newcommand{\mymacro}[1]|%
- |{$\xintFrac{#1}=[\xintFtoCs{#1}]$\vtop to 6pt{}}|}%
-Next, we use the following code:
-\centeredline{|$\xintFrac{49171/18089}\to{}$|}%
-\centeredline{|\xintListWithSep {,
- }{\xintApply{\mymacro}{\xintFtoCv{49171/18089}}}|}
-It produces:\par
-\noindent$ \xintFrac{49171/18089}\to {}$\xintListWithSep {,
- }{\xintApply{\mymacro}{\xintFtoCv{49171/18089}}}.
-
-
-\def\cn #1{\xintiPow {2}{#1}}%
-
-The macro \csbxint{CntoF} allows to specify the coefficients as
-functions of the index. The values to which expand the
-coefficient function do not have to be integers. \centeredline{|\def\cn
- #1{\xintiPow {2}{#1}}% 2^n|}%
- \centeredline{|\[\xintFrac{\xintCntoF {6}{\cn}}=\xintCFrac
- [l]{\xintCntoF {6}{\cn}}\]|}%
-\[\xintFrac{\xintCntoF {6}{\cn}}=\xintCFrac [l]{\xintCntoF
- {6}{\cn}}\]
-Notice the use of the optional argument |[l]| to \csa{xintCFrac}. Other
-possibilities are |[r]| and (default) |[c]|.
-\def\cn #1{\xintPow {2}{-#1}}%
-\centeredline{|\def\cn #1{\xintPow {2}{-#1}}% 1/2^n|}%
-\centeredline{%
-|\[\xintFrac{\xintCntoF {6}{\cn}} = \xintGCFrac [r]{\xintCntoGC {6}{\cn}}|}%
-\centeredline{| = [\xintFtoCs {\xintCntoF {6}{\cn}}]\]|}%
-\[\xintFrac{\xintCntoF {6}{\cn}}=\xintGCFrac [r]{\xintCntoGC {6}{\cn}}=
- [\xintFtoCs {\xintCntoF {6}{\cn}}]\]
-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. The following initial portion of a generalized continued
-fraction for $\pi$:
-\def\an #1{\the\numexpr 2*#1+1\relax }%
-\def\bn #1{\the\numexpr (#1+1)*(#1+1)\relax }%
-\[ \xintFrac{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}} =
- \cfrac{4}{\xintGCFrac{\xintGCntoGC {5}{\an}{\bn}}} =
-\xintTrunc {10}{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}}\dots\]
-was obtained with this code:
-\dverb|@
-\def\an #1{\the\numexpr 2*#1+1\relax }%
-\def\bn #1{\the\numexpr (#1+1)*(#1+1)\relax }%
-\[ \xintFrac{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}} =
- \cfrac{4}{\xintGCFrac{\xintGCntoGC {5}{\an}{\bn}}} =
-\xintTrunc {10}{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}}\dots\]|
-
-We see that the quality of approximation is not fantastic compared to the simple
-continued fraction of $\pi$ with about as many terms:
-\dverb|@
-\[ \xintFrac{\xintCstoF{3,7,15,1,292,1,1}}=
- \xintGCFrac{3+1/7+1/15+1/1+1/292+1/1+1/1}=
- \xintTrunc{10}{\xintCstoF{3,7,15,1,292,1,1}}\dots\]|
-\[ \xintFrac{\xintCstoF{3,7,15,1,292,1,1}}=
-\xintGCFrac{3+1/7+1/15+1/1+1/292+1/1+1/1}=
-\xintTrunc{10}{\xintCstoF{3,7,15,1,292,1,1}}\dots\]
-
-\hypertarget{e-convergents}{To}
-conclude this overview of most of the package functionalities, let us explore
-the convergents of Euler's number $e$.
-\dverb|@
-\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\texttt{\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}}}}|
-
-\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}
-
-
-\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\digitstt{\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}}}}
-
-% \def\testmacro #1{\xintTrunc {30}{\xintAdd {1[0]}{#1}}\xintAdd {1[0]}{#1}}
-% \pdfresettimer
-% \edef\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).
-\dverb|@
-\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|
-
-\oodef\z {\xintCntoF {199}{\cn}}%
-
-\begingroup\parindent 0pt \leftskip 2.5cm
-\indent\llap {Numerator = }\digitstt{\printnumber{\xintNumerator\z}}\par
-\indent\llap {Denominator = }\digitstt{\printnumber{\xintDenominator\z}}\par
-\indent\llap
- {Expansion = }\digitstt{\printnumber{\xintTrunc{268}\z}\dots}\par\endgroup
-
-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 $-1$.
-
-\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.
-
-\subsection{\csbh{xintGCFrac}}\label{xintGCFrac}
-
-\csa{xintGCFrac}|{a+b/c+d/e+f/g+h/...}|\etype{f} uses similarly |\cfrac| to
-typeset a
-generalized continued fraction in inline format. It admits the same optional
-argument as \csa{xintCFrac}.
-\centeredline{|\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}\]|}
-\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}\]
-As can be seen this is typesetting macro, although it does proceed to the
-evaluation of the coefficients themselves. See \csbxint{GCtoF} if you are
-impatient to see this fraction computed. Numerators and denominators are made
-arguments to the
-\csbxint{Frac} macro.
-
-\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
-\centeredline{|\xintGCtoGCx :;{1+2/3+4/5+6/7}| gives \xintGCtoGCx
- :;{1+2/3+4/5+6/7}} Plain \TeX{}+|amstex| users may be interested in:\par
-\dverb|@
-$$\xintGCtoGCx {+\cfrac}{\\}{a+b/...}\endcfrac$$
-$$\xintGCtoGCx {+\cfrac\xintFwOver}{\\\xintFwOver}{a+b/...}\endcfrac$$|
-
-
-\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|.
-\centeredline{%
- |\[ \xintSignedFrac{-5262046/89233} = [\xintFtoCs{-5262046/89233}]\]|}%
-\[ \xintSignedFrac{-5262046/89233} = [\xintFtoCs{-5262046/89233}]\]
-
-
-\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|, withing group braces and separated with
-the help of |sep|. \centeredline{|$$\xintFtoCx {+\cfrac1\\ }{f}\endcfrac$$|}
-will display the continued fraction in |\cfrac| format, with Plain \TeX{} and
-|amstex|.
-
-\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.
-\centeredline{|566827/208524=\xintFtoGC {566827/208524}|}%
-\centeredline{566827/208524=\xintFtoGC {566827/208524}}
-
-\subsection{\csbh{xintFtoCC}}\label{xintFtoCC}
-
-\csa{xintFtoCC}|{f}|\etype{\Ff} returns the `centered' continued fraction of
-|f|, in `inline format'. \centeredline{|566827/208524=\xintFtoCC
- {566827/208524}|}%
-\centeredline{566827/208524=\xintFtoCC {566827/208524}} \centeredline{%
- |\[\xintFrac{566827/208524} = \xintGCFrac{\xintFtoCC{566827/208524}}\]|}%
-\[\xintFrac{566827/208524} = \xintGCFrac{\xintFtoCC{566827/208524}}\]
-
-\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}. \centeredline{%
- |\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCv{5211/3748}}}\]|}%
-\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCv{5211/3748}}}\]
-
-\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}. \centeredline{%
- |\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCCv{5211/3748}}}\]|}%
-\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCCv{5211/3748}}}\]
-
-\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.
-\centeredline{|\[\xintGCFrac {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}|}%
-\centeredline{|=\xintSignedFrac{\xintCstoF {-1,3,-5,7,-9,11,-13}}|}%
-\centeredline{|=\xintSignedFrac{\xintGCtoF
- {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}}\]|}%
-\[\xintGCFrac {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}=
-\xintSignedFrac{\xintCstoF {-1,3,-5,7,-9,11,-13}}
-=\xintSignedFrac{\xintGCtoF {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}}\]
-\centeredline{|\xintGCFrac{{1/2}+1/{1/3}+1/{1/4}+1/{1/5}}= |}%
-\centeredline{| \xintFrac{\xintCstoF {1/2,1/3,1/4,1/5}}|}%
-\[\xintGCFrac{{1/2}+1/{1/3}+1/{1/4}+1/{1/5}}=
-\xintFrac{\xintCstoF {1/2,1/3,1/4,1/5}}\] 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{xintCstoCv}}\label{xintCstoCv}
-
-\csa{xintCstoCv}|{a,b,c,d,...,z}|\etype{f} returns the list of the corresponding
-convergents. 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.
-\centeredline{|\xintListWithSep:{\xintCstoCv{1,2,3,4,5,6}}|}%
-\centeredline{\digitstt{\xintListWithSep:{\xintCstoCv{1,2,3,4,5,6}}}}
-\centeredline{|\xintListWithSep:{\xintCstoCv{1,1/2,1/3,1/4,1/5,1/6}}|}%
-\centeredline{\digitstt{\xintListWithSep:{\xintCstoCv{1,1/2,1/3,1/4,1/5,1/6}}}}
-% j'ai retiré les [0] à partir de la version 1.09b, le 3 octobre 2013.
-\centeredline{|\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintCstoCv|}%
- \centeredline{|{\xintPow {-.3}{-5},7.3/4.57,\xintCstoF{3/4,9,-1/3}}}}\]|}%
-\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintCstoCv
- {\xintPow {-.3}{-5},7.3/4.57,\xintCstoF{3/4,9,-1/3}}}}\]
-
-
-\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.
-\centeredline{|\[\xintGCFrac {\xintCstoGC {-1,1/2,-1/3,1/4,-1/5}}|}%
-\centeredline{|=\xintSignedFrac {\xintCstoF {-1,1/2,-1/3,1/4,-1/5}}\]|}%
-\[\xintGCFrac {\xintCstoGC {-1,1/2,-1/3,1/4,-1/5}} =
-\xintSignedFrac{\xintCstoF {-1,1/2,-1/3,1/4,-1/5}}\]
-
-\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. \dverb|@
-\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}} =
-\xintFrac{\xintGCtoF {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}} =
-\xintFrac{\xintIrr{\xintGCtoF
- {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}}}\]|
-\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}} =
-\xintFrac{\xintGCtoF {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}} =
-\xintFrac{\xintIrr{\xintGCtoF {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}}}\]
-\dverb|@
-\[ \xintGCFrac{{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}} =
- \xintFrac{\xintGCtoF {{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}}} \]|
-\[ \xintGCFrac{{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}} =
- \xintFrac{\xintGCtoF {{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}}} \]
-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{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|.
-\dverb|@
-\[\xintListWithSep{,}{\xintApply\xintFrac
- {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}\]
-\[\xintListWithSep{,}{\xintApply\xintFrac{\xintApply\xintIrr
- {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}}\]|
-\[\xintListWithSep{,}{\xintApply\xintFrac
- {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}\]
-\[\xintListWithSep{,}{\xintApply\xintFrac{\xintApply\xintIrr
- {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}}\]
-
-\subsection{\csbh{xintCntoF}}\label{xintCntoF}
-
-\def\macro #1{\the\numexpr 1+#1*#1\relax}
-
-\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|. \centeredline{%
- |\def\macro #1{\the\numexpr 1+#1*#1\relax}\xintCntoF {5}{\macro}|}%
-\centeredline{\digitstt{\xintCntoF {5}{\macro}}}
-
-\subsection{\csbh{xintGCntoF}}\label{xintGCntoF}
-
-\def\coeffA #1{\the\numexpr #1+4-3*((#1+2)/3)\relax }%
-\def\coeffB #1{\xintMON{#1}}% (-1)^n
-
-\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|.
-\[\xintGCFrac{\xintGCntoGC {6}{\coeffA}{\coeffB}}
-= \xintFrac{\xintGCntoF {6}{\coeffA}{\coeffB}}\]
-There is also \csbxint{GCntoGC} to get the `inline format' continued
-fraction. The previous display was obtained with:
-\centeredline{|\def\coeffA #1{\the\numexpr #1+4-3*((#1+2)/3)\relax }%|}%
-\centeredline{|\def\coeffB #1{\xintMON{#1}}% (-1)^n|}%
-\centeredline{|\[\xintGCFrac{\xintGCntoGC {6}{\coeffA}{\coeffB}}|}%
-\centeredline{| = \xintFrac{\xintGCntoF {6}{\coeffA}{\coeffB}}\]|}
-
-
-\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|. \centeredline{%
- |\def\macro #1{\the\numexpr 1+#1*#1\relax}|}%
-\centeredline{|\xintCntoCs {5}{\macro}|\digitstt{->\xintCntoCs {5}{\macro}}}%
-\centeredline{|\[\xintFrac{\xintCntoF {5}{\macro}}=\xintCFrac{\xintCntoF
- {5}{\macro}}\]|}%
-\[ \xintFrac{\xintCntoF
- {5}{\macro}}=\xintCFrac{\xintCntoF {5}{\macro}}\]
-
-\subsection{\csbh{xintCntoGC}}\label{xintCntoGC}
-
-\def\macro #1{\the\numexpr\ifodd#1 -1-#1\else1+#1\fi\relax/%
- \the\numexpr 1+#1*#1\relax}
-%
-\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. \centeredline{%
- |\def\macro #1{\the\numexpr\ifodd#1 -1-#1\else1+#1\fi\relax/%|}%
- \centeredline{|\the\numexpr 1+#1*#1\relax}|}%
- \centeredline{|\edef\x{\xintCntoGC {5}{\macro}}\meaning\x|}%
- \centeredline{\edef\x{\xintCntoGC {5}{\macro}}\digitstt{\meaning\x}}%
- \centeredline{|\[\xintGCFrac{\xintCntoGC {5}{\macro}}\]|}%
-\[\xintGCFrac{\xintCntoGC {5}{\macro}}\]
-
-\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|. As shown, the coefficients are enclosed into added pairs
-of braces, and may thus be fractions. \dverb|@ \def\an #1{\the\numexpr
- #1*#1*#1+1\relax}%
-\def\bn #1{\the\numexpr \xintiiMON{#1}*(#1+1)\relax}%
-$\xintGCntoGC {5}{\an}{\bn}}=\xintGCFrac {\xintGCntoGC {5}{\an}{\bn}} =
-\displaystyle\xintFrac {\xintGCntoF {5}{\an}{\bn}}$\par|
-
-\def\an #1{\the\numexpr #1*#1*#1+1\relax}%
-\def\bn #1{\the\numexpr \xintiiMON{#1}*(#1+1)\relax}%
-$\xintGCntoGC {5}{\an}{\bn}=\xintGCFrac {\xintGCntoGC {5}{\an}{\bn}}
- = \displaystyle\xintFrac {\xintGCntoF {5}{\an}{\bn}}$\par
-
-
-
-\subsection{\csbh{xintiCstoF}, \csbh{xintiGCtoF}, \csbh{xint\-iCstoCv}, \csbh{xintiGCtoCv}}\label{xintiCstoF}
-\label{xintiGCtoF}
-\label{xintiCstoCv}
-\label{xintiGCtoCv}
-
-The same as the corresponding macros without the `i', but for
-integer-only input. Infinitesimally faster; to notice the higher
-efficiency one would need to use them with an input having (at least)
-hundreds of coefficients.
-
-
-\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 withing
-braces. \dverb|@ \edef\x {\xintGCtoGC {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac
- {6}+\xintCstoF {2,-7,-5}/16}} \meaning\x|
-
-\edef\x {\xintGCtoGC
- {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}+\xintCstoF {2,-7,-5}/16}}
-\digitstt{\meaning\x}
-
-To be honest I have, it seems, forgotten why I wrote this macro in the
-first place.
-
-% will be used by the \lverb things
-
-\def\givesomestretch{%
-\fontdimen2\font=0.33333\fontdimen6\font
-\fontdimen3\font=0.16666\fontdimen6\font
-\fontdimen4\font=0.11111\fontdimen6\font
-}%
-\def\MacroFont{\ttfamily\small\givesomestretch\hyphenchar\font45
- \baselineskip12pt\relax }
-
-
-\ifnum\NoSourceCode=1
-\bigskip
-\begin{framed}
- \ttfamily\small\givesomestretch\hyphenchar\font45 This documentation
- has been compiled without the source code. To produce the
- documentation with the source code included, run "tex xint.dtx" to
- generate xint.tex (if not already available), then thrice latex on
- xint.tex and finally dvipdfmx on xint.dvi (ignore the dvipdfmx
- warnings; see also
- \autoref{sec:install}).
-\end{framed}
-\fi
-
-\makeatletter
-\StopEventually{\end{document}\endinput}
-
-\def\storedlinecounts {}
-\def\StoreCodelineNo #1{\edef\storedlinecounts{%
- \unexpanded\expandafter{\storedlinecounts}%
- {{#1}{\the\c@CodelineNo}}}\c@CodelineNo\z@ }
-
-\makeatother
-
-\newgeometry{hmarginratio=4:3,hscale=0.75}
-
-
-\etocdepthtag.toc {implementation}
-
-\MakePercentIgnore
-%
-% \catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
-% \let</dtx>\relax
-% \def<*xinttools>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
-%</dtx>
-%<*xinttools>
-% \def\MARGEPAGENO{2.5em}
-% \section {Package \xinttoolsnameimp implementation}
-% \label{sec:toolsimp}
-%
-% Release |1.09g| splits off |xinttools.sty| from |xint.sty|.
-%
-% \localtableofcontents
-%
-% \subsection{Catcodes, \protect\eTeX{} and reload detection}
-%
-% The method for package identification and reload detection is copied verbatim
-% from the packages by \textsc{Heiko Oberdiek} (with some modifications starting
-% with release |1.09b|).
-%
-% The method for catcodes was also inspired by these packages, we proceed
-% slightly differently.
-%
-% 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 |xinttools.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 |xinttools.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 % }
- \catcode64=11 % @
- \catcode95=11 % _
- \catcode35=6 % #
- \catcode44=12 % ,
- \catcode45=12 % -
- \catcode46=12 % .
- \catcode58=12 % :
- \expandafter\let\expandafter\x\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{xinttools}{\numexpr not available, aborting input}%
- \aftergroup\endinput
- \else
- \ifx\x\relax % plain-TeX, first loading
- \else
- \def\empty {}%
- \ifx\x\empty % LaTeX, first loading,
- % variable is initialized, but \ProvidesPackage not yet seen
- \else
- \y{xinttools}{I was already loaded, aborting input}%
- \aftergroup\endinput
- \fi
- \fi
- \fi
- \def\ChangeCatcodesIfInputNotAborted
- {%
- \endgroup
- \def\XINT_storecatcodes
- {% takes care of all, to allow more economical code in modules
- \catcode34=\the\catcode34 % " xintbinhex, and 1.09k 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_storecatcodes\noexpand\endinput %
- }%
- \def\XINT_setcatcodes
- {%
- \catcode61=12 % =
- \catcode32=10 % space
- \catcode13=5 % ^^M
- \endlinechar=13 %
- \catcode123=1 % {
- \catcode125=2 % }
- \catcode95=11 % _ (replaces @ everywhere, starting with 1.06b)
- \catcode35=6 % #
- \catcode44=12 % ,
- \catcode45=12 % -
- \catcode46=12 % .
- \catcode58=11 % : (made letter for error cs)
- \catcode60=12 % <
- \catcode62=12 % >
- \catcode43=12 % +
- \catcode42=12 % *
- \catcode40=12 % (
- \catcode41=12 % )
- \catcode47=12 % /
- \catcode96=12 % ` (for ubiquitous \romannumeral-`0 and some \catcode )
- \catcode94=11 % ^
- \catcode36=3 % $
- \catcode91=12 % [
- \catcode93=12 % ]
- \catcode33=11 % !
- \catcode64=11 % @
- \catcode38=12 % &
- \catcode124=12 % |
- \catcode63=11 % ?
- \catcode34=12 % " missing from v < 1.09k although needed in xintbinhex
- }%
- \XINT_setcatcodes
- }%
-\ChangeCatcodesIfInputNotAborted
-\def\XINTsetupcatcodes {% for use by other modules
- \edef\XINT_restorecatcodes_endinput
- {%
- \XINT_storecatcodes\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 {xinttools}%
- [2014/02/05 v1.09ka Expandable and non-expandable utilities (jfB)]%
-% \end{macrocode}
-% \subsection{Token management, constants}
-% \lverb|In 1.09e \xint_undef replaced everywhere by \xint_bye.
-% Release 1.09h makes most everything \long.|
-% \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}%
-\xint_firstofone{\let\XINT_sptoken= } %<- space here!
-\long\def\xint_firstoftwo #1#2{#1}%
-\long\def\xint_secondoftwo #1#2{#2}%
-\long\def\xint_firstoftwo_thenstop #1#2{ #1}%
-\long\def\xint_secondoftwo_thenstop #1#2{ #2}%
-\def\xint_minus_thenstop { -}%
-\long\def\xint_gob_til_R #1\R {}%
-\long\def\xint_gob_til_W #1\W {}%
-\long\def\xint_gob_til_Z #1\Z {}%
-\long\def\xint_bye #1\xint_bye {}%
-\let\xint_relax\relax
-\def\xint_brelax {\xint_relax }%
-\long\def\xint_gob_til_xint_relax #1\xint_relax {}%
-\long\def\xint_afterfi #1#2\fi {\fi #1}%
-\chardef\xint_c_ 0
-\chardef\xint_c_i 1 % 1.09k did not have it, but needed in \xintSeq
-\chardef\xint_c_viii 8
-\newtoks\XINT_toks
-% \end{macrocode}
-% \subsection{ \csh{xintodef}, \csh{xintgodef}, \csh{odef}}
-% \lverb|1.09i. For use in \xintAssign. No parameter text! 1.09j uses \xint...
-% rather than \XINT_.... \xintAssign [o] will use the preexisting \odef if there
-% was one before xint' loading.|
-% \begin{macrocode}
-\def\xintodef #1{\expandafter\def\expandafter#1\expandafter }%
-\ifdefined\odef\else\let\odef\xintodef\fi
-\def\xintgodef {\global\xintodef }%
-% \end{macrocode}
-% \subsection{ \csh{xintoodef}, \csh{xintgoodef}, \csh{oodef}}
-% \lverb|1.09i. Can be prefixed with \global. No parameter text. The alternative
-% $\
-% $null \def\oodef #1#{\def\XINT_tmpa{#1}%$\
-% $null $quad $quad $quad \expandafter\expandafter\expandafter\expandafter$\
-% $null $quad $quad $quad \expandafter\expandafter\expandafter\def$\
-% $null $quad $quad $quad \expandafter\expandafter\expandafter\XINT_tmpa$\
-% $null $quad $quad $quad \expandafter\expandafter\expandafter }%$\
-% could not be prefixed by \global. Anyhow, macro parameter tokens would have to
-% somehow not be seen by expanded stuff, except if designed for it.
-% \xintAssign [oo] (etc...) uses the pre-existing \oodef if there was one. |
-% \begin{macrocode}
-\def\xintoodef #1{\expandafter\expandafter\expandafter\def
- \expandafter\expandafter\expandafter#1%
- \expandafter\expandafter\expandafter }%
-\ifdefined\oodef\else\let\oodef\xintoodef\fi
-\def\xintgoodef {\global\xintoodef }%
-% \end{macrocode}
-% \subsection{ \csh{xintfdef}, \csh{xintgfdef}, \csh{fdef}}
-% \lverb|1.09i. No parameter text! |
-% \begin{macrocode}
-\def\xintfdef #1#2{\expandafter\def\expandafter#1\expandafter
- {\romannumeral-`0#2}}%
-\ifdefined\fdef\else\let\fdef\xintfdef\fi
-\def\xintgfdef {\global\xintfdef }% should be \global\fdef if \fdef pre-exists?
-% \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{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.
-%
-% As in some other places, 1.09e replaces \Z by \xint_bye, although here it is
-% just for coherence of notation as \Z would be perfectly safe. 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 8%
- #2\XINT_revwbr_finish_c 7%
- #3\XINT_revwbr_finish_c 6%
- #4\XINT_revwbr_finish_c 5%
- #5\XINT_revwbr_finish_c 4%
- #6\XINT_revwbr_finish_c 3%
- #7\XINT_revwbr_finish_c 2%
- \R\XINT_revwbr_finish_c 1\Z
-}%
-\def\XINT_revwbr_finish_c #1#2\Z
-{%
- \expandafter\expandafter\expandafter
- \space
- \csname xint_gobble_\romannumeral #1\endcsname
-}%
-% \end{macrocode}
-% \subsection{\csh{xintLength}}
-% \lverb|\xintLength does NOT expand its argument.$\
-% 1.09g adds the missing \xintlength, which was previously called \XINT_length,
-% and suppresses \XINT_Length$\
-% 1.06: improved code is roughly 20$% faster than the one from earlier
-% versions. 1.09a, \xintnum inserted. 1.09e: \Z->\xint_bye as this is called
-% from \xintNthElt, and there it was necessary not to use \Z. Later use of \Z
-% and \W perfectly safe here.|
-% \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\expandafter {\the\numexpr #1+8\relax}%
-}%
-\def\XINT_length_finish_a\xint_relax
- \expandafter\XINT_length_loop\expandafter #1#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 8%
- #2\XINT_length_finish_c 7%
- #3\XINT_length_finish_c 6%
- #4\XINT_length_finish_c 5%
- #5\XINT_length_finish_c 4%
- #6\XINT_length_finish_c 3%
- #7\XINT_length_finish_c 2%
- \W\XINT_length_finish_c 1\Z
-}%
-\edef\XINT_length_finish_c #1#2\Z #3%
- {\noexpand\expandafter\space\noexpand\the\numexpr #3-#1\relax}%
-% \end{macrocode}
-% \subsection{\csh{xintZapFirstSpaces}}
-% \lverb|1.09f, written [2013/11/01].|
-% \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\space\space\noexpand\xint_bye\xint_relax }%
-\xint_firstofone {\long\def\XINT_zapbsp_a #1 } %<- space token here
-{%
-% \end{macrocode}
-% \lverb|If the original #1 started with a space, here #1 will be in fact empty,
-% so the effect will be to remove precisely one space from the original, because
-% the first two space tokens are matched to the ones of the macro parameter
-% text. If the original #1 did not start with a space then the #1 will be this
-% original #1, with its added first space, up to the first <sp><sp> found. The
-% added initial space will stop later the \romannumeral0. And in
-% \xintZapLastSpaces we also carried along a space in order to be able to mix
-% tne two codes in \xintZapSpaces. Testing for \emptiness of #1 is NOT done with
-% an \if test because #1 may contain \if, \fi things (one could use a
-% \detokenize method), and also because xint.sty has a style of its own for
-% doing these things...|
-% \begin{macrocode}
- \XINT_zapbsp_again? #1\xint_bye\XINT_zapbsp_b {#1}%
-% \end{macrocode}
-% \lverb|The #1 above is thus either empty, or it starts with a (char 32) space
-% token followed with a non (char 32) space token and at any rate #1 is
-% protected from brace stripping. It is assumed that the initial input does not
-% contain space tokens of other than 32 as character code.|
-% \begin{macrocode}
-}%
-\long\def\XINT_zapbsp_again? #1{\xint_bye #1\XINT_zapbsp_again }%
-% \end{macrocode}
-% \lverb|In the "empty" situation above, here #1=\xint_bye, else #1 could be
-% some brace things, but unbracing will anyhow not reveal any \xint_bye. When we
-% do below \XINT_zapbsp_again we recall that we have stripped two spaces out of
-% <sp><original #1>, so we have one <sp> less in #1, and when we loop we better
-% not forget to re-insert one initial <sp>.|
-% \begin{macrocode}
-\edef\XINT_zapbsp_again\XINT_zapbsp_b #1{\noexpand\XINT_zapbsp_a\space }%
-% \end{macrocode}
-% \lverb|We now have now gotten rid of the initial spaces, but #1 perhaps extend
-% only to some initial chunk which was delimited by <sp><sp>.|
-% \begin{macrocode}
-\long\def\XINT_zapbsp_b #1#2\xint_relax
- {\XINT_zapbsp_end? #2\XINT_zapbsp_e\empty #2{#1}}%
-% \end{macrocode}
-% \lverb|If the initial chunk up to <sp><sp> (after stripping away the first
-% spaces) was maximal, then #2 above is some spaces followed by \xint_bye, so in
-% the \XINT_zapbsp_end? below it appears as \xint_bye, else the #1 below will
-% not be nor give rise after brace removal to \xint_bye. And then the original
-% \xint_bye in #2 will have the effect that all is swallowed and we continue
-% with \XINT_zapbsp_e. If the chunk was maximal, then the #2 above contains as
-% many space tokens as there were originally at the end.|
-% \begin{macrocode}
-\long\def\XINT_zapbsp_end? #1{\xint_bye #1\XINT_zapbsp_end }%
-% \end{macrocode}
-% \lverb|The #2 starts with a space which stops the \romannumeral.
-% The #1 contains the same number of space tokens there was originally.|
-% \begin{macrocode}
-\long\def\XINT_zapbsp_end\XINT_zapbsp_e\empty #1\xint_bye #2{#2#1}%
-% \end{macrocode}
-% \lverb|&
-% Here the initial chunk was not maximal. So we need to get a second piece
-% all the way up to \xint_bye, we take this opportunity to remove the two
-% initially added ending space tokens. We inserted an \empty to prevent brace
-% removal. The \expandafter get rid of the \empty.|
-% \begin{macrocode}
-\xint_firstofone{\long\def\XINT_zapbsp_e #1 } \xint_bye
- {\expandafter\XINT_zapbsp_f \expandafter{#1}}%
-% \end{macrocode}
-% \lverb|Let's not forget when we glue to reinsert the two intermediate space
-% tokens. |
-% \begin{macrocode}
-\long\edef\XINT_zapbsp_f #1#2{#2\space\space #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 {\space}\noexpand\empty
- #1\space\space\noexpand\xint_bye \xint_relax}%
-% \end{macrocode}
-% \lverb|This creates a delimited macro with two space tokens:|
-% \begin{macrocode}
-\xint_firstofone {\long\def\XINT_zapesp_a #1#2 } %<- second space here
- {\expandafter\XINT_zapesp_b\expandafter{#2}{#1}}%
-% \end{macrocode}
-% \lverb|The \empty from \xintzaplastspaces is to prevent brace removal in the
-% #2 above. The \expandafter chain removes it.|
-% \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 reached the ending space tokens, #3 is a bunch of spaces followed
-% by \xint_bye. So the #1 below will be \xint_bye. In all other cases #1 can not
-% be \xint_bye nor can it give birth to it via brace stripping.|
-% \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. It started
-% with a space token which stops the \romannumeral0. The reason for the space is
-% the recycling of this code in \xintZapSpaces.|
-% \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. We might wonder why in
-% \XINT_zapesp_b we scooped everything up to the end, rather than trying to test
-% if the next thing was a bunch of spaces followed by \xint_bye\xint_relax. But
-% how can we expandably examine what comes next? if we pick up something as
-% undelimited parameter token we risk brace removal and we will never know about
-% it so we cannot reinsert correctly; the only way is to gather a delimited
-% macro parameter and be sure some token will be inside to forbid brace removal.
-% I do not see (so far) any other way than scooping everything up to the end.
-% Anyhow, 99$% of the use cases will NOT have <sp><sp> inside!.|
-% \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].+
-% \begin{macrocode}
-\def\xintZapSpaces {\romannumeral0\xintzapspaces }%
-% \end{macrocode}
-% \lverb|We start like \xintZapStartSpaces.|
-% \begin{macrocode}
-\long\edef\xintzapspaces #1%
- {\noexpand\XINT_zapsp_a \space #1\space\space\noexpand\xint_bye\xint_relax}%
-% \end{macrocode}
-% \lverb|&
-% Once the loop stripping the starting spaces is done, we plug into the
-% \xintZapLast$-Spaces code via \XINT_zapesp_b. As our #1 will always have an
-% initial space, this is why we arranged code of \xintZapLastSpaces to do the
-% same.|
-% \begin{macrocode}
-\xint_firstofone {\long\def\XINT_zapsp_a #1 } %<- space token here
-{%
- \XINT_zapsp_again? #1\xint_bye\XINT_zapesp_b {#1}{}%
-}%
-\long\def\XINT_zapsp_again? #1{\xint_bye #1\XINT_zapsp_again }%
-\long\edef\XINT_zapsp_again\XINT_zapesp_b #1#2{\noexpand\XINT_zapsp_a\space }%
-% \end{macrocode}
-% \subsection{\csh{xintZapSpacesB}}
-% \lverb+1.09f, written [2013/11/01].+
-% \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 expanded (protect the first item with a space if
-% it is not to be 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 through
-% \xintZapSpacesB to strip off 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 8%
- #2\XINT_csvtol_finish_c 7%
- #3\XINT_csvtol_finish_c 6%
- #4\XINT_csvtol_finish_c 5%
- #5\XINT_csvtol_finish_c 4%
- #6\XINT_csvtol_finish_c 3%
- #7\XINT_csvtol_finish_c 2%
- \R\XINT_csvtol_finish_c 1\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|&
-% \xintListWithSep {\sep}{{a}{b}...{z}} returns a \sep b \sep .... \sep z$\
-% Included in release 1.04. The 'sep' can be \par's: the macro
-% xintlistwithsep etc... are all declared long. 'sep' does not have to be a
-% single token. It is not expanded. The list may be a macro and it is expanded.
-% 1.06 modifies the `feature' of returning sep if the list is empty: the output
-% is now empty in that case. (sep was not used for a one element list, but
-% strangely it was for a zero-element list).
-%
-% Use of \Z as delimiter was objectively an error, which I fix here in 1.09e,
-% now the code uses \xint_bye.|
-% \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|&
-% \xintNthElt {i}{{a}{b}...{z}} (or `tokens' abcd...z) returns the i th
-% element (one pair of braces removed). The list is first expanded.
-% First included in release 1.06. With 1.06a, a value of i = 0 (or negative)
-% makes the macro return the length. This is different from \xintLen which is
-% for numbers (checks sign) and different from \xintLength which does not first
-% expand its argument. With 1.09b, only i=0 gives the length, negative values
-% return the i th element from the end. 1.09c has some slightly less quick
-% initial preparation (if #2 is very long, not good to have it twice), I wanted
-% to respect the noexpand directive in all cases, and the alternative would be
-% to define more macros.
-%
-% At some point I turned the \W's into \xint_relax's but forgot to modify
-% accordingly \XINT_nthelt_finish. So in case the index is larger than the
-% number of items the macro returned was an \xint_relax token rather than
-% nothing. Fixed in 1.09e. I also take the opportunity of this fix to replace
-% uses of \Z by \xint_bye. (and as a result I must do the change also in
-% \XINT_length_loop and related macros).
-% |
-% \begin{macrocode}
-\def\xintNthElt {\romannumeral0\xintnthelt }%
-\def\xintNthEltNoExpand {\romannumeral0\xintntheltnoexpand }%
-\def\xintnthelt #1%
-{%
- \expandafter\XINT_nthelt_a\expandafter {\the\numexpr #1}%
-}%
-\def\xintntheltnoexpand #1%
-{%
- \expandafter\XINT_ntheltnoexpand_a\expandafter {\the\numexpr #1}%
-}%
-\long\def\XINT_nthelt_a #1#2%
-{%
- \ifnum #1<0
- \xint_afterfi{\expandafter\XINT_nthelt_c\expandafter
- {\romannumeral0\xintrevwithbraces {#2}}{-#1}}%
- \else
- \xint_afterfi{\expandafter\XINT_nthelt_c\expandafter
- {\romannumeral-`0#2}{#1}}%
- \fi
-}%
-\long\def\XINT_ntheltnoexpand_a #1#2%
-{%
- \ifnum #1<0
- \xint_afterfi{\expandafter\XINT_nthelt_c\expandafter
- {\romannumeral0\xintrevwithbracesnoexpand {#2}}{-#1}}%
- \else
- \xint_afterfi{\expandafter\XINT_nthelt_c\expandafter
- {#2}{#1}}%
- \fi
-}%
-\long\def\XINT_nthelt_c #1#2%
-{%
- \ifnum #2>\xint_c_
- \expandafter\XINT_nthelt_loop_a
- \else
- \expandafter\XINT_length_loop
- \fi {#2}#1\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
- \expandafter\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 #1-8}%
-}%
-\def\XINT_nthelt_silentend #1\xint_bye { }%
-\def\XINT_nthelt_getit #1%
-{%
- \expandafter\expandafter\expandafter\XINT_nthelt_finish
- \csname xint_gobble_\romannumeral\numexpr#1-1\endcsname
-}%
-\long\edef\XINT_nthelt_finish #1#2\xint_bye
- {\noexpand\xint_gob_til_xint_relax #1\noexpand\expandafter\space
- \noexpand\xint_gobble_iii\xint_relax\space #1}%
-% \end{macrocode}
-% \subsection{\csh{xintApply}}
-% \lverb|&
-% \xintApply {\macro}{{a}{b}...{z}} returns {\macro{a}}...{\macro{b}}
-% where each instance of \macro is ff-expanded. The list is first
-% expanded and may thus be a macro. Introduced with release 1.04.
-%
-% Modified in 1.09e to not use \Z but rather \xint_bye.|
-% \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 expanded using \romannumeral-`0. The second
-% argument may be a macro as it is first expanded itself (fully). No braces
-% are added: this allows for example a non-expandable \def in \macro, without
-% having to do \gdef. The list is first expanded. Introduced with release 1.06b.
-% Define \macro to start with a space if it is not expandable or its execution
-% should be delayed only when all of \macro{a}...\macro{z} is ready.
-%
-% Modified in 1.09e to use \xint_bye rather than \Z.|
-% \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. Here also, let's use
-% \xint_bye rather than \Z as delimiter (1.09e; necessary change as #1 is used
-% prior to being expanded, thus \Z might very well arise here as a macro).|
-% \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 expand fully unbraced tokens and swallow arising
-% space tokens until the dust settles. For treating cases
-% {<blank>\x<blank>\y...}, with guaranteed expansion of the \x (which may itself
-% give space tokens), a simpler approach is possible with doubled
-% \romannumeral-`0, this is what I first did, but it had the feature that
-% <sptoken><sptoken>\x would not expand the \x. At any rate, <sptoken>'s before
-% the list terminator z were all correctly moved out of the way, hence the stuff
-% was robust for use in (the then current versions of) \xintApplyInline and
-% \xintFor. Although *two* space tokens would need devilishly prepared input,
-% nevertheless I decided to also survive that, so here the method is a bit more
-% complicated. But it simplifies things on the caller side.|
-% \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 expands (fully) its second (list) argument
-% first, which may thus be encapsulated in a macro.
-%
-% Release 1.09c has a new \xintApplyInline: the new version, while not
-% expandable, is designed to survive when the applied macro closes a group, as
-% is the case in alignemnts when it contains a $& or \\. It uses catcode 3 Z as
-% list terminator. Don't use it among the list items.
-%
-% 1.09d: the bug which was discovered in \xintFor* regarding space tokens at the
-% very end of the item list also was in \xintApplyInline. The new version will
-% expand unbraced item elements and this is in fact convenient to simulate
-% insertion of lists in others.
-%
-% 1.09e: the applied macro is allowed to be long, with items (or the first fixed
-% arguments of he macro, passed together with it as #1 to \xintApplyInline)
-% containing explicit \par's. (1.09g: some missing \long's added)
-%
-% 1.09f: terminator used to be z, now Z (still catcode 3).
-%|
-% \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, expanded fully.
-%
-% 1.09d: [2013/10/22] \xintFor* crashed when a space token was at the very end
-% of the list. It is crucial in this code to not let the ending Z be picked up
-% as a macro parameter without knowing in advance that it is its turn. So, we
-% conscientiously clean out of the way space tokens, but also we ff-expand with
-% \romannumeral-`0 (unbraced) items, a process which may create new space
-% tokens, so it is iterated. As unbraced items are expanded, it is easy to
-% simulate insertion of a list in another.
-% Unbraced items consecutive to an even (non-zero) number of space tokens will
-% not get expanded.
-%
-% 1.09e: [2013/10/29] does this better, no difference between an even or odd
-% number of explicit consecutive space tokens. Normal situations anyhow only
-% create at most one space token, but well. There was a feature in \xintFor (not
-% \xintFor*) from 1.09c that it treated an empty list as a list with one, empty,
-% item. This feature is kept in 1.09e, knowingly... Also, macros are made long,
-% hence the iterated text may contain \par and also the looped over items. I
-% thought about providing some macro expanding to the loop count, but as the
-% \xintFor is not expandable anyhow, there is no loss of generality if the
-% iterated commands do themselves the bookkeeping using a count or a LaTeX
-% counter, and deal with nesting or other problems. I can't do *everything*!
-%
-% 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. Compatibility
-% with \XINT_forever, the necessity to prevent unwanted brace stripping, and
-% shared code with \xintFor*, make this all a delicate balancing act. 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.
-%
-% The 1.09f \xintFor and \xintFor* modified the value of \count 255
-% which was silly, 1.09g used \XINT_count, but requiring a \count only
-% for that was also silly, 1.09h just uses \numexpr (all of that was only to
-% get rid simply of a possibly space in #2...).
-%
-% 1.09ka [2014/02/05] corrects the following bug: \xintBreakFor and
-% \xintBreakForAndDo could not be used in the last iteration.|
-% \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 #1%
-% {%
-% \def\XINT_flet_macro {#1}\XINT_flet_zapsp
-% }%
-\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 has
-% \xintZapSpacesB which helps getting rid of spaces for the \xintrationals case
-% (the other cases end up inside a \numexpr, or \dimexpr, so not necessary).|
-% \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: I don't know yet if {a}{b} is better for the user or worse than
-% (a,b). I prefer the former. I am not very motivated to deal with spaces in the
-% (a,b) approach which is the one (currently) followed here.
-%
-% [2013/11/02] 1.09f: I may not have been very motivated in 1.09c, but since
-% then I developped the \xintZapSpaces/\xintZapSpacesB tools (much to my
-% satisfaction). Based on this, and better parameter texts, \xintForpair and its
-% cousins now handle spaces very satisfactorily (this relies partly on the new
-% \xintCSVtoList which makes use of \xintZapSpacesB). 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+1\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+2\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+3\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,$\
-% \xintAssignArray {a}{b}..{z}\to\U
-%
-% version 1.01 corrects an oversight in 1.0 related to the value of
-% \escapechar at the time of using \xintAssignArray or \xintRelaxArray
-% These macros are non-expandable.
-%
-% In version 1.05a I suddenly see some incongruous \expandafter's in (what is
-% called now) \XINT_assignarray_end_c, which I remove.
-%
-% Release 1.06 modifies the macros created by \xintAssignArray to feed their
-% argument to a \numexpr. Release 1.06a detects an incredible typo in 1.01, (bad
-% copy-paste from
-% \xintRelaxArray) which caused \xintAssignArray to use #1 rather than the #2 as
-% in the correct earlier 1.0 version!!! This went through undetected because
-% \xint_arrayname, although weird, was still usable: the probability to
-% overwrite something was almost zero. The bug got finally revealed doing
-% \xintAssignArray {}{}{}\to\Stuff.
-%
-% With release 1.06b an empty argument (or expanding to empty) to
-% \xintAssignArray is ok.
-%
-% 1.09h simplifies the coding of \xintAssignArray (no more _end_a, _end_b,
-% etc...), and no use of a \count register anymore, and uses \xintiloop in
-% \xintRelaxArray. Furthermore, macros are made long.
-%
-% 1.09i allows an optional parameter \xintAssign [oo] for example, then \oodef
-% rather than \edef is used. Idem for \xintAssignArray. However in the latter
-% case, the global variant is not available, one should use \globaldefs for
-% that.
-%
-% 1.09j: I decide that the default behavior of \xintAssign should be to use
-% \def, not \edef when assigning to a cs an item of the list. This is a
-% breaking change but I don't think anybody on earth is using xint anyhow.
-% Also use of the optional parameter was broken if it was [], [g], [e], [x] as
-% the corresponding \XINT_... macros had not been defined (in the initial
-% version I did not have the XINT_ prefix; then I added it in case \oodef was
-% pre-existing and thus was not redefined by the package which instead had
-% \XINT_oodef, now \xintoodef.)|
-% \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<*xint>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
-%</xinttools>
-%<*xint>
-%
-% \StoreCodelineNo {xinttools}
-%
-% \section{Package \xintnameimp implementation}
-% \label{sec:xintimp}
-%
-% With release |1.09a| all macros doing arithmetic operations and a few more
-% 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. Simplifies the
-% discussion in the documentation too.
-%
-% \localtableofcontents
-%
-% \subsection{Catcodes, \protect\eTeX{} and reload detection}
-%
-% The code for reload detection is copied from \textsc{Heiko
-% Oberdiek}'s packages, and adapted here to check for previous
-% loading of the master \xintname package.
-%
-% The method for catcodes is slightly different, but still
-% 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\space { }%
- \let\z\endgroup
- \expandafter\let\expandafter\x\csname ver@xint.sty\endcsname
- \expandafter\let\expandafter\w\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{xint}{\numexpr not available, aborting input}%
- \aftergroup\endinput
- \else
- \ifx\x\relax % plain-TeX, first loading of xint.sty
- \ifx\w\relax % but xinttools.sty not yet loaded.
- \y{xint}{now issuing \string\input\space xinttools.sty}%
- \def\z{\endgroup\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 % xinttools.sty not yet loaded.
- \y{xint}{now issuing \string\RequirePackage{xinttools}}%
- \def\z{\endgroup\RequirePackage{xinttools}}%
- \fi
- \else
- \y{xint}{I was already loaded, aborting input}%
- \aftergroup\endinput
- \fi
- \fi
- \fi
-\z%
-% \end{macrocode}
-% \subsection{Confirmation of \xinttoolsnameimp loading}
-% \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 % :
- \ifdefined\PackageInfo
- \def\y#1#2{\PackageInfo{#1}{#2}}%
- \else
- \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
- \fi
- \def\empty {}%
- \expandafter\let\expandafter\w\csname ver@xinttools.sty\endcsname
- \ifx\w\relax % Plain TeX, user gave a file name at the prompt
- \y{xint}{Loading of package xinttools failed, aborting input}%
- \aftergroup\endinput
- \fi
- \ifx\w\empty % LaTeX, user gave a file name at the prompt
- \y{xint}{Loading of package xinttools failed, aborting input}%
- \aftergroup\endinput
- \fi
-\endgroup%
-% \end{macrocode}
-% \subsection{Catcodes}
-% \begin{macrocode}
-\XINTsetupcatcodes%
-% \end{macrocode}
-% \subsection{Package identification}
-% \begin{macrocode}
-\XINT_providespackage
-\ProvidesPackage{xint}%
- [2014/02/05 v1.09ka Expandable operations on long numbers (jfB)]%
-% \end{macrocode}
-% \subsection{Token management, constants}
-% \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}%
-\def\xint_gob_til_zero #10{}%
-\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_UDzerominusfork #10-#2#3\krof {#2}%
-\def\xint_UDsignsfork #1--#2#3\krof {#2}%
-% \chardef\xint_c_ 0 % already done in xinttools
-% \chardef\xint_c_i 1 % already done in xinttools
-\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 % will be done in xintfrac
-% \chardef\xinf_c_vii 7 % will be done in xintfrac
-% \chardef\xint_c_viii 8 % already done in xinttools
-\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{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{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}%
-\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}%
-\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\_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$\
-% Délimiteur pour _main: \W\W\W\W\W\W\W\Z avec SEPT \W|
-% \begin{macrocode}
-\def\XINT_cuz #1%
-{%
- \XINT_cuz_loop #1\W\W\W\W\W\W\W\Z%
-}%
-\def\XINT_cuz_loop #1#2#3#4#5#6#7#8%
-{%
- \xint_gob_til_W #8\xint_cuz_end_a\W
- \xint_gob_til_Z #8\xint_cuz_end_A\Z
- \XINT_cuz_check_a {#1#2#3#4#5#6#7#8}%
-}%
-\def\xint_cuz_end_a #1\XINT_cuz_check_a #2%
-{%
- \xint_cuz_end_b #2%
-}%
-\edef\xint_cuz_end_b #1#2#3#4#5\Z
-{%
- \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax
-}%
-\def\xint_cuz_end_A \Z\XINT_cuz_check_a #1{ 0}%
-\def\XINT_cuz_check_a #1%
-{%
- \expandafter\XINT_cuz_check_b\the\numexpr #1\relax
-}%
-\def\XINT_cuz_check_b #1%
-{%
- \xint_gob_til_zero #1\xint_cuz_backtoloop 0\XINT_cuz_stop #1%
-}%
-\def\XINT_cuz_stop #1\W #2\Z{ #1}%
-\def\xint_cuz_backtoloop 0\XINT_cuz_stop 0{\XINT_cuz_loop }%
-% \end{macrocode}
-% \subsection{\csh{xintIsOne}}
-% \lverb|&
-% Added in 1.03. Attention: \XINT_isOne does not do any expansion. Release 1.09a
-% defines \xintIsOne which is more user-friendly. Will be modified if xintfrac
-% is loaded. |
-% \begin{macrocode}
-\def\xintIsOne {\romannumeral0\xintisone }%
-\def\xintisone #1{\expandafter\XINT_isone\romannumeral0\xintnum{#1}\W\Z }%
-\def\XINT_isOne #1{\romannumeral0\XINT_isone #1\W\Z }%
-\def\XINT_isone #1#2%
-{%
- \xint_gob_til_one #1\XINT_isone_b 1%
- \expandafter\space\expandafter 0\xint_gob_til_Z #2%
-}%
-\def\XINT_isone_b #1\xint_gob_til_Z #2%
-{%
- \xint_gob_til_W #2\XINT_isone_yes \W
- \expandafter\space\expandafter 0\xint_gob_til_Z
-}%
-\def\XINT_isone_yes #1\Z { 1}%
-% \end{macrocode}
-% \subsection{\csh{xintNum}}
-% \lverb|&
-% For example \xintNum {----+-+++---+----000000000000003}$\
-% 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).|
-% \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+0\relax
-}%
-\def\XINT_num_NumEight #1#2#3#4#5#6#7#8%
-{%
- \ifnum \numexpr #1#2#3#4#5#6#7#8+0= 0
- \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|
-% \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-\z@
- 0#1\m@ne
- 0-\@ne
- \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{xintSgnFork}}
-% \lverb|Expandable three-way fork added in 1.07. The argument #1 must expand
-% to -1,0 or 1. 1.09i has _afterstop, renamed _thenstop later, for efficiency.|
-% \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{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{xintifSgn}}
-% \lverb|Expandable three-way fork added in 1.09a. Branches expandably
-% depending on whether <0, =0, >0. Choice of branch guaranteed in two steps.
-%
-% The use of \romannumeral0\xintsgn rather than \xintSgn is for matters related
-% to the transformation of the ternary operator : in \xintNewExpr. I hope I have
-% explained there the details because right now off hand I can't recall why.
-%
-% 1.09i has \xint_firstofthreeafterstop (now _thenstop) etc for faster
-% expansion.|
-% \begin{macrocode}
-\def\xintifSgn {\romannumeral0\xintifsgn }%
-\def\xintifsgn #1%
-{%
- \ifcase \romannumeral0\xintsgn{#1}
- \expandafter\xint_secondofthree_thenstop
- \or\expandafter\xint_thirdofthree_thenstop
- \else\expandafter\xint_firstofthree_thenstop
- \fi
-}%
-% \end{macrocode}
-% \subsection{\csh{xintifZero}, \csh{xintifNotZero}}
-% \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. |
-% \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
-}%
-% \end{macrocode}
-% \subsection{\csh{xintifOne}}
-% \lverb|added in 1.09i.|
-% \begin{macrocode}
-\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{xint\-ifFalseAelseB}}
-% \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}}
-% \lverb|&
-% 1.09e
-% \xintifCmp {n}{m}{if n<m}{if n=m}{if n>m}.|
-% \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
-}%
-% \end{macrocode}
-% \subsection{\csh{xintifEq}}
-% \lverb|&
-% 1.09a
-% \xintifEq {n}{m}{YES if n=m}{NO if n<>m}.|
-% \begin{macrocode}
-\def\xintifEq {\romannumeral0\xintifeq }%
-\def\xintifeq #1#2%
-{%
- \if0\xintCmp{#1}{#2}%
- \expandafter\xint_firstoftwo_thenstop
- \else\expandafter\xint_secondoftwo_thenstop
- \fi
-}%
-% \end{macrocode}
-% \subsection{\csh{xintifGt}}
-% \lverb|&
-% 1.09a \xintifGt {n}{m}{YES if n>m}{NO if n<=m}.|
-% \begin{macrocode}
-\def\xintifGt {\romannumeral0\xintifgt }%
-\def\xintifgt #1#2%
-{%
- \if1\xintCmp{#1}{#2}%
- \expandafter\xint_firstoftwo_thenstop
- \else\expandafter\xint_secondoftwo_thenstop
- \fi
-}%
-% \end{macrocode}
-% \subsection{\csh{xintifLt}}
-% \lverb|&
-% 1.09a \xintifLt {n}{m}{YES if n<m}{NO if n>=m}. Restyled in 1.09i|
-% \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
-}%
-% \end{macrocode}
-% \subsection{\csh{xintifOdd}}
-% \lverb|1.09e. Restyled in 1.09i.|
-% \begin{macrocode}
-\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{xintOpp}}
-% \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{xintAbs}}
-% \lverb|Release 1.09a has now \xintiabs which does \xintnum (contrarily to some
-% other i-macros, but similarly as \xintiAdd etc...) and this is
-% inherited by DecSplit, by Sqr, and macros of xintgcd.sty.|
-% \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}
-% \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{xintAdd}}
-% \lverb|Release 1.09a has \xintnum added into \xintiAdd.|
-% \begin{macrocode}
-\def\xintiiAdd {\romannumeral0\xintiiadd }%
-\def\xintiiadd #1%
-{%
- \expandafter\xint_iiadd\expandafter{\romannumeral-`0#1}%
-}%
-\def\xint_iiadd #1#2%
-{%
- \expandafter\XINT_add_fork \romannumeral-`0#2\Z #1\Z
-}%
-\def\xintiAdd {\romannumeral0\xintiadd }%
-\def\xintiadd #1%
-{%
- \expandafter\xint_add\expandafter{\romannumeral0\xintnum{#1}}%
-}%
-\def\xint_add #1#2%
-{%
- \expandafter\XINT_add_fork \romannumeral0\xintnum{#2}\Z #1\Z
-}%
-\let\xintAdd\xintiAdd \let\xintadd\xintiadd
-\def\XINT_Add #1#2{\romannumeral0\XINT_add_fork #2\Z #1\Z }%
-\def\XINT_add #1#2{\XINT_add_fork #2\Z #1\Z }%
-% \end{macrocode}
-% \lverb|ADDITION
-% Ici #1#2 vient du *deuxième* argument de \xintAdd et #3#4 donc du *premier*
-% [algo plus efficace lorsque le premier est plus long que le second]|
-% \begin{macrocode}
-\def\XINT_add_fork #1#2\Z #3#4\Z
-{%
- \xint_UDzerofork
- #1\XINT_add_secondiszero
- #3\XINT_add_firstiszero
- 0
- {\xint_UDsignsfork
- #1#3\XINT_add_minusminus % #1 = #3 = -
- #1-\XINT_add_minusplus % #1 = -
- #3-\XINT_add_plusminus % #3 = -
- --\XINT_add_plusplus
- \krof }%
- \krof
- {#2}{#4}#1#3%
-}%
-\def\XINT_add_secondiszero #1#2#3#4{ #4#2}%
-\def\XINT_add_firstiszero #1#2#3#4{ #3#1}%
-% \end{macrocode}
-% \lverb|#1 vient du *deuxième* et #2 vient du *premier*|
-% \begin{macrocode}
-\def\XINT_add_minusminus #1#2#3#4%
-{%
- \expandafter\xint_minus_thenstop%
- \romannumeral0\XINT_add_pre {#2}{#1}%
-}%
-\def\XINT_add_minusplus #1#2#3#4%
-{%
- \XINT_sub_pre {#4#2}{#1}%
-}%
-\def\XINT_add_plusminus #1#2#3#4%
-{%
- \XINT_sub_pre {#3#1}{#2}%
-}%
-\def\XINT_add_plusplus #1#2#3#4%
-{%
- \XINT_add_pre {#4#2}{#3#1}%
-}%
-\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{xintSub}}
-% \lverb|Release 1.09a has \xintnum added into \xintiSub.|
-% \begin{macrocode}
-\def\xintiiSub {\romannumeral0\xintiisub }%
-\def\xintiisub #1%
-{%
- \expandafter\xint_iisub\expandafter{\romannumeral-`0#1}%
-}%
-\def\xint_iisub #1#2%
-{%
- \expandafter\XINT_sub_fork \romannumeral-`0#2\Z #1\Z
-}%
-\def\xintiSub {\romannumeral0\xintisub }%
-\def\xintisub #1%
-{%
- \expandafter\xint_sub\expandafter{\romannumeral0\xintnum{#1}}%
-}%
-\def\xint_sub #1#2%
-{%
- \expandafter\XINT_sub_fork \romannumeral0\xintnum{#2}\Z #1\Z
-}%
-\def\XINT_Sub #1#2{\romannumeral0\XINT_sub_fork #2\Z #1\Z }%
-\def\XINT_sub #1#2{\XINT_sub_fork #2\Z #1\Z }%
-\let\xintSub\xintiSub \let\xintsub\xintisub
-% \end{macrocode}
-% \lverb|&
-% SOUSTRACTION
-% #3#4-#1#2:
-% #3#4 vient du *premier*
-% #1#2 vient du *second*|
-% \begin{macrocode}
-\def\XINT_sub_fork #1#2\Z #3#4\Z
-{%
- \xint_UDsignsfork
- #1#3\XINT_sub_minusminus
- #1-\XINT_sub_minusplus % attention, #3=0 possible
- #3-\XINT_sub_plusminus % attention, #1=0 possible
- --{\xint_UDzerofork
- #1\XINT_sub_secondiszero
- #3\XINT_sub_firstiszero
- 0\XINT_sub_plusplus
- \krof }%
- \krof
- {#2}{#4}#1#3%
-}%
-\def\XINT_sub_secondiszero #1#2#3#4{ #4#2}%
-\def\XINT_sub_firstiszero #1#2#3#4{ -#3#1}%
-\def\XINT_sub_plusplus #1#2#3#4%
-{%
- \XINT_sub_pre {#4#2}{#3#1}%
-}%
-\def\XINT_sub_minusminus #1#2#3#4%
-{%
- \XINT_sub_pre {#1}{#2}%
-}%
-\def\XINT_sub_minusplus #1#2#3#4%
-{%
- \xint_gob_til_zero #4\xint_sub_mp0\XINT_add_pre {#4#2}{#1}%
-}%
-\def\xint_sub_mp0\XINT_add_pre #1#2{ #2}%
-\def\XINT_sub_plusminus #1#2#3#4%
-{%
- \xint_gob_til_zero #3\xint_sub_pm0\expandafter\xint_minus_thenstop%
- \romannumeral0\XINT_add_pre {#2}{#3#1}%
-}%
-\def\xint_sub_pm #1\XINT_add_pre #2#3{ -#2}%
-\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{xintCmp}}
-% \lverb|Release 1.09a has \xintnum inserted into \xintCmp. Unnecessary
-% \xintiCmp suppressed in 1.09f.|
-% \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\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{xintIsZero}, \csh{xintIsNotZero}}
-% \lverb|1.09a. restyled in 1.09i.|
-% \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}%
-% \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. Unused and useless \xintiGeq
-% removed in 1.09e.
-% 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{xintMax}}
-% \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.|
-% \begin{macrocode}
-\def\xintiMax {\romannumeral0\xintimax }%
-\def\xintimax #1%
-{%
- \expandafter\xint_max\expandafter {\romannumeral0\xintnum{#1}}%
-}%
-\let\xintMax\xintiMax \let\xintmax\xintimax
-\def\xint_max #1#2%
-{%
- \expandafter\XINT_max_pre\expandafter {\romannumeral0\xintnum{#2}}{#1}%
-}%
-\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{xintMin}}
-% \lverb|\xintnum added New with 1.09a.|
-% \begin{macrocode}
-\def\xintiMin {\romannumeral0\xintimin }%
-\def\xintimin #1%
-{%
- \expandafter\xint_min\expandafter {\romannumeral0\xintnum{#1}}%
-}%
-\let\xintMin\xintiMin \let\xintmin\xintimin
-\def\xint_min #1#2%
-{%
- \expandafter\XINT_min_pre\expandafter {\romannumeral0\xintnum{#2}}{#1}%
-}%
-\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{xintMinof}}
-% \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{xintSum}}
-% \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{xintMul}}
-% \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{\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 }%
- \krof
- {#2}{#4}%
-}%
-\def\XINT_mul_zero #1#2{ 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{xintSqr}}
-% \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{xintPrd}}
-% \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}
-% \subsection{\csh{xintPow}}
-% \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}%
-}%
-\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}%
-}%
-% \end{macrocode}
-% \lverb+#1 = B, #2 = |A|+
-% \begin{macrocode}
-\def\XINT_pow_Anonneg_ #1#2%
-{%
- \ifcase\XINT_Cmp {#2}{1}
- \expandafter\XINT_pow_AisOne
- \or
- \expandafter\XINT_pow_AatleastTwo
- \else
- \expandafter\XINT_pow_AisZero
- \fi
- {#1}{#2}%
-}%
-\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{xintDivision}, \csh{xintQuo}, \csh{xintRem}}
-% \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 \xintDivision, like all other routines in xint (except
-% \xintSeq without optional parameter) still does not impact the input save
-% stack. One can have a produced quotient longer than 4x5000=20000 digits, and
-% no need to worry about \xintTrunc, \xintRound, \xintFloat, \xintFloatSqrt,
-% etc... and all other places using the division.
-%
-% However outputting to a file (which is basically the only thing one can do,
-% multiplying out two 20000 digits numbers already takes hours, for 100000 it
-% would be days if not weeks) 100000 digits is slow... the truncation routine
-% will add 100000 zeros (circa) and then trim them four by four. Definitely I
-% should do a routine XTrunc which will work by blocks of say 64, and
-% furthermore, being destined to be used in and \edef or a \write, it could be
-% much more efficient as it could simply be based on tail loop, which so far
-% nothing in xint does because I want things to expand fully under
-% \romannumeral-`0 (and don't imagine inserting chains of thousands of
-% \expandafter's...) in order to be nestable. Inside \xintexpr such style of
-% tail recursion leaving downstream things should definitely be implemented for
-% the routines for which it is possible as things get expanded inside
-% \csname..\endcsname. I don't do yet anything like this for 1.09j. |
-% \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\xintQuo {\romannumeral0\xintquo }%
-\def\xintRem {\romannumeral0\xintrem }%
-\def\xintquo {\expandafter\xint_firstoftwo_thenstop
- \romannumeral0\xintdivision }%
-\def\xintrem {\expandafter\xint_secondoftwo_thenstop
- \romannumeral0\xintdivision }%
-% \end{macrocode}
-% \lverb|#1 = A, #2 = B. On calcule le quotient et le reste dans la division
-% euclidienne de A par B.|
-% \begin{macrocode}
-\def\xintiiDivision {\romannumeral0\xintiidivision }%
-\def\xintiidivision #1%
-{%
- \expandafter\xint_iidivision\expandafter {\romannumeral-`0#1}%
-}%
-\def\xint_iidivision #1#2%
-{%
- \expandafter\XINT_div_fork \romannumeral-`0#2\Z #1\Z
-}%
-\def\xintDivision {\romannumeral0\xintdivision }%
-\def\xintdivision #1%
-{%
- \expandafter\xint_division\expandafter {\romannumeral0\xintnum{#1}}%
-}%
-\def\xint_division #1#2%
-{%
- \expandafter\XINT_div_fork \romannumeral0\xintnum{#2}\Z #1\Z
-}%
-% \end{macrocode}
-% \lverb|#1#2 = 2e input = diviseur = B.
-% #3#4 = 1er input = divisé = A.|
-% \begin{macrocode}
-\def\XINT_div_fork #1#2\Z #3#4\Z
-{%
- \xint_UDzerofork
- #1\XINT_div_BisZero
- #3\XINT_div_AisZero
- 0{\xint_UDsignfork
- #1\XINT_div_BisNegative % B < 0
- #3\XINT_div_AisNegative % A < 0, B > 0
- -\XINT_div_plusplus % B > 0, A > 0
- \krof }%
- \krof
- {#2}{#4}#1#3% #1#2=B, #3#4=A
-}%
-\edef\XINT_div_BisZero #1#2#3#4{\noexpand\xintError:DivisionByZero\space {0}{0}}%
-\def\XINT_div_AisZero #1#2#3#4{ {0}{0}}%
-% \end{macrocode}
-% \lverb|&
-% jusqu'à présent c'est facile.$\
-% minusplus signifie B < 0, A > 0$\
-% plusminus signifie B > 0, A < 0$\
-% Ici #3#1 correspond au diviseur B et #4#2 au divisé A.
-%
-% Cases with B<0 or especially A<0 are treated sub-optimally in terms of
-% post-processing, things get reversed which could have been produced directly
-% in the wanted order, but A,B>0 is given priority for optimization. I should
-% revise the next few macros, definitely.|
-% \begin{macrocode}
-\def\XINT_div_plusplus #1#2#3#4{\XINT_div_prepare {#3#1}{#4#2}}%
-% \end{macrocode}
-% \lverb|B = #3#1 < 0, A non nul positif ou négatif|
-% \begin{macrocode}
-\def\XINT_div_BisNegative #1#2#3#4%
-{%
- \expandafter\XINT_div_BisNegative_b
- \romannumeral0\XINT_div_fork #1\Z #4#2\Z
-}%
-\edef\XINT_div_BisNegative_b #1%
-{%
- \noexpand\expandafter\space\noexpand\expandafter
- {\noexpand\romannumeral0\noexpand\XINT_opp #1}%
-}%
-% \end{macrocode}
-% \lverb|B = #3#1 > 0, A =-#2< 0|
-% \begin{macrocode}
-\def\XINT_div_AisNegative #1#2#3#4%
-{%
- \expandafter\XINT_div_AisNegative_b
- \romannumeral0\XINT_div_prepare {#3#1}{#2}{#3#1}%
-}%
-\def\XINT_div_AisNegative_b #1#2%
-{%
- \if0\XINT_Sgn #2\Z
- \expandafter \XINT_div_AisNegative_Rzero
- \else
- \expandafter \XINT_div_AisNegative_Rpositive
- \fi
- {#1}{#2}%
-}%
-% \end{macrocode}
-% \lverb|en #3 on a une copie de B (à l'endroit)|
-% \begin{macrocode}
-\edef\XINT_div_AisNegative_Rzero #1#2#3%
-{%
- \noexpand\expandafter\space\noexpand\expandafter
- {\noexpand\romannumeral0\noexpand\XINT_opp #1}{0}%
-}%
-% \end{macrocode}
-% \lverb!#1 = quotient, #2 = reste, #3 = diviseur initial (à l'endroit)
-% remplace Reste par B - Reste, après avoir remplacé Q par -(Q+1)
-% de sorte que la formule a = qb + r, 0<= r < |b| est valable!
-% \begin{macrocode}
-\def\XINT_div_AisNegative_Rpositive #1%
-{%
- \expandafter \XINT_div_AisNegative_Rpositive_b \expandafter
- {\romannumeral0\xintiiopp{\xintInc {#1}}}%
-}%
-\def\XINT_div_AisNegative_Rpositive_b #1#2#3%
-{%
- \expandafter \xint_exchangetwo_keepbraces_thenstop \expandafter
- {\romannumeral0\XINT_sub {#3}{#2}}{#1}%
-}%
-% \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 négatif on ait renvoyé 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 déjà renversé, #2 se développe 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}
-% \lverb|&
-% &
-% -----------------------------------------------------------------$\
-% -----------------------------------------------------------------$\
-% DECIMAL OPERATIONS: FIRST DIGIT, LASTDIGIT, ODDNESS,
-% MULTIPLICATION BY TEN, QUOTIENT BY TEN, QUOTIENT OR
-% MULTIPLICATION BY POWER OF TEN, SPLIT OPERATION.|
-% \subsection{\csh{xintFDg}}
-% \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}}
-% \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 .|
-% \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{xintMON}, \csh{xintMMON}}
-% \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}}
-% \lverb|1.05 has \xintiOdd, whereas \xintOdd parses through \xintNum.
-% Inadvertently, 1.09a redefined \xintiLDg so \xintiOdd also parsed through
-% \xintNum. Anyway, having a \xintOdd and a \xintiOdd was silly. Removed in
-% 1.09f |
-% \begin{macrocode}
-\def\xintiiOdd {\romannumeral0\xintiiodd }%
-\def\xintiiodd #1%
-{%
- \ifodd\xintiiLDg{#1}
- \xint_afterfi{ 1}%
- \else
- \xint_afterfi{ 0}%
- \fi
-}%
-\def\xintOdd {\romannumeral0\xintodd }%
-\def\xintodd #1%
-{%
- \ifodd\xintLDg{#1}
- \xint_afterfi{ 1}%
- \else
- \xint_afterfi{ 0}%
- \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{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!
-% \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}
-% \subsection{\csh{xintiSqrt}, \csh{xintiSquareRoot}}
-% \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 xhich avoid this \xintnum overhead.
-%
-% 1.09j replaced the previous long \ifcase from \XINT_sqrt_c by some nested
-% \ifnum's.|
-% \begin{macrocode}
-\def\xintiSqrt {\romannumeral0\xintisqrt }%
-\def\xintisqrt
- {\expandafter\XINT_sqrt_post\romannumeral0\xintisquareroot }%
-\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 }%
-\def\xintiSquareRoot {\romannumeral0\xintisquareroot }%
-\def\xintisquareroot #1%
- {\expandafter\XINT_sqrt_checkin\romannumeral0\xintnum{#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 { 1.}%
-\edef\XINT_sqrt_isneg #1\Z {\noexpand\xintError:RootOfNegative\space 1.}%
-\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{xintIsTrue:csv}}
-% \lverb|1.09c. For use by \xinttheboolexpr.(inside \csname, no need for a
-% \romannumeral here). The macros may well be defined already here. I
-% make no advertisement because I have inserted no space parsing in the
-% :csv macros, as they will be used only with privately created comma
-% separated lists, having no space naturally. Nevertheless they exist
-% and can be used.|
-% \begin{macrocode}
-\def\xintIsTrue:csv #1{\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,\expandafter\XINT_:_f
- \else\expandafter\XINT_istrue:_d\fi #1}%
-\def\XINT_istrue:_d #1,%
- {\expandafter\XINT_istrue:_e\romannumeral0\xintisnotzero {#1},}%
-\def\XINT_istrue:_e #1,#2{\XINT_istrue:_b {#2,#1}}%
-\def\XINT_:_f ,#1#2^{\xint_gobble_i #1}%
-% \end{macrocode}
-% \subsection{\csh{xintANDof:csv}}
-% \lverb|1.09a. For use by \xintexpr (inside \csname, no need for a
-% \romannumeral here).|
-% \begin{macrocode}
-\def\xintANDof:csv #1{\expandafter\XINT_andof:_a\romannumeral-`0#1,,^}%
-\def\XINT_andof:_a {\expandafter\XINT_andof:_b\romannumeral-`0}%
-\def\XINT_andof:_b #1{\if #1,\expandafter\XINT_andof:_e
- \else\expandafter\XINT_andof:_c\fi #1}%
-\def\XINT_andof:_c #1,{\xintifTrueAelseB {#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}
-% \subsection{\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 {\expandafter\XINT_orof:_b\romannumeral-`0}%
-\def\XINT_orof:_b #1{\if #1,\expandafter\XINT_orof:_e
- \else\expandafter\XINT_orof:_c\fi #1}%
-\def\XINT_orof:_c #1,{\xintifTrueAelseB{#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}
-% \subsection{\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,{\expandafter\XINT_xorof:_b\romannumeral-`0#2,#1}%
-\def\XINT_xorof:_b #1{\if #1,\expandafter\XINT_:_e
- \else\expandafter\XINT_xorof:_c\fi #1}%
-\def\XINT_xorof:_c #1,#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_:_e ,#1#2^{#1}% allows empty list
-% \end{macrocode}
-% \subsection{\csh{xintiMaxof:csv}}
-% \lverb|1.09i. For use by \xintiiexpr.|
-% \begin{macrocode}
-\def\xintiMaxof:csv #1{\expandafter\XINT_imaxof:_b\romannumeral-`0#1,,}%
-\def\XINT_imaxof:_b #1,#2,{\expandafter\XINT_imaxof:_c\romannumeral-`0#2,{#1},}%
-\def\XINT_imaxof:_c #1{\if #1,\expandafter\XINT_of:_e
- \else\expandafter\XINT_imaxof:_d\fi #1}%
-\def\XINT_imaxof:_d #1,{\expandafter\XINT_imaxof:_b\romannumeral0\xintimax {#1}}%
-\def\XINT_of:_e ,#1,{#1}%
-\let\xintMaxof:csv\xintiMaxof:csv
-% \end{macrocode}
-% \subsection{\csh{xintiMinof:csv}}
-% \lverb|1.09i. For use by \xintiiexpr.|
-% \begin{macrocode}
-\def\xintiMinof:csv #1{\expandafter\XINT_iminof:_b\romannumeral-`0#1,,}%
-\def\XINT_iminof:_b #1,#2,{\expandafter\XINT_iminof:_c\romannumeral-`0#2,{#1},}%
-\def\XINT_iminof:_c #1{\if #1,\expandafter\XINT_of:_e
- \else\expandafter\XINT_iminof:_d\fi #1}%
-\def\XINT_iminof:_d #1,{\expandafter\XINT_iminof:_b\romannumeral0\xintimin {#1}}%
-\let\xintMinof:csv\xintiMinof:csv
-% \end{macrocode}
-% \subsection{\csh{xintiiSum:csv}}
-% \lverb|1.09i. For use by \xintiiexpr.|
-% \begin{macrocode}
-\def\xintiiSum:csv #1{\expandafter\XINT_iisum:_a\romannumeral-`0#1,,^}%
-\def\XINT_iisum:_a {\XINT_iisum:_b {0}}%
-\def\XINT_iisum:_b #1#2,{\expandafter\XINT_iisum:_c\romannumeral-`0#2,{#1}}%
-\def\XINT_iisum:_c #1{\if #1,\expandafter\XINT_:_e
- \else\expandafter\XINT_iisum:_d\fi #1}%
-\def\XINT_iisum:_d #1,#2{\expandafter\XINT_iisum:_b\expandafter
- {\romannumeral0\xintiiadd {#2}{#1}}}%
-\let\xintSum:csv\xintiiSum:csv
-% \end{macrocode}
-% \subsection{\csh{xintiiPrd:csv}}
-% \lverb|1.09i. For use by \xintiiexpr.|
-% \begin{macrocode}
-\def\xintiiPrd:csv #1{\expandafter\XINT_iiprd:_a\romannumeral-`0#1,,^}%
-\def\XINT_iiprd:_a {\XINT_iiprd:_b {1}}%
-\def\XINT_iiprd:_b #1#2,{\expandafter\XINT_iiprd:_c\romannumeral-`0#2,{#1}}%
-\def\XINT_iiprd:_c #1{\if #1,\expandafter\XINT_:_e
- \else\expandafter\XINT_iiprd:_d\fi #1}%
-\def\XINT_iiprd:_d #1,#2{\expandafter\XINT_iiprd:_b\expandafter
- {\romannumeral0\xintiimul {#2}{#1}}}%
-\let\xintPrd:csv\xintiiPrd:csv
-\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}
-%
-% The commenting is currently (\docdate) very sparse.
-%
-% \localtableofcontents
-% \subsection{Catcodes, \protect\eTeX{} and reload detection}
-%
-% The code for reload detection is copied from \textsc{Heiko
-% Oberdiek}'s packages, and adapted here to check for previous
-% loading of the master \xintname package.
-%
-% The method for catcodes is slightly different, but still
-% 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\space { }%
- \let\z\endgroup
- \expandafter\let\expandafter\x\csname ver@xintbinhex.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{xintbinhex}{\numexpr not available, aborting input}%
- \aftergroup\endinput
- \else
- \ifx\x\relax % plain-TeX, first loading of xintbinhex.sty
- \ifx\w\relax % but xint.sty not yet loaded.
- \y{xintbinhex}{now issuing \string\input\space xint.sty}%
- \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.
- \y{xintbinhex}{now issuing \string\RequirePackage{xint}}%
- \def\z{\endgroup\RequirePackage{xint}}%
- \fi
- \else
- \y{xintbinhex}{I was already loaded, aborting input}%
- \aftergroup\endinput
- \fi
- \fi
- \fi
-\z%
-% \end{macrocode}
-% \subsection{Confirmation of \xintnameimp loading}
-% \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 % :
- \ifdefined\PackageInfo
- \def\y#1#2{\PackageInfo{#1}{#2}}%
- \else
- \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
- \fi
- \def\empty {}%
- \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname
- \ifx\w\relax % Plain TeX, user gave a file name at the prompt
- \y{xintbinhex}{Loading of package xint failed, aborting input}%
- \aftergroup\endinput
- \fi
- \ifx\w\empty % LaTeX, user gave a file name at the prompt
- \y{xintbinhex}{Loading of package xint failed, aborting input}%
- \aftergroup\endinput
- \fi
-\endgroup%
-% \end{macrocode}
-% \subsection{Catcodes}
-% \begin{macrocode}
-\XINTsetupcatcodes%
-% \end{macrocode}
-% \subsection{Package identification}
-% \begin{macrocode}
-\XINT_providespackage
-\ProvidesPackage{xintbinhex}%
- [2014/02/05 v1.09ka 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 done in xint.sty
-% \chardef\xint_c_ii^vi 64 % already done 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{%
- \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}}%
-\xintApplyInline\XINT_tmpa
- {{0}{1}{2}{3}{4}{5}{6}{7}{8}{9}{10}{11}{12}{13}{14}{15}}%
-\def\XINT_tmpa #1{%
- \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}}%
-\xintApplyInline\XINT_tmpa
- {{0}{1}{2}{3}{4}{5}{6}{7}{8}{9}{10}{11}{12}{13}{14}{15}}%
-\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}
-%
-% The commenting is currently (\docdate) 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|.
-%
-% \localtableofcontents
-% \subsection{Catcodes, \protect\eTeX{} and reload detection}
-%
-% The code for reload detection is copied from \textsc{Heiko
-% Oberdiek}'s packages, and adapted here to check for previous
-% loading of the master \xintname package.
-%
-% The method for catcodes is slightly different, but still
-% 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\space { }%
- \let\z\endgroup
- \expandafter\let\expandafter\x\csname ver@xintgcd.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{xintgcd}{\numexpr not available, aborting input}%
- \aftergroup\endinput
- \else
- \ifx\x\relax % plain-TeX, first loading of xintgcd.sty
- \ifx\w\relax % but xint.sty not yet loaded.
- \y{xintgcd}{now issuing \string\input\space xint.sty}%
- \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.
- \y{xintgcd}{now issuing \string\RequirePackage{xint}}%
- \def\z{\endgroup\RequirePackage{xint}}%
- \fi
- \else
- \y{xintgcd}{I was already loaded, aborting input}%
- \aftergroup\endinput
- \fi
- \fi
- \fi
-\z%
-% \end{macrocode}
-% \subsection{Confirmation of \xintnameimp loading}
-% \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 % :
- \ifdefined\PackageInfo
- \def\y#1#2{\PackageInfo{#1}{#2}}%
- \else
- \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
- \fi
- \def\empty {}%
- \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname
- \ifx\w\relax % Plain TeX, user gave a file name at the prompt
- \y{xintgcd}{Loading of package xint failed, aborting input}%
- \aftergroup\endinput
- \fi
- \ifx\w\empty % LaTeX, user gave a file name at the prompt
- \y{xintgcd}{Loading of package xint failed, aborting input}%
- \aftergroup\endinput
- \fi
-\endgroup%
-% \end{macrocode}
-% \subsection{Catcodes}
-% \begin{macrocode}
-\XINTsetupcatcodes%
-% \end{macrocode}
-% \subsection{Package identification}
-% \begin{macrocode}
-\XINT_providespackage
-\ProvidesPackage{xintgcd}%
- [2014/02/05 v1.09ka Euclide algorithm with xint package (jfB)]%
-% \end{macrocode}
-% \subsection{\csh{xintGCD}}
-% The macros of |1.09a| benefits from the |\xintnum| which has been inserted
-% inside |\xintiabs| in \xintname;
-% this is a little overhead but is more convenient for the
-% user and also makes it easier to use into |\xintexpr|essions.
-% \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
-}%
-% \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{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{xintLCM}}
-% \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.|
-% \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\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{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{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}$\
-% {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}|
-% \begin{macrocode}
-\edef\xint_bezalg_end_ #1#2#3#4%
-{%
- \space {#1}{#3}{0}{1}{#2}{#4}{1}{0}%
-}%
-% \end{macrocode}
-% \subsection{\csh{xintTypesetEuclideAlgorithm}}
-% \lverb|&
-% TYPESETTING
-%
-% Organisation:
-%
-% {N}{A}{D}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0}$\
-% \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 #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 #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
-}%
-% \end{macrocode}
-% \subsection{\csh{xintGCDof:csv}}
-% \lverb|1.09a. For use by \xintexpr.|
-% \begin{macrocode}
-\def\xintGCDof:csv #1{\expandafter\XINT_gcdof:_b\romannumeral-`0#1,,}%
-\def\XINT_gcdof:_b #1,#2,{\expandafter\XINT_gcdof:_c\romannumeral-`0#2,{#1},}%
-\def\XINT_gcdof:_c #1{\if #1,\expandafter\XINT_of:_e
- \else\expandafter\XINT_gcdof:_d\fi #1}%
-\def\XINT_gcdof:_d #1,{\expandafter\XINT_gcdof:_b\romannumeral0\xintgcd {#1}}%
-% \end{macrocode}
-% \subsection{\csh{xintLCMof:csv}}
-% \lverb|1.09a. For use by \xintexpr.|
-% \begin{macrocode}
-\def\xintLCMof:csv #1{\expandafter\XINT_lcmof:_a\romannumeral-`0#1,,}%
-\def\XINT_lcmof:_a #1,#2,{\expandafter\XINT_lcmof:_c\romannumeral-`0#2,{#1},}%
-\def\XINT_lcmof:_c #1{\if#1,\expandafter\XINT_of:_e
- \else\expandafter\XINT_lcmof:_d\fi #1}%
-\def\XINT_lcmof:_d #1,{\expandafter\XINT_lcmof:_a\romannumeral0\xintlcm {#1}}%
-\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}
-%
-% The commenting is currently (\docdate) very sparse.
-%
-% \localtableofcontents
-% \subsection{Catcodes, \protect\eTeX{} and reload detection}
-%
-% The code for reload detection is copied from \textsc{Heiko
-% Oberdiek}'s packages, and adapted here to check for previous
-% loading of the master \xintname package.
-%
-% The method for catcodes is slightly different, but still
-% 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\space { }%
- \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.
- \y{xintfrac}{now issuing \string\input\space xint.sty}%
- \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.
- \y{xintfrac}{now issuing \string\RequirePackage{xint}}%
- \def\z{\endgroup\RequirePackage{xint}}%
- \fi
- \else
- \y{xintfrac}{I was already loaded, aborting input}%
- \aftergroup\endinput
- \fi
- \fi
- \fi
-\z%
-% \end{macrocode}
-% \subsection{Confirmation of \xintnameimp loading}
-% \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 % :
- \ifdefined\PackageInfo
- \def\y#1#2{\PackageInfo{#1}{#2}}%
- \else
- \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
- \fi
- \def\empty {}%
- \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname
- \ifx\w\relax % Plain TeX, user gave a file name at the prompt
- \y{xintfrac}{Loading of package xint failed, aborting input}%
- \aftergroup\endinput
- \fi
- \ifx\w\empty % LaTeX, user gave a file name at the prompt
- \y{xintfrac}{Loading of package xint failed, aborting input}%
- \aftergroup\endinput
- \fi
-\endgroup%
-% \end{macrocode}
-% \subsection{Catcodes}
-% \begin{macrocode}
-\XINTsetupcatcodes%
-% \end{macrocode}
-% \subsection{Package identification}
-% \begin{macrocode}
-\XINT_providespackage
-\ProvidesPackage{xintfrac}%
- [2014/02/05 v1.09ka Expandable operations on fractions (jfB)]%
-\chardef\xint_c_vi 6
-\chardef\xint_c_vii 7
-\chardef\xint_c_xviii 18
-% \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. |
-% \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.|
-% \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: these [n]'s and especially the possible /1 are truly annoying at
-% times.|
-% \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}}
-% \lverb|1.09a|
-% \begin{macrocode}
-\def\xintFloor {\romannumeral0\xintfloor }%
-\def\xintfloor #1{\expandafter\XINT_floor
- \romannumeral0\xintrawwithzeros {#1}.}%
-\def\XINT_floor #1/#2.{\xintiiquo {#1}{#2}}%
-% \end{macrocode}
-% \subsection{\csh{xintCeil}}
-% \lverb|1.09a|
-% \begin{macrocode}
-\def\xintCeil {\romannumeral0\xintceil }%
-\def\xintceil #1{\xintiiopp {\xintFloor {\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}}
-% \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 withing brackets as the other
-% \XINTinFloat... macros. |
-% \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\xintfE {\romannumeral0\xintfe }%
-\def\xintfe #1%
-{%
- \expandafter\XINT_fe \romannumeral0\XINT_infrac {#1}%
-}%
-\def\XINT_fe #1#2#3#4%
-{%
- \expandafter\XINT_e_end\expandafter{\the\numexpr #1+\xintNum{#4}}{#2}{#3}%
-}%
-\def\xintFloatE {\romannumeral0\xintfloate }%
-\def\xintfloate #1{\XINT_floate_chkopt #1\Z }%
-\def\XINT_floate_chkopt #1%
-{%
- \ifx [#1\expandafter\XINT_floate_opt
- \else\expandafter\XINT_floate_noopt
- \fi #1%
-}%
-\def\XINT_floate_noopt #1\Z
-{%
- \expandafter\XINT_floate_a\expandafter\XINTdigits
- \romannumeral0\XINT_infrac {#1}%
-}%
-\def\XINT_floate_opt [\Z #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\XINTinFloatfE {\romannumeral0\XINTinfloatfe }%
-\def\XINTinfloatfe [#1]#2%
-{%
- \expandafter\XINT_infloatfe_a\expandafter
- {\the\numexpr #1\expandafter}\romannumeral0\XINT_infrac {#2}%
-}%
-\def\XINT_infloatfe_a #1#2#3#4#5%
-{%
- \expandafter\expandafter\expandafter\XINT_infloat_a
- \expandafter\xint_exchangetwo_keepbraces\expandafter
- {\the\numexpr #2+\xintNum{#5}}{#1}{#3}{#4}\XINT_infloat_Q
-}%
-\def\xintiE {\romannumeral0\xintie }% for \xintiiexpr only
-\def\xintie #1%
-{%
- \expandafter\XINT_ie \romannumeral0\XINT_infrac {#1}% allows 3.123e3
-}%
-\def\XINT_ie #1#2#3#4% assumes #3=1 and uses \xint_dsh with its \numexpr
-{%
- \xint_dsh {#2}{0-(#1+#4)}% could have \xintNum{#4} for a bit more general
-}%
-% \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{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. |
-% \begin{macrocode}
-\def\xintNum {\romannumeral0\xintnum }%
-\def\xintnum #1{\expandafter\XINT_intcheck\romannumeral0\xintirr {#1}\Z }%
-\edef\XINT_intcheck #1/#2\Z
-{%
- \noexpand\if 0\noexpand\XINT_isOne {#2}\noexpand\xintError:NotAnInteger
- \noexpand\fi\space #1%
-}%
-% \end{macrocode}
-% \subsection{\csh{xintifInt}}
-% \lverb|1.09e. xintfrac.sty only.|
-% \begin{macrocode}
-\def\xintifInt {\romannumeral0\xintifint }%
-\def\xintifint #1{\expandafter\XINT_ifint\romannumeral0\xintirr {#1}\Z }%
-\def\XINT_ifint #1/#2\Z
-{%
- \if\XINT_isOne {#2}1%
- \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\XINT_tfrac_N
- 0-{\XINT_tfrac_P #1}%
- \krof
-}%
-\def\XINT_tfrac_zero #1\Z { 0/1[0]}%
-\def\XINT_tfrac_N {\expandafter\XINT_opp\romannumeral0\XINT_tfrac_P }%
-\def\XINT_tfrac_P #1/#2\Z
-{%
- \expandafter\XINT_rez_AB\romannumeral0\xintiirem{#1}{#2}\Z {0}{#2}%
-}%
-% \end{macrocode}
-% \subsection{\csh{XINTinFloatFrac}}
-% \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. |
-% \begin{macrocode}
-\def\XINTinFloatFrac {\romannumeral0\XINTinfloatfrac }%
-\def\XINTinfloatfrac [#1]#2%
-{%
- \expandafter\XINT_infloatfrac_a\expandafter
- {\romannumeral0\xinttfrac{#2}}{#1}%
-}%
-\def\XINT_infloatfrac_a #1#2{\XINTinFloat [#2]{#1}}%
-% \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.|
-% \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 autorisé !
-}%
-% \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 thinsg
-% 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, 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\Z }%
-\def\XINT_float_chkopt #1%
-{%
- \ifx [#1\expandafter\XINT_float_opt
- \else\expandafter\XINT_float_noopt
- \fi #1%
-}%
-\def\XINT_float_noopt #1\Z
-{%
- \expandafter\XINT_float_a\expandafter\XINTdigits
- \romannumeral0\XINT_infrac {#1}\XINT_float_Q
-}%
-\def\XINT_float_opt [\Z #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{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}}
-% \begin{macrocode}
-\def\xintAdd {\romannumeral0\xintadd }%
-\def\xintadd #1%
-{%
- \expandafter\xint_fadd\expandafter {\romannumeral0\XINT_infrac {#1}}%
-}%
-\def\xint_fadd #1#2{\expandafter\XINT_fadd_A\romannumeral0\XINT_infrac{#2}#1}%
-\def\XINT_fadd_A #1#2#3#4%
-{%
- \ifnum #4 > #1
- \xint_afterfi {\XINT_fadd_B {#1}}%
- \else
- \xint_afterfi {\XINT_fadd_B {#4}}%
- \fi
- {#1}{#4}{#2}{#3}%
-}%
-\def\XINT_fadd_B #1#2#3#4#5#6#7%
-{%
- \expandafter\XINT_fadd_C\expandafter
- {\romannumeral0\xintiimul {#7}{#5}}%
- {\romannumeral0\xintiiadd
- {\romannumeral0\xintiimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}%
- {\romannumeral0\xintiimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}%
- }%
- {#1}%
-}%
-\def\XINT_fadd_C #1#2#3%
-{%
- \expandafter\XINT_fadd_D\expandafter {#2}{#3}{#1}%
-}%
-\def\XINT_fadd_D #1#2{\XINT_outfrac {#2}{#1}}%
-% \end{macrocode}
-% \subsection{\csh{xintSub}}
-% \begin{macrocode}
-\def\xintSub {\romannumeral0\xintsub }%
-\def\xintsub #1%
-{%
- \expandafter\xint_fsub\expandafter {\romannumeral0\XINT_infrac {#1}}%
-}%
-\def\xint_fsub #1#2%
- {\expandafter\XINT_fsub_A\romannumeral0\XINT_infrac {#2}#1}%
-\def\XINT_fsub_A #1#2#3#4%
-{%
- \ifnum #4 > #1
- \xint_afterfi {\XINT_fsub_B {#1}}%
- \else
- \xint_afterfi {\XINT_fsub_B {#4}}%
- \fi
- {#1}{#4}{#2}{#3}%
-}%
-\def\XINT_fsub_B #1#2#3#4#5#6#7%
-{%
- \expandafter\XINT_fsub_C\expandafter
- {\romannumeral0\xintiimul {#7}{#5}}%
- {\romannumeral0\xintiisub
- {\romannumeral0\xintiimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}%
- {\romannumeral0\xintiimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}%
- }%
- {#1}%
-}%
-\def\XINT_fsub_C #1#2#3%
-{%
- \expandafter\XINT_fsub_D\expandafter {#2}{#3}{#1}%
-}%
-\def\XINT_fsub_D #1#2{\XINT_outfrac {#2}{#1}}%
-% \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}}
-% \begin{macrocode}
-\def\xintMul {\romannumeral0\xintmul }%
-\def\xintmul #1%
-{%
- \expandafter\xint_fmul\expandafter {\romannumeral0\XINT_infrac {#1}}%
-}%
-\def\xint_fmul #1#2%
- {\expandafter\XINT_fmul_A\romannumeral0\XINT_infrac {#2}#1}%
-\def\XINT_fmul_A #1#2#3#4#5#6%
-{%
- \expandafter\XINT_fmul_B
- \expandafter{\the\numexpr #1+#4\expandafter}%
- \expandafter{\romannumeral0\xintiimul {#6}{#3}}%
- {\romannumeral0\xintiimul {#5}{#2}}%
-}%
-\def\XINT_fmul_B #1#2#3%
-{%
- \expandafter \XINT_fmul_C \expandafter{#3}{#1}{#2}%
-}%
-\def\XINT_fmul_C #1#2{\XINT_outfrac {#2}{#1}}%
-% \end{macrocode}
-% \subsection{\csh{xintSqr}}
-% \begin{macrocode}
-\def\xintSqr {\romannumeral0\xintsqr }%
-\def\xintsqr #1%
-{%
- \expandafter\xint_fsqr\expandafter{\romannumeral0\XINT_infrac {#1}}%
-}%
-\def\xint_fsqr #1{\XINT_fmul_A #1#1}%
-% \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{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. Incredibly, it seems that 1.08b introduced a bug in
-% delimited arguments making the macro just non-functional when one of the input
-% was zero! I
-% did not detect this until working on release 1.09a, somehow I had not tested
-% that
-% \xintCmp just did NOT work! I must have done some last minute change... |
-% \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 was 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 was 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 was 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\Z }%
-\def\XINTinFloatAdd {\romannumeral0\XINTinfloatadd }%
-\def\XINTinfloatadd #1{\XINT_fladd_chkopt \XINTinfloat #1\Z }%
-\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\Z #3%
-{%
- #1[\XINTdigits]{\XINT_FL_Add {\XINTdigits+\xint_c_ii}{#2}{#3}}%
-}%
-\def\XINT_fladd_opt #1[\Z #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\Z }%
-\def\XINTinFloatSub {\romannumeral0\XINTinfloatsub }%
-\def\XINTinfloatsub #1{\XINT_flsub_chkopt \XINTinfloat #1\Z }%
-\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\Z #3%
-{%
- #1[\XINTdigits]{\XINT_FL_Add {\XINTdigits+\xint_c_ii}{#2}{\xintOpp{#3}}}%
-}%
-\def\XINT_flsub_opt #1[\Z #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\Z }%
-\def\XINTinFloatMul {\romannumeral0\XINTinfloatmul }%
-\def\XINTinfloatmul #1{\XINT_flmul_chkopt \XINTinfloat #1\Z }%
-\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\Z #3%
-{%
- #1[\XINTdigits]{\XINT_FL_Mul {\XINTdigits+\xint_c_ii}{#2}{#3}}%
-}%
-\def\XINT_flmul_opt #1[\Z #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\Z }%
-\def\XINTinFloatDiv {\romannumeral0\XINTinfloatdiv }%
-\def\XINTinfloatdiv #1{\XINT_fldiv_chkopt \XINTinfloat #1\Z }%
-\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\Z #3%
-{%
- #1[\XINTdigits]{\XINT_FL_Div {\XINTdigits+\xint_c_ii}{#2}{#3}}%
-}%
-\def\XINT_fldiv_opt #1[\Z #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{XINTinFloatSum}}
-% \lverb|1.09a: quick write-up, for use by \xintfloatexpr, will need to be
-% thought through again. Renamed (and slightly modified) in 1.09h. Should be
-% extended for optional precision. Should be rewritten for optimization. |
-% \begin{macrocode}
-\def\XINTinFloatSum {\romannumeral0\XINTinfloatsum }%
-\def\XINTinfloatsum #1{\expandafter\XINT_floatsum_a\romannumeral-`0#1\relax }%
-\def\XINT_floatsum_a #1{\expandafter\XINT_floatsum_b
- \romannumeral0\XINTinfloat[\XINTdigits]{#1}\Z }%
-\def\XINT_floatsum_b #1\Z #2%
- {\expandafter\XINT_floatsum_c\romannumeral-`0#2\Z {#1}\Z}%
-\def\XINT_floatsum_c #1%
- {\xint_gob_til_relax #1\XINT_floatsum_e\relax\XINT_floatsum_d #1}%
-\def\XINT_floatsum_d #1\Z
- {\expandafter\XINT_floatsum_b\romannumeral0\XINTinfloatadd {#1}}%
-\def\XINT_floatsum_e #1\Z #2\Z { #2}%
-% \end{macrocode}
-% \subsection{\csh{XINTinFloatPrd}}
-% \lverb|1.09a: quick write-up, for use by \xintfloatexpr, will need to be
-% thought through again. Renamed (and slightly modified) in 1.09h. Should be
-% extended for optional precision. Should be rewritten for optimization. |
-% \begin{macrocode}
-\def\XINTinFloatPrd {\romannumeral0\XINTinfloatprd }%
-\def\XINTinfloatprd #1{\expandafter\XINT_floatprd_a\romannumeral-`0#1\relax }%
-\def\XINT_floatprd_a #1{\expandafter\XINT_floatprd_b
- \romannumeral0\XINTinfloat[\XINTdigits]{#1}\Z }%
-\def\XINT_floatprd_b #1\Z #2%
- {\expandafter\XINT_floatprd_c\romannumeral-`0#2\Z {#1}\Z}%
-\def\XINT_floatprd_c #1%
- {\xint_gob_til_relax #1\XINT_floatprd_e\relax\XINT_floatprd_d #1}%
-\def\XINT_floatprd_d #1\Z
- {\expandafter\XINT_floatprd_b\romannumeral0\XINTinfloatmul {#1}}%
-\def\XINT_floatprd_e #1\Z #2\Z { #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\Z }%
-\def\XINTinFloatPow {\romannumeral0\XINTinfloatpow }%
-\def\XINTinfloatpow #1{\XINT_flpow_chkopt \XINTinfloat #1\Z }%
-\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\Z #3%
-{%
- \expandafter\XINT_flpow_checkB_start\expandafter
- {\the\numexpr #3\expandafter}\expandafter
- {\the\numexpr \XINTdigits}{#2}{#1[\XINTdigits]}%
-}%
-\def\XINT_flpow_opt #1[\Z #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\Z }%
-\def\XINTinFloatPower {\romannumeral0\XINTinfloatpower}%
-\def\XINTinfloatpower #1{\XINT_flpower_chkopt \XINTinfloat #1\Z }%
-\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\Z #3%
-{%
- \expandafter\XINT_flpower_checkB_start\expandafter
- {\the\numexpr \XINTdigits\expandafter}\expandafter
- {\romannumeral0\xintnum{#3}}{#2}{#1[\XINTdigits]}%
-}%
-\def\XINT_flpower_opt #1[\Z #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\Z }%
-\def\XINTinFloatSqrt {\romannumeral0\XINTinfloatsqrt }%
-\def\XINTinfloatsqrt #1{\XINT_flsqrt_chkopt \XINTinfloat #1\Z }%
-\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\Z
-{%
- #1[\XINTdigits]{\XINT_FL_sqrt \XINTdigits {#2}}%
-}%
-\def\XINT_flsqrt_opt #1[\Z #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]}%
-% \end{macrocode}
-% \subsection{\csh{XINTinFloatMaxof}}
-% \lverb|1.09a, for use by \xintNewFloatExpr. Name changed in 1.09h|
-% \begin{macrocode}
-\def\XINTinFloatMaxof {\romannumeral0\XINTinfloatmaxof }%
-\def\XINTinfloatmaxof #1{\expandafter\XINT_flmaxof_a\romannumeral-`0#1\relax }%
-\def\XINT_flmaxof_a #1{\expandafter\XINT_flmaxof_b
- \romannumeral0\XINTinfloat [\XINTdigits]{#1}\Z }%
-\def\XINT_flmaxof_b #1\Z #2%
- {\expandafter\XINT_flmaxof_c\romannumeral-`0#2\Z {#1}\Z}%
-\def\XINT_flmaxof_c #1%
- {\xint_gob_til_relax #1\XINT_flmaxof_e\relax\XINT_flmaxof_d #1}%
-\def\XINT_flmaxof_d #1\Z
- {\expandafter\XINT_flmaxof_b\romannumeral0\xintmax
- {\XINTinFloat [\XINTdigits]{#1}}}%
-\def\XINT_flmaxof_e #1\Z #2\Z { #2}%
-% \end{macrocode}
-% \subsection{\csh{XINTinFloatMinof}}
-% \lverb|1.09a, for use by \xintNewFloatExpr. Name changed in 1.09h|
-% \begin{macrocode}
-\def\XINTinFloatMinof {\romannumeral0\XINTinfloatminof }%
-\def\XINTinfloatminof #1{\expandafter\XINT_flminof_a\romannumeral-`0#1\relax }%
-\def\XINT_flminof_a #1{\expandafter\XINT_flminof_b
- \romannumeral0\XINTinfloat [\XINTdigits]{#1}\Z }%
-\def\XINT_flminof_b #1\Z #2%
- {\expandafter\XINT_flminof_c\romannumeral-`0#2\Z {#1}\Z}%
-\def\XINT_flminof_c #1%
- {\xint_gob_til_relax #1\XINT_flminof_e\relax\XINT_flminof_d #1}%
-\def\XINT_flminof_d #1\Z
- {\expandafter\XINT_flminof_b\romannumeral0\xintmin
- {\XINTinFloat [\XINTdigits]{#1}}}%
-\def\XINT_flminof_e #1\Z #2\Z { #2}%
-% \end{macrocode}
-% \subsection{\csh{xintRound:csv}}
-% \lverb|1.09a. For use by \xinttheiexpr.|
-% \begin{macrocode}
-\def\xintRound:csv #1{\expandafter\XINT_round:_a\romannumeral-`0#1,,^}%
-\def\XINT_round:_a {\XINT_round:_b {}}%
-\def\XINT_round:_b #1#2,%
- {\expandafter\XINT_round:_c\romannumeral-`0#2,{#1}}%
-\def\XINT_round:_c #1{\if #1,\expandafter\XINT_:_f
- \else\expandafter\XINT_round:_d\fi #1}%
-\def\XINT_round:_d #1,%
- {\expandafter\XINT_round:_e\romannumeral0\xintiround 0{#1},}%
-\def\XINT_round:_e #1,#2{\XINT_round:_b {#2,#1}}%
-% \end{macrocode}
-% \subsection{\csh{xintFloat:csv}}
-% \lverb|1.09a. For use by \xintthefloatexpr.|
-% \begin{macrocode}
-\def\xintFloat:csv #1{\expandafter\XINT_float:_a\romannumeral-`0#1,,^}%
-\def\XINT_float:_a {\XINT_float:_b {}}%
-\def\XINT_float:_b #1#2,%
- {\expandafter\XINT_float:_c\romannumeral-`0#2,{#1}}%
-\def\XINT_float:_c #1{\if #1,\expandafter\XINT_:_f
- \else\expandafter\XINT_float:_d\fi #1}%
-\def\XINT_float:_d #1,%
- {\expandafter\XINT_float:_e\romannumeral0\xintfloat {#1},}%
-\def\XINT_float:_e #1,#2{\XINT_float:_b {#2,#1}}%
-% \end{macrocode}
-% \subsection{\csh{xintSum:csv}}
-% \lverb|1.09a. For use by \xintexpr.|
-% \begin{macrocode}
-\def\xintSum:csv #1{\expandafter\XINT_sum:_a\romannumeral-`0#1,,^}%
-\def\XINT_sum:_a {\XINT_sum:_b {0/1[0]}}%
-\def\XINT_sum:_b #1#2,{\expandafter\XINT_sum:_c\romannumeral-`0#2,{#1}}%
-\def\XINT_sum:_c #1{\if #1,\expandafter\XINT_:_e
- \else\expandafter\XINT_sum:_d\fi #1}%
-\def\XINT_sum:_d #1,#2{\expandafter\XINT_sum:_b\expandafter
- {\romannumeral0\xintadd {#2}{#1}}}%
-% \end{macrocode}
-% \subsection{\csh{xintPrd:csv}}
-% \lverb|1.09a. For use by \xintexpr.|
-% \begin{macrocode}
-\def\xintPrd:csv #1{\expandafter\XINT_prd:_a\romannumeral-`0#1,,^}%
-\def\XINT_prd:_a {\XINT_prd:_b {1/1[0]}}%
-\def\XINT_prd:_b #1#2,{\expandafter\XINT_prd:_c\romannumeral-`0#2,{#1}}%
-\def\XINT_prd:_c #1{\if #1,\expandafter\XINT_:_e
- \else\expandafter\XINT_prd:_d\fi #1}%
-\def\XINT_prd:_d #1,#2{\expandafter\XINT_prd:_b\expandafter
- {\romannumeral0\xintmul {#2}{#1}}}%
-% \end{macrocode}
-% \subsection{\csh{xintMaxof:csv}}
-% \lverb|1.09a. For use by \xintexpr. Even with only one
-% argument, there does not seem to be really a motive for using \xintraw?|
-% \begin{macrocode}
-\def\xintMaxof:csv #1{\expandafter\XINT_maxof:_b\romannumeral-`0#1,,}%
-\def\XINT_maxof:_b #1,#2,{\expandafter\XINT_maxof:_c\romannumeral-`0#2,{#1},}%
-\def\XINT_maxof:_c #1{\if #1,\expandafter\XINT_of:_e
- \else\expandafter\XINT_maxof:_d\fi #1}%
-\def\XINT_maxof:_d #1,{\expandafter\XINT_maxof:_b\romannumeral0\xintmax {#1}}%
-% \end{macrocode}
-% \subsection{\csh{xintMinof:csv}}
-% \lverb|1.09a. For use by \xintexpr.|
-% \begin{macrocode}
-\def\xintMinof:csv #1{\expandafter\XINT_minof:_b\romannumeral-`0#1,,}%
-\def\XINT_minof:_b #1,#2,{\expandafter\XINT_minof:_c\romannumeral-`0#2,{#1},}%
-\def\XINT_minof:_c #1{\if #1,\expandafter\XINT_of:_e
- \else\expandafter\XINT_minof:_d\fi #1}%
-\def\XINT_minof:_d #1,{\expandafter\XINT_minof:_b\romannumeral0\xintmin {#1}}%
-% \end{macrocode}
-% \subsection{\csh{XINTinFloatMinof:csv}}
-% \lverb|1.09a. For use by \xintfloatexpr. Name changed in 1.09h|
-% \begin{macrocode}
-\def\XINTinFloatMinof:csv #1{\expandafter\XINT_flminof:_a\romannumeral-`0#1,,}%
-\def\XINT_flminof:_a #1,{\expandafter\XINT_flminof:_b
- \romannumeral0\XINTinfloat [\XINTdigits]{#1},}%
-\def\XINT_flminof:_b #1,#2,%
- {\expandafter\XINT_flminof:_c\romannumeral-`0#2,{#1},}%
-\def\XINT_flminof:_c #1{\if #1,\expandafter\XINT_of:_e
- \else\expandafter\XINT_flminof:_d\fi #1}%
-\def\XINT_flminof:_d #1,%
- {\expandafter\XINT_flminof:_b\romannumeral0\xintmin
- {\XINTinFloat [\XINTdigits]{#1}}}%
-% \end{macrocode}
-% \subsection{\csh{XINTinFloatMaxof:csv}}
-% \lverb|1.09a. For use by \xintfloatexpr. Name changed in 1.09h|
-% \begin{macrocode}
-\def\XINTinFloatMaxof:csv #1{\expandafter\XINT_flmaxof:_a\romannumeral-`0#1,,}%
-\def\XINT_flmaxof:_a #1,{\expandafter\XINT_flmaxof:_b
- \romannumeral0\XINTinfloat [\XINTdigits]{#1},}%
-\def\XINT_flmaxof:_b #1,#2,%
- {\expandafter\XINT_flmaxof:_c\romannumeral-`0#2,{#1},}%
-\def\XINT_flmaxof:_c #1{\if #1,\expandafter\XINT_of:_e
- \else\expandafter\XINT_flmaxof:_d\fi #1}%
-\def\XINT_flmaxof:_d #1,%
- {\expandafter\XINT_flmaxof:_b\romannumeral0\xintmax
- {\XINTinFloat [\XINTdigits]{#1}}}%
-% \end{macrocode}
-% \subsection{\csh{XINTinFloatSum:csv}}
-% \lverb|1.09a. For use by \xintfloatexpr. Renamed in 1.09h|
-% \begin{macrocode}
-\def\XINTinFloatSum:csv #1{\expandafter\XINT_floatsum:_a\romannumeral-`0#1,,^}%
-\def\XINT_floatsum:_a {\XINT_floatsum:_b {0[0]}}%
-\def\XINT_floatsum:_b #1#2,%
- {\expandafter\XINT_floatsum:_c\romannumeral-`0#2,{#1}}%
-\def\XINT_floatsum:_c #1{\if #1,\expandafter\XINT_:_e
- \else\expandafter\XINT_floatsum:_d\fi #1}%
-\def\XINT_floatsum:_d #1,#2{\expandafter\XINT_floatsum:_b\expandafter
- {\romannumeral0\XINTinfloatadd {#2}{#1}}}%
-% \end{macrocode}
-% \subsection{\csh{XINTinFloatPrd:csv}}
-% \lverb|1.09a. For use by \xintfloatexpr. Renamed in 1.09h|
-% \begin{macrocode}
-\def\XINTinFloatPred:csv #1{\expandafter\XINT_floatprd:_a\romannumeral-`0#1,,^}%
-\def\XINT_floatprd:_a {\XINT_floatprd:_b {1[0]}}%
-\def\XINT_floatprd:_b #1#2,%
- {\expandafter\XINT_floatprd:_c\romannumeral-`0#2,{#1}}%
-\def\XINT_floatprd:_c #1{\if #1,\expandafter\XINT_:_e
- \else\expandafter\XINT_floatprd:_d\fi #1}%
-\def\XINT_floatprd:_d #1,#2{\expandafter\XINT_floatprd:_b\expandafter
- {\romannumeral0\XINTinfloatmul {#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}
-%
-% The commenting is currently (\docdate) very sparse.
-%
-% \localtableofcontents
-% \subsection{Catcodes, \protect\eTeX{} and reload detection}
-%
-% The code for reload detection is copied from \textsc{Heiko
-% Oberdiek}'s packages, and adapted here to check for previous
-% loading of the \xintfracname package.
-%
-% The method for catcodes is slightly different, but still
-% 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\space { }%
- \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.
- \y{xintseries}{now issuing \string\input\space xintfrac.sty}%
- \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.
- \y{xintseries}{now issuing \string\RequirePackage{xintfrac}}%
- \def\z{\endgroup\RequirePackage{xintfrac}}%
- \fi
- \else
- \y{xintseries}{I was already loaded, aborting input}%
- \aftergroup\endinput
- \fi
- \fi
- \fi
-\z%
-% \end{macrocode}
-% \subsection{Confirmation of \xintfracnameimp loading}
-% \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 % :
- \ifdefined\PackageInfo
- \def\y#1#2{\PackageInfo{#1}{#2}}%
- \else
- \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
- \fi
- \def\empty {}%
- \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname
- \ifx\w\relax % Plain TeX, user gave a file name at the prompt
- \y{xintseries}{Loading of package xintfrac failed, aborting input}%
- \aftergroup\endinput
- \fi
- \ifx\w\empty % LaTeX, user gave a file name at the prompt
- \y{xintseries}{Loading of package xintfrac failed, aborting input}%
- \aftergroup\endinput
- \fi
-\endgroup%
-% \end{macrocode}
-% \subsection{Catcodes}
-% \begin{macrocode}
-\XINTsetupcatcodes%
-% \end{macrocode}
-% \subsection{Package identification}
-% \begin{macrocode}
-\XINT_providespackage
-\ProvidesPackage{xintseries}%
- [2014/02/05 v1.09ka 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.
-% 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\Z }%
-\def\XINT_flpowseries_chkopt #1%
-{%
- \ifx [#1\expandafter\XINT_flpowseries_opt
- \else\expandafter\XINT_flpowseries_noopt
- \fi
- #1%
-}%
-\def\XINT_flpowseries_noopt #1\Z #2%
-{%
- \expandafter\XINT_flpowseries\expandafter
- {\the\numexpr #1\expandafter}\expandafter
- {\the\numexpr #2}\XINTdigits
-}%
-\def\XINT_flpowseries_opt [\Z #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\Z }%
-\def\XINT_flpowseriesx_chkopt #1%
-{%
- \ifx [#1\expandafter\XINT_flpowseriesx_opt
- \else\expandafter\XINT_flpowseriesx_noopt
- \fi
- #1%
-}%
-\def\XINT_flpowseriesx_noopt #1\Z #2%
-{%
- \expandafter\XINT_flpowseriesx\expandafter
- {\the\numexpr #1\expandafter}\expandafter
- {\the\numexpr #2}\XINTdigits
-}%
-\def\XINT_flpowseriesx_opt [\Z #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}
-%
-% The commenting is currently (\docdate) very sparse.
-%
-% \localtableofcontents
-% \subsection{Catcodes, \protect\eTeX{} and reload detection}
-%
-% The code for reload detection is copied from \textsc{Heiko
-% Oberdiek}'s packages, and adapted here to check for previous
-% loading of the \xintfracname package.
-%
-% The method for catcodes is slightly different, but still
-% 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\space { }%
- \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.
- \y{xintcfrac}{now issuing \string\input\space xintfrac.sty}%
- \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.
- \y{xintcfrac}{now issuing \string\RequirePackage{xintfrac}}%
- \def\z{\endgroup\RequirePackage{xintfrac}}%
- \fi
- \else
- \y{xintcfrac}{I was already loaded, aborting input}%
- \aftergroup\endinput
- \fi
- \fi
- \fi
-\z%
-% \end{macrocode}
-% \subsection{Confirmation of \xintfracnameimp loading}
-% \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 % :
- \ifdefined\PackageInfo
- \def\y#1#2{\PackageInfo{#1}{#2}}%
- \else
- \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
- \fi
- \def\empty {}%
- \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname
- \ifx\w\relax % Plain TeX, user gave a file name at the prompt
- \y{xintcfrac}{Loading of package xintfrac failed, aborting input}%
- \aftergroup\endinput
- \fi
- \ifx\w\empty % LaTeX, user gave a file name at the prompt
- \y{xintcfrac}{Loading of package xintfrac failed, aborting input}%
- \aftergroup\endinput
- \fi
-\endgroup%
-% \end{macrocode}
-% \subsection{Catcodes}
-% \begin{macrocode}
-\XINTsetupcatcodes%
-% \end{macrocode}
-% \subsection{Package identification}
-% \begin{macrocode}
-\XINT_providespackage
-\ProvidesPackage{xintcfrac}%
- [2014/02/05 v1.09ka 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\Z
-}%
-\def\XINT_cfrac_opt_a #1%
-{%
- \ifx[#1\XINT_cfrac_opt_b\fi \XINT_cfrac_noopt #1%
-}%
-\def\XINT_cfrac_noopt #1\Z
-{%
- \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z
- \relax\relax
-}%
-\def\XINT_cfrac_opt_b\fi\XINT_cfrac_noopt [\Z #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\Z }%
-\def\XINT_gcfrac_opt_a #1%
-{%
- \ifx[#1\XINT_gcfrac_opt_b\fi \XINT_gcfrac_noopt #1%
-}%
-\def\XINT_gcfrac_noopt #1\Z
-{%
- \XINT_gcfrac #1+\W/\relax\relax
-}%
-\def\XINT_gcfrac_opt_b\fi\XINT_gcfrac_noopt [\Z #1]%
-{%
- \fi\csname XINT_gcfrac_opt#1\endcsname
-}%
-\def\XINT_gcfrac_optl #1%
-{%
- \XINT_gcfrac #1+\W/\relax\hfill
-}%
-\def\XINT_gcfrac_optc #1%
-{%
- \XINT_gcfrac #1+\W/\relax\relax
-}%
-\def\XINT_gcfrac_optr #1%
-{%
- \XINT_gcfrac #1+\W/\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_W #3\XINT_gcfrac_endloop\W
- \XINT_gcfrac_loop {{#3}{#2}#1}%
-}%
-\def\XINT_gcfrac_endloop\W\XINT_gcfrac_loop #1#2#3%
-{%
- \XINT_gcfrac_T #2#3#1\Z\Z
-}%
-\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_Z #5\XINT_gcfrac_end\Z\XINT_gcfrac_U
- #1#2{\xintFrac{#5}%
- \ifcase\xintSgn{#4}
- +\or+\else-\fi
- \cfrac{#1\xintFrac{\xintAbs{#4}}#2}{#3}}%
-}%
-\def\XINT_gcfrac_end\Z\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{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+\W/}%
-\def\XINT_gctgcx_loop_a #1#2#3#4+#5/%
-{%
- \xint_gob_til_W #5\XINT_gctgcx_end\W
- \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\W\XINT_gctgcx_loop_b #1#2#3#4{ #1}%
-% \end{macrocode}
-% \subsection{\csh{xintFtoCs}}
-% \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,}}%
-\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,}%
-}%
-\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{xintFtoGC}}
-% \begin{macrocode}
-\def\xintFtoGC {\romannumeral0\xintftogc }%
-\def\xintftogc {\xintftocx {+1/}}%
-% \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{xintFtoCv}}
-% \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{xintCstoF}}
-% \begin{macrocode}
-\def\xintCstoF {\romannumeral0\xintcstof }%
-\def\xintcstof #1%
-{%
- \expandafter\XINT_cstf_prep \romannumeral-`0#1,\W,%
-}%
-\def\XINT_cstf_prep
-{%
- \XINT_cstf_loop_a 1001%
-}%
-\def\XINT_cstf_loop_a #1#2#3#4#5,%
-{%
- \xint_gob_til_W #5\XINT_cstf_end\W
- \expandafter\XINT_cstf_loop_b
- \romannumeral0\xintrawwithzeros {#5}.{#1}{#2}{#3}{#4}%
-}%
-\def\XINT_cstf_loop_b #1/#2.#3#4#5#6%
-{%
- \expandafter\XINT_cstf_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_cstf_loop_c #1#2%
-{%
- \expandafter\XINT_cstf_loop_d\expandafter {\expandafter{#2}{#1}}%
-}%
-\def\XINT_cstf_loop_d #1#2%
-{%
- \expandafter\XINT_cstf_loop_e\expandafter {\expandafter{#2}#1}%
-}%
-\def\XINT_cstf_loop_e #1#2%
-{%
- \expandafter\XINT_cstf_loop_a\expandafter{#2}#1%
-}%
-\def\XINT_cstf_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,\W,%
-}%
-\def\XINT_icstf_prep
-{%
- \XINT_icstf_loop_a 1001%
-}%
-\def\XINT_icstf_loop_a #1#2#3#4#5,%
-{%
- \xint_gob_til_W #5\XINT_icstf_end\W
- \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+\W/%
-}%
-\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_W #2\XINT_gctf_end\W
- \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+\W/%
-}%
-\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_W #4\XINT_igctf_end\W
- \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{xintCstoCv}}
-% \begin{macrocode}
-\def\xintCstoCv {\romannumeral0\xintcstocv }%
-\def\xintcstocv #1%
-{%
- \expandafter\XINT_cstcv_prep \romannumeral-`0#1,\W,%
-}%
-\def\XINT_cstcv_prep
-{%
- \XINT_cstcv_loop_a {}1001%
-}%
-\def\XINT_cstcv_loop_a #1#2#3#4#5#6,%
-{%
- \xint_gob_til_W #6\XINT_cstcv_end\W
- \expandafter\XINT_cstcv_loop_b
- \romannumeral0\xintrawwithzeros {#6}.{#2}{#3}{#4}{#5}{#1}%
-}%
-\def\XINT_cstcv_loop_b #1/#2.#3#4#5#6%
-{%
- \expandafter\XINT_cstcv_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_cstcv_loop_c #1#2%
-{%
- \expandafter\XINT_cstcv_loop_d\expandafter {\expandafter{#2}{#1}}%
-}%
-\def\XINT_cstcv_loop_d #1#2%
-{%
- \expandafter\XINT_cstcv_loop_e\expandafter {\expandafter{#2}#1}%
-}%
-\def\XINT_cstcv_loop_e #1#2%
-{%
- \expandafter\XINT_cstcv_loop_f\expandafter{#2}#1%
-}%
-\def\XINT_cstcv_loop_f #1#2#3#4#5%
-{%
- \expandafter\XINT_cstcv_loop_g\expandafter
- {\romannumeral0\xintrawwithzeros {#1/#2}}{#5}{#1}{#2}{#3}{#4}%
-}%
-\def\XINT_cstcv_loop_g #1#2{\XINT_cstcv_loop_a {#2{#1}}}% 1.09b removes [0]
-\def\XINT_cstcv_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,\W,%
-}%
-\def\XINT_icstcv_prep
-{%
- \XINT_icstcv_loop_a {}1001%
-}%
-\def\XINT_icstcv_loop_a #1#2#3#4#5#6,%
-{%
- \xint_gob_til_W #6\XINT_icstcv_end\W
- \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+\W/%
-}%
-\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_W #3\XINT_gctcv_end\W
- \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+\W/%
-}%
-\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_W #4\XINT_igctcv_end_a\W
- \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{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 { 0/1[0]}%
- \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 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 { 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 1.06 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\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 strangely here, removed
- \else
- \xint_afterfi {\expandafter\XINT_cntcs_loop\expandafter
- {\the\numexpr #1-1\expandafter}\expandafter
- {\expandafter{\romannumeral-`0#2{#1}}}{#2}}%
- \fi
-}%
-\def\XINT_cntcs_loop #1#2#3%
-{%
- \ifnum #1>-1 \else \XINT_cntcs_exit \fi
- \expandafter\XINT_cntcs_loop\expandafter
- {\the\numexpr #1-1\expandafter }\expandafter
- {\expandafter{\romannumeral-`0#3{#1}},#2}{#3}%
-}%
-\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,{ }%
-% \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.|
-% \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 #1-1\expandafter}\expandafter
- {\expandafter{\romannumeral-`0#2{#1}}}{#2}}%
- \fi
-}%
-\def\XINT_cntgc_loop #1#2#3%
-{%
- \ifnum #1>-1 \else \XINT_cntgc_exit \fi
- \expandafter\XINT_cntgc_loop\expandafter
- {\the\numexpr #1-1\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 #1-1\expandafter}\expandafter
- {\expandafter{\romannumeral-`0#2{#1}}}{#2}{#3}}%
- \fi
-}%
-\def\XINT_gcntgc_loop #1#2#3#4%
-{%
- \ifnum #1>-1 \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 #3-1\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,\W,%
-}%
-\def\XINT_cstc_prep #1,{\XINT_cstc_loop_a {{#1}}}%
-\def\XINT_cstc_loop_a #1#2,%
-{%
- \xint_gob_til_W #2\XINT_cstc_end\W
- \XINT_cstc_loop_b {#1}{#2}%
-}%
-\def\XINT_cstc_loop_b #1#2{\XINT_cstc_loop_a {#1+1/{#2}}}%
-\def\XINT_cstc_end\W\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+\W/%
-}%
-\def\XINT_gctgc_start {\XINT_gctgc_loop_a {}}%
-\def\XINT_gctgc_loop_a #1#2+#3/%
-{%
- \xint_gob_til_W #3\XINT_gctgc_end\W
- \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\W\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}
-%
-% 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]|.
-%
-% Another peculiarity is that the input is allowed to contain (but only where
-% the scanner looks for a number or fraction) material within braces |{...}|.
-% This will be expanded completely and must give an integer, decimal number or
-% fraction (not in scientific notation). Conversely any explict fraction
-% |A/B[n]| \emph{with the brackets} or macro expanding to such a thing
-% \textbf{must} be enclosed within such braces: square brackets are not
-% acceptable by the expression parser.
-%
-% These two things are a bit \emph{experimental} and perhaps I will opt for
-% another approach at a later stage. To circumvent the potential hash-table
-% impact of the |\.=a/b[n]| I have provided the macro creators |\xintNewExpr|
-% and |\xintNewFloatExpr|.
-%
-% 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 modied) 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: the first one a catcode 11
-% exclamation mark, the second one an error generating macro, the third one a
-% printing macro and the fourth is |\.=a/b[n]|. The prefix |\xintthe| makes the
-% output printable by killing the first two tokens.
-%
-% Version |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|.
-%
-% Version |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.
-%
-% Version |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.
-%
-% Version |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.
-%
-% Version |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.
-%
-% Version |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).
-%
-% \localtableofcontents
-% \subsection{Catcodes, \protect\eTeX{} and reload detection}
-%
-% The code for reload detection is copied from \textsc{Heiko
-% Oberdiek}'s packages, and adapted here to check for previous
-% loading of the \xintfracname package.
-%
-% The method for catcodes is slightly different, but still
-% 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\space { }%
- \let\z\endgroup
- \expandafter\let\expandafter\x\csname ver@xintexpr.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{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.
- \y{xintexpr}{now issuing \string\input\space xintfrac.sty}%
- \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.
- \y{xintexpr}{now issuing \string\RequirePackage{xintfrac}}%
- \def\z{\endgroup\RequirePackage{xintfrac}}%
- \fi
- \else
- \y{xintexpr}{I was already loaded, aborting input}%
- \aftergroup\endinput
- \fi
- \fi
- \fi
-\z%
-% \end{macrocode}
-% \subsection{Confirmation of \xintfracnameimp loading}
-% \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 % :
- \ifdefined\PackageInfo
- \def\y#1#2{\PackageInfo{#1}{#2}}%
- \else
- \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}%
- \fi
- \def\empty {}%
- \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname
- \ifx\w\relax % Plain TeX, user gave a file name at the prompt
- \y{xintexpr}{Loading of package xintfrac failed, aborting input}%
- \aftergroup\endinput
- \fi
- \ifx\w\empty % LaTeX, user gave a file name at the prompt
- \y{xintexpr}{Loading of package xintfrac failed, aborting input}%
- \aftergroup\endinput
- \fi
-\endgroup%
-% \end{macrocode}
-% \subsection{Catcodes}
-% \begin{macrocode}
-\XINTsetupcatcodes%
-% \end{macrocode}
-% \subsection{Package identification}
-% \begin{macrocode}
-\XINT_providespackage
-\ProvidesPackage{xintexpr}%
- [2014/02/05 v1.09k Expandable expression parser (jfB)]%
-% \end{macrocode}
-% \subsection{Encapsulation in pseudo cs names, helper macros}
-% \lverb|1.09i uses .= for encapsulation, thus allowing \escapechar to be
-% anything (all previous releases were with ., so \escapechar 46 was forbidden).
-% Besides, the \edef definition has \space already expanded, perhaps this will
-% compensate a tiny bit the time penalty of `.=' viz `.' in unlocking... well
-% not really, I guess. (for no special reason 1.09k uses some \expandafter's
-% rather than \edef+\noexpand's for the definition of \XINT_expr_lock)|
-% \begin{macrocode}
-\def\xint_gob_til_! #1!{}% nota bene: this ! has catcode 11
-\expandafter\def\expandafter
-\XINT_expr_lock\expandafter#\expandafter1\expandafter !\expandafter
- {\expandafter\expandafter\space\csname .=#1\endcsname }%
-\def\XINT_expr_unlock {\expandafter\XINT_expr_unlock_a\string }%
-\def\XINT_expr_unlock_a #1.={}%
-\def\XINT_expr_unexpectedtoken {\xintError:ignored }%
-\def\XINT_newexpr_setprefix #1>{\noexpand\romannumeral-`0}%
-\def\xint_UDxintrelaxfork #1\xint_relax #2#3\krof {#2}%
-% \end{macrocode}
-% \subsection{\csh{xintexpr}, \csh{xinttheexpr}, \csh{xintthe}, ...}
-% \lverb|\xintthe is defined with a parameter, I guess I wanted to make sure no
-% stray space tokens could cause a problem.
-%
-% With 1.09i, \xintiexpr replaces
-% \xintnumexpr which is kept for compatibility but will be removed at some
-% point. Should perhaps issue a warning, but well, people can also read the
-% documentation. Also 1.09i removes \xinttheeval.
-%
-% 1.09i has re-organized the material here.
-%
-% 1.09j modifies the mechanism of \XINT_expr_usethe and
-% \XINT_expr_print, etc... in order for \xintexpr-essions to be usable
-% within \edef'initions. I hesitated quite a bit with adding
-% \xint_protect in front of the \.=digits macros, which will in
-% 99.99999$% of use cases supposed all have \relax meaning; and it is a
-% bit of a pain, really, it is quite a pain to add these extra tokens
-% only for \edef contexts and for situations which will never occur...
-% well no damn'it let's *NOT* add this extra \xint_protect. Just one
-% before the printing macro (which can not be \protected, else \xintthe
-% could not work).|
-% \begin{macrocode}
-\def\xint_protect {\noexpand\xint_protect\noexpand }% 1.09j
-\def\XINT_expr_done {!\XINT_expr_usethe\xint_protect\XINT_expr_print }%
-\let\XINT_iiexpr_done \XINT_expr_done
-\def\XINT_iexpr_done {!\XINT_expr_usethe\xint_protect\XINT_iexpr_print }%
-\def\XINT_flexpr_done {!\XINT_expr_usethe\xint_protect\XINT_flexpr_print }%
-\def\XINT_boolexpr_done {!\XINT_expr_usethe\xint_protect\XINT_boolexpr_print }%
-\protected\def\XINT_expr_usethe #1#2#3% modified in 1.09j
- {\xintError:missing_xintthe!\show#3missing xintthe (see log)!}%
-\def\xintthe #1{\romannumeral-`0\expandafter\xint_gobble_iii\romannumeral-`0#1}%
-\let\XINT_expr_print \XINT_expr_unlock
-\def\XINT_iexpr_print #1{\xintRound:csv {\XINT_expr_unlock #1}}%
-\def\XINT_flexpr_print #1{\xintFloat:csv {\XINT_expr_unlock #1}}%
-\def\XINT_boolexpr_print #1{\xintIsTrue:csv{\XINT_expr_unlock #1}}%
-\def\xintexpr {\romannumeral0\xinteval }%
-\def\xintfloatexpr {\romannumeral0\xintfloateval }%
-\def\xintiiexpr {\romannumeral0\xintiieval }%
-\def\xinteval
- {\expandafter\XINT_expr_until_end_a \romannumeral-`0\XINT_expr_getnext }%
-\def\xintfloateval
- {\expandafter\XINT_flexpr_until_end_a\romannumeral-`0\XINT_expr_getnext }%
-\def\xintiieval
- {\expandafter\XINT_iiexpr_until_end_a\romannumeral-`0\XINT_expr_getnext }%
-\def\xinttheexpr
- {\romannumeral-`0\expandafter\xint_gobble_iii\romannumeral0\xinteval }%
-\def\xintthefloatexpr
- {\romannumeral-`0\expandafter\xint_gobble_iii\romannumeral0\xintfloateval }%
-\def\xinttheiiexpr
- {\romannumeral-`0\expandafter\xint_gobble_iii\romannumeral0\xintiieval }%
-\def\xintiexpr {\romannumeral0\expandafter\expandafter\expandafter
- \XINT_iexpr_done \expandafter\xint_gobble_iv\romannumeral0\xinteval }%
-\def\xinttheiexpr {\romannumeral-`0\expandafter\expandafter\expandafter
- \XINT_iexpr_print\expandafter\xint_gobble_iv\romannumeral0\xinteval }%
-\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 }%
-\let\xintnumexpr \xintiexpr % deprecated
-\let\xintthenumexpr\xinttheiexpr % deprecated
-% \end{macrocode}
-% \subsection{\csh{xintifboolexpr}, \csh{xintifboolfloatexpr}, csh{xintifbooliiexpr}}
-% \lverb|1.09c. Does not work with comma separated expressions. I could
-% make use \xintORof:csv (or AND, or XOR) to allow it, but don't know it the
-% overhead is worth it.
-%
-% 1.09i adds \xintifbooliiexpr |
-% \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{\csh{XINT\_get\_next}: looking for a number}
-% \lverb|June 14: 1.08b adds a second \romannumeral-`0 to \XINT_expr_getnext in
-% an attempt to solve a problem with space tokens stopping the \romannumeral and
-% thus preventing expansion of the following token. For example: 1+ \the\cnta
-% caused a problem, as `\the' was not expanded. I did not define
-% \XINT_expr_getnext as a macro with parameter (which would have cured
-% preventively this), precisely to try to recognize brace pairs. The second
-% \romannumeral-`0 is added for the same reason in other places.
-%
-% The get-next scans forward to find a number: after expansion of what comes
-% next, an opening parenthesis signals a parenthesized sub-expression, a ! with
-% catcode 11 signals there was there an \xintexpr.. \relax sub-expression (now
-% evaluated), a minus is a prefix operator, a plus is silently ignored, a digit
-% or decimal point signals to start gathering a number, braced material {...} is
-% allowed and will be directly fed into a \csname..\endcsname for complete
-% expansion which must delivers a (fractional) number, possibly ending in [n];
-% explicit square brackets must be enclosed into such braces. Once a number
-% issues from the previous procedures, it is a locked into a
-% \csname...\endcsname, and the flow then proceeds with \XINT_expr_getop which
-% will scan for an infix or postfix operator following the number.
-%
-% A special r\^ole is played by underscores _ for use with \xintNewExpr
-% to input macro parameters.
-%
-% Release 1.09a implements functions; the idea is that a letter (actually,
-% anything not otherwise recognized!) triggers the function name gatherer, the
-% comma is promoted to a binary operator of priority intermediate between
-% parentheses and infix operators. The code had some other revisions in order
-% for all the _getnext and _getop macros to now be shared by \xintexpr and
-% \xintfloatexpr.
-%
-% 1.09i now allows direct insertion of \count's, \dimen's and \skip's which will
-% be unpacked using \number.
-%
-% 1.09i speeds up a bit the recognition of a braced thing: the case of a single
-% braced control sequence makes a third expansion mandatory, let's do it
-% immediately and not wait. So macros got shuffled and modified a bit.
-%
-% \XINT_expr_unpackvariable does not insert a [0] for compatibility with
-% \xintiiexpr. A [0] would have made a bit faster \xintexpr macros when dealing
-% with an unpacked count control sequence, as without it the \xintnum will be
-% used in the parsing by xintfrac macros when the number is used. But [0] is not
-% accepted by most macros ultimately called by \xintiiexpr.|
-% \begin{macrocode}
-\def\XINT_expr_getnext
-{%
- \expandafter\XINT_expr_getnext_checkforbraced_a
- \romannumeral-`0\romannumeral-`0%
-}%
-\def\XINT_expr_getnext_checkforbraced_a #1% was done later in <1.09i
-{%
- \expandafter\XINT_expr_getnext_checkforbraced_b\expandafter
- {\romannumeral-`0#1}%
-}%
-\def\XINT_expr_getnext_checkforbraced_b #1%
-{%
- \XINT_expr_getnext_checkforbraced_c #1\xint_relax\Z {#1}%
-}%
-\def\XINT_expr_getnext_checkforbraced_c #1#2%
-{%
- \xint_UDxintrelaxfork
- #1\XINT_expr_getnext_wasemptyorspace
- #2\XINT_expr_getnext_gotonetoken_wehope
- \xint_relax\XINT_expr_getnext_gotbracedstuff
- \krof
-}% doubly braced things are not acceptable, will cause errors.
-\def\XINT_expr_getnext_wasemptyorspace #1{\XINT_expr_getnext }%
-\def\XINT_expr_getnext_gotbracedstuff #1\xint_relax\Z #2%
-{%
- \expandafter\XINT_expr_getop\csname .=#2\endcsname
-}%
-\def\XINT_expr_getnext_gotonetoken_wehope\Z #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_countdimenetc_fork
- \else
- \expandafter\expandafter\expandafter
- \XINT_expr_getnext_onetoken_fork\expandafter\string
- \fi
- #1%
-}%
-\def\XINT_expr_subexpr !#1\fi !{\expandafter\XINT_expr_getop\xint_gobble_iii }%
-\def\XINT_expr_countdimenetc_fork #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
- \XINT_expr_unpackvariable
- \fi\fi\fi\fi\fi\fi
- \expandafter\XINT_expr_getnext\number #1%
-}%
-\def\XINT_expr_unpackvariable\fi\fi\fi\fi\fi\fi\expandafter\XINT_expr_getnext
- \number #1{\fi\fi\fi\fi\fi\fi
- \expandafter\XINT_expr_getop\csname .=\number#1\endcsname }%
-% \end{macrocode}
-% \lverb|1.09a: In order to have this code shared by \xintexpr and
-% \xintfloatexpr, I have moved to the until macros the responsability to choose
-% expr or floatexpr, hence here, the opening parenthesis for example can not be
-% triggered directly as it would not know in which context it works. Hence the
-% \xint_c_xviii ({}. And also the mechanism of \xintNewExpr has been modified to
-% allow use of #.
-%
-% 1.09i also has \xintiiexpr. |
-% \begin{macrocode}
-\begingroup
-\lccode`*=`#
-\lowercase{\endgroup
-\def\XINT_expr_sixwayfork #1(-.+*#2#3\krof {#2}%
-\def\XINT_expr_getnext_onetoken_fork #1%
-{% The * is in truth catcode 12 #. For (hacking) use with \xintNewExpr.
- \XINT_expr_sixwayfork
- #1-.+*{\xint_c_xviii ({}}% back to until for oparen triggering
- (#1.+*{-}%
- (-#1+*{\XINT_expr_scandec_II .}%
- (-.#1*{\XINT_expr_getnext }%
- (-.+#1{\XINT_expr_scandec_II }%
- (-.+*{\XINT_expr_scan_dec_or_func #1}%
- \krof
-}}%
-% \end{macrocode}
-% \subsection{\csh{XINT\_expr\_scan\_dec\_or\_func}: collecting an integer or
-% decimal number or hexa-decimal number or function name}
-% \lverb|\XINT_expr_scanfunc_b rewritten in 1.09i. And 1.09k adds hexadecimal
-% numbers to the syntax, with " as prefix, and possibly a fractional part.
-% Naturally to postfix with an E in scientific notation, one would need to
-% surround the hexadecimal number in parentheses to avoid ambiguities; or
-% rather, just use a lowercase e. By the way, if I allowed only lowercase e for
-% scientific notation I could possibly fuse together the scanning in the dec and
-% hexa cases; up to some loss of syntax control in the dec case.|
-% \begin{macrocode}
-\def\XINT_expr_scan_dec_or_func #1% this #1 has necessarily here catcode 12
-{%
- \ifnum \xint_c_ix<1#1
- \expandafter\XINT_expr_scandec_I
- \else
- \if #1"\expandafter\expandafter\expandafter\XINT_expr_scanhex_I
- \else % We assume we are dealing with a function name!!
- \expandafter\expandafter\expandafter\XINT_expr_scanfunc
- \fi
- \fi #1%
-}%
-\def\XINT_expr_scanfunc
-{%
- \expandafter\XINT_expr_func\romannumeral-`0\XINT_expr_scanfunc_c
-}%
-\def\XINT_expr_scanfunc_c #1%
-{%
- \expandafter #1\romannumeral-`0\expandafter
- \XINT_expr_scanfunc_a\romannumeral-`0\romannumeral-`0%
-}%
-\def\XINT_expr_scanfunc_a #1% please no braced things here!
-{%
- \ifcat #1\relax % missing opening parenthesis, probably
- \expandafter\XINT_expr_scanfunc_panic
- \else
- \xint_afterfi{\expandafter\XINT_expr_scanfunc_b \string #1}%
- \fi
-}%
-\def\xint_UDparenfork #1()#2#3\krof {#2}%
-\def\XINT_expr_scanfunc_b #1%
-{%
- \xint_UDparenfork
- #1){(}% and then \XINT_expr_func
- (#1{(}% and then \XINT_expr_func (this is for bool/toggle names)
- (){\XINT_expr_scanfunc_c #1}%
- \krof
-}%
-\def\XINT_expr_scanfunc_panic {\xintError:bigtroubleahead(0\relax }%
-\def\XINT_expr_func #1(% common to expr and flexpr and iiexpr
-{%
- \xint_c_xviii @{#1}% functions have the highest priority.
-}%
-% \end{macrocode}
-% \lverb|Scanning for a number of fraction. Once gathered, lock it and do
-% _getop. 1.09i modifies \XINT_expr_scanintpart_a (splits _aa) and also
-% \XINT_expr_scanfracpart_a in
-% order for the tacit multiplication of \count's and \dimen's to be compatible
-% with escape-char=a digit.
-%
-% 1.09j further extends for recognition of an \xint..expr and then insertion
-% of a * (which is done in \XINT_expr_getop_a).|
-% \begin{macrocode}
-\def\XINT_expr_scandec_I
-{%
- \expandafter\XINT_expr_getop\romannumeral-`0\expandafter
- \XINT_expr_lock\romannumeral-`0\XINT_expr_scanintpart_b
-}%
-\def\XINT_expr_scandec_II
-{%
- \expandafter\XINT_expr_getop\romannumeral-`0\expandafter
- \XINT_expr_lock\romannumeral-`0\XINT_expr_scanfracpart_b
-}%
-\def\XINT_expr_scanintpart_a #1% Please no braced material: 123{FORBIDDEN}
-{% careful that ! has catcode letter here
- \ifcat #1\relax\else
- \ifx !#1\else
- \expandafter\expandafter\expandafter
- \xint_thirdofthree
- \fi\fi
- \xint_firstoftwo !% this stops the scan
- {\expandafter\XINT_expr_scanintpart_aa\string }#1%
-}%
-\def\XINT_expr_scanintpart_aa #1%
-{%
- \ifnum \xint_c_ix<1#1
- \expandafter\XINT_expr_scanintpart_b
- \else
- \if .#1%
- \expandafter\expandafter\expandafter
- \XINT_expr_scandec_transition
- \else % gather what we got so far, leave catcode 12 #1 in stream
- \expandafter\expandafter\expandafter !% ! of catcode 11, space needed
- \fi
- \fi
- #1%
-}%
-\def\XINT_expr_scanintpart_b #1%
-{%
- \expandafter #1\romannumeral-`0\expandafter
- \XINT_expr_scanintpart_a\romannumeral-`0\romannumeral-`0%
-}%
-\def\XINT_expr_scandec_transition .%
-{%
- \expandafter.\romannumeral-`0\expandafter
- \XINT_expr_scanfracpart_a\romannumeral-`0\romannumeral-`0%
-}%
-\def\XINT_expr_scanfracpart_a #1%
-{%
- \ifcat #1\relax\else
- \ifx !#1\else
- \expandafter\expandafter\expandafter
- \xint_thirdofthree
- \fi\fi
- \xint_firstoftwo !% this stops the scan
- {\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
- \expandafter !%
- \fi
- #1%
-}%
-\def\XINT_expr_scanfracpart_b #1%
-{%
- \expandafter #1\romannumeral-`0\expandafter
- \XINT_expr_scanfracpart_a\romannumeral-`0\romannumeral-`0%
-}%
-% \end{macrocode}
-% \lverb|1.09k [2014/01/21]: added scanning for an hexadecimal number, possibly
-% with a "hexa-decimal" part, only with uppercase ABCDEF (xintbinhex.sty works
-% with ABCDEF, as tex itself requires uppercase letters after ", thus at least I
-% feel comfortable with not bothering allowing abcdef... which would be possible
-% but would complicate things; although perhaps there could be some use for
-% lowercase. If needed, can be implemented, but I will probably long be dead
-% when an archivist droid will be the first around circa 2500 AD to read these
-% lines).
-%
-% For compatibility with \xintiiexpr, the [] thing is incorporated only if there
-% the parser encounters a . indicating a fractional part (this fractional part
-% may be empty). Thus for (infinitesimally) faster further processing by
-% \xintexpr, "ABC.+ etc... is better than "ABC+ etc... on the other hand the
-% initial processing with a . followed by an empty fractional part adds its bit
-% of overhead... The . is not allowed in \xintiiexpr, as it will provoke
-% insertion of [0] which is incompatible with it.|
-% \begin{macrocode}
-\def\XINT_expr_scanhex_I #1%
-{%
- \expandafter\XINT_expr_getop\romannumeral-`0\expandafter
- \XINT_expr_lock\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\else
- \ifx !#1\else
- \expandafter\expandafter\expandafter
- \xint_thirdofthree
- \fi\fi
- \xint_firstoftwo {.I;!}%
- {\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%
-{%
- \expandafter #1\romannumeral-`0\expandafter
- \XINT_expr_scanhexI_a\romannumeral-`0\romannumeral-`0%
-}%
-\def\XINT_expr_scanhex_transition .%
-{%
- \expandafter.\expandafter.\romannumeral-`0\expandafter
- \XINT_expr_scanhexII_a\romannumeral-`0\romannumeral-`0%
-}%
-\def\XINT_expr_scanhexII_a #1%
-{%
- \ifcat #1\relax\else
- \ifx !#1\else
- \expandafter\expandafter\expandafter
- \xint_thirdofthree
- \fi\fi
- \xint_firstoftwo {;!}% this stops the scan
- {\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%
-{%
- \expandafter #1\romannumeral-`0\expandafter
- \XINT_expr_scanhexII_a\romannumeral-`0\romannumeral-`0%
-}%
-% \end{macrocode}
-% \subsection{\csh{XINT\_expr\_getop}: looking for an operator}
-% \lverb|June 14 (1.08b): I add here a second \romannumeral-`0, because
-% \XINT_expr_getnext and others try to expand the next token
-% but without grabbing it.
-%
-% This finds the next infix operator or closing parenthesis or postfix
-% exclamation mark !
-% or expression end. It then leaves in the token flow
-% <precedence> <operator> <locked number>. The <precedence> is generally
-% a character command which thus stops expansion and gives back control to an
-% \XINT_expr_until_<op> command; or it is the minus sign which will be
-% converted by a suitable \XINT_expr_checkifprefix_<p> into an operator
-% with a given inherited precedence. Earlier releases than 1.09c used tricks for
-% the postfix !, ?, :, with <precedence> being in fact a macro to act
-% immediately, and then re-activate \XINT_expr_getop.
-%
-% In versions earlier than 1.09a the <operator> was already made in to a control
-% sequence; but now it is a left as a token and will be (generally) converted by
-% the until macro which knows if it is in a \xintexpr or an \xintfloatexpr. (or
-% an \xintiiexpr, since 1.09i)
-%
-% 1.09i allows \count's, \dimen's, \skip's with tacit multiplication.
-%
-% 1.09j extends the mechanism of tacit multiplication to the case of a sub
-% xintexpression in its various variants. Careful that our ! has catcode 11 so
-% \ifx! would be a disaster...
-%
-% 1.09k extends tacit multiplication to the case of an encountered opening
-% parenthesis.
-%
-% |
-% \begin{macrocode}
-\def\XINT_expr_getop #1% this #1 is the current locked computed value
-{% full expansion of next token, first swallowing a possible space
- \expandafter\XINT_expr_getop_a\expandafter #1%
- \romannumeral-`0\romannumeral-`0%
-}%
-\def\XINT_expr_getop_a #1#2%
-{% if a control sequence is found, must be either \relax or register|variable
- \ifcat #2\relax\expandafter\xint_firstoftwo
- \else \expandafter\xint_secondoftwo
- \fi
- {\ifx #2\relax\expandafter\xint_firstofthree
- \else\expandafter\xint_secondofthree % tacit multiplication
- \fi }%
- {\ifx !#2\expandafter\xint_secondofthree % tacit multiplication
- \else % 1.09k adds tacit multiplication in front of (
- \if (#2\expandafter\expandafter\expandafter\xint_secondofthree
- \else
- \expandafter\expandafter\expandafter\xint_thirdofthree
- \fi
- \fi }%
- {\XINT_expr_foundend #1}%
- {\XINT_expr_foundop *#1#2}%
- {\XINT_expr_foundop #2#1}%
-}%
-\def\XINT_expr_foundend {\xint_c_ \relax }% \relax is a place holder here.
-\def\XINT_expr_foundop #1% then becomes <prec> <op> and is followed by <\.=f>
-{% 1.09a: no control sequence \XINT_expr_op_#1, code common to expr/flexpr
- \ifcsname XINT_expr_precedence_#1\endcsname
- \expandafter\xint_afterfi\expandafter
- {\csname XINT_expr_precedence_#1\endcsname #1}%
- \else
- \XINT_expr_unexpectedtoken
- \expandafter\XINT_expr_getop
- \fi
-}%
-% \end{macrocode}
-% \subsection{Parentheses}
-% \lverb|1.09a removes some doubling of \romannumeral-`\0 from 1.08b
-% which served no useful purpose here (I think...). |
-% \begin{macrocode}
-\def\XINT_tmpa #1#2#3#4#5%
-{%
- \def#1##1%
- {%
- \xint_UDsignfork
- ##1{\expandafter#1\romannumeral-`0#3}%
- -{#2##1}%
- \krof
- }%
- \def#2##1##2%
- {%
- \ifcase ##1\expandafter #4%
- \or\xint_afterfi{%
- \XINT_expr_extra_closing_paren
- \expandafter #1\romannumeral-`0\XINT_expr_getop
- }%
- \else
- \xint_afterfi{\expandafter#1\romannumeral-`0\csname XINT_#5_op_##2\endcsname }%
- \fi
- }%
-}%
-\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\expandafter\endcsname
- \csname XINT_#1_done\endcsname
- {#1}%
-}%
-\def\XINT_expr_extra_closing_paren {\xintError:removed }%
-\def\XINT_tmpa #1#2#3#4#5#6%
-{%
- \def #1{\expandafter #3\romannumeral-`0\XINT_expr_getnext }%
- \let #2#1%
- \def #3##1{\xint_UDsignfork
- ##1{\expandafter #3\romannumeral-`0#5}%
- -{#4##1}%
- \krof }%
- \def #4##1##2%
- {%
- \ifcase ##1\expandafter \XINT_expr_missing_cparen
- \or \expandafter \XINT_expr_getop
- \else \xint_afterfi
- {\expandafter #3\romannumeral-`0\csname XINT_#6_op_##2\endcsname }%
- \fi
- }%
-}%
-\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}%
-}%
-\def\XINT_expr_missing_cparen {\xintError:inserted \xint_c_ \XINT_expr_done }%
-\expandafter\let\csname XINT_expr_precedence_)\endcsname \xint_c_i
-\expandafter\let\csname XINT_flexpr_precedence_)\endcsname \xint_c_i
-\expandafter\let\csname XINT_iiexpr_precedence_)\endcsname \xint_c_i
-\expandafter\let\csname XINT_expr_op_)\endcsname \XINT_expr_getop
-\expandafter\let\csname XINT_flexpr_op_)\endcsname\XINT_expr_getop
-\expandafter\let\csname XINT_iiexpr_op_)\endcsname\XINT_expr_getop
-% \end{macrocode}
-% \subsection{The \csh{XINT\_expr\_until\_<op>} macros for boolean operators,
-% comparison operators, arithmetic operators, scientfic notation.}
-% \lverb|Extended in 1.09a with comparison and boolean operators.
-% 1.09i adds \xintiiexpr and incorporates optional part [\XINTdigits] for a tiny
-% bit faster float operations now already equipped with their optional
-% argument|.
-% \begin{macrocode}
-\def\XINT_tmpb #1#2#3#4#5#6%#7%
-{%
- \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}%{#7}%
-}%
-\def\XINT_tmpc #1#2#3#4#5#6#7#8#9%
-{%
- \def #1##1% \XINT_expr_op_<op>
- {% 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
- {\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
- {% 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#9{\XINT_expr_unlock ##1}{\XINT_expr_unlock ##4}\endcsname }%
- \fi
- }%
- \let #7#5%
-}%
-\def\XINT_tmpa #1{\XINT_tmpb {expr}{xint}#1{}}%
-\xintApplyInline {\XINT_tmpa }{%
- {|{iii}{vi}{OR}}%
- {&{iv}{vi}{AND}}%
- {<{v}{vi}{Lt}}%
- {>{v}{vi}{Gt}}%
- {={v}{vi}{Eq}}%
- {+{vi}{vi}{Add}}%
- {-{vi}{vi}{Sub}}%
- {*{vii}{vii}{Mul}}%
- {/{vii}{vii}{Div}}%
- {^{viii}{viii}{Pow}}%
- {e{ix}{ix}{fE}}%
- {E{ix}{ix}{fE}}%
-}%
-\def\XINT_tmpa #1{\XINT_tmpb {flexpr}{xint}#1{}}%
-\xintApplyInline {\XINT_tmpa }{%
- {|{iii}{vi}{OR}}%
- {&{iv}{vi}{AND}}%
- {<{v}{vi}{Lt}}%
- {>{v}{vi}{Gt}}%
- {={v}{vi}{Eq}}%
-}%
-\def\XINT_tmpa #1{\XINT_tmpb {flexpr}{XINTinFloat}#1{[\XINTdigits]}}%
-\xintApplyInline {\XINT_tmpa }{%
- {+{vi}{vi}{Add}}%
- {-{vi}{vi}{Sub}}%
- {*{vii}{vii}{Mul}}%
- {/{vii}{vii}{Div}}%
- {^{viii}{viii}{Power}}%
- {e{ix}{ix}{fE}}%
- {E{ix}{ix}{fE}}%
-}%
-\def\XINT_tmpa #1{\XINT_tmpb {iiexpr}{xint}#1{}}%
-\xintApplyInline {\XINT_tmpa }{%
- {|{iii}{vi}{OR}}%
- {&{iv}{vi}{AND}}%
- {<{v}{vi}{Lt}}%
- {>{v}{vi}{Gt}}%
- {={v}{vi}{Eq}}%
- {+{vi}{vi}{iiAdd}}%
- {-{vi}{vi}{iiSub}}%
- {*{vii}{vii}{iiMul}}%
- {/{vii}{vii}{iiQuo}}%
- {^{viii}{viii}{iiPow}}%
- {e{ix}{ix}{iE}}%
- {E{ix}{ix}{iE}}%
-}%
-% \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_,_a
- {%
- \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{\csh{XINT\_expr\_op\_-<level>}: minus as prefix inherits its
-% precedence level}
-% \lverb|1.09i: \xintiiexpr must use \xintiiOpp (or at least \xintiOpp, but that
-% would be a waste; however impacts round and trunc as I allow them).|
-% \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 conditional}
-% \lverb|New with 1.09a. Modified in 1.09c to have less precedence than
-% functions. Code is cleaner as it does not play tricks with _precedence. There
-% is no associated until macro, because action is immediate once activated (only
-% a previously scanned function can delay activation).|
-% \begin{macrocode}
-\let\XINT_expr_precedence_? \xint_c_x
-\def \XINT_expr_op_? #1#2#3%
-{%
- \xintifZero{\XINT_expr_unlock #1}%
- {\XINT_expr_getnext #3}%
- {\XINT_expr_getnext #2}%
-}%
-\let\XINT_flexpr_op_?\XINT_expr_op_?
-\let\XINT_iiexpr_op_?\XINT_expr_op_?
-% \end{macrocode}
-% \subsection{: as three-way conditional}
-% \lverb|New with 1.09a. Modified in 1.09c to have less precedence than
-% functions. |
-% \begin{macrocode}
-\let\XINT_expr_precedence_: \xint_c_x
-\def \XINT_expr_op_: #1#2#3#4%
-{%
- \xintifSgn {\XINT_expr_unlock #1}%
- {\XINT_expr_getnext #2}%
- {\XINT_expr_getnext #3}%
- {\XINT_expr_getnext #4}%
-}%
-\let\XINT_flexpr_op_:\XINT_expr_op_:
-\let\XINT_iiexpr_op_:\XINT_expr_op_:
-% \end{macrocode}
-% \subsection{! as postfix factorial operator}
-% \lverb|The factorial is currently the exact one, there is no float version.
-% Starting with 1.09c, it has lower priority than functions, it is not executed
-% immediately anymore. The code is cleaner and does not abuse _precedence, but
-% does assign it a true level. There is no until macro, because the factorial
-% acts on what precedes 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{Functions}
-% \lverb|New with 1.09a. Names of ..Float..:csv macros have been changed in
-% 1.09h |
-% \begin{macrocode}
-\def\XINT_tmpa #1#2#3#4{%
- \def #1##1%
- {%
- \ifcsname XINT_expr_onlitteral_##1\endcsname
- \expandafter\XINT_expr_funcoflitteral
- \else
- \expandafter #2%
- \fi {##1}%
- }%
- \def #2##1%
- {%
- \ifcsname XINT_#4_func_##1\endcsname
- \xint_afterfi
- {\expandafter\expandafter\csname XINT_#4_func_##1\endcsname}%
- \else \csname xintError:unknown `##1\string'\endcsname
- \xint_afterfi{\expandafter\XINT_expr_func_unknown}%
- \fi
- \romannumeral-`0#3%
- }%
-}%
-\xintFor #1 in {expr,flexpr,iiexpr} \do {%
- \expandafter\XINT_tmpa
- \csname XINT_#1_op_@\expandafter\endcsname
- \csname XINT_#1_op_@@\expandafter\endcsname
- \csname XINT_#1_oparen\endcsname {#1}%
-}%
-\def\XINT_expr_funcoflitteral #1%
-{%
- \expandafter\expandafter\csname XINT_expr_onlitteral_#1\endcsname
- \romannumeral-`0\XINT_expr_scanfunc
-}%
-\def\XINT_expr_onlitteral_bool #1#2#3{\expandafter\XINT_expr_getop
- \csname .=\xintBool{#3}\endcsname }%
-\def\XINT_expr_onlitteral_togl #1#2#3{\expandafter\XINT_expr_getop
- \csname .=\xintToggle{#3}\endcsname }%
-\def\XINT_expr_func_unknown #1#2#3% 1.09i removes [0], because \xintiiexpr
- {\expandafter #1\expandafter #2\csname .=0\endcsname }%
-\def\XINT_expr_func_reduce #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\xintIrr {\XINT_expr_unlock #3}\endcsname
-}%
-\let\XINT_flexpr_func_reduce\XINT_expr_func_reduce
-% \XINT_iiexpr_func_reduce not defined
-\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
- .=\XINTinFloatFrac [\XINTdigits]{\XINT_expr_unlock #3}\endcsname
-}%
-% \XINT_iiexpr_func_frac not defined
-\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]%
- {\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_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_floor #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\xintFloor {\XINT_expr_unlock #3}\endcsname
-}%
-\let\XINT_flexpr_func_floor\XINT_expr_func_floor
-\let\XINT_iiexpr_func_floor\XINT_expr_func_floor
-\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
-\let\XINT_iiexpr_func_ceil\XINT_expr_func_ceil
-\def\XINT_expr_twoargs #1,#2,{{#1}{#2}}%
-\def\XINT_expr_func_quo #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname .=%
- \expandafter\expandafter\expandafter\xintQuo
- \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\xintRem
- \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_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_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_oneortwo #1#2,#3,#4.%
-{%
- \if\relax#4\relax\expandafter\xint_firstoftwo\else
- \expandafter\xint_secondoftwo\fi
- {#1{0}}{#1{#3}}{#2}%
-}%
-\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_argandopt #1,#2,#3.%
-{%
- \if\relax#3\relax\expandafter\xint_firstoftwo\else
- \expandafter\xint_secondoftwo\fi
- {[\XINTdigits]}{[\xintNum {#2}]}{#1}%
-}%
-\def\XINT_expr_func_float #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname .=%
- \expandafter\XINTinFloat
- \romannumeral-`0\expandafter\XINT_expr_argandopt
- \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname
-}%
-\let\XINT_flexpr_func_float\XINT_expr_func_float
-% \XINT_iiexpr_func_float not defined
-\def\XINT_expr_func_sqrt #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname .=%
- \expandafter\XINTinFloatSqrt
- \romannumeral-`0\expandafter\XINT_expr_argandopt
- \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname
-}%
-\let\XINT_flexpr_func_sqrt\XINT_expr_func_sqrt
-\def\XINT_iiexpr_func_sqrt #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\xintiSqrt {\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
-\let\XINT_iiexpr_func_gcd\XINT_expr_func_gcd
-\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
-\let\XINT_iiexpr_func_lcm\XINT_expr_func_lcm
-\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
- .=\xintiMaxof: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
- .=\xintiMinof: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
-}%
-\def\XINT_expr_func_sum #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\xintSum:csv{\XINT_expr_unlock #3}\endcsname
-}%
-\def\XINT_flexpr_func_sum #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\XINTinFloatSum:csv{\XINT_expr_unlock #3}\endcsname
-}%
-\def\XINT_iiexpr_func_sum #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\xintiiSum:csv{\XINT_expr_unlock #3}\endcsname
-}%
-\def\XINT_expr_func_prd #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\xintPrd:csv{\XINT_expr_unlock #3}\endcsname
-}%
-\def\XINT_flexpr_func_prd #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\XINTinFloatPrd:csv{\XINT_expr_unlock #3}\endcsname
-}%
-\def\XINT_iiexpr_func_prd #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\xintiiPrd:csv{\XINT_expr_unlock #3}\endcsname
-}%
-\let\XINT_expr_func_add\XINT_expr_func_sum
-\let\XINT_expr_func_mul\XINT_expr_func_prd
-\let\XINT_flexpr_func_add\XINT_flexpr_func_sum
-\let\XINT_flexpr_func_mul\XINT_flexpr_func_prd
-\let\XINT_iiexpr_func_add\XINT_iiexpr_func_sum
-\let\XINT_iiexpr_func_mul\XINT_iiexpr_func_prd
-\def\XINT_expr_func_? #1#2#3%
-{%
- \expandafter #1\expandafter #2\csname
- .=\xintIsNotZero {\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
- .=\xintIsZero {\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
- .=\xintIsZero {\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,{\xintifNotZero{#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,{\xintifSgn{#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
-% \end{macrocode}
-% \subsection{\csh{xintNewExpr}, \csh{xintNewFloatExpr}\dots}
-% \lverb|&
-% Rewritten in 1.09a. Now, the parameters of the formula are entered in the
-% usual way by the user, with # not _. And _ is assigned to make macros
-% not expand. This way, : is freed, as we now need it for the ternary operator.
-% (on numeric data; if use with macro parameters, should be coded with the
-% functionn ifsgn , rather)
-%
-% Code unified in 1.09c, and \xintNewNumExpr, \xintNewBoolExpr added. 1.09i
-% renames \xintNewNumExpr to \xintNewIExpr, and defines \xintNewIIExpr.|
-% \begin{macrocode}
-\def\XINT_newexpr_print #1{\ifnum\xintNthElt{0}{#1}>1
- \expandafter\xint_firstoftwo
- \else
- \expandafter\xint_secondoftwo
- \fi
- {_xintListWithSep,{#1}}{\xint_firstofone#1}}%
-\xintForpair #1#2 in {(fl,Float),(i,iRound0),(bool,IsTrue)}\do {%
- \expandafter\def\csname XINT_new#1expr_print\endcsname
- ##1{\ifnum\xintNthElt{0}{##1}>1
- \expandafter\xint_firstoftwo
- \else
- \expandafter\xint_secondoftwo
- \fi
- {_xintListWithSep,{\xintApply{_xint#2}{##1}}}
- {_xint#2##1}}}%
-\toks0 {}%
-\xintFor #1 in {Bool,Toggle,Floor,Ceil,iRound,Round,iTrunc,Trunc,TFrac,%
- Lt,Gt,Eq,AND,OR,IsNotZero,IsZero,ifNotZero,ifSgn,%
- Irr,Num,Abs,Sgn,Opp,Quo,Rem,Add,Sub,Mul,Sqr,Div,Pow,Fac,fE,iSqrt,%
- iiAdd,iiSub,iiMul,iiSqr,iiPow,iiQuo,iiRem,iiSgn,iiAbs,iiOpp,iE}\do
- {\toks0
- \expandafter{\the\toks0\expandafter\def\csname xint#1\endcsname {_xint#1}}}%
-\xintFor #1 in {,Sqrt,Add,Sub,Mul,Div,Power,fE,Frac}\do
- {\toks0
- \expandafter{\the\toks0\expandafter\def\csname XINTinFloat#1\endcsname
- {_XINTinFloat#1}}}%
-\xintFor #1 in {GCDof,LCMof,Maxof,Minof,ANDof,ORof,XORof,Sum,Prd,%
- iMaxof,iMinof,iiSum,iiPrd}\do
- {\toks0
- \expandafter{\the\toks0\expandafter\def\csname xint#1:csv\endcsname
- ####1{_xint#1{\xintCSVtoListNonStripped {####1}}}}}%
-\xintFor #1 in {Maxof,Minof,Sum,Prd}\do
- {\toks0
- \expandafter{\the\toks0\expandafter\def\csname XINTinFloat#1:csv\endcsname
- ####1{_XINTinFloat#1{\xintCSVtoListNonStripped {####1}}}}}%
-\expandafter\def\expandafter\XINT_expr_protect\expandafter{\the\toks0
- \def\XINTdigits {_XINTdigits}%
- \def\XINT_expr_print ##1{\expandafter\XINT_newexpr_print\expandafter
- {\romannumeral0\xintcsvtolistnonstripped{\XINT_expr_unlock ##1}}}%
- \def\XINT_flexpr_print ##1{\expandafter\XINT_newflexpr_print\expandafter
- {\romannumeral0\xintcsvtolistnonstripped{\XINT_expr_unlock ##1}}}%
- \def\XINT_iexpr_print ##1{\expandafter\XINT_newiexpr_print\expandafter
- {\romannumeral0\xintcsvtolistnonstripped{\XINT_expr_unlock ##1}}}%
- \def\XINT_boolexpr_print ##1{\expandafter\XINT_newboolexpr_print\expandafter
- {\romannumeral0\xintcsvtolistnonstripped{\XINT_expr_unlock ##1}}}%
-}%
-\toks0 {}%
-\def\xintNewExpr {\xint_NewExpr\xinttheexpr }%
-\def\xintNewFloatExpr {\xint_NewExpr\xintthefloatexpr }%
-\def\xintNewIExpr {\xint_NewExpr\xinttheiexpr }%
-\let\xintNewNumExpr\xintNewIExpr
-\def\xintNewIIExpr {\xint_NewExpr\xinttheiiexpr }%
-\def\xintNewBoolExpr {\xint_NewExpr\xinttheboolexpr }%
-% \end{macrocode}
-% \lverb|1.09i has added \escapechar 92, as \meaning is used in \XINT_NewExpr,
-% and a non existent escape-char would be a problem with \scantokens. Also
-% \catcode32 is set to 10 in \xintexprSafeCatcodes for being extra-safe.|
-% \begin{macrocode}
-\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
- \escapechar92
- \XINT_NewExpr #1%
-}%
-\catcode`* 13
-\def\XINT_NewExpr #1#2%
-{%
- \def\XINT_tmpa ##1##2##3##4##5##6##7##8##9{#2}%
- \XINT_expr_protect
- \lccode`*=`_ \lowercase {\def*}{!noexpand!}%
- \catcode`_ 13 \catcode`: 11 %\endlinechar -1 %not sure why I had that, \par?
- \everyeof {\noexpand }%
- \edef\XINT_tmpb ##1##2##3##4##5##6##7##8##9%
- {\scantokens
- \expandafter{\romannumeral-`0#1%
- \XINT_tmpa {####1}{####2}{####3}%
- {####4}{####5}{####6}%
- {####7}{####8}{####9}%
- \relax}}%
- \lccode`*=`\$ \lowercase {\def*}{####}%
- \catcode`\$ 13 \catcode`! 0 \catcode`_ 11 %
- \the\toks0
- {\scantokens\expandafter{\expandafter
- \XINT_newexpr_setprefix\meaning\XINT_tmpb}}%
- \endgroup
-}%
-\let\xintexprRestoreCatcodes\empty
-\def\xintexprSafeCatcodes
-{% for end user.
- \edef\xintexprRestoreCatcodes {%
- \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
- }% it's hard to know where to stop...
- \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 \texttt{#1:}& \digitstt{ #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:
- \digitstt{\xintiiSum{\xintApply\mymacro\storedlinecounts}}. Each
- package starts
- with circa \digitstt{80} lines dealing with catcodes, package identification
- and reloading management, also for Plain \TeX\strut. Version
- \texttt{\xintversion} of \texttt{\xintdate}.\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 {21378}
-\makeatletter\check@checksum\makeatother
-\Finale
-%% End of file xint.dtx
diff --git a/Master/texmf-dist/doc/generic/xint/xint.pdf b/Master/texmf-dist/doc/generic/xint/xint.pdf
index 4b2f17f993c..70e5dccdff3 100644
--- a/Master/texmf-dist/doc/generic/xint/xint.pdf
+++ b/Master/texmf-dist/doc/generic/xint/xint.pdf
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generated by cgit v1.2.3 (git 2.39.1) at 2024-12-25 08:08:26 +0000