% -*- coding: iso-latin-1; time-stamp-format: "%02d-%02m-%:y %02H:%02M:%02S %Z" -*-
% File: xint.dtx, package: 1.09n (2014/04/01), documentation: 2014/04/01
% 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: <01-04-2014 19:06:46 CEST>}
%</dtx>
%<*drv>
\def\xintdate    {2014/04/01} 
\def\xintversion {1.09n}
%</drv>
%%----------------------------------------------------------------
%% The xint bundle (version 1.09n of April 1st, 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.
%
% Whether compiling xint.tex or xint.dtx, the documentation is by default
% produced without inclusion of the source code. See instructions in the
% file xint.tex for changing this default.
%
% 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
%% or without source code accoding to value of \NoSourceCode toggle below.
%%
%% 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 1 % replace 1<space> by 0<space> for source code inclusion
\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 xint.tex}
\Msg{* then dvipdfmx on xint.dvi. Edit xint.tex to get the code}
\Msg{* source included.    (ignore the 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, 1 = dvipdfmx
        \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 run latex thrice on xint.tex^^J}%
\Msg{* then dvipdfmx on xint.dvi. Edit xint.tex to get the code^^J}%
\Msg{* source included.  (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
% % (j'avais été fainéant à l'époque, mais le problème c'est que donc je
% % ne contrôle alors plus directement les paramètres d'espacement verticaux ce
% % qui rend délicat de synchroniser par exemple les espacements horizontaux
% % pour section et subsection ou vertical à la fin, etc..) 
% \makeatother

\newif\ifindescription % 1 avril 2014
\indescriptiontrue
\def\sectioncouleur{{cyan}}

% attention à ce 22 hard codé. 23 maintenant,... 24; et 31 non 32... et ça
% continue de changer 
%
% 1er avril 2014, je fais un vrai style un peu grossier alors qu'avant
% j'utilisais savedsectionline, par paresse.
\etocsetstyle{section}{}
     {\normalfont}
     {% avant: 1 avril 2014, j'avais
      % \savedsectionline{\numberline{\expandafter\textcolor\sectioncouleur
      %                                 {\etocnumber}}\etocname}
      %                  {{\mdseries\etocpage}}%
         \addvspace{\medskipamount}%
         \noindent
         \rightskip 1.5em
         \parfillskip -1.5em\relax
      \ifnum\etocthenumber=23 \gdef\sectioncouleur{{joli}}\fi
      \ifnum\etocthenumber=31 \gdef\sectioncouleur{[named]{RoyalPurple}}\fi
      \bfseries 
      \makebox[2.3em][l]{\expandafter\textcolor\sectioncouleur {\etocnumber}}%
      \strut\etocname
      \mdseries\nobreak\hfill\nobreak\strut\makebox[1.5em][r]{\etocpage}\par
% 1 avril 2014
      \let\ETOCsectionnumber\etocthenumber
% je pourrais sans doute simplement modifier mon \gobbletodot utilisé dans le
% style de sous-section, mais c'est qu'il y a la main TOC et les autres. Je fais
% des styles communs à toutes.
      }%
     {}%

\def\MARGEPAGENO {1.5em}

\etocsetstyle{subsection}
    {\begingroup\normalfont
     \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
             {\ifindescription\expandafter\textcolor\sectioncouleur
               {\normalfont\bfseries\ETOCsectionnumber}\fi
              .\expandafter\gobbletodot\etocthenumber}}}%
     \strut\etocname\nobreak\leaders\etoctoclineleaders\hfill\nobreak
     \strut\makebox[1.5em][r]{\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}}}}
%-------------------------------------------------------------------------------
% 24 février 2014. J'ai besoin de me débarasser du \to
\def\NewWith #1{%
    \vadjust{\vskip-\dp\strutbox
             \smash{\hbox to 0pt {\hss\color[named]{PineGreen}%
                        \normalfont\small
                        \hsize 1.5cm\rightskip.5cm minus.5cm
                        \vtop{\noindent New with #1}\ }}%
             \vskip\dp\strutbox }\strut{}} 

\makeatother

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

% 7 mars 2013
% This macro allows to conveniently center a line inside a paragraph and still
% 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 [1]{%
    \phantomsection\section* {Contents}
    \etoctoccontentsline*{toctobookmark}{Contents}{1}%
}
   \etocsettagdepth {description}{subsection}
   \etocsettagdepth {commandsA}  {none}
   \etocsettagdepth {xintexpr}   {none}
   \etocsettagdepth {commandsB}  {none}
   \etocsettagdepth {implementation}{none}
\tableofcontents
%original: \newcommand*\etocabovetocskip{3.5ex plus 1ex minus .2ex}
\renewcommand*\etocabovetocskip{\medskipamount}
\etocmulticolstyle [2]{\raggedcolumns}
   \etocsettagdepth {description}{none}
   \etocsettagdepth {commandsA}  {section}
   \etocsettagdepth {xintexpr}   {section}
   \etocsettagdepth {commandsB}  {section}
\ifnum\NoSourceCode=0
   \etocsettagdepth {implementation}{section}
\else
   \etocsettagdepth {implementation}{none}
\fi
\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}\label{ssec:printnumber}

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

%
\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 (however it doesnt allow, for example to separate digits by
groups of three).\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...).


\subsection{FAQ}

\begin{description}
\item[Will \xintexprname implement \texttt{exp}, \texttt{log},
  \texttt{cos}, \texttt{sin} \dots at some point?]
I guess so.
\item[\xintseriesname already provides generic tools.] Right, although the
  casual user of the \xintname bundle will not quite know how to do variable
  reduction expandably in order to use some series or Pad\'e approximants.
  Besides I wrote the code at the beginning of the project and perhaps I could
  do it better now (I have not looked at it for a while). Anyhow, generic things
  do not help much if one wants to optimize.
\item[Optimizing? isn't \TeX's macro expansion mechanism intrinsically slow?]
  Intensive use of \csa{numexpr} and some token manipulation algorithms
  exploiting to the best I could \TeX{} macros with parameters grant \xintname a
  significant speed up in expandable arithmetic on big integers compared to
  previously available implementations. You can do some comparisons with
  multiplication on numbers with @100@ digits or division of one of @100@ digits
  by another of @50@ digits, for example. However expandability is antagonist of
  speed, and I agree it is not very exciting to optimize slow things. And I was
  disappointed last year to realize the slowness of \TeX's mouth when it has to
  keep hundreds of tokens in cheek to mix them later with new aliments. Believe
  me, I try not to think too much about the fact that the whole enterprise is
  made irrelevant by Lua\LaTeX's ability to access external libraries.
\item[Well, why isn't this \texttt{log} etc\dots thing done yet?] 
I have to decide on the maximal precision to achieve: @24@, @32@, @48@,
@64@,\dots ; to settle that I would need to implement some initial versions and
benchmark them. 
\item[Fair enough. That's the common lot. So why not yet?] I am a bit
  overworked. It is also an opportunity to think over
  the basic underlying mathematics, and will need devoted thinking for some not
  insignificant amount of time. So far I didn't find the time, or rather I found
  out good means to waste it sillily. I also anticipate that originality could
  very 
  well not pay off at all, so small is the window for the precision.
\item[Any chance this could be done in time for TL2014?] No,
  sorry.\newline Release |1.09m| of |[2014/02/26]| was the end of a cycle, and
  this |1.09n| of |[2014/04/01]| is only for a bug fix and inclusion of this
  |FAQ| in the documentation.
\end{description}

\section{Recent changes}

\footnotesize


\noindent Release |1.09n| (|[2014/04/01]|):
\begin{itemize}
\item the user manual does not include by default the source code anymore: the
  |\NoSourceCode| toggle in file |xint.tex| has to be set to @0@ before
  compilation to get source code inclusion.
\item bug fix in |\XINT_nthelt_finish| (this bug was introduced in |1.09i| of
  |2013/12/18| and showed up when the index |N| was larger
  than the number of elements of the list).
\end{itemize}

\noindent Releases |1.09m| (|[2014/02/26]|):
\begin{itemize}
\item new macros in \xinttoolsname: \csbxint{Keep} keeps the first |N|
  or last |N| elements of a list (sequence of braced items);
  \csbxint{Trim} cuts out either the first |N| or the last |N| elements
  from a list.
\item new macros in \xintcfracname: \csbxint{FGtoC} finds the
  initial partial quotients common to two numbers or fractions |f| and
  |g|; \csbxint{GGCFrac} is a clone of \csbxint{GCFrac} which however
  does not assume that the coefficients of the generalized continued
  fraction are numeric quantities. Some other minor changes.
\end{itemize}

\noindent Releases |1.09ka| (|[2014/02/05]|) and |1.09kb| (|[2014/02/13]|):
\begin{itemize}
\item bug fix (\xintexprname): an aloof modification done by |1.09i| to
  \csbxint{NewExpr} had resulted in a spurious trailing space present in the
  outputs of all macros created by |\xintNewExpr|, making nesting of such macros
  impossible.
\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}

\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}

For a more detailed change history, see \autoref{sec:releases}.

\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
\hyperref[ssec: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.

Whether compiling xint.tex or xint.dtx, the documentation is by default
produced without inclusion of the source code. See instructions in the
file xint.tex for changing this default.

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}
\indescriptionfalse

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

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

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


% \clearpage % attention à ce 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, maintaining existing braces and adding a brace pair around each
naked token encountered. Space tokens (in-between top level braces or naked
tokens) are gobbled. This macro is mainly thought out for use on a \meta{list}
of such braced material; with such a list as argument the \fexpan sion will only
hit against the first opening brace, hence do nothing, and the braced stuff may
thus be macros one does not want to expand.
\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 \verb+|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{xintKeep}}\label{xintKeep}

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

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

\csa{xintKeepNoExpand} does the same without first \fexpan ding its list
argument.
\centeredline {|\oodef\test {\xintKeep {17}{\xintKeep {-69}{\xintSeq
        {1}{100}}}}\meaning\test|}
\centeredline {\oodef\test {\xintKeep {17}{\xintKeep {-69}{\xintSeq
        {1}{100}}}}\meaning\test}


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

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

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

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


\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|
(beware that |\xintSeq {1}{0}| is thus not empty but |{1}{0}|, use
|\xintSeq [1]{1}{N}| if you want an empty sequence for |N| zero or
negative).


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

\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
\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
\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...
\end{verbatim}
\vspace{-2\baselineskip}
\setlength{\unitlength}{1cm}%
% le picture de LaTeX est tout de même assez génial!
\begin{picture}(0,0)
\put(5,-.5){\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}
\begin{verbatim}
\[\begin{pmatrix}
    \A11&\A12&\A13&\A14\\
    \A21&\A22&\A23&\A24\\
    \A31&\A32&\A33&\A34
  \end{pmatrix}
\times
  \begin{pmatrix}
    \B11&\B12&\B13\\
    \B21&\B22&\B23\\
    \B31&\B32&\B33\\
    \B41&\B42&\B43
  \end{pmatrix}
=
\begin{pmatrix}
    \C11&\C12&\C13\\
    \C21&\C22&\C23\\
    \C31&\C32&\C33
\end{pmatrix}\]
\[\begin{pmatrix}
    \C11&\C12&\C13\\
    \C21&\C22&\C23\\
    \C31&\C32&\C33
\end{pmatrix}^2 = \begin{pmatrix}
    \D11&\D12&\D13\\
    \D21&\D22&\D23\\
    \D31&\D32&\D33
\end{pmatrix}\]
\end{verbatim}
\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}|

We see that despite the fact that the first encountered tabulation character in
the first row close a group and thus erases |\Row| from \TeX's memory,
|\xintApplyInline| knows how to deal with this.

Using \csbxint{ApplyUnbraced} is an alternative: the difference is that
this would have prepared all rows first and only put them back into the
token stream once they are all assembled, whereas with |\xintApplyInline|
each row is constructed and immediately fed back into the token stream: when
one does things with numbers having hundreds of digits, one learns that
keeping on hold and shuffling around hundreds of tokens has an impact on
\TeX{}'s speed (make this ``thousands of tokens'' for the impact to be
noticeable).

One may nest various |\xintApplyInline|'s. For example (see the 
\hyperref[float]{table} \vpageref{float}):\par
\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}

Version |1.0| was released |2013/03/28|. This is \texttt{\xintversion} of
\texttt{\xintdate}. 

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| (|2013/04/14|) of the
\xintname bundle. The general rule of the bundle that each macro first expands
(what comes first, fully) each one of its arguments applies.


|f|\ntype{\Ff} stands for an integer or a fraction (see \autoref{sec:inputs}
for the accepted input formats) or something which expands to an integer or
fraction. It is possible to use in the numerator or the denominator of |f| count
registers and even expressions with infix arithmetic operators, under some rules
which are explained in the previous \hyperref[sec:useofcount]{Use of count
  registers} section.

As in the \hyperref[sec:xint]{xint.sty} documentation, |x|\ntype{\numx}
stands for something which will internally be embedded in a \csa{numexpr}.
It 
may thus be a count register or something like |4*\count 255 + 17|, etc..., but
must expand to an integer obeying the \TeX{} bound.

The fraction format on output is the scientific notation for the `float' macros,
and the |A/B[n]| format for all other fraction macros, with the exception of
\csbxint{Trunc}, {\color{blue}\string\xint\-Round} (which produce decimal
numbers) and \csbxint{Irr}, \csbxint{Jrr}, \csbxint{RawWithZeros} (which returns
an |A/B| with no trailing |[n]|, and prints the |B| even if it is |1|), and
\csbxint{PRaw} which does not print the |[n]| if |n=0| or the |B| if |B=1|. 

To be certain to print an integer output without trailing |[n]| nor fraction
slash, one should use either |\xintPRaw {\xintIrr {f}}| or |\xintNum {f}| when
it is already known that |f| evaluates to a (big) integer. For example
|\xintPRaw {\xintAdd {2/5}{3/5}}| gives a perhaps disappointing
\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| (|2013/05/25|)
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.

Macros created by |\xintNewExpr| can thus be nested:
\dverb|@
    \xintNewExpr \MyFunction [1]{reduce(2*#1^3 - #1^-2*3)}
    (1) \MyFunction {\MyFunction {2/3}}
    \xintNewFloatExpr \MyOtherFunction [1]{(#1+#1^-1)/(#1-#1^-1)}
    (2) \MyOtherFunction {1.234}
    (3) \MyOtherFunction {\MyOtherFunction {1.234}}|\newline
\xintNewExpr \MyFunction [1]{reduce(2*#1^3 - #1^-2*3)}
\xintNewFloatExpr \MyOtherFunction [1]{(#1+#1^-1)/(#1-#1^-1)}
(1) \digitstt{\MyFunction {\MyFunction {2/3}}}\newline
(2) \digitstt{\MyOtherFunction {1.234}}\newline
(3) \digitstt{\MyOtherFunction {\MyOtherFunction {1.234}}}

\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| (|2013/06/07|) 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| (|2013/03/28|) of the
\xintname bundle.

Since release |1.09a| the macros filter their inputs through the \csbxint{Num}
macro, so one can use count registers, or fractions as long as they reduce to
integers. 

%% \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}

This package was first released with version |1.03| (|2013/04/14|) of the
\xintname bundle.

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...

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| (|2013/04/25|) of the
\xintname bundle. It was kept almost unchanged until |1.09m| of |2014/02/26|
which brings some new macros: \csbxint{FtoC}, \csbxint{CtoF}, \csbxint{CtoCv},
dealing with sequences of braced partial quotients rather than comma separated
ones, \csbxint{FGtoC} which is to produce ``guaranteed'' coefficients of some
real number known approximately, and \csbxint{GGCFrac} for displaying arbitrary
material as a continued fraction; also, some changes to existing macros:
\csbxint{FtoCs} and \csbxint{CntoCs} insert spaces after the commas,
\csbxint{CstoF} and \csbxint{CstoCv} authorize spaces in the input also before
the commas.

This section contains:
\begin{enumerate}
\item an \hyperref[ssec:cfracoverview]{overview} of the package functionalities,
\item a description of each one of the package macros,
\item further illustration of their use via the study of the
  \hyperref[ssec:e-convergents]{convergents of $e$}.
\end{enumerate}


\localtableofcontents

\subsection{Package overview}\label{ssec:cfracoverview}

The package computes partial quotients and convergents of a fraction, or
conversely start from coefficients and obtain the corresponding fraction; three
macros \csbxint {CFrac}, \csbxint {GCFrac} and \csbxint {GGCFrac} are
for typesetting (the first two assume that the coefficients are numeric
quantities acceptable by the \xintfracname \csbxint{Frac} macro, the
last one will display arbitrary material), the others
can be nested (if applicable) or see their outputs further processed by other
macros from the \xintname bundle, particularly the macros of \xinttoolsname
dealing with sequences of braced items or comma separated lists.

A \emph{simple} continued fraction has coefficients
|[c0,c1,...,cN]| (usually called partial quotients, but I 
dislike this entrenched terminology), where |c0| is a positive or
negative integer and the others are positive integers. 


Typesetting is usually done via the |amsmath| macro |\cfrac|:
\centeredline{|\[ c_0 + \cfrac{1}{c_1+\cfrac1{c_2+\cfrac1{c_3+\cfrac1{\ddots}}}}\]|}
\[ c_0 + \cfrac{1}{c_1+\cfrac1{c_2+\cfrac1{c_3+\cfrac1{\ddots}}}}\]
Here is a concrete example:
%
\centeredline{|\[ \xintFrac {208341/66317}=\xintCFrac {208341/66317} \]|} 
%
\[ \xintFrac {208341/66317}=\xintCFrac {208341/66317}\]%
%
But it is the command \csbxint{CFrac} which did all the work of \emph{computing}
the continued fraction \emph{and} using |\cfrac| from |amsmath| to typeset
it.

A \emph{generalized} continued fraction has the same structure but the
numerators are not restricted to be $1$, and numbers used in the continued
fraction may be arbitrary, also fractions, irrationals, complex,
indeterminates.\footnote{\xintcfracname may be used with indeterminates, 
  for basic conversions from one inline format to another, but not for
  actual computations. See \csbxint{GGCFrac}.}
The \emph{centered} continued fraction is an
example:
\centeredline{|\[ \xintFrac {915286/188421}=\xintGCFrac
  {5+-1/7+1/39+-1/53+-1/13} \]|}
\[ \xintFrac {915286/188421}=\xintGCFrac {5+-1/7+1/39+-1/53+-1/13}
=\xintCFrac {915286/188421}\] 
 
The command \csbxint{GCFrac}, contrarily to
\csbxint{CFrac}, does not compute anything, it just typesets starting from a
generalized continued fraction in inline format, which in this example
was input literally.  We also used \csa{xintCFrac}
for comparison of the two types of continued fractions.

To let \TeX{} compute the centered continued fraction of |f| there is
\csbxint{FtoCC}:
 \centeredline{|\[\xintFrac {915286/188421}\to\xintFtoCC {915286/188421}\]|}
\[\xintFrac {915286/188421}\to\xintFtoCC {915286/188421}\]
The package macros are expandable and may be nested (naturally \csa{xintCFrac}
and \csa{xintGCFrac} must be at the top level, as they deal with typesetting).
Thus 
 \centeredline{|\[\xintGCFrac {\xintFtoCC{915286/188421}}\]|}
produces
\[\xintGCFrac {\xintFtoCC{915286/188421}}\]


The `inline' format expected on input by \csbxint{GCFrac} is
\centeredline{$a_0+b_0/a_1+b_1/a_2+b_2/a_3+\cdots
  +b_{n-2}/a_{n-1}+b_{n-1}/a_n$}%
Fractions among the coefficients are allowed but they must be enclosed
within braces. Signed integers may be left without braces (but the |+|
signs are mandatory). No spaces are allowed around the plus and fraction
symbols. The coefficients may themselves be macros, as long as these
macros are \fexpan dable.
%
\centeredline{|\xintGCFrac {1+-1/57+\xintPow {-3}{7}/\xintQuo
    {132}{25}}|}
\[ \xintFrac{\xintGCtoF  {1+-1/57+\xintPow {-3}{7}/\xintQuo {132}{25}}}=
   \xintGCFrac {1+-1/57+\xintPow {-3}{7}/\xintQuo {132}{25}}\]
To compute the actual fraction one has \csbxint{GCtoF}:
\centeredline{|\[\xintFrac{\xintGCtoF  {1+-1/57+\xintPow {-3}{7}/\xintQuo
      {132}{25}}}\]|}
\[\xintFrac{\xintGCtoF  {1+-1/57+\xintPow {-3}{7}/\xintQuo
      {132}{25}}}\]

For non-numeric input  there is \csbxint{GGCFrac}.
%
\centeredline{|\[\xintGGCFrac
  {a_0+b_0/a_1+b_1/a_2+b_2/\ddots+\ddots/a_{n-1}+b_{n-1}/a_n}\]|} 
%
\[\xintGGCFrac {a_0+b_0/a_1+b_1/a_2+b_2/\ddots+\ddots/a_{n-1}+b_{n-1}/a_n}\]
%

For regular continued fractions, there is a simpler comma separated format:
\centeredline
{|-7,6,19,1,33\to\xintFrac{\xintCstoF{-7,6,19,1,33}}=\xintCFrac{\xintCstoF{-7,6,19,1,33}}|}
%
\[
-7,6,19,1,33\to
\xintFrac{\xintCstoF{-7,6,19,1,33}}=\xintCFrac{\xintCstoF{-7,6,19,1,33}}\] 
%
The command \csbxint{FtoCs} produces from a fraction |f| the comma separated
list of its coefficients.
\centeredline{|\[\xintFrac{1084483/398959}=[\xintFtoCs{1084483/398959}]\]|}
\[\xintFrac{1084483/398959}=[\xintFtoCs{1084483/398959}]\]
If one prefers other separators, one can use the two arguments macros
\csbxint{FtoCx} whose first argument is the separator (which may consist of more
than one token) which is to be used.
\centeredline{|\[\xintFrac{2721/1001}=\xintFtoCx {+1/(}{2721/1001})\cdots)\]|}
\[\xintFrac{2721/1001}=\xintFtoCx {+1/(}{2721/1001})\cdots)\]

This allows under Plain  \TeX{} with |amstex| to obtain the same effect 
as with \LaTeX{}+|\amsmath|+\csbxint{CFrac}:
\centeredline{|$$\xintFwOver{2721/1001}=\xintFtoCx {+\cfrac1\\ }{2721/1001}\endcfrac$$|}


As a shortcut to \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}}}

To obtain the coefficients as a sequence of braced numbers, there is
\csbxint{FtoC} (this is a shortcut for |\xintFtoCx {}|). This list
(sequence) may then be manipulated using the various macros of \xinttoolsname
such as the non-expandable macro \csbxint{AssignArray} or the expandable
\csbxint{Apply} and \csbxint{ListWithSep}.

Conversely to go from such a sequence of braced coefficients to the
corresponding fraction there is \csbxint{CtoF}.

The `|\printnumber|' (\autoref{ssec:printnumber}) macro which we use in this
document to print long numbers can also be useful on long continued fractions.
%
\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

The various fractions obtained from the truncation of a continued fraction to
its initial terms are called the convergents. The commands of \xintcfracname
such as \csbxint{FtoCv}, \csbxint{FtoCCv}, and others which compute such
convergents, return them as a list of braced items, with no separator (as does
\csbxint {FtoC} for the partial quotients). Here is an example:

\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{\xintiiPow {2}{#1}}%

The macro \csbxint{CntoF} allows to specify the coefficients as a function given
by a one-parameter macro. The produced values do not have to be integers.
\centeredline{|\def\cn #1{\xintiiPow {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|@
\[ \xintFrac{\xintCstoF{3,7,15,1,292,1,1}}=
   \xintGCFrac{3+1/7+1/15+1/1+1/292+1/1+1/1}=
   \xintTrunc{10}{\xintCstoF{3,7,15,1,292,1,1}}\dots\]|
\[ \xintFrac{\xintCstoF{3,7,15,1,292,1,1}}=
\xintGCFrac{3+1/7+1/15+1/1+1/292+1/1+1/1}=
\xintTrunc{10}{\xintCstoF{3,7,15,1,292,1,1}}\dots\]

When studying the continued fraction of some real number, there is always
some doubt about how many terms are valid, when computed starting from some
approximation. If $f\leqslant x\leqslant g$ and $f, g$ both have the
same first $K$ partial quotients, then $x$ also has the same first $K$ quotients
and convergents. The macro \csbxint{FGtoC} outputs as a sequence of braced items
the common partial quotients of its two arguments. We can thus use it to produce
a sure list of valid convergents of $\pi$ for example, starting from some proven
lower and upper bound:

\dverb|@
$$\pi\to [\xintListWithSep{,}
         {\xintFGtoC {3.14159265358979323}{3.14159265358979324}}, \dots]$$
$\pi\to\xintListWithSep{,\allowbreak\;}
   {\xintApply{\xintFrac}
   {\xintCtoCv{\xintFGtoC {3.14159265358979323}{3.14159265358979324}}}}, \dots$|
$$\pi\to [\xintListWithSep{,}
         {\xintFGtoC {3.14159265358979323}{3.14159265358979324}}, \dots]$$
\noindent$\pi\to\xintListWithSep{,\allowbreak\;}
    {\xintApply{\xintFrac}
     {\xintCtoCv{\xintFGtoC {3.14159265358979323}{3.14159265358979324}}}},\;\dots$

\subsection{\csbh{xintCFrac}}\label{xintCFrac}

\csa{xintCFrac}|{f}|\ntype{\Ff} is a math-mode only, \LaTeX{} with |amsmath|
only, macro which first computes then displays with the help of |\cfrac| the
simple continued fraction corresponding to the given fraction. It admits an
optional argument which may be |[l]|, |[r]| or (the default) |[c]| to specify
the location of the one's in the numerators of the sub-fractions. Each
coefficient is typeset using the \csbxint{Frac} macro from the \xintfracname
package. This macro is \fexpan dable in the sense that it prepares expandably
the whole expression with the multiple |\cfrac|'s, but it is not completely
expandable naturally as |\cfrac| isn't.

\subsection{\csbh{xintGCFrac}}\label{xintGCFrac}

\csa{xintGCFrac}|{a+b/c+d/e+f/g+h/...+x/y}|\ntype{f} uses similarly |\cfrac|
to prepare the typesetting with the |amsmath| |\cfrac| (\LaTeX{}) of a
generalized continued fraction given in inline format (or as macro which
will \fexpan d to it). It admits the
same optional argument as \csa{xintCFrac}. Plain \TeX{} with |amstex|
users, see \csbxint{GCtoGCx}.
%
\centeredline{|\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}\]|}
\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}\]
%
This is mostly a typesetting macro, although it does provoke the
expansion of the coefficients. See \csbxint{GCtoF} if you are impatient
to see this specific fraction computed.

It admits an optional argument within square brackets which may be
either |[l]|, |[c]| or |[r]|. Default is |[c]| (numerators are centered).

Numerators and denominators are made arguments to the \csbxint{Frac}
macro. This allows them to be themselves fractions or anything \fexpan
dable giving numbers or fractions, but also means however that they can
not be arbitrary material, they can not contain color changing commands
for example. One of the reasons is that \csa{xintGCFrac} tries to
determine the signs of the numerators and chooses accordingly to use
$+$ or $-$.

\subsection{\csbh{xintGGCFrac}}\label{xintGGCFrac}

\csa{xintGGCFrac}|{a+b/c+d/e+f/g+h/...+x/y}|\ntype{f} is a clone of
\csbxint{GCFrac}, hence again \LaTeX{} specific with package
|amsmath|.\NewWith {1.09m}
It does not assume the coefficients to be numbers as understood by
\xintfracname. The macro can be used for displaying arbitrary content as
a continued fraction with |\cfrac|, using only plus signs though. Note
though that it will first \fexpan d its argument, which may be thus be
one of the \xintcfracname macros producing a (general) continued
fraction in inline format, see \csbxint{FtoCx} for an example. If this
expansion is not wished, it is enough to start the argument with a
space.
\centeredline{|\[\xintGGCFrac {1+q/1+q^2/1+q^3/1+q^4/1+q^5/\ddots}\]|}
\[\xintGGCFrac {1+q/1+q^2/1+q^3/1+q^4/1+q^5/\ddots}\]

\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}} 
%
The following can be used byt Plain \TeX{}+|amstex| users to obtain an
output similar as the ones produced by \csbxint{GCFrac} and
\csbxint{GGCFrac}:\par
\dverb|@
$$\xintGCtoGCx {+\cfrac}{\\}{a+b/...}\endcfrac$$
$$\xintGCtoGCx {+\cfrac\xintFwOver}{\\\xintFwOver}{a+b/...}\endcfrac$$|


\subsection{\csbh{xintFtoC}}\label{xintFtoC}

\csa{xintFtoC}|{f}|\etype{\Ff} computes the 
coefficients of the simple continued fraction of |f| and returns them as a list
(sequence) of braced items.\NewWith {1.09m}

\centeredline{%
  |\oodef\test{\xintFtoC{-5262046/89233}}\texttt{\meaning\test}|}%
\centeredline{\oodef\test{\xintFtoC{-5262046/89233}}\texttt{\meaning\test}}

\subsection{\csbh{xintFtoCs}}\label{xintFtoCs}

\csa{xintFtoCs}|{f}|\etype{\Ff} returns the comma separated list of the
coefficients of the simple continued fraction of |f|. Notice that starting with
|1.09m| a space follows each comma (mainly for usage in text mode, as in math
mode spaces are produced in the typeset output by \TeX{} itself).
\centeredline{%
  |\[ \xintSignedFrac{-5262046/89233} \to [\xintFtoCs{-5262046/89233}]\]|}%
\[ \xintSignedFrac{-5262046/89233} \to [\xintFtoCs{-5262046/89233}]\]

\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| separated with the
help of |sep|, which may be anything (and is kept unexpanded). For
example, with Plain \TeX{} and |amstex|, 
%
\centeredline{|$$\xintFtoCx {+\cfrac1\\ }{-5262046/89233}\endcfrac$$|} 
%
will display the continued fraction using
|\cfrac|. Each coefficient is inside a brace pair \hbox{|{ }|}, allowing
a macro to end the separator and fetch it as argument,
for example, again with Plain \TeX{} and |amstex|:
\dverb|@
  \def\highlight #1{\ifnum #1>200 \textcolor{red}{#1}\else #1\fi}
  $$\xintFtoCx {+\cfrac1\\ \highlight}{104348/33215}\endcfrac$$|
%

Due to the different and extremely cumbersome syntax of |\cfrac| under
\LaTeX{} it proves a bit tortuous to obtain there the same effect.
Actually, it is partly for this purpose that |1.09m| added \csbxint
{GGCFrac}. We thus use \csa{xintFtoCx} with a suitable separator, and\;
then the whole thing as argument to \csbxint{GGCFrac}:
%
 \dverb|@
\def\highlight #1{\ifnum #1>200 
                    \fcolorbox{blue}{white}{\boldmath\color{red}$#1$}%
                  \else #1\fi}
\[\xintGGCFrac {\xintFtoCx {+1/\highlight}{208341/66317}}\]|
\def\highlight #1{\ifnum #1>200 
                    \fcolorbox{blue}{white}{\boldmath\color{red}$#1$}%
                  \else #1\fi}
\[\xintGGCFrac {\xintFtoCx {+1/\highlight}{208341/66317}}\]

\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{xintFGtoC}}\label{xintFGtoC}

\csa{xintFGtoC}|{f}{g}|\etype{\Ff\Ff} computes the common initial coefficients
to 
two given fractions |f| and |g|. Notice\NewWith {1.09m} that any real number |f<x<g| or |f>x>g|
will then necessarily share with |f| and |g| these common initial coefficients
for its regular continued fraction. The coefficients are output as a sequence of
braced numbers. This list can then be manipulated via macros from
\xinttoolsname, or other macros of \xintcfracname.

\centeredline{%
  |\oodef\test{\xintFGtoC{-5262046/89233}{-5314647/90125}}\texttt{\meaning\test}|}%
\centeredline{\oodef\test{\xintFGtoC{-5262046/89233}{-5314647/90125}}\texttt{\meaning\test}}
\centeredline{%
  |\oodef\test{\xintFGtoC{3.141592653}{3.141592654}}\texttt{\meaning\test}|}%
\centeredline{%
  \oodef\test{\xintFGtoC{3.141592653}{3.141592654}}\texttt{\meaning\test}}%
\centeredline{%
  |\oodef\test{\xintFGtoC{3.1415926535897932384}{3.1415926535897932385}}\texttt{\meaning\test}|}%
\oodef\test{\xintFGtoC{3.1415926535897932384}{3.1415926535897932385}}%
\centeredline{%
  \texttt{\meaning\test}}%
% \centeredline{\xintRound {30}{\xintCstoF{\xintListWithSep{,}{\test}}}}
% \centeredline{\xintRound {30}{\xintCtoF{\test}}}
% \centeredline{\xintCtoF{\test}}
\centeredline{%
  |\oodef\test{\xintFGtoC{1.41421356237309}{1.4142135623731}}\texttt{\meaning\test}|}%
\oodef\test{\xintFGtoC{1.41421356237309}{1.4142135623731}}%
\centeredline{%
  \texttt{\meaning\test}}%
% \centeredline{\xintRound {30}{\xintCstoF{\xintListWithSep{,}{\test}}}}
% \centeredline{\xintRound {30}{\xintCtoF{\test}}}
% \centeredline{\xintCtoF{\test}}

\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{xintCstoF}}\label{xintCstoF}

\csa{xintCstoF}|{a,b,c,d,...,z}|\etype{f} computes the fraction corresponding to
the coefficients, which may be fractions or even macros expanding to such
fractions. The final fraction may then be highly reducible. Starting with
release |1.09m| spaces before commas are allowed and trimmed automatically
(spaces after commas were already silently handled in earlier releases).
\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{xintCtoF}}\label{xintCtoF}

\csa{xintCtoF}|{{a}{b}{c}...{z}}|\etype{f} computes the fraction corresponding
to the coefficients, which may be fractions or even macros.\NewWith {1.09m}
\centeredline{|\xintCtoF {\xintApply {\xintiiPow 3}{\xintSeq {1}{5}}}|}
% pour vérifier que l'expansion se fait bien:
%\centeredline{\digitstt{\xintCtoF {\xintApply { \xintiiPow 3}{\xintSeq {1}{5}}}}}
\centeredline{\digitstt{\xintCtoF {\xintApply {\xintiiPow 3}{\xintSeq {1}{5}}}}}
\centeredline{|\[ \xintFrac{14946960/4805083}=\xintCFrac {14946960/4805083}\]|}
\[ \xintFrac{14946960/4805083}=\xintCFrac {14946960/4805083}\]

In the example above the power of @3@ was already pre-computed via the expansion
done by |\xintApply|, but if we try with |\xintApply { \xintiiPow 3}| where the
space will stop this expansion, we can check that |\xintCtoF| will itself
provoke the needed coefficient expansion.
% ok


\subsection{\csbh{xintGCtoF}}\label{xintGCtoF}

\csa{xintGCtoF}|{a+b/c+d/e+f/g+......+v/w+x/y}|\etype{f} computes the fraction
defined by the inline generalized continued fraction. Coefficients may be
fractions but must then be put within braces. They can be macros. The plus signs
are mandatory. \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{xintCstoCv}}\label{xintCstoCv}

\csa{xintCstoCv}|{a,b,c,d,...,z}|\etype{f} returns the sequence of the
corresponding convergents, each one within braces.

It is allowed to use fractions as coefficients (the computed
convergents have then no reason to be the real convergents of the final
fraction). When the coefficients are integers, the convergents are irreducible
fractions, but otherwise it is not necessarily the case.
\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{xintCtoCv}}\label{xintCtoCv}

\csa{xintCtoCv}|{{a}{b}{c}...{z}}|\etype{f} returns the sequence of the
corresponding convergents, each one within braces.\NewWith {1.09m}
\centeredline{|\oodef\test{\xintCtoCv {11111111111}}\texttt{\meaning\test}|}
\centeredline{\oodef\test{\xintCtoCv {11111111111}}\texttt{\meaning\test}}

\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{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{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{|\xintCntoF {5}{\macro}|} 
\centeredline{\digitstt{\xintCntoF {5}{\macro}}} 
%
This example shows that the fraction is output with a trailing number in square
brackets (representing a power of ten), this is for consistency with what do
most macros of \xintfracname, and does not have to be always this annoying |[0]|
as the coefficients may for example be numbers in scientific notation. To avoid
these trailing square brackets, for example if the coefficients are known to be integers, there is always the possibility to filter the output via
\csbxint{PRaw}, or \csbxint{Irr} (the latter is overkill in the case of integer
coefficients, as the fraction is guaranteed to be irreducible then).

\subsection{\csbh{xintGCntoF}}\label{xintGCntoF}

\def\coeffA #1{\the\numexpr #1+4-3*((#1+2)/3)\relax }%
\def\coeffB #1{\the\numexpr \ifodd #1 -\fi 1\relax }% (-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|.
\centeredline{|\def\coeffA #1{\the\numexpr #1+4-3*((#1+2)/3)\relax }%|}%
\centeredline{|\def\coeffB #1{\the\numexpr \ifodd #1 -\fi 1\relax }% (-1)^n|}
\centeredline{|\[\xintGCFrac{\xintGCntoGC {6}{\coeffA}{\coeffB}}|}%
\centeredline{|  = \xintFrac{\xintGCntoF  {6}{\coeffA}{\coeffB}}\]|}
\[\xintGCFrac{\xintGCntoGC {6}{\coeffA}{\coeffB}}
= \xintFrac{\xintGCntoF {6}{\coeffA}{\coeffB}}\]
There is also \csbxint{GCntoGC} to get the `inline format' continued
fraction. 


\subsection{\csbh{xintCntoCs}}\label{xintCntoCs}

\csa{xintCntoCs}|{N}{\macro}|\etype{\numx f} produces the comma separated list
of the corresponding coefficients, from |n=0| to |n=N|. The |N| is given to a
|\numexpr|. \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|. The coefficients are enclosed into pairs
of braces, and may thus be fractions, the fraction slash will not be
confused in further processing by the continued fraction slashes.
%
\dverb|@ 
\def\an #1{\the\numexpr  #1*#1*#1+1\relax}%
\def\bn #1{\the\numexpr \ifodd#1 -\fi 1*(#1+1)\relax}%
$\xintGCntoGC {5}{\an}{\bn}}=\xintGCFrac {\xintGCntoGC {5}{\an}{\bn}} =
\displaystyle\xintFrac {\xintGCntoF {5}{\an}{\bn}}$\par|

\def\an #1{\the\numexpr #1*#1*#1+1\relax}%
\def\bn #1{\the\numexpr \ifodd #1 -\fi 1*(#1+1)\relax}%
$\xintGCntoGC {5}{\an}{\bn}=\xintGCFrac {\xintGCntoGC {5}{\an}{\bn}}
        = \displaystyle\xintFrac {\xintGCntoF  {5}{\an}{\bn}}$\par


\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{xintiCstoF}, \csbh{xintiGCtoF}, \csbh{xint\-iCstoCv}, \csbh{xintiGCtoCv}}\label{xintiCstoF}
\label{xintiGCtoF}
\label{xintiCstoCv}
\label{xintiGCtoCv}

Essentially\etype{f} the same as the corresponding macros without the
`i', but for integer-only input. Infinitesimally faster, mainly for
internal use by the package.

\subsection{\csbh{xintGCtoGC}}\label{xintGCtoGC}

\csa{xintGCtoGC}|{a+b/c+d/e+f/g+......+v/w+x/y}|\etype{f} expands (with the
usual meaning) each one of the coefficients and returns an inline continued
fraction of the same type, each expanded coefficient being enclosed 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 forgotten for which purpose I wrote this macro in the first
place.


\subsection{Euler's number \texorpdfstring{$e$}{e}}\label{ssec:e-convergents}

Let us explore
the convergents of Euler's number $e$.
\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 $-1$.


% 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 edit xint.tex to set the
  \string\NoSourceCode\space toggle to 0, then run thrice "latex" on
  xint.tex and finally dvipdfmx on xint.dvi.
\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}%
  [2014/04/01 v1.09n 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}%
\long\def\xint_firstoftwo  #1#2{#1}%
\long\def\xint_secondoftwo #1#2{#2}%
\long\def\xint_firstofone_thenstop  #1{ #1}%
\long\def\xint_firstoftwo_thenstop  #1#2{ #1}%
\long\def\xint_secondoftwo_thenstop #1#2{ #2}%
\def\xint_minus_thenstop { -}%
\def\xint_gob_til_zero #10{}% no need to make it long, so far
\def\xint_UDzerominusfork #10-#2#3\krof {#2}% id.
\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
\chardef\xint_c_ii   2
\chardef\xint_c_iii  3
\chardef\xint_c_iv   4
\chardef\xint_c_v    5
\chardef\xint_c_vi   6
\chardef\xint_c_vii  7
\chardef\xint_c_viii 8
\newtoks\XINT_toks
\xint_firstofone{\let\XINT_sptoken= } %<- space here!
%    \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. Very minor optimization in 1.09m.| 
%    \begin{macrocode}
\def\xintLength {\romannumeral0\xintlength }%
\long\def\xintlength #1%
{%
    \XINT_length_loop 
    0.#1\xint_relax\xint_relax\xint_relax\xint_relax
         \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\long\def\XINT_length_loop #1.#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_xint_relax #9\XINT_length_finish_a\xint_relax
    \expandafter\XINT_length_loop\the\numexpr #1+\xint_c_viii.%
}%
\def\XINT_length_finish_a\xint_relax\expandafter\XINT_length_loop
    \the\numexpr #1+\xint_c_viii.#2\xint_bye
{%
    \XINT_length_finish_b #2\W\W\W\W\W\W\W\Z {#1}%
}%
\def\XINT_length_finish_b #1#2#3#4#5#6#7#8\Z
{%
    \xint_gob_til_W 
            #1\XINT_length_finish_c \xint_c_
            #2\XINT_length_finish_c \xint_c_i
            #3\XINT_length_finish_c \xint_c_ii
            #4\XINT_length_finish_c \xint_c_iii
            #5\XINT_length_finish_c \xint_c_iv
            #6\XINT_length_finish_c \xint_c_v
            #7\XINT_length_finish_c \xint_c_vi
            \W\XINT_length_finish_c \xint_c_vii\Z
}%
\edef\XINT_length_finish_c #1#2\Z #3%
   {\noexpand\expandafter\space\noexpand\the\numexpr #3+#1\relax}%
%    \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+&
% First included in release 1.06.
%  
% \xintNthElt {i}{stuff expanding to {a}{b}...{z}} (or `tokens' abcd...z)returns
% the i th element (one pair of braces removed). The list is first expanded. The
% \xintNthEltNoExpand does no expansion of its second argument. Both variants
% expand the first argument inside \numexpr.
%
% With i = 0, the number of items is returned. This is different from \xintLen
% which is only for numbers (particularly, it checks the sign) and different
% from \xintLength which does not first expand its argument.
% 
% Negative values return the |i|th element from the end. Release 1.09m
% rewrote the initial bits of the code (which checked the sign of #1 and
% expanded or not #2), ome `improvements' made earlier in 1.09c were quite
% sub-efficient. Now uses \xint_UDzerominusfork, moved from xint.sty.
%
% A bug in \XINT_nthelt_finish was introduced in 1.09i (2013/12/18): in order to
% pre-expand \space, I used an \edef as in quite a few other places of the code.
% But I didn't pay attention that the earlier \xint_gobble_iii had to be
% transformed into a \xint_gobble_ii, as the \space was now already expanded to
% a <space token>. This bug meant that when the index N was > the length of the
% list the macro would gobble one token :(. Fixed in 1.09n.+
%    \begin{macrocode}
\def\xintNthElt         {\romannumeral0\xintnthelt }%
\def\xintNthEltNoExpand {\romannumeral0\xintntheltnoexpand }%
\def\xintnthelt #1#2%
{%
    \expandafter\XINT_nthelt_a\the\numexpr #1\expandafter.%
    \expandafter{\romannumeral-`0#2}%
}%
\def\xintntheltnoexpand #1%
{%
    \expandafter\XINT_nthelt_a\the\numexpr #1.%
}%
\def\XINT_nthelt_a #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\XINT_nthelt_bzero}%
        0#1{\XINT_nthelt_bneg {#2}}%
         0-{\XINT_nthelt_bpos {#1#2}}%
    \krof 
}%
\long\def\XINT_nthelt_bzero #1%
{%
     \XINT_length_loop 0.#1\xint_relax\xint_relax\xint_relax\xint_relax
                 \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\long\def\XINT_nthelt_bneg #1#2%
{%
     \expandafter\XINT_nthelt_loop_a\expandafter {\the\numexpr #1\expandafter}%
     \romannumeral0\xintrevwithbracesnoexpand {#2}%
          \xint_relax\xint_relax\xint_relax\xint_relax
          \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\long\def\XINT_nthelt_bpos #1#2%
{%
     \XINT_nthelt_loop_a {#1}#2\xint_relax\xint_relax\xint_relax\xint_relax
                    \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\def\XINT_nthelt_loop_a #1%
{%
    \ifnum #1>\xint_c_viii
        \expandafter\XINT_nthelt_loop_b
    \else
        \XINT_nthelt_getit
    \fi
    {#1}%
}%
\long\def\XINT_nthelt_loop_b #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_xint_relax #9\XINT_nthelt_silentend\xint_relax
    \expandafter\XINT_nthelt_loop_a\expandafter{\the\numexpr #1-\xint_c_viii}%
}%
\def\XINT_nthelt_silentend #1\xint_bye { }%
\def\XINT_nthelt_getit\fi #1%
{%
    \fi\expandafter\expandafter\expandafter\XINT_nthelt_finish
    \csname xint_gobble_\romannumeral\numexpr#1-\xint_c_i\endcsname
}%
\long\edef\XINT_nthelt_finish #1#2\xint_bye 
   {\noexpand\xint_gob_til_xint_relax #1\noexpand\expandafter\space
                               \noexpand\xint_gobble_ii\xint_relax\space #1}%
%    \end{macrocode}
% \subsection{\csh{xintKeep}}
% \lverb+&
% First included in release 1.09m.
%  
% \xintKeep {i}{stuff expanding to {a}{b}...{z}} (or `tokens' abcd...z,
% but each naked token ends up braced in the output) returns (in two
% expansion steps) the first i elements from the list, which is first
% f-expanded. The i is expanded inside \numexpr. Variant
% \xintKeepNoExpand does not expand the list argument.
%
% With i = 0, the empty sequence is returned. 
%
% With i<0, the last |i| elements are returned (in the same order as in
% the original list). 
%
% With |i| equal to or bigger than the length of the (f-expanded) list,
% the full list is returned.+
%    \begin{macrocode}
\def\xintKeep         {\romannumeral0\xintkeep }%
\def\xintKeepNoExpand {\romannumeral0\xintkeepnoexpand }%
\def\xintkeep #1#2%
{%
    \expandafter\XINT_keep_a\the\numexpr #1\expandafter.%
    \expandafter{\romannumeral-`0#2}%
}%
\def\xintkeepnoexpand #1%
{%
    \expandafter\XINT_keep_a\the\numexpr #1.%
}%
\def\XINT_keep_a #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\expandafter\space\xint_gobble_i }%
        0#1{\XINT_keep_bneg_a {#2}}%
         0-{\XINT_keep_bpos {#1#2}}%
    \krof 
}%
\long\def\XINT_keep_bneg_a #1#2%
{%
    \expandafter\XINT_keep_bneg_b \the\numexpr \xintLength{#2}-#1.{#2}%
}%
\def\XINT_keep_bneg_b #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\xint_firstofone_thenstop }%
        0#1{\xint_firstofone_thenstop }%
         0-{\XINT_trim_bpos {#1#2}}%
    \krof 
}%
\long\def\XINT_keep_bpos #1#2%
{%
     \XINT_keep_loop_a {#1}{}#2\xint_relax\xint_relax\xint_relax\xint_relax
          \xint_relax\xint_relax\xint_relax\xint_bye
}%
\def\XINT_keep_loop_a #1%
{%
    \ifnum #1>\xint_c_vi
        \expandafter\XINT_keep_loop_b
    \else
        \XINT_keep_finish
    \fi
    {#1}%
}%
\long\def\XINT_keep_loop_b #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_xint_relax #9\XINT_keep_enda\xint_relax
    \expandafter\XINT_keep_loop_c\expandafter{\the\numexpr #1-\xint_c_vii}%
    {{#3}{#4}{#5}{#6}{#7}{#8}{#9}}{#2}%
}%
\long\def\XINT_keep_loop_c #1#2#3{\XINT_keep_loop_a {#1}{#3#2}}%
\long\def\XINT_keep_enda\xint_relax
     \expandafter\XINT_keep_loop_c\expandafter #1#2#3#4\xint_bye
{%
     \XINT_keep_endb #4\W\W\W\W\W\W\Z #2{#3}%
}%
\def\XINT_keep_endb #1#2#3#4#5#6#7\Z
{%
    \xint_gob_til_W
    #1\XINT_keep_endc_
    #2\XINT_keep_endc_i
    #3\XINT_keep_endc_ii
    #4\XINT_keep_endc_iii
    #5\XINT_keep_endc_iv
    #6\XINT_keep_endc_v
    \W\XINT_keep_endc_vi\Z 
}%
\long\def\XINT_keep_endc_    #1\Z #2#3#4#5#6#7#8#9{ #9}%
\long\def\XINT_keep_endc_i   #1\Z #2#3#4#5#6#7#8#9{ #9{#2}}%
\long\def\XINT_keep_endc_ii  #1\Z #2#3#4#5#6#7#8#9{ #9{#2}{#3}}%
\long\def\XINT_keep_endc_iii #1\Z #2#3#4#5#6#7#8#9{ #9{#2}{#3}{#4}}%
\long\def\XINT_keep_endc_iv  #1\Z #2#3#4#5#6#7#8#9{ #9{#2}{#3}{#4}{#5}}%
\long\def\XINT_keep_endc_v   #1\Z #2#3#4#5#6#7#8#9{ #9{#2}{#3}{#4}{#5}{#6}}%
\long\def\XINT_keep_endc_vi\Z #1#2#3#4#5#6#7#8{ #8{#1}{#2}{#3}{#4}{#5}{#6}}%
\long\def\XINT_keep_finish\fi #1#2#3#4#5#6#7#8#9\xint_bye
{%
    \fi\XINT_keep_finish_loop_a {#1}{}{#3}{#4}{#5}{#6}{#7}{#8}\Z {#2}%
}%
\def\XINT_keep_finish_loop_a #1%
{%
    \xint_gob_til_zero #1\XINT_keep_finish_z0%
    \expandafter\XINT_keep_finish_loop_b\expandafter
       {\the\numexpr #1-\xint_c_i}%
}%
\long\def\XINT_keep_finish_z0%
     \expandafter\XINT_keep_finish_loop_b\expandafter #1#2#3\Z #4{ #4#2}%
\long\def\XINT_keep_finish_loop_b #1#2#3%
{%
    \xint_gob_til_xint_relax #3\XINT_keep_finish_exit\xint_relax
    \XINT_keep_finish_loop_c {#1}{#2}{#3}%
}%
\long\def\XINT_keep_finish_exit\xint_relax
         \XINT_keep_finish_loop_c #1#2#3\Z #4{ #4#2}%
\long\def\XINT_keep_finish_loop_c #1#2#3%
        {\XINT_keep_finish_loop_a {#1}{#2{#3}}}%
%    \end{macrocode}
% \subsection{\csh{xintTrim}}
% \lverb+&
% First included in release 1.09m.
%  
% \xintTrim {i}{stuff expanding to {a}{b}...{z}} (or `tokens' abcd...z,
% but each naked token ends up braced in the output) returns (in two
% expansion steps) the sequence with the first i elements omitted. The
% list is first f-expanded. The i is expanded inside \numexpr. Variant
% \xintTrimNoExpand does not expand the list argument.
%
% With i = 0, the original (expanded) list is returned. 
%
% With i<0, the last |i| elements from the tail are suppressed.
%
% With |i| equal to or bigger than the length of the (f-expanded) list,
% the empty list is returned.+
%    \begin{macrocode}
\def\xintTrim         {\romannumeral0\xinttrim }%
\def\xintTrimNoExpand {\romannumeral0\xinttrimnoexpand }%
\def\xinttrim #1#2%
{%
    \expandafter\XINT_trim_a\the\numexpr #1\expandafter.%
    \expandafter{\romannumeral-`0#2}%
}%
\def\xinttrimnoexpand #1%
{%
    \expandafter\XINT_trim_a\the\numexpr #1.%
}%
\def\XINT_trim_a #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\xint_firstofone_thenstop }%
        0#1{\XINT_trim_bneg_a {#2}}%
         0-{\XINT_trim_bpos {#1#2}}%
    \krof 
}%
\long\def\XINT_trim_bneg_a #1#2%
{%
    \expandafter\XINT_trim_bneg_b \the\numexpr \xintLength{#2}-#1.{#2}%
}%
\def\XINT_trim_bneg_b #1#2.%
{%
    \xint_UDzerominusfork
        #1-{\expandafter\space\xint_gobble_i }%
        0#1{\expandafter\space\xint_gobble_i }%
         0-{\XINT_keep_bpos {#1#2}}%
    \krof 
}%
\long\def\XINT_trim_bpos #1#2%
{%
     \XINT_trim_loop_a {#1}#2\xint_relax\xint_relax\xint_relax\xint_relax
                    \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye
}%
\def\XINT_trim_loop_a #1%
{%
    \ifnum #1>\xint_c_vii
        \expandafter\XINT_trim_loop_b
    \else
        \XINT_trim_finish
    \fi
    {#1}%
}%
\long\def\XINT_trim_loop_b #1#2#3#4#5#6#7#8#9%
{%
    \xint_gob_til_xint_relax #9\XINT_trim_silentend\xint_relax
    \expandafter\XINT_trim_loop_a\expandafter{\the\numexpr #1-\xint_c_viii}%
}%
\def\XINT_trim_silentend #1\xint_bye { }%
\def\XINT_trim_finish\fi #1%
{%
    \fi\expandafter\expandafter\expandafter\XINT_trim_finish_a
    \expandafter\expandafter\expandafter\space % avoids brace removal
    \csname xint_gobble_\romannumeral\numexpr#1\endcsname
}%
\long\def\XINT_trim_finish_a #1\xint_relax #2\xint_bye {#1}%
%    \end{macrocode}
% \subsection{\csh{xintApply}}
% \lverb|&
% \xintApply {\macro}{{a}{b}...{z}} returns {\macro{a}}...{\macro{b}}
% where each instance of \macro is 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+\xint_c_i\endcsname}%
  \let\xintifForLast\xint_secondoftwo
  \ifx #5j\expandafter\xint_firstoftwo
     \else\expandafter\xint_secondoftwo
  \fi
  {\let\xintifForLast\xint_firstoftwo
   \xintBreakForAndDo {\XINT_x \xint_gobble_i Z}}%
  \XINT_x
  \let\xintifForFirst\xint_secondoftwo\XINT_forpair_d #1{#2}%
}%
\long\def\xintForthree #1#2#3in#4#5#6%
{%
    \let\xintifForFirst\xint_firstoftwo
    \XINT_toks  {\XINT_forthree_d #2{#6}}%
    \expandafter\the\expandafter\XINT_toks #4jZ%
}%
\long\def\XINT_forthree_d #1#2#3(#4)#5%
{%
  \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}%
  \XINT_toks \expandafter{\romannumeral0\xintcsvtolist{ #4}}%
  \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname
      \the\XINT_toks \csname XINT_for_right\the\numexpr#1+\xint_c_ii\endcsname}%
  \let\xintifForLast\xint_secondoftwo
  \ifx #5j\expandafter\xint_firstoftwo
     \else\expandafter\xint_secondoftwo
  \fi
  {\let\xintifForLast\xint_firstoftwo
   \xintBreakForAndDo {\XINT_x \xint_gobble_i Z}}%
  \XINT_x
  \let\xintifForFirst\xint_secondoftwo\XINT_forthree_d #1{#2}%
}%
\long\def\xintForfour #1#2#3in#4#5#6%
{%
    \let\xintifForFirst\xint_firstoftwo
    \XINT_toks  {\XINT_forfour_d #2{#6}}%
    \expandafter\the\expandafter\XINT_toks #4jZ%
}%
\long\def\XINT_forfour_d #1#2#3(#4)#5%
{%
  \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}%
  \XINT_toks \expandafter{\romannumeral0\xintcsvtolist{ #4}}%
  \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname
     \the\XINT_toks \csname XINT_for_right\the\numexpr#1+\xint_c_iii\endcsname}%
  \let\xintifForLast\xint_secondoftwo
  \ifx #5j\expandafter\xint_firstoftwo
     \else\expandafter\xint_secondoftwo
  \fi
  {\let\xintifForLast\xint_firstoftwo
   \xintBreakForAndDo {\XINT_x \xint_gobble_i Z}}%
  \XINT_x
  \let\xintifForFirst\xint_secondoftwo\XINT_forfour_d #1{#2}%
}%
\catcode`Z 11
\catcode`j 11
%    \end{macrocode}
% \subsection{\csh{xintAssign}, \csh{xintAssignArray}, \csh{xintDigitsOf}}
% \lverb|&
% \xintAssign {a}{b}..{z}\to\A\B...\Z,$\
% \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}%
  [2014/04/01 v1.09n 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{}% moved to xinttools
\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}% moved to xinttools
\def\xint_UDsignsfork     #1--#2#3\krof {#2}%
\chardef\xint_c_ix     9
\chardef\xint_c_x     10
\chardef\xint_c_ii^v  32 % not used in xint, common to xintfrac and xintbinhex
\chardef\xint_c_ii^vi 64
\mathchardef\xint_c_ixixixix 9999
\mathchardef\xint_c_x^iv    10000
\newcount\xint_c_x^viii  \xint_c_x^viii 100000000
%    \end{macrocode}
% \subsection{\csh{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+\xint_c_\relax
}%
\def\XINT_num_NumEight #1#2#3#4#5#6#7#8%
{%
    \ifnum \numexpr #1#2#3#4#5#6#7#8+\xint_c_= \xint_c_
      \xint_afterfi {\expandafter\XINT_num_keepsign_a
                     \the\numexpr #1#2#3#4#5#6#7#81\relax}%
    \else
      \xint_afterfi {\expandafter\XINT_num_finish
                     \the\numexpr #1#2#3#4#5#6#7#8\relax}%
    \fi
}%
\def\XINT_num_keepsign_a #1%
{%
    \xint_gob_til_one#1\XINT_num_gobacktoloop 1\XINT_num_keepsign_b  
}%
\def\XINT_num_gobacktoloop 1\XINT_num_keepsign_b {\XINT_num_loop }%
\def\XINT_num_keepsign_b #1{\XINT_num_loop -}%
\def\XINT_num_finish #1\xint_relax #2\Z { #1}%
%    \end{macrocode}
% \subsection{\csh{xintSgn}, \csh{xintiiSgn}, \csh{XINT\_Sgn}, \csh{XINT\_cntSgn}}
% \lverb|&
% Changed in 1.05. Earlier code was unnecessarily strange. 1.09a with \xintnum
%
% 1.09i defines \XINT_Sgn and \XINT_cntSgn (was \XINT__Sgn in 1.09i) for reasons
% of internal optimizations|
%    \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
% consequences propagating to \xintTrunc, \xintRound, \xintFloat,
% \xintFloatSqrt, etc... and all other places using the division. See
% also \xintXTrunc in this context.
%
% & 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}%
  [2014/04/01 v1.09n 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}%
  [2014/04/01 v1.09n 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}%
  [2014/04/01 v1.09n Expandable operations on fractions (jfB)]%
\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. 
%
% 1.09m ceases here and elsewhere, also in \xintcfracname, to use \Z as
% delimiter in the code for the optional argument, as this is unsafe (it
% makes impossible to the user to employ \Z as argument to the macro).
% Replaced by \xint_relax. 1.09e had already done that in \xintSeq, but
% this should have been systematic. |
%    \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\xint_relax }%
\def\XINT_floate_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_floate_opt
       \else\expandafter\XINT_floate_noopt
    \fi  #1%
}%
\def\XINT_floate_noopt #1\xint_relax
{%
    \expandafter\XINT_floate_a\expandafter\XINTdigits
                \romannumeral0\XINT_infrac {#1}%
}%
\def\XINT_floate_opt [\xint_relax #1]#2%
{%
    \expandafter\XINT_floate_a\expandafter
    {\the\numexpr #1\expandafter}\romannumeral0\XINT_infrac {#2}%
}%
\def\XINT_floate_a #1#2#3#4#5%
{%
    \expandafter\expandafter\expandafter\XINT_float_a
    \expandafter\xint_exchangetwo_keepbraces\expandafter
            {\the\numexpr #2+#5}{#1}{#3}{#4}\XINT_float_Q
}%
\def\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 things
% and I even do an \xintnum to ensure a 0 if something comes out empty from
% \xintDecSplit).|
%    \begin{macrocode}
\def\xintXTrunc #1#2%
{%
    \expandafter\XINT_xtrunc_a\expandafter 
    {\the\numexpr #1\expandafter}\romannumeral0\xintraw {#2}%
}%
\def\XINT_xtrunc_a #1%
{%
    \expandafter\XINT_xtrunc_b\expandafter
    {\the\numexpr\ifnum#1<\xint_c_i \xint_c_i-\fi #1}%
}%
\def\XINT_xtrunc_b #1%
{%
    \expandafter\XINT_xtrunc_c\expandafter
    {\the\numexpr (#1+\xint_c_ii^v)/\xint_c_ii^vi-\xint_c_i}{#1}%
}%
\def\XINT_xtrunc_c #1#2%
{%
    \expandafter\XINT_xtrunc_d\expandafter
    {\the\numexpr #2-\xint_c_ii^vi*#1}{#1}{#2}%
}%
\def\XINT_xtrunc_d #1#2#3#4/#5[#6]%
{%
    \XINT_xtrunc_e #4.{#6}{#5}{#3}{#2}{#1}%
}%
% #1=numerator.#2=N,#3=B,#4=D,#5=Blocs,#6=extra
\def\XINT_xtrunc_e #1%
{%
    \xint_UDzerominusfork
        #1-\XINT_xtrunc_zero
        0#1\XINT_xtrunc_N
        0-{\XINT_xtrunc_P #1}%
    \krof
}%
\def\XINT_xtrunc_zero .#1#2#3#4#5%
{%
    0.\romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                                  {\the\numexpr #5}{}\Z {}%
    \xintiloop [#4+-1]
    \ifnum \xintiloopindex>\xint_c_
    0000000000000000000000000000000000000000000000000000000000000000%
    \repeat
}%
\def\XINT_xtrunc_N {-\XINT_xtrunc_P }%
\def\XINT_xtrunc_P #1.#2%
{%
    \ifnum #2<\xint_c_ 
        \expandafter\XINT_xtrunc_negN_Q
    \else
        \expandafter\XINT_xtrunc_Q
    \fi  {#2}{#1}.%
}%
\def\XINT_xtrunc_negN_Q #1#2.#3#4#5#6%
{%
    \expandafter\XINT_xtrunc_negN_R
    \romannumeral0\XINT_div_prepare {#3}{#2}{#3}{#1}{#4}%
}%
% #1=Q, #2=R, #3=B, #4=N<0, #5=D 
\def\XINT_xtrunc_negN_R #1#2#3#4#5%
{%
    \expandafter\XINT_xtrunc_negN_S\expandafter
    {\the\numexpr -#4}{#5}{#2}{#3}{#1}%
}%
\def\XINT_xtrunc_negN_S #1#2%
{%
    \expandafter\XINT_xtrunc_negN_T\expandafter
    {\the\numexpr #2-#1}{#1}{#2}%
}%
\def\XINT_xtrunc_negN_T #1%
{%
    \ifnum \xint_c_<#1
      \expandafter\XINT_xtrunc_negNA
    \else
      \expandafter\XINT_xtrunc_negNW
    \fi {#1}%
}%
% #1=D-|N|>0, #2=|N|, #3=D, #4=R, #5=B, #6=Q
\def\XINT_xtrunc_unlock #10.{ }%
\def\XINT_xtrunc_negNA #1#2#3#4#5#6%
{%
   \expandafter\XINT_xtrunc_negNB\expandafter
   {\romannumeral0\expandafter\expandafter\expandafter
    \XINT_xtrunc_unlock\expandafter\string
    \csname\XINT_xtrunc_b {#1}#4/#5[0]\expandafter\endcsname
    \expandafter}\expandafter 
    {\the\numexpr\xintLength{#6}-#2}{#6}%
}%
\def\XINT_xtrunc_negNB #1#2#3{\XINT_xtrunc_negNC {#2}{#3}#1}%
\def\XINT_xtrunc_negNC #1%
{%
    \ifnum \xint_c_ < #1
      \expandafter\XINT_xtrunc_negNDa
    \else
      \expandafter\XINT_xtrunc_negNE
    \fi {#1}%
}%
\def\XINT_xtrunc_negNDa #1#2%
{%
    \expandafter\XINT_xtrunc_negNDb%
    \romannumeral0\XINT_split_fromleft_loop {#1}{}#2\W\W\W\W\W\W\W\W\Z
}%
\def\XINT_xtrunc_negNDb #1#2{#1.#2}%
\def\XINT_xtrunc_negNE #1#2%
{%
    0.\romannumeral0\XINT_dsx_zeroloop {-#1}{}\Z {}#2%
}%
% #1=D-|N|<=0, #2=|N|, #3=D, #4=R, #5=B, #6=Q
\def\XINT_xtrunc_negNW #1#2#3#4#5#6%
{%
    \expandafter\XINT_xtrunc_negNX\expandafter
    {\romannumeral0\xintnum{\xintDecSplitL {-#1}{#6}}}{#3}%
}%
\def\XINT_xtrunc_negNX #1#2%
{%  
    \expandafter\XINT_xtrunc_negNC\expandafter
    {\the\numexpr\xintLength {#1}-#2}{#1}%
}%
\def\XINT_xtrunc_Q #1%
{%
    \expandafter\XINT_xtrunc_prepare_I
    \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z 
}%
\def\XINT_xtrunc_prepare_I #1.#2#3%
{%
    \expandafter\XINT_xtrunc_prepareB_aa\expandafter
    {\romannumeral0\xintlength {#2}}{#2}{#1}%
}%
\def\XINT_xtrunc_prepareB_aa #1%
{%
    \ifnum #1=\xint_c_i
      \expandafter\XINT_xtrunc_prepareB_onedigit
    \else
      \expandafter\XINT_xtrunc_prepareB_PaBa
    \fi
    {#1}%
}%
\def\XINT_xtrunc_prepareB_onedigit #1#2%
{%
    \ifcase#2
    \or\expandafter\XINT_xtrunc_BisOne
    \or\expandafter\XINT_xtrunc_BisTwo
    \else\expandafter\XINT_xtrunc_prepareB_PaBe
    \fi {000}{0}{4}{#2}%
}%
\def\XINT_xtrunc_BisOne #1#2#3#4#5#6#7%
{%
    #5.\romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                                  {\the\numexpr #7}{}\Z {}%
    \xintiloop [#6+-1]
    \ifnum \xintiloopindex>\xint_c_
    0000000000000000000000000000000000000000000000000000000000000000%
    \repeat
}%
\def\XINT_xtrunc_BisTwo #1#2#3#4#5#6#7%
{%
    \xintHalf {#5}.\ifodd\xintiiLDg{#5} 5\else 0\fi 
    \romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                                  {\the\numexpr #7-\xint_c_i}{}\Z {}%
    \xintiloop [#6+-1]
    \ifnum \xintiloopindex>\xint_c_
    0000000000000000000000000000000000000000000000000000000000000000%
    \repeat
}%   
\def\XINT_xtrunc_prepareB_PaBa #1#2%
{%
    \expandafter\XINT_xtrunc_Pa\expandafter
    {\romannumeral0\XINT_xtrunc_prepareB_a {#1}{#2}}%
}%
\def\XINT_xtrunc_prepareB_a #1%
{%
  \expandafter\XINT_xtrunc_prepareB_c\expandafter
  {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}%
}%
\def\XINT_xtrunc_prepareB_c #1#2%
{%
    \csname XINT_xtrunc_prepareB_d\romannumeral\numexpr#1-#2\endcsname
    {#1}%
}%
\def\XINT_xtrunc_prepareB_d    {\XINT_xtrunc_prepareB_e {}{0000}}%
\def\XINT_xtrunc_prepareB_di   {\XINT_xtrunc_prepareB_e {0}{000}}%
\def\XINT_xtrunc_prepareB_dii  {\XINT_xtrunc_prepareB_e {00}{00}}%
\def\XINT_xtrunc_prepareB_diii {\XINT_xtrunc_prepareB_e {000}{0}}%
\def\XINT_xtrunc_prepareB_PaBe #1#2#3#4%
{%
    \expandafter\XINT_xtrunc_Pa\expandafter
    {\romannumeral0\XINT_xtrunc_prepareB_e {#1}{#2}{#3}{#4}}%
}%
\def\XINT_xtrunc_prepareB_e #1#2#3#4%
{%
    \ifnum#3=\xint_c_iv\expandafter\XINT_xtrunc_prepareLittleB_f
                   \else\expandafter\XINT_xtrunc_prepareB_f 
    \fi
    #4#1{#3}{#2}{#1}%
}%
\def\XINT_xtrunc_prepareB_f #1#2#3#4#5#{%
    \expandafter\space
    \expandafter\XINT_div_prepareB_g
     \the\numexpr  #1#2#3#4+\xint_c_i\expandafter
    .\the\numexpr (#1#2#3#4+\xint_c_i)/\xint_c_ii\expandafter
    .\romannumeral0\xintreverseorder {#1#2#3#4#5}.{#1#2#3#4}%
}%
\def\XINT_xtrunc_prepareLittleB_f #1#{%
    \expandafter\space\expandafter
    \XINT_div_prepareB_g \the\numexpr  #1/\xint_c_ii.{}.{}.{#1}%
}%
\def\XINT_xtrunc_Pa #1#2%
{%
    \expandafter\XINT_xtrunc_Pb\romannumeral0#1{#2}{#1}%
}%
\def\XINT_xtrunc_Pb #1#2#3#4{#1.\XINT_xtrunc_A {#4}{#2}{#3}}%
\def\XINT_xtrunc_A #1%
{%
    \unless\ifnum #1>\xint_c_ \XINT_xtrunc_transition\fi 
    \expandafter\XINT_xtrunc_B\expandafter{\the\numexpr #1-\xint_c_i}%
}%
\def\XINT_xtrunc_B #1#2#3%
{%
    \expandafter\XINT_xtrunc_D\romannumeral0#3%
    {#20000000000000000000000000000000000000000000000000000000000000000}%
    {#1}{#3}%
}%
\def\XINT_xtrunc_D #1#2#3%
{%
    \romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                  {\the\numexpr \xint_c_ii^vi-\xintLength{#1}}{}\Z {}#1%
    \XINT_xtrunc_A {#3}{#2}%
}%
\def\XINT_xtrunc_transition\fi
    \expandafter\XINT_xtrunc_B\expandafter #1#2#3#4%
{%
    \fi
    \ifnum #4=\xint_c_ \XINT_xtrunc_abort\fi
    \expandafter\XINT_xtrunc_x\expandafter
    {\romannumeral0\XINT_dsx_zeroloop {#4}{}\Z {#2}}{#3}{#4}%
}%
\def\XINT_xtrunc_x #1#2%
{%
    \expandafter\XINT_xtrunc_y\romannumeral0#2{#1}%
}%
\def\XINT_xtrunc_y #1#2#3%
{%
    \romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter
                  {\the\numexpr #3-\xintLength{#1}}{}\Z {}#1%
}%
\def\XINT_xtrunc_abort\fi\expandafter\XINT_xtrunc_x\expandafter #1#2#3{\fi}%
%    \end{macrocode}
% \subsection{\csh{xintDigits}}
% \lverb|&
% The mathchardef used to be called \XINT_digits, but for reasons originating in
% \xintNewExpr, release 1.09a uses \XINTdigits without underscore.|
%    \begin{macrocode}
\mathchardef\XINTdigits 16
\def\xintDigits #1#2%
   {\afterassignment \xint_gobble_i \mathchardef\XINTdigits=}%
\def\xinttheDigits {\number\XINTdigits }%
%    \end{macrocode}
% \subsection{\csh{xintFloat}}
% \lverb|1.07. Completely re-written in 1.08a, with spectacular speed
% gains. The earlier version was seriously silly when dealing with
% inputs having a big power of ten. Again some modifications in 1.08b
% for a better treatment of cases with long explicit numerators or
% denominators.
% 
% Here again some inner macros used the \xintiquo with extra \xintnum overhead
% in 1.09a, 1.09f reinstalled use of \xintiiquo without this overhead.|
%    \begin{macrocode}
\def\xintFloat   {\romannumeral0\xintfloat }%
\def\xintfloat #1{\XINT_float_chkopt #1\xint_relax }%
\def\XINT_float_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_float_opt
       \else\expandafter\XINT_float_noopt
    \fi  #1%
}%
\def\XINT_float_noopt #1\xint_relax
{%
    \expandafter\XINT_float_a\expandafter\XINTdigits
    \romannumeral0\XINT_infrac {#1}\XINT_float_Q
}%
\def\XINT_float_opt [\xint_relax #1]#2%
{%
    \expandafter\XINT_float_a\expandafter 
    {\the\numexpr #1\expandafter}%
    \romannumeral0\XINT_infrac {#2}\XINT_float_Q
}%
\def\XINT_float_a #1#2#3% #1=P, #2=n, #3=A, #4=B
{%
    \XINT_float_fork #3\Z {#1}{#2}% #1 = precision, #2=n
}%
\def\XINT_float_fork #1%
{%
    \xint_UDzerominusfork
     #1-\XINT_float_zero
     0#1\XINT_float_J
      0-{\XINT_float_K #1}%
    \krof
}%
\def\XINT_float_zero #1\Z #2#3#4#5{ 0.e0}%
\def\XINT_float_J {\expandafter\xint_minus_thenstop\romannumeral0\XINT_float_K }%
\def\XINT_float_K #1\Z #2% #1=A, #2=P, #3=n, #4=B
{%
    \expandafter\XINT_float_L\expandafter
    {\the\numexpr\xintLength{#1}\expandafter}\expandafter
    {\the\numexpr #2+\xint_c_ii}{#1}{#2}%
}%
\def\XINT_float_L #1#2%
{%
    \ifnum #1>#2
      \expandafter\XINT_float_Ma
    \else
      \expandafter\XINT_float_Mc
    \fi {#1}{#2}%
}%
\def\XINT_float_Ma #1#2#3%
{%
    \expandafter\XINT_float_Mb\expandafter
    {\the\numexpr #1-#2\expandafter\expandafter\expandafter}%
    \expandafter\expandafter\expandafter
    {\expandafter\xint_firstoftwo
     \romannumeral0\XINT_split_fromleft_loop {#2}{}#3\W\W\W\W\W\W\W\W\Z
     }{#2}%
}%
\def\XINT_float_Mb #1#2#3#4#5#6% #2=A', #3=P+2, #4=P, #5=n, #6=B
{%
   \expandafter\XINT_float_N\expandafter
   {\the\numexpr\xintLength{#6}\expandafter}\expandafter
   {\the\numexpr #3\expandafter}\expandafter
   {\the\numexpr #1+#5}%
   {#6}{#3}{#2}{#4}%
}% long de B, P+2, n', B, |A'|=P+2, A', P
\def\XINT_float_Mc #1#2#3#4#5#6%
{%
   \expandafter\XINT_float_N\expandafter
   {\romannumeral0\xintlength{#6}}{#2}{#5}{#6}{#1}{#3}{#4}%
}% long de B, P+2, n, B, |A|, A, P
\def\XINT_float_N #1#2%
{%
    \ifnum #1>#2
      \expandafter\XINT_float_O
    \else
      \expandafter\XINT_float_P
    \fi {#1}{#2}%
}%
\def\XINT_float_O #1#2#3#4%
{%
    \expandafter\XINT_float_P\expandafter
    {\the\numexpr #2\expandafter}\expandafter
    {\the\numexpr #2\expandafter}\expandafter
    {\the\numexpr #3-#1+#2\expandafter\expandafter\expandafter}%
    \expandafter\expandafter\expandafter
    {\expandafter\xint_firstoftwo
     \romannumeral0\XINT_split_fromleft_loop {#2}{}#4\W\W\W\W\W\W\W\W\Z
     }%
}% |B|,P+2,n,B,|A|,A,P
\def\XINT_float_P #1#2#3#4#5#6#7#8%
{%
    \expandafter #8\expandafter {\the\numexpr #1-#5+#2-\xint_c_i}%
    {#6}{#4}{#7}{#3}%
}% |B|-|A|+P+1,A,B,P,n
\def\XINT_float_Q #1%
{%
    \ifnum #1<\xint_c_
      \expandafter\XINT_float_Ri
    \else
      \expandafter\XINT_float_Rii
    \fi {#1}%
}%
\def\XINT_float_Ri #1#2#3%
{%
    \expandafter\XINT_float_Sa
    \romannumeral0\xintiiquo {#2}%
         {\XINT_dsx_addzerosnofuss {-#1}{#3}}\Z {#1}%
}%
\def\XINT_float_Rii #1#2#3%
{%
    \expandafter\XINT_float_Sa
    \romannumeral0\xintiiquo
         {\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}\Z {#1}%
}%
\def\XINT_float_Sa #1%
{%
    \if #19%
        \xint_afterfi {\XINT_float_Sb\XINT_float_Wb }%
    \else
        \xint_afterfi {\XINT_float_Sb\XINT_float_Wa }%
    \fi #1%
}%
\def\XINT_float_Sb #1#2\Z #3#4%
{%
    \expandafter\XINT_float_T\expandafter
    {\the\numexpr #4+\xint_c_i\expandafter}%
    \romannumeral-`0\XINT_lenrord_loop 0{}#2\Z\W\W\W\W\W\W\W\Z #1{#3}{#4}%
}%
\def\XINT_float_T #1#2#3%
{%
    \ifnum #2>#1
      \xint_afterfi{\XINT_float_U\XINT_float_Xb}%
    \else
      \xint_afterfi{\XINT_float_U\XINT_float_Xa #3}%
    \fi
}%
\def\XINT_float_U #1#2%
{%
    \ifnum #2<\xint_c_v
      \expandafter\XINT_float_Va
    \else
      \expandafter\XINT_float_Vb
    \fi #1%
}%
\def\XINT_float_Va #1#2\Z #3%
{%
    \expandafter#1%
    \romannumeral0\expandafter\XINT_float_Wa
    \romannumeral0\XINT_rord_main {}#2%
      \xint_relax
        \xint_bye\xint_bye\xint_bye\xint_bye
        \xint_bye\xint_bye\xint_bye\xint_bye
      \xint_relax \Z
}%
\def\XINT_float_Vb #1#2\Z #3%
{%
    \expandafter #1%
    \romannumeral0\expandafter #3%
    \romannumeral0\XINT_addm_A 0{}1000\W\X\Y\Z #2000\W\X\Y\Z \Z
}%
\def\XINT_float_Wa #1{ #1.}%
\def\XINT_float_Wb #1#2%
    {\if #11\xint_afterfi{ 10.}\else\xint_afterfi{ #1.#2}\fi }%
\def\XINT_float_Xa #1\Z #2#3#4%
{%
    \expandafter\XINT_float_Y\expandafter
    {\the\numexpr #3+#4-#2}{#1}%
}%
\def\XINT_float_Xb #1\Z #2#3#4%
{%
    \expandafter\XINT_float_Y\expandafter
    {\the\numexpr #3+#4+\xint_c_i-#2}{#1}%
}%
\def\XINT_float_Y #1#2{ #2e#1}%
%    \end{macrocode}
% \subsection{\csh{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\xint_relax }%
\def\XINTinFloatAdd    {\romannumeral0\XINTinfloatadd }%
\def\XINTinfloatadd  #1{\XINT_fladd_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_fladd_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_fladd_opt
       \else\expandafter\XINT_fladd_noopt
    \fi  #1#2%
}%
\def\XINT_fladd_noopt #1#2\xint_relax #3%
{%
    #1[\XINTdigits]{\XINT_FL_Add {\XINTdigits+\xint_c_ii}{#2}{#3}}%
}%
\def\XINT_fladd_opt #1[\xint_relax #2]#3#4%
{%
    #1[#2]{\XINT_FL_Add {#2+\xint_c_ii}{#3}{#4}}%
}%
\def\XINT_FL_Add #1#2%
{%
    \expandafter\XINT_FL_Add_a\expandafter{\the\numexpr #1\expandafter}%
    \expandafter{\romannumeral0\XINTinfloat [#1]{#2}}%
}%
\def\XINT_FL_Add_a #1#2#3%
{%
    \expandafter\XINT_FL_Add_b\romannumeral0\XINTinfloat [#1]{#3}#2{#1}%
}%
\def\XINT_FL_Add_b #1%
{%
    \xint_gob_til_zero #1\XINT_FL_Add_zero 0\XINT_FL_Add_c #1%
}%
\def\XINT_FL_Add_c #1[#2]#3%
{%
    \xint_gob_til_zero #3\XINT_FL_Add_zerobis 0\XINT_FL_Add_d #1[#2]#3%
}%
\def\XINT_FL_Add_d #1[#2]#3[#4]#5%
{%
    \ifnum \numexpr #2-#4-#5>\xint_c_i 
       \expandafter \xint_secondofthree_thenstop
    \else
       \ifnum \numexpr #4-#2-#5>\xint_c_i 
              \expandafter\expandafter\expandafter\xint_thirdofthree_thenstop
       \fi
    \fi
    \xintadd {#1[#2]}{#3[#4]}%
}%
\def\XINT_FL_Add_zero 0\XINT_FL_Add_c 0[0]#1[#2]#3{#1[#2]}%
\def\XINT_FL_Add_zerobis 0\XINT_FL_Add_d #1[#2]0[0]#3{#1[#2]}%
%    \end{macrocode}
% \subsection{\csh{xintFloatSub}, \csh{XINTinFloatSub}}
% \lverb|1.07|
%    \begin{macrocode}
\def\xintFloatSub {\romannumeral0\xintfloatsub }%
\def\xintfloatsub    #1{\XINT_flsub_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatSub {\romannumeral0\XINTinfloatsub }%
\def\XINTinfloatsub #1{\XINT_flsub_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flsub_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flsub_opt
       \else\expandafter\XINT_flsub_noopt
    \fi  #1#2%
}%
\def\XINT_flsub_noopt #1#2\xint_relax #3%
{%
    #1[\XINTdigits]{\XINT_FL_Add {\XINTdigits+\xint_c_ii}{#2}{\xintOpp{#3}}}%
}%
\def\XINT_flsub_opt #1[\xint_relax #2]#3#4%
{%
    #1[#2]{\XINT_FL_Add {#2+\xint_c_ii}{#3}{\xintOpp{#4}}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFloatMul}, \csh{XINTinFloatMul}}
% \lverb|1.07|
%    \begin{macrocode}
\def\xintFloatMul    {\romannumeral0\xintfloatmul}%
\def\xintfloatmul    #1{\XINT_flmul_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatMul {\romannumeral0\XINTinfloatmul }%
\def\XINTinfloatmul #1{\XINT_flmul_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flmul_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flmul_opt
       \else\expandafter\XINT_flmul_noopt
    \fi  #1#2%
}%
\def\XINT_flmul_noopt #1#2\xint_relax #3%
{%
    #1[\XINTdigits]{\XINT_FL_Mul {\XINTdigits+\xint_c_ii}{#2}{#3}}%
}%
\def\XINT_flmul_opt #1[\xint_relax #2]#3#4%
{%
    #1[#2]{\XINT_FL_Mul {#2+\xint_c_ii}{#3}{#4}}%
}%
\def\XINT_FL_Mul #1#2%
{%
    \expandafter\XINT_FL_Mul_a\expandafter{\the\numexpr #1\expandafter}%
    \expandafter{\romannumeral0\XINTinfloat [#1]{#2}}%
}%
\def\XINT_FL_Mul_a #1#2#3%
{%
    \expandafter\XINT_FL_Mul_b\romannumeral0\XINTinfloat [#1]{#3}#2%
}%
\def\XINT_FL_Mul_b #1[#2]#3[#4]{\xintE{\xintiiMul {#1}{#3}}{#2+#4}}%
%    \end{macrocode}
% \subsection{\csh{xintFloatDiv}, \csh{XINTinFloatDiv}}
% \lverb|1.07|
%    \begin{macrocode}
\def\xintFloatDiv    {\romannumeral0\xintfloatdiv}%
\def\xintfloatdiv    #1{\XINT_fldiv_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatDiv  {\romannumeral0\XINTinfloatdiv }%
\def\XINTinfloatdiv  #1{\XINT_fldiv_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_fldiv_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_fldiv_opt
       \else\expandafter\XINT_fldiv_noopt
    \fi  #1#2%
}%
\def\XINT_fldiv_noopt #1#2\xint_relax #3%
{%
    #1[\XINTdigits]{\XINT_FL_Div {\XINTdigits+\xint_c_ii}{#2}{#3}}%
}%
\def\XINT_fldiv_opt #1[\xint_relax #2]#3#4%
{%
    #1[#2]{\XINT_FL_Div {#2+\xint_c_ii}{#3}{#4}}%
}%
\def\XINT_FL_Div #1#2%
{%
    \expandafter\XINT_FL_Div_a\expandafter{\the\numexpr #1\expandafter}%
    \expandafter{\romannumeral0\XINTinfloat [#1]{#2}}%
}%
\def\XINT_FL_Div_a #1#2#3%
{%
    \expandafter\XINT_FL_Div_b\romannumeral0\XINTinfloat [#1]{#3}#2%
}%
\def\XINT_FL_Div_b #1[#2]#3[#4]{\xintE{#3/#1}{#4-#2}}%
%    \end{macrocode}
% \subsection{\csh{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\xint_relax }%
\def\XINTinFloatPow {\romannumeral0\XINTinfloatpow }%
\def\XINTinfloatpow #1{\XINT_flpow_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flpow_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flpow_opt
       \else\expandafter\XINT_flpow_noopt
    \fi
     #1#2%
}%
\def\XINT_flpow_noopt  #1#2\xint_relax #3%
{%
   \expandafter\XINT_flpow_checkB_start\expandafter
                {\the\numexpr #3\expandafter}\expandafter
                {\the\numexpr \XINTdigits}{#2}{#1[\XINTdigits]}%
}%
\def\XINT_flpow_opt #1[\xint_relax #2]#3#4%
{%
   \expandafter\XINT_flpow_checkB_start\expandafter
               {\the\numexpr #4\expandafter}\expandafter
               {\the\numexpr #2}{#3}{#1[#2]}%
}%
\def\XINT_flpow_checkB_start #1{\XINT_flpow_checkB_a #1\Z }%
\def\XINT_flpow_checkB_a #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_flpow_BisZero
      0#1{\XINT_flpow_checkB_b 1}%
       0-{\XINT_flpow_checkB_b 0#1}%
    \krof
}%
\def\XINT_flpow_BisZero \Z #1#2#3{#3{1/1[0]}}%
\def\XINT_flpow_checkB_b #1#2\Z #3%
{%
    \expandafter\XINT_flpow_checkB_c \expandafter
    {\romannumeral0\xintlength{#2}}{#3}{#2}#1%
}%
\def\XINT_flpow_checkB_c #1#2%
{%
    \expandafter\XINT_flpow_checkB_d \expandafter
    {\the\numexpr \expandafter\xintLength\expandafter
                  {\the\numexpr #1*20/\xint_c_iii }+#1+#2+\xint_c_i }%
}%
\def\XINT_flpow_checkB_d #1#2#3#4%
{%
    \expandafter \XINT_flpow_a
    \romannumeral0\XINTinfloat [#1]{#4}{#1}{#2}#3%
}%
\def\XINT_flpow_a #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_flpow_zero
      0#1{\XINT_flpow_b 1}%
       0-{\XINT_flpow_b 0#1}%
    \krof
}%
\def\XINT_flpow_b #1#2[#3]#4#5%
{%
    \XINT_flpow_loopI {#5}{#2[#3]}{\romannumeral0\XINTinfloatmul [#4]}%
    {#1*\ifodd #5 1\else 0\fi}%
}%
\def\XINT_flpow_zero [#1]#2#3#4#5% 
% xint is not equipped to signal infinity, the 2^31 will provoke
% deliberately a number too big and arithmetic overflow in \XINT_float_Xb
{%
    \if #41\xint_afterfi {\xintError:DivisionByZero #5{1[2147483648]}}%
    \else \xint_afterfi {#5{0[0]}}\fi
}%
\def\XINT_flpow_loopI #1%
{%
    \ifnum #1=\xint_c_i\XINT_flpow_ItoIII\fi
    \ifodd #1
       \expandafter\XINT_flpow_loopI_odd
    \else
       \expandafter\XINT_flpow_loopI_even
    \fi
    {#1}%
}%
\def\XINT_flpow_ItoIII\fi #1\fi #2#3#4#5%
{%
    \fi\expandafter\XINT_flpow_III\the\numexpr #5\relax #3%
}%    
\def\XINT_flpow_loopI_even #1#2#3%
{%
    \expandafter\XINT_flpow_loopI\expandafter
    {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter 
    {#3{#2}{#2}}{#3}%
}%
\def\XINT_flpow_loopI_odd #1#2#3%
{%
    \expandafter\XINT_flpow_loopII\expandafter
    {\the\numexpr #1/\xint_c_ii-\xint_c_i\expandafter}\expandafter
    {#3{#2}{#2}}{#3}{#2}%
}%
\def\XINT_flpow_loopII #1%
{%
    \ifnum #1 = \xint_c_i\XINT_flpow_IItoIII\fi
    \ifodd #1
       \expandafter\XINT_flpow_loopII_odd
    \else
       \expandafter\XINT_flpow_loopII_even
    \fi
    {#1}%
}%
\def\XINT_flpow_loopII_even #1#2#3%
{%
    \expandafter\XINT_flpow_loopII\expandafter
    {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter
    {#3{#2}{#2}}{#3}%
}%
\def\XINT_flpow_loopII_odd #1#2#3#4%
{%
    \expandafter\XINT_flpow_loopII_odda\expandafter
    {#3{#2}{#4}}{#1}{#2}{#3}%
}%
\def\XINT_flpow_loopII_odda #1#2#3#4%
{%
    \expandafter\XINT_flpow_loopII\expandafter
    {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter
    {#4{#3}{#3}}{#4}{#1}%
}%
\def\XINT_flpow_IItoIII\fi #1\fi #2#3#4#5#6%
{%
    \fi\expandafter\XINT_flpow_III\the\numexpr #6\expandafter\relax
    #4{#3}{#5}%
}%
\def\XINT_flpow_III #1#2[#3]#4%
{%
    \expandafter\XINT_flpow_IIIend\expandafter
    {\the\numexpr\if #41-\fi#3\expandafter}%
    \xint_UDzerofork
        #4{{#2}}%
         0{{1/#2}}%
    \krof #1%
}%
\def\XINT_flpow_IIIend #1#2#3#4%
{%
    \xint_UDzerofork
    #3{#4{#2[#1]}}%
     0{#4{-#2[#1]}}%
    \krof
}%
%    \end{macrocode}
% \subsection{\csh{xintFloatPower}, \csh{XINTinFloatPower}}
% \lverb|1.07. The core loop has been re-organized in 1.09j for some slight
% efficiency gain. |
%    \begin{macrocode}
\def\xintFloatPower {\romannumeral0\xintfloatpower}%
\def\xintfloatpower #1{\XINT_flpower_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatPower {\romannumeral0\XINTinfloatpower}%
\def\XINTinfloatpower #1{\XINT_flpower_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flpower_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flpower_opt
       \else\expandafter\XINT_flpower_noopt
    \fi
     #1#2%
}%
\def\XINT_flpower_noopt  #1#2\xint_relax #3%
{%
   \expandafter\XINT_flpower_checkB_start\expandafter
                {\the\numexpr \XINTdigits\expandafter}\expandafter
                {\romannumeral0\xintnum{#3}}{#2}{#1[\XINTdigits]}%
}%
\def\XINT_flpower_opt #1[\xint_relax #2]#3#4%
{%
   \expandafter\XINT_flpower_checkB_start\expandafter
               {\the\numexpr #2\expandafter}\expandafter
               {\romannumeral0\xintnum{#4}}{#3}{#1[#2]}%
}%
\def\XINT_flpower_checkB_start #1#2{\XINT_flpower_checkB_a #2\Z {#1}}%
\def\XINT_flpower_checkB_a #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_flpower_BisZero
      0#1{\XINT_flpower_checkB_b 1}%
       0-{\XINT_flpower_checkB_b 0#1}%
    \krof
}%
\def\XINT_flpower_BisZero \Z #1#2#3{#3{1/1[0]}}%
\def\XINT_flpower_checkB_b #1#2\Z #3%
{%
    \expandafter\XINT_flpower_checkB_c \expandafter
    {\romannumeral0\xintlength{#2}}{#3}{#2}#1%
}%
\def\XINT_flpower_checkB_c #1#2%
{%
    \expandafter\XINT_flpower_checkB_d \expandafter
    {\the\numexpr \expandafter\xintLength\expandafter
                  {\the\numexpr #1*20/\xint_c_iii }+#1+#2+\xint_c_i }%
}%
\def\XINT_flpower_checkB_d #1#2#3#4%
{%
    \expandafter \XINT_flpower_a
    \romannumeral0\XINTinfloat [#1]{#4}{#1}{#2}#3%
}%
\def\XINT_flpower_a #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_flpow_zero
      0#1{\XINT_flpower_b 1}%
       0-{\XINT_flpower_b 0#1}%
    \krof
}%
\def\XINT_flpower_b #1#2[#3]#4#5%
{%
    \XINT_flpower_loopI {#5}{#2[#3]}{\romannumeral0\XINTinfloatmul [#4]}%
    {#1*\xintiiOdd {#5}}%
}%
\def\XINT_flpower_loopI #1%
{%
    \if1\XINT_isOne {#1}\XINT_flpower_ItoIII\fi
    \if1\xintiiOdd{#1}%
       \expandafter\expandafter\expandafter\XINT_flpower_loopI_odd
    \else
       \expandafter\expandafter\expandafter\XINT_flpower_loopI_even
    \fi
    \expandafter {\romannumeral0\xinthalf{#1}}%
}%
\def\XINT_flpower_ItoIII\fi #1\fi\expandafter #2#3#4#5%
{%
    \fi\expandafter\XINT_flpow_III \the\numexpr #5\relax #3%
}%    
\def\XINT_flpower_loopI_even #1#2#3%
{%
    \expandafter\XINT_flpower_toI\expandafter {#3{#2}{#2}}{#1}{#3}%
}%
\def\XINT_flpower_loopI_odd #1#2#3%
{%
    \expandafter\XINT_flpower_toII\expandafter {#3{#2}{#2}}{#1}{#3}{#2}%
}%
\def\XINT_flpower_toI  #1#2{\XINT_flpower_loopI  {#2}{#1}}%
\def\XINT_flpower_toII #1#2{\XINT_flpower_loopII {#2}{#1}}%
\def\XINT_flpower_loopII #1%
{%
    \if1\XINT_isOne {#1}\XINT_flpower_IItoIII\fi
    \if1\xintiiOdd{#1}%
       \expandafter\expandafter\expandafter\XINT_flpower_loopII_odd
    \else
       \expandafter\expandafter\expandafter\XINT_flpower_loopII_even
    \fi
    \expandafter {\romannumeral0\xinthalf{#1}}%
}%
\def\XINT_flpower_loopII_even #1#2#3%
{%
    \expandafter\XINT_flpower_toII\expandafter
    {#3{#2}{#2}}{#1}{#3}%
}%
\def\XINT_flpower_loopII_odd #1#2#3#4%
{%
    \expandafter\XINT_flpower_loopII_odda\expandafter
    {#3{#2}{#4}}{#2}{#3}{#1}%
}%
\def\XINT_flpower_loopII_odda #1#2#3#4%
{%
    \expandafter\XINT_flpower_toII\expandafter
    {#3{#2}{#2}}{#4}{#3}{#1}%
}%
\def\XINT_flpower_IItoIII\fi #1\fi\expandafter #2#3#4#5#6%
{%
    \fi\expandafter\XINT_flpow_III\the\numexpr #6\expandafter\relax
    #4{#3}{#5}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFloatSqrt}, \csh{XINTinFloatSqrt}}
% \lverb|1.08|
%    \begin{macrocode}
\def\xintFloatSqrt     {\romannumeral0\xintfloatsqrt }%
\def\xintfloatsqrt   #1{\XINT_flsqrt_chkopt \xintfloat #1\xint_relax }%
\def\XINTinFloatSqrt   {\romannumeral0\XINTinfloatsqrt }%
\def\XINTinfloatsqrt #1{\XINT_flsqrt_chkopt \XINTinfloat #1\xint_relax }%
\def\XINT_flsqrt_chkopt #1#2%
{%
    \ifx [#2\expandafter\XINT_flsqrt_opt
       \else\expandafter\XINT_flsqrt_noopt
    \fi  #1#2%
}%
\def\XINT_flsqrt_noopt #1#2\xint_relax
{%
    #1[\XINTdigits]{\XINT_FL_sqrt \XINTdigits {#2}}%
}%
\def\XINT_flsqrt_opt #1[\xint_relax #2]#3%
{%
    #1[#2]{\XINT_FL_sqrt {#2}{#3}}%
}%
\def\XINT_FL_sqrt #1%
{%
    \ifnum\numexpr #1<\xint_c_xviii 
        \xint_afterfi {\XINT_FL_sqrt_a\xint_c_xviii}%
    \else
        \xint_afterfi {\XINT_FL_sqrt_a {#1+\xint_c_i}}%
    \fi
}%
\def\XINT_FL_sqrt_a #1#2%
{%     
    \expandafter\XINT_FL_sqrt_checkifzeroorneg
    \romannumeral0\XINTinfloat [#1]{#2}%
}%
\def\XINT_FL_sqrt_checkifzeroorneg #1%
{%
    \xint_UDzerominusfork
     #1-\XINT_FL_sqrt_iszero
     0#1\XINT_FL_sqrt_isneg
      0-{\XINT_FL_sqrt_b #1}%
    \krof
}%
\def\XINT_FL_sqrt_iszero #1[#2]{0[0]}%
\def\XINT_FL_sqrt_isneg  #1[#2]{\xintError:RootOfNegative 0[0]}%
\def\XINT_FL_sqrt_b #1[#2]%
{%
    \ifodd #2
        \xint_afterfi{\XINT_FL_sqrt_c 01}%
    \else
        \xint_afterfi{\XINT_FL_sqrt_c {}0}%
    \fi
    {#1}{#2}%
}%
\def\XINT_FL_sqrt_c #1#2#3#4%
{%
    \expandafter\XINT_flsqrt\expandafter {\the\numexpr #4-#2}{#3#1}%
}%
\def\XINT_flsqrt #1#2%
{%
    \expandafter\XINT_sqrt_a
    \expandafter{\romannumeral0\xintlength {#2}}\XINT_flsqrt_big_d {#2}{#1}%
}%
\def\XINT_flsqrt_big_d #1#2% 
{%
   \ifodd #2
     \expandafter\expandafter\expandafter\XINT_flsqrt_big_eB
   \else
     \expandafter\expandafter\expandafter\XINT_flsqrt_big_eA
   \fi
   \expandafter {\the\numexpr (#2-\xint_c_i)/\xint_c_ii }{#1}%
}%
\def\XINT_flsqrt_big_eA  #1#2#3%
{%
    \XINT_flsqrt_big_eA_a #3\Z {#2}{#1}{#3}%
}%
\def\XINT_flsqrt_big_eA_a #1#2#3#4#5#6#7#8#9\Z 
{%
    \XINT_flsqrt_big_eA_b {#1#2#3#4#5#6#7#8}%
}%
\def\XINT_flsqrt_big_eA_b #1#2%
{%
    \expandafter\XINT_flsqrt_big_f
    \romannumeral0\XINT_flsqrt_small_e {#2001}{#1}%
}%
\def\XINT_flsqrt_big_eB #1#2#3%
{%
    \XINT_flsqrt_big_eB_a #3\Z {#2}{#1}{#3}%
}%
\def\XINT_flsqrt_big_eB_a #1#2#3#4#5#6#7#8#9%
{%
    \XINT_flsqrt_big_eB_b {#1#2#3#4#5#6#7#8#9}%
}%
\def\XINT_flsqrt_big_eB_b #1#2\Z #3%
{%
    \expandafter\XINT_flsqrt_big_f
    \romannumeral0\XINT_flsqrt_small_e {#30001}{#1}%
}%
\def\XINT_flsqrt_small_e #1#2%
{%
   \expandafter\XINT_flsqrt_small_f\expandafter
   {\the\numexpr #1*#1-#2-\xint_c_i}{#1}%
}%
\def\XINT_flsqrt_small_f #1#2%
{%
   \expandafter\XINT_flsqrt_small_g\expandafter
   {\the\numexpr (#1+#2)/(2*#2)-\xint_c_i }{#1}{#2}%
}%
\def\XINT_flsqrt_small_g #1%
{%
    \ifnum #1>\xint_c_
       \expandafter\XINT_flsqrt_small_h
    \else
       \expandafter\XINT_flsqrt_small_end
    \fi
    {#1}%
}%
\def\XINT_flsqrt_small_h #1#2#3%
{%
    \expandafter\XINT_flsqrt_small_f\expandafter
    {\the\numexpr #2-\xint_c_ii*#1*#3+#1*#1\expandafter}\expandafter
    {\the\numexpr #3-#1}%
}%
\def\XINT_flsqrt_small_end #1#2#3%
{%
    \expandafter\space\expandafter
    {\the\numexpr \xint_c_i+#3*\xint_c_x^iv-
                      (#2*\xint_c_x^iv+#3)/(\xint_c_ii*#3)}%
}%
\def\XINT_flsqrt_big_f #1%
{%
    \expandafter\XINT_flsqrt_big_fa\expandafter 
    {\romannumeral0\xintiisqr {#1}}{#1}%
}%
\def\XINT_flsqrt_big_fa #1#2#3#4%
{%
    \expandafter\XINT_flsqrt_big_fb\expandafter
    {\romannumeral0\XINT_dsx_addzerosnofuss 
                     {\numexpr #3-\xint_c_viii\relax}{#2}}%
    {\romannumeral0\xintiisub
      {\XINT_dsx_addzerosnofuss 
           {\numexpr \xint_c_ii*(#3-\xint_c_viii)\relax}{#1}}{#4}}%
    {#3}%
}%
\def\XINT_flsqrt_big_fb #1#2%
{%
    \expandafter\XINT_flsqrt_big_g\expandafter {#2}{#1}%
}%
\def\XINT_flsqrt_big_g #1#2%
{%
    \expandafter\XINT_flsqrt_big_j
    \romannumeral0\xintiidivision
    {#1}{\romannumeral0\XINT_dbl_pos #2\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W }{#2}%
}%
\def\XINT_flsqrt_big_j #1%
{%
    \if0\XINT_Sgn #1\Z
        \expandafter \XINT_flsqrt_big_end_a
    \else \expandafter \XINT_flsqrt_big_k
    \fi {#1}%
}%
\def\XINT_flsqrt_big_k #1#2#3%
{%
    \expandafter\XINT_flsqrt_big_l\expandafter
    {\romannumeral0\XINT_sub_pre {#3}{#1}}%
    {\romannumeral0\xintiiadd {#2}{\romannumeral0\XINT_sqr {#1}}}%
}%
\def\XINT_flsqrt_big_l #1#2%
{%
   \expandafter\XINT_flsqrt_big_g\expandafter
   {#2}{#1}%
}%
\def\XINT_flsqrt_big_end_a #1#2#3#4#5%
{%
   \expandafter\XINT_flsqrt_big_end_b\expandafter
   {\the\numexpr -#4+#5/\xint_c_ii\expandafter}\expandafter
   {\romannumeral0\xintiisub 
    {\XINT_dsx_addzerosnofuss {#4}{#3}}%
    {\xintHalf{\xintiiQuo{\XINT_dsx_addzerosnofuss {#4}{#2}}{#3}}}}%
}%
\def\XINT_flsqrt_big_end_b #1#2{#2[#1]}%
%    \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}%
  [2014/04/01 v1.09n 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\xint_relax }%
\def\XINT_flpowseries_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_flpowseries_opt
       \else\expandafter\XINT_flpowseries_noopt
    \fi
    #1%
}%
\def\XINT_flpowseries_noopt  #1\xint_relax #2%
{%
    \expandafter\XINT_flpowseries\expandafter 
    {\the\numexpr #1\expandafter}\expandafter
    {\the\numexpr #2}\XINTdigits 
}%
\def\XINT_flpowseries_opt [\xint_relax #1]#2#3%
{%
    \expandafter\XINT_flpowseries\expandafter 
    {\the\numexpr #2\expandafter}\expandafter
    {\the\numexpr #3\expandafter}{\the\numexpr #1}%
}%
\def\XINT_flpowseries #1#2#3#4#5%
{%
   \ifnum #2<#1
      \xint_afterfi { 0.e0}%
   \else
      \xint_afterfi
        {\expandafter\XINT_flpowseries_loop_pre\expandafter
           {\romannumeral0\XINTinfloatpow [#3]{#5}{#1}}%
          {#1}{#5}{#2}{#4}{#3}%
        }%
   \fi
}%
\def\XINT_flpowseries_loop_pre #1#2#3#4#5#6%
{%
    \ifnum #4>#2 \else\XINT_flpowseries_dont_i \fi
    \expandafter\XINT_flpowseries_loop_i\expandafter 
    {\the\numexpr #2+\xint_c_i\expandafter}\expandafter
    {\romannumeral0\XINTinfloatmul [#6]{#5{#2}}{#1}}%
    {#1}{#3}{#4}{#5}{#6}%
}%
\def\XINT_flpowseries_dont_i \fi\expandafter\XINT_flpowseries_loop_i
    {\fi \expandafter\XINT_flpowseries_dont_ii }%
\def\XINT_flpowseries_dont_ii #1#2#3#4#5#6#7{\xintfloat [#7]{#2}}%
\def\XINT_flpowseries_loop_i #1#2#3#4#5#6#7%
{%
    \ifnum #5>#1 \else \XINT_flpowseries_exit_i \fi
    \expandafter\XINT_flpowseries_loop_ii\expandafter
    {\romannumeral0\XINTinfloatmul [#7]{#3}{#4}}%
    {#1}{#4}{#2}{#5}{#6}{#7}%
}%
\def\XINT_flpowseries_loop_ii #1#2#3#4#5#6#7%
{%
    \expandafter\XINT_flpowseries_loop_i\expandafter
    {\the\numexpr #2+\xint_c_i\expandafter}\expandafter
    {\romannumeral0\XINTinfloatadd [#7]{#4}%
                        {\XINTinfloatmul [#7]{#6{#2}}{#1}}}%
    {#1}{#3}{#5}{#6}{#7}%
}%
\def\XINT_flpowseries_exit_i\fi\expandafter\XINT_flpowseries_loop_ii
    {\fi \expandafter\XINT_flpowseries_exit_ii }%
\def\XINT_flpowseries_exit_ii #1#2#3#4#5#6#7%
{%
    \xintfloatadd [#7]{#4}{\XINTinfloatmul [#7]{#6{#2}}{#1}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFloatPowerSeriesX}}
% \lverb|1.08a|
%    \begin{macrocode}
\def\xintFloatPowerSeriesX {\romannumeral0\xintfloatpowerseriesx }%
\def\xintfloatpowerseriesx #1{\XINT_flpowseriesx_chkopt #1\xint_relax }%
\def\XINT_flpowseriesx_chkopt #1%
{%
    \ifx [#1\expandafter\XINT_flpowseriesx_opt
       \else\expandafter\XINT_flpowseriesx_noopt
    \fi
    #1%
}%
\def\XINT_flpowseriesx_noopt  #1\xint_relax #2%
{%
    \expandafter\XINT_flpowseriesx\expandafter 
    {\the\numexpr #1\expandafter}\expandafter
    {\the\numexpr #2}\XINTdigits 
}%
\def\XINT_flpowseriesx_opt [\xint_relax #1]#2#3%
{%
    \expandafter\XINT_flpowseriesx\expandafter 
    {\the\numexpr #2\expandafter}\expandafter
    {\the\numexpr #3\expandafter}{\the\numexpr #1}%
}%
\def\XINT_flpowseriesx #1#2#3#4#5%
{%
   \ifnum #2<#1
      \xint_afterfi { 0.e0}%
   \else
      \xint_afterfi
        {\expandafter \XINT_flpowseriesx_pre \expandafter
         {\romannumeral-`0#5}{#1}{#2}{#4}{#3}%
        }%
   \fi
}%
\def\XINT_flpowseriesx_pre #1#2#3#4#5%
{%
    \expandafter\XINT_flpowseries_loop_pre\expandafter
       {\romannumeral0\XINTinfloatpow [#5]{#1}{#2}}%
       {#2}{#1}{#3}{#4}{#5}%
}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xintseries>\relax
%\def<*xintcfrac>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xintseries>
%<*xintcfrac>
%
% \StoreCodelineNo {xintseries}
%
% \section{Package \xintcfracnameimp implementation}
% \label{sec:cfracimp}
% 
% The commenting is currently (\docdate) very sparse. Release |1.09m|
% (|2014/02/26|) has modified a few things: |\xintFtoCs| and
% |\xintCntoCs| insert spaces after the commas, |\xintCstoF| and
% |\xintCstoCv| authorize spaces in the input also before the commas,
% |\xintCntoCs| does not brace the produced coefficients, new macros
% |\xintFtoC|, |\xintCtoF|, |\xintCtoCv|, |\xintFGtoC|, and
% |\xintGGCFrac|. All uses of |\W| and many instances of |\Z| as
% delimiters removed, this was in some cases not very safe (for example
% in the treatment of the optional arguments to some macros). Actually I
% have also replaced everywhere else in the bundle the use of |\Z| in
% the treatment of macros with optional arguments with the safer
% |\xint_relax| (the more recent |\xintSeq| already used |\xint_bye|).
%
% \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}%
  [2014/04/01 v1.09n Expandable continued fractions with xint package (jfB)]%
%    \end{macrocode}
% \subsection{\csh{xintCFrac}}
%    \begin{macrocode}
\def\xintCFrac {\romannumeral0\xintcfrac }%
\def\xintcfrac #1%
{%
    \XINT_cfrac_opt_a #1\xint_relax
}%
\def\XINT_cfrac_opt_a #1%
{% 
    \ifx[#1\XINT_cfrac_opt_b\fi \XINT_cfrac_noopt #1%
}%
\def\XINT_cfrac_noopt #1\xint_relax
{%
    \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z
    \relax\relax
}%
\def\XINT_cfrac_opt_b\fi\XINT_cfrac_noopt [\xint_relax #1]%
{%
    \fi\csname XINT_cfrac_opt#1\endcsname
}%
\def\XINT_cfrac_optl #1%
{%
    \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z
    \relax\hfill
}%
\def\XINT_cfrac_optc #1%
{%
    \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z
    \relax\relax
}%
\def\XINT_cfrac_optr #1%
{%
    \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z
    \hfill\relax
}%
\def\XINT_cfrac_A #1/#2\Z
{%
    \expandafter\XINT_cfrac_B\romannumeral0\xintiidivision {#1}{#2}{#2}%
}%
\def\XINT_cfrac_B #1#2%
{%
    \XINT_cfrac_C #2\Z {#1}%
}%
\def\XINT_cfrac_C #1%
{%
    \xint_gob_til_zero #1\XINT_cfrac_integer 0\XINT_cfrac_D #1%
}%
\def\XINT_cfrac_integer 0\XINT_cfrac_D 0#1\Z #2#3#4#5{ #2}%
\def\XINT_cfrac_D #1\Z #2#3{\XINT_cfrac_loop_a {#1}{#3}{#1}{{#2}}}%  
\def\XINT_cfrac_loop_a 
{%
    \expandafter\XINT_cfrac_loop_d\romannumeral0\XINT_div_prepare
}%
\def\XINT_cfrac_loop_d #1#2%
{%
    \XINT_cfrac_loop_e #2.{#1}%
}%
\def\XINT_cfrac_loop_e #1%
{%
    \xint_gob_til_zero #1\xint_cfrac_loop_exit0\XINT_cfrac_loop_f #1%
}%
\def\XINT_cfrac_loop_f #1.#2#3#4%
{%
    \XINT_cfrac_loop_a {#1}{#3}{#1}{{#2}#4}%
}%
\def\xint_cfrac_loop_exit0\XINT_cfrac_loop_f #1.#2#3#4#5#6%
   {\XINT_cfrac_T #5#6{#2}#4\Z }%
\def\XINT_cfrac_T #1#2#3#4%
{%
  \xint_gob_til_Z #4\XINT_cfrac_end\Z\XINT_cfrac_T #1#2{#4+\cfrac{#11#2}{#3}}%
}%
\def\XINT_cfrac_end\Z\XINT_cfrac_T #1#2#3%
{%
    \XINT_cfrac_end_b #3%
}%
\def\XINT_cfrac_end_b \Z+\cfrac#1#2{ #2}%
%    \end{macrocode}
% \subsection{\csh{xintGCFrac}}
%    \begin{macrocode}
\def\xintGCFrac {\romannumeral0\xintgcfrac }%
\def\xintgcfrac #1{\XINT_gcfrac_opt_a #1\xint_relax }%
\def\XINT_gcfrac_opt_a #1%
{% 
    \ifx[#1\XINT_gcfrac_opt_b\fi \XINT_gcfrac_noopt #1%
}%
\def\XINT_gcfrac_noopt #1\xint_relax
{%
    \XINT_gcfrac #1+\xint_relax/\relax\relax
}%
\def\XINT_gcfrac_opt_b\fi\XINT_gcfrac_noopt [\xint_relax #1]%
{%
    \fi\csname XINT_gcfrac_opt#1\endcsname
}%
\def\XINT_gcfrac_optl #1%
{%
    \XINT_gcfrac #1+\xint_relax/\relax\hfill
}%
\def\XINT_gcfrac_optc #1%
{%
    \XINT_gcfrac #1+\xint_relax/\relax\relax
}%
\def\XINT_gcfrac_optr #1%
{%
    \XINT_gcfrac #1+\xint_relax/\hfill\relax
}%
\def\XINT_gcfrac 
{%
    \expandafter\XINT_gcfrac_enter\romannumeral-`0%
}%
\def\XINT_gcfrac_enter {\XINT_gcfrac_loop {}}%
\def\XINT_gcfrac_loop #1#2+#3/%
{%
    \xint_gob_til_xint_relax #3\XINT_gcfrac_endloop\xint_relax
    \XINT_gcfrac_loop {{#3}{#2}#1}%
}%
\def\XINT_gcfrac_endloop\xint_relax\XINT_gcfrac_loop #1#2#3%
{%
    \XINT_gcfrac_T #2#3#1\xint_relax\xint_relax
}%
\def\XINT_gcfrac_T #1#2#3#4{\XINT_gcfrac_U #1#2{\xintFrac{#4}}}%
\def\XINT_gcfrac_U #1#2#3#4#5%
{%
    \xint_gob_til_xint_relax #5\XINT_gcfrac_end\xint_relax\XINT_gcfrac_U 
              #1#2{\xintFrac{#5}%
               \ifcase\xintSgn{#4}
               +\or+\else-\fi
               \cfrac{#1\xintFrac{\xintAbs{#4}}#2}{#3}}%
}%
\def\XINT_gcfrac_end\xint_relax\XINT_gcfrac_U #1#2#3%
{%
    \XINT_gcfrac_end_b #3%
}%
\def\XINT_gcfrac_end_b #1\cfrac#2#3{ #3}%
%    \end{macrocode}
% \subsection{\csh{xintGGCFrac}}
% \lverb|New with 1.09m|
%    \begin{macrocode}
\def\xintGGCFrac {\romannumeral0\xintggcfrac }%
\def\xintggcfrac #1{\XINT_ggcfrac_opt_a #1\xint_relax }%
\def\XINT_ggcfrac_opt_a #1%
{% 
    \ifx[#1\XINT_ggcfrac_opt_b\fi \XINT_ggcfrac_noopt #1%
}%
\def\XINT_ggcfrac_noopt #1\xint_relax
{%
    \XINT_ggcfrac #1+\xint_relax/\relax\relax
}%
\def\XINT_ggcfrac_opt_b\fi\XINT_ggcfrac_noopt [\xint_relax #1]%
{%
    \fi\csname XINT_ggcfrac_opt#1\endcsname
}%
\def\XINT_ggcfrac_optl #1%
{%
    \XINT_ggcfrac #1+\xint_relax/\relax\hfill
}%
\def\XINT_ggcfrac_optc #1%
{%
    \XINT_ggcfrac #1+\xint_relax/\relax\relax
}%
\def\XINT_ggcfrac_optr #1%
{%
    \XINT_ggcfrac #1+\xint_relax/\hfill\relax
}%
\def\XINT_ggcfrac
{%
    \expandafter\XINT_ggcfrac_enter\romannumeral-`0%
}%
\def\XINT_ggcfrac_enter {\XINT_ggcfrac_loop {}}%
\def\XINT_ggcfrac_loop #1#2+#3/%
{%
    \xint_gob_til_xint_relax #3\XINT_ggcfrac_endloop\xint_relax
    \XINT_ggcfrac_loop {{#3}{#2}#1}%
}%
\def\XINT_ggcfrac_endloop\xint_relax\XINT_ggcfrac_loop #1#2#3%
{%
    \XINT_ggcfrac_T #2#3#1\xint_relax\xint_relax
}%
\def\XINT_ggcfrac_T #1#2#3#4{\XINT_ggcfrac_U #1#2{#4}}%
\def\XINT_ggcfrac_U #1#2#3#4#5%
{%
    \xint_gob_til_xint_relax #5\XINT_ggcfrac_end\xint_relax\XINT_ggcfrac_U 
              #1#2{#5+\cfrac{#1#4#2}{#3}}%
}%
\def\XINT_ggcfrac_end\xint_relax\XINT_ggcfrac_U #1#2#3%
{%
    \XINT_ggcfrac_end_b #3%
}%
\def\XINT_ggcfrac_end_b #1\cfrac#2#3{ #3}%
%    \end{macrocode}
% \subsection{\csh{xintGCtoGCx}}
%    \begin{macrocode}
\def\xintGCtoGCx {\romannumeral0\xintgctogcx }%
\def\xintgctogcx #1#2#3%
{%
    \expandafter\XINT_gctgcx_start\expandafter {\romannumeral-`0#3}{#1}{#2}%
}%
\def\XINT_gctgcx_start #1#2#3{\XINT_gctgcx_loop_a {}{#2}{#3}#1+\xint_relax/}%
\def\XINT_gctgcx_loop_a #1#2#3#4+#5/%
{%
    \xint_gob_til_xint_relax #5\XINT_gctgcx_end\xint_relax
    \XINT_gctgcx_loop_b {#1{#4}}{#2{#5}#3}{#2}{#3}%
}%
\def\XINT_gctgcx_loop_b #1#2%
{%
    \XINT_gctgcx_loop_a {#1#2}%
}%  
\def\XINT_gctgcx_end\xint_relax\XINT_gctgcx_loop_b #1#2#3#4{ #1}%
%    \end{macrocode}
% \subsection{\csh{xintFtoCs}}
% \lverb|Modified in 1.09m: a space is added after the inserted commas.|
%    \begin{macrocode}
\def\xintFtoCs {\romannumeral0\xintftocs }%
\def\xintftocs #1%
{%
    \expandafter\XINT_ftc_A\romannumeral0\xintrawwithzeros {#1}\Z
}%
\def\XINT_ftc_A #1/#2\Z
{%
    \expandafter\XINT_ftc_B\romannumeral0\xintiidivision {#1}{#2}{#2}%
}%
\def\XINT_ftc_B #1#2%
{%
    \XINT_ftc_C #2.{#1}%
}%
\def\XINT_ftc_C #1%
{%
    \xint_gob_til_zero #1\XINT_ftc_integer 0\XINT_ftc_D #1%
}%
\def\XINT_ftc_integer 0\XINT_ftc_D 0#1.#2#3{ #2}%
\def\XINT_ftc_D #1.#2#3{\XINT_ftc_loop_a {#1}{#3}{#1}{#2, }}% 1.09m adds a space 
\def\XINT_ftc_loop_a 
{%
    \expandafter\XINT_ftc_loop_d\romannumeral0\XINT_div_prepare
}%
\def\XINT_ftc_loop_d #1#2%
{%
    \XINT_ftc_loop_e #2.{#1}%
}%
\def\XINT_ftc_loop_e #1%
{%
    \xint_gob_til_zero #1\xint_ftc_loop_exit0\XINT_ftc_loop_f #1%
}%
\def\XINT_ftc_loop_f #1.#2#3#4%
{%
    \XINT_ftc_loop_a {#1}{#3}{#1}{#4#2, }% 1.09m has an added space here
}%
\def\xint_ftc_loop_exit0\XINT_ftc_loop_f #1.#2#3#4{ #4#2}%
%    \end{macrocode}
% \subsection{\csh{xintFtoCx}}
%    \begin{macrocode}
\def\xintFtoCx {\romannumeral0\xintftocx }%
\def\xintftocx #1#2%
{%
    \expandafter\XINT_ftcx_A\romannumeral0\xintrawwithzeros {#2}\Z {#1}%
}%
\def\XINT_ftcx_A #1/#2\Z
{%
    \expandafter\XINT_ftcx_B\romannumeral0\xintiidivision {#1}{#2}{#2}%
}%
\def\XINT_ftcx_B #1#2%
{%
    \XINT_ftcx_C #2.{#1}%
}%
\def\XINT_ftcx_C #1%
{%
    \xint_gob_til_zero #1\XINT_ftcx_integer 0\XINT_ftcx_D #1%
}%
\def\XINT_ftcx_integer 0\XINT_ftcx_D 0#1.#2#3#4{ #2}%
\def\XINT_ftcx_D #1.#2#3#4{\XINT_ftcx_loop_a {#1}{#3}{#1}{{#2}#4}{#4}}%  
\def\XINT_ftcx_loop_a 
{%
    \expandafter\XINT_ftcx_loop_d\romannumeral0\XINT_div_prepare
}%
\def\XINT_ftcx_loop_d #1#2%
{%
    \XINT_ftcx_loop_e #2.{#1}%
}%
\def\XINT_ftcx_loop_e #1%
{%
    \xint_gob_til_zero #1\xint_ftcx_loop_exit0\XINT_ftcx_loop_f #1%
}%
\def\XINT_ftcx_loop_f #1.#2#3#4#5%
{%
    \XINT_ftcx_loop_a {#1}{#3}{#1}{#4{#2}#5}{#5}%
}%
\def\xint_ftcx_loop_exit0\XINT_ftcx_loop_f #1.#2#3#4#5{ #4{#2}}%
%    \end{macrocode}
% \subsection{\csh{xintFtoC}}
% \lverb|New in 1.09m: this is the same as \xintFtoCx with empty separator. I
% had temporarily during preparation of 1.09m removed braces from \xintFtoCx,
% but I recalled later why that was useful (see doc), thus let's just here do
% \xintFtoCx {}|
%    \begin{macrocode}
\def\xintFtoC {\romannumeral0\xintftoc }%
\def\xintftoc {\xintftocx {}}%
%    \end{macrocode}
% \subsection{\csh{xintFtoGC}}
%    \begin{macrocode}
\def\xintFtoGC {\romannumeral0\xintftogc }%
\def\xintftogc {\xintftocx {+1/}}%
%    \end{macrocode}
% \subsection{\csh{xintFGtoC}}
% \lverb|New with 1.09m of 2014/02/26. Computes the common initial coefficients
% for the two fractions f and g, and outputs them as a sequence of braced
% items.|
%    \begin{macrocode}
\def\xintFGtoC {\romannumeral0\xintfgtoc}%
\def\xintfgtoc#1%
{%
    \expandafter\XINT_fgtc_a\romannumeral0\xintrawwithzeros {#1}\Z
}%
\def\XINT_fgtc_a #1/#2\Z #3%
{%
    \expandafter\XINT_fgtc_b\romannumeral0\xintrawwithzeros {#3}\Z #1/#2\Z { }%
}%
\def\XINT_fgtc_b #1/#2\Z
{%
    \expandafter\XINT_fgtc_c\romannumeral0\xintiidivision {#1}{#2}{#2}%
}%
\def\XINT_fgtc_c #1#2#3#4/#5\Z
{%
    \expandafter\XINT_fgtc_d\romannumeral0\xintiidivision 
                                         {#4}{#5}{#5}{#1}{#2}{#3}%
}%
\def\XINT_fgtc_d #1#2#3#4%#5#6#7%
{%
    \xintifEq {#1}{#4}{\XINT_fgtc_da {#1}{#2}{#3}{#4}}%
                      {\xint_thirdofthree}%
}%
\def\XINT_fgtc_da #1#2#3#4#5#6#7%
{%
     \XINT_fgtc_e {#2}{#5}{#3}{#6}{#7{#1}}%
}%
\def\XINT_fgtc_e #1%
{%
    \xintifZero {#1}{\expandafter\xint_firstofone\xint_gobble_iii}%
                    {\XINT_fgtc_f {#1}}%
}%
\def\XINT_fgtc_f #1#2%
{%
   \xintifZero {#2}{\xint_thirdofthree}{\XINT_fgtc_g {#1}{#2}}%
}%
\def\XINT_fgtc_g #1#2#3%
{%
    \expandafter\XINT_fgtc_h\romannumeral0\XINT_div_prepare {#1}{#3}{#1}{#2}%
}%
\def\XINT_fgtc_h #1#2#3#4#5%
{%
    \expandafter\XINT_fgtc_d\romannumeral0\XINT_div_prepare 
                       {#4}{#5}{#4}{#1}{#2}{#3}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFtoCC}}
%    \begin{macrocode}
\def\xintFtoCC {\romannumeral0\xintftocc }%
\def\xintftocc #1%
{%
    \expandafter\XINT_ftcc_A\expandafter {\romannumeral0\xintrawwithzeros {#1}}%
}%
\def\XINT_ftcc_A #1%
{%
    \expandafter\XINT_ftcc_B
    \romannumeral0\xintrawwithzeros {\xintAdd {1/2[0]}{#1[0]}}\Z {#1[0]}%
}%
\def\XINT_ftcc_B #1/#2\Z 
{%
    \expandafter\XINT_ftcc_C\expandafter {\romannumeral0\xintiiquo {#1}{#2}}%
}%
\def\XINT_ftcc_C #1#2%
{%
    \expandafter\XINT_ftcc_D\romannumeral0\xintsub {#2}{#1}\Z {#1}%
}%
\def\XINT_ftcc_D #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_ftcc_integer
      0#1\XINT_ftcc_En
       0-{\XINT_ftcc_Ep #1}%
    \krof
}%
\def\XINT_ftcc_Ep #1\Z #2%
{%
    \expandafter\XINT_ftcc_loop_a\expandafter
    {\romannumeral0\xintdiv {1[0]}{#1}}{#2+1/}%
}%
\def\XINT_ftcc_En #1\Z #2%
{%
    \expandafter\XINT_ftcc_loop_a\expandafter
    {\romannumeral0\xintdiv {1[0]}{#1}}{#2+-1/}%
}%
\def\XINT_ftcc_integer #1\Z #2{ #2}%
\def\XINT_ftcc_loop_a #1%
{%
    \expandafter\XINT_ftcc_loop_b
    \romannumeral0\xintrawwithzeros {\xintAdd {1/2[0]}{#1}}\Z {#1}%
}%
\def\XINT_ftcc_loop_b #1/#2\Z
{%
    \expandafter\XINT_ftcc_loop_c\expandafter 
    {\romannumeral0\xintiiquo {#1}{#2}}%
}%
\def\XINT_ftcc_loop_c #1#2%
{%
    \expandafter\XINT_ftcc_loop_d
    \romannumeral0\xintsub {#2}{#1[0]}\Z {#1}%
}%
\def\XINT_ftcc_loop_d #1%
{%
    \xint_UDzerominusfork
      #1-\XINT_ftcc_end
      0#1\XINT_ftcc_loop_N
       0-{\XINT_ftcc_loop_P #1}%
    \krof
}%
\def\XINT_ftcc_end #1\Z #2#3{ #3#2}%
\def\XINT_ftcc_loop_P #1\Z #2#3%
{%
    \expandafter\XINT_ftcc_loop_a\expandafter
    {\romannumeral0\xintdiv {1[0]}{#1}}{#3#2+1/}%
}%
\def\XINT_ftcc_loop_N #1\Z #2#3%
{%
    \expandafter\XINT_ftcc_loop_a\expandafter
    {\romannumeral0\xintdiv {1[0]}{#1}}{#3#2+-1/}%
}%
%    \end{macrocode}
% \subsection{\csh{xintCtoF}, \csh{xintCstoF}}
% \lverb|1.09m uses \xintCSVtoList on the argument of \xintCstoF to allow
% spaces also before the commas. And the original \xintCstoF code became the
% one of the new \xintCtoF dealing with a braced rather than comma separated
% list.|
%    \begin{macrocode}
\def\xintCstoF {\romannumeral0\xintcstof }%
\def\xintcstof #1%
{%
    \expandafter\XINT_ctf_prep \romannumeral0\xintcsvtolist{#1}\xint_relax
}%
\def\xintCtoF {\romannumeral0\xintctof }%
\def\xintctof #1%
{%
    \expandafter\XINT_ctf_prep \romannumeral-`0#1\xint_relax
}%
\def\XINT_ctf_prep
{%
    \XINT_ctf_loop_a 1001%
}%
\def\XINT_ctf_loop_a #1#2#3#4#5%
{%
    \xint_gob_til_xint_relax #5\XINT_ctf_end\xint_relax
    \expandafter\XINT_ctf_loop_b
    \romannumeral0\xintrawwithzeros {#5}.{#1}{#2}{#3}{#4}%
}%
\def\XINT_ctf_loop_b #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_ctf_loop_c\expandafter
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}%
}%
\def\XINT_ctf_loop_c #1#2%
{%
    \expandafter\XINT_ctf_loop_d\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_ctf_loop_d #1#2%
{%
    \expandafter\XINT_ctf_loop_e\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_ctf_loop_e #1#2%
{%
    \expandafter\XINT_ctf_loop_a\expandafter{#2}#1%
}%
\def\XINT_ctf_end #1.#2#3#4#5{\xintrawwithzeros {#2/#3}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintiCstoF}}
%    \begin{macrocode}
\def\xintiCstoF {\romannumeral0\xinticstof }%
\def\xinticstof #1%
{%
    \expandafter\XINT_icstf_prep \romannumeral-`0#1,\xint_relax,%
}%
\def\XINT_icstf_prep
{%
    \XINT_icstf_loop_a 1001%
}%
\def\XINT_icstf_loop_a #1#2#3#4#5,%
{%
    \xint_gob_til_xint_relax #5\XINT_icstf_end\xint_relax
    \expandafter
    \XINT_icstf_loop_b \romannumeral-`0#5.{#1}{#2}{#3}{#4}%
}%
\def\XINT_icstf_loop_b #1.#2#3#4#5%
{%
    \expandafter\XINT_icstf_loop_c\expandafter
    {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}%
    {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}%
    {#2}{#3}%
}%
\def\XINT_icstf_loop_c #1#2%
{%
    \expandafter\XINT_icstf_loop_a\expandafter {#2}{#1}%
}%
\def\XINT_icstf_end#1.#2#3#4#5{\xintrawwithzeros {#2/#3}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintGCtoF}}
%    \begin{macrocode}
\def\xintGCtoF {\romannumeral0\xintgctof }%
\def\xintgctof #1%
{%
    \expandafter\XINT_gctf_prep \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_gctf_prep
{%
    \XINT_gctf_loop_a 1001%
}%
\def\XINT_gctf_loop_a #1#2#3#4#5+%
{%
    \expandafter\XINT_gctf_loop_b
    \romannumeral0\xintrawwithzeros {#5}.{#1}{#2}{#3}{#4}%
}%
\def\XINT_gctf_loop_b #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_gctf_loop_c\expandafter
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}%
}%
\def\XINT_gctf_loop_c #1#2%
{%
    \expandafter\XINT_gctf_loop_d\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctf_loop_d #1#2%
{%
    \expandafter\XINT_gctf_loop_e\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctf_loop_e #1#2%
{%
    \expandafter\XINT_gctf_loop_f\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctf_loop_f #1#2/%
{%
    \xint_gob_til_xint_relax #2\XINT_gctf_end\xint_relax
    \expandafter\XINT_gctf_loop_g 
    \romannumeral0\xintrawwithzeros {#2}.#1%
}%
\def\XINT_gctf_loop_g #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_gctf_loop_h\expandafter
    {\romannumeral0\XINT_mul_fork #1\Z #6\Z }%
    {\romannumeral0\XINT_mul_fork #1\Z #5\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
}%
\def\XINT_gctf_loop_h #1#2%
{%
    \expandafter\XINT_gctf_loop_i\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctf_loop_i #1#2%
{%
    \expandafter\XINT_gctf_loop_j\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctf_loop_j #1#2%
{%
    \expandafter\XINT_gctf_loop_a\expandafter {#2}#1%
}%
\def\XINT_gctf_end #1.#2#3#4#5{\xintrawwithzeros {#2/#3}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintiGCtoF}}
%    \begin{macrocode}
\def\xintiGCtoF {\romannumeral0\xintigctof }%
\def\xintigctof #1%
{%
    \expandafter\XINT_igctf_prep \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_igctf_prep
{%
    \XINT_igctf_loop_a 1001%
}%
\def\XINT_igctf_loop_a #1#2#3#4#5+%
{%
    \expandafter\XINT_igctf_loop_b
    \romannumeral-`0#5.{#1}{#2}{#3}{#4}%
}%
\def\XINT_igctf_loop_b #1.#2#3#4#5%
{%
    \expandafter\XINT_igctf_loop_c\expandafter
    {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}%
    {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}%
    {#2}{#3}%
}%
\def\XINT_igctf_loop_c #1#2%
{%
    \expandafter\XINT_igctf_loop_f\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_igctf_loop_f #1#2#3#4/%
{%
    \xint_gob_til_xint_relax #4\XINT_igctf_end\xint_relax
    \expandafter\XINT_igctf_loop_g 
    \romannumeral-`0#4.{#2}{#3}#1%
}%
\def\XINT_igctf_loop_g #1.#2#3%
{%
    \expandafter\XINT_igctf_loop_h\expandafter
    {\romannumeral0\XINT_mul_fork #1\Z #3\Z }%
    {\romannumeral0\XINT_mul_fork #1\Z #2\Z }%
}%
\def\XINT_igctf_loop_h #1#2%
{%
    \expandafter\XINT_igctf_loop_i\expandafter {#2}{#1}%
}%
\def\XINT_igctf_loop_i #1#2#3#4%
{%
    \XINT_igctf_loop_a {#3}{#4}{#1}{#2}%
}%
\def\XINT_igctf_end #1.#2#3#4#5{\xintrawwithzeros {#4/#5}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintCtoCv}, \csh{xintCstoCv}}
% \lverb|1.09m uses \xintCSVtoList on the argument of \xintCstoCv to allow
% spaces also before the commas. The original \xintCstoCv code became the
% one of the new \xintCtoF dealing with a braced rather than comma separated
% list.|
%    \begin{macrocode}
\def\xintCstoCv {\romannumeral0\xintcstocv }%
\def\xintcstocv #1%
{%
    \expandafter\XINT_ctcv_prep\romannumeral0\xintcsvtolist{#1}\xint_relax
}%
\def\xintCtoCv {\romannumeral0\xintctocv }%
\def\xintctocv #1%
{%
    \expandafter\XINT_ctcv_prep\romannumeral-`0#1\xint_relax
}%
\def\XINT_ctcv_prep
{%
    \XINT_ctcv_loop_a {}1001%
}%
\def\XINT_ctcv_loop_a #1#2#3#4#5#6%
{%
    \xint_gob_til_xint_relax #6\XINT_ctcv_end\xint_relax
    \expandafter\XINT_ctcv_loop_b
    \romannumeral0\xintrawwithzeros {#6}.{#2}{#3}{#4}{#5}{#1}%
}%
\def\XINT_ctcv_loop_b #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_ctcv_loop_c\expandafter
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}%
}%
\def\XINT_ctcv_loop_c #1#2%
{%
    \expandafter\XINT_ctcv_loop_d\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_ctcv_loop_d #1#2%
{%
    \expandafter\XINT_ctcv_loop_e\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_ctcv_loop_e #1#2%
{%
    \expandafter\XINT_ctcv_loop_f\expandafter{#2}#1%
}%
\def\XINT_ctcv_loop_f #1#2#3#4#5%
{%
    \expandafter\XINT_ctcv_loop_g\expandafter 
    {\romannumeral0\xintrawwithzeros {#1/#2}}{#5}{#1}{#2}{#3}{#4}%
}%
\def\XINT_ctcv_loop_g #1#2{\XINT_ctcv_loop_a {#2{#1}}}% 1.09b removes [0]
\def\XINT_ctcv_end #1.#2#3#4#5#6{ #6}%
%    \end{macrocode}
% \subsection{\csh{xintiCstoCv}}
%    \begin{macrocode}
\def\xintiCstoCv {\romannumeral0\xinticstocv }%
\def\xinticstocv #1%
{%
    \expandafter\XINT_icstcv_prep \romannumeral-`0#1,\xint_relax,%
}%
\def\XINT_icstcv_prep
{%
    \XINT_icstcv_loop_a {}1001%
}%
\def\XINT_icstcv_loop_a #1#2#3#4#5#6,%
{%
    \xint_gob_til_xint_relax #6\XINT_icstcv_end\xint_relax
    \expandafter
    \XINT_icstcv_loop_b \romannumeral-`0#6.{#2}{#3}{#4}{#5}{#1}%
}%
\def\XINT_icstcv_loop_b #1.#2#3#4#5%
{%
    \expandafter\XINT_icstcv_loop_c\expandafter
    {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}%
    {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}%
    {{#2}{#3}}%
}%
\def\XINT_icstcv_loop_c #1#2%
{%
    \expandafter\XINT_icstcv_loop_d\expandafter {#2}{#1}%
}%
\def\XINT_icstcv_loop_d #1#2%
{%
    \expandafter\XINT_icstcv_loop_e\expandafter
    {\romannumeral0\xintrawwithzeros {#1/#2}}{{#1}{#2}}%
}%
\def\XINT_icstcv_loop_e #1#2#3#4{\XINT_icstcv_loop_a {#4{#1}}#2#3}%
\def\XINT_icstcv_end #1.#2#3#4#5#6{ #6}%  1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintGCtoCv}}
%    \begin{macrocode}
\def\xintGCtoCv {\romannumeral0\xintgctocv }%
\def\xintgctocv #1%
{%
    \expandafter\XINT_gctcv_prep \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_gctcv_prep
{%
    \XINT_gctcv_loop_a {}1001%
}%
\def\XINT_gctcv_loop_a #1#2#3#4#5#6+%
{%
    \expandafter\XINT_gctcv_loop_b
    \romannumeral0\xintrawwithzeros {#6}.{#2}{#3}{#4}{#5}{#1}%
}%
\def\XINT_gctcv_loop_b #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_gctcv_loop_c\expandafter
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}%
    {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}%
}%
\def\XINT_gctcv_loop_c #1#2%
{%
    \expandafter\XINT_gctcv_loop_d\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctcv_loop_d #1#2%
{%
    \expandafter\XINT_gctcv_loop_e\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctcv_loop_e #1#2%
{%
    \expandafter\XINT_gctcv_loop_f\expandafter {#2}#1%
}%
\def\XINT_gctcv_loop_f #1#2%
{%
    \expandafter\XINT_gctcv_loop_g\expandafter
    {\romannumeral0\xintrawwithzeros {#1/#2}}{{#1}{#2}}%
}%
\def\XINT_gctcv_loop_g #1#2#3#4%
{%
    \XINT_gctcv_loop_h {#4{#1}}{#2#3}% 1.09b removes [0]
}%
\def\XINT_gctcv_loop_h #1#2#3/%
{%
    \xint_gob_til_xint_relax #3\XINT_gctcv_end\xint_relax
    \expandafter\XINT_gctcv_loop_i
    \romannumeral0\xintrawwithzeros {#3}.#2{#1}%
}%
\def\XINT_gctcv_loop_i #1/#2.#3#4#5#6%
{%
    \expandafter\XINT_gctcv_loop_j\expandafter
    {\romannumeral0\XINT_mul_fork #1\Z #6\Z }%
    {\romannumeral0\XINT_mul_fork #1\Z #5\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #4\Z }%
    {\romannumeral0\XINT_mul_fork #2\Z #3\Z }%
}%
\def\XINT_gctcv_loop_j #1#2%
{%
    \expandafter\XINT_gctcv_loop_k\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_gctcv_loop_k #1#2%
{%
    \expandafter\XINT_gctcv_loop_l\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctcv_loop_l #1#2%
{%
    \expandafter\XINT_gctcv_loop_m\expandafter {\expandafter{#2}#1}%
}%
\def\XINT_gctcv_loop_m #1#2{\XINT_gctcv_loop_a {#2}#1}%
\def\XINT_gctcv_end #1.#2#3#4#5#6{ #6}%
%    \end{macrocode}
% \subsection{\csh{xintiGCtoCv}}
%    \begin{macrocode}
\def\xintiGCtoCv {\romannumeral0\xintigctocv }%
\def\xintigctocv #1%
{%
    \expandafter\XINT_igctcv_prep \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_igctcv_prep
{%
    \XINT_igctcv_loop_a {}1001%
}%
\def\XINT_igctcv_loop_a #1#2#3#4#5#6+%
{%
    \expandafter\XINT_igctcv_loop_b
    \romannumeral-`0#6.{#2}{#3}{#4}{#5}{#1}%
}%
\def\XINT_igctcv_loop_b #1.#2#3#4#5%
{%
    \expandafter\XINT_igctcv_loop_c\expandafter
    {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}%
    {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}%
    {{#2}{#3}}%
}%
\def\XINT_igctcv_loop_c #1#2%
{%
    \expandafter\XINT_igctcv_loop_f\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_igctcv_loop_f #1#2#3#4/%
{%
    \xint_gob_til_xint_relax #4\XINT_igctcv_end_a\xint_relax
    \expandafter\XINT_igctcv_loop_g 
    \romannumeral-`0#4.#1#2{#3}%
}%
\def\XINT_igctcv_loop_g #1.#2#3#4#5%
{%
    \expandafter\XINT_igctcv_loop_h\expandafter
    {\romannumeral0\XINT_mul_fork #1\Z #5\Z }%
    {\romannumeral0\XINT_mul_fork #1\Z #4\Z }%
    {{#2}{#3}}%
}%
\def\XINT_igctcv_loop_h #1#2%
{%
    \expandafter\XINT_igctcv_loop_i\expandafter {\expandafter{#2}{#1}}%
}%
\def\XINT_igctcv_loop_i #1#2{\XINT_igctcv_loop_k #2{#2#1}}%
\def\XINT_igctcv_loop_k #1#2%
{%
    \expandafter\XINT_igctcv_loop_l\expandafter
    {\romannumeral0\xintrawwithzeros {#1/#2}}%
}%
\def\XINT_igctcv_loop_l #1#2#3{\XINT_igctcv_loop_a {#3{#1}}#2}%1.09i removes [0]
\def\XINT_igctcv_end_a #1.#2#3#4#5%
{%
    \expandafter\XINT_igctcv_end_b\expandafter
    {\romannumeral0\xintrawwithzeros {#2/#3}}%
}%
\def\XINT_igctcv_end_b #1#2{ #2{#1}}% 1.09b removes [0]
%    \end{macrocode}
% \subsection{\csh{xintFtoCv}}
% \lverb|Still uses \xinticstocv \xintFtoCs rather than \xintctocv \xintFtoC.|
%    \begin{macrocode}
\def\xintFtoCv {\romannumeral0\xintftocv }%
\def\xintftocv #1%
{%
    \xinticstocv {\xintFtoCs {#1}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintFtoCCv}}
%    \begin{macrocode}
\def\xintFtoCCv {\romannumeral0\xintftoccv }%
\def\xintftoccv #1%
{%
    \xintigctocv {\xintFtoCC {#1}}%
}%
%    \end{macrocode}
% \subsection{\csh{xintCntoF}}
% \lverb|&
% Modified in 1.06 to give the N first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.|
%    \begin{macrocode}
\def\xintCntoF {\romannumeral0\xintcntof }%
\def\xintcntof #1%
{%
    \expandafter\XINT_cntf\expandafter {\the\numexpr #1}%
}%
\def\XINT_cntf #1#2%
{%
   \ifnum #1>\xint_c_
      \xint_afterfi {\expandafter\XINT_cntf_loop\expandafter 
                     {\the\numexpr #1-1\expandafter}\expandafter
                     {\romannumeral-`0#2{#1}}{#2}}%
   \else
      \xint_afterfi 
         {\ifnum #1=\xint_c_
              \xint_afterfi {\expandafter\space \romannumeral-`0#2{0}}%
          \else \xint_afterfi { }% 1.09m now returns nothing.
          \fi}%
   \fi
}%
\def\XINT_cntf_loop #1#2#3%
{%
    \ifnum #1>\xint_c_ \else \XINT_cntf_exit \fi
    \expandafter\XINT_cntf_loop\expandafter
    {\the\numexpr #1-1\expandafter }\expandafter
    {\romannumeral0\xintadd {\xintDiv {1[0]}{#2}}{#3{#1}}}%
    {#3}%
}%
\def\XINT_cntf_exit \fi 
    \expandafter\XINT_cntf_loop\expandafter
    #1\expandafter #2#3%
{%
    \fi\xint_gobble_ii #2%
}%
%    \end{macrocode}
% \subsection{\csh{xintGCntoF}}
% \lverb|Modified in 1.06 to give the N argument first to a \numexpr rather
% than expanding twice. I just use \the\numexpr and maintain the previous code
% after that.|
%    \begin{macrocode}
\def\xintGCntoF {\romannumeral0\xintgcntof }%
\def\xintgcntof #1%
{%
    \expandafter\XINT_gcntf\expandafter {\the\numexpr #1}%
}%
\def\XINT_gcntf #1#2#3%
{%
   \ifnum #1>\xint_c_
      \xint_afterfi {\expandafter\XINT_gcntf_loop\expandafter 
                     {\the\numexpr #1-1\expandafter}\expandafter
                     {\romannumeral-`0#2{#1}}{#2}{#3}}%
   \else
      \xint_afterfi 
         {\ifnum #1=\xint_c_
              \xint_afterfi {\expandafter\space\romannumeral-`0#2{0}}%
          \else \xint_afterfi { }% 1.09m now returns nothing rather than 0/1[0]
          \fi}%
   \fi
}%
\def\XINT_gcntf_loop #1#2#3#4%
{%
    \ifnum #1>\xint_c_ \else \XINT_gcntf_exit \fi
    \expandafter\XINT_gcntf_loop\expandafter
    {\the\numexpr #1-1\expandafter }\expandafter
    {\romannumeral0\xintadd {\xintDiv {#4{#1}}{#2}}{#3{#1}}}%
    {#3}{#4}%
}%
\def\XINT_gcntf_exit \fi 
    \expandafter\XINT_gcntf_loop\expandafter
    #1\expandafter #2#3#4%
{%
    \fi\xint_gobble_ii #2%
}%
%    \end{macrocode}
% \subsection{\csh{xintCntoCs}}
% \lverb|Modified in 1.09m: added spaces after the commas in the produced list.
% Moreover the coefficients are not braced anymore. A slight induced limitation
% is that the macro argument should not contain some explicit comma (cf. 
% \XINT_cntcs_exit_b), hence \xintCntoCs {\macro,} with \def\macro,#1{<stuff>}
% would crash. Not a very serious limitation, I believe. |
%    \begin{macrocode}
\def\xintCntoCs {\romannumeral0\xintcntocs }%
\def\xintcntocs #1%
{%
    \expandafter\XINT_cntcs\expandafter {\the\numexpr #1}%
}%
\def\XINT_cntcs #1#2%
{%
   \ifnum #1<0
      \xint_afterfi { }% 1.09i: a 0/1[0] was here, now the macro returns nothing
   \else       
      \xint_afterfi {\expandafter\XINT_cntcs_loop\expandafter 
                     {\the\numexpr #1-\xint_c_i\expandafter}\expandafter
                     {\romannumeral-`0#2{#1}}{#2}}% produced coeff not braced
   \fi
}%
\def\XINT_cntcs_loop #1#2#3%
{%
    \ifnum #1>-\xint_c_i \else \XINT_cntcs_exit \fi
    \expandafter\XINT_cntcs_loop\expandafter
    {\the\numexpr #1-\xint_c_i\expandafter}\expandafter 
    {\romannumeral-`0#3{#1}, #2}{#3}% space added, 1.09m
}%
\def\XINT_cntcs_exit \fi 
    \expandafter\XINT_cntcs_loop\expandafter
    #1\expandafter #2#3%
{%
    \fi\XINT_cntcs_exit_b #2%
}%
\def\XINT_cntcs_exit_b #1,{}% romannumeral stopping space already there
%    \end{macrocode}
% \subsection{\csh{xintCntoGC}}
% \lverb|&
% Modified in 1.06 to give the N first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.
%
% 1.09m maintains the braces, as the coeff are allowed to be fraction and the
% slash can not be naked in the GC format, contrarily to what happens in
% \xintCntoCs. Also the separators given to \xintGCtoGCx may then fetch the
% coefficients as argument, as they are braced.|
%    \begin{macrocode}
\def\xintCntoGC {\romannumeral0\xintcntogc }%
\def\xintcntogc #1%
{%
    \expandafter\XINT_cntgc\expandafter {\the\numexpr #1}%
}%
\def\XINT_cntgc #1#2%
{%
   \ifnum #1<0
      \xint_afterfi { }% 1.09i there was as strange 0/1[0] here, removed
   \else       
      \xint_afterfi {\expandafter\XINT_cntgc_loop\expandafter 
                     {\the\numexpr #1-\xint_c_i\expandafter}\expandafter
                     {\expandafter{\romannumeral-`0#2{#1}}}{#2}}%
   \fi
}%
\def\XINT_cntgc_loop #1#2#3%
{%
    \ifnum #1>-\xint_c_i \else \XINT_cntgc_exit \fi
    \expandafter\XINT_cntgc_loop\expandafter
    {\the\numexpr #1-\xint_c_i\expandafter }\expandafter 
    {\expandafter{\romannumeral-`0#3{#1}}+1/#2}{#3}%
}%
\def\XINT_cntgc_exit \fi 
    \expandafter\XINT_cntgc_loop\expandafter
    #1\expandafter #2#3%
{%
    \fi\XINT_cntgc_exit_b #2%
}%
\def\XINT_cntgc_exit_b #1+1/{ }%
%    \end{macrocode}
% \subsection{\csh{xintGCntoGC}}
% \lverb|&
% Modified in 1.06 to give the N first to a \numexpr rather than expanding
% twice. I just use \the\numexpr and maintain the previous code after that.|
%    \begin{macrocode}
\def\xintGCntoGC {\romannumeral0\xintgcntogc }%
\def\xintgcntogc #1%
{%
    \expandafter\XINT_gcntgc\expandafter {\the\numexpr #1}%
}%
\def\XINT_gcntgc #1#2#3%
{%
   \ifnum #1<0
      \xint_afterfi { }% 1.09i now returns nothing
   \else       
      \xint_afterfi {\expandafter\XINT_gcntgc_loop\expandafter 
                     {\the\numexpr #1-\xint_c_i\expandafter}\expandafter
                     {\expandafter{\romannumeral-`0#2{#1}}}{#2}{#3}}%
   \fi
}%
\def\XINT_gcntgc_loop #1#2#3#4%
{%
    \ifnum #1>-\xint_c_i \else \XINT_gcntgc_exit \fi
    \expandafter\XINT_gcntgc_loop_b\expandafter
    {\expandafter{\romannumeral-`0#4{#1}}/#2}{#3{#1}}{#1}{#3}{#4}%
}%
\def\XINT_gcntgc_loop_b #1#2#3%
{%
    \expandafter\XINT_gcntgc_loop\expandafter
    {\the\numexpr #3-\xint_c_i \expandafter}\expandafter
    {\expandafter{\romannumeral-`0#2}+#1}%
}%
\def\XINT_gcntgc_exit \fi
    \expandafter\XINT_gcntgc_loop_b\expandafter #1#2#3#4#5%
{%
    \fi\XINT_gcntgc_exit_b #1%
}%
\def\XINT_gcntgc_exit_b #1/{ }%
%    \end{macrocode}
% \subsection{\csh{xintCstoGC}}
%    \begin{macrocode}
\def\xintCstoGC {\romannumeral0\xintcstogc }%
\def\xintcstogc #1%
{%
    \expandafter\XINT_cstc_prep \romannumeral-`0#1,\xint_relax,%
}%
\def\XINT_cstc_prep #1,{\XINT_cstc_loop_a {{#1}}}%
\def\XINT_cstc_loop_a #1#2,%
{%
    \xint_gob_til_xint_relax #2\XINT_cstc_end\xint_relax
    \XINT_cstc_loop_b {#1}{#2}%
}%
\def\XINT_cstc_loop_b #1#2{\XINT_cstc_loop_a {#1+1/{#2}}}%
\def\XINT_cstc_end\xint_relax\XINT_cstc_loop_b #1#2{ #1}%
%    \end{macrocode}
% \subsection{\csh{xintGCtoGC}}
%    \begin{macrocode}
\def\xintGCtoGC {\romannumeral0\xintgctogc }%
\def\xintgctogc #1%
{%
    \expandafter\XINT_gctgc_start \romannumeral-`0#1+\xint_relax/%
}%
\def\XINT_gctgc_start {\XINT_gctgc_loop_a {}}%
\def\XINT_gctgc_loop_a #1#2+#3/%
{%
    \xint_gob_til_xint_relax #3\XINT_gctgc_end\xint_relax
    \expandafter\XINT_gctgc_loop_b\expandafter
    {\romannumeral-`0#2}{#3}{#1}%
}%
\def\XINT_gctgc_loop_b #1#2%
{%
    \expandafter\XINT_gctgc_loop_c\expandafter
    {\romannumeral-`0#2}{#1}%
}%
\def\XINT_gctgc_loop_c #1#2#3%
{%
    \XINT_gctgc_loop_a {#3{#2}+{#1}/}%
}%
\def\XINT_gctgc_end\xint_relax\expandafter\XINT_gctgc_loop_b
{%
    \expandafter\XINT_gctgc_end_b
}%
\def\XINT_gctgc_end_b #1#2#3{ #3{#1}}%
\XINT_restorecatcodes_endinput%
%    \end{macrocode}
%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11
%\let</xintcfrac>\relax
%\def<*xintexpr>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 }
%</xintcfrac>
%<*xintexpr>
%
% \StoreCodelineNo {xintcfrac}
%
% \section{Package \xintexprnameimp implementation}
% \label{sec:exprimp}
% 
% 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).
%
% Release |1.09kb| fixes the bug introduced in |\xintNewExpr| in |1.09i| of
% December 2013: an |\endlinechar -1| was removed, but without it there is a
% spurious trailing space token in the outputs of the created macros, and
% nesting is then impossible.
%
% \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}%
  [2014/04/01 v1.09n 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 % 1.09i, 2013/12/18 not sure why I had that? removed.
    % 2014/02/13: you idiot, if not then spurious extra ending space
    % token makes impossible nesting of created macros!
    \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 {21871}
\makeatletter\check@checksum\makeatother
\Finale
%% End of file xint.dtx