diff options
Diffstat (limited to 'Master/texmf-dist/doc/generic')
| -rw-r--r-- | Master/texmf-dist/doc/generic/xint/README | 161 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/generic/xint/xint.dtx | 24759 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/generic/xint/xint.pdf | bin | 998916 -> 1000698 bytes |
3 files changed, 161 insertions, 24759 deletions
diff --git a/Master/texmf-dist/doc/generic/xint/README b/Master/texmf-dist/doc/generic/xint/README new file mode 100644 index 00000000000..f34411fd82e --- /dev/null +++ b/Master/texmf-dist/doc/generic/xint/README @@ -0,0 +1,161 @@ +The xint bundle +Release 1.09kb (2014/02/13). Documentation date: 2014/02/13 + +Copyright (C) 2013-2014 by Jean-Francois Burnol +License: LaTeX Project Public License 1.3c or later. + +Contents: Abstract, Installation, License. + +Abstract +======== + +xinttools is loaded by xint (hence by all other packages of the +bundle, too): it provides utilities of independent interest such as +expandable and non-expandable loops. + +xint implements with expandable TEX macros additions, subtractions, +multiplications, divisions and powers with arbitrarily long numbers. + +xintfrac extends the scope of xint to decimal numbers, to numbers in +scientific notation and also to fractions with arbitrarily long such +numerators and denominators separated by a forward slash. + +xintexpr extends xintfrac with an expandable parser + \xintexpr . . . \relax +of expressions involving arithmetic operations in infix notation on +decimal numbers, fractions, numbers in scientific notation, with +parentheses, factorial symbol, function names, comparison operators, +logic operators, twofold and threefold way conditionals, +sub-expressions, macros expanding to the previous items. + +Further modules: + +xintbinhex is for conversions to and from binary and hexadecimal +bases. + +xintseries provides some basic functionality for computing in an +expandable manner partial sums of series and power series with +fractional coefficients. + +xintgcd implements the Euclidean algorithm and its typesetting. + +xintcfrac deals with the computation of continued fractions. + +Most macros, and all of those doing computations, work purely by +expansion without assignments, and may thus be used almost everywhere +in TeX. The packages may be used with any flavor of TeX supporting the +e-TeX extensions. LaTeX users will use \usepackage and others \input +to load the package components. + +Installation +============ + +A. Installation using xint.tds.zip: +----------------------------------- + +obtain xint.tds.zip from CTAN: + http://mirror.ctan.org/install/macros/generic/xint.tds.zip + +cd to the download repertory and issue + unzip xint.tds.zip -d <TEXMF> +for example: (assuming standard access rights, so sudo needed) + sudo unzip xint.tds.zip -d /usr/local/texlive/texmf-local + sudo mktexlsr + +On Mac OS X, installation into user home folder: + unzip xint.tds.zip -d ~/Library/texmf + +B. Installation after file extractions: +--------------------------------------- + +obtain xint.dtx, xint.ins and the README from CTAN: + http://www.ctan.org/pkg/xint + +- "tex xint.ins" generates the style files +(pre-existing files in the same repertory will be overwritten). + +- without xint.ins: "tex or latex or pdflatex or xelatex xint.dtx" +will also generate the style files (and xint.ins). + +xint.tex is also extracted, use it for the documentation: + +- with latex+dvipdfmx: latex xint.tex thrice then dvipdfmx xint.dvi +Ignore dvipdfmx warnings, but if the pdf file has problems with fonts +(possibly from an old dvipdfmx), use then rather pdflatex or xelatex. + +- with pdflatex or xelatex: run it directly thrice on xint.dtx, or run +it on xint.tex after having edited the suitable toggle therein. + +When compiling xint.tex, the documentation is by default produced +with the source code included. See instructions in the file for +changing this default. + +When compiling directly xint.dtx, the documentation is produced +without the source code (latex+dvips or pdflatex or xelatex). + +Finishing the installation: (on first installation the destination +repertories may need to be created) + + xinttools.sty | + xint.sty | + xintfrac.sty | + xintexpr.sty | --> TDS:tex/generic/xint/ + xintbinhex.sty | + xintgcd.sty | + xintseries.sty | + xintcfrac.sty | + + xint.dtx --> TDS:source/generic/xint/ + xint.ins --> TDS:source/generic/xint/ + xint.tex --> TDS:source/generic/xint/ + + xint.pdf --> TDS:doc/generic/xint/ + README --> TDS:doc/generic/xint/ + +Depending on the TDS destination and the TeX installation, it may be +necessary to refresh the TeX installation filename database (mktexlsr) + +C. Usage: +--------- + +Usage with LaTeX: \usepackage{xinttools} + \usepackage{xint} % (loads xinttools) + \usepackage{xintfrac} % (loads xint) + \usepackage{xintexpr} % (loads xintfrac) + + \usepackage{xintbinhex} % (loads xint) + \usepackage{xintgcd} % (loads xint) + \usepackage{xintseries} % (loads xintfrac) + \usepackage{xintcfrac} % (loads xintfrac) + +Usage with TeX: \input xinttools.sty\relax + \input xint.sty\relax % (loads xinttools) + \input xintfrac.sty\relax % (loads xint) + \input xintexpr.sty\relax % (loads xintfrac) + + \input xintbinhex.sty\relax % (loads xint) + \input xintgcd.sty\relax % (loads xint) + \input xintseries.sty\relax % (loads xintfrac) + \input xintcfrac.sty\relax % (loads xintfrac) + +License +======= + + This work may be distributed and/or modified under the + conditions of the LaTeX Project Public License, either + version 1.3c of this license or (at your option) any later + version. This version of this license is in + http://www.latex-project.org/lppl/lppl-1-3c.txt + and the latest version of this license is in + http://www.latex-project.org/lppl.txt + and version 1.3 or later is part of all distributions of + LaTeX version 2005/12/01 or later. + +This work consists of the source file xint.dtx and of its derived files: +xinttools.sty xint.sty, xintfrac.sty, xintexpr.sty, xintbinhex.sty, +xintgcd.sty, xintseries.sty, xintcfrac.sty, as well as xint.ins, xint.tex +and the documentation xint.pdf (or xint.dvi). + +The author of this work is Jean-Francois Burnol <jfbu at free dot fr>. +This work has the LPPL maintenance status `author-maintained'. + diff --git a/Master/texmf-dist/doc/generic/xint/xint.dtx b/Master/texmf-dist/doc/generic/xint/xint.dtx deleted file mode 100644 index 6f868a9a451..00000000000 --- a/Master/texmf-dist/doc/generic/xint/xint.dtx +++ /dev/null @@ -1,24759 +0,0 @@ -% -*- coding: iso-latin-1; time-stamp-format: "%02d-%02m-%:y %02H:%02M:%02S %Z" -*- -% File: xint.dtx, package: 1.09ka (2014/02/05), documentation: 2014/02/05 -% License: LaTeX Project Public License 1.3c or later. -% Copyright (C) 2013-2014 by Jean-Francois Burnol <jfbu at free dot fr> -%<*dtx> -\def\lasttimestamp{Time-stamp: <05-02-2014 21:53:23 CET>} -%</dtx> -%<*drv> -\def\xintdate {2014/02/05} -\def\xintversion {1.09ka} -%</drv> -%%---------------------------------------------------------------- -%% The xint bundle (version 1.09ka of February 5, 2014) -%<xinttools>%% xinttools: Expandable and non-expandable utilities -%<xint>%% xint: Expandable operations on long numbers -%<xintfrac>%% xintfrac: Expandable operations on fractions -%<xintexpr>%% xintexpr: Expandable expression parser -%<xintbinhex>%% xintbinhex: Expandable binary and hexadecimal conversions -%<xintgcd>%% xintgcd: Euclidean algorithm with xint package -%<xintseries>%% xintseries: Expandable partial sums with xint package -%<xintcfrac>%% xintcfrac: Expandable continued fractions with xint package -%% Copyright (C) 2013-2014 by Jean-Francois Burnol -%%---------------------------------------------------------------- -% Installation -% ============ -% -% A. Installation using xint.tds.zip: -% ----------------------------------- -% -% obtain xint.tds.zip from CTAN: -% http://mirror.ctan.org/install/macros/generic/xint.tds.zip -% -% cd to the download repertory and issue -% unzip xint.tds.zip -d <TEXMF> -% for example: (assuming standard access rights, so sudo needed) -% sudo unzip xint.tds.zip -d /usr/local/texlive/texmf-local -% sudo mktexlsr -% -% On Mac OS X, installation into user home folder: -% unzip xint.tds.zip -d ~/Library/texmf -% -% B. Installation after file extractions: -% --------------------------------------- -% -% obtain xint.dtx, xint.ins and the README from CTAN: -% http://www.ctan.org/pkg/xint -% -% - "tex xint.ins" generates the style files -% (pre-existing files in the same repertory will be overwritten). -% -% - without xint.ins: "tex or latex or pdflatex or xelatex xint.dtx" -% will also generate the style files (and xint.ins). -% -% xint.tex is also extracted, use it for the documentation: -% -% - with latex+dvipdfmx: latex xint.tex thrice then dvipdfmx xint.dvi -% Ignore dvipdfmx warnings, but if the pdf file has problems with fonts -% (possibly from an old dvipdfmx), use then rather pdflatex or xelatex. -% -% - with pdflatex or xelatex: run it directly thrice on xint.dtx, or run -% it on xint.tex after having edited the suitable toggle therein. -% -% When compiling xint.tex, the documentation is by default produced -% with the source code included. See instructions in the file for -% changing this default. -% -% When compiling directly xint.dtx, the documentation is produced -% without the source code (latex+dvips or pdflatex or xelatex). -% -% Finishing the installation: (on first installation the destination -% repertories may need to be created) -% -% xinttools.sty | -% xint.sty | -% xintfrac.sty | -% xintexpr.sty | --> TDS:tex/generic/xint/ -% xintbinhex.sty | -% xintgcd.sty | -% xintseries.sty | -% xintcfrac.sty | -% -% xint.dtx --> TDS:source/generic/xint/ -% xint.ins --> TDS:source/generic/xint/ -% xint.tex --> TDS:source/generic/xint/ -% -% xint.pdf --> TDS:doc/generic/xint/ -% README --> TDS:doc/generic/xint/ -% -% Depending on the TDS destination and the TeX installation, it may be -% necessary to refresh the TeX installation filename database (mktexlsr) -% -% C. Usage: -% --------- -% -% Usage with LaTeX: \usepackage{xinttools} -% \usepackage{xint} % (loads xinttools) -% \usepackage{xintfrac} % (loads xint) -% \usepackage{xintexpr} % (loads xintfrac) -% -% \usepackage{xintbinhex} % (loads xint) -% \usepackage{xintgcd} % (loads xint) -% \usepackage{xintseries} % (loads xintfrac) -% \usepackage{xintcfrac} % (loads xintfrac) -% -% Usage with TeX: \input xinttools.sty\relax -% \input xint.sty\relax % (loads xinttools) -% \input xintfrac.sty\relax % (loads xint) -% \input xintexpr.sty\relax % (loads xintfrac) -% -% \input xintbinhex.sty\relax % (loads xint) -% \input xintgcd.sty\relax % (loads xint) -% \input xintseries.sty\relax % (loads xintfrac) -% \input xintcfrac.sty\relax % (loads xintfrac) -% -% License -% ======= -% -% This work may be distributed and/or modified under the -% conditions of the LaTeX Project Public License, either -% version 1.3c of this license or (at your option) any later -% version. This version of this license is in -% http://www.latex-project.org/lppl/lppl-1-3c.txt -% and the latest version of this license is in -% http://www.latex-project.org/lppl.txt -% and version 1.3 or later is part of all distributions of -% LaTeX version 2005/12/01 or later. -% -% This work consists of the source file xint.dtx and of its derived files: -% xinttools.sty xint.sty, xintfrac.sty, xintexpr.sty, xintbinhex.sty, -% xintgcd.sty, xintseries.sty, xintcfrac.sty, as well as xint.ins, xint.tex -% and the documentation xint.pdf (or xint.dvi). -% -% The author of this work is Jean-Francois Burnol <jfbu at free dot fr>. -% This work has the LPPL maintenance status `author-maintained'. -% -%<*dtx> -\iffalse -%</dtx> -%<*drv>---------------------------------------------------------------------- -%% This is a generated file. Run latex thrice on this file xint.tex then -%% run dvipdfmx on xint.dvi to produce the documentation xint.pdf, with -%% source code included. (ignore the dvipdfmx warnings) -%% -%% Customize as desired the class options and the two toggles below. -%% -%% See xint.dtx for the copyright and the conditions for distribution -%% and/or modification of this work. -%% -\NeedsTeXFormat{LaTeX2e} -\ProvidesFile{xint.tex}% -[\xintdate\space v\xintversion\space driver file for xint documentation (jfB)]% -\PassOptionsToClass{a4paper,fontsize=11pt}{scrdoc} -\chardef\Withdvipdfmx 1 % replace 1<space> by 0<space> for using latex/pdflatex -\chardef\NoSourceCode 0 % replace 0<space> by 1<space> for no source code -\input xint.dtx -%%% Local Variables: -%%% mode: latex -%%% End: -%</drv>---------------------------------------------------------------------- -%<*ins>------------------------------------------------------------------------- -%% This is a generated file. -%% "tex xint.ins" extracts from xint.dtx: -%% xinttools.sty, xint.sty, xintfrac.sty, xintexpr.sty, xintbinhex.sty, -%% xintgcd.sty, xintseries.sty and xintcfrac.sty as well as xint.tex -%% (for typesetting the documentation). -%% -%% See xint.dtx for the copyright and the conditions for distribution -%% and/or modification of this work. -%% -\input docstrip.tex -\askforoverwritefalse -\generate{\nopreamble -\file{xint.tex}{\from{xint.dtx}{drv}} -\usepreamble\defaultpreamble -\file{xinttools.sty}{\from{xint.dtx}{xinttools}} -\file{xint.sty}{\from{xint.dtx}{xint}} -\file{xintbinhex.sty}{\from{xint.dtx}{xintbinhex}} -\file{xintgcd.sty}{\from{xint.dtx}{xintgcd}} -\file{xintfrac.sty}{\from{xint.dtx}{xintfrac}} -\file{xintseries.sty}{\from{xint.dtx}{xintseries}} -\file{xintcfrac.sty}{\from{xint.dtx}{xintcfrac}} -\file{xintexpr.sty}{\from{xint.dtx}{xintexpr}}} -\catcode32=13\relax% active space -\let =\space% -\Msg{************************************************************************} -\Msg{*} -\Msg{* To finish the installation you have to move the following} -\Msg{* files into a directory searched by TeX:} -\Msg{*} -\Msg{* xinttools.sty} -\Msg{* xint.sty} -\Msg{* xintbinhex.sty} -\Msg{* xintgcd.sty} -\Msg{* xintfrac.sty} -\Msg{* xintseries.sty} -\Msg{* xintcfrac.sty} -\Msg{* xintexpr.sty} -\Msg{*} -\Msg{* To produce the documentation run latex thrice on file xint.tex} -\Msg{* and then run dvipdfmx on file xint.dvi (ignore dvipdfmx warnings)} -\Msg{*} -\Msg{* Happy TeXing!} -\Msg{*} -\Msg{************************************************************************} -\endbatchfile -%</ins>------------------------------------------------------------------------- -%<*dtx> -\fi % end of \iffalse block -\def\striptimestamp #1 <#2 #3 #4>{#2 at #3 #4} -\def\getdocdate #1 <#2-#3-#4 #5>{#4/#3/#2} -\edef\docdate{\expandafter\getdocdate\lasttimestamp} -\edef\dtxtimestamp{\expandafter\striptimestamp\lasttimestamp} -\chardef\noetex 0 -\expandafter\ifx\csname numexpr\endcsname\relax \chardef\noetex 1 \fi -\ifnum\noetex=1 \chardef\extractfiles 0 % extract files, then stop -\else - \expandafter\ifx\csname ProvidesFile\endcsname\relax - \chardef\extractfiles 0 % etex etc.. on xint.dtx - \else % latex/pdflatex on xint.tex or on xint.dtx - \expandafter\ifx\csname Withdvipdfmx\endcsname\relax - % latex run is on etoc.dtx, we will extract all files - \chardef\extractfiles 1 % 1 = extract all and typeset doc - \chardef\Withdvipdfmx 0 % 0 = pdflatex or latex+dvips - \chardef\NoSourceCode 1 % - \NeedsTeXFormat{LaTeX2e}% - \PassOptionsToClass{a4paper,11pt}{scrdoc}% - \else % latex run is on etoc.tex, - \chardef\extractfiles 2 % no extractions - \fi - \ProvidesFile{xint.dtx}% - [bundle source (\xintversion, \xintdate) and documentation (\docdate)]% - \fi -\fi -\ifnum\extractfiles<2 % extract files -\def\MessageDeFin{\newlinechar10 \let\Msg\message -\Msg{^^J}% -\Msg{********************************************************************^^J}% -\Msg{*^^J}% -\Msg{* To finish the installation you have to move the following^^J}% -\Msg{* files into a directory searched by TeX:^^J}% -\Msg{*^^J}% -\Msg{*\space\space\space\space xinttools.sty^^J}% -\Msg{*\space\space\space\space xint.sty^^J}% -\Msg{*\space\space\space\space xintbinhex.sty^^J}% -\Msg{*\space\space\space\space xintgcd.sty^^J}% -\Msg{*\space\space\space\space xintfrac.sty^^J}% -\Msg{*\space\space\space\space xintseries.sty^^J}% -\Msg{*\space\space\space\space xintcfrac.sty^^J}% -\Msg{*\space\space\space\space xintexpr.sty^^J}% -\Msg{*^^J}% -\Msg{* To produce the documentation with source code included run latex^^J}% -\Msg{* thrice on file xint.tex and then dvipdfmx on xint.dvi^^J}% -\Msg{* \space\space\space\space(ignore the dvipdfmx warnings)^^J}% -\Msg{*^^J}% -\Msg{* Happy TeXing!^^J}% -\Msg{*^^J}% -\Msg{********************************************************************^^J}% -}% -\begingroup - \input docstrip.tex - \askforoverwritefalse - \generate{\nopreamble - \file{xint.ins}{\from{xint.dtx}{ins}} - \file{xint.tex}{\from{xint.dtx}{drv}} - \usepreamble\defaultpreamble - \file{xinttools.sty}{\from{xint.dtx}{xinttools}} - \file{xint.sty}{\from{xint.dtx}{xint}} - \file{xintbinhex.sty}{\from{xint.dtx}{xintbinhex}} - \file{xintgcd.sty}{\from{xint.dtx}{xintgcd}} - \file{xintfrac.sty}{\from{xint.dtx}{xintfrac}} - \file{xintseries.sty}{\from{xint.dtx}{xintseries}} - \file{xintcfrac.sty}{\from{xint.dtx}{xintcfrac}} - \file{xintexpr.sty}{\from{xint.dtx}{xintexpr}}} -\endgroup -\fi % end of file extraction -\ifnum\extractfiles=0 -% direct tex/etex/xetex/etc on xint.dtx, files now extracted, stop - \MessageDeFin\expandafter\end -\fi -% no use of docstrip to extract files if latex compilation was on etoc.tex -\ifdefined\MessageDeFin\AtEndDocument{\MessageDeFin}\fi -%------------------------------------------------------------------------------- -\documentclass {scrdoc} -\ifnum\NoSourceCode=1 \OnlyDescription\fi -\makeatletter -\ifnum\Withdvipdfmx=1 - \@for\@tempa:=hyperref,bookmark,graphicx,xcolor\do - {\PassOptionsToPackage{dvipdfmx}\@tempa} - % - \PassOptionsToPackage{dvipdfm}{geometry} - \PassOptionsToPackage{bookmarks=true}{hyperref} - \PassOptionsToPackage{dvipdfmx-outline-open}{hyperref} - \PassOptionsToPackage{dvipdfmx-outline-open}{bookmark} - % - \def\pgfsysdriver{pgfsys-dvipdfm.def} -\else - \PassOptionsToPackage{bookmarks=true}{hyperref} -\fi -\makeatother - -\pagestyle{headings} -\makeatletter -% January 4, 2014 -% took me a while to pinpoint yesterday evening the origin of the problem, if -% only I had visited -% http://www.komascript.de/release3.12 immediately! -% -% as I subscribe to c.t.tex and d.c.t.tex I thought a problem with KOMA scrartcl -% would have been mentioned there, if as crippling as is this one, so I -% initially thought something related to TOCs had changed in KOMA and that etoc -% was now incompatible, and thus I started examining this, until finally -% understanding this had nothing to do with the TOC but originated in a -% buggy \sectionmark, revealed with pagestyle headings. -% -% This morning I see this is fixed in the experimental archive -% http://www.komascript.de/~mkohm/texlive-KOMA/archive/ and appears in the -% CHANGELOG as r1584. It is a bit hard for me to understand why such a typo with -% big consequences is not yet fixed in the CTAN distributed version. I did waste -% 90 minutes on that, at a time I was concentrating on xint things. Bugs are -% unavoidable, especially typos like this originating from modifying earlier -% code, but this tiny typo is severely annoying to users (*) and in my humble -% opinion a CTAN update should have been done sooner. Ok, this was a -% turn-of-year time... -% -% (*) compiling old documents is broken, and one sometimes does not want to -% modify the source files. -% -\def\buggysectionmark #1{% KOMA 3.12 as released to CTAN December 2013 - \if@twoside\expandafter\markboth\else\expandafter\markright\fi - {\MakeMarkcase{\ifnumbered{section}{\sectionmarkformat\fi}{}#1}}{}} -\ifx\buggysectionmark\sectionmark -\def\sectionmark #1{% - \if@twoside\expandafter\markboth\else\expandafter\markright\fi - {\MakeMarkcase{\ifnumbered{section}{\sectionmarkformat}{}#1}}{}} -\fi -\makeatother - -\usepackage[T1]{fontenc} -\usepackage[latin1]{inputenc} - -%\usepackage{array} -\usepackage{multicol} - -%---- GEOMETRY WILL BE CHANGED FOR SOURCE CODE SECTIONS -\usepackage[hscale=0.66,vscale=0.75]{geometry} - -\usepackage{xintexpr} - -\usepackage{xintbinhex} -\usepackage{xintgcd} -\usepackage{xintseries} -\usepackage{xintcfrac} - -\usepackage{amsmath} % for \cfrac in the documentation -\usepackage{varioref} - -\usepackage{etoolbox} - -\usepackage{etoc}[2013/10/16] % I need \etocdepthtag.toc - -%---- USE OF ETOC FOR THE TABLES OF CONTENTS - -\def\gobbletodot #1.{} -\makeatletter -\let\savedsectionline\l@section -\makeatother - -\def\sectioncouleur{{cyan}} - -% attention à ce 22 hard codé. 23 maintenant,... 24; et 31 non 32... -% et ça continue de changer - -\etocsetstyle{section}{} - {} - {\ifnum\etocthenumber=23 \gdef\sectioncouleur{{joli}}\fi - \ifnum\etocthenumber=31 \gdef\sectioncouleur{[named]{RoyalPurple}}\fi - \savedsectionline{\numberline{\expandafter\textcolor\sectioncouleur - {\etocnumber}}\etocname} - {{\mdseries\etocpage}}% - }% cf l@section en classe scrartcl - {}% - -\def\MARGEPAGENO {1.5em} -\etocsetstyle{subsection} - {\begingroup - \setlength{\premulticols}{0pt} - \setlength{\multicolsep}{0pt} - \setlength{\columnsep}{1em} - \setlength{\columnseprule}{.4pt} - \raggedcolumns % only added for 1.08a, I should have done it long time ago! - \begin{multicols}{2} - \leftskip 2.3em - \rightskip \MARGEPAGENO plus 2em minus 1em % 18 octobre 2013 - \parfillskip -\MARGEPAGENO\relax - } - {} - {\noindent - \llap{\makebox[2.3em][l] - {\ttfamily\bfseries\etoclink - {.\expandafter\gobbletodot\etocthenumber}}}% - \strut - \etocname\nobreak\leaders\etoctoclineleaders\hfill\nobreak - \strut\makebox[1.5em][r]{\normalfont\small\etocpage}\endgraf } - {\end{multicols}\endgroup }% - -\makeatother - -\addtocontents{toc}{\protect\hypersetup{hidelinks}} -% je rends le @ actif... après begin document... (donc ok pour aux) -\addtocontents{toc}{\protect\makeatother} - -%--- TXFONTS: TXTT WILL BE MADE SMALLER AND WILL ALLOW HYPHENATION -\usepackage{txfonts} -\usepackage{pifont} - -% malheureusement, comme j'utilise des diacritiques dans mes -% parties commentées, imprimées verbatim, je ne pourrai pas -% utiliser dvipdfmx qui a un problème avec txtt - -\DeclareFontFamily{T1}{txtt}{} -\DeclareFontShape{T1}{txtt}{m}{n}{ %medium - <->s*[.96] t1xtt% -}{} -\DeclareFontShape{T1}{txtt}{m}{sc}{ %cap & small cap - <->s*[.96] t1xttsc% -}{} -\DeclareFontShape{T1}{txtt}{m}{sl}{ %slanted - <->s*[.96] t1xttsl% -}{} -\DeclareFontShape{T1}{txtt}{m}{it}{ %italic - <->ssub * txtt/m/sl% -}{} -\DeclareFontShape{T1}{txtt}{m}{ui}{ %unslanted italic - <->ssub * txtt/m/sl% -}{} -\DeclareFontShape{T1}{txtt}{bx}{n}{ %bold extended - <->t1xbtt% -}{} -\DeclareFontShape{T1}{txtt}{bx}{sc}{ %bold extended cap & small cap - <->t1xbttsc% -}{} -\DeclareFontShape{T1}{txtt}{bx}{sl}{ %bold extended slanted - <->t1xbttsl% -}{} -\DeclareFontShape{T1}{txtt}{bx}{it}{ %bold extended italic - <->ssub * txtt/bx/sl% -}{} -\DeclareFontShape{T1}{txtt}{bx}{ui}{ %bold extended unslanted italic - <->ssub * txtt/bx/sl% -}{} -\DeclareFontShape{T1}{txtt}{b}{n}{ %bold - <->ssub * txtt/bx/n% -}{} -\DeclareFontShape{T1}{txtt}{b}{sc}{ %bold cap & small cap - <->ssub * txtt/bx/sc% -}{} -\DeclareFontShape{T1}{txtt}{b}{sl}{ %bold slanted - <->ssub * txtt/bx/sl% -}{} -\DeclareFontShape{T1}{txtt}{b}{it}{ %bold italic - <->ssub * txtt/bx/it% -}{} -\DeclareFontShape{T1}{txtt}{b}{ui}{ %bold unslanted italic - <->ssub * txtt/bx/ui% -}{} - -\def\digitstt {\bgroup\fontfamily {lmtt}\selectfont\let\next=} - -\usepackage{xspace} -%\usepackage[dvipsnames]{color} -\usepackage[dvipsnames]{xcolor} -\usepackage{framed} - -\definecolor{joli}{RGB}{225,95,0} -\definecolor{JOLI}{RGB}{225,95,0} -\definecolor{BLUE}{RGB}{0,0,255} -\definecolor{niceone}{RGB}{38,128,192} - -% for the quick sort algorithm illustration -\definecolor{LEFT}{RGB}{216,195,88} -\definecolor{RIGHT}{RGB}{208,231,153} -\definecolor{INERT}{RGB}{199,200,194} -\definecolor{PIVOT}{RGB}{109,8,57} - -\usepackage[para]{footmisc} - -\usepackage[english]{babel} -\usepackage[autolanguage,np]{numprint} -\AtBeginDocument{\npthousandsep{,\hskip .5pt plus .1pt minus .1pt}} - - -\usepackage[pdfencoding=pdfdoc]{hyperref} -\hypersetup{% -linktoc=all,% -breaklinks=true,% -colorlinks=true,% -urlcolor=niceone,% -linkcolor=blue,% -pdfauthor={Jean-Fran\c cois Burnol},% -pdftitle={The xint bundle},% -pdfsubject={Arithmetic with TeX},% -pdfkeywords={Expansion, arithmetic, TeX},% -pdfstartview=FitH,% -pdfpagemode=UseOutlines} -\usepackage{bookmark} - -\usepackage{picture} % permet d'utiliser des unités dans les dimensions de la - % picture et dans \put -\usepackage{graphicx} -\usepackage{eso-pic} - - -%---- \MyMarginNote: a simple macro for some margin notes with no fuss -% je m'aperçois que je peux l'utiliser dans les footnotes... -\makeatletter -\def\MyMarginNote {\@ifnextchar[\@MyMarginNote{\@MyMarginNote[]}}% -% 18 janvier 2014, j'ai besoin d'un raccourci. -\let\inmarg\MyMarginNote -\def\@MyMarginNote [#1]#2{% - \vadjust{\vskip-\dp\strutbox - \smash{\hbox to 0pt - {\color[named]{PineGreen}\normalfont\small - \hsize 1.5cm\rightskip.5cm minus.5cm - \hss\vtop{\noindent #2}\ $\to$#1\ }}% - \vskip\dp\strutbox }\strut{}} -\def\MyMarginNoteWithBrace #1{% - \vadjust{\vskip-\dp\strutbox - \smash{\hbox to 0pt - {\color[named]{PineGreen}\normalfont\small - \hss #1\ $\Bigg\{$\ }}% - \vskip\dp\strutbox }\strut{}} -\def\IMPORTANT {\MyMarginNoteWithBrace {IMPORTANT!}} -% 26 novembre 2013: -\def\etype #1{% - \vadjust{\vskip-\dp\strutbox - \smash{\hbox to 0pt {\hss\color[named]{PineGreen}% - \itshape \xintListWithSep{\,}{#1}\ $\star$\quad }}% - \vskip\dp\strutbox }\strut{}} -\def\retype #1{% - \vadjust{\vskip-\dp\strutbox - \smash{\hbox to 0pt {\hss\color[named]{PineGreen}% - \itshape \xintListWithSep{\,}{#1}\ \ding{73}\quad }}% - \vskip\dp\strutbox }\strut{}} -\def\ntype #1{% - \vadjust{\vskip-\dp\strutbox - \smash{\hbox to 0pt {\hss\color[named]{PineGreen}% - \itshape \xintListWithSep{\,}{#1}\quad }}% - \vskip\dp\strutbox }\strut{}} -\def\Numf {{\vbox{\halign{\hfil##\hfil\cr \footnotesize - \upshape Num\cr - \noalign{\hrule height 0pt \vskip1pt\relax} - \itshape f\cr}}}} -\def\Ff {{\vbox{\halign{\hfil##\hfil\cr \footnotesize - \upshape Frac\cr - \noalign{\hrule height 0pt \vskip1pt\relax} - \itshape f\cr}}}} -\def\numx {{\vbox{\halign{\hfil##\hfil\cr \footnotesize - \upshape num\cr - \noalign{\hrule height 0pt \vskip1pt\relax} - \itshape x\cr}}}} -\makeatother - -%---- \centeredline: OUR OWN LITTLE MACRO FOR CENTERING LINES - -% 7 mars 2013 -% This macro allows to conveniently center a line inside a paragraph and still -% use therein \verb or other commands changing catcodes. -% A proposito, the \LaTeX \centerline uses \hsize and not \linewidth ! -% (which in my humble opinion is bad) - -% \ignorespaces ajouté le 9 juin. - -\makeatletter -\newcommand*\centeredline {% - \ifhmode \\\relax - \def\centeredline@{\hss\egroup\hskip\z@skip\ignorespaces }% - \else - \def\centeredline@{\hss\egroup }% - \fi - \afterassignment\@centeredline - \let\next=} -\def\@centeredline - {\hbox to \linewidth \bgroup \hss \bgroup \aftergroup\centeredline@ } -\makeatother - -%---- MODIFIED \verb, and verbatim like `environments' FITS BETTER OUR USE OF IT -% le \verb de doc.sty est très chiant car il a retiré \verbatim@font pour mettre -% un \ttfamily hard-coded à la place. [en fin de compte j'utilise dorénavant le -% vocable \MicroFont plutôt que \verbatim@font] -% -% à propos \do@noligs: -% macro:#1->\catcode `#1\active \begingroup \lccode `\~=`#1\relax \lowercase -% {\endgroup \def ~{\leavevmode \kern \z@ \char `#1}} -% ne manque-t-il pas un espace après le \char `#1? En effet! ça me pose des -% problèmes lorsque l'espace a catcode 10!! Ils ont voulu optimiser et gagner -% un token mais du coup ça en limite l'employabilité. -% -\def\MicroFont {\ttfamily\hyphenchar\font45 } -\def\MacroFont {\ttfamily\baselineskip12pt\relax} -\makeatletter - -% \makestarlowast ajouté le 8 juin 2013 - -% 18 octobre 2013, hyphénation dans les blocs verbatim -\def\dobackslash -{% - \catcode92 \active - \begingroup \lccode `\~=92\relax - \lowercase {\endgroup \def ~{\hskip \z@\@plus.1pt\@minus.1pt \char 92 }}% -}% -\def\dobraces -{% - \catcode123 \active - \begingroup \lccode `\~=123\relax - \lowercase {\endgroup \def ~{\hskip \z@\@plus.1pt\@minus.1pt - \char 123 }}% - \catcode125 \active - \begingroup \lccode `\~=125\relax - \lowercase {\endgroup \def ~{\char 125 \hskip \z@\@plus.1pt\@minus.1pt }}% -}% -% modif de \do@noligs: \char`#1} --> \char`#1 } -\def\do@noligs #1% -{% - \catcode `#1\active - \begingroup \lccode `\~=`#1\relax - \lowercase {\endgroup \def ~{\leavevmode \kern \z@ \char `#1 }}% -}% -% *** \verb utilise \MicroFont -\def\verb -{% - \relax \ifmmode\hbox\else\leavevmode\null\fi - \bgroup \MicroFont - \let\do\do@noligs \verbatim@nolig@list - \let\do\@makeother \dospecials \catcode32 10 - \dobackslash - \dobraces - \makestarlowast \@jfverb -}% -% -\long\def\lverb % pour utilisation dans la partie implémentation -% *** \lverb utilise \MacroFont (comme \verbatim) -{% - \relax\par\smallskip\noindent\null - \begingroup - \let\par\@@par\hbadness 100 \hfuzz 100pt\relax - \hsize .85\hsize - \MacroFont - \bgroup - \aftergroup\@@par \aftergroup\endgroup \aftergroup\medskip - \let\do\do@noligs \verbatim@nolig@list - \let\do\@makeother \dospecials - \catcode32 10 \catcode`\% 9 \catcode`\& 14 \catcode`\$ 0 - \@jfverb -} -% et voilà. Comme quoi, on peut aussi faire sans \trivlist si on veut. -% Voir aussi la re-définition de \MacroFont au moment du \StopEventually -% -% *** \dverb utilise \MacroFont (comme \verbatim) -% -% J'ai parfois besoin d'un caractère de contrôle, j'avais dans les premières -% versions de cette doc utilisé & ou $ mais ceci est devenu très peu commode -% lorsque j'ai commencé à insérer des tabular. Finalement j'ai fait sans, mais -% je prends aujourd'hui " qui par miracle est compatible aux emplois de \dverb -% dans la doc, et va me permettre par exemple d'en colorier des parties, via -% méthode sioux pour disposer des { et } temporairement. -% -\long\def\dverb % pour utilisation dans le manuel de l'utilisateur -{% - \relax\par\smallskip - \bgroup - \parindent0pt - \def\par{\@@par\leavevmode\null}% - \let\do\do@noligs \verbatim@nolig@list - \let\do\@makeother \dospecials - \def\"{\begingroup\catcode123 1 \catcode 125 2 \dverbescape}% - \catcode`\@ 14 \catcode`\" 0 \makestarlowast - \MacroFont \obeylines \@vobeyspaces - \@jfverb -} -\def\dverbescape #1;!{#1\endgroup } - -\def\@jfverb #1{\catcode`#1\active - \lccode`\~`#1\lowercase{\let~\egroup}}% -\makeatother - -\catcode`\_=11 - -\def\csa_aux #1{\ttfamily\hyphenchar\font45 \char`\\% - \scantokens{#1}\endgroup } -\def\csb_aux #1{\hyperref[\detokenize{xint#1}]{\ttfamily - \hyphenchar\font45 \char`\\\mbox{xint}\-% - \scantokens{#1}}\endgroup } - -\DeclareRobustCommand\csa {\begingroup\catcode`\_=11 - \everyeof{\noexpand}\endlinechar -1 - \makeatother - \makestarlowast - \csa_aux } -\DeclareRobustCommand\csbnolk {\begingroup\catcode`\_=11 - \everyeof{\noexpand}\endlinechar -1 - \makestarlowast - \makeatother - \color{blue}% - \csa_aux } -\DeclareRobustCommand\csbxint {\begingroup\catcode`\_=11 - \everyeof{\noexpand}\endlinechar -1 - \makestarlowast - \makeatother - \csb_aux } -\catcode`\_=8 - -\newcommand\csh[1]{\texorpdfstring{\csa{#1}}{\textbackslash #1}} -\newcommand\csbh[1]{\texorpdfstring{\csbnolk{#1}}{\textbackslash #1}} - -% emploi de \xintFor à partir de 1.09c -% There were some color leaks in 1.09i from dvipdfmx (not pdflatex) compilation, -% due to missing braces around use of \color, I have now added them. -\xintForpair #1#2 in -{(xinttools,tools),(xint,xint),(xintbinhex,binhex),(xintgcd,gcd),% - (xintfrac,frac),(xintseries,series),(xintcfrac,cfrac),(xintexpr,expr)} -\do -{% - \expandafter\def\csname #1name\endcsname - {\texorpdfstring - {\hyperref[sec:#2]% - {{\color{joli}\bfseries\ttfamily\hyphenchar\font45 #1}}} - {#1}% - \xspace }% - \expandafter\def\csname #1nameimp\endcsname - {\texorpdfstring - {\hyperref[sec:#2imp]% - {{\color[named]{RoyalPurple}% - \bfseries\ttfamily\hyphenchar\font45 #1}}} - {#1}% - \xspace }% -}% - -\frenchspacing -\renewcommand\familydefault\sfdefault - -%---- QUICK WAY TO PRINT LONG THINGS, IN PARTICULAR, BUT NOT EXCLUSIVELY, LONG -% NUMBERS -\def\allowsplits #1% -{% - \ifx #1\relax \else #1\hskip 0pt plus 1pt\relax - \expandafter\allowsplits\fi -}% -\def\printnumber #1% first ``fully'' expands its argument. -{\expandafter\allowsplits \romannumeral-`0#1\relax }% - - -%--- counts used in particular in the samples from the documentation of the -% xintseries.sty package -\newcount\cnta -\newcount\cntb -\newcount\cntc - -%--- printing (systematically) * in a lowered position in the various verbatim -% blocks using txtt. - -\def\lowast{\raisebox{-.25\height}{*}} -\begingroup - \catcode`* 13 - \gdef\makestarlowast {\let*\lowast\catcode`\*\active}% -\endgroup - -% 22 octobre 2013 -\newcommand\fexpan {\textit{f}-expan} - -% December 7, 2013. Expandably computing a big Fibonacci number -% with the help of TeX+\numexpr+\xintexpr, (c) Jean-François Burnol -\catcode`_ 11 -% -% ajouté 7 janvier 2014 au xint.dtx pour 1.07j. -% -% Le 17 janvier je me décide de simplifier l'algorithme car l'original ne tenait -% pas compte de la relation toujours vraie A=B+C dans les matrices symétriques -% utilisées en sous-main [[A,B],[B,C]]. -% -% la version ici est celle avec les * omis: car multiplication tacite devant les -% sous-expressions depuis 1.09j, et aussi devant les parenthèses depuis 1.09k. -% (pour tester) -\def\Fibonacci #1{% - \expandafter\Fibonacci_a\expandafter - {\the\numexpr #1\expandafter}\expandafter - {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval 0\relax}} -% -\def\Fibonacci_a #1{% - \ifcase #1 - \expandafter\Fibonacci_end_i - \or - \expandafter\Fibonacci_end_ii - \else - \ifodd #1 - \expandafter\expandafter\expandafter\Fibonacci_b_ii - \else - \expandafter\expandafter\expandafter\Fibonacci_b_i - \fi - \fi {#1}% -}% -\def\Fibonacci_b_i #1#2#3{\expandafter\Fibonacci_a\expandafter - {\the\numexpr #1/2\expandafter}\expandafter - {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval (2#2-#3)#3\relax}% -}% end of Fibonacci_b_i -\def\Fibonacci_b_ii #1#2#3#4#5{\expandafter\Fibonacci_a\expandafter - {\the\numexpr (#1-1)/2\expandafter}\expandafter - {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval (2#2-#3)#3\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval #2#4+#3#5\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval #2#5+#3(#4-#5)\relax}% -}% end of Fibonacci_b_ii -\def\Fibonacci_end_i #1#2#3#4#5{\xintthe#5} -\def\Fibonacci_end_ii #1#2#3#4#5{\xinttheiiexpr #2#5+#3(#4-#5)\relax} -\catcode`_ 8 - -\def\Fibo #1.{\Fibonacci {#1}} - -\begin{document}\thispagestyle{empty}\rmfamily -\pdfbookmark[1]{Title page}{TOP} -\makeatletter - -\begingroup\lccode`\~=`@ -\lowercase{\endgroup\def~}{\begingroup\fontfamily{lmtt}\selectfont - \let\do\@makeother\dospecials - \catcode`\@ \active - \jfendshrtverb } -\catcode`\@ \active -\def\jfendshrtverb #1@{#1\endgroup } - -% nice background added for 1.09j release, January 7, 2014. -% superbe, non? moi très content! -% bon je peaufine ce background le 17 janvier, c'est hard-coded mais je ne veux -% pas y passer plus de temps (ce qui est amusant c'est que j'ai constaté a -% posteriori qu'il y a 17 chiffres par lignes donc 1 chiffre avec son padding = -% 1cm... -% *\message{\xinttheexpr round(\dimexpr 8cm\relax/17,3)\relax} -% 877496.353 -\def\specialprintone #1% -{% - \ifx #1\relax \else \makebox[877496sp]{#1}\hskip 0pt plus 2sp\relax - \expandafter\specialprintone\fi -}% -\def\specialprintnumber #1% first ``fully'' expands its argument. -{\expandafter\specialprintone \romannumeral-`0#1\relax }% - -\AddToShipoutPicture*{% - \put(10.5cm,14.85cm) - {\makebox(0,0) - {\resizebox{17cm}{!}{\vbox - {\hsize 8cm\Huge\baselineskip.8\baselineskip\color{black!10}% - \digitstt{\specialprintnumber{F(1250)=}% - \specialprintnumber{\Fibonacci{1250}}}\par}}% - } - }% -} - -% Octobre 2013: le & est devenu assez pénible puisque j'en ai besoin dans mes -% exemples de \xintApplyInline/\xintFor avec des tabulars. Donc je décide -% (après avoir temporairement fait des choses un peu lourdes avec \lverb) de -% le remplacer par @ car il n'y en a quasi pas dans la partie user manual; -% idem pour \dverb. Cependant je dois faire attention avec un @ actif par -% exemple dans les tables de matières. Bon on va voir. - -{\normalfont\Large\parindent0pt \parfillskip 0pt\relax - \leftskip 2cm plus 1fil \rightskip 2cm plus 1fil - The \xintname bundle\par}% -{\centering - \textsc{Jean-François Burnol}\par - \footnotesize \ttfamily - jfbu (at) free (dot) fr\par - Package version: \xintversion\ (\xintdate)% - \let\thefootnote\empty - \footnote{Documentation generated from the - source file with timestamp ``\dtxtimestamp''.}\par -} -\setcounter{footnote}{0} - -\bigskip - -% comme \dverb ne fait pas un \par à la fin, il y a un problème avec le -% \baselineskip si on ne le spécifie pas en plus; il faudra que je voie si -% vraiment j'utilise \dverb sans terminer un paragraphe, il doit y avoir au plus -% quelque cas. -\begingroup\footnotesize\def\MacroFont {\ttfamily\baselineskip10pt\relax} -\baselineskip 10pt -\dverb|@ -\input xintexpr.sty -% December 7, 2013. Expandably computing a big Fibonacci number -% using TeX+\numexpr+\xintexpr, (c) Jean-François Burnol -% January 17, 2014: algorithm modified to be more economical in computations. - -\catcode`_ 11 - -\def\Fibonacci #1{% - \expandafter\Fibonacci_a\expandafter - {\the\numexpr #1\expandafter}\expandafter - {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval 0\relax}} - -\def\Fibonacci_a #1{% - \ifcase #1 - \expandafter\Fibonacci_end_i - \or - \expandafter\Fibonacci_end_ii - \else - \ifodd #1 - \expandafter\expandafter\expandafter\Fibonacci_b_ii - \else - \expandafter\expandafter\expandafter\Fibonacci_b_i - \fi - \fi {#1}% -}% - -\def\Fibonacci_b_i #1#2#3{\expandafter\Fibonacci_a\expandafter - {\the\numexpr #1/2\expandafter}\expandafter - {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval (2*#2-#3)*#3\relax}% -}% end of Fibonacci_b_i - -\def\Fibonacci_b_ii #1#2#3#4#5{\expandafter\Fibonacci_a\expandafter - {\the\numexpr (#1-1)/2\expandafter}\expandafter - {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval (2*#2-#3)*#3\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval #2*#4+#3*#5\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval #2*#5+#3*(#4-#5)\relax}% -}% end of Fibonacci_b_ii - -\def\Fibonacci_end_i #1#2#3#4#5{\xintthe#5} -\def\Fibonacci_end_ii #1#2#3#4#5{\xinttheiiexpr #2*#5+#3*(#4-#5)\relax} - -\catcode`_ 8 - -% This \Fibonacci macro is designed to compute *one* Fibonacci number, not a -% whole sequence of them. Let's reap the fruits of our work: - -\message{F(1250)=\Fibonacci {1250}} -\bye |\ttfamily\% see \autoref{ssec:fibonacci} for some explanations and -more.\par -\endgroup - -\clearpage - -% \pagebreak[3] - -\pdfbookmark[1]{Abstract}{ABSTRACT} - -\begin{addmargin}{1cm}\footnotesize - \begin{center} \bfseries\large Description of the packages\par\smallskip - \end{center}\medskip -\makeatletter -\renewenvironment{description} - {\list{}{\topsep\z@ \parsep\z@ \labelwidth\z@ \itemindent-\leftmargin - \let\makelabel\descriptionlabel}} - {\endlist} -\makeatother -\begin{description} -\item[\xinttoolsname] is loaded by \xintname (hence by all other packages of the - bundle, too): it provides utilities of independent interest such as expandable - and non-expandable loops. - -\item[\xintname] implements with expandable \TeX{} macros additions, - subtractions, multiplications, divisions and powers with arbitrarily long - numbers. - -\item[\xintfracname] extends the scope of \xintname to decimal numbers, to - numbers in scientific notation and also to fractions with arbitrarily - long such numerators and denominators separated by a forward slash. - -\item[\xintexprname] extends \xintfracname with an expandable parser |\xintexpr - . . . \relax| of expressions involving arithmetic operations in infix notation - on decimal numbers, fractions, numbers in scientific notation, with - parentheses, factorial symbol, function names, comparison operators, logic - operators, twofold and threefold way conditionals, sub-expressions, macros - expanding to the previous items. -\end{description} - -\noindent Further modules: -% -\begin{description} -\item[\xintbinhexname] is for conversions to and from binary and - hexadecimal bases. - -\item[\xintseriesname] provides some basic functionality for computing in an - expandable manner partial sums of series and power series with fractional - coefficients. - -\item[\xintgcdname] implements the Euclidean algorithm and its typesetting. - -\item[\xintcfracname] deals with the computation of continued fractions. -\end{description} - - Most macros, and all of those doing computations, work purely by expansion - without assignments, and may thus be used almost everywhere in \TeX{}. - - The packages may be used with any flavor of \TeX{} supporting the \eTeX{} - extensions. \LaTeX{} users will use |\usepackage| and others |\input| to - load the package components. - -\end{addmargin} - -\bigskip - -% \clearpage -% 18 octobre 2013, je remets la TOC ici. - -% je ne veux pas non plus que la main toc se liste elle-même donc je passe pour -% elle aussi à \section* - -\etocsetlevel{toctobookmark}{6} % 9 octobre 2013, je fais des petits tricks. - -% 18 novembre 2013, je n'inclus plus la TOC détaillée de xintexpr. Je -% reconfigure la TOC. - -\etocsettocdepth {subsection} - -\renewcommand*{\etocbelowtocskip}{0pt} -\renewcommand*{\etocinnertopsep}{0pt} -\renewcommand*{\etoctoclineleaders} - {\hbox{\normalfont\normalsize\hbox to 1ex {\hss.\hss}}} -\etocmulticolstyle [2]{% - \phantomsection\section* {Contents} - \etoctoccontentsline*{toctobookmark}{Contents}{1}% -} - - \etocsettagdepth {description}{section} - \etocsettagdepth {commandsA} {none} - \etocsettagdepth {xintexpr} {none} - \etocsettagdepth {commandsB} {none} - \etocsettagdepth {implementation}{none} -\tableofcontents -\etocmulticolstyle [2]{\raggedcolumns}{} - \etocsettagdepth {description}{none} - \etocsettagdepth {commandsA} {section} - \etocsettagdepth {xintexpr} {section} - \etocsettagdepth {commandsB} {section} - \etocsettagdepth {implementation}{section} -\tableofcontents -\medskip - -% pour la suite: -\etocignoredepthtags -\etocmulticolstyle [1]{% - \phantomsection\section* {Contents} - \etoctoccontentsline*{toctobookmark}{Contents}{2}% -} - -\etocdepthtag.toc {description} - -% \pdfbookmark[1]{Snapshot}{SNAPSHOT} - -\section{Read me first}\label{sec:quickintro} - -This section provides recommended reading on first discovering the package; -complete details are given later in the manual. - -{\etocdefaultlines\etocsettocstyle{}{}\localtableofcontents} - -\subsection{Presentation of the package} - -The components of the \xintname bundle provide macros dedicated to -\emph{expandable} computations on numbers exceeding the \TeX{} (and \eTeX{}) -limit of \digitstt{\number"7FFFFFFF}. - -The \eTeX{} extensions must be enabled; this is the case in modern -distributions by default, except if \TeX{} is invoked under the name -|tex| in command line (|etex| should be used then, or |pdftex| in |DVI| -output mode). All components may be used as regular \LaTeX{} packages -or, with any other format based on \TeX{}, loaded directly via -\string\input{} (e.g. |\input -xint.sty\relax|). -% -% {\makeatother\footnote{\csa{empty}, \csa{space}, \csa{z@}, -% \csa{@ne}, and \csa{m@ne} should have the same meaning as in Plain and -% \LaTeX.}} -% -Each package automatically loads those not already loaded -it depends on. - -The \xintname bundle consists of the three principal components \xintname, -\xintfracname (which loads \xintname), and \xintexprname (which loads -\xintfracname), and four additional modules. The macros of the \xintname bundle -not dealing directly with the manipulation of big numbers belong to a package -\xinttoolsname (automatically loaded by all others), which is of independent -interest. - -\subsection{User interface} - -The user interface for executing operations on numbers is via macros such as -\csbxint{Add} or \csbxint{Mul} which have two arguments, or via expressions -\csbxint{expr}|..\relax| which use infix notations such as |+|, |-|, |*|, |/|, -and |^| for the basic operations, and recognize functions of one or more comma -separated arguments (such as |max|, or |round|, or |sqrt|), parentheses, logic -operators of conjunction |&|, disjunction \verb+|+, as well as two-way |?| and -three-way |:| conditionals and more. - -In the latter case the contents are expanded completely from left to right until -the ending |\relax| is found and swallowed, and spaces and even (to some extent) -catcodes do not matter. In the former (macro) case the arguments are each -subjected to the process of \fexpan sion: repeated expansion of the first token -until finding something unexpandable (or being stopped by a space token). - -Conversely this process of \fexpan sion always provokes the complete expansion -of the package macros and \csbxint{expr}|..\relax| also will expand completely -under \fexpan sion, but to a private format; the \csbxint{the} prefix allows the -computation result either to be passed as argument to one of the package -macros,\footnote{the \csa{xintthe} prefix \fexpan ds the \csa{xintexpr}-ession - then unlocks it from its private format; it should not be used for - sub-expressions inside a bigger one as its is more efficient for the - expression parser to keep the result in the private format.} or also end up on -the printed page (or in an auxiliary file). - -To recapitulate: all macros dealing with computations (1.)~\emph{expand - completely under the sole process of repeated expansion of the first token, - (and two expansions suffice)},\footnote{see in \autoref{sec:expansions} for - more details.} (2.)~\emph{apply this \fexpan sion to each one of their - arguments.} Hence they can be nested one within the other up to arbitrary -depths. Conditional evaluations either within the macro arguments themselves, or -with branches defined in terms of these macros are made possible via macros such -as as \csbxint{ifSgn} or \csbxint{ifCmp}. - -There is no notion of \emph{declaration of a variable} to \xintname, -\xintfracname, or \xintexprname. The user employs the |\def|, |\edef|, or -|\newcommand| (in \LaTeX) as usual, for example: -% -\centeredline{|\def\x{17} \def\y{35} \edef\z{\xintMul {\x}{\y}}|} -% -As a faster alternative to |\edef| (when hundreds of digits are involved), the -package provides |\oodef| which only expands twice its argument. - -The \xintexprname package has a private internal -representation for the evaluated computation result. With -% -\centeredline{|\oodef\z {\xintexpr 3.141^17\relax}|} -% -the macro |\z| is already fully evaluated (two expansions were applied, and this -is enough), and can be reused in other |\xintexpr|-essions, such as for example -% -\centeredline{|\xintexpr \z+1/\z\relax|} -% -But to print it, or to use it as argument to one of the package macros, -it must be prefixed by |\xintthe| (a synonym for |\xintthe\xintexpr| is -\csbxint{theexpr}). Application of this |\xintthe| prefix outputs the -value in the \xintfracname semi-private internal format -|A/B[N]|,\footnote{there is also the notion of \csbxint{floatexpr}, for - which the output format after the action of \csa{xintthe} is a number in - floating point scientific notation.} representing the fraction -$(A/B)\times 10^N$. The example above produces a somewhat large output: -\digitstt{\oodef\z {\xintexpr 3.141^17\relax}% - \printnumber {\xinttheexpr \z+1/\z\relax }} - - \begin{framed} - By default, computations done by the macros of \xintfracname or within - |\xintexpr..\relax| are exact. Inputs containing decimal points or - scientific parts do not make the package switch to a `floating-point' mode. - The inputs, however long, are converted into exact internal representations. -% - % Floating point evaluations are done via special macros containing - % `Float' in their names, or inside |\xintfloatexpr|-essions. - \end{framed} - -% -The |A/B[N]| shape is the output format of most \xintfracname macros, it -benefits from accelerated parsing when used on input, compared to the normal -user syntax which has no |[N]| part. An example of valid user input for a -fraction is -% -\centeredline{|-123.45602e78/+765.987e-123|} -% -where both the decimal parts, the scientific exponent parts, and the whole -denominator are optional components. The corresponding semi-private form in this -case would be -% -\centeredline{\digitstt{\xintRaw{-123.45602e78/+765.987e-123}}} -% -The optional forward slash |/| introducing a denominator is not an operation, -but a denomination for a fractional input. Reduction to the irreducible form -must be asked for explicitely via the \csbxint{Irr} macro or the |reduce| -function within |\xintexpr..\relax|. Elementary operations on fractions work -blindly (addition does not even check for equality of the denominators and -multiply them automatically) and do none of the simplifications which -could be obvious to (some) human beings. - - -\subsection{Space and time, floating point macros} - -The size of the manipulated numbers is limited by two -factors:\footnote{there is an intrinsic limit of - \digitstt{\number"7FFFFFFF} on the number of digits, but it is - irrelevant, in view of the other limiting factors.} (1.)~\emph{the - available memory as configured in the |tex| executable}, -(2.)~\emph{the \emph{time} necessary to fully expand the computations - themselves}. The most limiting factor is the second one, the time -needed (for multiplication and division, and even more for powers) -explodes with increasing input sizes long before the computations could -get limited by constraints on \TeX's available memory: -computations with @100@ digits are still reasonably fast, but the -situation then deteriorates swiftly, as it takes of the order of seconds (on my -laptop) for the package to multiply exactly two numbers each of @1000@ digits -and it would take hours for numbers each of @20000@ digits.\footnote{Perhaps - some faster routines could emerge from an approach which, while maintaining - expandability would renounce at \fexpan dability (without impacting the input - save stack). There is one such routine \csbxint{XTrunc} which is able to write - to a file (or inside an \csa{edef}) tens of thousands of digits of a - (reasonably-sized) fraction.} - -To address this issue, floating -point macros are provided to work with a given arbitrary precision. The default -size for significands is @16@ digits. Working with significands of @24@, @32@, -@48@, @64@, or even @80@ digits is well within the reach of the package. But -routine multiplications and divisions will become too slow if the precision goes -into the hundreds, although the syntax to set it (|\xintDigits:=P;|) allows -values up to @32767@.\footnote{for a one-shot conversion of a fraction to float - format, or one addition, a precision exceeding \digitstt{32767} may be passed - as optional argument to the used macro.} The exponents may be as big as -\digitstt{$\pm$\number"7FFFFFFF}.\footnote{almost\dots{} as inner manipulations - may either add or subtract the precision value to the exponent, arithmetic - overflow may occur if the exponents are a bit to close to the \TeX{} bound - \digitstt{$\pm$\number"7FFFFFFF}.} - -Here is such a floating point computation: \centeredline{|\xintFloatPower [48] - {1.1547}{\xintiiPow {2}{35}}|} which thus computes -$(1.1547)^{2^{35}}=(1.1547)^{\xintiiPow {2}{35}}$ to be approximately -\centeredline{\digitstt{\np{\xintFloatPower [48] {1.1547}{\xintiiPow - {2}{35}}}}} -% -Notice that @2^35@ exceeds \TeX's bound, but \csa{xintFloatPower} allows it, -what counts is the exponent of the result which, while dangerously close to -@2^31@ is not quite there yet. The printing of the result was done via the -|\numprint| command from the \href{http://ctan.org/pkg/numprint}{numprint} -package\footnote{\url{http://ctan.org/pkg/numprint}}. - -The same computation can be done via the non-expandable assignment -|\xintDigits:=48;| and then \centeredline{|\xintthefloatexpr - 1.1547^(2^35)\relax|} Notice though that |2^35| will be evaluated as a -floating point number, and if the floating point precision had been too -low, this computation would have given an inexact value. It is safer, -and also more efficient to code this as: -% -\centeredline{|\xintthefloatexpr 1.1547^\xintiiexpr 2^35\relax\relax|} -% -The |\xintiiexpr| is a cousin of |\xintexpr| which is big integer-only and skips -the overhead of fraction management. Notice on this example that being -embedded inside the |floatexpr|-ession has nil influence on the -|iiexpr|-ession: expansion proceeds in exactly the same way as if it had -been at the `top' level. - - -\xintexprname provides \emph{no} implementation of the |IEEE| standard: -no |NaN|s, signed infinities, signed zeroes, error traps, \dots; what is -achieved though is exact rounding for the basic operations. The only -non-algebraic operation currently implemented is square root extraction. -The power functions (there are three of them: \csbxint{Pow} to which |^| -is mapped in |\xintexpr..\relax|, \csbxint{FloatPower} for |^| in -|\xintfloatexpr..relax|, and \csbxint{FloatPow} which is slighty faster -but limits the exponent to the \TeX{} bound) allow only integral -exponents. - - -\subsection{Printing big numbers on the page} - -When producing very long numbers there is the question of printing them on - the page, without going beyond the page limits. In this document, I have most - of the time made use of these macros (not provided by the package:) - -% -\begingroup\baselineskip11pt\def\MacroFont{\small\ttfamily\baselineskip11pt\relax }% -\dverb|@ -\def\allowsplits #1{\ifx #1\relax \else #1\hskip 0pt plus 1pt\relax - \expandafter\allowsplits\fi}% -\def\printnumber #1{\expandafter\allowsplits \romannumeral-`0#1\relax }% -% \printnumber thus first ``fully'' expands its argument.| -\par\endgroup -% -An alternative (\autoref{fn:np}) is to suitably configure the thousand separator -with the \href{http://ctan.org/pkg/numprint}{numprint} package (does not work in -math mode; I also tried \href{http://ctan.org/pkg/siunitx}{siunitx} but even in -text mode could not get it to break numbers accross lines). Recently I became -aware of the -\href{http://ctan.org/pkg/seqsplit}{seqsplit} -package\footnote{\url{http://ctan.org/pkg/seqsplit}} -which can be used to achieve this splitting accross lines, and does work -in inline math mode.\par - -\subsection{Expandable implementations of mathematical algorithms} - -Another use of the |\xintexpr|-essions is illustrated with the algorithm on the -title page: it shows how one may chain expandable evaluations, almost as if one -were using the |\numexpr| facilities.\footnote{The implementation uses the - (already once-expanded) integer only variant \csa{xintiiexpr} as \csa{romannumeral0}\csa{xintiieval..}\csa{relax}.} -Notice that the @47@th Fibonacci number is \digitstt{\Fibonacci {47}} thus -already too big for \TeX{} and \eTeX{}, a difficulty which our front page showed -how to overcome (see \autoref{ssec:fibonacci} for more). The |\Fibonacci| macro -is completely expandable hence can be used for example within |\message| to -write to the log and terminal. - -It is even \fexpan dable (although not in only two steps, this could be added -but does not matter here), thus if we are interested in knowing how many digits -@F(1250)@ has, suffices to use |\xintLen {\Fibonacci {1250}}| (which expands to -\digitstt{\xintLen {\Fibonacci {1250}}}), or if we want to check the formula -@gcd(F(1859),F(1573))=F(gcd(1859,1573))=F(143)@, we only need\footnote{The - \csa{xintGCD} macro is provided by the \xintgcdname package.} -\centeredline{|\xintGCD{\Fibonacci{1859}}{\Fibonacci{1573}}=\Fibonacci{\xintGCD{1859}{1573}}|} -\centeredline{\digitstt{\printnumber{\xintGCD{\Fibonacci{1859}}{\Fibonacci{1573}}}=\printnumber{\Fibonacci{\xintGCD{1859}{1573}}}}} - -The |\Fibonacci| macro expanded its |\xintGCD{1859}{1573}| argument via the -services of |\numexpr|: this step allows only things obeying the \TeX{} bound, -naturally! (but \digitstt{F(\xintiiPow2{31}}) would be rather big anyhow...). - - -\section{Recent changes} - -\footnotesize - -\noindent Release |1.09ka| (|[2014/02/05]|): -\begin{itemize} -\item bug fix (\xinttoolsname): \csbxint{BreakFor} and \csbxint{BreakForAndDo} - were buggy when used in the last iteration of an |\xintFor| loop. -\item bug fix (\xinttoolsname): \csbxint{Seq} from |1.09k| needed a |\chardef| - which was missing from |xinttools.sty|, it was in |xint.sty|. -\end{itemize} - -\noindent Release |1.09k| (|[2014/01/21]|): -\begin{itemize} -\item inside |\xintexpr..\relax| (and its variants) tacit multiplication - is implied when a number or operand is followed directly with an - opening parenthesis, -\item the |"| for denoting (arbitrarily big) hexadecimal numbers is recognized - by |\xintexpr| and its variants (package \xintbinhexname is required); a - fractional hexadecimal part introduced by a - dot |.| is allowed. -\item re-organization of the first sections of the user manual. -\item bug fix: forgotten loading time |"| catcode sanity check has been added. -\end{itemize} - -For a more detailed change history, see \autoref{sec:releases}. Main recent -additions: \smallskip - -\noindent Release |1.09j| (|[2014/01/09]|): -\begin{itemize} -\item the core division routines have been re-written for some (limited) - efficiency gain, more pronounced for small divisors. As a result the - \hyperlink{Machin1000}{computation of one thousand digits of $\pi$} - is close to three times faster than with earlier releases. -\item a new macro \csbxint{XTrunc} is designed to produce thousands or even tens - of thousands of digits of the decimal expansion of a fraction. -\item the tacit multiplication done in \csbxint{expr}|..\relax| on encountering - a count register or variable, or a |\numexpr|, while scanning a (decimal) - number, is extended to the case of a sub |\xintexpr|-ession. -\item \csbxint{expr} can now be used in an |\edef| with no |\xintthe| - prefix. -\end{itemize} - -\noindent Release |1.09i| (|[2013/12/18]|): -\begin{itemize} -\item \csbxint{iiexpr} is a variant of \csbxint{expr} which is optimized to deal - only with (long) integers, |/| does a euclidean quotient. -\item |\xintnumexpr|, |\xintthenumexpr|, |\xintNewNumExpr| are renamed, - respectively, \csbxint{iexpr}, \csbxint{theiexpr}, \csbxint{NewIExpr}. The - earlier denominations are kept but to be removed at some point. -\item it is now possible within |\xintexpr...\relax| and its variants to use - count, dimen, and skip registers or variables without explicit |\the/\number|: - the parser inserts automatically |\number| and a tacit multiplication is - implied when a register or variable immediately follows a number or fraction. -\item \xinttoolsname defines \hyperref[odef]{\ttfamily\char92odef}, - \hyperref[oodef]{\ttfamily\char92oodef}, - \hyperref[fdef]{\ttfamily\char92fdef}. These tools are provided for the case - one uses the package macros in a non-expandable context, particularly - \hyperref[oodef]{\ttfamily\char92oodef} which expands twice the macro - replacement text and is thus a faster alternative to |\edef|. This can be - significant when repeatedly making |\def|-initions expanding to hundreds of - digits. -\end{itemize} - - -\noindent Release |1.09h| (|[2013/11/28]|): -\begin{itemize} -\item all macros of \xinttoolsname for which it makes sense are now - declared |\long|. -\end{itemize} - -\noindent Release |1.09g| (|[2013/11/22]|): -\begin{itemize} -\item package \xinttoolsname is detached from \xintname, to make tools such as - \csbxint{For}, \csbxint{ApplyUnbraced}, and \csbxint{iloop} available without - the \xintname overhead. -\item new expandable nestable loops \csbxint{loop} and \csbxint{iloop}. -\end{itemize} - -\noindent Release |1.09f| (|[2013/11/04]|): -\begin{itemize} -\item new \csbxint{ZapFirstSpaces}, \csbxint{ZapLastSpaces}, - \csbxint{ZapSpaces}, \csbxint{ZapSpacesB}, for expandably stripping away - leading and/or ending spaces. -\item \csbxint{CSVtoList} by default uses \csbxint{ZapSpacesB} to strip away - spaces around commas (or at the start and end of the comma separated list). -\item also the \csbxint{For} loop will strip out all spaces around commas and at - the start and the end of its list argument; and similarly for - \csbxint{Forpair}, \csbxint{Forthree}, \csbxint{Forfour}. -\item \csbxint{For} \emph{et al.} accept all macro parameters - from |#1| to |#9|. -\end{itemize} - - -\noindent Release |1.09e| (|[2013/10/29]|): -\begin{itemize} -\item new \csbxint{integers}, \csbxint{dimensions}, \csbxint{rationals} for - infinite \csbxint{For} loops, interrupted with \csbxint{BreakFor} and - \csbxint{BreakForAndDo}. -\item new \csbxint{ifForFirst}, \csbxint{ifForLast} for the \csa{xintFor} and - \csa{xintFor*} loops, -\item the \csa{xintFor} and \csa{xintFor*} loops are now |\long|, the - replacement text and the items may contain explicit |\par|'s. -\item new conditionals \csbxint{ifCmp}, \csbxint{ifInt}, \csbxint{ifOdd}. -\item the documentation has been enriched with various additional examples, - such as the \hyperref[ssec:quicksort]{the quick sort algorithm illustrated} or - the computation of prime numbers (\autoref{ssec:primesI}, - \autoref{ssec:primesII}, \autoref{ssec:primesIII}). -\end{itemize} - -\noindent Release |1.09c| (|[2013/10/09]|): -\begin{itemize} -\item added \hyperlink{item:bool}{|bool|} and \hyperlink{item:bool}{|togl|} to - the - \csbxint{expr} syntax; also added \csbxint{boolexpr} and \csbxint{ifboolexpr}. -\item \csbxint{For} is a new type of loop, whose replacement text inserts the - comma separated values or list items via macro parameters, rather than - encapsulated in macros; the loops are nestable up to four levels (nine - levels since |1.09f|), - and their replacement texts are allowed to close groups as happens with the - tabulation in alignments, -\item \csbxint{ApplyInline} has been enhanced in order to be usable for - generating rows (partially or completely) in an alignment, -\item new command \csbxint{Seq} to generate (expandably) arithmetic sequences of - (short) integers, -\end{itemize} - - -\noindent Release |1.09a| (|[2013/09/24]|): -\begin{itemize} -\item \csbxint{expr}|..\relax| and - \csbxint{floatexpr}|..\relax| admit functions in their - syntax, with comma separated values as arguments, among them \texttt{reduce, - sqr, sqrt, abs, sgn, floor, ceil, quo, rem, round, trunc, float, gcd, lcm, - max, min, sum, prd, add, mul, not, all, any, xor}. -\item comparison (|<|, |>|, |=|) and logical (\verb$|$, |&|) operators. -\item \csbxint{NewExpr} now works with the standard macro parameter character - |#|. -\item both regular |\xintexpr|-essions and commands defined by |\xintNewExpr| - will work with comma separated lists of expressions, -\item new commands \csbxint{Floor}, \csbxint{Ceil}, \csbxint{Maxof}, - \csbxint{Minof} (package \xintfracname), \csbxint{GCDof}, \csbxint{LCM}, - \csbxint{LCMof} (package \xintgcdname), \csbxint{ifLt}, \csbxint{ifGt}, - \csbxint{ifSgn}, \csbxint{ANDof}, ... -\item The arithmetic macros from package \xintname now filter their operands via - \csbxint{Num} which means that they may use directly count registers and - |\numexpr|-essions without having to prefix them by |\the|. This is thus - similar to the situation holding previously but with \xintfracname loaded. -\end{itemize} - -See \autoref{sec:releases} for more. - -\normalsize - - - -\section{Some examples} - -The main initial goal is to allow computations with integers and fractions of -arbitrary sizes. - -Here are some examples. The first one uses only the base module \xintname, the -next two require the \xintfracname package, which deals with fractions. Then two -examples with the \xintgcdname package, one with the \xintseriesname package, -and finally a computation with a float. Some inputs are simplified by the use -of the \xintexprname package. - -{\color{magenta}@123456^99@: }\\ -{\color[named]{Purple}\csa{xintiPow}|{123456}{99}|}: \digitstt{\printnumber{\xintiPow {123456}{99}}} - -{\color{magenta}1234/56789 with 1500 digits after the decimal point: }\\ -{\color[named]{Purple}\csa{xintTrunc}|{1500}{1234/56789}\dots|}: -\digitstt{\printnumber {\xintTrunc {1500}{1234/56789}}\dots } - -{\color{magenta}@0.99^{-100}@ with 200 digits after the decimal point:}\\ -{\color[named]{Purple}\csa{xinttheexpr trunc}|(.99^-100,200)\relax\dots|}: -\digitstt{\printnumber{\xinttheexpr trunc(.99^-100,200)\relax}\dots } - - -{\color{magenta}% - Computation of a Bezout identity with |7^200-3^200| and |2^200-1|:}\par -{\color[named]{Purple} -\dverb|@ -\xintAssign \xintBezout {\xinttheiexpr 7^200-3^200\relax} - {\xinttheiexpr 2^200-1\relax}\to\A\B\U\V\D|% -\centeredline {|\U$\times$(7^200-3^200)+\xintiOpp\V$\times$(2^200-1)=\D|}}% -% -\xintAssign \xintBezout {\xinttheiexpr 7^200-3^200\relax} - {\xinttheiexpr 2^200-1\relax}\to\A\B\U\V\D -\digitstt -{\printnumber\U$\times$(@7^200-3^200@)+% - \printnumber{\xintiOpp\V}$\times$(@2^200-1@)=\printnumber\D} - -\textcolor{magenta}{The Euclide algorithm applied to -\np{22206980239027589097} and \np{8169486210102119256}:}% -\footnote{this example is computed tremendously faster than the - other ones, but we had to limit the space taken by the output.}\par -{\color[named]{Purple} -\noindent|\xintTypesetEuclideAlgorithm -{22206980239027589097}{8169486210102119256}|\endgraf} -\xintTypesetEuclideAlgorithm -{22206980239027589097}{8169486210102119256} \smallskip - -{\color{magenta}$\sum_{n=1}^{500} (4n^2 - 9)^{-2}$ with each term rounded to - twelve digits, and the sum to nine digits:} {\color[named]{Purple}% - |\def\coeff #1%|\\ - | {\xintiRound {12}{1/\xintiSqr{\the\numexpr 4*#1*#1-9\relax }[0]}}|\\ - |\xintRound {9}{\xintiSeries {1}{500}{\coeff}[-12]}|:} \def\coeff #1% -{\xintiRound {12}{1/\xintiSqr{\the\numexpr 4*#1*#1-9\relax }[0]}} -\digitstt{\xintRound {9}{\xintiSeries {1}{500}{\coeff}[-12]}}\endgraf - -The complete series, extended to -infinity, has value -$\frac{\pi^2}{144}-\frac1{162}={}$% -\digitstt{\np{0.06236607994583659534684445}\dots}\,% -\footnote{\label{fn:np}This number is typeset using the - \href{http://www.ctan.org/pkg/numprint}{numprint} package, with - \texttt{\detokenize{\npthousandsep{,\hskip 1pt plus .5pt minus .5pt}}}. - But the breaking across - lines works only in text mode. The number itself was (of course...) computed - initially with \xintname, with 30 digits of $\pi$ as input. - See - \hyperref[ssec:Machin]{{how {\xintname} may compute $\pi$ - from scratch}}.} I also used (this is a lengthier computation -than the one above) \xintseriesname to evaluate the sum with \np{100000} terms, -obtaining 16 -correct decimal digits for the complete sum. The -coefficient macro must be redefined to avoid a |\numexpr| overflow, as -|\numexpr| inputs must not exceed @2^31-1@; my choice -was: -{\color[named]{Purple}\dverb|@ -\def\coeff #1% -{\xintiRound {22}{1/\xintiSqr{\xintiMul{\the\numexpr 2*#1-3\relax} - {\the\numexpr 2*#1+3\relax}}[0]}} -|% -}% - - -{\color{magenta}Computation of $2^{\np{999999999}}$ with |24| significant - figures:}\\ -|\numprint{|{\color[named]{Purple}|\xintFloatPow [24] {2}{999999999}|}|}| expands to: -\centeredline{\digitstt{\np{\xintFloatPow [24] {2}{999999999}}}} where the -|\numprint| macro from the \hyperref[fn:np]{eponym package} was used. - -\edef\x{\xintQuo{\xintPow {2}{1000}}{\xintFac{100}}} -\edef\y{\xintLen{\x}} - -As an example of chaining package macros, let us consider the following -code snippet within a file with filename |myfile.tex|: -\dverb|@ -\newwrite\outstream -\immediate\openout\outstream \jobname-out\relax -\immediate\write\outstream {\xintQuo{\xintPow{2}{1000}}{\xintFac{100}}} -% \immediate\closeout\outstream -|% -The tex run creates a file |myfile-out.tex|, and then writes to it the quotient -from the euclidean division of @2^{1000}@ by @100!@. The number of digits is -|\xintLen{\xintQuo{\xintPow{2}{1000}}{\xintFac{100}}}| which expands (in two -steps) and tells us that @[2^{1000}/100!]@ has {\y} digits. This is not so many, -let us print them here: \digitstt{\printnumber\x}. - -For the sake of typesetting this documentation and not have big numbers -extend into the margin and go beyond the page physical limits, I use -these commands (not provided by the package): -\dverb|@ -\def\allowsplits #1{\ifx #1\relax \else #1\hskip 0pt plus 1pt \relax - \expandafter\allowsplits\fi}% -\def\printnumber #1% first ``fully'' expands its argument. -{\expandafter\allowsplits \romannumeral-`0#1\relax }| - -The |\printnumber| macro is not part of the package and would need additional -thinking for more general use.\footnote{as explained in \hyperref[fn:np]{a - previous footnote}, the |numprint| package may also be used, in text mode - only (as the thousand separator seemingly ends up typeset in a |\string\hbox| - when in math mode).} It may be used like this: -% -\centeredline{|\printnumber {\xintQuo{\xintPow {2}{1000}}{\xintFac{100}}}|} -or as |\printnumber\mynumber| or |\printnumber{\mynumber}| if -|\mynumber| was previously defined via a |\newcommand|, or a |\def|: -% -\centeredline{% - |\def\mynumber {\xintQuo {\xintPow {2}{1000}}{\xintFac{100}}}|}% - - -Just to show off (again), let's print 300 digits (after the decimal point) of -the decimal expansion of @0.7^{-25}@:\footnote{the |\string\np| typesetting - macro is from the |numprint| package.} -\centeredline{|\np {\xinttheexpr trunc(.7^-25,300)\relax}\dots|} - \digitstt{\np {\xinttheexpr trunc(.7^-25,300)\relax}\dots } - -This computation is with \csbxint{theexpr} from package \xintexprname, which -allows to use standard infix notations and function names to access the package -macros, such as here |trunc| which corresponds to the \xintfracname macro -\csbxint{Trunc}. The fraction |.7^-25| is first evaluated \emph{exactly}; for -some more complex inputs, such as |.7123045678952^-243|, the exact evaluation -before truncation would be expensive, and (assuming one needs twenty digits) one -would rather use floating mode: \centeredline{|\xintDigits:=20; - \np{\xintthefloatexpr .7123045678952^-243\relax}|}% -\xintDigits:=20;% -\centeredline{|.7123045678952^-243|${}\approx{}$% - \digitstt{\np{\xintthefloatexpr .7123045678952^-243\relax }}} The exponent -|-243| didn't have to be put inside parentheses, contrarily to what happens with -some professional computational software. -% 6.342,022,117,488,416,127,3 10^35 -% maple n'aime pas ^-243 il veut les parenthèses, bon et il donne, en Digits -% = 24: 0.634202211748841612732270 10^36 - -\xintDigits:=16; - - -\section {Further illustrative examples within this document} -\label{sec:awesome} - - -The utilities provided by \xinttoolsname (\autoref{sec:tools}), some -completely expandable, others not, are of independent interest. Their -use is illustrated through various examples: among those, it is shown in -\autoref{ssec:quicksort} how to implement in a completely expandable way -the \hyperref[quicksort]{Quick Sort algorithm} and also how to -illustrate it graphically. Other examples include some dynamically -constructed alignments with automatically computed prime number cells: -one using a completely expandable prime test and \csbxint{ApplyUnbraced} -(\autoref{ssec:primesI}), another one with \csbxint{For*} -(\autoref{ssec:primesIII}). - -One has also a \hyperref[edefprimes]{computation of primes - within an \csa{edef}} (\autoref{xintiloop}), with the help of -\csbxint{iloop}. Also with \csbxint{iloop} an -\hyperref[ssec:factorizationtable]{automatically generated table of - factorizations} (\autoref{ssec:factorizationtable}). - -The title page fun with Fibonacci numbers is continued in -\autoref{ssec:fibonacci} with \csbxint{For*} joining the game. - -The computations of \hyperref[ssec:Machin]{ $\pi$ and $\log 2$} -(\autoref{ssec:Machin}) using \xintname and the computation of the -\hyperlink{e-convergents}{convergents of $e$} with the further help of -the \xintcfracname package are among further examples. -There is also an example of an \hyperref[xintXTrunc]{interactive - session}, where results are output to the log or to a file. - -Almost all of the computational results interspersed through the -documentation are not hard-coded in the source of the document but just written -there using the package macros, and were selected to not impact too much the -compilation time. - - -\section{General overview} - -The main characteristics are: -\begin{enumerate} -\item exact algebra on arbitrarily big numbers, integers as well as fractions, -\item floating point variants with user-chosen precision, -\item implemented via macros compatible with expansion-only - context. -\end{enumerate} - -`Arbitrarily big': this means with less than - |2^31-1|\digitstt{=\number"7FFFFFFF} digits, as most of the macros will - have to compute the length of the inputs and these lengths must be treatable - as \TeX{} integers, which are at most \digitstt{\number "7FFFFFFF} - in absolute value. - This is a distant irrelevant upper bound, as no such thing can fit - in \TeX's memory! And besides, -the true limitation is from the \emph{time} taken by the -expansion-compatible algorithms, as will be commented upon soon. - -As just recalled, ten-digits numbers starting with a @3@ already exceed the -\TeX{} bound on integers; and \TeX{} does not have a native processing of -floating point numbers (multiplication by a decimal number of a dimension -register is allowed --- this is used for example by the -\href{http://mirror.ctan.org/graphics/pgf/base}{pgf} basic math -engine.) - -\TeX{} elementary operations on numbers are done via the non-expandable -\emph{advance, multiply, \emph{and} divide} assignments. This was changed with -\eTeX{}'s |\numexpr| which does expandable computations using standard infix -notations with \TeX{} integers. But \eTeX{} did not modify the \TeX{} bound on -acceptable integers, and did not add floating point support. - -The \href{http://www.ctan.org/pkg/bigintcalc}{bigintcalc} package by -\textsc{Heiko Oberdiek} provided expandable operations (using some of |\numexpr| -possibilities, when available) on arbitrarily big integers, beyond the \TeX{} -bound. The present package does this again, using more of |\numexpr| (\xintname -requires the \eTeX{} extensions) for higher speed, and also on fractions, not -only integers. Arbitrary precision floating points operations are a derivative, -and not the initial design goal.\footnote{currently (|v1.08|), the only - non-elementary operation implemented for floating point numbers is the - square-root extraction; no signed infinities, signed zeroes, |NaN|'s, error - trapes\dots, have been - implemented, only the notion of `scientific notation with a given number of - significant figures'.}${}^{\text{,\,}}$\footnote{multiplication of two floats - with |P=\string\xinttheDigits| digits is first done exactly then rounded to - |P| digits, rather than using a specially tailored multiplication for floating - point numbers which would be more efficient (it is a waste to evaluate fully - the multiplication result with |2P| or |2P-1| digits.)} - -The \LaTeX3 project has implemented expandably floating-point computations with -16 significant figures -(\href{http://www.ctan.org/pkg/l3kernel}{l3fp}), including -special functions such as exp, log, sine and cosine.\footnote{at the time of - writing the \href{http://www.ctan.org/pkg/l3kernel}{l3fp} - (exactly represented) floating point numbers have their exponents limited to - $\pm$\digitstt{9999}.} - -The \xintname package can be used for @24@, @40@, etc\dots{} significant figures -but one rather quickly (not much beyond @100@ figures) hits against a -`wall' created by the constraint of expandability: currently, multiplying out -two one-hundred digits numbers takes circa @80@ or @90@ times longer than for -two ten-digits numbers, which is reasonable, but multiplying out two -one-thousand digits numbers takes more than @500@ times longer than for two one -hundred-digits numbers. This shows that the algorithm is drifting from quadratic -to cubic in that range. On my laptop multiplication of two @1000@-digits numbers -takes some seconds, so it can not be done routinely in a -document.\footnote{without entering into too much technical details, the source - of this `wall' is that when dealing with two long operands, when one wants to - pick some digits from the second one, one has to jump above all digits - constituting the first one, which can not be stored away: expandability - forbids assignments to memory storage. One may envision some sophisticated - schemes, dealing with this problem in less naive ways, trying to move big - chunks of data higher up in the input stream and come back to it later, - etc...; but each `better' algorithm adds overhead for the smaller inputs. For - example, I have another version of addition which is twice faster on inputs - with 500 digits or more, but it is slightly less efficient for 50 digits or - less. This `wall' dissuaded me to look into implementing `intelligent' - multiplication which would be sub-quadratic in a model where storing and - retrieving from memory do not cost much.} - -The conclusion perhaps could be that it is in the end lucky that the speed gains -brought by \xintname for expandable operations on big numbers do open some -non-empty range of applicability in terms of the number of kept digits for -routine floating point operations. - -The second conclusion, somewhat depressing after all the hard work, is -that if one really wants to do computations with \emph{hundreds} of digits, one -should drop the expandability requirement. And indeed, as clearly -demonstrated long ago by the \href{http://www.ctan.org/pkg/pi}{pi - computing file} by \textsc{D. Roegel} one can program \TeX{} to -compute with many digits at a much higher speed than what \xintname -achieves: but, direct access to memory storage in one form or another -seems a necessity for this kind of speed and one has to renounce at the -complete expandability.\footnote{I could, naturally, be proven - wrong!}\,\footnote{The Lua\TeX{} project possibly makes endeavours - such as \xintname appear even more insane that they are, in truth.} - - -% \section{Missing things} - - -% `Arbitrary-precision' floating-point -% operations are currently limited to the basic four operations, the power -% function with integer exponent, and the extraction of square-roots. - - -\section{Origins of the package} - -Package |bigintcalc| by \textsc{Heiko Oberdiek} already -provides expandable arithmetic operations on ``big integers'', -exceeding the \TeX{} limits (of @2^{31}-1@), so why another\footnote{this section was written before the - \xintfracname package; the author is not aware of another package allowing - expandable computations with arbitrarily big fractions.} -one? - -I got started on this in early March 2013, via a thread on the -|c.t.tex| usenet group, where \textsc{Ulrich D\,i\,e\,z} used the -previously cited package together with a macro (|\ReverseOrder|) -which I had contributed to another thread.\footnote{the - \csa{ReverseOrder} could be avoided in that circumstance, but it - does play a crucial r\^ole here.} What I had learned in this -other thread thanks to interaction with \textsc{Ulrich D\,i\,e\,z} and -\textsc{GL} on expandable manipulations of tokens motivated me to -try my hands at addition and multiplication. - -I wrote macros \csa{bigMul} and \csa{bigAdd} which I posted to the -newsgroup; they appeared to work comparatively fast. These first -versions did not use the \eTeX{} \csa{numexpr} primitive, they worked -one digit at a time, having previously stored carry-arithmetic in -1200 macros. - -I noticed that the |bigintcalc| package used\csa{numexpr} -if available, but (as far as I could tell) not -to do computations many digits at a time. Using \csa{numexpr} for -one digit at a time for \csa{bigAdd} and \csa{bigMul} slowed them -a tiny bit but avoided cluttering \TeX{} memory with the 1200 -macros storing pre-computed digit arithmetic. I wondered if some speed -could be gained by using \csa{numexpr} to do four digits at a time -for elementary multiplications (as the maximal admissible number -for \csa{numexpr} has ten digits). - -The present package is the result of this initial questioning. - -% \begin{framed}\centering -% \xintname requires the \eTeX{} extensions. -% \end{framed} - - - -\section{Expansion matters} -\label{sec:expansions} - - -By convention in this manual \fexpan sion (``full expansion'' or ``full first -expansion'') is the process of expanding repeatedly the first token seen until -hitting against something not further expandable like an unexpandable -\TeX-primitive or an opening brace |{| or a character (inactive). For - those familiar with \LaTeX3 (which is not used by \xintname) this is what is - called in its documentation full expansion. Technically, macro arguments in - \xintname which are submitted to such a \fexpan sion are so via prefixing them - with |\romannumeral-`0|. An explicit or implicit space token stops such an - expansion and is gobbled. - -% -Most of the package macros, and all those dealing with computations, are -expandable in the strong sense that they expand to their final result via this -\fexpan sion. Again copied from \LaTeX3 documentation conventions, this will be -signaled in the description of the macro by a \etype{}star in the margin. -All\footnote{except \csbxint{loop} and \csbxint{iloop}.} -expandable macros of the \xintname packages completely expand in two steps. - -Furthermore the macros dealing with computations, as well as many utilities from -\xinttoolsname, apply this process of \fexpan sion to their arguments. Again -from \LaTeX3's conventions this will be signaled by a% -% -\ntype{{\setbox0 \hbox{\Ff}\hbox to \wd0 {\hss f\hss}}} -% -margin annotation. Some additional parsing -which is done by most macros of \xintname is indicated with a -variant\ntype{\Numf{\kern.5cm}}; and the extended fraction parsing done by most -macros of \xintfracname has its own symbol\ntype{\Ff}. When the argument has a -priori to obey the \TeX{} bound of \digitstt{\number"7FFFFFFF} it is -systematically fed to a |\numexpr..\relax| hence the expansion is then a -\emph{complete} one, signaled with an \ntype{\numx}\emph{x} in the margin. This -means not only complete expansion, but also that spaces are ignored, infix -algebra is possible, count registers are allowed, etc\dots - -The \csbxint{ApplyInline} and \csbxint{For*}\ntype{{\lowast f}} macros from -\xinttoolsname apply a special iterated \fexpan sion, which gobbles spaces, to -all those items which are found \emph{unbraced} from left to right in the list -argument; this is denoted specially as here in the margin. Some other macros -such as \csbxint{Sum}\ntype{f{$\to$}{\lowast\Ff}} from \xintfracname first do an -\fexpan sion, then treat each found (braced or not) item (skipping spaces -between such items) via the general fraction input parsing, this is signaled as -here in the margin where the signification of the \lowast{} is thus a bit -different from the previous case. - -A few macros from \xinttoolsname do not expand, or expand only once their -argument\ntype{n{{\color{black}\upshape, resp.}} o}. This is also -signaled in the margin with notations \`a la \LaTeX3. - -As the computations are done by \fexpan dable macros which \fexpan d their -argument they may be chained up to arbitrary depths and still produce expandable -macros. - -Conversely, wherever the package expects on input a ``big'' integers, or a -``fraction'', \fexpan sion of the argument \emph{must result in a complete - expansion} for this argument to be acceptable.\footnote{this is not quite as - stringent as claimed here, see \autoref{sec:useofcount} for more details.} -The -main exception is inside \csbxint{expr}|...\relax| where everything will be -expanded from left to right, completely. - -Summary of important expansion aspects: -\begin{enumerate} -\item the macros \fexpan d their arguments, this means that they expand - the first token seen (for each argument), then expand, etc..., until something - un-expandable - such as a\strut{} digit or a brace is hit against. This example - \centeredline{|\def\x{98765}\def\y{43210}|% - |\xintAdd {\x}{\x\y}|} is \emph{not} a legal construct, as the |\y| will - remain untouched by expansion and not get converted into the digits which - are expected by the sub-routines of |\xintAdd|. It is a |\numexpr| - which will expand it and an arithmetic overflow will arise as |9876543210| - exceeds the \TeX{} bounds. - - \begingroup\slshape - With \csbxint{theexpr} one could write |\xinttheexpr \x+\x\y\relax|, or - |\xintAdd\x{\xinttheexpr\x\y\relax}|.\hfill - \endgroup - -\item\label{fn:expansions} using |\if...\fi| constructs \emph{inside} the - package macro arguments requires suitably mastering \TeX niques - (|\expandafter|'s and/or swapping techniques) to ensure that the \fexpan sion - will indeed absorb the \csa{else} or closing \csa{fi}, else some error will - arise in further processing. Therefore it is highly recommended to use the - package provided conditionals such as \csbxint{ifEq}, \csbxint{ifGt}, - \csbxint{ifSgn}, \csbxint{ifOdd}\dots, or, for \LaTeX{} users and when dealing - with short integers the - \href{http://www.ctan.org/pkg/etoolbox}{etoolbox}\footnote{\url{http://www.ctan.org/pkg/etoolbox}} - expandable conditionals (for small integers only) such as \texttt{\char92 - ifnumequal}, \texttt{\char92 ifnumgreater}, \dots . Use of - \emph{non-expandable} things such as \csa{ifthenelse} is impossible inside the - arguments of \xintname macros. - - \begingroup\slshape - One can use naive |\if..\fi| things inside an \csbxint{theexpr}-ession - and cousins, as long as the test is - expandable, for example\upshape -\centeredline{|\xinttheiexpr\ifnum3>2 143\else 33\fi 0^2\relax|$\to$\digitstt{\xinttheiexpr \ifnum3>2 143\else 33\fi 0^2\relax =1430\char`\^2}} - \endgroup - -\item after the definition |\def\x {12}|, one can not use - {\color{blue}|-\x|} as input to one of the package macros: the \fexpan sion - will act only on the minus sign, hence do nothing. The only way is to use the - \csbxint{Opp} macro, or perhaps here rather \csbxint{iOpp} which does - maintains integer format on output, as they replace a number with - its opposite. - - \begingroup\slshape - Again, this is otherwise inside an \csbxint{theexpr}-ession or - \csbxint{thefloatexpr}-ession. There, the - minus sign may prefix macros which will expand to numbers (or parentheses - etc...) - \endgroup - -\def\x {12}% -\def\AplusBC #1#2#3{\xintAdd {#1}{\xintMul {#2}{#3}}}% - - -\item \label{item:xpxp} With the definition \centeredline{% - |\def\AplusBC #1#2#3{\xintAdd {#1}{\xintMul {#2}{#3}}}|} one obtains an - expandable macro producing the expected result, not in two, but rather in - three steps: a first expansion is consumed by the macro expanding to its - definition. As the package macros expand their arguments until no more is - possible (regarding what comes first), this |\AplusBC| may be used inside - them: {|\xintAdd {\AplusBC {1}{2}{3}}{4}|} does work and returns - \digitstt{\xintAdd {\AplusBC {1}{2}{3}}{4}}. - - If, for some reason, it is important to create a macro expanding in two steps - to its final value, one may either do: -\smallskip\centeredline {|\def\AplusBC - #1#2#3{|{\color{blue}|\romannumeral-`0\xintAdd |}|{#1}{\xintMul {#2}{#3}}}|}or use the \emph{lowercase} form of - \csa{xintAdd}: \smallskip\centeredline {|\def\AplusBC - #1#2#3{|{\color{blue}|\romannumeral0\xintadd |}|{#1}{\xintMul {#2}{#3}}}|} - - and then \csa{AplusBC} will share the same properties as do the - other \xintname `primitive' macros. - - -\end{enumerate} - -The |\romannumeral0| and |\romannumeral-`0| things above look like an invitation -to hacker's territory; if it is not important that the macro expands in two -steps only, there is no reason to follow these guidelines. Just chain -arbitrarily the package macros, and the new ones will be completely expandable -and usable one within the other. - -Since release |1.07| the \csbxint{NewExpr} command automatizes the creation of -such expandable macros: \centeredline{|\xintNewExpr\AplusBC[3]{#1+#2*#3}|} -creates the |\AplusBC| macro doing the above and expanding in two expansion -steps. - - -\section{User interface} - -Maintaining complete expandability is not for the faint of heart as it excludes -doing macro definitions in the midst of the computation; in many cases, one does -not need complete expandability, and definitions are allowed. In such contexts, -there is no declaration for the user to be made to the package of a ``typed -variable'' such as a long integer, or a (long) fraction, or possibly an -|\xintexpr|-ession. Rather, the user has at its disposals the general tools of -the \TeX{} language: |\def| or (in \LaTeX) |\newcommand|, and |\edef|. - -The \xinttoolsname package provides |\oodef| which expands twice the replacement -text, hence forces complete expansion when the top level of this replacement -text is a call to one of the \xintname bundle macros, its arguments being -themselves chains of such macros. There is also |\fdef| which will apply \fexpan -sion to the replacement text. Both are in such uses faster alternatives to -|\edef|. - -This section will explain the various inputs which are recognized by the package -macros and the format for their outputs. Inputs have mainly five possible -shapes: -\begin{enumerate} -\item expressions which will end up inside a |\numexpr..\relax|, -\item long integers in the strict format (no |+|, no leading zeroes, a count - register or variable must be prefixed by |\the| or |\number|) -\item long integers in the general format allowing both |-| and |+| signs, then - leading zeroes, and a count register or variable without prefix is allowed, -\item fractions with numerators and denominators as in the - previous item, or also decimal numbers, possibly in scientific notation (with - a lowercase |e|), and - also optionally the semi-private |A/B[N]| format, -\item and finally expandable material understood by the |\xintexpr| parser. -\end{enumerate} -Outputs are mostly of the following types: -\begin{enumerate} -\item long integers in the strict format, -\item fractions in the |A/B[N]| format where |A| and |B| are both strict long - integers, and |B| is positive, -\item numbers in scientific format (with a lowercase |e|), -\item the private |\xintexpr| format which needs the |\xintthe| prefix in order - to end up on the printed page (or get expanded in the log) - or be used as argument to the package macros. -\end{enumerate} - -{\etocdefaultlines\etocsettocstyle{}{}\localtableofcontents} - - - -\subsection {Input formats}\label{sec:inputs} - -% \edef\z {\xintAdd -% {+--0367.8920280/-++278.289287}{-109.2882/+270.12898}} - -Some macro arguments are by nature `short' integers,\ntype{\numx} \emph{i.e.} -less than (or equal to) in absolute value \np{\number "7FFFFFFF}. This is -generally the case for arguments which serve to count or index something. They -will be embedded in a |\numexpr..\relax| hence on input one may even use count -registers or variables and expressions with infix operators. Notice though that -|-(..stuff..)| is surprisingly not legal in the |\numexpr| syntax! - -But \xintname is mainly devoted to big numbers; -the allowed input formats for `long numbers' and `fractions' are: -\begin{enumerate} -\item the strict format\ntype{f} is for some macros of \xintname which only - \fexpan d their arguments. After this \fexpan sion the input should be a - string of digits, optionally preceded by a unique minus sign. The first digit - can be zero only if the number is zero. A plus sign is not accepted. |-0| is - not legal in the strict format. A count register can serve as argument of such - a macro only if prefixed by |\the| or |\number|. Most macros of \xintname are - like \csbxint{Add} and accept the extended format described in the next item; - they may have a `strict' variant such as \csbxint{iiAdd} which remains - available even with \xintfracname loaded, for optimization purposes. -\item the macro \csbxint{Num} normalizes into strict format an input having - arbitrarily many minus and plus signs, followed by a string of zeroes, then - digits:\centeredline{|\xintNum - {+-+-+----++-++----00000000009876543210}|\digitstt{=\xintNum - {+-+-+----++-++----0000000009876543210}}} The extended integer - format\ntype{\Numf} is thus for the arithmetic macros of \xintname which - automatically parse their arguments via this \csbxint{Num}.\footnote{A - \LaTeX{} \texttt{\char 92value\{countername\}} is accepted as macro - argument.} -\item the fraction format\ntype{\Ff} is what is expected by the macros of - \xintfracname: a fraction is constituted of a numerator |A| and optionally a - denominator |B|, separated by a forward slash |/| and |A| and |B| may be - macros which will be automatically given to \csbxint{Num}. Each of |A| and |B| - may be decimal numbers (the decimal mark must be a |.|). Here is an - example:\footnote{the square brackets one sees in various outputs are - explained - near the end of this section.} \centeredline{|\xintAdd - {+--0367.8920280/-++278.289287}{-109.2882/+270.12898}|}% - Scientific notation is accepted for both numerator and - denominator of a fraction, and is produced on output by \csbxint{Float}: - \centeredline{|\xintAdd{10.1e1}{101.010e3}|% - \digitstt{=\xintAdd{10.1e1}{101.010e3}}} - \centeredline{|\xintFloatAdd{10.1e1}{101.010e3}|% - \digitstt{=\xintFloatAdd{10.1e1}{101.010e3}}} - \centeredline{|\xintPow {2}{100}|% - \digitstt{=\xintPow {2}{100}}} - \centeredline{|\xintFloat{\xintPow {2}{100}}|% - \digitstt{=\xintFloat{\xintPow {2}{100}}}} - \centeredline{|\xintFloatPow {2}{100}|% - \digitstt{=\xintFloatPow {2}{100}}} -% -Produced fractions having a denominator equal to one are, as a general rule, -nevertheless printed as fractions. In math mode \csbxint{Frac} will remove such -dummy denominators, and in inline text mode one has \csbxint{PRaw} with the -similar effect. -% -\centeredline{|\xintPRaw{\xintAdd{10.1e1}{101.010e3}}|% - \digitstt{=\xintPRaw{\xintAdd{10.1e1}{101.010e3}}}} -\centeredline{|\xintRaw{1.234e5/6.789e3}|% - \digitstt{=\xintRaw{1.234e5/6.789e3}}}% -\item the \hyperref[xintexpr]{expression format} is for inclusion in an - \csbxint{expr}|...\relax|, it uses infix notations, function names, complete - expansion, and is described in its devoted section - (\autoref{sec:exprsummaryII}). -\end{enumerate} -Generally speaking, there should be no spaces among the digits in the inputs -(in arguments to the package macros). -Although most would be harmless in most macros, there are some cases -where spaces could break havoc. So the best is to avoid them entirely. - -This is entirely otherwise inside an |\xintexpr|-ession, where spaces are -ignored (except when they occur inside arguments to some macros, thus -escaping the |\xintexpr| parser). See the \hyperref[sec:expr]{documentation}. - - - -Even with \xintfracname loaded, some macros by their nature can not accept -fractions on input. Those parsing their inputs through \csbxint{Num} will accept -a fraction reducing to an integer. For example |\xintQuo {100/2}{12/3}| works, -because its arguments are, after simplification, integers. -% -% In this -% documentation, I often say ``numbers or fractions'', although at times the -% vocable ``numbers'' by itself may also include ``fractions''; and ``decimal -% numbers'' are counted among ``fractions''. - -With \xintfracname loaded, a number may be empty or start directly with a -decimal point: \centeredline{|\xintRaw{}=\xintRaw{.}|\digitstt{=\xintRaw{}}} -\centeredline{|\xintPow{-.3/.7}{11}|\digitstt{=\xintPow{-.3/+.7}{11}}} -\centeredline{|\xinttheexpr (-.3/.7)^11\relax|% - \digitstt{=\xinttheexpr (-.3/.7)^11\relax}} It is also licit to use |\A/\B| as -input if each of |\A| and |\B| expands (in the sense previously described) to a -``decimal number'' as examplified above by the numerators and denominators -(thus, possibly with a `scientific' exponent part, with a lowercase `e'). Or one -may have just one macro |\C| which expands to such a ``fraction with optional -decimal points'', or mixed things such as |\A 245/7.77|, where the numerator -will be the concatenation of the expansion of |\A| and |245|. But, as explained -already |123\A| is a no-go, \emph{except inside an |\string\xintexpr|-ession}! - -The scientific notation is necessarily (except in |\xintexpr..\relax|) with a -lowercase |e|. It may appear both at the numerator and at the denominator of a -fraction. \centeredline{|\xintRaw - {+--+1253.2782e++--3/---0087.123e---5}|\digitstt{=\xintRaw - {+--+1253.2782e++--3/---0087.123e---5}}} - -Arithmetic macros of \xintname which parse their arguments automatically through -\csbxint{Num} are signaled by a special -symbol%\ntype{\Numf{\unskip\kern\dimexpr\FrameSep+\FrameRule\relax}} -\ntype{\Numf} in the margin. This symbol also means that these arguments may -contain to some extent infix algebra with count registers, see the section -\hyperref[sec:useofcount]{Use of count registers}. - - - With \xintfracname loaded the symbol \smash{\Numf} means that a fraction is - accepted if it is a whole number in disguise; and for macros accepting the - full fraction format with no restriction there is the corresponding symbol - in the margin\ntype{\Ff}. - - -The \xintfracname macros generally output -their result in |A/B[n]| format, representing the fraction |A/B| times |10^n|. - -This format with a trailing |[n]| (possibly, |n=0|) is accepted on input -but it presupposes that the numerator and denominator |A| and |B| are in -the strict integer format described above. So |16000/289072[17]| or -|3[-4]| are authorized and it is even possible to use |\A/\B[17]| if -|\A| expands to |16000| and |\B| to |289072|, or |\A| if |\A| expands to -|3[-4]|. However, NEITHER the numerator NOR the denominator may then -have a decimal point\IMPORTANT{}. And, for this format, ONLY the -numerator may carry a UNIQUE minus sign (and no superfluous leading -zeroes; and NO plus sign). - -It is allowed for user input but the parsing is minimal and it is mandatory to -follow the above rules. This reduced flexibility, compared to the format without -the square brackets, allows nesting package macros without too much speed -impact. - -\subsection{Output formats} - - -With package \xintfracname loaded, the routines \csbxint{Add}, \csbxint{Sub}, -\csbxint{Mul}, \csbxint{Pow}, \csbxint{Sum}, \csbxint{Prd} are modified to allow -fractions on input,\footnote{the power function does not accept a fractional - exponent. Or rather, does not expect, and errors will result if one is - provided.}\,\footnote{macros \csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul}, - \csbxint{iPow}, are the original ones dealing only with integers. They are - available as synonyms, also when \xintfracname is not loaded. With - \xintfracname loaded they accept on input also fractions, if these fractions - reduce to integers, and then the output format is the original \xintname's - one. The macros \csbxint{iiAdd}, \csbxint{iiSub}, \csbxint{iiMul}, - \csbxint{iiPow}, \csbxint{iiSum}, \csbxint{iiPrd} are strictly integer-only: - they skip the overhead of parsing their arguments via - \csbxint{Num}.}\,\footnote{also \csbxint{Cmp}, \csbxint{Sgn}, \csbxint{Geq}, - \csbxint{Opp}, \csbxint{Abs}, \csbxint{Max}, \csbxint{Min} are extended to - fractions; and the last four have the integer-only variants \csbxint{iOpp}, - \csbxint{iAbs}, \csbxint{iMax}, \csbxint{iMin}.}\,\footnote{and \csbxint{Fac}, - \csbxint{Quo}, \csbxint{Rem}, \csbxint{Division}, \csbxint{FDg}, - \csbxint{LDg}, \csbxint{Odd}, \csbxint{MON}, \csbxint{MMON} all accept a - fractional input as long as it reduces to an integer.} and produce on output a -fractional number |f=A/B[n]| where |A| and |B| are integers, with |B| positive, -and |n| is a ``short'' integer. -% -% (\emph{i.e} less in absolute value than |2^{31}-9|). -% -This represents |(A/B)| times |10^n|. The fraction |f| may be, and -generally is, reducible, and |A| and |B| may well end up with zeroes (\emph{i.e.} -|n| does not contain all powers of 10). Conversely, this format is accepted on -input (and is parsed more quickly than fractions containing decimal points; the -input may be a number without denominator).\footnote{at each stage of the - computations, the sum of |n| and the length of |A|, or of the absolute value - of |n| and the length of |B|, must be kept less than - |2\string^\string{31\string}-9|.} - -Thus loading \xintfracname not only relaxes the format of the inputs; it -also modifies the format of the outputs: except when a fraction is -filtered on output by \csbxint{Irr} or \csbxint{RawWithZeros}, or -\csbxint{PRaw}, or by the truncation or rounding macros, or is given as -argument in math mode to \csbxint{Frac}, the output format is normally -of the \fbox{|A/B[n]|} form (which stands for |(A/B)|$\times$|10^n|). -The |A| and |B| may end in zeroes (\emph{i.e}, |n| does not represent all -powers of ten), and will generally have a common factor. The denominator -|B| is always strictly positive. - -A macro \csbxint{Frac} is provided for the typesetting (math-mode only) -of such a `raw' output. The command \csbxint{Frac} is not accepted as -input to the package macros, it is for typesetting only (in math mode). - -The macro \csbxint{Raw} prints the fraction -directly from its internal representation in |A/B[n]| form. The macro -\csbxint{PRaw} does the same but without printing the |[n]| if |n=0| and without -printing |/1| if |B=1|. - -% To convert the trailing |[n]| into explicit zeroes either at the -% numerator or the denominator, use \csbxint{RawWithZeros}. In both cases -% the |B| is printed even if it has value |1|. Conversely (sort of), the -% macro \csbxint{REZ} puts all powers of ten into the |[n]| (REZ stands -% for remove zeroes). Here also, the |B| is printed even if it has value -% |1|. - -The macro \csbxint{Irr} reduces the fraction to its irreducible form -|C/D| (without a trailing |[0]|), and it prints the |D| even if |D=1|. - -The macro \csbxint{Num} from package \xintname is extended: it now does -like \csbxint{Irr}, raises an error if the fraction did not reduce to an -integer, and outputs the numerator. This macro should be used when one -knows that necessarily the result of a computation is an integer, and -one wants to get rid of its denominator |/1| which would be left by -\csa{xintIrr} (or one can use \csbxint{PRaw} on top of \csbxint{Irr}). - - -% The macro \csbxint{Trunc}|{N}{f}| prints\footnote{`prints' does not at all mean -% that this macro is designed for typesetting; I am just using the verb here in -% analogy to the effect of the functioning of a computing software in console -% mode. The package does not provide any `printing' facility, besides its -% rudimentary \csbxint{Frac} and \csbxint{FwOver} math-mode only macros. To deal -% with really long numbers, some macros are necessary as \TeX{} by default will -% print a long number on a single line extending beyond the page limits. The -% \csa{printnumber} command used in this documentation is just one way to -% address this problem, some other method should be used if it is important that -% digits occupy the same width always.} the decimal expansion of |f| with |N| -% digits after the decimal point.\footnote{the current release does not provide a -% macro to get the period of the decimal expansion.} Currently, it does not -% verify that |N| is non-negative and strange things could happen with a negative -% |N|. A negative |f| is no problem, needless to say. When the original -% fraction is negative and its truncation has only zeroes, it is printed as -% |-0.0...0|, with |N| zeroes following the decimal point: -% \centeredline{|\xintTrunc {5}{\xintPow {-13}{-9}}|\digitstt{=\xintTrunc -% {5}{\xintPow {-13}{-9}}}}% -% \centeredline{|\xintTrunc {20}{\xintPow {-13}{-9}}|\digitstt{=\xintTrunc -% {20}{\xintPow {-13}{-9}}}} The output always contains a decimal point (even -% for |N=0|) followed by |N| digits, except when the original fraction was zero. -% In that case the output is |0|, with no decimal point. \centeredline{|\xintTrunc -% {10}{\xintSum {{1/2}{1/3}{1/5}{-31/30}}}|% -% \digitstt{=\xintTrunc {10}{\xintSum {{1/2}{1/3}{1/5}{-31/30}}}}} - -% \edef\z {\xintPow {1.01}{100}} - -% The macro \csbxint{iTrunc}|{N}{f}| is like \csa{xintTrunc}|{N}{f}| -% followed by multiplication by |10^N|. Thus, it outputs an integer -% in a format acceptable by the integer-only macros. -% To get the integer part of the decimal expansion of |f|, use -% |\xintiTrunc{0}{f}|: \centeredline{|\xintiTrunc {0}{\xintPow -% {1.01}{100}}|\digitstt{=\xintiTrunc {0}\z}}% -% \centeredline{|\xintiTrunc {0}{\xintPow{0.123}{-10}}|\digitstt{=\xintiTrunc -% {0}{\xintPow{0.123}{-10}}}} - -See also the documentations of \csbxint{Trunc}, \csbxint{iTrunc}, -\csbxint{XTrunc}, \csbxint{Round}, \csbxint{iRound} and -\csbxint{Float}. - -The \csbxint{iAdd}, \csbxint{iSub}, \csbxint{iMul}, \csbxint{iPow}, and -some others accept fractions on input under -the condition that they are (big) integers in disguise and then output a -(possibly big) integer, without fraction slash nor trailing |[n]|. - -The \csbxint{iiAdd}, \csbxint{iiSub}, \csbxint{iiMul}, \csbxint{iiPow}, and -some others with `\textcolor{blue}{ii}' in their names accept on input -only integers in strict format (skipping the overhead of the -\csbxint{Num} parsing) and output naturally a -(possibly big) integer, without fraction slash nor trailing |[n]|. - - -\subsection{Multiple outputs}\label{sec:multout} - -Some macros have an output consisting of more than one number or -fraction, each one is then returned within braces. Examples of -multiple-output macros are \csbxint{Division} which gives first the -quotient and then the remainder of euclidean division, \csbxint{Bezout} -from the \xintgcdname package which outputs five numbers, -\csbxint{FtoCv} from the \xintcfracname package which returns the list -of the convergents of a fraction, ... \autoref{sec:assign} and -\autoref{sec:utils} mention utilities, expandable or not, to cope with -such outputs. - -Another type of multiple outputs is when using commas inside -\csbxint{expr}|..\relax|: -\centeredline{|\xinttheiexpr 10!,2^20,lcm(1000,725)\relax|% - $\to$\digitstt{\xinttheiexpr 10!,2^20,lcm(1000,725)\relax}} - - -\section{Use of \TeX{} registers and variables} - -{\etocdefaultlines\etocsettocstyle{}{}\localtableofcontents} - -\subsection{Use of count registers}\label{sec:useofcount} - -Inside |\xintexpr..\relax| and its variants, a count register or count control -sequence is automatically unpacked using |\number|, with tacit multiplication: -|1.23\counta| is like |1.23*\number\counta|. There -is a subtle difference between count \emph{registers} and count -\emph{variables}. In |1.23*\counta| the unpacked |\counta| variable defines a -complete operand thus |1.23*\counta 7| is a syntax error. But |1.23*\count0| -just replaces |\count0| by |\number\count0| hence |1.23*\count0 7| is like -|1.23*57| if |\count0| contains the integer value |5|. - -Regarding now the package macros, there is first the case of arguments having to -be short integers: this means that they are fed to a |\numexpr...\relax|, hence -submitted to a \emph{complete expansion} which must deliver an integer, and -count registers and even algebraic expressions with them like -|\mycountA+\mycountB*17-\mycountC/12+\mycountD| are admissible arguments (the -slash stands here for the rounded integer division done by |\numexpr|). This -applies in particular to the number of digits to truncate or round with, to the -indices of a series partial sum, \dots - -The macros allowing the extended format for long numbers or dealing with -fractions will \emph{to some extent} allow the direct use of count -registers and even infix algebra inside their arguments: a count -register |\mycountA| or |\count 255| is admissible as numerator or also as -denominator, with no need to be prefixed by |\the| or |\number|. It is possible -to have as argument an algebraic expression as would be acceptable by a -|\numexpr...\relax|, under this condition: \emph{each of the numerator and - denominator is expressed with at most \emph{eight} - tokens}.\footnote{Attention! there is no problem with a \LaTeX{} - \csa{value}\texttt{\{countername\}} if if comes first, but if it comes later - in the input it will not get expanded, and braces around the name will be - removed and chaos\IMPORTANT{} will ensues inside a \csa{numexpr}. One should - enclose the whole input in \csa{the}\csa{numexpr}|...|\csa{relax} in such - cases.} The slash for rounded division in a |\numexpr| should be written with -braces |{/}| to not be confused with the \xintfracname delimiter between -numerator and denominator (braces will be removed internally). Example: -|\mycountA+\mycountB{/}17/1+\mycountA*\mycountB|, or |\count 0+\count -2{/}17/1+\count 0*\count 2|, but in the latter case the numerator has the -maximal allowed number of tokens (the braced slash counts for only one). -\centeredline{|\cnta 10 \cntb 35 \xintRaw - {\cnta+\cntb{/}17/1+\cnta*\cntb}|\digitstt{->\cnta 10 \cntb 35 \xintRaw - {\cnta+\cntb{/}17/1+\cnta*\cntb}}} For longer algebraic expressions using -count registers, there are two possibilities: -\begin{enumerate} -\item encompass each of the numerator and denominator in |\the\numexpr...\relax|, -\item encompass each of the numerator and denominator in |\numexpr {...}\relax|. -\end{enumerate} -\dverb|@ -\cnta 100 \cntb 10 \cntc 1 -\xintPRaw {\numexpr {\cnta*\cnta+\cntb*\cntb+\cntc*\cntc+ - 2*\cnta*\cntb+2*\cnta*\cntc+2*\cntb*\cntc}\relax/% - \numexpr {\cnta*\cnta+\cntb*\cntb+\cntc*\cntc}\relax }| -\cnta 100 \cntb 10 \cntc 1 -\centeredline{\digitstt{\xintPRaw {\numexpr - {\cnta*\cnta+\cntb*\cntb+\cntc*\cntc+ - 2*\cnta*\cntb+2*\cnta*\cntc+2*\cntb*\cntc}\relax/% - \numexpr {\cnta*\cnta+\cntb*\cntb+\cntc*\cntc}\relax }}} -The braces would not be accepted - as regular -|\numexpr|-syntax: and indeed, they - are removed at some point in the processing. - - -\subsection{Dimensions} -\label{sec:Dimensions} - -\meta{dimen} variables can be converted into (short) integers suitable for the -\xintname macros by prefixing them with |\number|. This transforms a dimension -into an explicit short integer which is its value in terms of the |sp| unit -(@1/65536@\,|pt|). -When |\number| is applied to a \meta{glue} variable, the stretch and shrink -components are lost. - -For \LaTeX{} users: a length is a \meta{glue} variable, prefixing a -length command defined by \csa{newlength} with \csa{number} will thus discard -the |plus| and |minus| glue components and return the dimension component as -described above, and usable in the \xintname bundle macros. - -This conversion is done automatically inside an -|\xintexpr|-essions, with tacit multiplication implied if prefixed by some -(integral or decimal) number. - -One may thus compute areas or volumes with no limitations, in units of |sp^2| -respectively |sp^3|, do arithmetic with them, compare them, etc..., and possibly -express some final result back in another unit, with the suitable conversion -factor and a rounding to a given number of decimal places. - -A \hyperref[tableofdimensions]{table of dimensions} illustrates that the -internal values used by \TeX{} do not correspond always to the closest rounding. -For example a millimeter exact value in terms of |sp| units is -\digitstt{72.27/10/2.54*65536=\xinttheexpr trunc(72.27/10/2.54*65536,3)\relax - ...} and \TeX{} uses internally \digitstt{\number\dimexpr 1mm\relax}|sp| (it -thus appears that \TeX{} truncates to get an integral multiple of the |sp| -unit). - - -% impossible avec le \ignorespaces mis par LaTeX de faire \number\dimexpr -% idem à la fin avec \unskip, si je veux xinttheexpr -\begin{figure*}[ht!] -\phantomsection\label{tableofdimensions} -\begingroup\let\ignorespaces\empty - \let\unskip\empty - \def\T{\expandafter\TT\number\dimexpr} - \def\TT#1!{\gdef\tempT{#1}} - \def\E{\expandafter\expandafter\expandafter - \EE\xintexpr reduce(} - \def\EE#1!{\gdef\tempE{#1}} -\centeredline{\begin{tabular}{% - >{\bfseries\strut}c% - c% - >{\E}c<{)\relax!}@{}% - >{\xintthe\tempE}r@{${}={}$}% - >{\xinttheexpr trunc(\tempE,3)\relax...}l% - >{\T}c<{!}@{}% - >{\tempT}r% - >{\xinttheexpr round(100*(\tempT-\tempE)/\tempE,4)\relax\%}c} - \hline - Unit&% - definition&% - \omit &% - \multicolumn{2}{c}{Exact value in \texttt{sp} units\strut}&% - \omit &% - \omit\parbox{2cm}{\centering\strut\TeX's value in \texttt{sp} units\strut}&% - \omit\parbox{2cm}{\centering\strut Relative error\strut}\\\hline - cm&0.01 m&72.27/2.54*65536&&&1cm&&\\ - mm&0.001 m&72.27/10/2.54*65536&&&1mm&&\\ - in&2.54 cm&72.27*65536&&&1in&&\\ - pc&12 pt&12*65536&&&1pc&&\\ - pt&1/72.27 in&65536&&&1pt&&\\ - bp&1/72 in&72.27*65536/72&&&1bp&&\\ - \omit\hfil\llap{3}bp\hfil&1/24 in&72.27*65536/24&&&3bp&&\\ - \omit\hfil\llap{12}bp\hfil&1/6 in&72.27*65536/6&&&12bp&&\\ - \omit\hfil\llap{72}bp\hfil&1 in&72.27*65536&&&72bp&&\\ - dd&1238/1157 pt&1238/1157*65536&&&1dd&&\\ - \omit\hfil\llap{11}dd\hfil&11*1238/1157 pt&11*1238/1157*65536&&&11dd&&\\ - \omit\hfil\llap{12}dd\hfil&12*1238/1157 pt&12*1238/1157*65536&&&12dd&&\\ - sp&1/65536 pt&1&&&1sp&&\\\hline - \multicolumn{8}{c}{\bfseries\large\TeX{} \strut dimensions}\\\hline -\end{tabular}} -\endgroup -\end{figure*} - -There is something quite amusing with the Didot point. According to the \TeX -Book, @1157@\,|dd|=@1238@\,|pt|. The actual internal value of @1@\,|dd| in \TeX{} is @70124@\,|sp|. We can use \xintcfracname to display the list of -centered convergents of the fraction @70124/65536@: -\centeredline{|\xintListWithSep{, }{\xintFtoCCv{70124/65536}}|} -% -\xintFor* #1 in {\xintFtoCCv{70124/65536}}\do {@#1@, }and we don't find -@1238/1157@ therein, but another approximant @1452/1357@! - -And indeed multiplying @70124/65536@ by @1157@, and respectively @1357@, we find -the approximations (wait for more, later): -\centeredline{``@1157@\,|dd|''\digitstt{=\xinttheexpr trunc(1157\dimexpr - 1dd\relax/\dimexpr 1pt\relax,12)\relax}\dots|pt|} -\centeredline{``@1357@\,|dd|''\digitstt{=\xinttheexpr trunc(1357\dimexpr - 1dd\relax/\dimexpr 1pt\relax,12)\relax}\dots|pt|} -and we seemingly discover that @1357@\,|dd|=@1452@\,|pt| is \emph{far more - accurate} than -the \TeX Book formula @1157@\,|dd|=@1238@\,|pt|~! -The formula to compute @N@\,|dd| was -% -\centeredline{|\xinttheexpr trunc(N\dimexpr 1dd\relax/\dimexpr - 1pt\relax,12)\relax}|} -% - -What's the catch? The catch is that \TeX{} \emph{does not} compute @1157@\,|dd| -like we just did: -\centeredline{@1157@\,|dd|=|\number\dimexpr 1157dd\relax/65536|% - \digitstt{=\xintTrunc{12}{\number\dimexpr 1157dd\relax/65536}}\dots|pt|} -\centeredline{@1357@\,|dd|=|\number\dimexpr 1357dd\relax/65536|% - \digitstt{=\xintTrunc{12}{\number\dimexpr 1357dd\relax/65536}}\dots|pt|} -We thus discover that \TeX{} (or rather here, e-\TeX{}, but one can check that -this works the same in \TeX82), uses indeed @1238/1157@ as a conversion factor, -and necessarily intermediate computations are done with more precision than is -possible with only integers less than @2^31@ (or @2^30@ for dimensions). Hence -the @1452/1357@ ratio is irrelevant, a misleading artefact of the necessary -rounding (or, as we see, truncating) for one |dd| as an integral number of -|sp|'s. - -Let us now -use |\xintexpr| to compute the value of the Didot point in millimeters, if -the above rule is exactly verified: \centeredline{|\xinttheexpr - trunc(1238/1157*25.4/72.27,12)\relax|% - \digitstt{=\xinttheexpr trunc(1238/1157*25.4/72.27,12)\relax}|...mm|} This -fits very well with the possible values of the Didot point as listed in the -\href{http://en.wikipedia.org/wiki/Point_%28typography%29#Didot}{Wikipedia Article}. -% -The value @0.376065@\,|mm| is said to be the \emph{the traditional value in - European printers' offices}. So the @1157@\,|dd|=@1238@\,|pt| rule refers to -this Didot point, or more precisely to the \emph{conversion factor} to be used -between this Didot and \TeX{} points. - -The actual value in millimeters of exactly one Didot point as implemented in -\TeX{} is -% -\centeredline -{|\xinttheexpr trunc(\dimexpr 1dd\relax/65536/72.27*25.4,12)\relax|} -\centeredline{% -\digitstt{=\xinttheexpr trunc(\dimexpr - 1dd\relax/65536/72.27*25.4,12)\relax}|...mm|} -The difference of circa @5@\AA\ is arguably tiny! - -% 543564351/508000000 - -By the way the \emph{European printers' offices \emph{(dixit Wikipedia)} Didot} is thus exactly -\centeredline{|\xinttheexpr reduce(.376065/(25.4/72.27))\relax|% - \digitstt{=\xinttheexpr reduce(.376065/(25.4/72.27))\relax}\,|pt|} -and the centered convergents of this fraction are \xintFor* #1 in -{\xintFtoCCv{543564351/508000000}}\do {@#1@\xintifForLast{.}{, }} We do recover -the @1238/1157@ therein! - -% As a final comment on the \hyperref[tableofdimensions]{table of dimensions}, we -% conclude that the ``Relative Error'' column is misleading as these relative -% errors by necessity decrease for integer multiples of the given dimension units. -% This was already indicated by the \textbf{72bp} row. - -% To conclude our comments on the -% \hyperref[tableofdimensions]{table of dimensions}, the big point, now known as -% \emph{Desktop Publishing Point} is less accurately implemented in \TeX{} than -% other units. Let us test for example the relation @1@\,|in|@=72@\,|bp|, the difference is -% % -% \centeredline{|\number\numexpr\dimexpr1in\relax-72*\dimexpr1bp\relax\relax|% -% \digitstt{=\number\numexpr\dimexpr1in\relax-72*\dimexpr1bp\relax\relax}\,|sp|} -% \centeredline{|\number\dimexpr1in-72bp\relax|% -% \digitstt{=\number\dimexpr1in-72bp\relax}\,|sp|} -% on the other hand -% \centeredline{|\xinttheexpr reduce(\dimexpr1in\relax-72.27*\dimexpr1pt\relax)\relax|} -% \centeredline -% \digitstt{=\xinttheexpr reduce(\dimexpr1in\relax-72.27*\dimexpr1pt\relax)\relax}\,|sp|=@-0.72@\,|sp|} -% \centeredline -% {\digitstt{=\number\dimexpr1in-72.27pt\relax}\,|sp|=@-0.72@\,|sp|} - - - -\section{\csh{ifcase}, \csh{ifnum}, ... constructs}\label{sec:ifcase} - -When using things such as |\ifcase \xintSgn{\A}| one has to make sure to leave -a space after the closing brace for \TeX{} to -stop its scanning for a number: once \TeX{} has finished expanding -|\xintSgn{\A}| and has so far obtained either |1|, |0|, or |-1|, a -space (or something `unexpandable') must stop it looking for more -digits. Using |\ifcase\xintSgn\A| without the braces is very dangerous, -because the blanks (including the end of line) following |\A| will be -skipped and not serve to stop the number which |\ifcase| is looking for. -With |\def\A{1}|: -\dverb|@ -\ifcase \xintSgn\A 0\or OK\else ERROR\fi ---> gives ERROR -\ifcase \xintSgn\A\space 0\or OK\else ERROR\fi ---> gives OK -\ifcase \xintSgn{\A} 0\or OK\else ERROR\fi ---> gives OK| -% \def\A{1} -% \ifcase \xintSgn\A 0\or OK\else ERROR\fi\ -% \ifcase \xintSgn{\A} 0\or OK\else ERROR\fi - -In order to use successfully |\if...\fi| constructions either as arguments to -the \xintname bundle expandable macros, or when building up a completely -expandable macro of one's own, one needs some \TeX nical expertise (see also -\autoref{fn:expansions} on page~\pageref{fn:expansions}). - -It is thus much to be recommended to opt rather for already existing expandable -branching macros, such as the ones which are provided by \xintname: -\csbxint{SgnFork}, \csbxint{ifSgn}, \csbxint{ifZero}, \csbxint{ifOne}, -\csbxint{ifNotZero}, \csbxint{ifTrueAelseB}, \csbxint{ifCmp}, \csbxint{ifGt}, -\csbxint{ifLt}, \csbxint{ifEq}, \csbxint{ifOdd}, and \csbxint{ifInt}. See their -respective documentations. All these conditionals always have either two or -three branches, and empty brace pairs |{}| for unused branches should not be -forgotten. - -If these tests are to be applied to standard \TeX{} short integers, it is more -efficient to use (under \LaTeX{}) the equivalent conditional tests from the -\href{http://www.ctan.org/pkg/etoolbox}{etoolbox}\footnote{\url{http://www.ctan.org/pkg/etoolbox}} -package. - - - -\section{Assignments}\label{sec:assign} - -\xintAssign \xintBezout{357}{323}\to\tmpA\tmpB\tmpU\tmpV\tmpD - -It might not be necessary to maintain at all times complete expandability. A -devoted syntax is provided to make these things more efficient, for example when -using the \csa{xintDivision} macro which computes both quotient and remainder at -the same time: -\centeredline{\csbxint{Assign}\csa{xintDivision}|{100}{3}|\csbnolk{to}|\A\B|} -\centeredline{\csbxint{Assign}\csa{xintDivision}% - |{\xintiPow {2}{1000}}{\xintFac{100}}|\csbnolk{to}|\A\B|} gives -\xintAssign\xintDivision{\xintiPow {2}{1000}}{\xintFac{100}}\to\A\B -|\meaning\A|\digitstt{: \expandafter\allowsplits\meaning\A\relax} and -|\meaning\B|\digitstt{: \expandafter\allowsplits\meaning\B\relax}. - -% -Another example (which uses \csbxint{Bezout} from the \xintgcdname package): -\centeredline{\csbxint{Assign}\csa{xintBezout}|{357}{323}|% -\csbnolk{to}|\A\B\U\V\D|} is equivalent to setting |\A| to \digitstt{\tmpA}, -|\B| to \digitstt{\tmpB}, |\U| to \digitstt{\tmpU}, |\V| to \digitstt{\tmpV}, -and |\D| to \digitstt{\tmpD}. And indeed -\digitstt{(\tmpU)$\times$\tmpA-(\tmpV)$\times$\tmpB$=$% -\xintiSub{\xintiMul\tmpU\tmpA}{\xintiMul\tmpV\tmpB}} is a Bezout Identity. - -Thus, what |\xintAssign| does is to first apply an -\hyperref[sec:expansions]{\fexpan sion} to what comes next; it then defines one -after the other (using |\def|; an optional argument allows to modify the -expansion type, see \autoref{xintAssign} for details), the macros found after -|\to| to correspond to the successive braced contents (or single tokens) located -prior to |\to|. - -\xintAssign -\xintBezout{3570902836026}{200467139463}\to\tmpA\tmpB\tmpU\tmpV\tmpD - -\centeredline{\csbxint{Assign}\csa{xintBezout}|{3570902836026}{200467139463}|% - \csbnolk{to}|\A\B\U\V\D|} -\noindent -gives then |\U|\digitstt{: - \expandafter\allowsplits\meaning\tmpU\relax}, - |\V|\digitstt{: - \expandafter\allowsplits\meaning\tmpV\relax} and |\D|\digitstt{=\tmpD}. - -% -In situations when one does not know in advance the number of items, one has -\csbxint{AssignArray} or its synonym \csbxint{DigitsOf}: -\centeredline{\csbxint{DigitsOf}\csa{xintiPow}|{2}{100}|\csbnolk{to}\csa{DIGITS}} -This defines \csa{DIGITS} to be macro with one parameter, \csa{DIGITS}|{0}| -gives the size |N| of the array and \csa{DIGITS}|{n}|, for |n| from |1| to |N| -then gives the |n|th element of the array, here the |n|th digit of @2^{100}@, -from the most significant to the least significant. As usual, the generated -macro \csa{DIGITS} is completely expandable (in two steps). As it wouldn't make -much sense to allow indices exceeding the \TeX{} bounds, the macros created by -\csbxint{AssignArray} put their argument inside a \csa{numexpr}, so it is -completely expanded and may be a count register, not necessarily prefixed by -|\the| or |\number|. Consider the following code snippet: -% -\dverb+@ -\newcount\cnta -\newcount\cntb -\begingroup -\xintDigitsOf\xintiPow{2}{100}\to\DIGITS -\cnta = 1 -\cntb = 0 -\loop -\advance \cntb \xintiSqr{\DIGITS{\cnta}} -\ifnum \cnta < \DIGITS{0} -\advance\cnta 1 -\repeat - -|2^{100}| (=\xintiPow {2}{100}) has \DIGITS{0} digits and the sum of -their squares is \the\cntb. These digits are, from the least to -the most significant: \cnta = \DIGITS{0} -\loop \DIGITS{\cnta}\ifnum \cnta > 1 \advance\cnta -1 , \repeat. -\endgroup -+ - -\edef\z{\xintiPow {2}{100}} - -\begingroup -\xintDigitsOf\z\to\DIGITS -\cnta = 1 -\cntb = 0 -\loop -\advance \cntb \xintiSqr{\DIGITS{\cnta}} -\ifnum \cnta < \DIGITS{0} -\advance\cnta 1 -\repeat - -@2^{100}@ (=\z) has \DIGITS{0} digits and the sum of -their squares is \the\cntb. These digits are, from the least to -the most significant: \cnta = \DIGITS{0} -\loop \DIGITS{\cnta}\ifnum \cnta > 1 \advance\cnta -1 , \repeat. -\endgroup - -% We used a group in order to release the memory taken by the -% \csa{DIGITS} array: indeed internally, besides \csa{DIGITS} itself, -% additional macros are defined which are \csa{DIGITS0}, \csa{DIGITS00}, -% \csa{DIGITS1}, \csa{DIGITS2}, ..., \csa{DIGITSN}, where |N| is the size of -% the array (which is the value returned by |\DIGITS{0}|; the digits -% are parts of the names not arguments). - -% The command \csbxint{RelaxArray}\csa{DIGITS} sets all these macros to -% \csa{relax}, but it was simpler to put everything withing a group. - -Warning: \csbxint{Assign}, \csbxint{AssignArray} and \csbxint{DigitsOf} -\emph{do not do any check} on whether the macros they define are already -defined. - -% In the example above, we deliberately broke all rules of complete expandability, -% but had we wanted to compute the sum of the digits, not the sum of the squares, -% we could just have written: \csbxint{iiSum}|{\xintiPow{2}{100}}|\digitstt{=% -% \xintiiSum\z}. Indeed, \csa{xintiiSum} is usually used on braced items as in -% \centeredline{% -% \csbxint{iiSum}|{{123}{-345}{\xintFac{7}}{\xintiOpp{\xintRem{3347}{591}}}}|% -% \digitstt{=% -% \xintiiSum{{123}{-345}{\xintFac{7}}{\xintiOpp{\xintRem{3347}{591}}}}}} but -% in the previous example each digit of @2^{100}@ was treated as one item due to -% the rules of \TeX{} for parsing macro arguments. - -% Note: |{-\xintRem{3347}{591}}| would not be a valid input, because -% the expansion will apply only to the minus sign and leave -% unaffected the |\xintRem|. So we used \csa{xint}\-|iOpp| which replaces -% a number with its opposite. - - -% As a last example with \csa{xintAssignArray} here is one line -% extracted from the source code of the \xintgcdname macro -% \csbxint{TypesetEuclideAlgorithm}: -% \centeredline{|\xintAssignArray\xintEuclideAlgorithm -% {#1}{#2}\to\U|} -% This is done inside a group. After this command |\U{1}| contains -% the number |N| of steps of the algorithm (not to be confused with -% |\U{0}=2N+4| which is the number of elements in the |\U| array), -% and the GCD is to be found in |\U{3}|, a convenient location -% between |\U{2}| and |\U{4}| which are (absolute values of the -% expansion of) the -% initial inputs. Then follow |N| quotients and remainders -% from the first to the last step of the algorithm. The -% \csa{xintTypesetEuclideAlgorithm} macro organizes this data -% for typesetting: this is just an example of one way to do it. - -\section{Utilities for expandable manipulations}\label{sec:utils} - -The package now has more utilities to deal expandably with `lists of things', -which were treated un-expandably in the previous section with \csa{xintAssign} -and \csa{xintAssignArray}: \csbxint{ReverseOrder} and \csbxint{Length} since the -first release, \csbxint{Apply} and \csbxint{ListWithSep} since |1.04|, -\csbxint{RevWithBraces}, \csbxint{CSVtoList}, \csbxint{NthElt} since |1.06|, -\csbxint{ApplyUnbraced}, since |1.06b|, \csbxint{loop} and \csbxint{iloop} since -|1.09g|.\footnote{All these utilities, as well as \csbxint{Assign}, - \csbxint{AssignArray} and the \csbxint{For} loops are now available from the - \xinttoolsname package, independently of the big integers facilities of - \xintname.} - -\edef\z{\xintiPow {2}{100}} - -As an example the following code uses only expandable operations: -\dverb+@ -|2^{100}| (=\xintiPow {2}{100}) has \xintLen{\xintiPow {2}{100}}} digits -and the sum of their squares is -\xintiiSum{\xintApply {\xintiSqr}{\xintiPow {2}{100}}}. -These digits are, from the least to the most significant: -\xintListWithSep {, }{\xintRev{\xintiPow {2}{100}}}. The thirteenth most -significant digit is \xintNthElt{13}{\xintiPow {2}{100}}. The seventh -least significant one is \xintNthElt{7}{\xintRev{\xintiPow {2}{100}}}. -+ -|2^{100}| (=\z) has \xintLen{\z} digits and the sum of -their squares is \xintiiSum{\xintApply\xintiSqr\z}. These digits are, from the -least to the most significant: \xintListWithSep {, }{\xintRev\z}. The -thirteenth most -significant digit is \xintNthElt{13}{\z}. The seventh -least significant one is \xintNthElt{7}{\xintRev\z}. - -It would be more efficient to do once and for all -|\oodef\z{\xintiPow {2}{100}}|, and then use |\z| in place of - |\xintiPow {2}{100}| everywhere as this would spare the CPU some repetitions. - -Expandably computing primes is done in \autoref{xintSeq}. - - -\section{A new kind of for loop} - -As part of the \hyperref[sec:tools]{utilities} coming with the \xinttoolsname -package, there is a new kind of for loop, \csbxint{For}. Check it out -(\autoref{xintFor}). - -\section{A new kind of expandable loop} - -Also included in \xinttoolsname, \csbxint{iloop} is an expandable loop giving -access to an iteration index, without using count registers which would break -expandability. Check it out (\autoref{xintiloop}). - -\section{Exceptions (error messages)} - -In situations such as division by zero, the package will insert in the -\TeX{} processing an undefined control sequence (we copy this method -from the |bigintcalc| package). This will trigger the writing to the log -of a message signaling an undefined control sequence. The name of the -control sequence is the message. The error is raised \emph{before} the -end of the expansion so as to not disturb further processing of the -token stream, after completion of the operation. Generally the problematic -operation will output a zero. Possible such error message control -sequences: -\dverb|@ -\xintError:ArrayIndexIsNegative -\xintError:ArrayIndexBeyondLimit -\xintError:FactorialOfNegativeNumber -\xintError:FactorialOfTooBigNumber -\xintError:DivisionByZero -\xintError:NaN -\xintError:FractionRoundedToZero -\xintError:NotAnInteger -\xintError:ExponentTooBig -\xintError:TooBigDecimalShift -\xintError:TooBigDecimalSplit -\xintError:RootOfNegative -\xintError:NoBezoutForZeros -\xintError:ignored -\xintError:removed -\xintError:inserted -\xintError:bigtroubleahead -\xintError:unknownfunction| - -\section{Common input errors when using the package macros} - -\edef\x{\xintMul {3}{5}/\xintMul{7}{9}} - -Here is a list of common input errors. Some will cause compilation errors, -others are more annoying as they may pass through unsignaled. -\begin{itemize} -\item using |-| to prefix some macro: |-\xintiSqr{35}/271|.\footnote{to the - contrary, this \emph{is} - allowed inside an |\string\xintexpr|-ession.} -\item using one pair of braces too many |\xintIrr{{\xintiPow {3}{13}}/243}| (the - computation goes through with no error signaled, but the result is completely - wrong). -\item using |[]| and decimal points at the same time |1.5/3.5[2]|, or with a - sign in the denominator |3/-5[7]|. The scientific notation has no such - restriction, the two inputs |1.5/-3.5e-2| and |-1.5e2/3.5| are equivalent: - |\xintRaw{1.5/-3.5e-2}|\digitstt{=\xintRaw{1.5/-3.5e-2}}, - |\xintRaw{-1.5e2/3.5}|\digitstt{=\xintRaw{-1.5e2/3.5}}. -\item specifying numerators and - denominators with macros producing fractions when \xintfracname is loaded: - |\edef\x{|\allowbreak|\xintMul {3}{5}/\xintMul{7}{9}}|. This expands to - \texttt{\x} which is - invalid on input. Using this |\x| in a fraction macro will most certainly - cause a compilation error, with its usual arcane and undecipherable - accompanying message. The fix here would be to use |\xintiMul|. The simpler - alternative with package \xintexprname: - |\xinttheexpr 3*5/(7*9)\relax|. -\item generally speaking, using in a context expecting an integer (possibly - restricted to the \TeX{} bound) a macro or expression which returns a - fraction: |\xinttheexpr 4/2\relax| outputs \digitstt{\xinttheexpr 4/2\relax}, - not @2@. Use |\xintNum {\xinttheexpr 4/2\relax}| or |\xinttheiexpr 4/2\relax| - (which rounds the result to the nearest integer, here, the result is already - an integer) or |\xinttheiiexpr 4/2\relax| (but |/| therein is euclidean - quotient, which on positive operands is like truncating to the integer part, - not rounding). -\end{itemize} - - -\section{Package namespace} - -Inner macros of \xinttoolsname, \xintname, \xintfracname, \xintexprname, -\xintbinhexname, \xintgcdname, \xintseriesname, and \xintcfracname{} all begin -either with |\XINT_| or with |\xint_|.\footnote{starting with release |1.06b| - the style files use for macro names a more modern underscore |\_| rather than - the \texttt{\char`\@} sign. A handful of private macros starting with - |\string\XINT| do not have the underscore for technical reasons: - \csa{XINTsetupcatcodes}, \csa{XINTdigits} and macros with names starting with - |XINTinFloat| or |XINTinfloat|.} The package public commands all start with -|\xint|. Some other control sequences are used only as delimiters, and left -undefined, they may have been defined elsewhere, their meaning doesn't matter -and is not touched. - -\xinttoolsname defines \hyperref[odef]{\ttfamily\char92odef}, -\hyperref[oodef]{\ttfamily\char92oodef}, \hyperref[fdef]{\ttfamily\char92fdef}, -but only if macros with these names do not already exist (|\xintoodef| etc... -are defined anyhow for use in \csbxint{Assign} and \csbxint{AssignArray}). - -{\makeatother The \xintname packages presuppose that the \csa{space}, -\csa{empty}, |\m@ne|, |\z@| and |\@ne| control sequences -have their meanings as in Plain \TeX{} or \LaTeX2e.} - - -\section{Loading and usage} - -\dverb|@ -Usage with LaTeX: \usepackage{xinttools} - \usepackage{xint} % (loads xinttools) - \usepackage{xintfrac} % (loads xint) - \usepackage{xintexpr} % (loads xintfrac) - - \usepackage{xintbinhex} % (loads xint) - \usepackage{xintgcd} % (loads xint) - \usepackage{xintseries} % (loads xintfrac) - \usepackage{xintcfrac} % (loads xintfrac) - -Usage with TeX: \input xinttools.sty\relax - \input xint.sty\relax % (loads xinttools) - \input xintfrac.sty\relax % (loads xint) - \input xintexpr.sty\relax % (loads xintfrac) - - \input xintbinhex.sty\relax % (loads xint) - \input xintgcd.sty\relax % (loads xint) - \input xintseries.sty\relax % (loads xintfrac) - \input xintcfrac.sty\relax % (loads xintfrac) -| - -We have added, directly copied from packages by \textsc{Heiko Oberdiek}, a -mechanism of re-load and \eTeX{} detection, especially for Plain \TeX{}. As -\eTeX{} is required, the executable |tex| can not be used, |etex| or |pdftex| -(version |1.40| or later) or ..., must be invoked. Each package refuses to be -loaded twice and automatically loads the other components on which it has -dependencies.\footnote{exception: \xintexprname needs the user to explicitely - load \xintgcdname, resp. \xintbinhexname, if use is to be made in - \csa{xintexpr} of the \texttt{lcm} and \texttt{gcd} functions, and, resp., - hexadecimal numbers.} - -Also initially inspired from the \textsc{Heiko Oberdiek} packages we have -included a complete catcode protection mecanism. The packages may be loaded in -any catcode configuration satisfying these requirements: the percent is of -category code comment character, the backslash is of category code escape -character, digits have category code other and letters have category code -letter. Nothing else is assumed, and the previous configuration is restored -after the loading of each one of the packages. - -This is for the loading of the packages. - -For the input of numbers as macro arguments the minus sign must have its -standard category code (``\emph{other}''). Similarly the slash used for -fractions must have its standard category code. And the square brackets, if made -use of in the input, also must be of category code \emph{other}. The `e' of the -scientific notation must be of category code \emph{letter}. - -All these requirements (which are anyhow satisfied by default) are -relaxed for the contents of an |\xintexpr|-ession: spaces are gobbled, -catcodes mostly do not matter, the |e| of scientific notation may be |E| -(on input) \dots{} - - -\section{Installation}\label{sec:install} - -\begingroup -\def\MacroFont {\ttfamily\small\baselineskip11pt\relax\catcode`\"=12 } -\dverb!@ -A. Installation using xint.tds.zip: ------------------------------------ - -obtain xint.tds.zip from CTAN: - http://mirror.ctan.org/install/macros/generic/xint.tds.zip - -cd to the download repertory and issue - unzip xint.tds.zip -d <TEXMF> -for example: (assuming standard access rights, so sudo needed) - sudo unzip xint.tds.zip -d /usr/local/texlive/texmf-local - sudo mktexlsr - -On Mac OS X, installation into user home folder: - unzip xint.tds.zip -d ~/Library/texmf - -B. Installation after file extractions: ---------------------------------------- - -obtain xint.dtx, xint.ins and the README from CTAN: - http://www.ctan.org/pkg/xint - -- "tex xint.ins" generates the style files -(pre-existing files in the same repertory will be overwritten). - -- without xint.ins: "tex or latex or pdflatex or xelatex xint.dtx" -will also generate the style files (and xint.ins). - -xint.tex is also extracted, use it for the documentation: - -- with latex+dvipdfmx: latex xint.tex thrice then dvipdfmx xint.dvi -Ignore dvipdfmx warnings, but if the pdf file has problems with fonts -(possibly from an old dvipdfmx), use then rather pdflatex or xelatex. - -- with pdflatex or xelatex: run it directly thrice on xint.dtx, or run -it on xint.tex after having edited the suitable toggle therein. - -When compiling xint.tex, the documentation is by default produced -with the source code included. See instructions in the file for -changing this default. - -When compiling directly xint.dtx, the documentation is produced -without the source code (latex+dvips or pdflatex or xelatex). - -Finishing the installation: (on first installation the destination -repertories may need to be created) - - xinttools.sty | - xint.sty | - xintfrac.sty | - xintexpr.sty | --> TDS:tex/generic/xint/ - xintbinhex.sty | - xintgcd.sty | - xintseries.sty | - xintcfrac.sty | - - xint.dtx --> TDS:source/generic/xint/ - xint.ins --> TDS:source/generic/xint/ - xint.tex --> TDS:source/generic/xint/ - - xint.pdf --> TDS:doc/generic/xint/ - README --> TDS:doc/generic/xint/ - -Depending on the TDS destination and the TeX installation, it may be -necessary to refresh the TeX installation filename database (mktexlsr)! -\endgroup - -\section{The \csh{xintexpr} math parser (I)} -\label{sec:exprsummary} - -% 27 octobre 2013 plus de problème avec &... il n'est plus actif (ouf) -\xintexprSafeCatcodes -\newcommand\formula[3]{\xinttheexpr round((#1 & (#2 | #3)) * (355/113*#3 - - (#1 - #2/2)^2), 8)\relax } -\xintexprRestoreCatcodes - - -Here is some random formula, defining a \LaTeX{} command with three parameters, -\centeredline{\verb$\newcommand\formula[3]$} -\centeredline{\verb${\xinttheexpr round((#1 & (#2 | #3)) * (355/113*#3 - (#1 - - #2/2)^2), 8) \relax}$} - -\smallskip - -Let |a=#1|, |b=#2|, |c=#3| be the parameters. The first term is the logical -operation |a and (b or c)| where a number or fraction has truth value @1@ if it -is non-zero, and @0@ otherwise. So here it means that |a| must be non-zero as -well as |b| or |c|, for this first operand to be @1@, else the formula returns -@0@. This multiplies a second term which is algebraic. Finally the result (where -all intermediate computations are done \emph{exactly}) is rounded to a value -with @8@ digits after the decimal mark, and printed. -\centeredline{|\formula - {771.3/9.1}{1.51e2}{37.73}| expands to - \digitstt{\formula {771.3/9.1}{1.51e2}{37.73}}} -Note that |#1|, |#2|, and |#3| are not protected by parentheses in the -definition of |\formula|, this is something to keep in mind if for example we -want to use |2+5| as third argument: it should be |(2+5)| then. - - -\begingroup % 9 octobre pour une meilleure gestion de l'indentation -\leftmargini 0pt -\list\labelitemi{\def\makelabel#1{\hss\llap{#1}}\listparindent\parindent - \labelwidth\parindent - \itemindent\labelwidth}% -% -\item as everything gets expanded, the characters - \verb$+,-,*,/,^,!,&,|,?,:,<,>,=,(,),"$ and the comma, which may appear - in the |infix| syntax, should not (if actually used in the expression) be - active (for example from serving as - shorthands for some language in the |Babel| system). - The command \csbxint{exprSafeCatcodes} resets these characters to their - standard catcodes and \csbxint{exprRestoreCatcodes} restores the status - prevailing at the time of the previous \csa{xintexprSafeCatcodes}. -\item many expressions have equivalent macro formulations written without - |\xinttheexpr|.\footnote{Not everything allows a straightforward reformulation - because the package macros only \fexpan d their arguments while - \csa{xintexpr} expands everything from left to right.} Here for |\formula| - we could have used: \centeredline {|\xintRound {8}{\xintMul {\xintAND - {#1}{\xintOR {#2}{#3}}}{\xintSub |} \centeredline {| {\xintMul - {355/113}{#3}}{\xintPow {\xintSub {#1}{\xintDiv {#2}{2}}}{2}}}}|} with - the inherent difficulty of keeping up with braces and everything... -\item if such a formula is used thousands of times in a document (for plots?), - this could impact some parts of the \TeX{} program memory (for technical - reasons explained in \autoref{sec:expr}). So, a utility \csbxint{NewExpr} - is provided to do the work of translating an |\xintexpr|-ession with - parameters into a chain of macro evaluations.\footnote{As its makes some macro - definitions, it is not an expandable command. It does not need protection - against active characters as it does it itself.} With - \centeredline{|\xintNewExpr\formula[3]|} - \centeredline{\verb${ round((#1 & (#2 | #3)) * (355/113*#3 - (#1 - #2/2)^2), - 8) }$} - one gets a command |\formula| with three parameters and meaning: - -\xintNewExpr\formula[3] -{ round((#1 & (#2 | #3)) * (355/113*#3 - (#1 - #2/2)^2), - 8) } - -{\centering\ttfamily - -\meaning\formula - -}This does the same thing as the hand-written version from the previous item -(but expands in only two steps).\footnote{But the hand-written version as well - as the \csa{xintNewExpr} generated one differ from the original \csa{formula} - command which allowed each of its argument to use all the operators and - functions recognized by \csa{xintexpr}, and this aspect is lost. To recover it - the arguments themselves should be passed as \csa{xinttheexpr..\char92relax} - to the defined macro.} The use -even thousands of times of such an |\xintNewExpr|-generated |\formula| has no -memory impact. -\item count registers and |\numexpr|-essions are accepted (LaTeX{}'s counters - can be inserted using |\value|) without needing |\the| or |\number| as prefix. - Also dimen registers and control sequences, skip registers and control - sequences (\LaTeX{}'s lengths), |\dimexpr|-essions, |\glueexpr|-essions are - automatically unpacked using |\number|, discarding the stretch and shrink - components and giving the dimension value in |sp| units (@1/65536@th of a - \TeX{} point). Furthermore, tacit multiplication is implied, when the - register, variable, or expression if immediately prefixed by a (decimal) - number. -\item tacit multiplication (the parser inserts a |*|) applies when the parser is - currently scanning the digits of a number (or its decimal part), or is looking - for an infix operator, and: (1.)\inmarg{v1.09i}~\emph{encounters a register, - variable or \eTeX{} expression (as described in the previous item)}, - (2.)\inmarg{v1.09j}~\emph{encounters a sub-\csa{xintexpr}-ession}, or - (3.)\inmarg{\\ v1.09k}~\emph{encounters an opening parenthesis.} -\item so far only |\xinttheexpr| was mentioned: there is also |\xintexpr| which, - like a |\numexpr|, needs a prefix which is called \csbxint{the}. Thus - \csbxint{theexpr} as was done in the definition of |\formula| is equivalent to - \csbxint{the}|\xintexpr|. -\item This latter form is convenient when one has defined for - example: -% -\centeredline{|\def\x {\xintexpr \a + \b \relax}| or |\edef\x {\xintexpr \a+\b\relax}|} -% -One may then do |\xintthe\x|, either for printing the result -on the page or use it in some other package macros. The |\edef| does the -computation but keeps it in an internal private format. -Naturally, the |\edef| is only possible if |\a| and |\b| are already defined. -\item in both cases (the `yet-to-be computed' and the -`already computed') |\x| can then be inserted in other expressions, as -for example -% -\centeredline {|\edef\y {\xintexpr \x^3\relax}|} -% -This would have worked also with |\x| defined as |\def\x {(\a+\b)}| but -|\edef\x| would not have been an option then, and |\x| could have been used only -inside an |\xintexpr|-ession, whereas the previous |\x| can also be used as -|\xintthe\x| in any context triggering the expansion of |\xintthe|. -\item sometimes one needs an integer, not a fraction or decimal number. The - |round| function rounds to the nearest integer, and |\xintexpr - round(...)\relax| has an alternative and equivalent syntax as \csbxint{iexpr}| - ... \relax|. There is also \csbxint{theiexpr}. The rounding is applied to the - final result only, intermediate computations are not rounded. -\item \csbxint{iiexpr}|..\relax| and \csbxint{theiiexpr}|..\relax| deal only - with (long) integers and skip the overhead of the fraction internal format. - The infix operator |/| does euclidean division, thus |2+5/3| will not be - treated exactly but be like |2+1|. -\item there is also \csbxint{boolexpr}| ... \relax| and \csbxint{theboolexpr}| - ... \relax|. Same as |\xintexpr| with the final result converted to - @1@ - if it is not zero. See also \csbxint{ifboolexpr} - (\autoref{xintifboolexpr}) and the \hyperlink{item:bool}{discussion} - of the |bool| and |togl| functions in \autoref{sec:exprsummary}. Here is an - example: -\begingroup -\def\MacroFont {\ttfamily\parskip0pt \parindent 15pt \baselineskip 12pt \relax } -\dverb!@ -\xintNewBoolExpr \AssertionA[3]{ #1 & (#2|#3) } -\xintNewBoolExpr \AssertionB[3]{ #1 | (#2) } -\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) } -\xintNewBoolExpr \AssertionC[3]{ xor(#1,#2,#3) } -\xintFor #1 in {0,1} \do {% - \xintFor #2 in {0,1} \do {% - \xintFor #3 in {0,1} \do {% - \centeredline{#1 AND (#2 OR #3) is \AssertionA {#1}{#2}{#3}\hfil - #1 OR (#2 AND #3) is \AssertionB {#1}{#2}{#3}\hfil - #1 XOR #2 XOR #3 is \AssertionC {#1}{#2}{#3}}}}} - -% -\item there is also \csbxint{floatexpr}| ... \relax| where the algebra is done - in floating point approximation (also for each intermediate result). Use the - syntax |\xintDigits:=N;| to set the precision. Default: @16@ digits. - \centeredline{|\xintthefloatexpr 2^100000\relax:| \digitstt{\xintthefloatexpr - 2^100000\relax }} The square-root operation can be used in |\xintexpr|, it - is computed as a float with the precision set by |\xintDigits| or by the - optional second argument: \centeredline{|\xinttheexpr sqrt(2,60)\relax|:} - \centeredline{\digitstt{\xinttheexpr sqrt(2,60)\relax }} Notice the |a/b[n]| - notation: usually the denominator |b| even if |1| gets printed; it does not - show here because the square root is computed by a version of - \csbxint{FloatSqrt} which for efficiency when used in such expressions outputs - the result in a format |d_1 d_2 .... d_P [N]| equivalent to the usual float - output format |d_1.d_2...d_P e<expon.>|. To get a float - format, it is easier to use an |\xintfloatexpr|, but the precision must be set - using the non expandable |\xintDigits:=60;| assignment, there is no optional - parameter possible currently to |\xintfloatexpr|: -% -\centeredline{|\xintDigits:=60;\xintthefloatexpr sqrt(2)\relax|} -\centeredline{\digitstt{\xintDigits:=60;\xintthefloatexpr sqrt(2)\relax}} -% -Or, without manipulating |\xintDigits|, another option to convert to float a -computation done by an |\xintexpr|: -\centeredline{|\xintFloat[60]{\xinttheexpr sqrt(2,60)\relax}|} -\centeredline{\digitstt{\xintFloat[60]{\xinttheexpr sqrt(2,60)\relax}}} -% -Floats - are quickly indispensable when using the power function (which can only have - an integer exponent), as exact results will easily have hundreds, if not - thousands, of digits. -% -\centeredline{|\xintDigits:=48; - \xintthefloatexpr 2^100000\relax|: } -\centeredline{\xintDigits:=48;\digitstt{\xintthefloatexpr 2^100000\relax}} -% -\item hexadecimal \TeX{} number\inmarg{New with 1.09k!} denotations - (\emph{i.e.}, with a |"| prefix) are recognized by the |\xintexpr| parser and - its variants. Except in |\xintiiexpr|, a (possibly empty) fractional part - with the dot |.| as ``hexadecimal'' mark is allowed. -% -\centeredline{|\xinttheexpr "FEDCBA9876543210\relax|$\to$\digitstt{\xinttheexpr - "FEDCBA9876543210\relax}} -\centeredline{|\xinttheiexpr 16^5-("F75DE.0A8B9+"8A21.F5746+16^-5)\relax|$\to$\digitstt{\xinttheiexpr 16^5-("F75DE.0A8B9+"8A21.F5746+16^-5)\relax}} -% -Letters must be uppercased, as with standard - \TeX{} hexadecimal denotations. Loading the \xintbinhexname package is required - for this functionality. -\endlist -\endgroup - -\section{The \csh{xintexpr} math parser (II)} -\label{sec:exprsummaryII} - -An expression is built with infix operators (including comparison and boolean -operators), parentheses, functions, and the two branching operators |?| and |:|. -The parser expands everything from the left to the right and everything may thus -be revealed step by step by expansion of macros. Spaces anywhere are allowed. - -Note that |2^-10| is perfectly accepted input, no need for parentheses; -operators of power |^|, division |/|, and subtraction |-| are all -left-associative: |2^4^8| is evaluated as |(2^4)^8|. The minus sign as prefix -has various precedence levels: |\xintexpr -3-4*-5^-7\relax| evaluates as -|(-3)-(4*(-(5^(-7))))| and |-3^-4*-5-7| as |(-((3^(-4))*(-5)))-7|. - -If one uses directly macros within |\xintexpr..\relax|, rather than the -operators or the functions which are described next, one should take into -account that: -\begin{enumerate} -\item the parser will not see the macro arguments, (but they may themselves be - set-up as |\xinttheexpr...\relax|), -\item the output format of most \xintfracname macros is |A/B[N]|, and square - brackets are \emph{not understood by the parser}. One \emph{must} enclose the - macro and its arguments inside a brace pair |{..}|, which will be recognized - and treated specially, -\item a macro outputting numbers in scientific notation |x.yEz| (either with a - lowercase |e| or uppercase |E|), must \emph{not} be enclosed - in a brace pair, this is the exact opposite of the |A/B[N]| case; scientific - numbers, explicit or implicit, should just be inserted directly in the - expression. -\end{enumerate} - -One may insert a sub-|\xintexpr|-expression within a larger one. Each one of -|\xintexpr|, |\xintiexpr|, |\xintfloatexpr|, |\xintboolexpr| may be inserted in -another one. On the other hand the integer only |\xintiiexpr| will generally -choke on a sub-|\xintexpr| as the latter (except if it did not do any operation -or had an overall top level |round| or |trunc| or |?(..)| or\dots) produces (in -internal format) an |A/B[N]| which the strictly integer only \csbxint{iiexpr} -does not understand. See \autoref{xintiiexpr} for more information. - -Here is, listed from the highest priority to the lowest, the complete list of -operators and functions. Functions are at the top level of priority. Next are -the postfix operators: |!| for the factorial, |?| and |:| are two-fold way and -three-fold way branching constructs. Also at the top level of priority the |e| -and |E| of the scientific notation and the |"|\inmarg{\string" is new in 1.09k} -for hexadecimal numbers, then power, multiplication/division, -addition/subtraction, comparison, and logical operators. At the lowest level: -commas then parentheses. - - -The |\relax| at the end of an expression is \emph{mandatory}. - - % 1.09c ajoute bool et togl - % 1.09a: - % reduce, - % sqr, sqrt, abs, sgn, floor, ceil, quo, rem, round, trunc, float, gcd, lcm, - % max, min, sum, prd, add, mul, not, all, any, xor - % ?, !, if, ifsgn, ?, :. - -\newcommand\ctexttt [1]{\begingroup\color[named]{DarkOrchid}\ttfamily\bfseries - #1\endgroup} - -\begingroup % 9 octobre pour la gestion de l'indentation et couleurs -\leftmargini 0pt -\leftmarginii .5\parindent -\list\labelitemi{\def\makelabel#1{\hss\llap{#1}}\listparindent\parindent - \labelwidth\parindent - \itemindent\labelwidth}% -\item - Functions are at the same top level of priority. All functions even - |?| and |!| (as prefix) require parentheses around their argument - (possibly a comma separated list). - \begin{framed} - \ctexttt{floor, ceil, frac, reduce, sqr, abs, sgn, ?, !, not, bool, - togl, round, trunc, float, sqrt, quo, rem, if, ifsgn, all, any, - xor, add (=sum), mul (=prd), max, min, gcd, lcm.} - - |quo| and |rem| - operate only on integers; |gcd| and |lcm| also and require - \xintgcdname loaded; |togl| requires the |etoolbox| package; |all|, |any|, - |xor|, |add|, |mul|, |max| and |min| are functions with arbitrarily many - comma separated arguments. - \end{framed} - \begin{description} - \item[functions with one (numeric) argument] (numeric: any expression leading - to an integer, decimal number, fraction, or floating number in scientific - notation) \ctexttt{floor, ceil, frac, reduce, sqr, abs, sgn, ?, !, not}. The - |?(x)| function returns the truth value, @1@ if |x<>0|, @0@ if |x=0|. The - |!(x)| is the logical not. The |reduce| function puts the fraction in - irreducible form. The |frac| function is fractional part, - same sign as the number:\newline - \null\quad\quad|\xinttheexpr - frac(-3.57)\relax|$\to$\digitstt{\xinttheexpr frac(-3.57)\relax}\newline - \null\quad\quad|\xinttheexpr - trunc(frac(-3.57),2)\relax|$\to$\digitstt{\xinttheexpr - trunc(frac(-3.57),2)\relax}\newline - \null\quad\quad|\xintthefloatexpr - frac(-3.57)\relax|$\to$\digitstt{\xintthefloatexpr - frac(-3.57)\relax}.\newline - Like - the other functions |!| and |?| \emph{must} use parentheses. - - \item[functions with one (alphabetical) argument] \hypertarget{item:bool} - {\ctexttt{bool,togl}}. - |bool(name)| returns @1@ if the \TeX{} conditional |\ifname| would - act as |\iftrue| and @0@ otherwise. This works with conditionals - defined by |\newif| (in \TeX{} or \LaTeX{}) or with primitive - conditionals such as |\ifmmode|. For example: - \centeredline{|\xintifboolexpr{25*4-if(bool(mmode),100,75)}{YES}{NO}|} - will return $\xintifboolexpr{25*4-if(bool(mmode),100,75)}{YES}{NO}$ - if executed in math mode (the computation is then $100-100=0$) and - \xintifboolexpr{25*4-if(bool(mmode),100,75)}{YES}{NO} if not (the - \ctexttt{if} conditional is described below; the - \csbxint{ifboolexpr} test automatically encapsulates its first - argument in an |\xintexpr| and follows the first branch if the - result is non-zero (see \autoref{xintifboolexpr})). - - The alternative syntax |25*4-\ifmmode100\else75\fi| could have been used - here, the usefulness of |bool(name)| lies in the availability in the - |\xintexpr| syntax of the logic operators of conjunction |&|, inclusive - disjunction \verb+|+, negation |!| (or |not|), of the multi-operands - functions |all|, |any|, |xor|, of the two branching operators |if| and - |ifsgn| (see also |?| and |:|), which allow arbitrarily complicated - combinations of various |bool(name)|. - - Similarly |togl(name)| returns @1@ - if the \LaTeX{} package - \href{http://www.ctan.org/pkg/etoolbox}{etoolbox}\footnote{\url{http://www.ctan.org/pkg/etoolbox}} - has been used to define a toggle named |name|, and this toggle is - currently set to |true|. Using |togl| in an |\xintexpr..\relax| - without having loaded - \href{http://www.ctan.org/pkg/etoolbox}{etoolbox} will result in an - error from |\iftoggle| being a non-defined macro. If |etoolbox| is - loaded but |togl| is used on a name not recognized by |etoolbox| the - error message will be of the type ``ERROR: Missing |\endcsname| - inserted.'', with further information saying that |\protect| should - have not been encountered (this |\protect| comes from the expansion - of the non-expandable |etoolbox| error message). - - When |bool| or |togl| is encountered by the |\xintexpr| parser, the argument - enclosed in a parenthesis pair is expanded as usual from left to right, - token by token, until the closing parenthesis is found, but everything is - taken literally, no computations are performed. For example |togl(2+3)| will - test the value of a toggle declared to |etoolbox| with name |2+3|, and not - |5|. Spaces are gobbled in this process. It is impossible to use |togl| on - such names containing spaces, but |\iftoggle{name with spaces}{1}{0}| will - work, naturally, as its expansion will pre-empt the |\xintexpr| scanner. - - There isn't in |\xintexpr...| a |test| function available analogous to the - |test{\ifsometest}| construct from the |etoolbox| package; but any - \emph{expandable} |\ifsometest| can be inserted directly in an - |\xintexpr|-ession as |\ifsometest10| (or |\ifsometest{1}{0}|), for example - |if(\ifsometest{1}{0},YES,NO)| (see the |if| operator below) works. - - A straight |\ifsometest{YES}{NO}| would do the same more - efficiently, the point - of |\ifsometest10| is to allow arbitrary boolean combinations using - the (described later) \verb+&+ and \verb+|+ logic operators: - \verb+\ifsometest10 & \ifsomeothertest10 | \ifsomethirdtest10+, etc... |YES| - or |NO| above stand for material compatible with the - |\xintexpr| parser syntax. - - See also \csbxint{ifboolexpr}, in this context. - \item[functions with one mandatory and a second optional argument] - \ctexttt{round, trunc,\\ float, sqrt}. For - example |round(2^9/3^5,12)=|\digitstt{\xinttheexpr round(2^9/3^5,12)\relax.} - The |sqrt| is available also in |\xintexpr|, not only in |\xintfloatexpr|. - The second optional argument is the required float precision. - \item[functions with two arguments] - \ctexttt{quo, rem}. These functions are integer only, they give the quotient - and remainder in Euclidean division (more generally one can use - the |floor| function; related: the |frac| function). - \item[the if conditional (twofold way)] \ctexttt{if}|(cond,yes,no)| checks if - |cond| is true or false and takes the corresponding branch. Any non zero - number or fraction is logical true. The zero value is logical false. Both - ``branches'' are evaluated (they are not really branches but just numbers). - See also the |?| operator. - \item[the ifsgn conditional (threefold way)] - \ctexttt{ifsgn}|(cond,<0,=0,>0)| checks the sign of |cond| and - proceeds correspondingly. All three are evaluated. See also the |:| - operator. - \item[functions with an arbitrary number of arguments] \ctexttt{all, any, - xor, add (=sum), mul (=prd), max, min, gcd, lcm}: |gcd| and |lcm| are - integer-only and require the \xintgcdname package. Currently, the |and| and - |or| keywords are left undefined by the package, which uses rather |all| - and |any|. They must have at least one argument. - \end{description} -\item The three postfix operators \ctexttt{!, ?, :}. - \begin{description} - \item[{\color[named]{DarkOrchid}!}] computes the factorial of an integer. |sqrt(36)!| evaluates to |6!| - (\digitstt{=\np{\xinttheexpr sqrt(36)!\relax}}) and not to the square root of - |36!| (\digitstt{$\approx$\np{\xintthefloatexpr sqrt(36!)\relax}}). This is - the exact - factorial even when used inside |\xintfloatexpr|. -\item[{\color[named]{DarkOrchid}?}] is used as |(cond)?{yes}{no}|. It evaluates the (numerical) condition - (any non-zero value counts as |true|, zero counts as |false|). It then acts as - a macro with two mandatory arguments within braces (hence this escapes from - the parser scope, the braces can not be hidden in a macro), chooses the - correct branch \emph{without evaluating the wrong one}. Once the braces are - removed, the parser scans and expands the uncovered material so for example - \centeredline{|\xinttheiexpr (3>2)?{5+6}{7-1}2^3\relax|} is legal and - computes |5+62^3=|\digitstt{\xinttheiexpr(3>2)?{5+(6}{7-(1}2^3)\relax}. Note - though that it would be better practice to include here the |2^3| inside the - branches. The contents of the branches may be arbitrary as long as once glued - to what is next the syntax is respected: {|\xintexpr - (3>2)?{5+(6}{7-(1}2^3)\relax| also works.} Differs thus from the |if| - conditional in two ways: the false branch is not at all computed, and the - number scanner is still active on exit, more digits may follow. -\item[{\color[named]{DarkOrchid}:}] is used as |(cond):{<0}{=0}{>0}|. |cond| is anything, its sign is - evaluated (it is not necessary to use |sgn(cond):{<}{=}{>}|) and depending on - the sign the correct branch is un-braced, the two others are swallowed. The - un-braced branch will then be parsed as usual. Differs from the |ifsgn| - conditional as the two false branches are not evaluated and furthermore the - number scanner is still active on exit. - \centeredline{|\def\x{0.33}\def\y{1/3}|} \centeredline{|\xinttheexpr - (\x-\y):{sqrt}{0}{1/}(\y-\x)\relax|% - \digitstt{=\def\x{0.33}\def\y{1/3}\xinttheexpr - (\x-\y):{sqrt}{0}{1/}(\y-\x)\relax }} - \end{description} -\item \def\MicroFont{\color[named]{DarkOrchid}\ttfamily\bfseries} - The |.| as decimal mark; the number scanner treats it as an inherent, - optional and unique component of a being formed number. One can do things - such as {\def\MicroFont{\ttfamily}|\xinttheexpr - .^2+2^.\relax|$\to$\digitstt{\xinttheexpr .^2+2^.\relax} (which is - |0^2+2^0|)}. -\item The |"| for hexadecimal numbers: it is treated with highest priority, - allowed only at locations where the parser expects to start forming a numeric - operand, once encountered it triggers the hexadecimal scanner which looks for - successive hexadecimal digits (as usual skipping spaces and expanding forward - everything) possibly a unique optional dot (allowed directly in front) and - then an optional (possibly empty) fractional part. The dot and fractional part - are not allowed in {\def\MicroFont{\ttfamily}|\xintiiexpr..\relax|}. The |"| - functionality requires that the user loaded \xintbinhexname (there is no - warning, but an ``undefined control sequence'' error will naturally results if - the package has not been loaded). -\item - % - The |e| and |E| for scientific notation. They are treated as infix operators - of highest priority: this means that they serve as an end marker (possibly - arising from macro expansion) for the scanned number, and then will pre-empt - the number coming next, either explicit, or arising from expansion, from - parenthesized material, from a sub-expression etc..., to serve as exponent. - \begingroup - \def\MicroFont{\ttfamily}% - From - the rules above, inside |\xintexpr|, |1e3-1| - is \digitstt{\xinttheexpr 1e3-1\relax}, |1e3^2| is \digitstt{\xinttheexpr - 1e3^2\relax}, and |"Ae("A+"F)^"A| - is \digitstt{\xinttheexpr "Ae("A+"F)^"A\relax}.\endgroup -\item The power operator |^|. It is left associative: -\begingroup\def\MicroFont{\ttfamily}% -|\xinttheiexpr 2^2^3\relax| evaluates to \xinttheiexpr 2^2^3\relax, not -\xinttheiexpr 2^(2^3)\relax. Note that if the float precision is too low, -iterated powers withing |\xintfloatexpr..\relax| may fail: for example with the -default setting |(1+1e-8)^(12^16)| will be computed with |12^16| approximated -from its @16@ most significant digits but it has @18@ digits -(\digitstt{={\xintiiPow{12}{16}}}), hence the result is wrong: -% -\centeredline{$\np{\xintthefloatexpr (1+1e-8)^(12^16)\relax }$} -% -One should code -% -\centeredline{|\xintthe\xintfloatexpr (1+1e-8)^\xintiiexpr 12^20\relax \relax|} -% -to obtain the correct floating point evaluation -% -\centeredline{$\np{1.00000001}^{12^{16}}\approx\np{\xintthefloatexpr - (1+1e-8)^\xintiiexpr 12^16\relax\relax }$}% -% -\endgroup -\item Multiplication and division \raisebox{-.3\height}{|*|}, |/|. The division - is left associative, too: \begingroup\def\MicroFont{\ttfamily}% - |\xinttheiexpr 100/50/2\relax| evaluates to - \xinttheiexpr 100/50/2\relax, not \xinttheiexpr 100/(50/2)\relax.\endgroup -\item Addition and subtraction |+|, |-|. Again, |-| is left - associative: \begingroup\def\MicroFont{\ttfamily}% - |\xinttheiexpr 100-50-2\relax| evaluates to - \xinttheiexpr 100-50-2\relax, not \xinttheiexpr 100-(50-2)\relax.\endgroup -\item Comparison operators |<|, |>|, |=| (currently, no @<=@, @>=@, - \dots ). -\item Conjunction (logical and): |&|. (no @&&@) -\item Inclusive disjunction (logical or): \verb$|$. (no @||@) -\item The comma |,|. \def\MicroFont{\ttfamily}% - With |\xinttheiexpr 2^3, 3^4, 5^6\relax| one obtains as output - \xinttheiexpr 2^3,3^4,5^6\relax{} (no space after the commas on output). -\item The parentheses. -\endlist -\endgroup - -See \autoref{ssec:countinexpr} for count and dimen registers and variables. - - -\section{Change log for earlier releases} -\label{sec:releases} - -% peut-être je devrais mettre ici le dernier aussi? - -\footnotesize - -\noindent Release |1.09j| (|[2014/01/09]|): -\begin{itemize} -\item the core division routines have been re-written for some (limited) - efficiency gain, more pronounced for small divisors. As a result the - \hyperlink{Machin1000}{computation of one thousand digits of $\pi$} - is close to three times faster than with earlier releases. -\item some various other small improvements, particularly in the power routines. -\item a new macro \csbxint{XTrunc} is designed to produce thousands or even tens - of thousands of digits of the decimal expansion of a fraction. Although - completely expandable it has its use limited to inside an |\edef|, |\write|, - |\message|, \dots. It - can thus not be nested as argument to another package macro. -\item the tacit multiplication done in \csbxint{expr}|..\relax| on encountering - a count register or variable, or a |\numexpr|, while scanning a (decimal) - number, is extended to the case of a sub |\xintexpr|-ession. -\item \csbxint{expr} can now be used in an |\edef| with no |\xintthe| - prefix; it will execute completely the computation, and the error - message about a missing |\xintthe| will be inhibited. Previously, in - the absence of |\xintthe|, expansion could only be a full one (with - |\romannumeral-`0|), not a complete one (with |\edef|). Note that this - differs from the behavior of the non-expandable |\numexpr|: |\the| or - |\number| are needed not only to print but also to trigger the - computation, whereas |\xintthe| is mandatory only for the printing step. -\item the default behavior of \csbxint {Assign} is changed, it now does not do - any further expansion beyond the initial full-expansion which provided the - list of items to be assigned to macros. -\item bug-fix: |1.09i| did an unexplainable change to |\XINT_infloat_zero| which - broke the floating point routines for vanishing operands =:((( -\item dtx bug-fix: the |1.09i .ins| file produced a buggy |.tex| file. -\end{itemize} - -\noindent Release |1.09i| (|[2013/12/18]|): -\begin{itemize} -\item \csbxint{iiexpr} is a variant of \csbxint{expr} which is optimized to deal - only with (long) integers, |/| does a euclidean quotient. -\item |\xintnumexpr|, |\xintthenumexpr|, |\xintNewNumExpr| are renamed, - respectively, \csbxint{iexpr}, \csbxint{theiexpr}, \csbxint{NewIExpr}. The - earlier denominations are kept but to be removed at some point. -\item it is now possible within |\xintexpr...\relax| and its variants to use - count, dimen, and skip registers or variables without explicit |\the/\number|: - the parser inserts automatically |\number| and a tacit multiplication is - implied when a register or variable immediately follows a number or fraction. - Regarding dimensions and |\number|, see the further discussion in - \autoref{sec:Dimensions}. -\item new conditional \csbxint{ifOne}; |\xintifTrueFalse| renamed to - \csbxint{ifTrueAelseB}; new macros \csbxint{TFrac} (`fractional part', mapped - to function |frac| in |\xintexpr|-essions), \csbxint{FloatE}. -\item \csbxint{Assign} admits an optional argument to specify the expansion - type to be used: |[]| (none, default), |[o]| (once), |[oo]| (twice), |[f]| - (full), |[e]| (|\edef|),... to define the macros -\item related to the previous item, \xinttoolsname defines - \hyperref[odef]{\ttfamily\char92odef}, - \hyperref[oodef]{\ttfamily\char92oodef}, - \hyperref[fdef]{\ttfamily\char92fdef} (if the names have already been - assigned, it uses |\xintoodef| etc...). These tools are provided for the - case one uses the package macros in a non-expandable context, particularly - \hyperref[oodef]{\ttfamily\char92oodef} which expands twice the macro - replacement text and is thus a faster alternative to |\edef| taking into - account that the \xintname bundle macros expand already completely in only - two steps. This can be significant when repeatedly making |\def|-initions - expanding to hundreds of digits. -\item some across the board slight efficiency improvement as a result of - modifications of various types to ``fork'' macros and ``branching - conditionals'' which are used internally. -\item bug-fix: |\xintAND| and |\xintOR| inserted a space token in some cases and - did not expand as promised in two steps (bug dating back to |1.09a| I think; - this bug was without consequences when using |&| and \verb+|+ in - \csa{xintexpr-}essions, it affected only the macro form) - |:-((|. -\item bug-fix: \csbxint{FtoCCv} still ended fractions with the |[0]|'s which - were supposed to have been removed since release |1.09b|. -\end{itemize} - -\noindent Release |1.09h| (|[2013/11/28]|): -\begin{itemize} -\item parts of the documentation have been re-written or re-organized, - particularly the discussion of expansion issues and of input and - output formats. -\item the expansion types of macro arguments are documented in the margin of the - macro descriptions, with conventions mainly taken over from those in the - \LaTeX3 documentation. -\item a dependency of \xinttoolsname on \xintname (inside \csbxint{Seq}) has - been removed. -\item \csbxint{TypesetEuclideAlgorithm} and \csbxint{TypesetBezoutAlgorithm} - have been slightly modified (regarding indentation). -\item macros \csa{xintiSum} and \csa{xintiPrd} are renamed to \csbxint{iiSum} - and \csbxint{iiPrd}. -\item a count register used in |1.09g| in the \csbxint{For} loops for parsing - purposes has been removed and replaced by use of a |\numexpr|. -\item the few uses of |\loop| have been replaced by |\xintloop/\xintiloop|. -\item all macros of \xinttoolsname for which it makes sense are now - declared |\long|. -\end{itemize} - -\noindent Release |1.09g| (|[2013/11/22]|): -\begin{itemize} -\item package \xinttoolsname is detached from \xintname, to make tools such as - \csbxint{For}, \csbxint{ApplyUnbraced}, and \csbxint{iloop} available without - the \xintname overhead. -\item new expandable nestable loops \csbxint{loop} and \csbxint{iloop}. -\item bugfix: \csbxint{For} and \csbxint{For*} do not modify anymore the value - of |\count 255|. -\end{itemize} - -\noindent Release |1.09f| (|[2013/11/04]|): -\begin{itemize} -\item new \csbxint{ZapFirstSpaces}, \csbxint{ZapLastSpaces}, - \csbxint{ZapSpaces}, \csbxint{ZapSpacesB}, for expandably stripping away - leading and/or ending spaces. -\item \csbxint{CSVtoList} by default uses \csbxint{ZapSpacesB} to strip away - spaces around commas (or at the start and end of the comma separated list). -\item also the \csbxint{For} loop will strip out all spaces around commas and at - the start and the end of its list argument; and similarly for - \csbxint{Forpair}, \csbxint{Forthree}, \csbxint{Forfour}. -\item \csbxint{For} \emph{et al.} accept all macro parameters - from - |#1| to |#9|. -\item for reasons of inner coherence some macros previously with one extra `|i|' - in their names (e.g. \csa{xint\-iMON}) now have a doubled `|ii|' - (\csbxint{iiMON}) to indicate that they skip the overhead of parsing their - inputs via \csbxint{Num}. Macros with a \emph{single} `|i|' such as - \csbxint{iAdd} are those which maintain the non-\xintfracname output format - for big integers, but do parse their inputs via \csbxint{Num} (since release - |1.09a|). They too may have doubled-|i| variants for matters of programming - optimization when working only with (big) integers and not fractions or - decimal numbers. -\end{itemize} - - -\noindent Release |1.09e| (|[2013/10/29]|): -\begin{itemize} -\item new \csbxint{integers}, \csbxint{dimensions}, \csbxint{rationals} for - infinite \csbxint{For} loops, interrupted with \csbxint{BreakFor} and - \csbxint{BreakForAndDo}. -\item new \csbxint{ifForFirst}, \csbxint{ifForLast} for the \csa{xintFor} and - \csa{xintFor*} loops, -\item the \csa{xintFor} and \csa{xintFor*} loops are now |\long|, the - replacement text and the items may contain explicit |\par|'s. -\item bug fix, the \csbxint{For} loop (not \csbxint{For*}) did not correctly - detect an - empty list. -\item new conditionals \csbxint{ifCmp}, \csbxint{ifInt}, \csbxint{ifOdd}. -\item bug fix, |\xintiSqrt {0}| crashed. |:-((| -\item the documentation has been enriched with various additional examples, - such as the \hyperref[ssec:quicksort]{the quick sort algorithm illustrated} or - the computation of prime numbers (\autoref{ssec:primesI}, - \autoref{ssec:primesII}, \autoref{ssec:primesIII}). -\item the documentation explains with more details various expansion related - issues, particularly in relation to conditionals. -\end{itemize} - -\noindent Release |1.09d| (|[2013/10/22]|):\nobreak -\begin{itemize} -\item \csbxint{For*} is modified to gracefully handle a space token (or - more than one) located at the - very end of its list argument (as in for example |\xintFor* #1 in - {{a}{b}{c}<space>} \do {stuff}|; - spaces at other locations were already harmless). Furthermore this new -version \fexpan ds the un-braced list items. After -|\def\x{{1}{2}}| and |\def\y{{a}\x {b}{c}\x }|, |\y| will appear to -\csbxint{For*} exactly as if it had been defined as -|\def\y{{a}{1}{2}{b}{c}{1}{2}}|. -\item same bug fix in \csbxint{ApplyInline}. -\end{itemize} - -\noindent Release |1.09c| (|[2013/10/09]|): -\begin{itemize} -\item added \hyperlink{item:bool}{|bool|} and \hyperlink{item:bool}{|togl|} to - the - \csbxint{expr} syntax; also added \csbxint{boolexpr} and \csbxint{ifboolexpr}. -\item added |\xintNewNumExpr| (now \csbxint{NewIExpr} and \csbxint{NewBoolExpr}, -\item \csbxint{For} is a new type of loop, whose replacement text inserts the - comma separated values or list items via macro parameters, rather than - encapsulated in macros; the loops are nestable up to four levels (nine - levels since |1.09f|) and their replacement texts are allowed to close - groups as happens with the tabulation in alignments, -\item \csbxint{Forpair}, \csbxint{Forthree}, \csbxint{Forfour} are experimental - variants of \csbxint{For}, -\item \csbxint{ApplyInline} has been enhanced in order to be usable for - generating rows (partially or completely) in an alignment, -\item new command \csbxint{Seq} to generate (expandably) arithmetic sequences of - (short) integers, -\item the factorial |!| and branching |?|, |:|, operators (in - \csbxint{expr}|...\relax|) have now less precedence than a function name - located just before: |func(x)!| is the factorial of |func(x)|, not |func(x!)|, -\item again various improvements and changes in the documentation. -\end{itemize} - -\noindent Release |1.09b| (|[2013/10/03]|): -\begin{itemize} -\item various improvements in the documentation, -\item more economical catcode management and re-loading handling, -\item removal of all those |[0]|'s previously forcefully added at the end of - fractions by various macros of \xintcfracname, -\item \csbxint{NthElt} with a negative index returns from the tail of the list, -\item new macro \csbxint{PRaw} to have something like what |\xintFrac| does in - math - mode; i.e. a |\xintRaw| which does not print the denominator if it is one. -\end{itemize} - -\noindent Release |1.09a| (|[2013/09/24]|): -\begin{itemize} -\item \csbxint{expr}|..\relax| and - \csbxint{floatexpr}|..\relax| admit functions in their - syntax, with comma separated values as arguments, among them \texttt{reduce, - sqr, sqrt, abs, sgn, floor, ceil, quo, rem, round, trunc, float, gcd, lcm, - max, min, sum, prd, add, mul, not, all, any, xor}. -\item comparison (|<|, |>|, |=|) and logical (\verb$|$, |&|) operators. -\item the command |\xintthe| which converts |\xintexpr|essions into printable - format (like |\the| with |\numexpr|) is more efficient, for example one can do - |\xintthe\x| if |\x| was def'ined to be an |\xintexpr..\relax|: -\centeredline{|\def\x{\xintexpr 3^57\relax}\def\y{\xintexpr \x^(-2)\relax}|} -\centeredline{|\def\z{\xintexpr - \y-3^-114\relax}|\hspace{1cm}|\xintthe\z=|\begingroup -\def\x{\xintexpr 3^57\relax}\def\y{\xintexpr \x^(-2)\relax}% -\def\z{\xintexpr \y-3^-114\relax}\digitstt{\xintthe\z}\endgroup} -\item |\xintnumexpr .. \relax| (now renamed \csbxint{iexpr}) is |\xintexpr - round( .. ) \relax|. -\item \csbxint{NewExpr} now works with the standard macro parameter character - |#|. -\item both regular |\xintexpr|-essions and commands defined by |\xintNewExpr| - will work with comma separated lists of expressions, -\item new commands \csbxint{Floor}, \csbxint{Ceil}, \csbxint{Maxof}, - \csbxint{Minof} (package \xintfracname), \csbxint{GCDof}, \csbxint{LCM}, - \csbxint{LCMof} (package \xintgcdname), \csbxint{ifLt}, \csbxint{ifGt}, - \csbxint{ifSgn}, \csbxint{ANDof}, ... -\item The arithmetic macros from package \xintname now filter their operands via - \csbxint{Num} which means that they may use directly count registers and - |\numexpr|-essions without having to prefix them by |\the|. This is thus - similar to the situation holding previously but with \xintfracname loaded. -\item a bug introduced in |1.08b| made \csbxint{Cmp} crash when one of its - arguments was zero. |:-((| -\end{itemize} - - -\noindent Release |1.08b| (|[2013/06/14]|): -\begin{itemize} -\item Correction of a problem with spaces inside |\xintexpr|-essions. -\item Additional improvements to the handling of floating point numbers. -\item The macros of \xintfracname allow to use count registers in their - arguments in ways which were not previously documented. See - \hyperref[sec:useofcount]{Use of count registers}. -\end{itemize} - -\noindent Release |1.08a| (|[2013/06/11]|): -\begin{itemize} -\item Improved efficiency of the basic conversion from exact - fractions to floating point numbers, - with ensuing speed gains especially for the power function macros - \csbxint{FloatPow} and \csbxint{FloatPower}, -\item Better management by the \xintfracname macros \csbxint{Cmp}, - \csbxint{Max}, \csbxint{Min} and \csbxint{Geq} of inputs having big powers - of ten in them. -\item Macros for floating point numbers added to the \xintseriesname package. -\end{itemize} - -\noindent Release |1.08| (|[2013/06/07]|): -\begin{itemize} -\item Extraction of square roots, for floating point numbers - (\csbxint{FloatSqrt}), and also in - a version adapted to integers (\csbxint{iSqrt}). -\item New package \xintbinhexname providing \hyperref[sec:binhex]{conversion - routines} to and from binary and hexadecimal bases. -\end{itemize} - -\noindent Release |1.07| (|[2013/05/25)]|): -\begin{itemize} -\item The \xintfracname macros accept numbers written in scientific notation, - the \csbxint{Float} command serves to output its argument with a given number - |D| of significant figures. The value of |D| is either given as optional - argument to \csbxint{Float} or set with |\xintDigits := D;|. The default value - is |16|. -\item The \xintexprname package is a new core constituent (which loads - automatically \xintfracname and \xintname) and implements the expandable - expanding parsers \centeredline{\csbxint{expr}| . . . \relax|, - and its variant - \csbxint{floatexpr}| . . . \relax|} allowing on input formulas using the - standard form with infix - operators |+|, |-|, |*|, |/|, and |^|, and arbitrary levels of - parenthesizing. Within a float expression the operations are executed - according to the current value of \csbxint{Digits}. Within an - |\xintexpr|-ession the binary operators are computed exactly. -\item The floating point precision |D| is set (this is a -local assignment to a |\mathchar| variable) with |\xintDigits := D;| and queried -with |\xinttheDigits|. It may be set to anything up to |32767|.\footnote{but - values higher than 100 or 200 will presumably give too slow evaluations.} The -macro incarnations of the binary operations admit an optional argument which -will replace pointwise |D|; this argument may exceed the |32767| bound. -\item To write the |\xintexpr| parser I benefited from the commented source of - the -\LaTeX3 parser; the |\xintexpr| parser has its own features and peculiarities. -See \hyperref[sec:expr]{its documentation}. -\end{itemize} - -Initial release |1.0| was on |2013/03/28|. - - -% \noindent Historians debate the early history of the \xintname bundle, whose -% details will need patient reconstruction from the scattered archeological -% remnants. It has been established that the initial release |1.0| was on -% |2013/03/28|, although only closer scrutiny of the CTAN logs could help -% completely exclude possibility of an earlier |0.9|. - - - -\normalsize - - -\etocdepthtag.toc {commandsA} - -\section{Commands of the \xinttoolsname package} -\label{sec:tools} - -\def\n{\string{N\string}} -\def\m{\string{M\string}} -\def\x{\string{x\string}} - -These utilities used to be provided within the \xintname package; since |1.09g| -they have been moved to an independently usable package \xinttoolsname, which -has none of the \xintname facilities regarding big numbers. Whenever relevant -release |1.09h| has made the macros |\long| so they accept |\par| tokens on -input. - -First the completely expandable utilities up to \csbxint{iloop}, then the non -expandable utilities. - -This section contains various concrete examples and ends with a -\hyperref[ssec:quicksort]{completely expandable implementation of the Quick Sort - algorithm} together with a graphical illustration of its action. - -\clearpage - -\localtableofcontents - - -\subsection{\csbh{xintReverseOrder}}\label{xintReverseOrder} - -\csa{xintReverseOrder}\marg{list}\etype{n} does not do any expansion of its -argument and just reverses the order of the tokens in the \meta{list}. Braces -are removed once and the enclosed material, now unbraced, does not get -reverted. Unprotected spaces (of any character code) are gobbled. -\centeredline{|\xintReverseOrder{\xintDigitsOf\xintiPow {2}{100}\to\Stuff}|} -\centeredline{gives: - \ttfamily{\string\Stuff\string\to1002\string\xintiPow\string\xintDigitsOf}} - -\subsection{\csbh{xintRevWithBraces}}\label{xintRevWithBraces} - -%{\small New in release |1.06|.\par} - -\edef\X{\xintRevWithBraces{12345}} -\edef\y{\xintRevWithBraces\X} -\expandafter\def\expandafter\w\expandafter - {\romannumeral0\xintrevwithbraces{{\A}{\B}{\C}{\D}{\E}}} - -% -\csa{xintRevWithBraces}\marg{list}\etype{f} first does the \fexpan sion of its -argument then it reverses the order of the tokens, or braced material, it -encounters, adding a pair of braces to each (thus, maintaining brace pairs -already existing). Spaces (in-between external brace pairs) are gobbled. This -macro is mainly thought out for use on a \meta{list} of such braced material; -with such a list as argument the \fexpan sion will only hit against the first -opening brace, hence do nothing, and the braced stuff may thus be macros one -does not want to expand. \centeredline{|\edef\x{\xintRevWithBraces{12345}}|} -\centeredline{|\meaning\x:|\ttfamily{\meaning\X}} -\centeredline{|\edef\y{\xintRevWithBraces\x}|}% -\centeredline{|\meaning\y:|\ttfamily{\meaning\y}} The examples above could be -defined with |\edef|'s because the braced material did not contain macros. -Alternatively: \centeredline{|\expandafter\def\expandafter\w\expandafter|}% -\centeredline{|{\romannumeral0\xintrevwithbraces{{\A}{\B}{\C}{\D}{\E}}}|} -\centeredline{|\meaning\w:|\ttfamily{\meaning\w}} The macro -\csa{xintReverseWithBracesNoExpand}\etype{n} does the same job without the -initial -expansion of its argument. - -\subsection{\csbh{xintLength}}\label{xintLength} - -\csa{xintLength}\marg{list}\etype{n} does not do \emph{any} expansion of its -argument and just counts how many tokens there are (possibly none). So to use it -to count things in the replacement text of a macro one should do -|\expandafter\xintLength\expandafter{\x}|. One may also use it inside macros as -|\xintLength{#1}|. Things enclosed in braces count as one. Blanks between tokens -are not counted. See \csbxint{NthElt}|{0}| for a variant which first \fexpan ds -its argument. \centeredline{|\xintLength {\xintiPow - {2}{100}}|\digitstt{=\xintLength {\xintiPow{2}{100}}}} -\centeredline{${}\neq{}$|\xintLen {\xintiPow {2}{100}}|\digitstt{=\xintLen - {\xintiPow{2}{100}}}} - -\subsection{\csbh{xintZapFirstSpaces}, \csbh{xintZapLastSpaces}, \csbh{xintZapSpaces}, \csbh{xintZapSpacesB}} -\label{xintZapFirstSpaces} -\label{xintZapLastSpaces} -\label{xintZapSpaces} -\label{xintZapSpacesB} -%{\small New with release |1.09f|.\par} - -\csa{xintZapFirstSpaces}\marg{stuff}\etype{n} does not do \emph{any} expansion -of its -argument, nor brace removal of any sort, nor does it alter \meta{stuff} in -anyway apart from stripping away all \emph{leading} spaces. - -This macro will be mostly of interest to programmers who will know what I will -now be talking about. \emph{The essential points, naturally, are the complete - expandability and the fact that no brace removal nor any other alteration is - done to the input.} - -\TeX's input scanner already converts consecutive blanks into single space -tokens, but \csa{xintZapFirstSpaces} handles successfully also inputs with -consecutive multiple space tokens. -However, it is assumed that \meta{stuff} does not contain (except inside braced -sub-material) space tokens of character code distinct from @32@. - -It expands in two steps, and if the goal is to apply it to the -expansion text of |\x| to define |\y|, then one should do: -|\expandafter\def\expandafter\y\expandafter - {\romannumeral0\expandafter\xintzapfirstspaces\expandafter{\x}}|. - -Other use case: inside a macro as |\edef\x{\xintZapFirstSpaces {#1}}| assuming -naturally that |#1| is compatible with such an |\edef| once the leading spaces -have been stripped. - -\begingroup -\def\x { \a { \X } { \b \Y } } -\centeredline{|\xintZapFirstSpaces { \a { \X } { \b \Y } }->|% -\digitstt{\color{magenta}{}\expandafter\detokenize\expandafter -{\romannumeral0\expandafter\xintzapfirstspaces\expandafter{\x}}}+++} -\endgroup - -\medskip - -\noindent\csbxint{ZapLastSpaces}\marg{stuff}\etype{n} does not do \emph{any} expansion of -its argument, nor brace removal of any sort, nor does it alter \meta{stuff} in -anyway apart from stripping away all \emph{ending} spaces. The same remarks as -for \csbxint{ZapFirstSpaces} apply. - -% ATTENTION à l'\ignorespaces fait par \color! -\begingroup -\def\x { \a { \X } { \b \Y } } -\centeredline{|\xintZapLastSpaces { \a { \X } { \b \Y } }->|% -\digitstt{\color{magenta}{}\expandafter\detokenize\expandafter -{\romannumeral0\expandafter\xintzaplastspaces\expandafter{\x}}}+++} -\endgroup - -\medskip - -\noindent\csbxint{ZapSpaces}\marg{stuff}\etype{n} does not do \emph{any} -expansion of its -argument, nor brace removal of any sort, nor does it alter \meta{stuff} in -anyway apart from stripping away all \emph{leading} and all \emph{ending} -spaces. The same remarks as for \csbxint{ZapFirstSpaces} apply. - -\begingroup -\def\x { \a { \X } { \b \Y } } -\centeredline{|\xintZapSpaces { \a { \X } { \b \Y } }->|% -\digitstt{\color{magenta}{}\expandafter\detokenize\expandafter -{\romannumeral0\expandafter\xintzapspaces\expandafter{\x}}}+++} -\endgroup - -\medskip - -\noindent\csbxint{ZapSpacesB}\marg{stuff}\etype{n} does not do \emph{any} -expansion of -its argument, nor does it alter \meta{stuff} in anyway apart from stripping away -all leading and all ending spaces and possibly removing one level of braces if -\meta{stuff} had the shape |<spaces>{braced}<spaces>|. The same remarks as for -\csbxint{ZapFirstSpaces} apply. - -\begingroup -\def\x { \a { \X } { \b \Y } } -\centeredline{|\xintZapSpacesB { \a { \X } { \b \Y } }->|% -\digitstt{\color{magenta}{}\expandafter\detokenize\expandafter -{\romannumeral0\expandafter\xintzapspacesb\expandafter{\x}}}+++} -\def\x { { \a { \X } { \b \Y } } } -\centeredline{|\xintZapSpacesB { { \a { \X } { \b \Y } } }->|% -\digitstt{\color{magenta}{}\expandafter\detokenize\expandafter -{\romannumeral0\expandafter\xintzapspacesb\expandafter{\x}}}+++} -\endgroup - The spaces here at the start and end of the output come from the braced - material, and are not removed (one would need a second application for that; - recall though that the \xintname zapping macros do not expand their argument). - -\subsection{\csbh{xintCSVtoList}} -\label{xintCSVtoList} -\label{xintCSVtoListNoExpand} - -% {\small New with release |1.06|. Starting with |1.09f|, \fbox{\emph{removes -% spaces around commas}!}\par} - -\csa{xintCSVtoList}|{a,b,c...,z}|\etype{f} returns |{a}{b}{c}...{z}|. A -\emph{list} is by -convention in this manual simply a succession of tokens, where each braced thing -will count as one item (``items'' are defined according to the rules of \TeX{} -for fetching undelimited parameters of a macro, which are exactly the same rules -as for \LaTeX{} and command arguments [they are the same things]). The word -`list' in `comma separated list of items' has its usual linguistic meaning, -and then an ``item'' is what is delimited by commas. - -So \csa{xintCSVtoList} takes on input a `comma separated list of items' and -converts it into a `\TeX{} list of braced items'. The argument to -|\xintCSVtoList| may be a macro: it will first be -\hyperref[sec:expansions]{\fexpan ded}. Hence the item before the first comma, -if it is itself a macro, will be expanded which may or may not be a good thing. -A space inserted at the start of the first item serves to stop that expansion -(and disappears). The macro \csbxint{CSVtoListNoExpand}\etype{n} does the same -job without -the initial expansion of the list argument. - -Apart from that no expansion of the items is done and the list items may thus be -completely arbitrary (and even contain perilous stuff such as unmatched |\if| -and |\fi| tokens). - -Contiguous spaces and tab characters, are collapsed by \TeX{} -into single spaces. All such spaces around commas\footnote{and multiple space - tokens are not a problem; but those at the top level (not hidden inside - braces) \emph{must} be of character code |32|.} \fbox{are removed}, as well as -the spaces at the start and the spaces at the end of the list.\footnote{let us - recall that this is all done completely expandably... There is absolutely no - alteration of any sort of the item apart from the stripping of initial and - final space tokens (of character code |32|) and brace removal if and only if - the item apart from intial and final spaces (or more generally multiple |char - 32| space tokens) is braced.} The items may contain explicit |\par|'s or -empty lines (converted by the \TeX{} input parsing into |\par| tokens). - -\begingroup - -\edef\X{\xintCSVtoList { 1 ,{ 2 , 3 , 4 , 5 }, a , {b,T} U , { c , d } , { {x , - y} } }} - -\centeredline{|\xintCSVtoList { 1 ,{ 2 , 3 , 4 , 5 }, a , {b,T} U , { c , d } , - { {x , y} } }|} -\centeredline{|->|% -{\makeatletter\digitstt{\expandafter\strip@prefix\meaning\X}}} - -One sees on this example how braces protect commas from -sub-lists to be perceived as delimiters of the top list. Braces around an entire -item are removed, even when surrounded by spaces before and/or after. Braces for -sub-parts of an item are not removed. - -We observe also that there is a slight difference regarding the brace stripping -of an item: if the braces were not surrounded by spaces, also the initial and -final (but no other) spaces of the \emph{enclosed} material are removed. This is -the only situation where spaces protected by braces are nevertheless removed. - -From the rules above: for an empty argument (only spaces, no braces, no comma) -the output is -\digitstt{\expandafter\detokenize\expandafter{\romannumeral0\xintcsvtolist { }}} -(a list with one empty item), -for ``|<opt. spaces>{}<opt. -spaces>|'' the output is -\digitstt{\expandafter\detokenize\expandafter - {\romannumeral0\xintcsvtolist { {} }}} -(again a list with one empty item, the braces were removed), -for ``|{ }|'' the output is -\digitstt{\expandafter\detokenize\expandafter - {\romannumeral0\xintcsvtolist {{ }}}} -(again a list with one empty item, the braces were removed and then -the inner space was removed), -for ``| { }|'' the output is -\digitstt{\expandafter\detokenize\expandafter -{\romannumeral0\xintcsvtolist { { }}}} (again a list with one empty item, the initial space served only to stop the expansion, so this was like ``|{ }|'' as input, the braces were removed and the inner space was stripped), -for ``\texttt{\ \{\ \ \}\ }'' the output is -\digitstt{\expandafter\detokenize\expandafter -{\romannumeral0\xintcsvtolist { { } }}} (this time the ending space of the first -item meant that after brace removal the inner spaces were kept; recall though -that \TeX{} collapses on input consecutive blanks into one space token), -for ``|,|'' the output consists of two consecutive -empty items -\digitstt{\expandafter\detokenize\expandafter{\romannumeral0\xintcsvtolist - {,}}}. Recall that on output everything is braced, a |{}| is an ``empty'' -item. -% -Most of the above is mainly irrelevant for every day use, apart perhaps from the -fact to be noted that an empty input does not give an empty output but a -one-empty-item list (it is as if an ending comma was always added at the end of -the input). - -\def\y { \a,\b,\c,\d,\e} -\expandafter\def\expandafter\Y\expandafter{\romannumeral0\xintcsvtolist{\y}} -\def\t {{\if},\ifnum,\ifx,\ifdim,\ifcat,\ifmmode} -\expandafter\def\expandafter\T\expandafter{\romannumeral0\xintcsvtolist{\t}} - -\centeredline{|\def\y{ \a,\b,\c,\d,\e} \xintCSVtoList\y->|% - {\makeatletter\digitstt{\expandafter\strip@prefix\meaning\Y}}} -\centeredline{|\def\t {{\if},\ifnum,\ifx,\ifdim,\ifcat,\ifmmode}|} \centeredline -{|\xintCSVtoList\t->|\makeatletter\digitstt{\expandafter\strip@prefix\meaning\T}} -The results above were automatically displayed using \TeX's primitive -\csa{meaning}, which adds a space after each control sequence name. These spaces -are not in the actual braced items of the produced lists. The first items |\a| -and |\if| were either preceded by a space or braced to prevent expansion. The -macro \csa{xintCSVtoListNoExpand} would have done the same job without the -initial expansion of the list argument, hence no need for such protection but if -|\y| is defined as |\def\y{\a,\b,\c,\d,\e}| we then must do: -\centeredline{|\expandafter\xintCSVtoListNoExpand\expandafter {\y}|} Else, we -may have direct use: \centeredline{|\xintCSVtoListNoExpand - {\if,\ifnum,\ifx,\ifdim,\ifcat,\ifmmode}|} -\centeredline{|->|\digitstt{\expandafter\detokenize\expandafter - {\romannumeral0\xintcsvtolistnoexpand - {\if,\ifnum,\ifx,\ifdim,\ifcat,\ifmmode}}}} -% -Again these spaces are an artefact from the use in the source of the document of -\csa{meaning} (or rather here, \csa{detokenize}) to display the result of using -\csa{xintCSVtoListNoExpand} (which is done for real in this document -source). - -For the similar conversion from comma separated list to braced items list, but -without removal of spaces around the commas, there is -\csa{xintCSVtoListNonStripped}\etype{f} and -\csa{xintCSVtoListNonStrippedNoExpand}\etype{n}. - -\endgroup - -\subsection{\csbh{xintNthElt}}\label{xintNthElt} - -% {\small New in release |1.06|. With |1.09b| negative indices count from the tail.\par} - -\def\macro #1{\the\numexpr 9-#1\relax} - -\csa{xintNthElt\x}\marg{list}\etype{\numx f} gets (expandably) the |x|th braced -item of the \meta{list}. An unbraced item token will be returned as is. The list -itself may be a macro which is first \fexpan ded. \centeredline{|\xintNthElt - {3}{{agh}\u{zzz}\v{Z}}| is \texttt{\xintNthElt {3}{{agh}\u{zzz}\v{Z}}}} -\centeredline{|\xintNthElt {3}{{agh}\u{{zzz}}\v{Z}}| is - \texttt{\expandafter\expandafter\expandafter - \detokenize\expandafter\expandafter\expandafter {\xintNthElt - {3}{{agh}\u{{zzz}}\v{Z}}}}} \centeredline{|\xintNthElt - {2}{{agh}\u{{zzz}}\v{Z}}| is \texttt{\expandafter\expandafter\expandafter - \detokenize\expandafter\expandafter\expandafter {\xintNthElt - {2}{{agh}\u{{zzz}}\v{Z}}}}} \centeredline{|\xintNthElt {37}{\xintFac - {100}}|\digitstt{=\xintNthElt {37}{\xintFac {100}}} is the thirty-seventh - digit of @100!@.} \centeredline{|\xintNthElt {10}{\xintFtoCv - {566827/208524}}|\digitstt{=\xintNthElt {10}{\xintFtoCv {566827/208524}}}} -is the tenth convergent of @566827/208524@ (uses \xintcfracname package). -\centeredline{|\xintNthElt {7}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}|% - \digitstt{=\xintNthElt {7}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}}}% -\centeredline{|\xintNthElt {0}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}|% - \digitstt{=\xintNthElt {0}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}}} -\centeredline{|\xintNthElt {-3}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}|% - \digitstt{=\xintNthElt {-3}{\xintCSVtoList {1,2,3,4,5,6,7,8,9}}}} If |x=0|, -the macro returns the \emph{length} of the expanded list: this is not equivalent -to \csbxint{Length} which does no pre-expansion. And it is different from -\csbxint{Len} which is to be used only on integers or fractions. - -If |x<0|, the macro returns the \texttt{|x|}th element from the end of the list. - \centeredline{|\xintNthElt - {-5}{{{agh}}\u{zzz}\v{Z}}| is \texttt{\expandafter\expandafter\expandafter - \detokenize - \expandafter\expandafter\expandafter{\xintNthElt - {-5}{{{agh}}\u{zzz}\v{Z}}}}} - - -The macro \csa{xintNthEltNoExpand}\etype{\numx n} does the same job but without first -expanding the list argument: |\xintNthEltNoExpand {-4}{\u\v\w T\x\y\z}| is -\xintNthEltNoExpand {-4}{\a\b\c\u\v\w T\x\y\z}. - -In cases where |x| is larger (in absolute value) than the length of the list -then |\xintNthElt| returns nothing. - -\subsection{\csbh{xintListWithSep}}\label{xintListWithSep} - -%{\small New with release |1.04|.\par} - -\def\macro #1{\the\numexpr 9-#1\relax} - -\csa{xintListWithSep}|{sep}|\marg{list}\etype{nf} inserts the given separator -|sep| in-between all items of the given list of braced items: this separator may -be a macro (or multiple tokens) but will not be expanded. The second argument -also may be itself a macro: it is \fexpan ded. Applying \csa{xintListWithSep} -removes the braces from the list items (for example |{1}{2}{3}| turns into -\digitstt{\xintListWithSep,{123}} via |\xintListWithSep{,}{{1}{2}{3}}|). An -empty input gives an empty output, a singleton gives a singleton, the separator -is used starting with at least two elements. Using an empty separator has the -net effect of unbracing the braced items constituting the \meta{list} (in such -cases the new list may thus be longer than the original). -\centeredline{|\xintListWithSep{:}{\xintFac - {20}}|\digitstt{=\xintListWithSep{:}{\xintFac {20}}}} - -The macro \csa{xintListWithSepNoExpand}\etype{nn} does the same -job without the initial expansion. - -\subsection{\csbh{xintApply}}\label{xintApply} - -%{\small New with release |1.04|.\par} - -\def\macro #1{\the\numexpr 9-#1\relax} - -\csa{xintApply}|{\macro}|\marg{list}\etype{ff} expandably applies the one -parameter command |\macro| to each item in the \meta{list} given as second -argument and returns a new list with these outputs: each item is given one after -the other as parameter to |\macro| which is expanded at that time (as usual, -\emph{i.e.} fully for what comes first), the results are braced and output -together as a succession of braced items (if |\macro| is defined to start with a -space, the space will be gobbled and the |\macro| will not be expanded; it is -allowed to have its own arguments, the list items serve as last arguments to -|\macro|). Hence |\xintApply{\macro}{{1}{2}{3}}| returns -|{\macro{1}}{\macro{2}}{\macro{3}}| where all instances of |\macro| have been -already \fexpan ded. - -Being expandable, |\xintApply| is useful for example inside alignments where -implicit groups make standard loops constructs usually fail. In such situation -it is often not wished that the new list elements be braced, see -\csbxint{ApplyUnbraced}. The |\macro| does not have to be expandable: -|\xintApply| will try to expand it, the expansion may remain partial. - -The \meta{list} may -itself be some macro expanding (in the previously described way) to the list of -tokens to which the command |\macro| will be applied. For example, if the -\meta{list} expands to some positive number, then each digit will be replaced by -the result of applying |\macro| on it. \centeredline{|\def\macro #1{\the\numexpr - 9-#1\relax}|} \centeredline{|\xintApply\macro{\xintFac - {20}}|\digitstt{=\xintApply\macro{\xintFac {20}}}} - -The macro \csa{xintApplyNoExpand}\etype{fn} does the same job without the first -initial expansion which gave the \meta{list} of braced tokens to which |\macro| -is applied. - -\subsection{\csbh{xintApplyUnbraced}}\label{xintApplyUnbraced} - -%{\small New in release |1.06b|.\par} - -\def\macro #1{\expandafter\def\csname myself#1\endcsname {#1}} -\xintApplyUnbraced\macro{{elta}{eltb}{eltc}} - -\csa{xintApplyUnbraced}|{\macro}|\marg{list}\etype{ff} is like \csbxint{Apply}. -The difference is that after having expanded its list argument, and applied -|\macro| in turn to each item from the list, it reassembles the outputs without -enclosing them in braces. The net effect is the same as doing -\centeredline{|\xintListWithSep {}{\xintApply {\macro}|\marg{list}|}|} This is -useful for preparing a macro which will itself define some other macros or make -assignments, as the scope will not be limited by brace pairs. -% -\dverb|@ - \def\macro #1{\expandafter\def\csname myself#1\endcsname {#1}} - \xintApplyUnbraced\macro{{elta}{eltb}{eltc}} - \meaning\myselfelta: "meaning"myselfelta - \meaning\myselfeltb: "meaning"myselfeltb - \meaning\myselfeltc: "meaning"myselfeltc| - -% -The macro \csa{xintApplyUnbracedNoExpand}\etype{fn} does the same job without -the first initial expansion which gave the \meta{list} of braced tokens to which -|\macro| is applied. - -\subsection{\csbh{xintSeq}}\label{xintSeq} -%{\small New with release |1.09c|.\par} - -\csa{xintSeq}|[d]{x}{y}|\etype{{{\upshape[\numx]}}\numx\numx} generates expandably |{x}{x+d}...| up to and -possibly including |{y}| if |d>0| or down to and including |{y}| if |d<0|. -Naturally |{y}| is omitted if |y-x| is not a multiple of |d|. If |d=0| the macro -returns |{x}|. If |y-x| and |d| have opposite signs, the macro returns nothing. -If the optional argument |d| is omitted it is taken to be the sign of |y-x|. - - -The current implementation is only for (short) integers; possibly, a future -variant could allow big integers and fractions, although one already has -access to similar -functionality using \csbxint{Apply} to get any arithmetic sequence of long -integers. Currently thus, |x| and |y| are expanded inside a -|\numexpr| so they may be count registers or a \LaTeX{} |\value{countername}|, -or arithmetic with such things. - -\centeredline{|\xintListWithSep{,\hskip2pt - plus 1pt minus 1pt }{\xintSeq {12}{-25}}|} -\noindent\digitstt{\xintListWithSep{,\hskip2pt plus 1pt minus 1pt }{\xintSeq - {12}{-25}}} -\centeredline{|\xintiiSum{\xintSeq [3]{1}{1000}}|\digitstt{=\xintiiSum{\xintSeq [3]{1}{1000}}}} - -\textbf{Important:} for reasons of efficiency, this macro, when not given the -optional argument |d|, works backwards, leaving in the token stream the already -constructed integers, from the tail down (or up). But this will provoke a -failure of \IMPORTANT{} the |tex| run if the number of such items exceeds the -input stack -limit; on my installation this limit is at @5000@. - -However, when given the optional argument |d| (which may be @+1@ or -@-1@), the macro proceeds differently and does not put stress on the input stack -(but is significantly slower for sequences with thousands of integers, -especially if they are somewhat big). For -example: |\xintSeq [1]{0}{5000}| works and |\xintiiSum{\xintSeq [1]{0}{5000}}| -returns the correct value \digitstt{\xintHalf{\xintiMul{5000}{5001}}}. - -The produced integers are with explicit litteral digits, so if used in |\ifnum| -or other tests they should be properly terminated\footnote{a \csa{space} will - stop the \TeX{} scanning of a number and be gobbled in the process, - maintaining expandability if this is required; the \csa{relax} stops the - scanning but is not gobbled and remains afterwards as a token.}. - -\subsection{Completely expandable prime test}\label{ssec:primesI} - -Let us now construct a completely expandable macro which returns @1@ if its -given input is prime and @0@ if not: -\dverb|@ -\def\remainder #1#2{\the\numexpr #1-(#1/#2)*#2\relax } -\def\IsPrime #1% - {\xintANDof {\xintApply {\remainder {#1}}{\xintSeq {2}{\xintiSqrt{#1}}}}}| - -This uses \csbxint{iSqrt} and assumes its input is at least @5@. Rather than -\xintname's own \csbxint{Rem} we used a quicker |\numexpr| expression as we -are dealing with short integers. Also we used \csbxint{ANDof} which will -return @1@ only if all the items are non-zero. The macro is a bit -silly with an even input, ok, let's enhance it to detect an even input: -\dverb|@ -\def\IsPrime #1% - {\xintifOdd {#1} - {\xintANDof % odd case - {\xintApply {\remainder {#1}} - {\xintSeq [2]{3}{\xintiSqrt{#1}}}% - }% - } - {\xintifEq {#1}{2}{1}{0}}% - }| - -We used the \xintname provided expandable tests (on big integers or fractions) -in oder for |\IsPrime| to be \fexpan dable. - -Our integers are short, but without |\expandafter|'s with -\makeatletter|\@firstoftwo|\catcode`@ \active, or some other related techniques, -direct use of |\ifnum..\fi| tests is dangerous. So to make the macro more -efficient we are going to use the expandable tests provided by the package -\href{http://ctan.org/pkg/etoolbox}{etoolbox}\footnote{\url{http://ctan.org/pkg/etoolbox}}. -The macro becomes: -% -\dverb|@ -\def\IsPrime #1% - {\ifnumodd {#1} - {\xintANDof % odd case - {\xintApply {\remainder {#1}}{\xintSeq [2]{3}{\xintiSqrt{#1}}}}} - {\ifnumequal {#1}{2}{1}{0}}}| - -In the odd case however we have to assume the integer is at least @7@, as -|\xintSeq| generates an empty list if |#1=3| or |5|, and |\xintANDof| returns -@1@ when supplied an empty list. Let us ease up a bit |\xintANDof|'s work by -letting it work on only @0@'s and @1@'s. We could use: -% -\dverb|@ -\def\IsNotDivisibleBy #1#2% - {\ifnum\numexpr #1-(#1/#2)*#2=0 \expandafter 0\else \expandafter1\fi}|% - -\noindent -where the |\expandafter|'s are crucial for this macro to be \fexpan dable and -hence work within the applied \csbxint{ANDof}. Anyhow, now that we have loaded -\href{http://ctan.org/pkg/etoolbox}{etoolbox}, we might as well use: -% -\dverb|@ -\newcommand{\IsNotDivisibleBy}[2]{\ifnumequal{#1-(#1/#2)*#2}{0}{0}{1}} -|% -Let us enhance our prime macro to work also on the small primes: -\dverb|@ -\newcommand{\IsPrime}[1] % returns 1 if #1 is prime, and 0 if not - {\ifnumodd {#1} - {\ifnumless {#1}{8} - {\ifnumequal{#1}{1}{0}{1}}% 3,5,7 are primes - {\xintANDof - {\xintApply - { \IsNotDivisibleBy {#1}}{\xintSeq [2]{3}{\xintiSqrt{#1}}}}% - }}% END OF THE ODD BRANCH - {\ifnumequal {#1}{2}{1}{0}}% EVEN BRANCH -}| - -The input is still assumed positive. There is a deliberate blank before -\csa{IsNotDivisibleBy} to use this feature of \csbxint{Apply}: a space stops the -expansion of the applied macro (and disappears). This expansion will be done by -\csbxint{ANDof}, which has been designed to skip everything as soon as it finds -a false (i.e. zero) input. This way, the efficiency is considerably improved. - -We did generate via the \csbxint{Seq} too many potential divisors though. Later -sections give two variants: one with \csbxint{iloop} (\autoref{ssec:primesII}) -which is still expandable and another one (\autoref{ssec:primesIII}) which is a -close variant of the |\IsPrime| code above but with the \csbxint{For} loop, thus -breaking expandability. The \hyperref[ssec:primesII]{xintiloop variant} does not -first evaluate the integer square root, the \hyperref[ssec:primesIII]{xintFor - variant} still does. I did not compare their efficiencies. - -% Hmm, if one really needs to compute primes fast, sure I do applaud using -% \xintname, but, well, there is some slight -% overhead\MyMarginNoteWithBrace{funny private joke} in using \TeX{} for these -% things (something like a factor @1000@? not tested\dots) compared to accessing -% to the |CPU| ressources via standard compiled code from a standard programming -% language\dots - -Let us construct with this expandable primality test a table of the prime -numbers up to @1000@. We need to count how many we have in order to know how -many tab stops one shoud add in the last row.\footnote{although a tabular row - may have less tabs than in the preamble, there is a problem with the - \char`\|\space\space - vertical rule, if one does that.} There is some subtlety for this -last row. Turns out to be better to insert a |\\| only when we know for sure we -are starting a new row; this is how we have designed the |\OneCell| macro. And -for the last row, there are many ways, we use again |\xintApplyUnbraced| but -with a macro which gobbles its argument and replaces it with a tabulation -character. The \csbxint{For*} macro would be more elegant here. -% -\dverb?@ -\newcounter{primecount} -\newcounter{cellcount} -\newcommand{\NbOfColumns}{13} -\newcommand{\OneCell}[1]{% - \ifnumequal{\IsPrime{#1}}{1} - {\stepcounter{primecount} - \ifnumequal{\value{cellcount}}{\NbOfColumns} - {\\\setcounter{cellcount}{1}#1} - {&\stepcounter{cellcount}#1}% - } % was prime - {}% not a prime, nothing to do -} -\newcommand{\OneTab}[1]{&} -\begin{tabular}{|*{\NbOfColumns}{r}|} -\hline -2 \setcounter{cellcount}{1}\setcounter{primecount}{1}% - \xintApplyUnbraced \OneCell {\xintSeq [2]{3}{999}}% - \xintApplyUnbraced \OneTab - {\xintSeq [1]{1}{\the\numexpr\NbOfColumns-\value{cellcount}\relax}}% - \\ -\hline -\end{tabular} -There are \arabic{primecount} prime numbers up to 1000.? - -The table has been put in \hyperref[primesupto1000]{float} which appears -\vpageref{primesupto1000}. -We had to be careful to use in the last row \csbxint{Seq} with its optional -argument |[1]| so as to not generate a decreasing sequence from |1| to |0|, but -really an empty sequence in case the row turns out to already have all its -cells (which doesn't happen here but would with a number of columns dividing -@168@). -% -\newcommand{\IsNotDivisibleBy}[2]{\ifnumequal{#1-(#1/#2)*#2}{0}{0}{1}} - -\newcommand{\IsPrime}[1] - {\ifnumodd {#1} - {\ifnumless {#1}{8} - {\ifnumequal{#1}{1}{0}{1}}% 3,5,7 are primes - {\xintANDof - {\xintApply - { \IsNotDivisibleBy {#1}}{\xintSeq [2]{3}{\xintiSqrt{#1}}}}% - }}% END OF THE ODD BRANCH - {\ifnumequal {#1}{2}{1}{0}}% EVEN BRANCH -} - -\newcounter{primecount} -\newcounter{cellcount} -\newcommand{\NbOfColumns}{13} -\newcommand{\OneCell}[1] - {\ifnumequal{\IsPrime{#1}}{1} - {\stepcounter{primecount} - \ifnumequal{\value{cellcount}}{\NbOfColumns} - {\\\setcounter{cellcount}{1}#1} - {&\stepcounter{cellcount}#1}% - } % was prime - {}% not a prime nothing to do -} -\newcommand{\OneTab}[1]{&} -\begin{figure*}[ht!] - \centering - \phantomsection\label{primesupto1000} - \begin{tabular}{|*{\NbOfColumns}{r}|} - \hline - 2\setcounter{cellcount}{1}\setcounter{primecount}{1}% - \xintApplyUnbraced \OneCell {\xintSeq [2]{3}{999}}% - \xintApplyUnbraced \OneTab - {\xintSeq [1]{1}{\the\numexpr\NbOfColumns-\value{cellcount}\relax}}% - \\ - \hline - \end{tabular} -\smallskip -\centeredline{There are \arabic{primecount} prime numbers up to 1000.} -\end{figure*} - -\subsection{\csbh{xintloop}, \csbh{xintbreakloop}, \csbh{xintbreakloopanddo}, \csbh{xintloopskiptonext}} -\label{xintloop} -\label{xintbreakloop} -\label{xintbreakloopanddo} -\label{xintloopskiptonext} -% {\small New with release |1.09g|. Release |1.09h| -% makes them long macros.\par} - -|\xintloop|\meta{stuff}|\if<test>...\repeat|\retype{} is an expandable loop -compatible with nesting. However to break out of the loop one almost always need -some un-expandable step. The cousin \csbxint{iloop} is \csbxint{loop} with an -embedded expandable mechanism allowing to exit from the loop. The iterated -commands may contain |\par| tokens or empty lines. - -If a sub-loop is to be used all the material from the start of the main loop and -up to the end of the entire subloop should be braced; these braces will be -removed and do not create a group. The simplest to allow the nesting of one or -more sub-loops is to brace everything between \csa{xintloop} and \csa{repeat}, -being careful not to leave a space between the closing brace and |\repeat|. - -As this loop and \csbxint{iloop} will primarily be of interest to experienced -\TeX{} macro programmers, my description will assume that the user is -knowledgeable enough. Some examples in this document will be perhaps more -illustrative than my attemps at explanation of use. - -One can abort the loop with \csbxint{breakloop}; this should not be used inside -the final test, and one should expand the |\fi| from the corresponding test -before. One has also \csbxint{breakloopanddo} whose first argument will be -inserted in the token stream after the loop; one may need a macro such as -|\xint_afterfi| to move the whole thing after the |\fi|, as a simple -|\expandafter| will not be enough. - -One will usually employ some count registers to manage the exit test from the -loop; this breaks expandability, see \csbxint{iloop} for an expandable integer -indexed loop. Use in alignments will be complicated by the fact that cells -create groups, and also from the fact that any encountered unexpandable material -will cause the \TeX{} input scanner to insert |\endtemplate| on each encountered -|&| or |\cr|; thus |\xintbreakloop| may not work as expected, but the situation -can be resolved via |\xint_firstofone{&}| or use of |\TAB| with |\def\TAB{&}|. -It is thus simpler for alignments to use rather than \csbxint{loop} either the -expandable \csbxint{ApplyUnbraced} or the non-expandable but alignment -compatible \csbxint{ApplyInline}, \csbxint{For} or \csbxint{For*}. - -As an example, let us suppose we have two macros |\A|\marg{i}\marg{j} and -|\B|\marg{i}\marg{j} behaving like (small) integer valued matrix entries, and we -want to define a macro |\C|\marg{i}\marg{j} giving the matrix product (|i| and -|j| may be count registers). We will assume that |\A[I]| expands to the number -of rows, |\A[J]| to the number of columns and want the produced |\C| to act in -the same manner. The code is very dispendious in use of |\count| registers, not -optimized in any way, not made very robust (the defined macro can not have the -same name as the first two matrices for example), we just wanted to quickly -illustrate use of the nesting capabilities of |\xintloop|.\footnote{for a more sophisticated implementation of matrix multiplication, inclusive of determinants, inverses, and display utilities, with entries big integers or decimal numbers or even fractions see \url{http://tex.stackexchange.com/a/143035/4686} from November 11, 2013.} -\begingroup -\makeatother -\begin{verbatim} -\newcount\rowmax \newcount\colmax \newcount\summax -\newcount\rowindex \newcount\colindex \newcount\sumindex -\newcount\tmpcount -\makeatletter -\def\MatrixMultiplication #1#2#3{% - \rowmax #1[I]\relax - \colmax #2[J]\relax - \summax #1[J]\relax - \rowindex 1 - \xintloop % loop over row index i - {\colindex 1 - \xintloop % loop over col index k - {\tmpcount 0 - \sumindex 1 - \xintloop % loop over intermediate index j - \advance\tmpcount \numexpr #1\rowindex\sumindex*#2\sumindex\colindex\relax - \ifnum\sumindex<\summax - \advance\sumindex 1 - \repeat }% - \expandafter\edef\csname\string#3{\the\rowindex.\the\colindex}\endcsname - {\the\tmpcount}% - \ifnum\colindex<\colmax - \advance\colindex 1 - \repeat }% - \ifnum\rowindex<\rowmax - \advance\rowindex 1 - \repeat - \expandafter\edef\csname\string#3{I}\endcsname{\the\rowmax}% - \expandafter\edef\csname\string#3{J}\endcsname{\the\colmax}% - \def #3##1{\ifx[##1\expandafter\Matrix@helper@size - \else\expandafter\Matrix@helper@entry\fi #3{##1}}% -}% -\def\Matrix@helper@size #1#2#3]{\csname\string#1{#3}\endcsname }% -\def\Matrix@helper@entry #1#2#3% - {\csname\string#1{\the\numexpr#2.\the\numexpr#3}\endcsname }% -\def\A #1{\ifx[#1\expandafter\A@size - \else\expandafter\A@entry\fi {#1}}% -\def\A@size #1#2]{\ifx I#23\else4\fi}% 3rows, 4columns -\def\A@entry #1#2{\the\numexpr #1+#2-1\relax}% not pre-computed... -\def\B #1{\ifx[#1\expandafter\B@size - \else\expandafter\B@entry\fi {#1}}% -\def\B@size #1#2]{\ifx I#24\else3\fi}% 4rows, 3columns -\def\B@entry #1#2{\the\numexpr #1-#2\relax}% not pre-computed... -\makeatother -\MatrixMultiplication\A\B\C \MatrixMultiplication\C\C\D % etc... -\[\begin{pmatrix} - \A11&\A12&\A13&\A14\\ - \A21&\A22&\A23&\A24\\ - \A31&\A32&\A33&\A34 - \end{pmatrix} -\times - \begin{pmatrix} - \B11&\B12&\B13\\ - \B21&\B22&\B23\\ - \B31&\B32&\B33\\ - \B41&\B42&\B43 - \end{pmatrix} -= -\begin{pmatrix} - \C11&\C12&\C13\\ - \C21&\C22&\C23\\ - \C31&\C32&\C33 -\end{pmatrix}\] -\[\begin{pmatrix} - \C11&\C12&\C13\\ - \C21&\C22&\C23\\ - \C31&\C32&\C33 -\end{pmatrix}^2 = \begin{pmatrix} - \D11&\D12&\D13\\ - \D21&\D22&\D23\\ - \D31&\D32&\D33 -\end{pmatrix}\] -\end{verbatim} -\newcount\rowmax \newcount\colmax \newcount\summax -\newcount\rowindex \newcount\colindex \newcount\sumindex -\newcount\tmpcount -\makeatletter -\def\MatrixMultiplication #1#2#3{% - \rowmax #1[I]\relax - \colmax #2[J]\relax - \summax #1[J]\relax - \rowindex 1 - \xintloop % loop over row index i - {\colindex 1 - \xintloop % loop over col index k - {\tmpcount 0 - \sumindex 1 - \xintloop % loop over intermediate index j - \advance\tmpcount \numexpr #1\rowindex\sumindex*#2\sumindex\colindex\relax - \ifnum\sumindex<\summax - \advance\sumindex 1 - \repeat }% - \expandafter\edef\csname\string#3{\the\rowindex.\the\colindex}\endcsname - {\the\tmpcount}% - \ifnum\colindex<\colmax - \advance\colindex 1 - \repeat }% - \ifnum\rowindex<\rowmax - \advance\rowindex 1 - \repeat - \expandafter\edef\csname\string#3{I}\endcsname{\the\rowmax}% - \expandafter\edef\csname\string#3{J}\endcsname{\the\colmax}% - \def #3##1{\ifx[##1\expandafter\Matrix@helper@size - \else\expandafter\Matrix@helper@entry\fi #3{##1}}% -}% -\def\Matrix@helper@size #1#2#3]{\csname\string#1{#3}\endcsname }% -\def\Matrix@helper@entry #1#2#3% - {\csname\string#1{\the\numexpr#2.\the\numexpr#3}\endcsname }% -\def\A #1{\ifx[#1\expandafter\A@size - \else\expandafter\A@entry\fi {#1}}% -\def\A@size #1#2]{\ifx I#23\else4\fi}% 3rows, 4columns -\def\A@entry #1#2{\the\numexpr #1+#2-1\relax}% not pre-computed... -\def\B #1{\ifx[#1\expandafter\B@size - \else\expandafter\B@entry\fi {#1}}% -\def\B@size #1#2]{\ifx I#24\else3\fi}% 4rows, 3columns -\def\B@entry #1#2{\the\numexpr #1-#2\relax}% not pre-computed... -\makeatother -\MatrixMultiplication\A\B\C \MatrixMultiplication\C\C\D -\setlength{\unitlength}{1cm}% -% le picture de LaTeX est tout de même assez génial! -\begin{picture}(0,0) -\put(5,11){\vtop{\hsize8cm -\[\begin{pmatrix} - \A11&\A12&\A13&\A14\\ - \A21&\A22&\A23&\A24\\ - \A31&\A32&\A33&\A34 - \end{pmatrix} -\times - \begin{pmatrix} - \B11&\B12&\B13\\ - \B21&\B22&\B23\\ - \B31&\B32&\B33\\ - \B41&\B42&\B43 - \end{pmatrix} -= -\begin{pmatrix} - \C11&\C12&\C13\\ - \C21&\C22&\C23\\ - \C31&\C32&\C33 -\end{pmatrix}\] -\[\begin{pmatrix} - \C11&\C12&\C13\\ - \C21&\C22&\C23\\ - \C31&\C32&\C33 -\end{pmatrix}^2 = \begin{pmatrix} - \D11&\D12&\D13\\ - \D21&\D22&\D23\\ - \D31&\D32&\D33 -\end{pmatrix}\]\MatrixMultiplication\C\D\E -\[\begin{pmatrix} - \C11&\C12&\C13\\ - \C21&\C22&\C23\\ - \C31&\C32&\C33 -\end{pmatrix}^3 = \begin{pmatrix} - \E11&\E12&\E13\\ - \E21&\E22&\E23\\ - \E31&\E32&\E33 -\end{pmatrix}\]\MatrixMultiplication\C\E\F -\[\begin{pmatrix} - \C11&\C12&\C13\\ - \C21&\C22&\C23\\ - \C31&\C32&\C33 -\end{pmatrix}^4 = \begin{pmatrix} - \F11&\F12&\F13\\ - \F21&\F22&\F23\\ - \F31&\F32&\F33 -\end{pmatrix}\]}} -\end{picture}\par -\endgroup - -\kern-2\baselineskip - -\subsection{\csbh{xintiloop}, \csbh{xintiloopindex}, \csbh{xintouteriloopindex}, - \csbh{xintbreakiloop}, \csbh{xintbreakiloopanddo}, \csbh{xintiloopskiptonext}, -\csbh{xintiloopskipandredo}} -\label{xintiloop} -\label{xintbreakiloop} -\label{xintbreakiloopanddo} -\label{xintiloopskiptonext} -\label{xintiloopskipandredo} -\label{xintiloopindex} -\label{xintouteriloopindex} -%{\small New with release |1.09g|.\par} - -\csa{xintiloop}|[start+delta]|\meta{stuff}|\if<test> ... \repeat|\retype{} is a -completely expandable nestable loop. complete expandability depends naturally on -the actual iterated contents, and complete expansion will not be achievable -under a sole \fexpan sion, as is indicated by the hollow star in the margin; -thus the loop can be used inside an |\edef| but not inside arguments to the -package macros. It can be used inside an |\xintexpr..\relax|. - -This loop benefits via \csbxint{iloopindex} to (a limited access to) the integer -index of the iteration. The starting value |start| (which may be a |\count|) and -increment |delta| (\emph{id.}) are mandatory arguments. A space after the -closing square bracket is not significant, it will be ignored. Spaces inside the -square brackets will also be ignored as the two arguments are first given to a -|\numexpr...\relax|. Empty lines and explicit |\par| tokens are accepted. - -As with \csbxint{loop}, this tool will mostly be of interest to advanced users. -For nesting, one puts inside braces all the -material from the start (immediately after |[start+delta]|) and up to and -inclusive of the inner loop, these braces will be removed and do not create a -loop. In case of nesting, \csbxint{outeriloopindex} gives access to the index of -the outer loop. If needed one could write on its model a macro giving access to -the index of the outer outer loop (or even to the |nth| outer loop). - - -The \csa{xintiloopindex} and \csa{xintouteriloopindex} can not be used inside -braces, and generally speaking this means they should be expanded first when -given as argument to a macro, and that this macro receives them as delimited -arguments, not braced ones. Or, but naturally this will break expandability, one -can assign the value of \csa{xintiloopindex} to some |\count|. Both -\csa{xintiloopindex} and \csa{xintouteriloopindex} extend to the litteral -representation of the index, thus in |\ifnum| tests, if it comes last one has to -correctly end the macro with a |\space|, or encapsulate it in a -|\numexpr..\relax|. - -When the repeat-test of the loop is, for example, |\ifnum\xintiloopindex<10 -\repeat|, this means that the last iteration will be with |\xintiloopindex=10| -(assuming |delta=1|). There is also |\ifnum\xintiloopindex=10 \else\repeat| to -get the last iteration to be the one with |\xintiloopindex=10|. - -One has \csbxint{breakiloop} and \csbxint{breakiloopanddo} to abort the loop. -The syntax of |\xintbreakiloopanddo| is a bit surprising, the sequence of tokens -to be executed after breaking the loop is not within braces but is delimited by -a dot as in: -% -\centeredline{|\xintbreakiloopanddo <afterloop>.etc.. etc... \repeat|} -% -The reason is that one may wish to use the then current value of -|\xintiloopindex| in |<afterloop>| but it can't be within braces at the time it -is evaluated. However, it is not that easy as |\xintiloopindex| must be expanded -before, so one ends up with code like this: -% -\centeredline -{|\expandafter\xintbreakiloopanddo\expandafter\macro\xintiloopindex.%|} -\centeredline{|etc.. etc.. \repeat|} -% -As moreover the |\fi| from the test leading to the decision of breaking out of -the loop must be cleared out of the way, the above should be -a branch of an expandable conditional test, else one needs something such -as: -\centeredline -{|\xint_afterfi{\expandafter\xintbreakiloopanddo\expandafter\macro\xintiloopindex.}%|} -\centeredline{|\fi etc..etc.. \repeat|} - - -There is \csbxint{iloopskiptonext} to abort the current iteration and skip to -the next, \hyperref[xintiloopskipandredo]{\ttfamily\hyphenchar\font45 \char92 - xintiloopskip\-and\-redo} to skip to the end of the current iteration and redo -it with the same value of the index (something else will have to change for this -not to become an eternal loop\dots ). - -Inside alignments, if the looped-over text contains a |&| or a |\cr|, any -un-expandable material before a \csbxint{iloopindex} will make it fail because -of |\endtemplate|; in such cases one can always either replace |&| by a macro -expanding to it or replace it by a suitable |\firstofone{&}|, and similarly for -|\cr|. - -\phantomsection\label{edefprimes} -As an example, let us construct an |\edef\z{...}| which will define |\z| to be a -list of prime numbers: -\dverb|@ -\edef\z -{\xintiloop [10001+2] - {\xintiloop [3+2] - \ifnum\xintouteriloopindex<\numexpr\xintiloopindex*\xintiloopindex\relax - \xintouteriloopindex, - \expandafter\xintbreakiloop - \fi - \ifnum\xintouteriloopindex=\numexpr - (\xintouteriloopindex/\xintiloopindex)*\xintiloopindex\relax - \else - \repeat - }% no space here - \ifnum \xintiloopindex < 10999 \repeat }% -\meaning\z| -\begingroup%\ttfamily -\edef\z -{\xintiloop [10001+2] - {\xintiloop [3+2] - \ifnum\xintouteriloopindex<\numexpr\xintiloopindex*\xintiloopindex\relax - \xintouteriloopindex, - \expandafter\xintbreakiloop - \fi - \ifnum\xintouteriloopindex=\numexpr - (\xintouteriloopindex/\xintiloopindex)*\xintiloopindex\relax - \else - \repeat - }% no space here - \ifnum \xintiloopindex < 10999 \repeat }% -\meaning\z and we should have taken some steps to not have a trailing comma, but -the point was to show that one can do that in an |\edef|\,! See also -\autoref{ssec:primesII} which extracts from this code its way of testing -primality. -\endgroup - - -Let us create an alignment where each row will contain all divisors of its -first entry. -\dverb|@ -\tabskip1ex -\halign{&\hfil#\hfil\cr - \xintiloop [1+1] - {\expandafter\bfseries\xintiloopindex & - \xintiloop [1+1] - \ifnum\xintouteriloopindex=\numexpr - (\xintouteriloopindex/\xintiloopindex)*\xintiloopindex\relax - \xintiloopindex&\fi - \ifnum\xintiloopindex<\xintouteriloopindex\space % \space is CRUCIAL - \repeat \cr }% - \ifnum\xintiloopindex<30 - \repeat }| - -% -\noindent We wanted this first entry in bold face, but |\bfseries| leads to -unexpandable tokens, so the |\expandafter| was necessary for |\xintiloopindex| -and |\xintouteriloopindex| not to be confronted with a hard to digest -|\endtemplate|. An alternative way of coding is: -% -\dverb|@ \tabskip1ex -\def\firstofone #1{#1}% -\halign{&\hfil#\hfil\cr - \xintiloop [1+1] - {\bfseries\xintiloopindex\firstofone{&}% - \xintiloop [1+1] \ifnum\xintouteriloopindex=\numexpr - (\xintouteriloopindex/\xintiloopindex)*\xintiloopindex\relax - \xintiloopindex\firstofone{&}\fi - \ifnum\xintiloopindex<\xintouteriloopindex\space % \space is CRUCIAL - \repeat \firstofone{\cr}}% - \ifnum\xintiloopindex<30 \repeat }| - -\noindent -Here is the output, thus obtained without any count register: -\begingroup\catcode`_ 11 -\begin{multicols}2 -\tabskip1ex -\halign{&\hfil#\hfil\cr - \xintiloop [1+1] - {\expandafter\bfseries\xintiloopindex & - \xintiloop [1+1] - \ifnum\xintouteriloopindex=\numexpr - (\xintouteriloopindex/\xintiloopindex)*\xintiloopindex\relax - \xintiloopindex&\fi - \ifnum\xintiloopindex<\xintouteriloopindex\space % CRUCIAL \space HERE - \repeat \cr }% - \ifnum\xintiloopindex<30 - \repeat -} -\end{multicols} -\endgroup - -\subsection{Another completely expandable prime test}\label{ssec:primesII} - -The |\IsPrime| macro from \autoref{ssec:primesI} checked expandably if a (short) -integer was prime, here is a partial rewrite using \csbxint{iloop}. We use the -|etoolbox| expandable conditionals for convenience, but not everywhere as -|\xintiloopindex| can not be evaluated while being braced. This is also the -reason why |\xintbreakiloopanddo| is delimited, and the next macro -|\SmallestFactor| which returns the smallest prime factor examplifies that. One -could write more efficient completely expandable routines, the aim here was only -to illustrate use of the general purpose \csbxint{iloop}. A little table giving -the first values of |\SmallestFactor| follows, its coding uses \csbxint{For}, -which is described later; none of this uses count registers. -% -\dverb?@ -\newcommand{\IsPrime}[1] % returns 1 if #1 is prime, and 0 if not - {\ifnumodd {#1} - {\ifnumless {#1}{8} - {\ifnumequal{#1}{1}{0}{1}}% 3,5,7 are primes - {\if - \xintiloop [3+2] - \ifnum#1<\numexpr\xintiloopindex*\xintiloopindex\relax - \expandafter\xintbreakiloopanddo\expandafter1\expandafter.% - \fi - \ifnum#1=\numexpr (#1/\xintiloopindex)*\xintiloopindex\relax - \else - \repeat 00\expandafter0\else\expandafter1\fi - }% - }% END OF THE ODD BRANCH - {\ifnumequal {#1}{2}{1}{0}}% EVEN BRANCH -}% -\catcode`_ 11 -\newcommand{\SmallestFactor}[1] % returns the smallest prime factor of #1>1 - {\ifnumodd {#1} - {\ifnumless {#1}{8} - {#1}% 3,5,7 are primes - {\xintiloop [3+2] - \ifnum#1<\numexpr\xintiloopindex*\xintiloopindex\relax - \xint_afterfi{\xintbreakiloopanddo#1.}% - \fi - \ifnum#1=\numexpr (#1/\xintiloopindex)*\xintiloopindex\relax - \xint_afterfi{\expandafter\xintbreakiloopanddo\xintiloopindex.}% - \fi - \iftrue\repeat - }% - }% END OF THE ODD BRANCH - {2}% EVEN BRANCH -}% -\catcode`_ 8 - \begin{tabular}{|c|*{10}c|} - \hline - \xintFor #1 in {0,1,2,3,4,5,6,7,8,9}\do {&\bfseries #1}\\ - \hline - \bfseries 0&--&--&2&3&2&5&2&7&2&3\\ - \xintFor #1 in {1,2,3,4,5,6,7,8,9}\do - {\bfseries #1% - \xintFor #2 in {0,1,2,3,4,5,6,7,8,9}\do - {&\SmallestFactor{#1#2}}\\}% - \hline - \end{tabular} -? -\catcode`_ 11 -\newcommand{\SmallestFactor}[1] % returns the smallest prime factor of #1>1 - {\ifnumodd {#1} - {\ifnumless {#1}{8} - {#1}% 3,5,7 are primes - {\xintiloop [3+2] - \ifnum#1<\numexpr\xintiloopindex*\xintiloopindex\relax - \xint_afterfi{\xintbreakiloopanddo#1.}% - \fi - \ifnum#1=\numexpr (#1/\xintiloopindex)*\xintiloopindex\relax - \xint_afterfi{\expandafter\xintbreakiloopanddo\xintiloopindex.}% - \fi - \iftrue\repeat - }% - }% END OF THE ODD BRANCH - {2}% EVEN BRANCH -}% -\catcode`_ 8 -{\centering - \begin{tabular}{|c|*{10}c|} - \hline - \xintFor #1 in {0,1,2,3,4,5,6,7,8,9}\do {&\bfseries #1}\\ - \hline - \bfseries 0&--&--&2&3&2&5&2&7&2&3\\ - \xintFor #1 in {1,2,3,4,5,6,7,8,9}\do - {\bfseries #1% - \xintFor #2 in {0,1,2,3,4,5,6,7,8,9}\do - {&\SmallestFactor{#1#2}}\\}% - \hline - \end{tabular}\par } - -\subsection{A table of factorizations} -\label{ssec:factorizationtable} - -As one more example with \csbxint{iloop} let us use an alignment to display the -factorization of some numbers. The loop will actually only play a minor r\^ole -here, just handling the row index, the row contents being almost entirely -produced via a macro |\factorize|. The factorizing macro does not use -|\xintiloop| as it didn't appear to be the convenient tool. As |\factorize| will -have to be used on |\xintiloopindex|, it has been defined as a delimited macro. - -To spare some fractions of a second in the compilation time of this document -(which has many many other things to do), \number"7FFFFFED{} and -\number"7FFFFFFF, which turn out to be prime numbers, are not given to -|factorize| but just typeset directly; this illustrates use of -\csbxint{iloopskiptonext}. - -\begingroup -\def\MacroFont{\ttfamily\baselineskip12pt\relax \catcode`\" 12 } -\dverb|@ -\tabskip1ex -\halign {\hfil\strut#\hfil&&\hfil#\hfil\cr\noalign{\hrule} - \xintiloop ["7FFFFFE0+1] - \expandafter\bfseries\xintiloopindex & - \ifnum\xintiloopindex="7FFFFFED - \number"7FFFFFED\cr\noalign{\hrule} - \expandafter\xintiloopskiptonext - \fi - \expandafter\factorize\xintiloopindex.\cr\noalign{\hrule} - \ifnum\xintiloopindex<"7FFFFFFE - \repeat - \bfseries \number"7FFFFFFF&\number "7FFFFFFF\cr\noalign{\hrule} -}|\par\smallskip -\endgroup - -The \hyperref[floatfactorize]{table} has been made into a float which appears -\vpageref{floatfactorize}. Here is now the code for factorization; the -conditionals use -the package provided |\xint_firstoftwo| and |\xint_secondoftwo|, one could have -employed rather \LaTeX{}'s own \texttt{\char92\string@firstoftwo} and -\texttt{\char92\string@secondoftwo}, or, simpler still in \LaTeX{} context, the -|\ifnumequal|, |\ifnumless| \dots, utilities from the package |etoolbox| which -do exactly that under the hood. Only \TeX{} acceptable numbers are treated here, -but it would be easy to make a translation and use the \xintname macros, thus -extending the scope to big numbers; naturally up to a cost in speed. - -The reason for some strange looking expressions is to avoid arithmetic overflow. - -\begingroup -\def\MacroFont{\ttfamily\baselineskip12pt\relax \catcode`\" 12 } -\dverb|@ -\catcode`_ 11 -\def\abortfactorize #1\xint_secondoftwo\fi #2#3{\fi} - -\def\factorize #1.{\ifnum#1=1 \abortfactorize\fi - % avoid overflow if #1="7FFFFFFF - \ifnum\numexpr #1-2=\numexpr ((#1/2)-1)*2\relax - \expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi - {2&\expandafter\factorize\the\numexpr#1/2.}% - {\factorize_b #1.3.}}% - -\def\factorize_b #1.#2.{\ifnum#1=1 \abortfactorize\fi - % this will avoid overflow which could result from #2*#2 - \ifnum\numexpr #1-(#2-1)*#2<#2 - #1\abortfactorize % this #1 is prime - \fi - % again, avoiding overflow as \numexpr integer division - % rounds rather than truncates. - \ifnum \numexpr #1-#2=\numexpr ((#1/#2)-1)*#2\relax - \expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi - {#2&\expandafter\factorize_b\the\numexpr#1/#2.#2.}% - {\expandafter\factorize_b\the\numexpr #1\expandafter.% - \the\numexpr #2+2.}}% -\catcode`_ 8| -\endgroup - -\catcode`_ 11 -\def\abortfactorize #1\xint_secondoftwo\fi #2#3{\fi} - -\def\factorize #1.{\ifnum#1=1 \abortfactorize\fi - \ifnum\numexpr #1-2=\numexpr ((#1/2)-1)*2\relax - \expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi - {2&\expandafter\factorize\the\numexpr#1/2.}% - {\factorize_b #1.3.}}% - -\def\factorize_b #1.#2.{\ifnum#1=1 \abortfactorize\fi - \ifnum\numexpr #1-(#2-1)*#2<#2 - #1\abortfactorize - \fi - \ifnum \numexpr #1-#2=\numexpr ((#1/#2)-1)*#2\relax - \expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi - {#2&\expandafter\factorize_b\the\numexpr#1/#2.#2.}% - {\expandafter\factorize_b\the\numexpr #1\expandafter.% - \the\numexpr #2+2.}}% -\catcode`_ 8 -\begin{figure*}[ht!] -\centering\phantomsection\label{floatfactorize} -\tabskip1ex -\centeredline{\vbox{\halign {\hfil\strut#\hfil&&\hfil#\hfil\cr\noalign{\hrule} - \xintiloop ["7FFFFFE0+1] - \expandafter\bfseries\xintiloopindex & - \ifnum\xintiloopindex="7FFFFFED - \number"7FFFFFED\cr\noalign{\hrule} - \expandafter\xintiloopskiptonext - \fi - \expandafter\factorize\xintiloopindex.\cr\noalign{\hrule} - \ifnum\xintiloopindex<"7FFFFFFE - \repeat - \bfseries \number"7FFFFFFF&\number "7FFFFFFF\cr\noalign{\hrule} -}}} -\centeredline{A table of factorizations} -\end{figure*} - - -\begin{framed} - The next utilities are not compatible with expansion-only context. -\end{framed} - -\subsection{\csbh{xintApplyInline}}\label{xintApplyInline} - -% {\small |1.09a|, enhanced in |1.09c| to be usable within alignments, and -% corrected in |1.09d| for a problem related to spaces at the very end of the -% list parameter.\par} - -\csa{xintApplyInline}|{\macro}|\marg{list}\ntype{o{\lowast f}} works non -expandably. It applies the one-parameter |\macro| to the first element of the -expanded list (|\macro| may have itself some arguments, the list item will be -appended as last argument), and is then re-inserted in the input stream after -the tokens resulting from this first expansion of |\macro|. The next item is -then handled. - -This is to be used in situations where one needs to do some repetitive -things. It is not expandable and can not be completely expanded inside a -macro definition, to prepare material for later execution, contrarily to what -\csbxint{Apply} or \csbxint{ApplyUnbraced} achieve. - -\dverb|@ -\def\Macro #1{\advance\cnta #1 , \the\cnta} -\cnta 0 -0\xintApplyInline\Macro {3141592653}.| -\def\Macro #1{\advance\cnta #1 , \the\cnta} -\cnta 0 -Output: 0\xintApplyInline\Macro {3141592653}. - - -The first argument |\macro| does not have to be an expandable macro. - -\csa{xintApplyInline} submits its second, token list parameter to an -\hyperref[sec:expansions]{\fexpan -sion}. Then, each \emph{unbraced} item will also be \fexpan ded. This provides -an easy way to insert one list inside another. \emph{Braced} items are not -expanded. Spaces in-between items are gobbled (as well as those at the start -or the end of the list), but not the spaces \emph{inside} the braced items. - -\csa{xintApplyInline}, despite being non-expandable, does survive to -contexts where the executed |\macro| closes groups, as happens inside -alignments with the tabulation character |&|. -This tabular for example:\par -\smallskip -\centeredline - {\begin{tabular}{ccc} - $N$ & $N^2$ & $N^3$ \\ \hline - \def\Row #1{ #1 & \xintiSqr {#1} & \xintiPow {#1}{3} \\ \hline }% - \xintApplyInline \Row {\xintCSVtoList{17,28,39,50,61}} - \end{tabular}} -\smallskip -% 38 = &, 43 = +, 36=$, 45 = - -was obtained from the following input: -\dverb|@ -\begin{tabular}{ccc} - $N$ & $N^2$ & $N^3$ \\ \hline - \def\Row #1{ #1 & \xintiSqr {#1} & \xintiPow {#1}{3} \\ \hline }% - \xintApplyInline \Row {\xintCSVtoList{17,28,39,50,61}} -\end{tabular}|% -Despite the fact that the first encountered tabulation character in the first -row close a group and thus erases |\Row| from \TeX's memory, |\xintApplyInline| -knows how to deal with this. - -Using \csbxint{ApplyUnbraced} is an alternative: the difference is that -this would have prepared all rows first and only put them back into the -token stream once they are all assembled, whereas with |\xintApplyInline| -each row is constructed and immediately fed back into the token stream: when -one does things with numbers having hundreds of digits, one learns that -keeping on hold and shuffling around hundreds of tokens has an impact on -\TeX{}'s speed (make this ``thousands of tokens'' for the impact to be -noticeable). - -One may nest various |\xintApplyInline|'s. For example (see the -\hyperref[float]{table} \vpageref{float}):\par -\dverb|@ -\def\Row #1{#1:\xintApplyInline {\Item {#1}}{0123456789}\\ }% -\def\Item #1#2{&\xintiPow {#1}{#2}}% -\begin{tabular}{ccccccccccc} - &0&1&2&3&4&5&6&7&8&9\\ \hline - \xintApplyInline \Row {0123456789} -\end{tabular}| -\begin{figure*}[ht!] - \centering\phantomsection\label{float} - \def\Row #1{#1:\xintApplyInline {\Item {#1}}{0123456789}\\ }% - \def\Item #1#2{&\xintiPow {#1}{#2}}% - \centeredline {\begin{tabular}{ccccccccccc} &0&1&2&3&4&5&6&7&8&9\\ \hline - \xintApplyInline \Row {0123456789} - \end{tabular}} -\end{figure*} - - -One could not move the definition of |\Item| inside the tabular, -as it would get lost after the first |&|. But this -works: -\dverb|@ -\begin{tabular}{ccccccccccc} - &0&1&2&3&4&5&6&7&8&9\\ \hline - \def\Row #1{#1:\xintApplyInline {&\xintiPow {#1}}{0123456789}\\ }% - \xintApplyInline \Row {0123456789} -\end{tabular}| - -A limitation is that, contrarily to what one may have expected, the -|\macro| for an |\xintApplyInline| can not be used to define -the |\macro| for a nested sub-|\xintApplyInline|. For example, -this does not work:\par -\dverb|@ - \def\Row #1{#1:\def\Item ##1{&\xintiPow {#1}{##1}}% - \xintApplyInline \Item {0123456789}\\ }% - \xintApplyInline \Row {0123456789} % does not work -|% -But see \csbxint{For}. - -\subsection{\csbh{xintFor}, \csbh{xintFor*}}\label{xintFor}\label{xintFor*} -% {\small New with |1.09c|. Extended in |1.09e| (\csbxint{BreakFor}, -% \csbxint{integers}, \dots). |1.09f| version handles all macro parameters up -% to -% |#9| and removes spaces around commas.\par} - -\csbxint{For}\ntype{on} is a new kind of for loop. Rather than using macros -for encapsulating list items, its behavior is more like a macro with parameters: -|#1|, |#2|, \dots, |#9| are used to represent the items for up to nine levels of -nested loops. Here is an example: -% -\dverb|@ -\xintFor #9 in {1,2,3} \do {% - \xintFor #1 in {4,5,6} \do {% - \xintFor #3 in {7,8,9} \do {% - \xintFor #2 in {10,11,12} \do {% - $$#9\times#1\times#3\times#2=\xintiiPrd{{#1}{#2}{#3}{#9}}$$}}}} -|% -This example illustrates that one does not have to use |#1| as the first one: -the order is arbitrary. But each level of nesting should have its specific macro -parameter. Nine levels of nesting is presumably overkill, but I did not know -where it was reasonable to stop. |\par| tokens are accepted in both the comma -separated list and the replacement text. - -\begin{framed} - A macro |\macro| whose definition uses internally an \csbxint{For} loop may be - used inside another \csbxint{For} loop even if the two loops both use the same - macro parameter. Note: the loop definition inside |\macro| must double - the character |#| as is the general rule in \TeX{} with definitions done - inside macros. - - The macros \csa{xintFor} and \csa{xintFor*} are not expandable, one can not - use them inside an |\edef|. But they may be used inside alignments (such as a - \LaTeX{} |tabular|), as will be shown in examples. -\end{framed} - -The spaces between the various declarative elements are all optional; -furthermore spaces around the commas or at the start and end of the list -argument are allowed, they will be removed. If an item must contain itself -commas, it should be braced to prevent these commas from being misinterpreted as -list separator. These braces will be removed during processing. The list -argument may be a macro |\MyList| expanding in one step to the comma separated -list (if it has no arguments, it does not have to be braced). It -will be expanded (only once) to reveal its comma separated items for processing, -comma separated items will not be expanded before being fed into the replacement -text as |#1|, or |#2|, etc\dots, only leading and trailing spaces are removed. - -A starred variant \csbxint{For*}\ntype{{\lowast f}n} deals with lists of braced -items, rather than comma separated items. It has also a distinct expansion -policy, which is detailed below. - -Contrarily to what happens in loops where the item is represented by a macro, -here it is truly exactly as when defining (in \LaTeX{}) a ``command'' with -parameters |#1|, etc... This may avoid the user quite a few troubles with -|\expandafter|s or other |\edef/\noexpand|s which one encounters at times when -trying to do things with \LaTeX's {\makeatother|\@for|} or other loops -which encapsulate the item in a macro expanding to that item. - -\begin{framed} - The non-starred variant \csbxint{For} deals with comma separated values - (\emph{spaces before and after the commas are removed}) and the comma - separated list may be a macro which is only expanded once (to prevent - expansion of the first item |\x| in a list directly input as |\x,\y,...| it - should be input as |{\x},\y,..| or |<space>\x,\y,..|, naturally all of that - within the mandatory braces of the \csa{xintFor \#n in \{list\}} syntax). The - items are not expanded, if the input is |<stuff>,\x,<stuff>| then |#1| will be - at some point |\x| not its expansion (and not either a macro with |\x| as - replacement text, just the token |\x|). Input such as |<stuff>,,<stuff>| - creates an empty |#1|, the iteration is not skipped. An empty list does lead - to the use of the replacement text, once, with an empty |#1| (or |#n|). Except - if the entire list is represented as a single macro with no parameters, - \fbox{it must be braced.} -\end{framed} - -\begin{framed} - The starred variant \csbxint{For*} deals with token lists (\emph{spaces - between braced items or single tokens are not significant}) and - \hyperref[sec:expansions]{\fexpan ds} each \emph{unbraced} list item. This - makes it easy to simulate concatenation of various list macros |\x|, |\y|, ... - If |\x| expands to |{1}{2}{3}| and |\y| expands to |{4}{5}{6}| then |{\x\y}| - as argument to |\xintFor*| has the same effect as |{{1}{2}{3}{4}{5}{6}}|% - \stepcounter{footnote}% - \makeatletter\hbox {\@textsuperscript {\normalfont \thefootnote - }}\makeatother. Spaces at the start, end, or in-between items are gobbled - (but naturally not the spaces which may be inside \emph{braced} items). Except - if the list argument is a single macro with no parameters, \fbox{it must be - braced.} Each item which is not braced will be fully expanded (as the |\x| - and |\y| in the example above). An empty list leads to an empty result. -\end{framed} -\begingroup\makeatletter -\def\@footnotetext #1{\insert\footins {\reset@font \footnotesize \interlinepenalty \interfootnotelinepenalty \splittopskip \footnotesep \splitmaxdepth \dp \strutbox \floatingpenalty \@MM \hsize \columnwidth \@parboxrestore \color@begingroup \@makefntext {\rule \z@ \footnotesep \ignorespaces #1\@finalstrut \strutbox }\color@endgroup }} -\addtocounter{footnote}{-1} -\edef\@thefnmark {\thefootnote} -\@footnotetext{braces around single token items - are optional so this is the same as \texttt{\{123456\}}.} -% \stepcounter{footnote} -% \edef\@thefnmark {\thefootnote} -% \@footnotetext{the \csa{space} will stop the \TeX{} scanning of a number and be -% gobbled in the process; the \csa{relax} stops the scanning but is not -% gobbled. Or one may do \csa{numexpr}\texttt{\#1}\csa{relax}, and then the -% \csa{relax} is gobbled.} -\endgroup -%\addtocounter{Hfootnote}{2} -\addtocounter{Hfootnote}{1} - -The macro \csbxint{Seq} which generates arithmetic sequences may only be used -with \csbxint{For*} (numbers from output of |\xintSeq| are braced, not separated -by commas). \centeredline{|\xintFor* #1 in {\xintSeq [+2]{-7}{+2}}\do {stuff - with #1}|} will have |#1=-7,-5,-3,-1, and 1|. The |#1| as issued from the -list produced by \csbxint{Seq} is the litteral representation as would be -produced by |\arabic| on a \LaTeX{} counter, it is not a count register. When -used in |\ifnum| tests or other contexts where \TeX{} looks for a number it -should thus be postfixed with |\relax| or |\space|. - -When nesting \csa{xintFor*} loops, using \csa{xintSeq} in the inner loops is -inefficient, as the arithmetic sequence will be re-created each time. A more -efficient style is: -% -\dverb|@ - \edef\innersequence {\xintSeq[+2]{-50}{50}}% - \xintFor* #1 in {\xintSeq {13}{27}} \do - {\xintFor* #2 in \innersequence \do {stuff with #1 and #2}% - .. some other macros .. }| - -This is a general remark applying for any nesting of loops, one should avoid -recreating the inner lists of arguments at each iteration of the outer loop. -However, in the example above, if the |.. some other macros ..| part -closes a group which was opened before the |\edef\innersequence|, then -this definition will be lost. An alternative to |\edef|, also efficient, -exists when dealing with arithmetic sequences: it is to use the -\csbxint{integers} keyword (described later) which simulates infinite -arithmetic sequences; the loops will then be terminated via a test |#1| -(or |#2| etc\dots) and subsequent use of \csbxint{BreakFor}. - - - -The \csbxint{For} loops are not completely expandable; but they may be nested -and used inside alignments or other contexts where the replacement text closes -groups. Here is an example (still using \LaTeX's tabular): - -\begingroup -\centeredline{\begin{tabular}{rccccc} - \xintFor #7 in {A,B,C} \do {% - #7:\xintFor* #3 in {abcde} \do {&($ #3 \to #7 $)}\\ }% -\end{tabular}} -\endgroup - -\dverb|@ -\begin{tabular}{rccccc} - \xintFor #7 in {A,B,C} \do {% - #7:\xintFor* #3 in {abcde} \do {&($ #3 \to #7 $)}\\ }% -\end{tabular}| - -When -inserted inside a macro for later execution the |#| characters must be -doubled.\footnote{sometimes what seems to be a macro argument isn't really; in - \csa{raisebox\{1cm\}\{}\csa{xintFor \#1 in \{a,b,c\} }\csa{do \{\#1\}\}} no - doubling should be done.} For example: -% -\dverb|@ -\def\T{\def\z {}% - \xintFor* ##1 in {{u}{v}{w}} \do {% - \xintFor ##2 in {x,y,z} \do {% - \expandafter\def\expandafter\z\expandafter {\z\sep (##1,##2)} }% - }% -}% -\T\def\sep {\def\sep{, }}\z |% -\def\T{\def\z {}% - \xintFor* ##1 in {{u}{v}{w}} \do {% - \xintFor ##2 in {x,y,z} \do {% - \expandafter\def\expandafter\z\expandafter {\z\sep (##1,##2)} }% - }}% -\centeredline{\T\def\sep {\def\sep{, }}\z} Similarly when the replacement text -of |\xintFor| defines a macro with parameters, the macro character |#| must be -doubled. - -It is licit to use inside an \csbxint{For} a |\macro| which itself has -been defined to use internally some other \csbxint{For}. The same macro -parameter |#1| can be used with no conflict (as mentioned above, in the -definition of |\macro| the |#| used in the \csbxint{For} declaration must be -doubled, as is the general rule in \TeX{} with things defined inside other -things). - -The iterated commands as well as the list items are allowed to contain explicit -|\par| tokens. Neither \csbxint{For} nor \csbxint{For*} create groups. The -effect is like piling up the iterated commands with each time |#1| (or |#2| ...) -replaced by an item of the list. However, contrarily to the completely -expandable \csbxint{ApplyUnbraced}, but similarly to the non completely -expandable \csbxint{ApplyInline} each iteration is executed first before looking -at the next |#1|\footnote{to be completely honest, both \csbxint{For} and - \csbxint{For*} intially scoop up both the list and the iterated commands; - \csbxint{For} scoops up a second time the entire comma separated list in order - to feed it to \csbxint{CSVtoList}. The starred variant \csbxint{For*} which - does not need this step will thus be a bit faster on equivalent inputs.} (and -the starred variant \csbxint{For*} keeps on expanding each unbraced item it -finds, gobbling spaces). - -\subsection{\csbh{xintifForFirst}, \csbh{xintifForLast}} -\label{xintifForFirst}\label{xintifForLast} -% {\small New in |1.09e|.\par} - - -\csbxint{ifForFirst}\,\texttt{\{YES branch\}\{NO branch\}}\etype{nn} - and \csbxint{ifForLast}\,\texttt{\{YES - branch\}\hskip 0pt plus 0.2em \{NO branch\}}\etype{nn} execute the |YES| or -|NO| branch -if the -\csbxint{For} -or \csbxint{For*} loop is currently in its first, respectively last, iteration. - -Designed to work as expected under nesting. Don't forget an empty brace pair -|{}| if a branch is to do nothing. May be used multiple times in the replacement -text of the loop. - -There is no such thing as an iteration counter provided by the \csa{xintFor} -loops; the user is invited to define if needed his own count register or -\LaTeX{} counter, for example with a suitable |\stepcounter| inside the -replacement text of the loop to update it. - -\subsection{ \csbh{xintBreakFor}, \csbh{xintBreakForAndDo}} -\label{xintBreakFor}\label{xintBreakForAndDo} -%{\small New in |1.09e|.\par} - -One may immediately terminate an \csbxint{For} or \csbxint{For*} loop with -\csbxint{BreakFor}. As the criterion for breaking will be decided on a -basis of some test, it is recommended to use for this test the syntax of -\href{http://ctan.org/pkg/ifthen}{ifthen}\footnote{\url{http://ctan.org/pkg/ifthen}} -or -\href{http://ctan.org/pkg/etoolbox}{etoolbox}\footnote{\url{http://ctan.org/pkg/etoolbox}} -or the \xintname own conditionals, rather than one of the various -|\if...\fi| of \TeX{}. Else (and this is without even mentioning all the various -pecularities of the -|\if...\fi| constructs), one has to carefully move the break after the closing -of -the conditional, typically with |\expandafter\xintBreakFor\fi|.\footnote{the - difficulties here are similar to those mentioned in \autoref{sec:ifcase}, - although less severe, as complete expandability is not to be maintained; hence - the allowed use of \href{http://ctan.org/pkg/ifthen}{ifthen}.} - -There is also \csbxint{BreakForAndDo}. Both are illustrated by various examples -in the next section which is devoted to ``forever'' loops. - - -\subsection{\csbh{xintintegers}, \csbh{xintdimensions}, \csbh{xintrationals}} -\label{xintegers}\label{xintintegers} -\label{xintdimensions}\label{xintrationals} -%{\small New in |1.09e|.\par} - -If the list argument to \csbxint{For} (or \csbxint{For*}, both are equivalent in -this context) is \csbxint{integers} (equivalently \csbxint{egers}) or more -generally \csbxint{integers}|[||start|\allowbreak|+|\allowbreak|delta||]| -(\emph{the whole within braces}!)\footnote{the |start+delta| optional - specification may have extra spaces around the plus sign of near the square - brackets, such spaces are removed. The same applies with \csa{xintdimensions} - and \csa{xintrationals}.}, then \csbxint{For} does an infinite iteration where -|#1| (or |#2|, \dots, |#9|) will run through the arithmetic sequence of (short) -integers with initial value |start| and increment |delta| (default values: -|start=1|, |delta=1|; if the optional argument is present it must contains both -of them, and they may be explicit integers, or macros or count registers). The -|#1| (or |#2|, \dots, |#9|) will stand for |\numexpr <opt sign><digits>\relax|, -and the litteral representation as a string of digits can thus be obtained as -\fbox{\csa{the\#1}} or |\number#1|. Such a |#1| can be used in an |\ifnum| test -with no need to be postfixed with a space or a |\relax| and one should -\emph{not} add them. - -If the list argument is \csbxint{dimensions} or more generally -\csbxint{dimensions}|[||start|\allowbreak|+|\allowbreak|delta||]| (\emph{within - braces}!), then -\csbxint{For} does an infinite iteration where |#1| (or |#2|, \dots, |#9|) will -run through the arithmetic sequence of dimensions with initial value -|start| and increment |delta|. Default values: |start=0pt|, |delta=1pt|; if -the optional argument is present it must contain both of them, and they may -be explicit specifications, or macros, or dimen registers, or length commands -in \LaTeX{} (the stretch and shrink components will be discarded). The |#1| -will be |\dimexpr <opt sign><digits>sp\relax|, from which one can get the -litteral (approximate) representation in points via |\the#1|. So |#1| can be -used anywhere \TeX{} expects a dimension (and there is no need in conditionals -to insert a |\relax|, and one should \emph{not} do it), and to print its value -one uses \fbox{\csa{the\#1}}. The chosen representation guarantees exact -incrementation with no rounding errors accumulating from converting into -points at each step. - - -% original definitions, a bit slow. - -% \def\DimToNum #1{\number\dimexpr #1\relax } -% % cube -% \xintNewNumExpr \FA [2] {{_DimToNum {$2}}^3/{_DimToNum {$1}}^2} %$ -% % square root -% \xintNewNumExpr \FB [2] {sqrt ({_DimToNum {$2}}*{_DimToNum {$1}})} -% \xintNewExpr \Ratio [2] {trunc({_DimToNum {$2}}/{_DimToNum{$1}},3)} - -% improved faster code (4 four times faster) - -\def\DimToNum #1{\the\numexpr \dimexpr#1\relax/10000\relax } -\def\FA #1#2{\xintDSH{-4}{\xintQuo {\xintiPow {\DimToNum {#2}}{3}}{\xintiSqr -{\DimToNum{#1}}}}} -\def\FB #1#2{\xintDSH {-4}{\xintiSqrt - {\xintiMul {\DimToNum {#2}}{\DimToNum{#1}}}}} -\def\Ratio #1#2{\xintTrunc {2}{\DimToNum {#2}/\DimToNum{#1}}} - -% a further 2.5 gain is made through using .25pt as horizontal step. -\begin{figure*}[ht!] -\phantomsection\hypertarget{graphic}{}% -\centeredline{% -\raisebox{-1cm}{\xintFor #1 in {\xintdimensions [0pt+.25pt]} \do - {\ifdim #1>2cm \expandafter\xintBreakFor\fi - {\color [rgb]{\Ratio {2cm}{#1},0,0}% - \vrule width .25pt height \FB {2cm}{#1}sp depth -\FA {2cm}{#1}sp }% - }% end of For iterated text -}% -\hspace{1cm}% -\scriptsize\def\MacroFont {\ttfamily\baselineskip8pt\relax} -\begin{minipage}{\dimexpr\linewidth-3cm-\parindent\relax} -\dverb|@ -\def\DimToNum #1{\number\dimexpr #1\relax } -\xintNewNumExpr \FA [2] {{_DimToNum {$2}}^3/{_DimToNum {$1}}^2} % cube -\xintNewNumExpr \FB [2] {sqrt ({_DimToNum {$2}}*{_DimToNum {$1}})} % sqrt -\xintNewExpr \Ratio [2] {trunc({_DimToNum {$2}}/{_DimToNum{$1}},3)} -\xintFor #1 in {\xintdimensions [0pt+.1pt]} \do - {\ifdim #1>2cm \expandafter\xintBreakFor\fi - {\color [rgb]{\Ratio {2cm}{#1},0,0}% - \vrule width .1pt height \FB {2cm}{#1}sp depth -\FA {2cm}{#1}sp }% - }% end of For iterated text -|\par -\end{minipage}} -\end{figure*} - -% attention, pour le \meaning dans cette note de base de page - -The\xintNewNumExpr \FA [2] {{_DimToNum {$2}}^3/{_DimToNum {$1}}^2} %$ -\hyperlink{graphic}{graphic}, with the code on its right\footnote{the somewhat - peculiar use of |\_| and |\$| is explained in \autoref{xintNewExpr}; they are - made necessary from the fact that the parameters are passed to a \emph{macro} - (\csa{DimToNum}) and not only to \emph{functions}, as are known to - \hyperref[sec:exprsummary]{\csa{xintexpr}}. But one can also define directly - the desired function, for example the constructed \csa{FA} turns out to have - meaning \texttt{\meaning\FA}, where the \csa{romannumeral} part is only to - ensure it expands in only two steps, and could be removed. A handwritten macro - would use here \csa{xintiPow} and not \csa{xintPow}, as we know it has to deal - with integers only. See the next footnote.}, is for illustration only, not -only because of pdf rendering artefacts when displaying adjacent rules (which do -\emph{not} show in |dvi| output as rendered by |xdvi|, and depend from your -viewer), but because not using anything but rules it is quite inefficient and -must do lots of computations to not confer a too ragged look to the borders. -With a width of |.5pt| rather than |.1pt| for the rules, one speeds up the -drawing by a factor of five, but the boundary is then visibly ragged. -\newbox\codebox -\begingroup\makeatletter -\def\x{% - \parindent0pt - \def\par{\@@par\leavevmode\null}% - \let\do\do@noligs \verbatim@nolig@list - \let\do\@makeother \dospecials - \catcode`\@ 14 \makestarlowast - \ttfamily \scriptsize\baselineskip 8pt \obeylines \@vobeyspaces - \catcode`\|\active - \lccode`\~`\|\lowercase{\let~\egroup}}% -\global\setbox\codebox \vbox\bgroup\x -\def\DimToNum #1{\the\numexpr \dimexpr#1\relax/10000\relax } % no need to be more precise! -\def\FA #1#2{\xintDSH {-4}{\xintQuo {\xintiPow {\DimToNum {#2}}{3}}{\xintiSqr {\DimToNum{#1}}}}} -\def\FB #1#2{\xintDSH {-4}{\xintiSqrt {\xintiMul {\DimToNum {#2}}{\DimToNum{#1}}}}} -\def\Ratio #1#2{\xintTrunc {2}{\DimToNum {#2}/\DimToNum{#1}}} -\xintFor #1 in {\xintdimensions [0pt+.25pt]} \do - {\ifdim #1>2cm \expandafter\xintBreakFor\fi - {\color [rgb]{\Ratio {2cm}{#1},0,0}% - \vrule width .25pt height \FB {2cm}{#1}sp depth -\FA {2cm}{#1}sp }% - }% end of For iterated text -|% -\endgroup -\footnote{to tell the whole truth we cheated and divided by |10| the - computation time through using the following definitions, together with a - horizontal step of |.25pt| rather than |.1pt|. The displayed original code - would make the slowest computation of all those done in this document using - the \xintname bundle macros!\par\smallskip - \noindent\box \codebox\par } - -If the list argument to \csbxint{For} (or \csbxint{For*}) is \csbxint{rationals} -or more generally -\csbxint{rationals}|[||start|\allowbreak|+|\allowbreak|delta||]| (\emph{within - braces}!), then \csbxint{For} does an infinite iteration where |#1| (or |#2|, -\dots, |#9|) will run through the arithmetic sequence of \xintfracname fractions -with initial value |start| and increment |delta| (default values: |start=1/1|, -|delta=1/1|). This loop works \emph{only with \xintfracname loaded}. if the -optional argument is present it must contain both of them, and they may be given -in any of the formats recognized by \xintfracname (fractions, decimal -numbers, numbers in scientific notations, numerators and denominators in -scientific notation, etc...) , or as macros or count registers (if they are -short integers). The |#1| (or |#2|, \dots, |#9|) will be an |a/b| fraction -(without a |[n]| part), where -the denominator |b| is the product of the denominators of -|start| and |delta| (for reasons of speed |#1| is not reduced to irreducible -form, and for another reason explained later |start| and |delta| are not put -either into irreducible form; the input may use explicitely \csa{xintIrr} to -achieve that). - -\begingroup\small -\noindent\dverb|@ -\xintFor #1 in {\xintrationals [10/21+1/21]} \do -{#1=\xintifInt {#1} - {\textcolor{blue}{\xintTrunc{10}{#1}}} - {\xintTrunc{10}{#1}}% in blue if an integer - \xintifGt {#1}{1.123}{\xintBreakFor}{, }% -}| - -\smallskip -\centeredline{\parbox{\dimexpr\linewidth-3em}{\xintFor #1 in {\xintrationals [10/21+1/21]} \do -{#1=\xintifInt {#1} - {\textcolor{blue}{\xintTrunc{10}{#1}}} - {\xintTrunc{10}{#1}}% display in blue if an integer - \xintifGt {#1}{1.123}{\xintBreakFor}{, }% - }}} -\endgroup - -\smallskip The example above confirms that computations are done exactly, and -illustrates that the two initial (reduced) denominators are not multiplied when -they are found to be equal. It is thus recommended to input |start| and |delta| -with a common smallest possible denominator, or as fixed point numbers with the -same numbers of digits after the decimal mark; and this is also the reason why -|start| and |delta| are not by default made irreducible. As internally the -computations are done with numerators and denominators completely expanded, one -should be careful not to input numbers in scientific notation with exponents in -the hundreds, as they will get converted into as many zeroes. - -\begingroup\footnotesize \def\MacroFont {\ttfamily\relax} -\noindent\dverb|@ -\xintFor #1 in {\xintrationals [0.000+0.125]} \do -{\edef\tmp{\xintTrunc{3}{#1}}% - \xintifInt {#1} - {\textcolor{blue}{\tmp}} - {\tmp}% - \xintifGt {#1}{2}{\xintBreakFor}{, }% - }| -\smallskip - -\centeredline{\parbox{\dimexpr.7\linewidth}{\raggedright -\xintFor #1 in {\xintrationals [0.000+0.125]} \do -{\edef\tmp{\xintTrunc{3}{#1}}% - \xintifInt {#1} - {\textcolor{blue}{\tmp}} - {\tmp}% - \xintifGt {#1}{2}{\xintBreakFor}{, }% - }}} - -\smallskip - -We see here that \csbxint{Trunc} outputs (deliberately) zero as @0@, not (here) -@0.000@, the idea being not to lose the information that the truncated thing was -truly zero. Perhaps this behavior should be changed? or made optional? Anyhow -printing of fixed points numbers should be dealt with via dedicated packages -such as |numprint| or |siunitx|.\par -\endgroup - - -\subsection{Another table of primes}\label{ssec:primesIII} - -As a further example, let us dynamically generate a tabular with the first @50@ -prime numbers after @12345@. First we need a macro to test if a (short) number -is prime. Such a completely expandable macro was given in \autoref{xintSeq}, -here we consider a variant which will be slightly more efficient. This new -|\IsPrime| has two parameters. The first one is a macro which it redefines to -expand to the result of the primality test applied to the second argument. For -convenience we use the \href{http://ctan.org/pkg/etoolbox}{etoolbox} wrappers to -various |\ifnum| tests, although here there isn't anymore the constraint of -complete expandability (but using explicit |\if..\fi| in tabulars has its -quirks); equivalent tests are provided by \xintname, but they have some overhead -as they are able to deal with arbitrarily big integers. - -\def\IsPrime #1#2% -{\edef\TheNumber {\the\numexpr #2}% positive integer - \ifnumodd {\TheNumber} - {\ifnumgreater {\TheNumber}{1} - {\edef\ItsSquareRoot{\xintiSqrt \TheNumber}% - \xintFor ##1 in {\xintintegers [3+2]}\do - {\ifnumgreater {##1}{\ItsSquareRoot} - {\def#1{1}\xintBreakFor} - {}% - \ifnumequal {\TheNumber}{(\TheNumber/##1)*##1} - {\def#1{0}\xintBreakFor } - {}% - }} - {\def#1{0}}}% 1 is not prime - {\ifnumequal {\TheNumber}{2}{\def#1{1}}{\def#1{0}}}% -}% - -\dverb|@ -\def\IsPrime #1#2% """color[named]{PineGreen}#1=\Result, #2=tested number (assumed >0).;! -{\edef\TheNumber {\the\numexpr #2}%"""color[named]{PineGreen} hence #2 may be a count or \numexpr.;! - \ifnumodd {\TheNumber} - {\ifnumgreater {\TheNumber}{1} - {\edef\ItsSquareRoot{\xintiSqrt \TheNumber}% - \xintFor """color{red}##1;! in {"""color{red}\xintintegers;! [3+2]}\do - {\ifnumgreater {"""color{red}##1;!}{\ItsSquareRoot} """color[named]{PineGreen}% "textcolor{red}{##1} is a \numexpr.;! - {\def#1{1}\xintBreakFor} - {}% - \ifnumequal {\TheNumber}{(\TheNumber/##1)*##1} - {\def#1{0}\xintBreakFor } - {}% - }} - {\def#1{0}}}% 1 is not prime - {\ifnumequal {\TheNumber}{2}{\def#1{1}}{\def#1{0}}}% -}| - -%\newcounter{primecount} -%\newcounter{cellcount} -\begin{figure*}[ht!] - \centering\phantomsection\label{primes} - \begin{tabular}{|*{7}c|} - \hline - \setcounter{primecount}{0}\setcounter{cellcount}{0}% - \xintFor #1 in {\xintintegers [12345+2]} \do - {\IsPrime\Result{#1}% - \ifnumgreater{\Result}{0} - {\stepcounter{primecount}% - \stepcounter{cellcount}% - \ifnumequal {\value{cellcount}}{7} - {\the#1 \\\setcounter{cellcount}{0}} - {\the#1 &}} - {}% - \ifnumequal {\value{primecount}}{50} - {\xintBreakForAndDo - {\multicolumn {6}{l|}{These are the first 50 primes after 12345.}\\}} - {}% - }\hline -\end{tabular} -\end{figure*} - -As we used \csbxint{For} inside a macro we had to double the |#| in its |#1| -parameter. Here is now the code which creates the prime table (the table has -been put in a \hyperref[primes]{float}, which appears -\vpageref[above]{primes}): -\dverb?@ -\newcounter{primecount} -\newcounter{cellcount} -\begin{figure*}[ht!] - \centering - \begin{tabular}{|*{7}c|} - \hline - \setcounter{primecount}{0}\setcounter{cellcount}{0}% - \xintFor """color{red}#1;! in {"""color{red}\xintintegers;! [12345+2]} \do -"""color[named]{PineGreen}% "textcolor{red}{#1} is a \numexpr.;! - {\IsPrime\Result{#1}% - \ifnumgreater{\Result}{0} - {\stepcounter{primecount}% - \stepcounter{cellcount}% - \ifnumequal {\value{cellcount}}{7} - {"""color{red}\the#1;! \\\setcounter{cellcount}{0}} - {"""color{red}\the#1;! &}} - {}% - \ifnumequal {\value{primecount}}{50} - {\xintBreakForAndDo - {\multicolumn {6}{l|}{These are the first 50 primes after 12345.}\\}} - {}% - }\hline -\end{tabular} -\end{figure*}? - -\subsection{Some arithmetic with Fibonacci numbers} -\label{ssec:fibonacci} - -Here is again the code employed on the title page to compute Fibonacci numbers: - -\begingroup\footnotesize\baselineskip10pt -\def\MacroFont {\ttfamily} -\dverb|@ -\def\Fibonacci #1{% \Fibonacci{N} computes F(N) with F(0)=0, F(1)=1. - \expandafter\Fibonacci_a\expandafter - {\the\numexpr #1\expandafter}\expandafter - {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval 1\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval 0\relax}} -% -\def\Fibonacci_a #1{% - \ifcase #1 - \expandafter\Fibonacci_end_i - \or - \expandafter\Fibonacci_end_ii - \else - \ifodd #1 - \expandafter\expandafter\expandafter\Fibonacci_b_ii - \else - \expandafter\expandafter\expandafter\Fibonacci_b_i - \fi - \fi {#1}% -}% * signs are omitted from the next macros, tacit multiplications -\def\Fibonacci_b_i #1#2#3{\expandafter\Fibonacci_a\expandafter - {\the\numexpr #1/2\expandafter}\expandafter - {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval (2#2-#3)#3\relax}% -}% end of Fibonacci_b_i -\def\Fibonacci_b_ii #1#2#3#4#5{\expandafter\Fibonacci_a\expandafter - {\the\numexpr (#1-1)/2\expandafter}\expandafter - {\romannumeral0\xintiieval sqr(#2)+sqr(#3)\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval (2#2-#3)#3\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval #2#4+#3#5\expandafter\relax\expandafter}\expandafter - {\romannumeral0\xintiieval #2#5+#3(#4-#5)\relax}% -}% end of Fibonacci_b_ii -\def\Fibonacci_end_i #1#2#3#4#5{\xintthe#5} -\def\Fibonacci_end_ii #1#2#3#4#5{\xinttheiiexpr #2#5+#3(#4-#5)\relax} -\def\Fibonacci_end_i #1#2#3#4#5{{#4}{#5}}% {F(N+1)}{F(N)} in \xintexpr format -\def\Fibonacci_end_ii #1#2#3#4#5% - {\expandafter - {\romannumeral0\xintiieval #2#4+#3#5\expandafter\relax - \expandafter}\expandafter - {\romannumeral0\xintiieval #2#5+#3(#4-#5)\relax}}% idem. -% \FibonacciN returns F(N) (in encapsulated format: needs \xintthe for printing) -\def\FibonacciN {\expandafter\xint_secondoftwo\romannumeral-`0\Fibonacci }% -|\par\endgroup - -\catcode`_ 11 -\def\Fibonacci_end_i #1#2#3#4#5{{#4}{#5}}% -\def\Fibonacci_end_ii #1#2#3#4#5% - {\expandafter - {\romannumeral0\xintiieval #2#4+#3#5\expandafter\relax - \expandafter}\expandafter - {\romannumeral0\xintiieval #2#5+#3(#4-#5)\relax}}% idem. -% \Fibonacci returns {F(N+1)}{F(N)} (both in \xintexpr encapsulation) -% \FibonacciN returns F(N) (also in encapsulated format) -\def\FibonacciN {\expandafter\xint_secondoftwo\romannumeral-`0\Fibonacci }% -\catcode`_ 8 - -% ok -% \def\Fibo #1.{\xintthe\FibonacciN {#1}}% to use \xintiloopindex... -% \message{\xintiloop [0+1] -% \expandafter\Fibo\xintiloopindex., -% \ifnum\xintiloopindex<49 \repeat \xintthe\FibonacciN{50}.} - -I have modified the ending, as I now want not only one specific value |F(N)| but -a pair of successive values which can serve as starting point of another routine -devoted to compute a whole sequence |F(N), F(N+1), F(N+2),....|. This pair is, -for efficiency, kept in the encapsulated internal \xintexprname format. -|\FibonacciN| outputs the single |F(N)|, also as an |\xintexpr|-ession, and -printing it will thus need the |\xintthe| prefix. - -\begingroup\footnotesize\sffamily\baselineskip 10pt\let\MacroFont\ttfamily -Here a code snippet which -checks the routine via a \string\message\ of the first @51@ Fibonacci -numbers (this is not an efficient way to generate a sequence of such -numbers, it is only for validating \csa{FibonacciN}). -% -\dverb|@ -\def\Fibo #1.{\xintthe\FibonacciN {#1}}% -\message{\xintiloop [0+1] \expandafter\Fibo\xintiloopindex., - \ifnum\xintiloopindex<49 \repeat \xintthe\FibonacciN{50}.}|\par -\endgroup - -The various |\romannumeral0\xintiieval| could very well all have been -|\xintiiexpr|'s but then we would have needed more |\expandafter|'s. -Indeed the order of expansion must be controlled for the whole thing to work, -and |\romannumeral0\xintiieval| is the first expanded form of |\xintiiexpr|. - -The way we use |\expandafter|'s to chain successive |\xintexpr| evaluations is -exactly analogous to well-known expandable techniques made possible by -|\numexpr|. - -\begin{framed} - There is a difference though: |\numexpr| is \emph{NOT} expandable, and to - force its expansion we must prefix it with |\the| or |\number|. On the other - hand |\xintexpr|, |\xintiexpr|, ..., (or |\xinteval|, |\xintieval|, ...) - expand fully when prefixed by |\romannumeral-`0|: the computation is fully - executed and its result encapsulated in a private format. - - Using |\xintthe| as prefix is necessary to print the result (this is like - |\the| for |\numexpr|), but it is not necessary to get the computation done - (contrarily to the situation with |\numexpr|). - - And, starting with release |1.09j|, it is also allowed to expand a non - |\xintthe| prefixed |\xintexpr|-ession inside an |\edef|: the private format - is now protected, hence the error message complaining about a missing - |\xintthe| will not be executed, and the integrity of the format will be - preserved. - - This new possibility brings some efficiency gain, when one writes - non-expandable algorithms using \xintexprname. If |\xintthe| is - employed inside |\edef| the number or fraction will be un-locked into - its possibly hundreds of digits and all these tokens will possibly - weigh on the upcoming shuffling of (braced) tokens. The private - encapsulated format has only a few tokens, hence expansion will - proceed a bit faster. - - \indent see footnote\footnotemark -\end{framed} - -\footnotetext{To be completely honest the examination by \TeX{} of all - successive digits was not avoided, as it occurs already in the locking-up of - the result, what is avoided is to spend time un-locking, and then have - the macros shuffle around possibly hundreds of digit tokens rather - than a few control words.\par - Technical note: I decided (somewhat hesitantly) for - reasons of optimization purposes to skip in the private \csa{xintexpr} - format a \csa{protect}-ion for the \csa{.=digits/digits[digits]} - control sequences used internally. Thus in the improbable case that - some macro package (such control sequence names are unavailable to the - casual user) has given a meaning to one such control sequence, there - is a possibility of a crash when embedding an \csa{xintexpr} without - \csa{xintthe} prefix in an \csa{edef} (the computations by themselves - do proceed perfectly correctly even if these control sequences have - acquired some non \csa{relax} meaning).} - -Our |\Fibonacci| expands completely under \fexpan sion, -so we can use \hyperref[fdef]{\ttfamily\char92fdef} rather than |\edef| in a -situation such as \centeredline {|\fdef \X {\FibonacciN {100}}|} but for the -reasons explained above, it is as efficient to employ |\edef|. And if we want -\centeredline{|\edef \Y {(\FibonacciN{100},\FibonacciN{200})}|,} then |\edef| is -necessary. - -Allright, so let's now give the code to generate a sequence of braced Fibonacci -numbers |{F(N)}{F(N+1)}{F(N+2)}...|, using |\Fibonacci| for the first -two and then using the standard recursion |F(N+2)=F(N+1)+F(N)|: - -\catcode`_ 11 -\def\FibonacciSeq #1#2{%#1=starting index, #2>#1=ending index - \expandafter\Fibonacci_Seq\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2-1}% -}% -\def\Fibonacci_Seq #1#2{% - \expandafter\Fibonacci_Seq_loop\expandafter - {\the\numexpr #1\expandafter}\romannumeral0\Fibonacci {#1}{#2}% -}% -\def\Fibonacci_Seq_loop #1#2#3#4{% standard Fibonacci recursion - {#3}\unless\ifnum #1<#4 \Fibonacci_Seq_end\fi - \expandafter\Fibonacci_Seq_loop\expandafter - {\the\numexpr #1+1\expandafter}\expandafter - {\romannumeral0\xintiieval #2+#3\relax}{#2}{#4}% -}% -\def\Fibonacci_Seq_end\fi\expandafter\Fibonacci_Seq_loop\expandafter - #1\expandafter #2#3#4{\fi {#3}}% -\catcode`_ 8 - -\begingroup\footnotesize\baselineskip10pt -\def\MacroFont {\ttfamily} -\dverb|@ -\catcode`_ 11 -\def\FibonacciSeq #1#2{%#1=starting index, #2>#1=ending index - \expandafter\Fibonacci_Seq\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2-1}% -}% -\def\Fibonacci_Seq #1#2{% - \expandafter\Fibonacci_Seq_loop\expandafter - {\the\numexpr #1\expandafter}\romannumeral0\Fibonacci {#1}{#2}% -}% -\def\Fibonacci_Seq_loop #1#2#3#4{% standard Fibonacci recursion - {#3}\unless\ifnum #1<#4 \Fibonacci_Seq_end\fi - \expandafter\Fibonacci_Seq_loop\expandafter - {\the\numexpr #1+1\expandafter}\expandafter - {\romannumeral0\xintiieval #2+#3\relax}{#2}{#4}% -}% -\def\Fibonacci_Seq_end\fi\expandafter\Fibonacci_Seq_loop\expandafter - #1\expandafter #2#3#4{\fi {#3}}% -\catcode`_ 8 -|\par\endgroup - -Deliberately and for optimization, this |\FibonacciSeq| macro is -completely expandable but not \fexpan dable. It would be easy to modify -it to be so. But I wanted to check that the \csbxint{For*} does apply -full expansion to what comes next each time it fetches an item from its -list argument. Thus, there is no need to generate lists of braced -Fibonacci numbers beforehand, as \csbxint{For*}, without using any -|\edef|, still manages to generate the list via iterated full expansion. - -I initially used only one |\halign| in a three-column |multicols| -environment, but |multicols| only knows to divide the page horizontally -evenly, thus I employed in the end one |\halign| for each column (I -could have then used a |tabular| as no column break was then needed). - - -\begin{figure*}[ht!] - \phantomsection\label{fibonacci} - \newcounter{index} - \fdef\Fibxxx{\FibonacciN {30}}% - \setcounter{index}{30}% -\centeredline{\tabskip 1ex -\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr - \xintFor* #1 in {\FibonacciSeq {30}{59}}\do - {\theindex &\xintthe#1 & - \xintRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}% -}\vrule -\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr - \xintFor* #1 in {\FibonacciSeq {60}{89}}\do - {\theindex &\xintthe#1 & - \xintRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}% -}\vrule -\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr - \xintFor* #1 in {\FibonacciSeq {90}{119}}\do - {\theindex &\xintthe#1 & - \xintRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}% -}}% -% -\centeredline{Some Fibonacci numbers together with their residues modulo - |F(30)|\digitstt{=\xintthe\Fibxxx}} -\end{figure*} - -\begingroup\footnotesize\baselineskip10pt -\def\MacroFont {\ttfamily} -\dverb|@ -\newcounter{index} -\tabskip 1ex - \fdef\Fibxxx{\FibonacciN {30}}% - \setcounter{index}{30}% -\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr - \xintFor* #1 in {\FibonacciSeq {30}{59}}\do - {\theindex &\xintthe#1 & - \xintRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}% -}\vrule -\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr - \xintFor* #1 in {\FibonacciSeq {60}{89}}\do - {\theindex &\xintthe#1 & - \xintRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}% -}\vrule -\vbox{\halign{\bfseries#.\hfil&#\hfil &\hfil #\cr - \xintFor* #1 in {\FibonacciSeq {90}{119}}\do - {\theindex &\xintthe#1 & - \xintRem{\xintthe#1}{\xintthe\Fibxxx}\stepcounter{index}\cr }}% -}% -|\par\endgroup - -This produces the Fibonacci numbers from |F(30)| to |F(119)|, and -computes also all the -congruence classes modulo |F(30)|. The output has -been put in a \hyperref[fibonacci]{float}, which appears -\vpageref[above]{fibonacci}. I leave to the mathematically inclined -readers the task to explain the visible patterns\dots |;-)|. - -\subsection{\csbh{xintForpair}, \csbh{xintForthree}, \csbh{xintForfour}}\label{xintForpair}\label{xintForthree}\label{xintForfour} -% {\small New in |1.09c|. The \csa{xintifForFirst} -% |1.09e| mechanism was missing and has been added for |1.09f|. The |1.09f| -% version handles better spaces and admits all (consecutive) macro -% parameters.\par} - -The syntax\ntype{on} is illustrated in this -example. The notation is the usual one for |n|-uples, with parentheses and -commas. Spaces around commas and parentheses are ignored. -% -\dverb|@ -\begin{tabular}{cccc} - \xintForpair #1#2 in { ( A , a ) , ( B , b ) , ( C , c ) } \do {% - \xintForpair #3#4 in { ( X , x ) , ( Y , y ) , ( Z , z ) } \do {% - $\Biggl($\begin{tabular}{cc} - -#1- & -#3-\\ - -#4- & -#2-\\ - \end{tabular}$\Biggr)$&}\\\noalign{\vskip1\jot}}% -\end{tabular}|% -\centeredline{\begin{tabular}{cccc} - \xintForpair #1#2 in { ( A , a ) , ( B , b ) , ( C , c ) } \do {% - \xintForpair #3#4 in { ( X , x ) , ( Y , y ) , ( Z , z ) } \do {% - $\Biggl($\begin{tabular}{cc} - -#1- & -#3-\\ - -#4- & -#2-\\ - \end{tabular}$\Biggr)$&}\\\noalign{\vskip1\jot}}% -\end{tabular}} - -\smallskip Only |#1#2|, |#2#3|, |#3#4|, \dots, |#8#9| are valid (no error check -is done on the input syntax, |#1#3| or similar all end up in errors). -One can nest with \csbxint{For}, for disjoint sets of macro parameters. There is -also \csa{xintForthree} (from |#1#2#3| to |#7#8#9|) and \csa{xintForfour} (from -|#1#2#3#4| to |#6#7#8#9|). |\par| tokens are accepted in both the comma -separated list and the replacement text. - -% These three macros |\xintForpair|, |\xintForthree| and |\xintForfour| are to -% be considered in experimental status, and may be removed, replaced or -% substantially modified at some later stage. - -\subsection{\csbh{xintAssign}}\label{xintAssign} -%\small{ |1.09i| adds optional parameter. |1.09j| has default optional -% parameter |[]| rather than |[e]|\par} - -\csa{xintAssign}\meta{braced things}\csa{to}% -\meta{as many cs as they are things} %\ntype{{(f$\to$\lowast [x)}{\lowast N}} -% -defines (without checking if something gets overwritten) the control sequences -on the right of \csa{to} to expand to the successive tokens or braced items -found one after the otehr on the on the left of \csa{to}. It is not expandable. - -A `full' expansion is first applied to the material in front of -\csa{xintAssign}, which may thus be a macro expanding to a list of braced items. - -\xintAssign \xintiPow {7}{13}\to\SevenToThePowerThirteen -\xintAssign \xintDivision{1000000000000}{133333333}\to\Q\R - -Special case: if after this initial expansion no brace is found immediately -after \csa{xintAssign}, it is assumed that there is only one control sequence -following |\to|, and this control sequence is then defined via -|\def| to expand to the material between -\csa{xintAssign} and \csa{to}. Other types of expansions are specified through -an optional parameter to \csa{xintAssign}, see \emph{infra}. -\centeredline{|\xintAssign \xintDivision{1000000000000}{133333333}\to\Q\R|} -\centeredline{|\meaning\Q: |\digitstt{\meaning\Q}, |\meaning\R:| - \digitstt{\meaning\R}} \centeredline{|\xintAssign \xintiPow - {7}{13}\to\SevenToThePowerThirteen|} -\centeredline{|\SevenToThePowerThirteen|\digitstt{=\SevenToThePowerThirteen}} -\centeredline{(same as |\edef\SevenToThePowerThirteen{\xintiPow {7}{13}}|)} - - -\noindent\csa{xintAssign}\MyMarginNote{Changed!} admits since |1.09i| an -optional parameter, for example |\xintAssign [e]...| or |\xintAssign [oo] -...|. The latter means that the definitions of the macros initially on the -right of |\to| will be made with \hyperref[oodef]{\ttfamily\char92oodef} which -expands twice the replacement text. The default is simply to make the -definitions with |\def|, corresponding to an empty optional paramter |[]|. -Possibilities: |[], [g], [e], [x], [o], [go], [oo], [goo], [f], [gf]|. - -In all cases, recall that |\xintAssign| starts with an \fexpan sion of what -comes next; this produces some list of tokens or braced items, and the -optional parameter only intervenes to decide the expansion type to be applied -then to each one of these items. - -\emph{Note:} prior to release |1.09j|, |\xintAssign| did an |\edef| by -default, but it now does |\def|. Use the optional parameter |[e]| to force use -of |\edef|. - -% This -% macro uses various \csa{edef}'s, thus is incompatible with expansion-only -% contexts. - -\subsection{\csbh{xintAssignArray}}\label{xintAssignArray} -% {\small Changed in release |1.06| to let the defined macro pass its -% argument through a |\numexpr...\relax|. |1.09i| adds optional -% parameter. \par} - -\xintAssignArray \xintBezout {1000}{113}\to\Bez - -\csa{xintAssignArray}\meta{braced - things}\csa{to}\csa{myArray} %\ntype{{(f$\to$\lowast x)}N} -% -first expands fully what comes immediately after |\xintAssignArray| and -expects to find a list of braced things |{A}{B}...| (or tokens). It then -defines \csa{myArray} as a macro with one parameter, such that \csa{myArray\x} -expands to give the |x|th braced thing of this original -list (the argument \texttt{\x} itself is fed to a |\numexpr| by |\myArray|, -and |\myArray| expands in two steps to its output). With |0| as parameter, -\csa{myArray}|{0}| returns the number |M| of elements of the array so that the -successive elements are \csa{myArray}|{1}|, \dots, \csa{myArray}|{M}|. -\centeredline{|\xintAssignArray \xintBezout {1000}{113}\to\Bez|} will set -|\Bez{0}| to \digitstt{\Bez0}, |\Bez{1}| to \digitstt{\Bez1}, |\Bez{2}| to -\digitstt{\Bez2}, |\Bez{3}| to \digitstt{\Bez3}, |\Bez{4}| to -\digitstt{\Bez4}, and |\Bez{5}| to \digitstt{\Bez5}: -\digitstt{(\Bez3)${}\times{}$\Bez1${}-{}$(\Bez4)${}\times{}$\Bez2${}={}$\Bez5.} -This macro is incompatible with expansion-only contexts. - -\csa{xintAssignArray}\MyMarginNote{Changed!} admits now an optional -parameter, for example |\xintAssignArray [e]...|. This means that the -definitions of the macros will be made with |\edef|. The default is -|[]|, which makes the definitions with |\def|. Other possibilities: |[], -[o], [oo], [f]|. Contrarily to \csbxint{Assign} one can not use the |g| -here to make the definitions global. For this, one should rather do -|\xintAssignArray| within a group starting with |\globaldefs 1|. - -Note that prior to release |1.09j| each item (token or braced material) was -submitted to an |\edef|, but the default is now to use |\def|. - -\subsection{\csbh{xintRelaxArray}}\label{xintRelaxArray} - -\csa{xintRelaxArray}\csa{myArray} %\ntype{N} -% -(globally) sets to \csa{relax} all macros which were defined by the previous -\csa{xintAssignArray} with \csa{myArray} as array macro. - -\subsection{\csbh{odef}, \csbh{oodef}, \csbh{fdef}} -\label{odef} -\label{oodef} -\label{fdef} - -\csa{oodef}|\controlsequence {<stuff>}| does -\dverb|@ - \expandafter\expandafter\expandafter\def - \expandafter\expandafter\expandafter\controlsequence - \expandafter\expandafter\expandafter{<stuff>}| - -% -This works only for a single -|\controlsequence|, with no parameter text, even without parameters. An -alternative would be: -\dverb|@ -\def\oodef #1#{\def\oodefparametertext{#1}% - \expandafter\expandafter\expandafter\expandafter - \expandafter\expandafter\expandafter\def - \expandafter\expandafter\expandafter\oodefparametertext - \expandafter\expandafter\expandafter }| -% - -\noindent -but it does not allow |\global| as prefix, and, besides, would have anyhow its -use (almost) limited to parameter texts without macro parameter tokens -(except if the expanded thing does not see them, or is designed to deal with -them). - -There is a similar macro |\odef| with only one expansion of the replacement text -|<stuff>|, and |\fdef| which expands fully |<stuff>| using |\romannumeral-`0|. - -These tools are provided as it is sometimes wasteful (from the point of view of -running time) to do an |\edef| when one knows that the contents expand in only -two steps for example, as is the case with all (except \csbxint{loop} and -\csbxint{iloop}) the expandable macros of the \xintname packages. -Each will be defined only if \xinttoolsname finds them currently undefined. They -can be prefixed with |\global|. - - -\subsection{The Quick Sort algorithm illustrated}\label{ssec:quicksort} - -First a completely expandable macro which sorts a list of numbers. The |\QSfull| -macro expands its list argument, which may thus be a macro; its items must -expand to possibly big integers (or also decimal numbers or fractions if using -\xintfracname), but if an item is expressed as a computation, this computation -will be redone each time the item is considered! If the numbers have many digits -(i.e. hundreds of digits...), the expansion of |\QSfull| is fastest if each -number, rather than being explicitely given, is represented as a single token -which expands to it in one step. - -If the interest is only in \TeX{} integers, then one should replace the macros -|\QSMore|, |QSEqual|, |QSLess| with versions using the -\href{http://ctan.org/pkg/etoolbox}{etoolbox} (\LaTeX{} only) |\ifnumgreater|, -|\ifnumequal| and |\ifnumless| conditionals rather than \csbxint{ifGt}, -\csbxint{ifEq}, \csbxint{ifLt}. - -\begingroup\makeatletter\let\check@percent\relax -\def\MacroFont{\small\baselineskip12pt \ttfamily } -\begin{verbatim} -% THE QUICK SORT ALGORITHM EXPANDABLY -\input xintfrac.sty -% HELPER COMPARISON MACROS -\def\QSMore #1#2{\xintifGt {#2}{#1}{{#2}}{ }} -% the spaces are there to stop the \romannumeral-`0 originating -% in \xintapplyunbraced when it applies a macro to an item -\def\QSEqual #1#2{\xintifEq {#2}{#1}{{#2}}{ }} -\def\QSLess #1#2{\xintifLt {#2}{#1}{{#2}}{ }} -% -\makeatletter -\def\QSfull {\romannumeral0\qsfull } -\def\qsfull #1{\expandafter\qsfull@a\expandafter{\romannumeral-`0#1}} -\def\qsfull@a #1{\expandafter\qsfull@b\expandafter {\xintLength {#1}}{#1}} -\def\qsfull@b #1{\ifcase #1 - \expandafter\qsfull@empty - \or\expandafter\qsfull@single - \else\expandafter\qsfull@c - \fi -}% -\def\qsfull@empty #1{ } % the space stops the \QSfull \romannumeral0 -\def\qsfull@single #1{ #1} -% for simplicity of implementation, we pick up the first item as pivot -\def\qsfull@c #1{\qsfull@ci #1\undef {#1}} -\def\qsfull@ci #1#2\undef {\qsfull@d {#1}}% #3 is the list, #1 its first item -\def\qsfull@d #1#2{\expandafter\qsfull@e\expandafter - {\romannumeral0\qsfull - {\xintApplyUnbraced {\QSMore {#1}}{#2}}}% - {\romannumeral0\xintapplyunbraced {\QSEqual {#1}}{#2}}% - {\romannumeral0\qsfull - {\xintApplyUnbraced {\QSLess {#1}}{#2}}}% -}% -\def\qsfull@e #1#2#3{\expandafter\qsfull@f\expandafter {#2}{#3}{#1}}% -\def\qsfull@f #1#2#3{\expandafter\space #2#1#3} -\makeatother -% EXAMPLE -\edef\z {\QSfull {{1.0}{0.5}{0.3}{1.5}{1.8}{2.0}{1.7}{0.4}{1.2}{1.4}% - {1.3}{1.1}{0.7}{1.6}{0.6}{0.9}{0.8}{0.2}{0.1}{1.9}}} -\tt\meaning\z -\def\a {3.123456789123456789}\def\b {3.123456789123456788} -\def\c {3.123456789123456790}\def\d {3.123456789123456787} -\expandafter\def\expandafter\z\expandafter - {\romannumeral0\qsfull {{\a}\b\c\d}}% \a is braced to not be expanded -\meaning\z -\end{verbatim} - -% THE QUICK SORT ALGORITHM EXPANDABLY -\def\QSMore #1#2{\xintifGt {#2}{#1}{{#2}}{ }} -% the spaces stop the \romannumeral-`0 done by \xintapplyunbraced each time -% it applies its macro argument to an item -\def\QSEqual #1#2{\xintifEq {#2}{#1}{{#2}}{ }} -\def\QSLess #1#2{\xintifLt {#2}{#1}{{#2}}{ }} -% -\def\QSfull {\romannumeral0\qsfull } -\def\qsfull #1{\expandafter\qsfull@a\expandafter{\romannumeral-`0#1}} -\def\qsfull@a #1{\expandafter\qsfull@b\expandafter {\xintLength {#1}}{#1}} -\def\qsfull@b #1{\ifcase #1 - \expandafter\qsfull@empty - \or\expandafter\qsfull@single - \else\expandafter\qsfull@c - \fi -}% -\def\qsfull@empty #1{ } % the space stops the \QSfull \romannumeral0 -\def\qsfull@single #1{ #1} -\def\qsfull@c #1{\qsfull@ci #1\undef {#1}} % we pick up the first as Pivot -\def\qsfull@ci #1#2\undef {\qsfull@d {#1}} -\def\qsfull@d #1#2{\expandafter\qsfull@e\expandafter - {\romannumeral0\qsfull - {\xintApplyUnbraced {\QSMore {#1}}{#2}}}% - {\romannumeral0\xintapplyunbraced {\QSEqual {#1}}{#2}}% - {\romannumeral0\qsfull - {\xintApplyUnbraced {\QSLess {#1}}{#2}}}% -}% -\def\qsfull@e #1#2#3{\expandafter\qsfull@f\expandafter {#2}{#3}{#1}}% -\def\qsfull@f #1#2#3{\expandafter\space #2#1#3} -\makeatother -% EXAMPLE -\edef\z {\QSfull {{1.0}{0.5}{0.3}{1.5}{1.8}{2.0}{1.7}{0.4}{1.2}{1.4}% - {1.3}{1.1}{0.7}{1.6}{0.6}{0.9}{0.8}{0.2}{0.1}{1.9}}} -\noindent Output:\par -\texttt{\printnumber{\meaning\z}} - -\def\a {3.123456789123456789}\def\b {3.123456789123456788} -\def\c {3.123456789123456790}\def\d {3.123456789123456787} -\expandafter\def\expandafter\z\expandafter - {\romannumeral0\qsfull {{\a}\b\c\d}}% \a is braced to not be expanded -\texttt{\printnumber{\meaning\z}} -\endgroup - - - -We then turn to a graphical illustration of the algorithm. For simplicity the -pivot is always chosen to be the first list item. We also show later how to -illustrate the variant which picks up the last item of each unsorted -chunk as pivot. - -\begingroup -\makeatletter -\let\check@percent\relax -% il utilise MacroFont -\def\MacroFont{\small\baselineskip 12pt \ttfamily } -\begin{verbatim} -\input xintfrac.sty % if Plain TeX -% -\definecolor{LEFT}{RGB}{216,195,88} -\definecolor{RIGHT}{RGB}{208,231,153} -\definecolor{INERT}{RGB}{199,200,194} -\definecolor{PIVOT}{RGB}{109,8,57} -% -\def\QSMore #1#2{\xintifGt {#2}{#1}{{#2}}{ }}% space will be gobbled -\def\QSEqual #1#2{\xintifEq {#2}{#1}{{#2}}{ }} -\def\QSLess #1#2{\xintifLt {#2}{#1}{{#2}}{ }} -% -\makeatletter -\def\QS@a #1{\expandafter \QS@b \expandafter {\xintLength {#1}}{#1}} -\def\QS@b #1{\ifcase #1 - \expandafter\QS@empty - \or\expandafter\QS@single - \else\expandafter\QS@c - \fi -}% -\def\QS@empty #1{} -\def\QS@single #1{\QSIr {#1}} -\def\QS@c #1{\QS@d #1!{#1}} % we pick up the first as pivot. -\def\QS@d #1#2!{\QS@e {#1}}% #1 = first element, #3 = list -\def\QS@e #1#2{\expandafter\QS@f\expandafter - {\romannumeral0\xintapplyunbraced {\QSMore {#1}}{#2}}% - {\romannumeral0\xintapplyunbraced {\QSEqual {#1}}{#2}}% - {\romannumeral0\xintapplyunbraced {\QSLess {#1}}{#2}}% -}% -\def\QS@f #1#2#3{\expandafter\QS@g\expandafter {#2}{#3}{#1}}% -% Here \QSLr, \QSIr, \QSr have been let to \relax, so expansion stops. -% #2= elements < pivot, #1 = elements = pivot, #3 = elements > pivot -\def\QS@g #1#2#3{\QSLr {#2}\QSIr {#1}\QSRr {#3}}% -% -\def\DecoLEFT #1{\xintFor* ##1 in {#1} \do {\colorbox{LEFT}{##1}}} -\def\DecoINERT #1{\xintFor* ##1 in {#1} \do {\colorbox{INERT}{##1}}} -\def\DecoRIGHT #1{\xintFor* ##1 in {#1} \do {\colorbox{RIGHT}{##1}}} -\def\DecoPivot #1{\begingroup\color{PIVOT}\advance\fboxsep-\fboxrule - \fbox{#1}\endgroup} -\def\DecoLEFTwithPivot #1{% - \xintFor* ##1 in {#1} \do - {\xintifForFirst {\DecoPivot {##1}}{\colorbox{LEFT}{##1}}}% -} -\def\DecoRIGHTwithPivot #1{% - \xintFor* ##1 in {#1} \do - {\xintifForFirst {\DecoPivot {##1}}{\colorbox{RIGHT}{##1}}}% -} -% -\def\QSinitialize #1{\def\QS@list{\QSRr {#1}}% - \let\QSRr\DecoRIGHT -% \QS@list \par -\par\centerline{\QS@list} -} -\def\QSoneStep {\let\QSLr\DecoLEFTwithPivot - \let\QSIr\DecoINERT - \let\QSRr\DecoRIGHTwithPivot -% \QS@list -\centerline{\QS@list} -% \par - \def\QSLr {\noexpand\QS@a}% - \let\QSIr\relax - \def\QSRr {\noexpand\QS@a}% - \edef\QS@list{\QS@list}% - \let\QSLr\relax - \let\QSRr\relax - \edef\QS@list{\QS@list}% - \let\QSLr\DecoLEFT - \let\QSIr\DecoINERT - \let\QSRr\DecoRIGHT -% \QS@list -\centerline{\QS@list} -% \par -} -\begingroup\offinterlineskip -\small -\QSinitialize {{1.0}{0.5}{0.3}{1.5}{1.8}{2.0}{1.7}{0.4}{1.2}{1.4}% - {1.3}{1.1}{0.7}{1.6}{0.6}{0.9}{0.8}{0.2}{0.1}{1.9}} -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\endgroup -\end{verbatim} - -\def\QSMore #1#2{\xintifGt {#2}{#1}{{#2}}{ }}% space will be gobbled -\def\QSEqual #1#2{\xintifEq {#2}{#1}{{#2}}{ }} -\def\QSLess #1#2{\xintifLt {#2}{#1}{{#2}}{ }} -% -\def\QS@a #1{\expandafter \QS@b \expandafter {\xintLength {#1}}{#1}} -\def\QS@b #1{\ifcase #1 - \expandafter\QS@empty - \or\expandafter\QS@single - \else\expandafter\QS@c - \fi -}% -\def\QS@empty #1{} -\def\QS@single #1{\QSIr {#1}} -\def\QS@c #1{\QS@d #1!{#1}} % we pick up the first as pivot. -\def\QS@d #1#2!{\QS@e {#1}}% #1 = first element, #3 = list -\def\QS@e #1#2{\expandafter\QS@f\expandafter - {\romannumeral0\xintapplyunbraced {\QSMore {#1}}{#2}}% - {\romannumeral0\xintapplyunbraced {\QSEqual {#1}}{#2}}% - {\romannumeral0\xintapplyunbraced {\QSLess {#1}}{#2}}% -}% -\def\QS@f #1#2#3{\expandafter\QS@g\expandafter {#2}{#3}{#1}}% -% #2= elements < pivot, #1 = elements = pivot, #3 = elements > pivot -% Here \QSLr, \QSIr, \QSr have been let to \relax, so expansion stops. -\def\QS@g #1#2#3{\QSLr {#2}\QSIr {#1}\QSRr {#3}}% -% -\def\DecoLEFT #1{\xintFor* ##1 in {#1} \do {\colorbox{LEFT}{##1}}} -\def\DecoINERT #1{\xintFor* ##1 in {#1} \do {\colorbox{INERT}{##1}}} -\def\DecoRIGHT #1{\xintFor* ##1 in {#1} \do {\colorbox{RIGHT}{##1}}} -\def\DecoPivot #1{\begingroup\color{PIVOT}\advance\fboxsep-\fboxrule - \fbox{#1}\endgroup} -\def\DecoLEFTwithPivot #1{% - \xintFor* ##1 in {#1} \do - {\xintifForFirst {\DecoPivot {##1}}{\colorbox{LEFT}{##1}}}% -} -\def\DecoRIGHTwithPivot #1{% - \xintFor* ##1 in {#1} \do - {\xintifForFirst {\DecoPivot {##1}}{\colorbox{RIGHT}{##1}}}% -} -% -\def\QSinitialize #1{\def\QS@list{\QSRr {#1}}% - \let\QSRr\DecoRIGHT -% \QS@list \par -\par\centerline{\QS@list} -} -\def\QSoneStep {\let\QSLr\DecoLEFTwithPivot - \let\QSIr\DecoINERT - \let\QSRr\DecoRIGHTwithPivot -% \QS@list -\centerline{\QS@list} -% \par - \def\QSLr {\noexpand\QS@a}% - \let\QSIr\relax - \def\QSRr {\noexpand\QS@a}% - \edef\QS@list{\QS@list}% - \let\QSLr\relax - \let\QSRr\relax - \edef\QS@list{\QS@list}% - \let\QSLr\DecoLEFT - \let\QSIr\DecoINERT - \let\QSRr\DecoRIGHT -% \QS@list -\centerline{\QS@list} -% \par -} - -\phantomsection\label{quicksort} -\begingroup\offinterlineskip -\small -\QSinitialize {{1.0}{0.5}{0.3}{1.5}{1.8}{2.0}{1.7}{0.4}{1.2}{1.4}% - {1.3}{1.1}{0.7}{1.6}{0.6}{0.9}{0.8}{0.2}{0.1}{1.9}} -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\endgroup - - -If one wants rather to have the pivot from the end of the yet to sort chunks, -then one should use the following variants: -\begin{verbatim} -\def\QS@c #1{\expandafter\QS@e\expandafter - {\romannumeral0\xintnthelt {-1}{#1}}{#1}% -}% -\def\DecoLEFTwithPivot #1{% - \xintFor* ##1 in {#1} \do - {\xintifForLast {\DecoPivot {##1}}{\colorbox{LEFT}{##1}}}% -} -\def\DecoRIGHTwithPivot #1{% - \xintFor* ##1 in {#1} \do - {\xintifForLast {\DecoPivot {##1}}{\colorbox{RIGHT}{##1}}}% -} -\def\QSinitialize #1{\def\QS@list{\QSLr {#1}}% - \let\QSLr\DecoLEFT -% \QS@list \par -\par\centerline{\QS@list} -} -\end{verbatim} -\def\QS@c #1{\expandafter\QS@e\expandafter - {\romannumeral0\xintnthelt {-1}{#1}}{#1}% -}% -\def\DecoLEFTwithPivot #1{% - \xintFor* ##1 in {#1} \do - {\xintifForLast {\DecoPivot {##1}}{\colorbox{LEFT}{##1}}}% -} -\def\DecoRIGHTwithPivot #1{% - \xintFor* ##1 in {#1} \do - {\xintifForLast {\DecoPivot {##1}}{\colorbox{RIGHT}{##1}}}% -} -\def\QSinitialize #1{\def\QS@list{\QSLr {#1}}% - \let\QSLr\DecoLEFT -% \QS@list \par -\par\centerline{\QS@list} -} -\begingroup\offinterlineskip -\small -\QSinitialize {{1.0}{0.5}{0.3}{1.5}{1.8}{2.0}{1.7}{0.4}{1.2}{1.4}% - {1.3}{1.1}{0.7}{1.6}{0.6}{0.9}{0.8}{0.2}{0.1}{1.9}} -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\QSoneStep -\endgroup - -\endgroup - -It is possible to modify this code to let it do \csa{QSonestep} repeatedly and -stop automatically when the sort is finished.\footnote{\url{http://tex.stackexchange.com/a/142634/4686}} - - -\section{Commands of the \xintname package} -\label{sec:xint} - - -In the description of the macros \texttt{\n} and \texttt{\m} stand for (long) -numbers within braces or for a control sequence possibly within braces and -\hyperref[sec:expansions]{\fexpan ding} to such a number (without the braces!), -or for material within braces which \fexpan ds to such a number, as is -acceptable on input by the \csbxint{Num} macro: a sequence of plus and minus -signs, followed by some string of zeroes, followed by digits. The margin -annotation for such an argument which is parsed by \csbxint{Num} is -\textcolor[named]{PineGreen}{\Numf}. Sometimes however only a -\textcolor[named]{PineGreen}{\emph{f}} symbol appears in the margin, signaling -that the input will not be parsed via \csbxint{Num}. - -The letter \texttt{x} (with margin annotation -\textcolor[named]{PineGreen}{\numx}) stands for something which will be inserted -in-between a |\numexpr| and a |\relax|. It will thus be completely expanded and -must give an integer obeying the \TeX{} bounds. Thus, it may be for example a -count register, or itself a \csa{numexpr} expression, or just a number written -explicitely with digits or something like |4*\count 255 + 17|, etc... - -For the rules regarding direct use of count registers or \csa{numexpr} -expression, in the argument to the package macros, see the -\hyperref[sec:useofcount]{Use of count} section. - -Some of these macros are extended by \xintfracname to accept fractions -on input, and, generally, to output a fraction. But this means that -additions, subtractions, multiplications output in fraction format; to -guarantee the integer format on output when the inputs are integers, the -original integer-only macros \csa{xintAdd}, \csa{xintSub}, -\csa{xintMul}, etc\dots are available under the names \csa{xintiAdd}, -\csa{xintiSub}, \csa{xintiMul}, \dots, also when \xintfracname is not -loaded. Even these originally integer-only macros will accept fractions -on input if \xintfracname is loaded as long as they are integers in -disguise; they produce on output integers without any forward -slash mark nor trailing |[n]|. - -But |\xintAdd| will output fractions |A/B[n]|, with |B| present even if its -value is one. See the \xintfracname \hyperref[sec:frac]{documentation} for -additional information. - -% on how macros of \xintname are modified after loading -% \xintfracname (or \xintexprname). - - -% \xintfracname will extend \csbxint{Num} for it to remove this unit -% denominator and convert the |[n]| part into explicit zeros; see also -% \csbxint{PRaw} which does not make the assumption that the fraction is an -% integer in disguise. - -% This is mandatory when the computation result is fetched -% into a context where \TeX{} expects a number (assuming it does not exceed -% @2^31@). See the also the \xintfracname \hyperref[sec:frac]{documentation} for -% more information on how macros of \xintname are modified after loading -% \xintfracname (or \xintexprname). - - -% Package \xintname also provides some general macro programming or token -% manipulation utilities (expandable as well as non-expandable), which are -% described in the next section (\autoref{sec:tools}). - -\localtableofcontents - -\subsection{\csbh{xintRev}} \label{xintRev} - -\csa{xintRev\n}\etype{f} will revert the order of the digits of the number, -keeping the optional sign. Leading zeroes -resulting from the operation are not removed (see the -\csa{xintNum} macro for this). This macro and all other -macros dealing with numbers first expand `fully' their arguments. -\centeredline{|\xintRev{-123000}|\digitstt{=\xintRev{-123000}}} -\centeredline{|\xintNum{\xintRev{-123000}}|% - \digitstt{=\xintNum{\xintRev{-123000}}}} - - -\subsection{\csbh{xintLen}}\label{xintiLen} - -\csa{xintLen\n}\etype{\Numf} returns the length of the number, not counting the -sign. \centeredline{|\xintLen{-12345678901234567890123456789}|\digitstt - {=\xintLen{-12345678901234567890123456789}}} Extended by \xintfracname to -fractions: the length of |A/B[n]| is the length of |A| plus the -length of |B| plus the absolute value of |n| and minus one (an integer input as -|N| is internally represented in a form equivalent to |N/1[0]| so the minus one -means that the extended \csa{xintLen} behaves the same as the original for -integers). \centeredline{|\xintLen{-1e3/5.425}|\digitstt - {=\xintLen{-1e3/5.425}}} The length is computed on the |A/B[n]| which would -have been returned by \csbxint{Raw}: |\xintRaw {-1e3/5.425}|\digitstt{=\xintRaw - {-1e3/5.425}}. - -Let's point out that the whole thing should sum up to -less than circa @2^{31}@, but this is a bit theoretical. - -|\xintLen| is only for numbers or fractions. See \csbxint{Length} for counting -tokens (or rather braced groups), more generally. - -\subsection{\csbh{xintDigitsOf}}\label{xintDigitsOf} - -This is a synonym for \csbxint{AssignArray},\ntype{fN} to be used to define -an array giving all the digits of a given (positive, else the minus sign will -be treated as first item) number. -\begingroup\xintDigitsOf\xintiPow {7}{500}\to\digits -\centeredline{|\xintDigitsOf\xintiPow {7}{500}\to\digits|} -\noindent @7^500@ has |\digits{0}=|\digits{0} digits, and the 123rd among them -(starting from the most significant) is -|\digits{123}=|\digits{123}. -\endgroup - -\subsection{\csbh{xintNum}}\label{xintiNum} - -\csa{xintNum\n}\etype{f} removes chains of plus or minus signs, followed by -zeroes. \centeredline{|\xintNum{+---++----+--000000000367941789479}|\digitstt - {=\xintNum{+---++----+--000000000367941789479}}} Extended by \xintfracname to -accept also a fraction on input, as long as it reduces to an integer after -division of the numerator by the denominator. -\centeredline{|\xintNum{123.48/-0.03}|\digitstt{=\xintNum{123.48/-0.03}}} - - -\subsection{\csbh{xintSgn}}\label{xintiiSgn} - -\csa{xintSgn\n}\etype{\Numf} returns 1 if the number is positive, 0 if it is -zero and -1 if it is negative. Extended by \xintfracname to fractions. -\csbxint{iiSgn} skips the \csbxint{Num} overhead.\etype{f} - -\subsection{\csbh{xintOpp}}\label{xintiOpp}\label{xintiiOpp} - -\csa{xintOpp\n}\etype{\Numf} return the opposite |-N| of the number |N|. -Extended by \xintfracname to fractions. \csa{xintiOpp} is a synonym not modified -by \xintfracname\footnote{here, and in all similar instances, this means that - the macro remains integer-only both on input and output, but it does accept on - input a fraction which in disguise is a (big) integer.}, and -\csa{xintiiOpp} skips the \csbxint{Num} overhead.\etype{f} - - -\subsection{\csbh{xintAbs}}\label{xintiAbs}\label{xintiiAbs} - -\csa{xintAbs\n}\etype{\Numf} returns the absolute value of the number. Extended -by \xintfracname to fractions. \csa{xintiAbs} is a synonym not modified -by \xintfracname, and \csa{xintiiAbs} skips the \csbxint{Num} overhead.\etype{f} - - -\subsection{\csbh{xintAdd}}\label{xintiAdd}\label{xintiiAdd} - -\csa{xintAdd\n\m}\etype{\Numf\Numf} returns the sum of the two numbers. Extended -by \xintfracname to fractions. \csa{xintiAdd} is a synonym not modified by -\xintfracname, and \csa{xintiiAdd} skips the \csbxint{Num} overhead.\etype{ff} - - - -\subsection{\csbh{xintSub}}\label{xintiSub}\label{xintiiSub} - -\csa{xintSub\n\m}\etype{\Numf\Numf} returns the difference |N-M|. Extended -by \xintfracname to fractions. \csa{xintiSub} is a synonym not modified by -\xintfracname, and \csa{xintiiSub} skips the \csbxint{Num} overhead.\etype{ff} - - -\subsection{\csbh{xintCmp}}\label{xintiCmp} - -\csa{xintCmp\n\m}\etype{\Numf\Numf} returns 1 if |N>M|, 0 if |N=M|, and -1 -if |N<M|. Extended by \xintfracname to fractions. - -\subsection{\csbh{xintEq}}\label{xintEq} -%{\small New with release |1.09a|.\par} - -\csa{xintEq\n\m}\etype{\Numf\Numf} returns 1 if |N=M|, 0 otherwise. Extended -by \xintfracname to fractions. - -\subsection{\csbh{xintGt}}\label{xintGt} -%{\small New with release |1.09a|.\par} - -\csa{xintGt\n\m}\etype{\Numf\Numf} returns 1 if |N|$>$|M|, 0 otherwise. -Extended by \xintfracname to fractions. - -\subsection{\csbh{xintLt}}\label{xintLt} -%{\small New with release |1.09a|.\par} - -\csa{xintLt\n\m}\etype{\Numf\Numf} returns 1 if |N|$<$|M|, 0 otherwise. -Extended by \xintfracname to fractions. - -\subsection{\csbh{xintIsZero}}\label{xintIsZero} -%{\small New with release |1.09a|.\par} - -\csa{xintIsZero\n}\etype{\Numf} returns 1 if |N=0|, 0 otherwise. -Extended by \xintfracname to fractions. - -\subsection{\csbh{xintNot}}\label{xintNot} -%{\small New with release |1.09c|.\par} - -\csa{xintNot}\etype{\Numf} is a synonym for \csa{xintIsZero}. - -\subsection{\csbh{xintIsNotZero}}\label{xintIsNotZero} -%{\small New with release |1.09a|.\par} - -\csa{xintIsNotZero\n}\etype{\Numf} returns 1 if |N<>0|, 0 otherwise. -Extended by \xintfracname to fractions. - -\subsection{\csbh{xintIsOne}}\label{xintIsOne} -%{\small New with release |1.09a|.\par} - -\csa{xintIsOne\n}\etype{\Numf} returns 1 if |N=1|, 0 otherwise. -Extended by \xintfracname to fractions. - -\subsection{\csbh{xintAND}}\label{xintAND} -%{\small New with release |1.09a|.\par} - -\csa{xintAND\n\m}\etype{\Numf\Numf} returns 1 if |N<>0| and |M<>0| and zero -otherwise. Extended by \xintfracname to fractions. - -\subsection{\csbh{xintOR}}\label{xintOR} -%{\small New with release |1.09a|.\par} - -\csa{xintOR\n\m}\etype{\Numf\Numf} returns 1 if |N<>0| or |M<>0| and zero -otherwise. Extended by \xintfracname to fractions. - - -\subsection{\csbh{xintXOR}}\label{xintXOR} -%{\small New with release |1.09a|.\par} - -\csa{xintXOR\n\m}\etype{\Numf\Numf} returns 1 if exactly one of |N| or |M| -is true (i.e. non-zero). Extended by \xintfracname to fractions. - -\subsection{\csbh{xintANDof}}\label{xintANDof} -%{\small New with release |1.09a|.\par} - -\csa{xintANDof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns 1 if all -are true (i.e. non zero) and zero otherwise. The list argument may be a macro, -it (or rather its first token) is \fexpan ded first (each item also is \fexpan -ded). Extended by \xintfracname to fractions. - -\subsection{\csbh{xintORof}}\label{xintORof} -%{\small New with release |1.09a|.\par} - -\csa{xintORof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns 1 if at -least one is true (i.e. does not vanish). The list argument may be a macro, it -is \fexpan ded first. Extended by \xintfracname to fractions. - - -\subsection{\csbh{xintXORof}}\label{xintXORof} -%{\small New with release |1.09a|.\par} - -\csa{xintXORof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns 1 if an odd -number of them are true (i.e. does not vanish). The list argument may be a -macro, it is \fexpan ded first. Extended by \xintfracname to fractions. - - -\subsection{\csbh{xintGeq}}\label{xintiGeq} - -\csa{xintGeq\n\m}\etype{\Numf\Numf} returns 1 if the \emph{absolute value} -of the first number is at least equal to the absolute value of the second -number. If \verb+|N|<|M|+ it returns 0. Extended by \xintfracname to fractions. -%(starting with release |1.07|) -Please note that the macro compares -\emph{absolute values}. - -\subsection{\csbh{xintMax}}\label{xintiMax} - -\csa{xintMax\n\m}\etype{\Numf\Numf} returns the largest of the two in the -sense of the order structure on the relative integers (\emph{i.e.} the -right-most number if they are put on a line with positive numbers on the right): -|\xintiMax {-5}{-6}|\digitstt{=\xintiMax{-5}{-6}}. Extended by \xintfracname to -fractions. \csa{xintiMax} is a synonym not modified by -\xintfracname. - -\subsection{\csbh{xintMaxof}}\label{xintMaxof} -%{\small New with release |1.09a|.\par} - -\csa{xintMaxof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns the maximum. -The list argument may be a macro, it is \fexpan ded first. Extended by -\xintfracname to fractions. \csa{xintiMaxof} is a -synonym not modified by \xintfracname. - - -\subsection{\csbh{xintMin}}\label{xintiMin} - -\csa{xintMin\n\m}\etype{\Numf\Numf} returns the smallest of the two in the -sense of the order structure on the relative integers (\emph{i.e.} the left-most -number if they are put on a line with positive numbers on the right): |\xintiMin -{-5}{-6}|\digitstt{=\xintiMin{-5}{-6}}. Extended by \xintfracname to fractions. -\csa{xintiMin} is a synonym not modified by -\xintfracname. - -\subsection{\csbh{xintMinof}}\label{xintMinof} -%{\small New with release |1.09a|.\par} - -\csa{xintMinof}|{{a}{b}{c}...}|\etype{f{$\to$}\lowast\Numf} returns the minimum. -The list argument may be a macro, it is \fexpan ded first. Extended by -\xintfracname to fractions. \csa{xintiMinof} is a synonym not modified by -\xintfracname. - - -\subsection{\csbh{xintSum}}\label{xintiiSum} - -\csa{xintSum}\marg{braced things}\etype{{\lowast f}} after expanding its -argument expects to find a sequence of tokens (or braced material). Each is -expanded (with the usual meaning), and the sum of all these numbers is returned. -Note: the summands are \emph{not} parsed by \csbxint{Num}. - -\csa{xintSum} is -extended by \xintfracname to fractions. The original, which accepts (after -\fexpan sion) only (big) integers in the strict format and produces a (big) -integer is available as \csa{xintiiSum}, also with \xintfracname loaded. - -\centeredline{% - \csa{xintiiSum}|{{123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}}|% - \digitstt{=\xintiiSum{{123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}}}} -\centeredline{\csa{xintiiSum}|{1234567890}|\digitstt{=\xintiiSum{1234567890}}} -An empty sum is no error and returns zero: |\xintiiSum -{}|\digitstt{=\xintiiSum {}}. A sum with only one term returns that -number: |\xintiiSum {{-1234}}|\digitstt{=\xintiiSum {{-1234}}}. -Attention that |\xintiiSum {-1234}| is not legal input and will make the -\TeX{} run fail. On the other hand |\xintiiSum -{1234}|\digitstt{=\xintiiSum{1234}}. Extended by \xintfracname to -fractions. - -% retiré de la doc le 22 octobre 2013 - -% \subsection{\csbh{xintSumExpr}}\label{xintiiSumExpr} - -% \csa{xintSumExpr}\meta{braced things}\csa{relax} is to what \csa{xintSum} -% expands. The argument is then expanded (with the usual meaning) and should give -% a list of braced quantities or macros, each one will be expanded in turn. -% \centeredline{% -% \csa{xintiiSumExpr}| {123}{-98763450}|% -% |{\xintFac{7}}{\xintiMul{3347}{591}}\relax|\digitstt{=% -% \xintiiSumExpr {123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}\relax}} - -% Note: I am not so happy with the name which seems to suggest that the -% |+| sign should be used instead of braces. Perhaps this will change -% in the future. - -% Extended by \xintfracname to fractions. - -\subsection{\csbh{xintMul}}\label{xintiMul}\label{xintiiMul} -%{\small Modified in release |1.03|.\par} - -\csa{xintMul\n\m}\etype{\Numf\Numf} returns the product of the two numbers. -% Starting with release |1.03| of \xintname, the macro checks the lengths of the -% two numbers and then activates its algorithm with the best (or at least, -% hoped-so) choice of which one to put first. This makes the macro a bit slower -% for numbers up to 50 digits, but may give substantial speed gain when one of the -% number has 100 digits or more. -Extended by \xintfracname to fractions. -\csa{xintiMul} is a synonym not modified by \xintfracname, and -\csa{xintiiMul} skips the \csbxint{Num} overhead.\etype{ff} - -\subsection{\csbh{xintSqr}}\label{xintiSqr}\label{xintiiSqr} - -\csa{xintSqr\n}\etype{\Numf} returns the square. Extended by \xintfracname to -fractions. \csa{xintiSqr} is a synonym not modified by -\xintfracname, and \csa{xintiiSqr} skips the \csbxint{Num} overhead.\etype{f} - - - -\subsection{\csbh{xintPrd}}\label{xintiiPrd} - -\csa{xintPrd}\marg{braced things}\etype{{\lowast f}} after expanding its -argument expects to find a sequence of (of braced items or unbraced -single tokens). Each is -expanded (with the usual meaning), and the product of all these numbers is -returned. Note: the operands are \emph{not} parsed by \csbxint{Num}. -\centeredline{% - \csa{xintiiPrd}|{{-9876}{\xintFac{7}}{\xintiMul{3347}{591}}}|% - \digitstt{=% - \xintiiPrd{{-9876}{\xintFac{7}}{\xintiMul{3347}{591}}}}} -\centeredline{\csa{xintiiPrd}|{123456789123456789}|\digitstt{=% - \xintiiPrd{123456789123456789}}} An empty product is no error and returns 1: -|\xintiiPrd {}|\digitstt{=\xintiiPrd {}}. A product reduced to a single term -returns this number: |\xintiiPrd {{-1234}}|\digitstt{=\xintiiPrd {{-1234}}}. -Attention that |\xintiiPrd {-1234}| is not legal input and will make the \TeX{} -compilation fail. On the other hand |\xintiiPrd {1234}|\digitstt{=\xintiiPrd - {1234}}. \centeredline{$\displaystyle 2^{200}3^{100}7^{100}$} -\centeredline{|=\xintiiPrd {{\xintiPow {2}{200}}{\xintiPow {3}{100}}{\xintiPow - {7}{100}}}|} -\digitstt{=\printnumber{\xintNum {\xinttheexpr - 2^200*3^100*7^100\relax }}} - -With \xintexprname, the above could be coded simply as \centeredline -{|\xinttheiiexpr 2^200*3^100*7^100\relax |} - -Extended by \xintfracname to fractions. The original, which accepts (after -\fexpan sion) only (big) integers in the strict format and produces a (big) -integer is available as \csbxint{iiPrd}, also with \xintfracname loaded. - - -% I temporarily remove mention of \xintPrdExpr from the documentation; I -% really dislike the name now. - -% \subsection{\csbh{xintPrdExpr}}\label{xintiiPrdExpr} - -% {\small Name change in |1.06a|! I apologize, but I suddenly decided that -% \csa{xintProductExpr} was a bad choice; so I just replaced it by the current -% name. \par} - -% \csa{xintPrdExpr}\marg{argument}\csa{relax} is to what \csa{xintPrd} expands -% ; its argument is expanded (with the usual meaning) and should give a list of -% braced numbers or macros. Each will be expanded when it is its turn. -% \centeredline{\csa{xintiiPrdExpr}| 123456789123456789\relax|\digitstt{=% -% \xintiiPrdExpr 123456789123456789\relax}} - -% Note: I am not so happy with the name which seems to suggest that the -% |*| sign should be used instead of braces. Perhaps this will change -% in the future. - -% Extended by \xintfracname to fractions. - -\subsection{\csbh{xintPow}}\label{xintiPow}\label{xintiiPow} - -\csa{xintPow\n\x}\etype{\Numf\numx} returns |N^x|. When |x| is zero, this is 1. -If |N| is zero and |x<0|, if \verb+|N|>1+ and |x<0| negative, or if \verb+|N|>1+ -and |x>100000|,\MyMarginNote{Changed!} then an error is raised. Indeed |2^50000| -already has \digitstt{\xintLen{\xintFloatPow [1]{2}{50000}}} digits; each exact -multiplication of two one thousand digits numbers already takes a few seconds, -and it would take hours for the expandable computation to conclude with two -numbers with each circa @15000@ digits. Perhaps some completely expandable but -not \fexpan dable variants could fare better? - -Extended by \xintfracname to fractions (\csbxint{Pow}) and to floats -(\csbxint{FloatPow} for which the exponent must still obey the \TeX{} bound and -\csbxint{FloatPower} which has no restriction at all on the size of the -exponent). Negative exponents do not then cause errors anymore. The float -version is able to deal with things such as |2^999999999| without any problem. -For example |\xintFloatPow[4]{2}{50000}|\digitstt{=\xintFloatPow[4]{2}{50000}} -and |\xintFloatPow[4]{2}{999999999}| -\digitstt{=\xintFloatPow[4]{2}{999999999}}.\footnote{On my laptop - |\string\xintiiPow \{2\}\{9999\}| obtains all |3010| digits in about ten or - eleven seconds. In contrast, the float versions for |8|, |16|, |24|, or even - more significant figures, do their jobs in less than one hundredth of a second - (|1.09j|; we used in the text only four significant digits only for reasons of - space, not time.) This is done without |log|/|exp| which are not (yet?) - implemented in \xintfracname. The \LaTeX3 - \href{http://www.ctan.org/pkg/l3kernel}{l3fp} package does this with - |log|/|exp| and is ten times faster, but allows only |16| significant - figures and the (exactly represented) floating point numbers must have their - exponents limited to $\pm$\digitstt{9999}.} - -\csa{xintiPow} is a synonym not modified by \xintfracname, and \csa{xintiiPow} -is an integer only variant skipping the \csbxint{Num} overhead\etype{f\numx}, it -produces the same result as \csa{xintiPow} with stricter assumptions on the -inputs, and is thus a tiny bit faster. - -Within an \csbxint{iiexpr}|..\relax| the infix operator |^| is mapped to -\csa{xintiiPow}; within an \csbxint{expr}-ession\MyMarginNote{corr. of the - previous doc.} it is mapped to \csbxint{Pow} -(as extended by \xintfracname); in \csbxint{floatexpr}, it is mapped to -\csbxint{FloatPower}. - - - - -\subsection{\csbh{xintSgnFork}}\label{xintSgnFork} -%{\small New with release |1.07|. See also \csbxint{ifSgn}.\par} - -\csa{xintSgnFork}\verb+{-1|0|1}+\marg{A}\marg{B}\marg{C}\etype{xnnn} expandably -chooses to execute either the \meta{A}, \meta{B} or \meta{C} code, -depending on its first argument. This first argument should be anything -expanding to either |-1|, |0| or |1| (a count register must be -prefixed by |\the| and a |\numexpr...\relax| also must be prefixed by -|\the|). This utility is provided to help construct expandable macros -choosing depending on a condition which one of the package macros to -use, or which values to confer to their arguments. - -\subsection{\csbh{xintifSgn}}\label{xintifSgn} -%{\small New with release |1.09a|.\par} - -Similar to \csa{xintSgnFork}\etype{\Numf nnn} except that the first argument may -expand to a (big) integer (or a fraction if \xintfracname is loaded), and it is -its sign which decides which of the three branches is taken. Furthermore this -first argument may be a count register, with no |\the| or |\number| prefix. - -\subsection{\csbh{xintifZero}}\label{xintifZero} -%{\small New with release |1.09a|.\par} - -\csa{xintifZero}\marg{N}\marg{IsZero}\marg{IsNotZero}\etype{\Numf nn} expandably -checks if the first mandatory argument |N| (a number, possibly a fraction if -\xintfracname is loaded, or a macro expanding to one such) is zero or not. It -then either executes the first or the second branch. Beware that both branches -must be present. - -\subsection{\csbh{xintifNotZero}}\label{xintifNotZero} -%{\small New with release |1.09a|.\par} - -\csa{xintifNotZero}\marg{N}\marg{IsNotZero}\marg{IsZero}\etype{\Numf nn} -expandably checks if the first mandatory argument |N| (a number, possibly a -fraction if \xintfracname is loaded, or a macro expanding to one such) is not -zero or is zero. It then either executes the first or the second branch. Beware -that both branches must be present. - -\subsection{\csbh{xintifOne}}\label{xintifOne} -%{\small New with release |1.09i|.\par} - -\csa{xintifOne}\marg{N}\marg{IsOne}\marg{IsNotOne}\etype{\Numf nn} expandably -checks if the first mandatory argument |N| (a number, possibly a fraction if -\xintfracname is loaded, or a macro expanding to one such) is one or not. It -then either executes the first or the second branch. Beware that both branches -must be present. - - -\subsection{\csbh{xintifTrueAelseB}, \csbh{xint\-ifFalseAelseB}} -\label{xintifTrueAelseB} -\label{xintifFalseAelseB} - -%\label{xintifFalseTrue} -%{\small New with release |1.09c|, renamed in |1.09e|.\par} - -\csa{xintifTrueAelseB}\marg{N}\marg{true branch}\marg{false branch}\etype{\Numf - nn} is a synonym for \csbxint{ifNotZero}. - -{\small -\noindent 1. with |1.09i|, the synonyms |\xintifTrueFalse| and |\xintifTrue| are - deprecated - and will be removed in next release.\par -\noindent 2. These macros have no lowercase versions, use |\xintifzero|, -|\xintifnotzero|.\par } - -\csa{xintifFalseAelseB}\marg{N}\marg{false branch}\marg{true branch}\etype{\Numf - nn} is a synonym for \csbxint{ifZero}. - - - - -\subsection{\csbh{xintifCmp}}\label{xintifCmp} -%{\small New with release |1.09e|.\par} - -\csa{xintifCmp}\marg{A}\marg{B}\marg{if A<B}\marg{if A=B}\marg{if - A>B}\etype{\Numf\Numf nnn} compares -its arguments and chooses accordingly the correct branch. - -\subsection{\csbh{xintifEq}}\label{xintifEq} -%{\small New with release |1.09a|.\par} - -\csa{xintifEq}\marg{A}\marg{B}\marg{YES}\marg{NO}\etype{\Numf\Numf nn} -checks equality of its two first arguments (numbers, or fractions if -\xintfracname is loaded) and does the |YES| or the |NO| branch. - -\subsection{\csbh{xintifGt}}\label{xintifGt} -%{\small New with release |1.09a|.\par} - -\csa{xintifGt}\marg{A}\marg{B}\marg{YES}\marg{NO}\etype{\Numf\Numf nn} checks if -$A>B$ and in that case executes the |YES| branch. Extended to fractions (in -particular decimal numbers) by \xintfracname. - -\subsection{\csbh{xintifLt}}\label{xintifLt} -%{\small New with release |1.09a|.\par} - -\csa{xintifLt}\marg{A}\marg{B}\marg{YES}\marg{NO}\etype{\Numf\Numf nn} -checks if $A<B$ and in that case executes the |YES| branch. Extended to -fractions (in particular decimal numbers) by \xintfracname. - -\begin{framed} - The macros described next are all integer-only on input. With \xintfracname - loaded their argument is first given to \csbxint{Num} and may thus be - a fraction, as long as it is in fact an integer in disguise. -\end{framed} - -\subsection{\csbh{xintifOdd}}\label{xintifOdd} -%{\small New with release |1.09e|.\par} - -\csa{xintifOdd}\marg{A}\marg{YES}\marg{NO}\etype{\Numf nn} checks if $A$ is and -odd integer and in that case executes the |YES| branch. - - -\subsection{\csbh{xintFac}}\label{xintiFac} - -\csa{xintFac\x}\etype{\numx} returns the factorial. It is an error if the -argument is negative or at least @10^5@.% avant 1.09j c'était 1000000. - -With \xintfracname loaded, the macro is modified to accept a fraction as -argument, as long as this fraction turns out to be an integer: |\xintFac -{66/3}|\digitstt{=\xintFac {66/3}}. \csa{xintiFac} is a synonym not modified by -the loading of \xintfracname. - -% the construct |\xintFac{\xintAdd {2}{3}}| will fail, -% use either |\xintFac{\xintiAdd {2}{3}}| or |\xintFac{\xintNum{\xintAdd -% {2}{3}}}|. - -% temps obsolètes, mettre à jour -% On my laptop @1000!@ (2568 digits) -% is computed in a little less than ten seconds, @2000!@ (5736 -% digits) is computed in a little less than one hundred seconds, and -% @3000!@ (which has 9131 digits) needs close to seven minutes\dots -% I have no idea how much time @10000!@ would need (do rather -% @9999!@ if you can, the algorithm has some overhead at the -% transition from @N=9999@ to @10000@ and higher; @10000!@ has 35660 -% digits). Not to mention @100000!@ which, from the Stirling formula, -% should have 456574 digits. - -\subsection{\csbh{xintDivision}}\label{xintDivision}\label{xintiiDivision} - -\csa{xintDivision\n\m}\etype{\Numf\Numf} returns |{quotient Q}{remainder R}|. -This is euclidean division: |N = QM + R|, |0|${}\leq{}$\verb+R < |M|+. So the -remainder is always non-negative and the formula |N = QM + R| always holds -independently of the signs of |N| or |M|. Division by zero is an error (even if -|N| vanishes) and returns |{0}{0}|. The variant \csa{xintiiDivision}\etype{ff} -skips the overhead of parsing via \csbxint{Num}. - -This macro is integer only (with \xintfracname loaded it accepts -fractions on input, but they must be integers in disguise) and not to be -confused with the \xintfracname macro \csbxint{Div} which divides one -fraction by another. - -\subsection{\csbh{xintQuo}}\label{xintQuo}\label{xintiiQuo} - -\csa{xintQuo\n\m}\etype{\Numf\Numf} returns the quotient from the euclidean -division. When both |N| and |M| are positive one has -\csa{xintQuo\n\m}|=\xintiTrunc {0}{N/M}| (using package \xintfracname). With -\xintfracname loaded it accepts fractions on input, but they must be integers in -disguise. The variant \csa{xintiiQuo}\etype{ff} -skips the overhead of parsing via \csbxint{Num}. - -\subsection{\csbh{xintRem}}\label{xintRem}\label{xintiiRem} - -\csa{xintRem\n\m}\etype{\Numf\Numf} returns the remainder from the euclidean -division. With \xintfracname loaded it accepts fractions on input, but they must -be integers in disguise. The variant \csa{xintiiRem}\etype{ff} -skips the overhead of parsing via \csbxint{Num}. - - -\subsection{\csbh{xintFDg}}\label{xintFDg}\label{xintiiFDg} - -\csa{xintFDg\n}\etype{\Numf} returns the first digit (most significant) of the -decimal expansion. The variant \csa{xintiiFDg}\etype{f} -skips the overhead of parsing via \csbxint{Num}. - -\subsection{\csbh{xintLDg}}\label{xintLDg}\label{xintiiLDg} - -\csa{xintLDg\n}\etype{\Numf} returns the least significant digit. When the -number is positive, this is the same as the remainder in the euclidean division -by ten. The variant \csa{xintiiLDg}\etype{f} -skips the overhead of parsing via \csbxint{Num}. - -\subsection{\csbh{xintMON}, \csbh{xintMMON}} -\label{xintMON}\label{xintMMON}\label{xintiiMON}\label{xintiiMMON} -%{\small New in version |1.03|.\par} - -\csa{xintMON\n}\etype{\Numf} returns |(-1)^N| and \csa{xintMMON\n} returns -|(-1)^{N-1}|. \centeredline{|\xintMON {-280914019374101929}|\digitstt{=\xintMON - {280914019374101929}}, |\xintMMON {-280914019374101929}|\digitstt{=\xintMMON - {280914019374101929}}} -The variants \csa{xintiiMON}\etype{f} and \csa{xintiiMMON} -skip the overhead of parsing via \csbxint{Num}. - -\subsection{\csbh{xintOdd}}\label{xintOdd}\label{xintiiOdd} - -\csa{xintOdd\n}\etype{\Numf} is 1 if the number is odd and 0 otherwise. The -variant \csa{xintiiOdd} skip the overhead of parsing via \csbxint{Num}.\etype{f} - - - -\subsection{\csbh{xintiSqrt}, \csbh{xintiSquareRoot}}\label{xintiSqrt} -\label{xintiSquareRoot} -%{\small New with |1.08|.\par} - -\xintAssign\xintiSquareRoot {17000000000000000000000000}\to\A\B - -\noindent\csa{xintiSqrt\n}\etype{\Numf} returns the largest integer whose -square is at most equal to |N|. \centeredline{|\xintiSqrt - {2000000000000000000000000000000000000}=|% - \digitstt{\xintiSqrt{2000000000000000000000000000000000000}}} -\centeredline{|\xintiSqrt {3000000000000000000000000000000000000}=|% - \digitstt{\xintiSqrt{3000000000000000000000000000000000000}}} -\centeredline{|\xintiSqrt {\xintDSH {-120}{2}}=|}% -\centeredline{\digitstt{\xintiSqrt {\xintDSH {-120}{2}}}} -\csa{xintiSquareRoot\n}\etype{\Numf} returns |{M}{d}| with |d>0|, |M^2-d=N| and -|M| smallest (hence |=1+|\csa{xint\-iSqrt}|{N}|). -\centeredline{|\xintAssign\xintiSquareRoot - {17000000000000000000000000}\to\A\B|}% -\centeredline{|\xintiSub{\xintiSqr\A}\B=\A^2-\B|}% -\centeredline{\digitstt{\xintiSub{\xintiSqr\A}\B=\A\string^2-\B}} A rational -approximation to $\sqrt{|N|}$ is $|M|-\frac{|d|}{|2M|}$ (this is a majorant and -the error is at most |1/2M|; if |N| is a perfect square |k^2| then |M=k+1| and -this gives |k+1/(2k+2)|, not |k|). - -Package \xintfracname has \csbxint{FloatSqrt} for square -roots of floating point numbers. - - -\begin{framed} - The macros described next are strictly for integer-only arguments. These - arguments are \emph{not} filtered via \csbxint{Num}. -\end{framed} - -\subsection{\csbh{xintInc}, \csbh{xintDec}} -\label{xintInc} -\label{xintDec} -%{\small New with |1.08|.\par} - -\csa{xintInc\n}\etype{f} is |N+1| and \csa{xintDec\n} is |N-1|. These macros -remain integer-only, even with \xintfracname loaded. - -\subsection{\csbh{xintDouble}, \csbh{xintHalf}} -\label{xintDouble} -\label{xintHalf} -%{\small New with |1.08|.\par} - -\csa{xintDouble\n}\etype{f} returns |2N| and \csa{xintHalf\n} is |N/2| rounded -towards zero. These macros remain integer-only, even with \xintfracname loaded. - -\subsection{\csbh{xintDSL}}\label{xintDSL} - -\csa{xintDSL\n}\etype{f} is decimal shift left, \emph{i.e.} multiplication by -ten. - -\subsection{\csbh{xintDSR}}\label{xintDSR} - -\csa{xintDSR\n}\etype{f} is decimal shift right, \emph{i.e.} it removes the last -digit (keeping the sign), equivalently it is the closest integer to |N/10| when -starting at zero. - -\subsection{\csbh{xintDSH}}\label{xintDSH} - -\csa{xintDSH\x\n}\etype{\numx f} is parametrized decimal shift. When |x| is -negative, it is like iterating \csa{xintDSL} \verb+|x|+ times (\emph{i.e.} -multiplication by @10^{-@|x|@}@). When |x| positive, it is like iterating -\csa{DSR} |x| times (and is more efficient), and for a non-negative |N| this is -thus the same as the quotient from the euclidean division by |10^x|. - -\subsection{\csbh{xintDSHr}, \csbh{xintDSx}}\label{xintDSHr}\label{xintDSx} -%{\small New in release |1.01|.\par} - -\csa{xintDSHr\x\n}\etype{\numx f} expects |x| to be zero or positive and it -returns then a value |R| which is correlated to the value |Q| returned by -\csa{xintDSH\x\n} in the following manner: -\begin{itemize} -\item if |N| is - positive or zero, |Q| and |R| are the quotient and remainder in - the euclidean division by |10^x| (obtained in a more efficient - manner than using \csa{xintDivision}), -\item if |N| is negative let - |Q1| and |R1| be the quotient and remainder in the euclidean - division by |10^x| of the absolute value of |N|. If |Q1| - does not vanish, then |Q=-Q1| and |R=R1|. If |Q1| vanishes, then - |Q=0| and |R=-R1|. -\item for |x=0|, |Q=N| and |R=0|. -\end{itemize} -So one has |N = 10^x Q + R| if |Q| turns out to be zero or -positive, and |N = 10^x Q - R| if |Q| turns out to be negative, -which is exactly the case when |N| is at most |-10^x|. - - -\csa{xintDSx\x\n}\etype{\numx f} for |x| negative is exactly as -\csa{xintDSH\x\n}, \emph{i.e.} multiplication by @10^{-@|x|@}@. For |x| zero or -positive it returns the two numbers |{Q}{R}| described above, each one within -braces. So |Q| is \csa{xintDSH\x\n}, and |R| is \csa{xintDSHr\x\n}, but computed -simultaneously. - -\begin{flushleft} - \xintAssign\xintDSx {-1}{-123456789}\to\M - \noindent{|\xintAssign\xintDSx {-1}{-123456789}\to\M|}\\ - |\meaning\M: |\digitstt{\meaning\M}.\\ - \xintAssign\xintDSx {-20}{1234567689}\to\M - {|\xintAssign\xintDSx {-20}{123456789}\to\M|}\\ - |\meaning\M: |\digitstt{\meaning\M}.\\ - \xintAssign\xintDSx{0}{-123004321}\to\Q\R - {|\xintAssign\xintDSx {0}{-123004321}\to\Q\R|}\\ - \noindent|\meaning\Q: |\digitstt{\meaning\Q}, |\meaning\R: - |\digitstt{\meaning\R.}\\ - |\xintDSH {0}{-123004321}|\digitstt{=\xintDSH {0}{-123004321}}, - |\xintDSHr {0}{-123004321}|\digitstt{=\xintDSHr {0}{-123004321}}\\ - \xintAssign\xintDSx {6}{-123004321}\to\Q\R - {|\xintAssign\xintDSx {6}{-123004321}\to\Q\R|}\\ - |\meaning\Q: |\digitstt{\meaning\Q}, - |\meaning\R: |\digitstt{\meaning\R.}\\ - |\xintDSH {6}{-123004321}|\digitstt{=\xintDSH {6}{-123004321}}, - |\xintDSHr {6}{-123004321}|\digitstt{=\xintDSHr {6}{-123004321}}\\ - \xintAssign\xintDSx {8}{-123004321}\to\Q\R - {|\xintAssign\xintDSx {8}{-123004321}\to\Q\R|}\\ - |\meaning\Q: |\digitstt{\meaning\Q}, - |\meaning\R: |\digitstt{\meaning\R.} \\ - |\xintDSH {8}{-123004321}|\digitstt{=\xintDSH {8}{-123004321}}, - |\xintDSHr {8}{-123004321}|\digitstt{=\xintDSHr {8}{-123004321}}\\ - \xintAssign\xintDSx {9}{-123004321}\to\Q\R - {|\xintAssign\xintDSx {9}{-123004321}\to\Q\R|}\\ - |\meaning\Q: |\digitstt{\meaning\Q}, - |\meaning\R: |\digitstt{\meaning\R.}\\ - |\xintDSH {9}{-123004321}|\digitstt{=\xintDSH {9}{-123004321}}, - |\xintDSHr {9}{-123004321}|\digitstt{=\xintDSHr {9}{-123004321}}\\ -\end{flushleft} - -\subsection{\csbh{xintDecSplit}}\label{xintDecSplit} - -%{\small This has been modified in release |1.01|.\par} - -\csa{xintDecSplit\x\n}\etype{\numx f} cuts the number into two pieces (each one -within a pair of enclosing braces). First the sign if present is \emph{removed}. -Then, for |x| positive or null, the second piece contains the |x| least -significant digits (\emph{empty} if |x=0|) and the first piece the remaining -digits (\emph{empty} when |x| equals or exceeds the length of |N|). Leading -zeroes in the second piece are not removed. When |x| is negative the first piece -contains the \verb+|x|+ most significant digits and the second piece the -remaining digits (\emph{empty} if @|x|@ equals or exceeds the length of |N|). -Leading zeroes in this second piece are not removed. So the absolute value of the -original number is always the concatenation of the first and second piece. - -{\footnotesize This macro's behavior for |N| non-negative is final and will not - change. I am still hesitant about what to do with the sign of a - negative |N|.\par} - - -\xintAssign\xintDecSplit {0}{-123004321}\to\L\R -\centeredline{|\xintAssign\xintDecSplit {0}{-123004321}\to\L\R|} -\noindent|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.} -\xintAssign\xintDecSplit {5}{-123004321}\to\L\R -\centeredline{|\xintAssign\xintDecSplit {5}{-123004321}\to\L\R|} -|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.} -\xintAssign\xintDecSplit {9}{-123004321}\to\L\R -\centeredline{|\xintAssign\xintDecSplit {9}{-123004321}\to\L\R|} -|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.} -\xintAssign\xintDecSplit {10}{-123004321}\to\L\R -\centeredline{|\xintAssign\xintDecSplit {10}{-123004321}\to\L\R|} -|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.} -\xintAssign\xintDecSplit {-5}{-12300004321}\to\L\R -\centeredline{|\xintAssign\xintDecSplit {-5}{-12300004321}\to\L\R|} -|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.} -\xintAssign\xintDecSplit {-11}{-12300004321}\to\L\R -\centeredline{|\xintAssign\xintDecSplit {-11}{-12300004321}\to\L\R|} -|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.} -\xintAssign\xintDecSplit {-15}{-12300004321}\to\L\R -\centeredline{|\xintAssign\xintDecSplit {-15}{-12300004321}\to\L\R|} -|\meaning\L: |\digitstt{\meaning\L}, |\meaning\R: |\digitstt{\meaning\R.} - -\subsection{\csbh{xintDecSplitL}}\label{xintDecSplitL} - -\csa{xintDecSplitL\x\n}\etype{\numx f} returns the first piece after the action -of \csa{xintDecSplit}. - -\subsection{\csbh{xintDecSplitR}}\label{xintDecSplitR} - -\csa{xintDecSplitR\x\n}\etype{\numx f} returns the second piece after the action -of \csa{xintDecSplit}. - - - -\section{Commands of the \xintfracname package} -\label{sec:frac} - -\def\x{\string{x\string}} - -This package was first included in release |1.03| of the \xintname bundle. The -general rule of the bundle that each macro first expands (what comes first, -fully) each one of its arguments applies. - - -|f|\ntype{\Ff} stands for an integer or a fraction (see \autoref{sec:inputs} -for the accepted input formats) or something which expands to an integer or -fraction. It is possible to use in the numerator or the denominator of |f| count -registers and even expressions with infix arithmetic operators, under some rules -which are explained in the previous \hyperref[sec:useofcount]{Use of count - registers} section. - -As in the \hyperref[sec:xint]{xint.sty} documentation, |x|\ntype{\numx} -stands for something which will internally be embedded in a \csa{numexpr}. -It -may thus be a count register or something like |4*\count 255 + 17|, etc..., but -must expand to an integer obeying the \TeX{} bound. - -The fraction format on output is the scientific notation for the `float' macros, -and the |A/B[n]| format for all other fraction macros, with the exception of -\csbxint{Trunc}, {\color{blue}\string\xint\-Round} (which produce decimal -numbers) and \csbxint{Irr}, \csbxint{Jrr}, \csbxint{RawWithZeros} (which returns -an |A/B| with no trailing |[n]|, and prints the |B| even if it is |1|), and -\csbxint{PRaw} which does not print the |[n]| if |n=0| or the |B| if |B=1|. - -To be certain to print an integer output without trailing |[n]| nor fraction -slash, one should use either |\xintPRaw {\xintIrr {f}}| or |\xintNum {f}| when -it is already known that |f| evaluates to a (big) integer. For example -|\xintPRaw {\xintAdd {2/5}{3/5}}| gives a perhaps disappointing -\digitstt{\xintPRaw {\xintAdd {2/5}{3/5}}}\footnote{yes, \csbxint{Add} blindly - multiplies denominators... }, whereas |\xintPRaw {\xintIrr {\xintAdd - {2/5}{3/5}}}| returns \digitstt{\xintPRaw {\xintIrr {\xintAdd - {2/5}{3/5}}}}. As we knew the result was an integer we could have used -|\xintNum {\xintAdd {2/5}{3/5}}=|\xintNum {\xintAdd {2/5}{3/5}}. - -Some macros (such as \csbxint{iTrunc}, -\csbxint{iRound}, and \csbxint{Fac}) always produce directly integers on output. - - -\localtableofcontents - -\subsection{\csbh{xintNum}}\label{xintNum} - -The macro\etype{f} is extended to accept a fraction on input. But this fraction -should reduce to an integer. If not an error will be raised. The original is -available as \csbxint{iNum}. It is imprudent to apply \csa{xintNum} to numbers -with a large power of ten given either in scientific notation or with the |[n]| -notation, as the macro will add the necessary zeroes to get an explicit integer. - -\subsection{\csbh{xintifInt}}\label{xintifInt} -%{\small New with release |1.09e|.\par} - -\csa{xintifInt}|{f}{YES branch}{NO branch}|\etype{\Ff nn} expandably chooses the -|YES| branch if |f| reveals itself after expansion and simplification to be an -integer. As with the other \xintname conditionals, both branches must be present -although one of the two (or both, but why then?) may well be an empty brace pair -|{}|. As will all other \xintname conditionals, spaces in-between the braced -things do not matter, but a space after the closing brace of the |NO| branch is -significant. - - -\subsection{\csbh{xintLen}}\label{xintLen} - -The original macro\etype{\Ff} is extended to accept a fraction on input. -\centeredline {|\xintLen {201710/298219}|\digitstt{=\xintLen {201710/298219}}, -|\xintLen {1234/1}|\digitstt{=\xintLen {1234/1}}, |\xintLen {1234}|% - \digitstt{=\xintLen {1234}}} - - -\subsection{\csbh{xintRaw}}\label{xintRaw} -%{\small New with release |1.04|.\par} -%{\small \color{red}MODIFIED IN |1.07|.\par} - -This macro `prints' the\etype{\Ff} -fraction |f| as it is received by the package after its parsing and -expansion, in a form |A/B[n]| equivalent to the internal -representation: the denominator |B| is always strictly positive and is -printed even if it has value |1|. -\centeredline{|\xintRaw{\the\numexpr 571*987\relax.123e-10/\the\numexpr - -201+59\relax e-7}=|}% -\centeredline{\digitstt{\xintRaw{\the\numexpr - 571*987\relax.123e-10/\the\numexpr -201+59\relax e-7}}} - -\subsection{\csbh{xintPRaw}}\label{xintPRaw} -%{\small New in |1.09b|.\par} - -|PRaw|\etype{\Ff} stands for ``pretty raw''. It does \emph{not} show the |[n]| -if |n=0| and does \emph{not} show the |B| if |B=1|. \centeredline{|\xintPRaw - {123e10/321e10}=|\digitstt{\xintPRaw {123e10/321e10}}, |\xintPRaw - {123e9/321e10}=|\digitstt{\xintPRaw {123e9/321e10}}} \centeredline{|\xintPRaw - {\xintIrr{861/123}}=|\digitstt{\xintPRaw{\xintIrr{861/123}}} \ vz.\ - |\xintIrr{861/123}=|\digitstt{\xintIrr{861/123}}} See also \csbxint{Frac} (or -\csbxint{FwOver}) for math mode. As is examplified above the \csbxint{Irr} macro -which puts the fraction into irreducible form does not remove the |/1| if the -fraction is an integer. One can use \csbxint{Num} for that, but there will be an -error message if the fraction was not an integer; so the combination -|\xintPRaw{\xintIrr{f}}| is the way to go. - -\subsection{\csbh{xintNumerator}}\label{xintNumerator} - -This returns\etype{\Ff} the numerator corresponding to the internal -representation of a fraction, with positive powers of ten converted into zeroes -of this numerator: \centeredline{|\xintNumerator - {178000/25600000[17]}|\digitstt{=\xintNumerator {178000/25600000[17]}}} -\centeredline{|\xintNumerator {312.289001/20198.27}|% - \digitstt{=\xintNumerator {312.289001/20198.27}}} -\centeredline{|\xintNumerator {178000e-3/256e5}|\digitstt{=\xintNumerator - {178000e-3/256e5}}} \centeredline{|\xintNumerator - {178.000/25600000}|\digitstt{=\xintNumerator {178.000/25600000}}} As shown by -the examples, no simplification of the input is done. For a result uniquely -associated to the value of the fraction first apply \csa{xintIrr}. - -\subsection{\csbh{xintDenominator}}\label{xintDenominator} - -This returns\etype{\Ff} the denominator corresponding to the internal -representation of the fraction:\footnote{recall that the |[]| construct excludes - presence of a decimal point.} \centeredline{|\xintDenominator - {178000/25600000[17]}|\digitstt{=\xintDenominator {178000/25600000[17]}}}% -\centeredline{|\xintDenominator {312.289001/20198.27}|% - \digitstt{=\xintDenominator {312.289001/20198.27}}} -\centeredline{|\xintDenominator {178000e-3/256e5}|\digitstt{=\xintDenominator - {178000e-3/256e5}}} \centeredline{|\xintDenominator - {178.000/25600000}|\digitstt{=\xintDenominator {178.000/25600000}}} As shown -by the examples, no simplification of the input is done. The denominator looks -wrong in the last example, but the numerator was tacitly multiplied by @1000@ -through the removal of the decimal point. For a result uniquely associated to -the value of the fraction first apply \csa{xintIrr}. - -\subsection{\csbh{xintRawWithZeros}}\label{xintRawWithZeros} -%{\small New name in |1.07| (former name |\xintRaw|).\par} - -This macro `prints'\etype{\Ff} the -fraction |f| (after its parsing and expansion) in |A/B| form, with |A| -as returned by \csa{xintNumerator}|{f}| and |B| as returned by -\csa{xintDenominator}|{f}|. -\centeredline{|\xintRawWithZeros{\the\numexpr 571*987\relax.123e-10/\the\numexpr - -201+59\relax e-7}=|}% -\centeredline{\digitstt{\xintRawWithZeros{\the\numexpr - 571*987\relax.123e-10/\the\numexpr -201+59\relax e-7}}} - - -\subsection{\csbh{xintREZ}}\label{xintREZ} - -This command\etype{\Ff} normalizes a fraction by removing the powers of ten from -its numerator and denominator: \centeredline{|\xintREZ - {178000/25600000[17]}|\digitstt{=\xintREZ {178000/25600000[17]}}} -\centeredline{|\xintREZ {1780000000000e30/2560000000000e15}|\digitstt{=\xintREZ - {1780000000000e30/2560000000000e15}}} As shown by the example, it does not -otherwise simplify the fraction. - - -\subsection{\csbh{xintFrac}}\label{xintFrac} - -This is a \LaTeX{} only command,\etype{\Ff} to be used in math mode only. It -will print a fraction, internally represented as something equivalent to -|A/B[n]| as |\frac {A}{B}10^n|. The power of ten is omitted when |n=0|, the -denominator is omitted when it has value one, the number being separated from -the power of ten by a |\cdot|. |$\xintFrac {178.000/25600000}$| gives $\xintFrac -{178.000/25600000}$, |$\xintFrac {178.000/1}$| gives $\xintFrac {178.000/1}$, -|$\xintFrac {3.5/5.7}$| gives $\xintFrac {3.5/5.7}$, and |$\xintFrac {\xintNum - {\xintFac{10}/|\allowbreak|\xintiSqr{\xintFac {5}}}}$| gives $\xintFrac -{\xintNum {\xintFac{10}/\xintiSqr{\xintFac {5}}}}$. As shown by the examples, -simplification of the input (apart from removing the decimal points and moving -the minus sign to the numerator) is not done automatically and must be the -result of macros such as |\xintIrr|, |\xintREZ|, or |\xintNum| (for fractions -being in fact integers.) - -\subsection{\csbh{xintSignedFrac}}\label{xintSignedFrac} - -%{\small New with release |1.04|.\par} - -This is as \csbxint{Frac}\etype{\Ff} except that a negative fraction has the -sign put in front, not in the numerator. \centeredline{|\[\xintFrac - {-355/113}=\xintSignedFrac {-355/113}\]|} -\[\xintFrac {-355/113}=\xintSignedFrac {-355/113}\] - -\subsection{\csbh{xintFwOver}}\label{xintFwOver} - -This does the same as \csa{xintFrac}\etype{\Ff} except that the \csa{over} -primitive is used for the fraction (in case the denominator is not one; and a -pair of braces contains the |A\over B| part). |$\xintFwOver {178.000/25600000}$| -gives $\xintFwOver {178.000/25600000}$, |$\xintFwOver {178.000/1}$| gives -$\xintFwOver {178.000/1}$, |$\xintFwOver {3.5/5.7}$| gives $\xintFwOver -{3.5/5.7}$, and |$\xintFwOver {\xintNum {\xintFac{10}/\xintiSqr{\xintFac - {5}}}}$| gives $\xintFwOver {\xintNum {\xintFac{10}/\xintiSqr{\xintFac - {5}}}}$. - -\subsection{\csbh{xintSignedFwOver}}\label{xintSignedFwOver} - -%{\small New with release |1.04|.\par} - -This is as \csbxint{FwOver}\etype{\Ff} except that a negative fraction has the -sign put in front, not in the numerator. \centeredline{|\[\xintFwOver - {-355/113}=\xintSignedFwOver {-355/113}\]|} -\[\xintFwOver {-355/113}=\xintSignedFwOver {-355/113}\] - - -\subsection{\csbh{xintIrr}}\label{xintIrr} - -This puts the fraction\etype{\Ff} into its unique irreducible form: -\centeredline{|\xintIrr {178.256/256.178}|% - \digitstt{=\xintIrr {178.256/256.178}}${}=\xintFrac{\xintIrr - {178.256/256.178}[0]}$}% -Note that the current implementation does not cleverly first factor powers of 2 -and 5, so input such as |\xintIrr {2/3[100]}| will make \xintfracname do the -Euclidean division of |2|\raisebox{.5ex}{|.|}|10^{100}| by |3|, which is a bit -stupid. - -Starting with release |1.08|, \csa{xintIrr} does not remove the trailing |/1| -when the output is an integer. This was deemed better for various (stupid?) -reasons and thus the output format is now \emph{always} |A/B| with |B>0|. Use -\csbxint{PRaw} on top of \csa{xintIrr} if it is needed to get rid of a possible -trailing |/1|. For display in math mode, use rather |\xintFrac{\xintIrr {f}}| or -|\xintFwOver{\xintIrr {f}}|. - -\subsection{\csbh{xintJrr}}\label{xintJrr} - -This also puts the fraction\etype{\Ff} into its unique irreducible form: -\centeredline{|\xintJrr {178.256/256.178}|% - \digitstt{=\xintJrr {178.256/256.178}}}% -This is faster than \csa{xintIrr} for fractions having some big common -factor in the numerator and the denominator.\par -{\centering |\xintJrr {\xintiPow{\xintFac {15}}{3}/\xintiiPrdExpr -{\xintFac{10}}{\xintFac{30}}{\xintFac{5}}\relax }|\digitstt{=% - \xintJrr {\xintiPow{\xintFac {15}}{3}/\xintiiPrdExpr -{\xintFac{10}}{\xintFac{30}}{\xintFac{5}}\relax }}\par} But to notice the -difference one would need computations with much bigger numbers than in this -example. -Starting with release |1.08|, \csa{xintJrr} does not remove the trailing |/1| -when the output is an integer. - - -\subsection{\csbh{xintTrunc}}\label{xintTrunc} - -\csa{xintTrunc}|{x}{f}|\etype{\numx\Ff} returns the integral part, a dot, and -then the first |x| digits of the decimal -expansion of the fraction |f|. The -argument |x| should be non-negative. - -In the special case when |f| evaluates to @0@, the output is @0@ with no decimal -point nor decimal digits, else the post decimal mark digits are always printed. -A non-zero negative |f| which is smaller in absolute value than |10^{-x}| will -give @-0.000...@. -\centeredline{|\xintTrunc - {16}{-803.2028/20905.298}|\digitstt{=\xintTrunc {16}{-803.2028/20905.298}}}% -\centeredline{|\xintTrunc {20}{-803.2028/20905.298}|\digitstt{=\xintTrunc - {20}{-803.2028/20905.298}}}% -\centeredline{|\xintTrunc {10}{\xintPow {-11}{-11}}|\digitstt{=\xintTrunc - {10}{\xintPow {-11}{-11}}}}% -\centeredline{|\xintTrunc {12}{\xintPow {-11}{-11}}|\digitstt{=\xintTrunc - {12}{\xintPow {-11}{-11}}}}% -\centeredline{|\xintTrunc {12}{\xintAdd {-1/3}{3/9}}|\digitstt{=\xintTrunc - {12}{\xintAdd {-1/3}{3/9}}}} The digits printed are exact up to and -including the last one. - -% The identity |\xintTrunc {x}{-f}=-\xintTrunc {x}{f}| -% holds.\footnote{Recall that |-\string\macro| is not valid as argument to any -% package macro, one must use |\string\xintOpp\string{\string\macro\string}| or -% |\string\xintiOpp\string{\string\macro\string}|, except inside -% |\string\xinttheexpr...\string\relax|.} - -\subsection{\csbh{xintiTrunc}}\label{xintiTrunc} - -\csa{xintiTrunc}|{x}{f}|\etype{\numx\Ff} returns the integer equal to |10^x| -times what \csa{xintTrunc}|{x}{f}| would produce. -% -\centeredline{|\xintiTrunc - {16}{-803.2028/20905.298}|\digitstt{=\xintiTrunc {16}{-803.2028/20905.298}}}% -\centeredline{|\xintiTrunc {10}{\xintPow {-11}{-11}}|\digitstt{=\xintiTrunc - {10}{\xintPow {-11}{-11}}}}% -\centeredline{|\xintiTrunc {12}{\xintPow {-11}{-11}}|\digitstt{=\xintiTrunc - {12}{\xintPow {-11}{-11}}}}% -The difference between \csa{xintTrunc}|{0}{f}| and \csa{xintiTrunc}|{0}{f}| is -that the latter never has the decimal mark always present in the former except -for |f=0|. And \csa{xintTrunc}|{0}{-0.5}| returns ``\digitstt{\xintTrunc - 0{-0.5}}'' whereas \csa{xintiTrunc}|{0}{-0.5}| simply returns -``\digitstt{\xintiTrunc 0{-0.5}}''. - -\subsection{\csbh{xintXTrunc}}\label{xintXTrunc} - -%{\small New with release |1.09j|.\par} - -\csa{xintXTrunc}|{x}{f}|\retype{\numx\Ff} is completely expandable but not -\fexpan dable, as is indicated by the hollow star in the margin. It can not be -used as argument to the other package macros, but is designed to be used inside -an |\edef|, or rather a |\write|. Here is an example session where the user -after some warming up checks that @1/66049=1/257^2@ has period @257*256=65792@ -(it is also checked here that this is indeed the smallest period). -% -\begingroup\small -\dverb|@ -xxx:_xint $ etex -jobname worksheet-66049 -This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013) - restricted \write18 enabled. -**\relax -entering extended mode - -*\input xintfrac.sty -(./xintfrac.sty (./xint.sty (./xinttools.sty))) -*\message{\xintTrunc {100}{1/71}}% Warming up! - -0.01408450704225352112676056338028169014084507042253521126760563380281690140845 -07042253521126760563380 -*\message{\xintTrunc {350}{1/71}}% period is 35 - -0.01408450704225352112676056338028169014084507042253521126760563380281690140845 -0704225352112676056338028169014084507042253521126760563380281690140845070422535 -2112676056338028169014084507042253521126760563380281690140845070422535211267605 -6338028169014084507042253521126760563380281690140845070422535211267605633802816 -901408450704225352112676056338028169 -*\edef\Z {\xintXTrunc {65792}{1/66049}}% getting serious... - -*\def\trim 0.{}\oodef\Z {\expandafter\trim\Z}% removing 0. - -*\edef\W {\xintXTrunc {131584}{1/66049}}% a few seconds - -*\oodef\W {\expandafter\trim\W} - -*\oodef\ZZ {\expandafter\Z\Z}% doubling the period - -*\ifx\W\ZZ \message{YES!}\else\message{BUG!}\fi % xint never has bugs... -YES! -*\message{\xintTrunc {260}{1/66049}}% check visually that 256 is not a period - -0.00001514027464458205271843631243470756559523989765174340262532362337052794137 -6856576178291874214598252812306015231116292449545034746930309315810988811337037 -6538630410755651107511090251177156353616254598858423291798513225029902042423049 -5541189117170585474420505 -*\edef\X {\xintXTrunc {257*128}{1/66049}}% infix here ok, less than 8 tokens - -*\oodef\X {\expandafter\trim\X}% we now have the first 257*128 digits - -*\oodef\XX {\expandafter\X\X}% was 257*128 a period? - -*\ifx\XX\Z \message{257*128 is a period}\else \message{257 * 128 not a period}\fi -257 * 128 not a period -*\immediate\write-1 {1/66049=0.\Z... (repeat)} - -*\oodef\ZA {\xintNum {\Z}}% we remove the 0000, or we could use next \xintiMul - -*\immediate\write-1 {10\string^65792-1=\xintiiMul {\ZA}{66049}} - -*% This was slow :( I should write a multiplication, still completely - -*% expandable, but not f-expandable, which could be much faster on such cases. - -*\bye -No pages of output. -Transcript written on worksheet-66049.log. -xxx:_xint $ | -\endgroup - -Using |\xintTrunc| rather than |\xintXTrunc| would be hopeless on such long -outputs (and even |\xintXTrunc| needed of the order of seconds to complete -here). But it is not worth it to use |\xintXTrunc| for less than hundreds of -digits. - -Fraction arguments to |\xintXTrunc| corresponding to a |A/B[N]| with a negative -|N| are treated somewhat less efficiently (additional memory impact) than for positive or zero |N|. This is because the algorithm tries to work with the -smallest denominator hence does not extend |B| with zeroes, and technical -reasons lead to the use of some tricks.\footnote{Technical note: I do not - provide an |\char92 xintXFloat| because this would almost certainly mean - having to clone the entire core division routines into a ``long division'' - variant. But this could have given another approach to the implementation of - |\char 92 xintXTrunc|, especially for the case of a negative |N|. Doing these - things with \TeX{} is an effort. Besides an - |\char 92 xintXFloat| would be interesting only if also for example the square - root routine was provided in an |X| version (I have not given thought to - that). If feasible |X| routines would be interesting in the |\char 92 - xintexpr| context where things are expanded inside |\char92 csname..\char92 - endcsname|.} - -Contrarily to \csbxint{Trunc}, in the case of the second argument revealing -itself to be exactly zero, \csbxint{XTrunc} will output @0.000...@, not @0@. -Also, the first argument must be at least @1@. - -\subsection{\csbh{xintRound}}\label{xintRound} - -%{\small New with release |1.04|.\par} - -\csa{xintRound}|{x}{f}|\etype{\numx\Ff} returns the start of the decimal -expansion of the fraction |f|, rounded to |x| digits precision after the decimal -point. The argument |x| should be non-negative. Only when |f| evaluates exactly -to zero does \csa{xintRound} return |0| without decimal point. When |f| is not -zero, its sign is given in the output, also when the digits printed are all -zero. \centeredline{|\xintRound {16}{-803.2028/20905.298}|\digitstt{=\xintRound - {16}{-803.2028/20905.298}}}% -\centeredline{|\xintRound {20}{-803.2028/20905.298}|\digitstt{=\xintRound - {20}{-803.2028/20905.298}}}% -\centeredline{|\xintRound {10}{\xintPow {-11}{-11}}|\digitstt{=\xintRound - {10}{\xintPow {-11}{-11}}}}% -\centeredline{|\xintRound {12}{\xintPow {-11}{-11}}|\digitstt{=\xintRound - {12}{\xintPow {-11}{-11}}}}% -\centeredline{|\xintRound {12}{\xintAdd {-1/3}{3/9}}|\digitstt{=\xintRound - {12}{\xintAdd {-1/3}{3/9}}}} The identity |\xintRound {x}{-f}=-\xintRound -{x}{f}| holds. And regarding $(-11)^{-11}$ here is some more of its expansion: -\centeredline{\digitstt{\xintTrunc {50}{\xintPow {-11}{-11}}\dots}} - -\subsection{\csbh{xintiRound}}\label{xintiRound} - -%{\small New with release |1.04|.\par} - -\csa{xintiRound}|{x}{f}|\etype{\numx\Ff} returns the integer equal to |10^x| -times what \csa{xintRound}|{x}{f}| would return. \centeredline{|\xintiRound - {16}{-803.2028/20905.298}|\digitstt{=\xintiRound {16}{-803.2028/20905.298}}}% -\centeredline{|\xintiRound {10}{\xintPow {-11}{-11}}|\digitstt{=\xintiRound - {10}{\xintPow {-11}{-11}}}}% -Differences between \csa{xintRound}|{0}{f}| and \csa{xintiRound}|{0}{f}|: the -former cannot be used inside integer-only macros, and the latter removes the -decimal point, and never returns |-0| (and removes all superfluous leading -zeroes.) - -\subsection{\csbh{xintFloor}}\label{xintFloor} -%{\small New with release |1.09a|.\par} - -|\xintFloor {f}|\etype{\Ff} returns the largest relative integer |N| with -|N|${}\leq{}$|f|. \centeredline{|\xintFloor {-2.13}|\digitstt{=\xintFloor - {-2.13}}, |\xintFloor {-2}|\digitstt{=\xintFloor {-2}}, |\xintFloor - {2.13}|\digitstt{=\xintFloor {2.13}}% -} - -\subsection{\csbh{xintCeil}}\label{xintCeil} -%{\small New with release |1.09a|.\par} - -|\xintCeil {f}|\etype{\Ff} returns the smallest relative integer |N| with -|N|${}>{}$|f|. \centeredline{|\xintCeil {-2.13}|\digitstt{=\xintCeil {-2.13}}, - |\xintCeil {-2}|\digitstt{=\xintCeil {-2}}, |\xintCeil - {2.13}|\digitstt{=\xintCeil {2.13}}% -} - -\subsection{\csbh{xintTFrac}}\label{xintTFrac} - -\csa{xintTFrac}|{f}|\etype{\Ff} returns the fractional part, -|f=trunc(f)+frac(f)|. -The |T| stands for `Trunc', and there could similar macros associated to -`Round', `Floor', and `Ceil'. Inside |\xintexpr..\relax|, the function |frac| is -mapped to \csa{xintTFrac}. Inside |\xint|\-|floatexpr..\relax|, |frac| first -applies -\csa{xintTFrac} to its argument (which may be in float format, or -an exact fraction), and only next makes the float conversion. -\centeredline{|\xintTFrac {1235/97}|\digitstt{=\xintTFrac {1235/97}}\quad - |\xintTFrac {-1235/97}|\digitstt{=\xintTFrac {-1235/97}}} -\centeredline{|\xintTFrac {1235.973}|\digitstt{=\xintTFrac {1235.973}}\quad - |\xintTFrac {-1235.973}|\digitstt{=\xintTFrac {-1235.973}}} -\centeredline{|\xintTFrac {1.122435727e5}|% - \digitstt{=\xintTFrac {1.122435727e5}}} - - -\subsection{\csbh{xintE}}\label{xintE} -%{\small New with |1.07|.} - -|\xintE {f}{x}|\etype{\Ff\numx} multiplies the fraction |f| by @10^x@. The -\emph{second} argument |x| must obey the \TeX{} bounds. Example: -\centeredline{|\count 255 123456789 \xintE {10}{\count 255}|\digitstt{->\count - 255 123456789 \xintE {10}{\count 255}}} Be careful that for obvious reasons -such gigantic numbers should not be given to \csbxint{Num}, or added to -something with a widely different order of magnitude, as the package always -works to get the \emph{exact} result. There is \emph{no problem} using them for -\emph{float} operations:\centeredline{|\xintFloatAdd - {1e1234567890}{1}|\digitstt{=\xintFloatAdd {1e1234567890}{1}}} - -\subsection{\csbh{xintFloatE}}\label{xintFloatE} -%{\small New with |1.097|.} - -|\xintFloatE [P]{f}{x}|\etype{{\upshape[\numx]}\Ff\numx} multiplies the input -|f| by @10^x@, and -converts it to float format according to the optional first argument or current -value of |\xintDigits|. -\centeredline{|\xintFloatE {1.23e37}{53}|\digitstt{=\xintFloatE {1.23e37}{53}}} - -\subsection{\csbh{xintDigits}, \csbh{xinttheDigits}}\label{xintDigits} - -%{\small New with release |1.07|.\par} - -The syntax |\xintDigits := D;| (where spaces do not matter) assigns the -value of |D| to the number of digits to be used by floating point -operations. The default is |16|. The maximal value is |32767|. The macro -|\xinttheDigits|\etype{} serves to print the current value. - -\subsection{\csbh{xintFloat}}\label{xintFloat} - -%{\small New with release |1.07|.\par} - -The macro |\xintFloat [P]{f}|\etype{{\upshape[\numx]}\Ff} has an optional argument |P| which replaces -the current value of |\xintDigits|. The (rounded truncation of the) fraction -|f| is then printed in scientific form, with |P| digits, -a lowercase |e| and an exponent |N|. The first digit is from |1| to |9|, it is -preceded by an optional minus sign and -is followed by a dot and |P-1| digits, the trailing zeroes -are not trimmed. In the exceptional case where the -rounding went to the next power of ten, the output is |10.0...0eN| -(with a sign, perhaps). The sole exception is for a zero value, which then gets -output as |0.e0| (in an \csa{xintCmp} test it is the only possible output of -\csa{xintFloat} or one of the `Float' macros which will test positive for -equality with zero). -\centeredline{|\xintFloat[32]{1234567/7654321}|% - \digitstt{=\xintFloat[32]{1234567/7654321}}} -% \pdfresettimer -\centeredline{|\xintFloat[32]{1/\xintFac{100}}|% - \digitstt{=\xintFloat[32]{1/\xintFac{100}}}} -% \the\pdfelapsedtime -% 992: plus rapide que ce que j'aurais cru.. - -The argument to \csa{xintFloat} may be an |\xinttheexpr|-ession, like the -other macros; only its final evaluation is submitted to \csa{xintFloat}: the -inner evaluations of chained arguments are not at all done in `floating' -mode. For this one must use |\xintthefloatexpr|. - - -\subsection{\csbh{xintAdd}}\label{xintAdd} - -The original\etype{\Ff\Ff} macro is extended to accept fractions on input. Its -output will now always be in the form |A/B[n]|. The original is available as -\csbxint{iAdd}. - -\subsection{\csbh{xintFloatAdd}}\label{xintFloatAdd} - -%{\small New with release |1.07|.\par} - -|\xintFloatAdd [P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Ff} first replaces |f| and -|g| with their float approximations, with 2 safety digits. It then adds exactly -and outputs in float format with precision |P| (which is optional) or -|\xintDigits| if |P| was absent, the result of this computation. - - -\subsection{\csbh{xintSub}}\label{xintSub} - -The original\etype{\Ff\Ff} macro is extended to accept fractions on input. Its -output will now always be in the form |A/B[n]|. The original is available as -\csbxint{iSub}. - -\subsection{\csbh{xintFloatSub}}\label{xintFloatSub} - -%{\small New with release |1.07|.\par} - -|\xintFloatSub [P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Ff} first replaces |f| and -|g| with their float approximations, with 2 safety digits. It then subtracts -exactly and outputs in float format with precision |P| (which is optional), or -|\xintDigits| if |P| was absent, the result of this computation. - - -\subsection{\csbh{xintMul}}\label{xintMul} - -The original\etype{\Ff\Ff} macro is extended to accept fractions on input. Its -output will now always be in the form |A/B[n]|. The original, only for big -integers, and outputting a big integer, is available as \csbxint{iMul}. - -\subsection{\csbh{xintFloatMul}}\label{xintFloatMul} - -%{\small New with release |1.07|.\par} - -|\xintFloatMul [P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Ff} first replaces |f| and -|g| with their float approximations, with 2 safety digits. It then multiplies -exactly and outputs in float format with precision |P| (which is optional), or -|\xintDigits| if |P| was absent, the result of this computation. - -\subsection{\csbh{xintSqr}}\label{xintSqr} - -The original\etype{\Ff} macro is extended to accept a fraction on input. Its -output will now always be in the form |A/B[n]|. The original which outputs only -big integers is available as \csbxint{iSqr}. - -\subsection{\csbh{xintDiv}}\label{xintDiv} - -\csa{xintDiv}|{f}{g}|\etype{\Ff\Ff} computes the fraction |f/g|. As with all -other computation macros, no simplification is done on the output, which is in -the form |A/B[n]|. - -\subsection{\csbh{xintFloatDiv}}\label{xintFloatDiv} - -%{\small New with release |1.07|.\par} - -|\xintFloatDiv [P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Ff} first replaces |f| and -|g| with their float approximations, with 2 safety digits. It then divides -exactly and outputs in float format with precision |P| (which is optional), or -|\xintDigits| if |P| was absent, the result of this computation. - - -\subsection{\csbh{xintFac}}\label{xintFac} -%{\small Modified in |1.08b| (to allow fractions on input).\par} - -The original\etype{\Numf} is extended to allow a fraction on input but this -fraction |f| must simplify to a integer |n| (non negative and at most |999999|, -but already |100000!| is prohibitively time-costly). On output |n!| (no trailing -|/1[0]|). The original macro (which has less overhead) is still available as -\csbxint{iFac}. - -\subsection{\csbh{xintPow}}\label{xintPow} - -\csa{xintPow}{|{f}{g}|}:\etype{\Ff\Numf} the original macro is extended to -accept fractions on input. The output will now always be in the form |A/B[n]| -(even when the exponent vanishes: |\xintPow -{2/3}{0}|\digitstt{=\xintPow{2/3}{0}}). The original is available as -\csbxint{iPow}. - -The exponent is allowed to be input as a fraction but it must simplify to an -integer: |\xintPow {2/3}{10/2}|\digitstt{=\xintPow {2/3}{10/2}}. This integer -will be checked to not exceed |100000|. Indeed |2^50000| already has -\digitstt{\xintLen {\xintFloatPow [1]{2}{50000}}} digits, and squaring such a -number would take hours (I think) with the expandable routine of \xintname. - -\subsection{\csbh{xintFloatPow}}\label{xintFloatPow} -%{\small New with |1.07|.\par} - -|\xintFloatPow [P]{f}{x}|\etype{{\upshape[\numx]}\Ff\numx} uses either the -optional argument |P| or the value of |\xintDigits|. It computes a floating -approximation to |f^x|. The precision |P| must be at least |1|, naturally. - -The exponent |x| will be fed to a |\numexpr|, hence count registers are accepted -on input for this |x|. And the absolute value \verb+|x|+ must obey the \TeX{} -bound. For larger exponents use the slightly slower routine \csbxint{FloatPower} -which allows the exponent to be a fraction simplifying to an integer and does -not limit its size. This slightly slower routine is the one to which |^| is -mapped inside |\xintthefloatexpr...\relax|. - - -The macro |\xintFloatPow| chooses dynamically an appropriate number of -digits for the intermediate computations, large enough to achieve the desired -accuracy (hopefully). - -\centeredline{|\xintFloatPow [8]{3.1415}{1234567890}|% - \digitstt{=\xintFloatPow [8]{3.1415}{1234567890}}} - - - -\subsection{\csbh{xintFloatPower}}\label{xintFloatPower} -%{\small New with |1.07|.\par} - -\csa{xintFloatPower}|[P]{f}{g}|\etype{{\upshape[\numx]}\Ff\Numf} computes a -floating point value |f^g| where the exponent |g| is not constrained to be at -most the \TeX{} bound \texttt{\number "7FFFFFFF}. It may even be a fraction -|A/B| but must simplify to a (possibly big) integer. -\centeredline{|\xintFloatPower [8]{1.000000000001}{1e12}|% - \digitstt{=\xintFloatPower [8]{1.000000000001}{1e12}}} -\centeredline{|\xintFloatPower [8]{3.1415}{3e9}|% - \digitstt{=\xintFloatPower [8]{3.1415}{3e9}}} Note that |3e9>2^31|. But the -number following |e| in the output must at any rate obey the \TeX{} -\digitstt{\number"7FFFFFFF} bound. - - -Inside an |\xintfloatexpr|-ession, \csa{xintFloatPower} is the function to which -|^| is mapped. The exponent may then be something like |(144/3/(1.3-.5)-37)| -which is, in disguise, an integer. - - -The intermediate multiplications are done with a higher precision than -|\xintDigits| or the optional |P| argument, in order for the -final result to hopefully have the desired accuracy. - -\subsection{\csbh{xintFloatSqrt}}\label{xintFloatSqrt} -%{\small New with |1.08|.\par} - -\csa{xintFloatSqrt}|[P]{f}|\etype{{\upshape[\numx]}\Ff} computes a floating -point approximation of $\sqrt{|f|}$, either using the optional precision |P| or -the value of |\xintDigits|. The computation is done for a precision of at least -17 figures (and the output is rounded if the asked-for precision was smaller). -\centeredline{|\xintFloatSqrt [50]{12.3456789e12}|}% -\centeredline{${}\approx{}$\digitstt{\xintFloatSqrt [50]{12.3456789e12}}}% -\centeredline{|\xintDigits:=50;\xintFloatSqrt {\xintFloatSqrt {2}}|}% -\centeredline{% - ${}\approx{}$\xintDigits:=50;\digitstt{\xintFloatSqrt {\xintFloatSqrt {2}}}} - -% maple: 0.351364182864446216166582311675807703715914271812431919843183 1O^7 -% 3.5136418286444621616658231167580770371591427181243e6 -% maple: 1.18920711500272106671749997056047591529297209246381741301900 -% 1.1892071150027210667174999705604759152929720924638e0 - - -\xintDigits:=16; - -% removed from doc october 22 - -% \subsection{\csbh{xintSum}, \csbh{xintSumExpr}}\label{xintSum} -% \label{xintSumExpr} - -\subsection{\csbh{xintSum}}\label{xintSum}\label{xintSumExpr} - -% The original commands are extended to accept fractions on input and produce -% fractions on output. Their outputs will now always be in the form |A/B[n]|. The -% originals are available as \csa{xintiiSum} and \csa{xintiiSumExpr}. - -The original\etype{f{$\to$}{\lowast\Ff}} command is extended to accept fractions -on input and produce fractions on output. The output will now always be in the -form |A/B[n]|. The original, for big integers only (in strict format), is -available as \csa{xintiiSum}. - - -% \subsection{\csbh{xintPrd}, \csbh{xintPrdExpr}}\label{xintPrd}\label{xintPrdExpr} - -\subsection{\csbh{xintPrd}}\label{xintPrd}\label{xintPrdExpr} - -The original\etype{f{$\to$}{\lowast\Ff}} is extended to accept fractions on -input and produce fractions on output. The output will now always be in the form -|A/B[n]|. The original, for big integers only (in strict format), is available -as \csa{xintiiPrd}. - -\subsection{\csbh{xintCmp}}\label{xintCmp} -%{\small Rewritten in |1.08a|.\par} - -The macro\etype{\Ff\Ff} is extended to fractions. Its output is still either -|-1|, |0|, or |1| with no forward slash nor trailing |[n]|. - -For choosing branches according to the result of comparing |f| and |g|, the -following syntax is recommended: |\xintSgnFork{\xintCmp{f}{g}}{code for - f<g}{code for f=g}{code for f>g}|. - -% Note that since release |1.08a| using this macro on inputs with large powers of -% tens does not take a quasi-infinite time, contrarily to the earlier, somewhat -% dumb version (the earlier version indirectly led to the creation of giant chains -% of zeroes in certain circumstances, causing a serious efficiency impact). - -\subsection{\csbh{xintIsOne}} -See \csbxint{IsOne}\etype{\Ff} (\autoref{xintIsOne}). - -\subsection{\csbh{xintGeq}}\label{xintGeq} -%{\small Rewritten in |1.08a|.\par} - -The macro\etype{\Ff\Ff} is extended to fractions. Beware that the comparison is -on the \emph{absolute values} of the fractions. Can be used as: -\verb+\xintSgnFork{\xintGeq{f}{g}}{}{code for |f|<|g|}{code for - |f|+$\geqslant$\verb+|g|}+ - - - -\subsection{\csbh{xintMax}}\label{xintMax} -%{\small Rewritten in |1.08a|.\par} - -The macro is extended to fractions.\etype{\Ff\Ff} But now |\xintMax {2}{3}| -returns \digitstt{\xintMax {2}{3}}. The original, for use with (possibly big) -integers only, is available as \csbxint{iMax}: |\xintiMax -{2}{3}=|\digitstt{\xintiMax {2}{3}}. - -\subsection{\csbh{xintMaxof}} -See \csbxint{Maxof} (\autoref{xintMaxof}).\etype{f{$\to$}{\lowast\Ff}} - -\subsection{\csbh{xintMin}}\label{xintMin} -%{\small Rewritten in |1.08a|.\par} - -The macro is extended to fractions.\etype{\Ff\Ff} The original, for (big) -integers only, is available as \csbxint{iMin}. - -\subsection{\csbh{xintMinof}} -See \csbxint{Minof} (\autoref{xintMinof}).\etype{f{$\to$}{\lowast\Ff}} - -\subsection{\csbh{xintAbs}}\label{xintAbs} - -The macro is extended to fractions.\etype{\Ff} The original, for (big) integers -only, is available as \csbxint{iAbs}. Note that |\xintAbs -{-2}|\digitstt{=\xintAbs {-2}} whereas |\xintiAbs {-2}|\digitstt{=\xintiAbs - {-2}}. - -\subsection{\csbh{xintSgn}}\label{xintSgn} - -The macro is extended to fractions.\etype{\Ff} Naturally, its output is still -either |-1|, |0|, or |1| with no forward slash nor trailing |[n]|. - -\subsection{\csbh{xintOpp}}\label{xintOpp} - -The macro is extended to fractions.\etype{\Ff} The original is available as -\csbxint{iOpp}. Note that |\xintOpp {3}| now outputs \digitstt{\xintOpp {3}} -whereas |\xintiOpp {3}| returns \digitstt{\xintiOpp {3}}. - -\subsection{\csbh{xintDivision}, \csbh{xint\-Quo}, \csbh{xint\-Rem}, - \csbh{xintFDg}, \csbh{xintLDg}, \csbh{xintMON}, \csbh{xintMMON}, \csbh{xintOdd}} - -These macros\etype{\Ff\Ff} accept a fraction on input if this fraction in fact -reduces to an integer (if not an |\xintError:NotAnInteger| will be -raised).\etype{{\textcolor{black}{\upshape or}}\Ff} There is no difference in -the format of the outputs, which are still (possibly big) integers without -fraction slash nor trailing |[n]|, the sole difference is in the extended range -of accepted inputs. - -All have variants whose names start with |xintii| rather than |xint|; these -variants accept on input only integers in the strict format (they do not use -\csbxint{Num}). They thus have less overhead, and may be used when one is -dealing exclusively with (big) integers. These variants are already available in -\xintname, there is no need for \xintfracname to be loaded. - -\centeredline{|\xintNum {1e80}|} -\centeredline{\digitstt{\xintNum{1e80}}} - - -\etocdepthtag.toc {xintexpr} - -\section{Expandable expressions with the \xintexprname package}% -\label{sec:expr} - -The \xintexprname package was first released with version |1.07| of the -\xintname bundle. It loads automatically \xintfracname, hence -also \xintname and \xinttoolsname. - -% Release |1.09a| has extended the scope of |\xintexpr|-essions: infix -% comparison operators (|<|, |>|, |=|), logical operators (|&|, \verb+|+), -% functions (|round|, |sqrt|, |max|, |all|, etc...), conditional ``branching'' -% (|if| and |?|, |ifsgn| and |:|). - -The syntax is described in \autoref{sec:exprsummary} and -\autoref{sec:exprsummaryII}. - -\localtableofcontents - - -\subsection{The \csbh{xintexpr} expressions}\label{xintexpr}% -\label{xinttheexpr}\label{xintthe} - - -An \xintexprname{}ession is a construct -\csbxint{expr}\meta{expandable\_expression}|\relax|\etype{x} where the expandable -expression is read and completely expanded from left to right. - -During this parsing, braced sub-content \marg{expandable} may be serving as a -macro parameter, or a branch of the |?| two-way and |:| three-way operators; -else it is treated in a special manner: -\begin{enumerate} -\item it is allowed to occur only at the spots where numbers are legal, -\item the \meta{expandable} contents is then completely expanded as if it were - put in a |\csname..\endcsname|,\footnote{well, actually it \emph{is} put in - such a \texttt{\char92csname..\char92endcsname}.} thus it escapes entirely - the parser scope and infix notations will not be recognized except if the - expanded macros know how to handle them by themselves, -\item and this complete expansion \emph{must} produce a number or a fraction, - possibly with decimal mark and trailing |[n]|, the scientific notation is - \emph{not} authorized. -\end{enumerate} -Braces are the only way to input some number or fraction with -a trailing |[n]|: square brackets are -\emph{not} accepted in a |\xintexpr...\relax| if not within such braces. - - -An |\xintexpr..\relax| \emph{must} end in a |\relax| (which will be absorbed). -Like a |\numexpr| expression, it is not printable as is, nor can it be directly -employed as argument to the other package macros. For this one must use one -of the two equivalent forms: -\begin{itemize} -\item \csbxint{theexpr}\meta{expandable\_expression}|\relax|\etype{x}, or -\item \csbxint{the}|\xintexpr|\meta{expandable\_expression}|\relax|.\etype{x} -\end{itemize} - -The computations are done \emph{exactly}, and with no simplification of the -result. The output format for the result can be coded inside the expression -through the use of one of the functions |round|, |trunc|, |float|, -|reduce|.\footnote{In |round| and |trunc| the second optional parameter is the - number of digits of the fractional part; in |float| it is the total number of - digits of the mantissa.} Here are some examples\par -\begingroup\raggedright\leftskip.5cm -{|\xinttheexpr 1/5!-1/7!-1/9!\relax|% - \digitstt{=\xinttheexpr 1/5!-1/7!-1/9!\relax}}\\ -{|\xinttheexpr round(1/5!-1/7!-1/9!,18)\relax|% - \digitstt{=\xinttheexpr round(1/5!-1/7!-1/9!,18)\relax}}\\ -{|\xinttheexpr float(1/5!-1/7!-1/9!,18)\relax|% - \digitstt{=\xinttheexpr float(1/5!-1/7!-1/9!,18)\relax}}\\ -{|\xinttheexpr reduce(1/5!-1/7!-1/9!)\relax|% - \digitstt{=\xinttheexpr reduce(1/5!-1/7!-1/9!)\relax}}\\ -{|\xinttheexpr 1.99^-2 - 2.01^-2 \relax|% - \digitstt{=\xinttheexpr 1.99^-2 - 2.01^-2 \relax}}\\ -{|\xinttheexpr round(1.99^-2 - 2.01^-2, 10)\relax|% - \digitstt{=\xinttheexpr round(1.99^-2 - 2.01^-2, 10) \relax}}\par -\endgroup - -\smallskip -\begingroup % 18 octobre, je reprends la méthode déjà utilisée au début du - % document le 9 octobre. -\leftmargini 0pt -\list\labelitemi{\def\makelabel#1{\hss\llap{#1}}\listparindent\parindent - \labelwidth\parindent - \itemindent\labelwidth}% -\item the expression may contain arbitrarily many levels of nested parenthesized - sub-expressions. -\item sub-contents giving numbers of fractions should be either - \begin{enumerate} - \item parenthesized, - \item a sub-expression |\xintexpr...\relax|, - \item or within braces. - \end{enumerate} - When a sub-expression is hit against in the midst of absorbing the - digits of a number, a |*| to force tacit multiplication is - inserted.\inmarg{1.09j}. Similarly, if it is an opening parenthesis - which is hit against.\inmarg{1.09k} - \item an expression can not be given as argument to the other package macros, - nor printed, for this one must use |\xinttheexpr...\relax| or - |\xintthe\xintexpr...\relax|. - \item one does not use |\xinttheexpr...\relax| as a sub-constituent of an - |\xintexpr...\relax| but simply |\xintexpr...\relax|; this is mainly because - most of the time |\xinttheexpr..\relax| will insert explicit square brackets - which are not parsable, as already mentioned, in the surrounding expression. -\item each \xintexprname{}ession is completely expandable and obtains - its result in two expansion steps. -\endlist -\endgroup - -In an algorithm implemented non-expandably, one may define macros to -expand to infix expressions to be used within others. One then has the -choice between parentheses or |\xintexpr...\relax|: |\def\x {(\a+\b)}| -or |\def\x {\xintexpr \a+\b\relax}|. The latter is the better choice as -it allows also to be prefixed with |\xintthe|. Furthemore, if |\a| and -|\b| are already defined |\oodef\x {\xintexpr \a+\b\relax}| will do the -computation on the spot. - -\subsection{\texorpdfstring{\texttt{\protect\string\numexpr}}{\textbackslash - numexpr} or \texorpdfstring{\texttt{\protect\string\dimexpr}}{\textbackslash - dimexpr} expressions, count and dimension registers and variables} -\label{ssec:countinexpr} - -Count registers, count control sequences, dimen registers, -dimen control sequences, skips and skip control sequences, |\numexpr|, -|\dimexpr|, |\glueexpr| can be inserted directly, they will be unpacked using -|\number| (which gives the internal value in terms of scaled points for the -dimensional variables: @1@\,|pt|${}={}$@65536@\,|sp|; stretch and shrink -components are thus discarded). Tacit multiplication is implied, when a -number or decimal number prefixes such a register or control sequence. - -\LaTeX{} lengths are skip control sequences and \LaTeX{} counters should be -inserted using |\value|. - -In the case of numbered registers like |\count255| or |\dimen0|, the resulting -digits will be re-parsed, so for example |\count255 0| is like |100| if -|\the\count255| would give |10|. Control sequences define complete numbers, thus -cannot be extended that way with more digits, on the other hand they are more -efficient as they avoid the re-parsing of their unpacked contents. - -A token list variable must be prefixed by |\the|, it will not be unpacked -automatically (the parser will actually try |\number|, and thus fail). Do not -use |\the| but only |\number| with a dimen or skip, as the |\xintexpr| parser -doesn't understand |pt| and its presence is a syntax error. To use a dimension -expressed in terms of points or other \TeX{} recognized units, incorporate it in -|\dimexpr...\relax|. - -If one needs to optimize, |1.72\dimexpr 3.2pt\relax| is less efficient -than |1.72*{\number\dimexpr 3.2pt\relax}| as the latter avoids re-parsing the -digits of the representation of the dimension as scaled points. -\centeredline{|\xinttheexpr 1.72\dimexpr 3.2pt\relax/2.71828\relax=|} -\centeredline{|\xinttheexpr 1.72*{\number\dimexpr 3.2pt\relax}/2.71828\relax|} -\centeredline{\digitstt{\xinttheexpr 1.72\dimexpr - 3.2pt\relax/2.71828\relax=\xinttheexpr 1.72*{\number\dimexpr - 3.2pt\relax}/2.71828\relax}} -Regarding how dimensional expressions are converted by \TeX{} into scaled points -see \autoref{sec:Dimensions}. - -\subsection{Catcodes and spaces} - -\subsubsection{\csbh{xintexprSafeCatcodes}} -\label{xintexprSafeCatcodes} -%{\small New with release |1.09a|.\par} - -Active characters will interfere with |\xintexpr|-essions. One may prefix them -with |\string| within |\xintexpr..\relax|, thus preserving expandability, or -there is the non-expandable \csa{xintexprSafeCatcodes} which can be issued -before the use of |\xintexpr|. This command sets (not globally) the catcodes of -the relevant characters to safe values. This is used internally by -\csbxint{NewExpr} (restoring the catcodes on exit), hence \csbxint{NewExpr} does -not have to be protected against active characters. - -\subsubsection{\csbh{xintexprRestoreCatcodes}}\label{xintexprRestoreCatcodes} -%{\small New with release |1.09a|.\par} - -Restores the catcodes to the earlier state. - -\bigskip - -Unbraced spaces inside an |\xinttheexpr...\relax| should mostly be -innocuous (except inside macro arguments). - -|\xintexpr| and |\xinttheexpr| are for the most part agnostic regarding -catcodes: -(unbraced) digits, binary operators, minus and plus signs as prefixes, dot as -decimal mark, parentheses, may be indifferently of catcode letter or other or -subscript or superscript, ..., it doesn't matter.\footnote{Furthermore, although - \csbxint{expr} uses \csa{string}, it is (we hope) escape-char agnostic.} - -The characters \verb[+,-,*,/,^,!,&,|,?,:,<,>,=,(,),"[, the dot and the comma -should not be active as everything is expanded along the way. If one of them is -active, it should be prefixed with |\string|. - -The |!| as either logical negation or postfix factorial operator must be a -standard (\emph{i.e.} catcode @12@) |!|, more precisely a catcode @11@ -exclamation point |!| must be avoided as it is used internally by |\xintexpr| -for various special purposes. - - -% In the case of the factorial, the macro -% |\xintFac| may be used rather than the postfix |!|, preferably within braces as -% this will avoid the subsequent slow scan digit by digit of its expansion (other -% macros from the \xintfracname package generally \emph{must} be used within a -% brace pair, as they expand to fractions |A/B[n]| with the trailing |[n]|; the -% |\xintFac| produces an integer with no |[n]| and braces are only optional, but -% preferable, as the scanner will get the job done faster.) - -% Sub-material within braces is treated technically in a different manner, and -% depending on the macros used therein may be more sensitive to the catcode of the -% five operations. - -Digits, slash, square brackets, minus sign, in the output from an |\xinttheexpr| -are all of catcode 12. For |\xintthefloatexpr| the `e' in the output is of -catcode 11. - -A macro with arguments will expand and grab its arguments before the -parser may get a chance to see them, so the situation with catcodes and spaces -is not the same within such macro arguments (or within braces used to protect -square brackets). - - -\subsection{Expandability, \csh{xinteval}} - -As is the case with all other package macros |\xintexpr| \fexpan ds (in two -steps) to its final (non-printable) result; and |\xinttheexpr| \fexpan ds (in -two steps) to the chain of digits (and possibly minus sign |-|, decimal mark -|.|, fraction slash |/|, scientific |e|, square brackets |[|, |]|) representing -the result. - -Starting with |1.09j|, an |\xintexpr..\relax| can be inserted without |\xintthe| -prefix inside an |\edef|, or a |\write|.\MyMarginNote{New with 1.09j!} It -expands to a private more compact representation (five tokens) than -|\xinttheexpr| or |\xintthe\xintexpr|. - -The material between |\xintexpr| and |relax| should contain only expandable -material; the exception is with brace pairs which, apart from their usual r\^ole -for macro arguments, are also allowed in places where the scanner expects a -numeric operand, the braced material should expand to some number (or fraction), -but scientific notation is not allowed. Conversely fractions in |A/B[N]| format -(either explicit or from macro expansion) must be enclosed in such a brace pair. - -The once expanded |\xintexpr| is |\romannumeral0\xinteval|. And there is -similarly |\xintieval|, |\xintiieval|, and |\xintfloateval|. For the other cases -one can use |\romannumeral-`0| as prefix. For an example of expandable -algorithms making use of chains of |\xinteval|-uations connected via -|\expandafter| see \autoref{ssec:fibonacci}.\MyMarginNote{New with 1.09j!} - -An expression can only be legally finished by a |\relax| token, which -will be absorbed. - - -\subsection{Memory considerations} - -The parser creates an undefined control sequence for each intermediate -computation (this does not refer to the intermediate steps needed in -the evaluations of the \csbxint{Add}, \csbxint{Mul}, etc... macros corresponding -to the infix operators, but only to each conversion of such an infix operator -into a computation). So, a moderately sized expression might create 10, or 20 -such control sequences. On my \TeX{} installation, the memory available for such -things is of circa \np{200000} multi-letter control words. So this means that a -document containing hundreds, perhaps even thousands of expressions will compile -with no problem. - -Besides the hash table, also \TeX{} main memory is impacted. Thus, if -\xintexprname is used for computing plots\footnote{this is not very - probable as so far \xintname does not include a mathematical library - with floating point calculations, but provides only the basic - operations of algebra.}, this may cause a problem. - - - -There is a solution.\footnote{which convinced me that I could stick with the - parser implementation despite its potential impact on the hash-table - and other parts of \TeX{}'s memory.} - -A -document can possibly do tens of thousands of evaluations only -if some formulas are being used repeatedly, for example inside loops, with -counters being incremented, or with data being fetched from a file. So it is the -same formula used again and again with varying numbers inside. - -With the \csbxint{NewExpr} command, it is possible to convert once and -for all an expression containing parameters into an expandable macro -with parameters. Only this initial definition of this macro actually -activates the \csbxint{expr} parser and will (very moderately) impact -the hash-table: once this unique parsing is done, a macro with -parameters is produced which is built-up recursively from the -\csbxint{Add}, \csbxint{Mul}, etc... macros, exactly as it would be -necessary to do without the facilities of the \xintexprname package. - -\subsection{The \csbh{xintNewExpr} command}\label{xintNewExpr} - -% This allows to define a completely expandable macro with parameters, expanding -% in two steps to its final evaluation, and corresponding to the given -% \xintname{}expression where the parameters are input using the usual -% macro-parameter: |#1|, ..., |#9|. - -The command is used -as:\centeredline{|\xintNewExpr{\myformula}[n]|\marg{stuff}, where} -\begin{itemize} -\item \meta{stuff} will be inserted inside |\xinttheexpr . . . \relax|, -\item |n| is an integer between zero and nine, inclusive, and tells how many - parameters will |\myformula| have (it is \emph{mandatory} even if - |n=0|\footnote{there is some use for \csa{xintNewExpr}|[0]| compared to an - \csa{edef} as \csa{xintNewExpr} has some built-in catcode protection.}) -\item the placeholders |#1|, |#2|, ..., |#n| are used inside \meta{stuff} - in their usual r\^ole. -\end{itemize} - -The macro |\myformula| is defined without checking if it -already exists, \LaTeX{} users might prefer to do first |\newcommand*\myformula -{}| to get a reasonable error message in case |\myformula| already exists. - -The definition of |\myformula| made by |\xintNewExpr| is global. The protection -against active characters is done automatically. - -It will be a completely expandable macro entirely built-up using |\xintAdd|, -|\xintSub|, |\xintMul|, |\xintDiv|, |\xintPow|, etc\dots as corresponds to the -expression written with the infix operators. - -\begin{framed} - A ``formula'' created by |\xintNewExpr| is thus a macro whose parameters are - given to a possibly very complicated combination of the various macros of - \xintname and \xintfracname; hence one can not use infix notation inside the - arguments, as in for example |\myformula {28^7-35^12}| which would have been - allowed by - \centeredline{|\def\myformula #1{\xinttheexpr (#1)^3\relax}|} - One will have to do |\myformula {\xinttheexpr 28^7-35^12\relax}|, or redefine - |\myformula| to have more parameters. -\end{framed} - -% The formula may contain besides the infix operators and macro -% parameters some arbitrary decimal numbers, fractions (within braces) and also -% macros. If these macros do not involve the parameters, nothing special needs to -% be done, they will be expanded once during the construction of the formula. But -% if the parameters are to be used within the macros themselves, then the macro -% should be code with an underscore |_| rather than a backslash |\|. - -\dverb|@ -@\xintNewExpr\myformA[4]{ #1 + #2 * #3^#4 } -@\xintNewExpr\myformB[3]{ (#1 + 1.75)^#2 + #3*2.7 } -@\xintNewExpr\myformC[3]{ #1*#1+#2*#2+#3*#3-(#1*#2+#2*#3+#3*#1) } -@\xintNewExpr\myformD[2]{ (1+1.5*#1)^#2 - (1+1.5*#2)^#1 } -@\xintNewExpr\myformE[2]{ -----((((((#1*10-5*#2)))))) } -@\xintNewExpr\myformF[4]{ -#1^-#2*-#3-#4 } -@\xintNewExpr\myformG[4]{ -#1*-#2^-#3-#4 } -\xintNewExpr\DET[9]{ #1*#5*#9+#2*#6*#7+#3*#4*#8-#1*#6*#8-#2*#4*#9-#3*#5*#7 }| - -% \xintNewExpr\myformA[4]{ #1 + #2 * #3^#4 } -% \xintNewExpr\myformB[3]{ (#1 + 1.75)^#2 + #3*2.7 } -% \xintNewExpr\myformC[3]{ #1*#1+#2*#2+#3*#3-(#1*#2+#2*#3+#3*#1) } -% \xintNewExpr\myformD[2]{ (1+1.5*#1)^#2 - (1+1.5*#2)^#1 } -% \xintNewExpr\myformE[2]{ -----((((((#1*10-5*#2)))))) } -% \xintNewExpr\myformF[4]{ -#1^-#2*-#3-#4 } -% \xintNewExpr\myformG[4]{ -#1*-#2^-#3-#4 } -\xintNewExpr\DET[9]{ #1*#5*#9+#2*#6*#7+#3*#4*#8-#1*#6*#8-#2*#4*#9-#3*#5*#7 } - -\ttfamily -% |\meaning\myformA:|\printnumber{\meaning\myformA}\endgraf -% |\meaning\myformB:|\printnumber{\meaning\myformB}\endgraf -% |\meaning\myformC:|\printnumber{\meaning\myformC}\endgraf -% |\meaning\myformD:|\printnumber{\meaning\myformD}\endgraf -% |\meaning\myformE:|\printnumber{\meaning\myformE}\endgraf -% |\meaning\myformF:|\printnumber{\meaning\myformF}\endgraf -% |\meaning\myformG:|\printnumber{\meaning\myformG}\endgraf -|\meaning\DET:|\printnumber{\meaning\DET}\endgraf - - -\centeredline{|\xintNum{\DET {1}{1}{1}{10}{-10}{5}{11}{-9}{6}}|% - \digitstt{=\xintNum{\DET {1}{1}{1}{10}{-10}{5}{11}{-9}{6}}}}% -\centeredline{|\xintNum{\DET {1}{2}{3}{10}{0}{-10}{21}{2}{-17}}|% - \digitstt{=\xintNum{\DET {1}{2}{3}{10}{0}{-10}{21}{2}{-17}}}} - - -\rmfamily - - -\emph{Remark:} |\meaning| has been used within the argument to a |\printnumber| -command, to avoid going into the right margin, but this zaps all spaces -originally in the output from |\meaning|. Here is as an illustration the raw -output of -|\meaning| on the previous example: - -\ttfamily -\meaning\DET -\rmfamily - -This is why |\printnumber| was used, to have breaks across lines. - -\subsubsection {Use of conditional operators} - -The |1.09a| conditional operators |?| and |:| cannot be parsed by |\xintNewExpr| -when they contain macro parameters |#1|,\dots, |#9| within their scope. However -replacing them with the functions |if| and, respectively |ifsgn|, the parsing -should succeed. And the created macro will \emph{not evaluate the branches - to be skipped}, thus behaving exactly like |?| and |:| would have in the -|\xintexpr|. - -\xintNewExpr\Formula [3]{ if((#1>#2) & (#2>#3), - sqrt(#1-#2)*sqrt(#2-#3), #1^2+#3/#2) } - -\centeredline{|\xintNewExpr\Formula [3]|} -\centeredline{|{ if((#1>#2) & (#2>#3), - sqrt(#1-#2)*sqrt(#2-#3), #1^2+#3/#2) }|} - -\ttfamily -\noindent|\meaning\Formula:|\printnumber{\meaning\Formula}\endgraf - -\rmfamily -This formula (with |\xintifNotZero|) will gobble the false branch. - -Remark: this -|\XINTinFloatSqrt| macro is a non-user package macro used internally within -|\xintexpr|-essions, it produces the result in |A[n]| form rather -than in scientific notation, and for reasons of the inner workings of -|\xintexpr|-essions, this is necessary; a hand-made macro would -have used instead the equivalent |\xintFloatSqrt|. - -Another example - -\xintNewExpr\myformula[3]{ ifsgn(#1,#2/#3,#2-#3,#2*#3) } -\centeredline{|\xintNewExpr\myformula [3]|} -\centeredline{|{ ifsgn(#1,#2/#3,#2-#3,#2*#3) }|} - -\ttfamily -\noindent\printnumber{\meaning\myformula}\endgraf - -\rmfamily -Again, this macro gobbles the false branches, as would have the operator |:| -inside an |\xintexpr|-ession. - - - -\subsubsection{Use of macros} - - -For macros to be inserted within such a created \xintname-formula command, there -are two cases: -\begin{itemize} -\item the macro does not involve the numbered parameters in its arguments: it - may then be left as is, and will be evaluated once during the construction of - the formula, -\item it does involve at least one of the parameters as argument. Then: - \begin{enumerate} - \item the whole thing (macro + argument) should be braced (not necessary if it - is already included into a braced group), - \item the macro should be coded with an underscore |_| in place of the - backslash |\|. - \item the parameters should be coded with a dollar sign |$1|, |$2|, etc... - \end{enumerate} -\end{itemize} - -Here is a silly example illustrating the general principle (the macros here have -equivalent functional forms which are more convenient; but some of the more -obscure package macros of \xintname dealing with integers do not have functions -pre-defined to be in correspondance with them): - -\dverb|@ -\xintNewExpr\myformI[2]{ {_xintRound{$1}{$2}} - {_xintTrunc{$1}{$2}} } -\meaning\myformI:| - -\xintNewExpr\myformI[2]{ {_xintRound{$1}{$2}} - {_xintTrunc{$1}{$2}} } -\ttfamily -\centeredline{\meaning\myformI} - -\dverb|@ -\xintNewIIExpr\formula [3]{rem(#1,quo({_the_numexpr $2_relax},#3))} -\meaning\formula:|%$ - -\xintNewIIExpr\formula [3]{rem(#1,quo({_the_numexpr $2_relax},#3))}%$ -\noindent{\meaning\formula}\endgraf - -\rmfamily - -\subsection{\csbh{xintiexpr}, \csbh{xinttheiexpr}} -\label{xintiexpr}\label{xinttheiexpr} -% \label{xintnumexpr}\label{xintthenumexpr} - -Equivalent\etype{x} to doing |\xintexpr round(...)\relax|. Thus, only the final -result is rounded to an integer. Half integers are rounded towards $+\infty$ for -positive numbers and towards $-\infty$ for negative ones. Can be used on comma -separated lists of expressions. - -Initially\MyMarginNote{|1.09i| warning} baptized |\xintnumexpr|, -|\xintthenumexpr| but -I am not too happy about this choice of name; one should keep in mind that -|\numexpr|'s integer division rounds, whereas in |\xintiexpr|, the |/| is an -exact fractional operation, and only the final result is rounded to an integer. - -So |\xintnumexpr|, |\xintthenumexpr| are deprecated, and although still provided -for the time being this might change in the future. - -\subsection{\csbh{xintiiexpr}, \csbh{xinttheiiexpr}} -\label{xintiiexpr}\label{xinttheiiexpr} - -This variant\etype{x} maps |/| to the euclidean quotient and deals almost only -with (long) integers. It uses the `ii' macros for addition, subtraction, -multiplication, power, square, sums, products, euclidean quotient and remainder. -The |round| and |trunc|, in the presence of the second optional argument, are -mapped to \csbxint{iRound}, respectively \csbxint{iTrunc}, hence they always -produce (long) integers. - -To input a fraction to |round|, |trunc|, |floor| or |ceil| one can -use braces, else the |/| will do the euclidean quotient. -The minus sign should be put together with the fraction: |round(-{30/18})| is -illegal (even if the fraction had been an integer), use -|round({-30/18})|\digitstt{=\xinttheiiexpr round({-30/18})\relax}. - -Decimal numbers are allowed only if postfixed immediately with |e| or |E|, the -number will then be truncated to an integer after multiplication by the power of -ten with exponent the number following |e| or |E|. -\centeredline{|\xinttheiiexpr 13.4567e3+10000123e-3\relax|% - \digitstt{=\xinttheiiexpr 13.4567e3+10000123e-3\relax}} -% - -A fraction within braces should be followed immediately by an |e| (or inside a -|round|, |trunc|, etc...) to convert it -into an integer as expected by the main operations. The truncation is only done -after the |e| action. - -The |reduce| function is not available and will raise un error. The |frac| -function also. The |sqrt| function is mapped to \csbxint{iSqrt}. - -Numbers in float notation, obtained via a macro like \csbxint{FloatSqrt}, are a -bit of a challenge: they can not be within braces (this has been mentioned -already, |e| is not legal within braces) and if not braced they will be -truncated when the parser meets the |e|. The way out of the dilemma is to use a -sub-expression: -\centeredline{|\xinttheiiexpr \xintFloatSqrt{2}\relax|% - \digitstt{=\xinttheiiexpr \xintFloatSqrt{2}\relax}} -\centeredline{|\xinttheiiexpr \xintexpr\xintFloatSqrt{2}\relax e10\relax|% - \digitstt{=\xinttheiiexpr \xintexpr\xintFloatSqrt{2}\relax e10\relax}} -\centeredline{|\xinttheiiexpr round(\xintexpr\xintFloatSqrt{2}\relax,10)\relax|% - \digitstt{=\xinttheiiexpr round(\xintexpr\xintFloatSqrt{2}\relax,10)\relax}} -(recall that |round| is mapped within |\xintiiexpr..\relax| to \csbxint{iRound} -which always outputs an integer). - -The whole point of \csbxint{iiexpr} is to gain some speed in integer only -algorithms, and the above explanations related to how to use fractions therein -are a bit peripheral. We observed of the order of @30@\% speed gain when dealing -with numbers with circa one hundred digits, but this gain decreases the longer -the manipulated numbers become and becomes negligible for numbers with thousand -digits: the overhead from parsing fraction format is little compared -to other expensive aspects of the expandable shuffling of tokens. - - -\subsection{\csbh{xintboolexpr}, - \csbh{xinttheboolexpr}}\label{xintboolexpr}\label{xinttheboolexpr} -%{\small New in |1.09c|.\par} - -Equivalent\etype{x} to doing |\xintexpr ...\relax| and returning @1@ if the -result does not vanish, and @0@ is the result is zero. As |\xintexpr|, this -can be used on comma separated lists of expressions, and will return a -comma separated list of @0@'s and @1@'s. - - -\subsection{\csbh{xintfloatexpr}, - \csbh{xintthe\-float\-expr}}\label{xintfloatexpr}\label{xintthefloatexpr} - -\csbxint{floatexpr}|...\relax|\etype{x} is exactly like |\xintexpr...\relax| but -with the four binary operations and the power function mapped to -\csa{xintFloatAdd}, \csa{xintFloatSub}, \csa{xintFloatMul}, \csa{xintFloatDiv} -and \csa{xintFloatPower}. The precision is from the current setting of -|\xintDigits| (it can not be given as an optional parameter). - -Currently, the factorial function hasn't yet a float version; so inside -|\xintthefloatexpr . . . \relax|, |n!| will be computed exactly. Perhaps this -will be improved in a future release. - -\xintDigits:= 9; - -Note that |1.000000001| and |(1+1e-9)| will not be equivalent for -|D=\xinttheDigits| set to nine or less. Indeed the addition implicit in |1+1e-9| -(and executed when the closing parenthesis is found) will provoke the rounding -to |1|. Whereas |1.000000001|, when found as operand of one of the four -elementary operations is kept with |D+2| digits, and even more for the power -function. \centeredline{|\xintDigits:= 9; \xintthefloatexpr - (1+1e-9)-1\relax|\digitstt{=\xintthefloatexpr (1+1e-9)-1\relax}} -\centeredline{|\xintDigits:= 9; \xintthefloatexpr - 1.000000001-1\relax|\digitstt{=\xintthefloatexpr 1.000000001-1\relax}} - -For the fun of it:\xintDigits:=20; |\xintDigits:=20;|% -\centeredline{|\xintthefloatexpr (1+1e-7)^1e7\relax|% - \digitstt{=\xintthefloatexpr (1+1e-7)^1e7\relax}} - -|\xintDigits:=36;|\xintDigits:=36; -\centeredline{|\xintthefloatexpr - ((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax|} -\centeredline{\digitstt{\xintthefloatexpr - ((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax}} -\centeredline{|\xintFloat{\xinttheexpr - ((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax}|} -\centeredline{\digitstt{\xintFloat - {\xinttheexpr((1/13+1/121)*(1/179-1/173))/(1/19-1/18)\relax}}} - -\xintDigits := 16; - -The latter result is the rounding of the exact result. The previous one has -rounding errors coming from the various roundings done for each -sub-expression. It was a bit funny to discover that |maple|, configured with -|Digits:=36;| and with decimal dots everywhere to let it input the numbers as -floats, gives exactly the same result with the same rounding errors -as does |\xintthefloatexpr|! - -Using |\xintthefloatexpr| only pays off compared to using |\xinttheexpr| -followed with |\xintFloat| if the computations turn out to involve hundreds of -digits. For elementary calculations with hand written numbers (not using the -scientific notation with exponents differing greatly) it will generally be more -efficient to use |\xinttheexpr|. The situation is quickly otherwise if one -starts using the Power function. Then, |\xintthefloat| is often useful; and -sometimes indispensable to achieve the (approximate) computation in reasonable -time. - -We can try some crazy things: -% -\centeredline{|\xintDigits:=12;\xintthefloatexpr 1.000000000000001^1e15\relax|} -% -\centeredline{\xintDigits:=12;% - \digitstt{\xintthefloatexpr 1.000000000000001^1e15\relax}} -% -Contrarily to some professional computing sofware which are our concurrents on -this market, the \digitstt{1.000000000000001} wasn't rounded to |1| despite the -setting of \csa{xintDigits}; it would have been if we had input it as -|(1+1e-15)|. - -% \xintDigits:=12; -% \pdfresettimer -% \edef\z{\xintthefloatexpr 1.000000000000001^1e15\relax}% -% \edef\temps{\the\pdfelapsedtime}% -% \xintRound {5}{\temps/65536}s\endgraf - - -\xintDigits := 16; % mais en fait \centeredline crée un groupe. - - -\subsection{\csbh{xintifboolexpr}}\label{xintifboolexpr} -%{\small New in |1.09c|.\par} - -\csh{xintifboolexpr}|{<expr>}{YES}{NO}|\etype{xnn} does |\xinttheexpr -<expr>\relax| and then executes the |YES| or the |NO| branch depending on -whether the outcome was non-zero or zero. |<expr>| can involove various |&| and -\verb+|+, parentheses, |all|, |any|, |xor|, the |bool| or |togl| operators, but -is not limited to them: the most general computation can be done, the test is on -whether the outcome of the computation vanishes or not. - -Will not work on an expression composed of comma separated sub-expressions. - -\subsection{\csbh{xintifboolfloatexpr}}\label{xintifboolfloatexpr} -%{\small New in |1.09c|.\par} - -\csh{xintifboolfloatexpr}|{<expr>}{YES}{NO}|\etype{xnn} does |\xintthefloatexpr -<expr>\relax| and then executes the |YES| or the |NO| branch depending on -whether the outcome was non zero or zero. - -\subsection{\csbh{xintifbooliiexpr}}\label{xintifbooliiexpr} -%{\small New in |1.09i|.\par} - -\csh{xintifbooliiexpr}|{<expr>}{YES}{NO}|\etype{xnn} does |\xinttheiiexpr -<expr>\relax| and then executes the |YES| or the |NO| branch depending on -whether the outcome was non zero or zero. - -\subsection{\csbh{xintNewFloatExpr}}\label{xintNewFloatExpr} - -This is exactly like \csbxint{NewExpr} except that the created formulas are -set-up to use |\xintthefloatexpr|. The precision used for numbers fetched as -parameters will be the one locally given by |\xintDigits| at the time of use of -the created formulas, not |\xintNewFloatExpr|. However, the numbers hard-wired -in the original expression will have been evaluated with the then current -setting for |\xintDigits|. - -\subsection{\csbh{xintNewIExpr}}\label{xintNewIExpr} -%{\small New in |1.09c|.\par } - -Like \csbxint{NewExpr} but using |\xinttheiexpr|. Former denomination was -|\xintNewNumExpr| which is deprecated and should not be used. - -\subsection{\csbh{xintNewIIExpr}}\label{xintNewIIExpr} -%{\small New in |1.09i|.\par } - -Like \csbxint{NewExpr} but using |\xinttheiiexpr|. - -\subsection{\csbh{xintNewBoolExpr}}\label{xintNewBoolExpr} -%{\small New in |1.09c|.\par } - -Like \csbxint{NewExpr} but using |\xinttheboolexpr|. - -\xintDigits:= 16; - -\subsection{Technicalities} - -As already mentioned \csa{xintNewExpr}|\myformula[n]| does not check the prior -existence of a macro |\myformula|. And the number of parameters |n| given as -mandatory argument withing square brackets should be (at least) equal -to the number of parameters in the expression. - -Obviously I should mention that \csa{xintNewExpr} itself can not be used in an -expansion-only context, as it creates a macro. - -The |\escapechar| setting may be arbitrary when using -|\xintexpr|. - -The format of the output of -|\xintexpr|\meta{stuff}|\relax| is a |!| (with catcode 11) followed by -|\XINT_expr_usethe| which prints an error message in the document and in -the log file if it is executed, then a |\xint_protect| token, a token -doing the actual printing and finally a token |\.=A/B[n]|. Using -|\xinttheexpr| means zapping the first three things, the fourth one will -then unlock |A/B[n]| from the (presumably undefined, but it does not -matter) control sequence |\.=A/B[n]|. - -Thanks to the release |1.09j| added |\xint_protect| token and the fact -that |\XINT_expr_usethe| is |\protected|, one can now use |\xintexpr| -inside an |\edef|, with no need of the |\xintthe| prefix. - -\begin{framed} - Note that |\xintexpr| is thus compatible with complete expansion, contrarily - to |\numexpr| which is non-expandable, if not prefixed by |\the| or |\number|, - and away from contexts where \TeX{} is building a number. See - \autoref{ssec:fibonacci} for some illustration. -% -% \MyMarginNote[\kern\dimexpr\FrameSep+\FrameRule\relax]{New with 1.09j!} -\end{framed} - -I decided to put all intermediate results (from each evaluation of an infix -operators, or of a parenthesized subpart of the expression, or from application -of the minus as prefix, or of the exclamation sign as postfix, or any -encountered braced material) inside |\csname...\endcsname|, as this can be done -expandably and encapsulates an arbitrarily long fraction in a single token (left -with undefined meaning), thus providing tremendous relief to the programmer in -his/her expansion control. - -\begin{framed} - As the |\xintexpr| computations corresponding to functions and infix - or postfix operators are done inside |\csname...\endcsname|, the - \fexpan dability could possibly be dropped and one could imagine - implementing the basic operations with expandable but not \fexpan - dable macros (as \csbxint{XTrunc}.) I have not investigated that - possibility. -\end{framed} - -% \begin{framed} -% This implementation and user interface are still to be considered -% \emph{experimental}. -% \end{framed} - -Syntax errors in the input such as using a one-argument function with two -arguments will generate low-level \TeX{} processing unrecoverable errors, with -cryptic accompanying message. - -Some other problems will give rise to `error messages' macros giving some -indication on the location and nature of the problem. Mainly, an attempt has -been made to handle gracefully missing or extraneous parentheses. - -When the scanner is looking for a number and finds something else not otherwise -treated, it assumes it is the start of the function name and will expand forward -in the hope of hitting an opening parenthesis; if none is found at least it -should stop when encountering the |\relax| marking the end of the expressions. - -Note that |\relax| is mandatory (contrarily to a |\numexpr|). - - -\subsection{Acknowledgements} - -I was greatly helped in my preparatory thinking, prior to producing such an -expandable parser, by the commented source of the -\href{http://www.ctan.org/pkg/l3kernel}{l3fp} package, specifically the -|l3fp-parse.dtx| file (in the version of April-May 2013). Also the source of the -|calc| package was instructive, despite the fact that here for |\xintexpr| the -principles are necessarily different due to the aim of achieving expandability. - - -\etocdepthtag.toc {commandsB} - -\section{Commands of the \xintbinhexname package} -\label{sec:binhex} - -This package was first included in the |1.08| release of \xintname. It -provides expandable conversions of arbitrarily long numbers -to and from binary and hexadecimal. - -The argument is first \fexpan ded. It then may start with an optional minus -sign (unique, of category code other), followed with optional leading zeroes -(arbitrarily many, category code other) and then ``digits'' (hexadecimal -letters may be of category code letter or other, and must be -uppercased). The optional (unique) minus sign (plus sign is not allowed) is -kept in the output. Leading zeroes are allowed, and stripped. The -hexadecimal letters on output are of category code letter, and -uppercased. - -% \clearpage - -\localtableofcontents - - - -\subsection{\csbh{xintDecToHex}}\label{xintDecToHex} - -Converts from decimal to hexadecimal.\etype{f} - -\texttt{\string\xintDecToHex \string{\printnumber{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}\string}}\endgraf\noindent\digitstt{->\printnumber{\xintDecToHex{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}}} - -\subsection{\csbh{xintDecToBin}}\label{xintDecToBin} - -Converts from decimal to binary.\etype{f} - -\texttt{\string\xintDecToBin \string{\printnumber{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}\string}}\endgraf\noindent\digitstt{->\printnumber{\xintDecToBin{2718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003}}} - -\subsection{\csbh{xintHexToDec}}\label{xintHexToDec} - -Converts from hexadecimal to decimal.\etype{f} - -\texttt{\string\xintHexToDec - \string{\printnumber{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}\string}}\endgraf\noindent -\digitstt{->\printnumber{\xintHexToDec{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}}} - -\subsection{\csbh{xintBinToDec}}\label{xintBinToDec} - -Converts from binary to decimal.\etype{f} - -\texttt{\string\xintBinToDec - \string{\printnumber{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}\string}}\endgraf\noindent -\digitstt{->\printnumber{\xintBinToDec{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}}} - -\subsection{\csbh{xintBinToHex}}\label{xintBinToHex} - -Converts from binary to hexadecimal.\etype{f} - -\texttt{\string\xintBinToHex - \string{\printnumber{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}\string}}\endgraf\noindent -\digitstt{->\printnumber{\xintBinToHex{100011010100100111001011111000110011010010100100110101001011100000101000111110111110100001010100000010111100100010100111000111110000010110001011111000100000110110001000111000100100010111010111011110010101101010111011000001011101100111000110100100111001011110100011011011100111001000110110001100000001100101001001101101011111100110111110110101100100100011000100000010100110001100011}}} - -\subsection{\csbh{xintHexToBin}}\label{xintHexToBin} - -Converts from hexadecimal to binary.\etype{f} - -\texttt{\string\xintHexToBin - \string{\printnumber{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}\string}}\endgraf\noindent -\digitstt{->\printnumber{\xintHexToBin{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}}} - - -\subsection{\csbh{xintCHexToBin}}\label{xintCHexToBin} - -Also converts from hexadecimal to binary.\etype{f} Faster on inputs with at -least one hundred hexadecimal digits. - -\texttt{\string\xintCHexToBin - \string{\printnumber{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}\string}}\endgraf\noindent -\digitstt{->\printnumber{\xintCHexToBin{11A9397C66949A97051F7D0A817914E3E0B17C41B11C48BAEF2B5760BB38D272F46DCE46C6032936BF37DAC918814C63}}} - - - -\section{Commands of the \xintgcdname package} -\label{sec:gcd} - - -This package was included in the original release |1.0| of the \xintname bundle. - -Since release |1.09a| the macros filter their inputs through the \csbxint{Num} -macro, so one can use count registers, or fractions as long as they reduce to -integers. - -%% \clearpage - -\localtableofcontents - -\subsection{\csbh{xintGCD}}\label{xintGCD} - -\csa{xintGCD\n\m}\etype{\Numf\Numf} computes the greatest common divisor. It is -positive, except when both |N| and |M| vanish, in which case the macro returns -zero. -\centeredline{\csa{xintGCD}|{10000}{1113}|\digitstt{=\xintGCD{10000}{1113}}} -\centeredline{|\xintGCD{123456789012345}{9876543210321}=|\digitstt - {\xintGCD{123456789012345}{9876543210321}}} - -\subsection{\csbh{xintGCDof}}\label{xintGCDof} -%{\small New with release |1.09a|.\par} - -\csa{xintGCDof}|{{a}{b}{c}...}|\etype{f{$\to$}{\lowast\Numf}} computes the greatest common divisor of all -integers |a|, |b|, \dots{} The list argument -may be a macro, it is \fexpan ded first and must contain at least one item. - - -\subsection{\csbh{xintLCM}}\label{xintLCM} -%{\small New with release |1.09a|.\par} - -\csa{xintGCD\n\m}\etype{\Numf\Numf} computes the least common multiple. It is -|0| if one of the two integers vanishes. - -\subsection{\csbh{xintLCMof}}\label{xintLCMof} -%{\small New with release |1.09a|.\par} - -\csa{xintLCMof}|{{a}{b}{c}...}|\etype{f{$\to$}{\lowast\Numf}} computes the least -common multiple of all integers |a|, |b|, \dots{} The list argument may be a -macro, it is \fexpan ded first and must contain at least one item. - -\subsection{\csbh{xintBezout}}\label{xintBezout} - -\xintAssign{{\xintBezout {10000}{1113}}}\to\X -\xintAssign {\xintBezout {10000}{1113}}\to\A\B\U\V\D - -\csa{xintBezout\n\m}\etype{\Numf\Numf} returns five numbers |A|, |B|, |U|, |V|, -|D| within braces. |A| is the first (expanded, as usual) input number, |B| the -second, |D| is the GCD, and \digitstt{UA - VB = D}. \centeredline{|\xintAssign - {{\xintBezout {10000}{1113}}}\to\X|} \centeredline{|\meaning\X: - |\digitstt{\meaning\X }.} -\noindent{|\xintAssign {\xintBezout {10000}{1113}}\to\A\B\U\V\D|}\\ -|\A: |\digitstt{\A }, -|\B: |\digitstt{\B }, -|\U: |\digitstt{\U }, -|\V: |\digitstt{\V }, -|\D: |\digitstt{\D }.\\ -\xintAssign {\xintBezout {123456789012345}{9876543210321}}\to\A\B\U\V\D -\noindent{|\xintAssign {\xintBezout {123456789012345}{9876543210321}}\to\A\B\U\V\D -|}\\ -|\A: |\digitstt{\A }, -|\B: |\digitstt{\B }, -|\U: |\digitstt{\U }, -|\V: |\digitstt{\V }, -|\D: |\digitstt{\D }. - - -\subsection{\csbh{xintEuclideAlgorithm}}\label{xintEuclideAlgorithm} - -\xintAssign {{\xintEuclideAlgorithm {10000}{1113}}}\to\X - -\def\restorebracecatcodes - {\catcode`\{=1 \catcode`\}=2 } - -\def\allowlistsplit - {\catcode`\{=12 \catcode`\}=12 \allowlistsplita } - -\def\allowlistsplitx {\futurelet\listnext\allowlistsplitxx } - -\def\allowlistsplitxx {\ifx\listnext\relax \restorebracecatcodes - \else \expandafter\allowlistsplitxxx \fi } -\begingroup -\catcode`\[=1 -\catcode`\]=2 -\catcode`\{=12 -\catcode`\}=12 -\gdef\allowlistsplita #1{[#1\allowlistsplitx {] -\gdef\allowlistsplitxxx {#1}% - [{#1}\hskip 0pt plus 1pt \allowlistsplitx ] -\endgroup - -\csa{xintEuclideAlgorithm\n\m}\etype{\Numf\Numf} applies the Euclide algorithm -and keeps a copy of all quotients and remainders. \centeredline{|\xintAssign - {{\xintEuclideAlgorithm {10000}{1113}}}\to\X|} - -|\meaning\X: |\digitstt{\expandafter\allowlistsplit - \meaning\X\relax .} - -The first token is the number of steps, the second is |N|, the -third is the GCD, the fourth is |M| then the first quotient and -remainder, the second quotient and remainder, \dots until the -final quotient and last (zero) remainder. - -\subsection{\csbh{xintBezoutAlgorithm}}\label{xintBezoutAlgorithm} - - -\xintAssign {{\xintBezoutAlgorithm {10000}{1113}}}\to\X - -\csa{xintBezoutAlgorithm\n\m}\etype{\Numf\Numf} applies the Euclide algorithm -and keeps a copy of all quotients and remainders. Furthermore it computes the -entries of the successive products of the 2 by 2 matrices -$\left(\vcenter{\halign {\,#&\,#\cr q & 1 \cr 1 & 0 \cr}}\right)$ formed from -the quotients arising in the algorithm. \centeredline{|\xintAssign - {{\xintEuclideAlgorithm {10000}{1113}}}\to\X|} - -|\meaning\X: |\digitstt{\expandafter\allowlistsplit\meaning\X \relax .} - -The first token is the number of steps, the second is |N|, then -|0|, |1|, the GCD, |M|, |1|, |0|, the first quotient, the first -remainder, the top left entry of the first matrix, the bottom left -entry, and then these four things at each step until the end. - - -\subsection{\csbh{xintTypesetEuclideAlgorithm}\texorpdfstring{\allowbreak\null\hspace*{.25cm}}{}}% -\label{xintTypesetEuclideAlgorithm} - -This macro is just an example of how to organize the data returned by -\csa{xintEuclideAlgorithm}.\ntype{\Numf\Numf} Copy the source code to a new -macro and modify it to what is needed. -\centeredline{|\xintTypesetEuclideAlgorithm {123456789012345}{9876543210321}|} -\xintTypesetEuclideAlgorithm {123456789012345}{9876543210321} - - -\subsection{\csbh{xintTypesetBezoutAlgorithm}}% -\label{xintTypesetBezoutAlgorithm} - -This macro is just an example of how to organize the data returned by -\csa{xintBezoutAlgorithm}.\ntype{\Numf\Numf} Copy the source code to a new -macro and modify it to what is needed. -\centeredline{|\xintTypesetBezoutAlgorithm {10000}{1113}|} -\xintTypesetBezoutAlgorithm {10000}{1113} - - -\section{Commands of the \xintseriesname package} -\label{sec:series} - -Some arguments to the package commands are macros which are expanded only later, -when given their parameters. The arguments serving as indices are systematically -given to a |\numexpr| expressions (new with |1.06|!) , hence \fexpan ded, -they may be count registers, etc... - -This package was first released with version |1.03| of the \xintname bundle. - -We use \Ff{} for the expansion type of various macro arguments, but if only -\xintname and not \xintfracname is loaded this should be more appropriately -\Numf. The macro \csbxint{iSeries} is special and expects summing big integers -obeying the strict format, even if \xintfracname is loaded. - -%% \clearpage - -\localtableofcontents - -\subsection{\csbh{xintSeries}}\label{xintSeries} - -\def\coeff #1{\xintiiMON{#1}/#1.5} % (-1)^n/(n+1/2) -\edef\w {\xintSeries {0}{50}{\coeff}} -\edef\z {\xintJrr {\w}[0]} - -\csa{xintSeries}|{A}{B}{\coeff}|\etype{\numx\numx\Ff} computes -$\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|$. The initial and final indices -must obey the |\numexpr| constraint of expanding to numbers at most |2^31-1|. -The |\coeff| macro must be a one-parameter \fexpan dable command, taking on -input an explicit number |n| and producing some number or fraction |\coeff{n}|; -it is expanded at the time it is -needed.\footnote{\label{fn:xintiiMON}\csbxint{iiMON} is like \csbxint{MON} but - does not parse its argument through \csbxint{Num}, for efficiency; other - macros of this type are \csbxint{iiAdd}, \csbxint{iiMul}, - \csbxint{iiSum}, \csbxint{iiPrd}, \csbxint{iiMMON}, - \csbxint{iiLDg}, \csbxint{iiFDg}, \csbxint{iiOdd}, \dots} -% -\dverb|@ -\def\coeff #1{\xintiiMON{#1}/#1.5} % (-1)^n/(n+1/2) -\edef\w {\xintSeries {0}{50}{\coeff}} % we want to re-use it -\edef\z {\xintJrr {\w}[0]} % the [0] for a microsecond gain. -% \xintJrr preferred to \xintIrr: a big common factor is suspected. -% But numbers much bigger would be needed to show the greater efficiency. -\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} = \xintFrac\z \]| -\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} = \xintFrac\z \] -For info, -before action by |\xintJrr| the inner representation of the result has a -denominator of |\xintLen {\xintDenominator\w}=|\xintLen -{\xintDenominator\w} digits. This troubled me as @101!!@ has only 81 -digits: |\xintLen {\xintQuo {\xintFac {101}}{\xintiMul {\xintiPow - {2}{50}}{\xintFac{50}}}}|\digitstt{=\xintLen {\xintQuo {\xintFac - {101}}{\xintiMul {\xintiPow {2}{50}}{\xintFac{50}}}}}. The -explanation lies in the too clever to be efficient |#1.5| trick. It -leads to a silly extra @5^{51}@ (which has \xintLen {\xintPow {5}{51}} -digits) in the denominator. See the explanations in the next section. - -\begin{framed} - Note: as soon as the coefficients look like factorials, it is more - efficient to use the \csbxint{RationalSeries} macro whose evaluation - will avoid a denominator build-up; indeed the raw operations of - addition and subtraction of fractions blindly multiply out - denominators. So the raw evaluation of $\sum_{n=0}^{|N|}1/n!$ with - \csa{xintSeries} will have a denominator equal to $\prod_{n=0}^{|N|} - n!$. Needless to say this makes it more difficult to compute the exact - value of this sum with |N=50|, for example, whereas with - \csbxint{RationalSeries} the denominator does not - get bigger than $50!$. - -\footnotesize - For info: by the way $\prod_{n=0}^{50} n!$ is easily computed by \xintname - and is a number with 1394 digits. And $\prod_{n=0}^{100} n!$ is also - computable by \xintname (24 seconds on my laptop for the brute force - iterated multiplication of all factorials, a - specialized routine would do it faster) and has 6941 digits (this - means more than two pages if printed...). Whereas $100!$ only has - 158 digits. -\end{framed} - -% \newcount\cntb -% \cnta 2 -% \loop -% \advance\cntb by \xintLen{\xintFac{\the\cnta}}% -% \ifnum\cnta < 50 -% \advance\cnta 1 -% \repeat -% \the\cntb - -% \cnta 2 -% \def\z{1} -% \pdfresettimer -% \loop -% \edef\z {\xintiMul\z{\xintFac{\the\cnta}}}% -% \ifnum\cnta < 100 -% \advance\cnta 1 -% \repeat -% \edef\temps{\the\pdfelapsedtime}% - -% \temps: \xintQuo\temps{\xintiMul{60}{65536}} minutes, -% \xintQuo{\xintRem\temps{\xintiMul{60}{65536}}}{65536} secondes et -% \xintiTrunc {2}{\xintRem\temps{65536}/65536} centièmes de secondes -% 1573518: 0 minutes, 24 secondes et 0 centièmes de secondes -% nota bene, marrant c'était 0,99 centièmes en fait. - -% \xintLen\z - -% \printnumber\z - -\setlength{\columnsep}{0pt} -\dverb|@ -\def\coeffleibnitz #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]} -\cnta 1 -\loop % in this loop we recompute from scratch each partial sum! -% we can afford that, as \xintSeries is fast enough. -\noindent\hbox to 2em{\hfil\texttt{\the\cnta.} }% - \xintTrunc {12} - {\xintSeries {1}{\cnta}{\coeffleibnitz}}\dots -\endgraf -\ifnum\cnta < 30 \advance\cnta 1 \repeat| -\begin{multicols}{3} - \def\coeffleibnitz #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]} \cnta 1 - \loop - \noindent\hbox to 2em{\hfil\digitstt{\the\cnta.} }% - \xintTrunc {12}{\xintSeries {1}{\cnta}{\coeffleibnitz}}\dots - \endgraf - \ifnum\cnta < 30 \advance\cnta 1 \repeat -\end{multicols} - -\subsection{\csbh{xintiSeries}}\label{xintiSeries} - -\def\coeff #1{\xintiTrunc {40} - {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}}% - - \csa{xintiSeries}|{A}{B}{\coeff}|\etype{\numx\numx f} computes - $\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|$ where |\coeff{n}| - must \fexpan d to a (possibly long) integer in the strict format. -\dverb|@ -\def\coeff #1{\xintiTrunc {40}{\xintMON{#1}/#1.5}}% -% better: -\def\coeff #1{\xintiTrunc {40} - {\the\numexpr 2*\xintiiMON{#1}\relax/\the\numexpr 2*#1+1\relax [0]}}% -% better still: -\def\coeff #1{\xintiTrunc {40} - {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}}% -% (-1)^n/(n+1/2) times 10^40, truncated to an integer. -\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx - \xintTrunc {40}{\xintiSeries {0}{50}{\coeff}[-40]}\dots\]| - -The |#1.5| trick to define the |\coeff| macro was neat, but |1/3.5|, for -example, turns internally into |10/35| whereas it would be more efficient to -have |2/7|. The second way of coding the wanted coefficient avoids a superfluous -factor of five and leads to a faster evaluation. The third way is faster, after -all there is no need to use \csbxint{MON} (or rather -\hyperref[fn:xintiiMON]{\csa{xintiiMON}} which has -less parsing overhead) on integers -obeying the \TeX{} bound. The denominator having no sign, we have added the -|[0]| as this speeds up (infinitesimally) the parsing. -\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx \xintTrunc -{40}{\xintiSeries {0}{50}{\coeff}[-40]}\] We should have cut out at -least the last two digits: truncating errors originating with the first -coefficients of the sum will never go away, and each truncation -introduces an uncertainty in the last digit, so as we have 40 terms, we -should trash the last two digits, or at least round at 38 digits. It is -interesting to compare with the computation where rounding rather than -truncation is used, and with the decimal -expansion of the exactly computed partial sum of the series: -\dverb|@ -\def\coeff #1{\xintiRound {40} % rounding at 40 - {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}}% -% (-1)^n/(n+1/2) times 10^40, rounded to an integer. -\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx - \xintTrunc {40}{\xintiSeries {0}{50}{\coeff}[-40]}\] -\def\exactcoeff #1% - {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}% -\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} - = \xintTrunc {50}{\xintSeries {0}{50}{\exactcoeff}}\dots\]| - -\def\coeff #1{\xintiRound {40} - {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}}% -% (-1)^n/(n+1/2) times 10^40, rounded to an integer. -\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx - \xintTrunc {40}{\xintiSeries {0}{50}{\coeff}[-40]}\] -\def\exactcoeff #1% - {\the\numexpr\ifodd #1 -2\else2\fi\relax/\the\numexpr 2*#1+1\relax [0]}% -\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} - = \xintTrunc {50}{\xintSeries {0}{50}{\exactcoeff}}\dots\] -This shows indeed that our sum of truncated terms -estimated wrongly the 39th and 40th digits of the exact result\footnote{as - the series - is alternating, we can roughly expect an error of $\sqrt{40}$ and the - last two digits are off by 4 units, which is not contradictory to our - expectations.} and that the sum of rounded terms fared a bit better. - -\subsection{\csbh{xintRationalSeries}}\label{xintRationalSeries} - -%{\small \hspace*{\parindent}New with release |1.04|.\par} - -\noindent \csa{xintRationalSeries}|{A}{B}{f}{\ratio}|\etype{\numx\numx\Ff\Ff} -evaluates $\sum_{\text{|n=A|}}^{\text{|n=B|}}|F(n)|$, where |F(n)| is specified -indirectly via the data of |f=F(A)| and the one-parameter macro |\ratio| which -must be such that |\macro{n}| expands to |F(n)/F(n-1)|. The name indicates that -\csa{xintRationalSeries} was designed to be useful in the cases where -|F(n)/F(n-1)| is a rational function of |n| but it may be anything expanding to -a fraction. The macro |\ratio| must be an expandable-only compatible command and -expand to its value after iterated full expansion of its first token. |A| and -|B| are fed to a |\numexpr| hence may be count registers or arithmetic -expressions built with such; they must obey the \TeX{} bound. The initial term -|f| may be a macro |\f|, it will be expanded to its value representing |F(A)|. - -\dverb|@ -\def\ratio #1{2/#1[0]}% 2/n, to compute exp(2) -\cnta 0 % previously declared count -\loop \edef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}% -\noindent$\sum_{n=0}^{\the\cnta} \frac{2^n}{n!}= - \xintTrunc{12}\z\dots= - \xintFrac\z=\xintFrac{\xintIrr\z}$\vtop to 5pt{}\endgraf -\ifnum\cnta<20 \advance\cnta 1 \repeat| - -\def\ratio #1{2/#1[0]}% 2/n, comes from the series of exp(2) -\cnta 0 -\loop -\edef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}% -\noindent$\sum_{n=0}^{\the\cnta} \frac{2^n}{n!}= - \xintTrunc{12}\z\dots= - \xintFrac\z=\xintFrac{\xintIrr\z}$\vtop to 5pt{}\endgraf -\ifnum\cnta<20 \advance\cnta 1 \repeat - -\medskip -Such computations would become quickly completely inaccessible via the -\csbxint{Series} macros, as the factorials in the denominators would get -all multiplied together: the raw addition and subtraction on fractions -just blindly multiplies denominators! Whereas \csa{xintRationalSeries} -evaluate the partial sums via a less silly iterative scheme. -\dverb|@ -\def\ratio #1{-1/#1[0]}% -1/n, comes from the series of exp(-1) -\cnta 0 % previously declared count -\loop -\edef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}% -\noindent$\sum_{n=0}^{\the\cnta} \frac{(-1)^n}{n!}= - \xintTrunc{20}\z\dots=\xintFrac{\z}=\xintFrac{\xintIrr\z}$ - \vtop to 5pt{}\endgraf -\ifnum\cnta<20 \advance\cnta 1 \repeat| - -\def\ratio #1{-1/#1[0]}% -1/n, comes from the series of exp(-1) -\cnta 0 % previously declared count - -\loop -\edef\z {\xintRationalSeries {0}{\cnta}{1}{\ratio }}% -\noindent$\sum_{n=0}^{\the\cnta} \frac{(-1)^n}{n!}= - \xintTrunc{20}\z\dots=\xintFrac{\z}=\xintFrac{\xintIrr\z}$ - \vtop to 5pt{}\endgraf -\ifnum\cnta<20 \advance\cnta 1 \repeat - - - \def\ratioexp #1#2{\xintDiv{#1}{#2}}% #1/#2 - -\medskip We can incorporate an indeterminate if we define |\ratio| to be -a macro with two parameters: |\def\ratioexp - #1#2{\xintDiv{#1}{#2}}|\texttt{\%}| x/n: x=#1, n=#2|. -Then, if |\x| expands to some fraction |x|, the -command \centeredline{|\xintRationalSeries {0}{b}{1}{\ratioexp{\x}}|} -will compute $\sum_{n=0}^{n=b} x^n/n!$:\par -\dverb|@ -\cnta 0 -\def\ratioexp #1#2{\xintDiv{#1}{#2}}% #1/#2 -\loop -\noindent -$\sum_{n=0}^{\the\cnta} (.57)^n/n! = \xintTrunc {50} - {\xintRationalSeries {0}{\cnta}{1}{\ratioexp{.57}}}\dots$ - \vtop to 5pt {}\endgraf -\ifnum\cnta<50 \advance\cnta 10 \repeat| - -\cnta 0 -\loop -\noindent -$\sum_{n=0}^{\the\cnta} (.57)^n/n! = \xintTrunc {50} - {\xintRationalSeries {0}{\cnta}{1}{\ratioexp{.57}}}\dots$ - \vtop to 5pt {}\endgraf -\ifnum\cnta<50 \advance\cnta 10 \repeat -Observe that in this last example the |x| was directly inserted; if it -had been a more complicated explicit fraction it would have been -worthwile to use |\ratioexp\x| with |\x| defined to expand to its value. -In the further situation where this fraction |x| is not explicit but -itself defined via a complicated, and time-costly, formula, it should be -noted that \csa{xintRationalSeries} will do again the evaluation of |\x| -for each term of the partial sum. The easiest is thus when |x| can be -defined as an |\edef|. If however, you are in an expandable-only context -and cannot store in a macro like |\x| the value to be used, a variant of -\csa{xintRationalSeries} is needed which will first evaluate this |\x| and then -use this result without recomputing it. This is \csbxint{RationalSeriesX}, -documented next. - -Here is a slightly more complicated evaluation: -\dverb|@ -\cnta 1 -\loop \edef\z {\xintRationalSeries - {\cnta} - {2*\cnta-1} - {\xintiPow {\the\cnta}{\cnta}/\xintFac{\cnta}} - {\ratioexp{\the\cnta}}}% -\edef\w {\xintRationalSeries {0}{2*\cnta-1}{1}{\ratioexp{\the\cnta}}}% -\noindent -$\sum_{n=\the\cnta}^{\the\numexpr 2*\cnta-1\relax} \frac{\the\cnta^n}{n!}/% - \sum_{n=0}^{\the\numexpr 2*\cnta-1\relax} \frac{\the\cnta^n}{n!} = - \xintTrunc{8}{\xintDiv\z\w}\dots$ \vtop to 5pt{}\endgraf -\ifnum\cnta<20 \advance\cnta 1 \repeat| - -\cnta 1 -\begin{multicols}{2} -\loop \edef\z {\xintRationalSeries - {\cnta} - {2*\cnta-1} - {\xintiPow {\the\cnta}{\cnta}/\xintFac{\cnta}} - {\ratioexp{\the\cnta}}}% -\edef\w {\xintRationalSeries {0}{2*\cnta-1}{1}{\ratioexp{\the\cnta}}}% -\noindent$\sum_{n=\the\cnta}^{\the\numexpr 2*\cnta-1\relax} \frac{\the\cnta^n}{n!}/% - \sum_{n=0}^{\the\numexpr 2*\cnta-1\relax} \frac{\the\cnta^n}{n!} = - \xintTrunc{8}{\xintDiv\z\w}\dots$ \vtop to 5pt{}\endgraf -\ifnum\cnta<20 \advance\cnta 1 \repeat -\end{multicols} - -\subsection{\csbh{xintRationalSeriesX}}\label{xintRationalSeriesX} - -%{\small \hspace*{\parindent}New with release |1.04|.\par} - -\noindent\csa{xintRationalSeriesX}|{A}{B}{\first}{\ratio}{\g}|% -\etype{\numx\numx\Ff\Ff f} is a parametrized version of \csa{xintRationalSeries} -where |\first| is now a one-parameter macro such that |\first{\g}| gives the -initial term and |\ratio| is a two-parameter macro such that |\ratio{n}{\g}| -represents the ratio of one term to the previous one. The parameter |\g| is -evaluated only once at the beginning of the computation, and can thus itself be -the yet unevaluated result of a previous computation. - -Let |\ratio| be such a two-parameter macro; note the subtle differences -between\centeredline{|\xintRationalSeries {A}{B}{\first}{\ratio{\g}}|} -\centeredline{and |\xintRationalSeriesX {A}{B}{\first}{\ratio}{\g}|.} First the -location of braces differ... then, in the former case |\first| is a -\emph{no-parameter} macro expanding to a fractional number, and in the latter, -it is a -\emph{one-parameter} macro which will use |\g|. Furthermore the |X| variant -will expand |\g| at the very beginning whereas the former non-|X| former variant -will evaluate it each time it needs it (which is bad if this -evaluation is time-costly, but good if |\g| is a big explicit fraction -encapsulated in a macro). - - -The example will use the macro \csbxint{PowerSeries} which computes -efficiently exact partial sums of power series, and is discussed in the -next section. -\dverb|@ -\def\firstterm #1{1[0]}% first term of the exponential series -% although it is the constant 1, here it must be defined as a -% one-parameter macro. Next comes the ratio function for exp: -\def\ratioexp #1#2{\xintDiv {#1}{#2}}% x/n -% These are the (-1)^{n-1}/n of the log(1+h) series: -\def\coefflog #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]}% -% Let L(h) be the first 10 terms of the log(1+h) series and -% let E(t) be the first 10 terms of the exp(t) series. -% The following computes E(L(a/10)) for a=1,...,12. -\cnta 0 -\loop -\noindent\xintTrunc {18}{% - \xintRationalSeriesX {0}{9}{\firstterm}{\ratioexp} - {\xintPowerSeries{1}{10}{\coefflog}{\the\cnta[-1]}}}\dots -\endgraf -\ifnum\cnta < 12 \advance \cnta 1 \repeat| - -\def\firstterm #1{1[0]}% first term of the exponential series -% although it is the constant 1, here it must be defined as a -% one-parameter macro. Next comes the ratio function for exp: -\def\ratioexp #1#2{\xintDiv {#1}{#2}}% x/n -% These are the (-1)^{n-1}/n of the log(1+h) series -\def\coefflog #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]}% -% Let L(h) be the first 10 terms of the log(1+h) series and -% let E(t) be the first 10 terms of the exp(t) series. -% The following computes E(L(a/12)) for a=1,..., 12. -\begin{multicols}{3}\raggedcolumns - \cnta 1 - \loop - \noindent\xintTrunc {18}{% - \xintRationalSeriesX {0}{9}{\firstterm}{\ratioexp} - {\xintPowerSeries{1}{10}{\coefflog}{\the\cnta[-1]}}}\dots - \endgraf - \ifnum\cnta < 12 \advance \cnta 1 \repeat -\end{multicols} - % to see how they look like... - % \loop - % \noindent\printnumber{% - % \xintRationalSeriesX {0}{9}{\firstterm}{\ratioexp} - % {\xintPowerSeries{1}{10}{\coefflog}{\the\cnta[-2]}}}\dots - % \endgraf - % \ifnum\cnta < 60 \advance \cnta 1 \repeat - -These completely exact operations rapidly create numbers with many digits. Let -us print in full the raw fractions created by the operation illustrated above: - -\edef\z{\xintRationalSeriesX {0}{9}{\firstterm} -{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1[-1]}}} - -|E(L(1[-1]))=|\digitstt{\printnumber{\z}} (length of numerator: -\xintLen {\xintNumerator \z}) - -\edef\z{\xintRationalSeriesX {0}{9}{\firstterm} -{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{12[-2]}}} - -|E(L(12[-2]))=|\digitstt{\printnumber{\z}} (length of numerator: -\xintLen {\xintNumerator \z}) - -\edef\z{\xintRationalSeriesX {0}{9}{\firstterm} -{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{123[-3]}}} - -|E(L(123[-3]))=|\digitstt{\printnumber{\z}} (length of numerator: -\xintLen {\xintNumerator \z}) - - -We see that the denominators here remain the same, as our input only had various -powers of ten as denominators, and \xintfracname efficiently assemble (some -only, as we can see) powers of ten. Notice that 1 more digit in an input -denominator seems to mean 90 more in the raw output. We can check that with some -other test cases: - - -\edef\z{\xintRationalSeriesX {0}{9}{\firstterm} -{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/7}}} - -|E(L(1/7))=|\digitstt{\printnumber{\z}} (length of numerator: -\xintLen {\xintNumerator \z}; length of denominator: -\xintLen {\xintDenominator \z}) - -\edef\z{\xintRationalSeriesX {0}{9}{\firstterm} -{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/71}}} - -|E(L(1/71))=|\digitstt{\printnumber{\z}} (length of numerator: -\xintLen {\xintNumerator \z}; length of denominator: -\xintLen {\xintDenominator \z}) - - -\edef\z{\xintRationalSeriesX {0}{9}{\firstterm} -{\ratioexp}{\xintPowerSeries{1}{10}{\coefflog}{1/712}}} - -|E(L(1/712))=|\digitstt{\printnumber{\z}} (length of numerator: -\xintLen {\xintNumerator \z}; length of denominator: -\xintLen {\xintDenominator \z}) - -% \pdfresettimer -% \edef\w{\xintDenominator{\xintIrr{\z}}} -% \the\pdfelapsedtime - -For info the last fraction put into irreducible form still has 288 digits in its -denominator.\footnote{putting this fraction in irreducible form takes more time - than is typical of the other computations in this document; so exceptionally I - have hard-coded the 288 in the document source.} Thus -decimal numbers such as |0.123| (equivalently -|123[-3]|) give less computing intensive tasks than fractions such as |1/712|: -in the case of decimal numbers the (raw) denominators originate in the -coefficients of the series themselves, powers of ten of the input within -brackets being treated separately. And even then the -numerators will grow with the size of the input in a sort of linear way, the -coefficient being given by the order of series: here 10 from the log and 9 from -the exp, so 90. One more digit in the input means 90 more digits in the -numerator of the output: obviously we can not go on composing such partial sums -of series and hope that \xintname will joyfully do all at the speed of light! -Briefly said, imagine that the rules of the game make the programmer like a -security guard at an airport scanning machine: a never-ending flux of passengers -keep on arriving and all you can do is re-shuffle the first nine of them, -organize marriages among some, execute some, move children farther back among -the first nine only. If a passenger comes along with many hand luggages, this -will slow down the process even if you move him to ninth position, because -sooner or later you will have to digest him, and the children will be big too. -There is no way to move some guy out of the file and to a discrete interrogatory -room for separate treatment or to give him/her some badge saying ``I left my -stuff in storage box 357''. - -Hence, truncating the output (or better, rounding) is the only way to go if one -needs a general calculus of special functions. This is why the package -\xintseriesname provides, besides \csbxint{Series}, \csbxint{RationalSeries}, or -\csbxint{PowerSeries} which compute \emph{exact} sums, also has -\csbxint{FxPtPowerSeries} for fixed-point computations. - -Update: release |1.08a| of \xintseriesname now includes a tentative naive -\csbxint{FloatPowerSeries}. - -\subsection{\csbh{xintPowerSeries}}\label{xintPowerSeries} - -\csa{xintPowerSeries}|{A}{B}{\coeff}{f}|\etype{\numx\numx\Ff\Ff} -evaluates the sum -$\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|\cdot |f|^{\text{|n|}}$. The -initial and final indices are given to a |\numexpr| expression. The |\coeff| -macro (which, as argument to \csa{xintPowerSeries} is expanded only at the time -|\coeff{n}| is needed) should be defined as a one-parameter expandable command, -its input will be an explicit number. - -The |f| can be either a fraction directly input or a macro |\f| expanding to -such a fraction. It is actually more efficient to encapsulate an explicit -fraction |f| in such a macro, if it has big numerators and denominators (`big' -means hundreds of digits) as it will then take less space in the processing -until being (repeatedly) used. - -This macro computes the \emph{exact} result (one can use it also for polynomial -evaluation). Starting with release |1.04| a Horner scheme for polynomial -evaluation is used, which has the advantage to avoid a denominator build-up -which was plaguing the |1.03| version. \footnote{with powers |f\string^k|, from - |k=0| to |N|, a denominator |d| of |f| became - |d\string^\string{1+2+\dots+N\string}|, which is bad. With the |1.04| method, - the part of the denominator originating from |f| does not accumulate to more - than |d\string^N|. } - -\begin{framed} - Note: as soon as the coefficients look like factorials, it is more efficient - to use the \csbxint{RationalSeries} macro whose evaluation, also based on a - similar Horner scheme, will avoid a denominator build-up originating in the - coefficients themselves. -\end{framed} - -\dverb|@ -\def\geom #1{1[0]} % the geometric series -\def\f {5/17[0]} -\[ \sum_{n=0}^{n=20} \Bigl(\frac 5{17}\Bigr)^n - =\xintFrac{\xintIrr{\xintPowerSeries {0}{20}{\geom}{\f}}} - =\xintFrac{\xinttheexpr (17^21-5^21)/12/17^20\relax}\]|% -\def\geom #1{1[0]} % the geometric series -\def\f {5/17[0]} % -\[ \sum_{n=0}^{n=20} \Bigl(\frac 5{17}\Bigr)^n - =\xintFrac{\xintIrr{\xintPowerSeries {0}{20}{\geom}{\f}}} - =\xintFrac{\xinttheexpr (17^21-5^21)/12/17^20\relax}\] - -\dverb|@ -\def\coefflog #1{1/#1[0]}% 1/n -\def\f {1/2[0]}% -\[ \log 2 \approx \sum_{n=1}^{20} \frac1{n\cdot 2^n} - = \xintFrac {\xintIrr {\xintPowerSeries {1}{20}{\coefflog}{\f}}}\] -\[ \log 2 \approx \sum_{n=1}^{50} \frac1{n\cdot 2^n} - = \xintFrac {\xintIrr {\xintPowerSeries {1}{50}{\coefflog}{\f}}}\]|% -\def\coefflog #1{1/#1[0]} % 1/n -\def\f {1/2[0]}% -\[ \log 2 \approx \sum_{n=1}^{20} \frac1{n\cdot 2^n} - = \xintFrac {\xintIrr {\xintPowerSeries - {1}{20}{\coefflog}{\f}}}\] -\[ \log 2 \approx \sum_{n=1}^{50} \frac1{n\cdot 2^n} - = \xintFrac {\xintIrr {\xintPowerSeries {1}{50}{\coefflog}{\f}}}\] -\dverb|@ -\cnta 1 % previously declared count -\loop % in this loop we recompute from scratch each partial sum! -% we can afford that, as \xintPowerSeries is fast enough. -\noindent\hbox to 2em{\hfil\texttt{\the\cnta.} }% - \xintTrunc {12} - {\xintPowerSeries {1}{\cnta}{\coefflog}{\f}}\dots -\endgraf -\ifnum \cnta < 30 \advance\cnta 1 \repeat| -\setlength{\columnsep}{0pt} -\begin{multicols}{3} - \cnta 1 % previously declared count - \loop % in this loop we recompute from scratch each partial sum! -% we can afford that, as \xintPowerSeries is fast enough. -\noindent\hbox to 2em{\hfil\texttt{\the\cnta.} }% - \xintTrunc {12}{\xintPowerSeries {1}{\cnta}{\coefflog}{\f}}\dots -\endgraf -\ifnum \cnta < 30 \advance\cnta 1 \repeat -\end{multicols} -\dverb|@ -%\def\coeffarctg #1{1/\the\numexpr\xintMON{#1}*(2*#1+1)\relax }% -\def\coeffarctg #1{1/\the\numexpr\ifodd #1 -2*#1-1\else2*#1+1\fi\relax }% -% the above gives (-1)^n/(2n+1). The sign being in the denominator, -% **** no [0] should be added ****, -% else nothing is guaranteed to work (even if it could by sheer luck) -% NOTE in passing this aspect of \numexpr: -% **** \numexpr -(1)\relax does not work!!! **** -\def\f {1/25[0]}% 1/5^2 -\[\mathrm{Arctg}(\frac15)\approx - \frac15\sum_{n=0}^{15} \frac{(-1)^n}{(2n+1)25^n} -= \xintFrac{\xintIrr {\xintDiv - {\xintPowerSeries {0}{15}{\coeffarctg}{\f}}{5}}}\]| - -\def\coeffarctg #1{1/\the\numexpr\ifodd #1 -2*#1-1\else2*#1+1\fi\relax }% -\def\f {1/25[0]}% 1/5^2 -\[\mathrm{Arctg}(\frac15)\approx - \frac15\sum_{n=0}^{15} \frac{(-1)^n}{(2n+1)25^n} -= \xintFrac{\xintIrr {\xintDiv - {\xintPowerSeries {0}{15}{\coeffarctg}{\f}}{5}}}\] - -\subsection{\csbh{xintPowerSeriesX}}\label{xintPowerSeriesX} - -%{\small\hspace*{\parindent}New with release |1.04|.\par} - -\noindent This is the same as \csbxint{PowerSeries}\ntype{\numx\numx\Ff\Ff} -apart -from the fact that the last parameter |f| is expanded once and for all before -being then used repeatedly. If the |f| parameter is to be an explicit big -fraction with many (dozens) digits, rather than using it directly it is slightly -better to have some macro |\g| defined to expand to the explicit fraction and -then use \csbxint{PowerSeries} with |\g|; but if |f| has not yet been evaluated -and will be the output of a complicated expansion of some |\f|, and if, due to -an expanding only context, doing |\edef\g{\f}| is no option, then -\csa{xintPowerSeriesX} should be used with |\f| as last parameter. -% -\dverb|@ -\def\ratioexp #1#2{\xintDiv {#1}{#2}}% x/n -% These are the (-1)^{n-1}/n of the log(1+h) series: -\def\coefflog #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]}% -% Let L(h) be the first 10 terms of the log(1+h) series and -% let E(t) be the first 10 terms of the exp(t) series. -% The following computes L(E(a/10)-1) for a=1,..., 12. -\cnta 1 -\loop -\noindent\xintTrunc {18}{% - \xintPowerSeriesX {1}{10}{\coefflog} - {\xintSub - {\xintRationalSeries {0}{9}{1[0]}{\ratioexp{\the\cnta[-1]}}} - {1}}}\dots -\endgraf -\ifnum\cnta < 12 \advance \cnta 1 \repeat| - -\cnta 0 -\def\ratioexp #1#2{\xintDiv {#1}{#2}}% x/n -% These are the (-1)^{n-1}/n of the log(1+h) series -\def\coefflog #1{\the\numexpr\ifodd #1 1\else-1\fi\relax/#1[0]}% -% Let L(h) be the first 10 terms of the log(1+h) series and -% let E(t) be the first 10 terms of the exp(t) series. -% The following computes L(E(a/10)-1) for a=1,..., 12. -\begin{multicols}{3}\raggedcolumns -\cnta 1 - \loop - \noindent\xintTrunc {18}{% - \xintPowerSeriesX {1}{10}{\coefflog} - {\xintSub - {\xintRationalSeries {0}{9}{1[0]}{\ratioexp{\the\cnta[-1]}}} - {1}}}\dots - \endgraf - \ifnum\cnta < 12 \advance \cnta 1 \repeat -\end{multicols} - - -\subsection{\csbh{xintFxPtPowerSeries}}\label{xintFxPtPowerSeries} - -\csa{xintFxPtPowerSeries}|{A}{B}{\coeff}{f}{D}|\etype{\numx\numx} -computes -$\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|\cdot |f|^{\,\text{|n|}}$ with each - term of the series truncated to |D| digits\etype{\Ff\Ff\numx} - after the decimal point. As - usual, |A| and |B| are completely expanded through their inclusion in a - |\numexpr| expression. Regarding |D| it will be similarly be expanded each - time it is used inside an \csa{xintTrunc}. The one-parameter macro |\coeff| - is similarly expanded at the time it is used inside the - computations. Idem for |f|. If |f| itself is some complicated macro it is - thus better to use the variant \csbxint{FxPtPowerSeriesX} which expands it - first and then uses the result of that expansion. - -The current (|1.04|) implementation is: the first power |f^A| is -computed exactly, then \emph{truncated}. Then each successive power is -obtained from the previous one by multiplication by the exact value of -|f|, and truncated. And |\coeff{n}|\raisebox{.5ex}{|.|}|f^n| is obtained -from that by multiplying by |\coeff{n}| (untruncated) and then -truncating. Finally the sum is computed exactly. Apart from that -\csa{xintFxPtPowerSeries} (where |FxPt| means `fixed-point') is like -\csa{xintPowerSeries}. - -There should be a variant for things of the type $\sum c_n \frac {f^n}{n!}$ to -avoid having to compute the factorial from scratch at each coefficient, the same -way \csa{xintFxPtPowerSeries} does not compute |f^n| from scratch at each |n|. -Perhaps in the next package release. - -\def\coeffexp #1{1/\xintFac {#1}[0]}% [0] for faster parsing -\def\f {-1/2[0]}% -\newcount\cnta - -\setlength{\multicolsep}{0pt} - -\begin{multicols}{3}[% -\centeredline{$e^{-\frac12}\approx{}$}]% -\cnta 0 -\noindent\loop -$\xintFxPtPowerSeries {0}{\cnta}{\coeffexp}{\f}{20}$\\ -\ifnum\cnta<19 -\advance\cnta 1 -\repeat\par -\end{multicols} -\dverb|@ -\def\coeffexp #1{1/\xintFac {#1}[0]}% 1/n! -\def\f {-1/2[0]}% [0] for faster input parsing -\cnta 0 % previously declared \count register -\noindent\loop -$\xintFxPtPowerSeries {0}{\cnta}{\coeffexp}{\f}{20}$\\ -\ifnum\cnta<19 \advance\cnta 1 \repeat\par -% One should **not** trust the final digits, as the potential truncation -% errors of up to 10^{-20} per term accumulate and never disappear! (the -% effect is attenuated by the alternating signs in the series). We can -% confirm that the last two digits (of our evaluation of the nineteenth -% partial sum) are wrong via the evaluation with more digits: | - -\centeredline{|\xintFxPtPowerSeries {0}{19}{\coeffexp}{\f}{25}=| -\digitstt{\xintFxPtPowerSeries {0}{19}{\coeffexp}{\f}{25}}} -\edef\z{\xintIrr {\xintPowerSeries {0}{19}{\coeffexp}{\f}}}% - -\texttt{\hyphenchar\font45 }% -It is no difficulty for \xintfracname to compute exactly, with the help -of \csa{xintPowerSeries}, the nineteenth partial sum, and to then give -(the start of) its exact decimal expansion: -\centeredline{|\xintPowerSeries {0}{19}{\coeffexp}{\f}| ${}= - \displaystyle\xintFrac{\z}$% - \vphantom{\vrule height 20pt depth 12pt}}% -\centeredline{${}=\xintTrunc {30}{\z}\dots$} Thus, one should always -estimate a priori how many ending digits are not reliable: if there are -|N| terms and |N| has |k| digits, then digits up to but excluding the -last |k| may usually be trusted. If we are optimistic and the series is -alternating we may even replace |N| with $\sqrt{|N|}$ to get the number |k| -of digits possibly of dubious significance. - - -\subsection{\csbh{xintFxPtPowerSeriesX}}\label{xintFxPtPowerSeriesX} - -%{\small\hspace*{\parindent}New with release |1.04|.\par} - -\noindent\csa{xintFxPtPowerSeriesX}|{A}{B}{\coeff}{\f}{D}|% -\ntype{\numx\numx} -computes, exactly as -\csa{xintFxPtPowerSeries}, the sum of -|\coeff{n}|\raisebox{.5ex}{|.|}|\f^n|\etype{\Ff\Ff\numx} from |n=A| to |n=B| with each term -of the series being \emph{truncated} to |D| digits after the decimal -point. The sole difference is that |\f| is first expanded and it -is the result of this which is used in the computations. - -% Let us illustrate this on the computation of |(1+y)^{5/3}| where -% |1+y=(1+x)^{3/5}| and each of the two binomial series is evaluated with ten -% terms, the results being computed with |8| digits after the decimal point, and -% @|f|<1/10@. - - -Let us illustrate this on the numerical exploration of the identity -\centeredline{|log(1+x) = -log(1/(1+x))|}% -Let |L(h)=log(1+h)|, and |D(h)=L(h)+L(-h/(1+h))|. Theoretically thus, -|D(h)=0| but we shall evaluate |L(h)| and |-h/(1+h)| keeping only 10 -terms of their respective series. We will assume @|h|<0.5@. With only -ten terms kept in the power series we do not have quite 3 digits -precision as @2^10=1024@. So it wouldn't make sense to evaluate things -more precisely than, say circa 5 digits after the decimal points. -\dverb|@ -\cnta 0 -\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}% (-1)^{n-1}/n -\def\coeffalt #1{\the\numexpr\ifodd#1 -1\else1\fi\relax [0]}% (-1)^n -\loop -\noindent \hbox to 2.5cm {\hss\texttt{D(\the\cnta/100): }}% -\xintAdd {\xintFxPtPowerSeriesX {1}{10}{\coefflog}{\the\cnta [-2]}{5}} - {\xintFxPtPowerSeriesX {1}{10}{\coefflog} - {\xintFxPtPowerSeriesX {1}{10}{\coeffalt}{\the\cnta [-2]}{5}} - {5}}\endgraf -\ifnum\cnta < 49 \advance\cnta 7 \repeat| - -\cnta 0 -\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}% (-1)^{n-1}/n -\def\coeffalt #1{\the\numexpr\ifodd#1 -1\else1\fi\relax [0]}% (-1)^n - - -\begin{multicols}2 -\loop -\noindent \hbox to 2.5cm {\hss\texttt{D(\the\cnta/100): }}% -\digitstt{\xintAdd {\xintFxPtPowerSeriesX {1}{10}{\coefflog}{\the\cnta [-2]}{5}} - {\xintFxPtPowerSeriesX {1}{10}{\coefflog} - {\xintFxPtPowerSeriesX {1}{10}{\coeffalt}{\the\cnta [-2]}{5}} - {5}}}\endgraf -\ifnum\cnta < 49 \advance\cnta 7 \repeat -\end{multicols} - -Let's say we evaluate functions on |[-1/2,+1/2]| with values more or less also -in |[-1/2,+1/2]| and we want to keep 4 digits of precision. So, roughly we need -at least 14 terms in series like the geometric or log series. Let's make this -15. Then it doesn't make sense to compute intermediate summands with more than 6 -digits precision. So we compute with 6 digits -precision but return only 4 digits (rounded) after the decimal point. -This result with 4 post-decimal points precision is then used as input -to the next evaluation. -\dverb|@ -\loop -\noindent \hbox to 2.5cm {\hss\texttt{D(\the\cnta/100): }}% -\xintRound{4} - {\xintAdd {\xintFxPtPowerSeriesX {1}{15}{\coefflog}{\the\cnta [-2]}{6}} - {\xintFxPtPowerSeriesX {1}{15}{\coefflog} - {\xintRound {4}{\xintFxPtPowerSeriesX {1}{15}{\coeffalt} - {\the\cnta [-2]}{6}}} - {6}}% - }\endgraf -\ifnum\cnta < 49 \advance\cnta 7 \repeat| - -\begin{multicols}2 -\loop -\noindent \hbox to 2.5cm {\hss\texttt{D(\the\cnta/100): }}% -\digitstt{\xintRound{4} - {\xintAdd {\xintFxPtPowerSeriesX {1}{15}{\coefflog}{\the\cnta [-2]}{6}} - {\xintFxPtPowerSeriesX {1}{15}{\coefflog} - {\xintRound {4}{\xintFxPtPowerSeriesX {1}{15}{\coeffalt} - {\the\cnta [-2]}{6}}} - {6}}% - }}\endgraf -\ifnum\cnta < 49 \advance\cnta 7 \repeat -\end{multicols} - - -Not bad... I have cheated a bit: the `four-digits precise' numeric -evaluations were left unrounded in the final addition. However the inner -rounding to four digits worked fine and made the next step faster than -it would have been with longer inputs. The morale is that one should not -use the raw results of \csa{xintFxPtPowerSeriesX} with the |D| digits -with which it was computed, as the last are to be considered garbage. -Rather, one should keep from the output only some smaller number of -digits. This will make further computations faster and not less precise. -I guess there should be some command to do this final truncating, or -better, rounding, at a given number |D'<D| of digits. Maybe for the next -release. - - -\subsection{\csbh{xintFloatPowerSeries}}\label{xintFloatPowerSeries} - -%{\small\hspace*{\parindent}New with |1.08a|.\par} - -\noindent\csa{xintFloatPowerSeries}|[P]{A}{B}{\coeff}{f}|% -\ntype{{\upshape[\numx]}\numx\numx} - computes -$\sum_{\text{|n=A|}}^{\text{|n=B|}}|\coeff{n}|\cdot |f|^{\,\text{|n|}}$ -with a floating point\etype{\Ff\Ff} -precision given by the optional parameter |P| or by the current setting of -|\xintDigits|. - -In the current, preliminary, version, no attempt has been made to try to -guarantee to the final result the precision |P|. Rather, |P| is used for all -intermediate floating point evaluations. So -rounding errors will make some of the last printed digits invalid. The -operations done are first the evaluation of |f^A| using \csa{xintFloatPow}, then -each successive power is obtained from this first one by multiplication by |f| -using \csa{xintFloatMul}, then again with \csa{xintFloatMul} this is multiplied -with |\coeff{n}|, and the sum is done adding one term at a time with -\csa{xintFloatAdd}. To sum up, this is just the naive transformation of -\csa{xintFxPtPowerSeries} from fixed point to floating point. - -\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}% - -\dverb+@ -\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}% -\xintFloatPowerSeries [8]{1}{30}{\coefflog}{-1/2[0]}+% -\centeredline{\digitstt{\xintFloatPowerSeries [8]{1}{30}{\coefflog}{-1/2[0]}}} - -\subsection{\csbh{xintFloatPowerSeriesX}}\label{xintFloatPowerSeriesX} - -%{\small\hspace*{\parindent}New with |1.08a|.\par} - -\noindent\csa{xintFloatPowerSeriesX}|[P]{A}{B}{\coeff}{f}|% -\ntype{{\upshape[\numx]}\numx\numx} -is like -\csa{xintFloatPowerSeries} with the difference that |f| is -expanded once\etype{\Ff\Ff} -and for all at the start of the computation, thus allowing -efficient chaining of such series evaluations. -\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}% - -\dverb+@ -\def\coeffexp #1{1/\xintFac {#1}[0]}% 1/n! (exact, not float) -\def\coefflog #1{\the\numexpr\ifodd#1 1\else-1\fi\relax/#1[0]}% -\xintFloatPowerSeriesX [8]{0}{30}{\coeffexp} - {\xintFloatPowerSeries [8]{1}{30}{\coefflog}{-1/2[0]}}+% -\centeredline{\digitstt{\xintFloatPowerSeriesX [8]{0}{30}{\coeffexp} - {\xintFloatPowerSeries [8]{1}{30}{\coefflog}{-1/2[0]}}}} - - -\subsection{Computing \texorpdfstring{$\log 2$}{log(2)} and \texorpdfstring{$\pi$}{pi}}\label{ssec:Machin} - -In this final section, the use of \csbxint{FxPtPowerSeries} (and -\csbxint{PowerSeries}) will be -illustrated on the (expandable... why make things simple when it is so easy to -make them difficult!) computations of the first digits of the decimal expansion -of the familiar constants $\log 2$ and $\pi$. - -Let us start with $\log 2$. We will get it from this formula (which is -left as an exercise): \centeredline{\digitstt{log(2)=-2\,log(1-13/256)-% - 5\,log(1-1/9)}}% -The number of terms to be kept in the log series, for a desired -precision of |10^{-D}| was roughly estimated without much theoretical -analysis. Computing exactly the partial sums with \csa{xintPowerSeries} -and then printing the truncated values, from |D=0| up to |D=100| showed -that it worked in terms of quality of the approximation. Because of -possible strings of zeroes or nines in the exact decimal expansion (in -the present case of $\log 2$, strings of zeroes around the fourtieth and -the sixtieth decimals), this -does not mean though that all digits printed were always exact. In -the end one always end up having to compute at some higher level of -desired precision to validate the earlier result. - -Then we tried with \csa{xintFxPtPowerSeries}: this is worthwile only for -|D|'s at least 50, as the exact evaluations are faster (with these -short-length |f|'s) for a lower -number of digits. And as expected the degradation in the quality of -approximation was in this range of the order of two or three digits. -This meant roughly that the 3+1=4 ending digits were wrong. Again, we ended -up having to compute with five more digits and compare with the earlier -value to validate it. We use truncation rather than rounding because our -goal is not to obtain the correct rounded decimal expansion but the -correct exact truncated one. - -% 693147180559945309417232121458176568075500134360255254120680009493 - -\dverb|@ -\def\coefflog #1{1/#1[0]}% 1/n -\def\xa {13/256[0]}% we will compute log(1-13/256) -\def\xb {1/9[0]}% we will compute log(1-1/9) -\def\LogTwo #1% -% get log(2)=-2log(1-13/256)- 5log(1-1/9) -{% we want to use \printnumber, hence need something expanding in two steps - % only, so we use here the \romannumeral0 method - \romannumeral0\expandafter\LogTwoDoIt \expandafter - % Nb Terms for 1/9: - {\the\numexpr #1*150/143\expandafter}\expandafter - % Nb Terms for 13/256: - {\the\numexpr #1*100/129\expandafter}\expandafter - % We print #1 digits, but we know the ending ones are garbage - {\the\numexpr #1\relax}% allows #1 to be a count register -}% -\def\LogTwoDoIt #1#2#3% -% #1=nb of terms for 1/9, #2=nb of terms for 13/256, -{% #3=nb of digits for computations, also used for printing - \xinttrunc {#3} % lowercase form to stop the \romannumeral0 expansion! - {\xintAdd - {\xintMul {2}{\xintFxPtPowerSeries {1}{#2}{\coefflog}{\xa}{#3}}} - {\xintMul {5}{\xintFxPtPowerSeries {1}{#1}{\coefflog}{\xb}{#3}}}% - }% -}% -\noindent $\log 2 \approx \LogTwo {60}\dots$\endgraf -\noindent\phantom{$\log 2$}${}\approx{}$\printnumber{\LogTwo {65}}\dots\endgraf -\noindent\phantom{$\log 2$}${}\approx{}$\printnumber{\LogTwo {70}}\dots\endgraf| - -\def\coefflog #1{1/#1[0]}% 1/n -\def\xa {13/256[0]}% we will compute log(1-13/256) -\def\xb {1/9[0]}% we will compute log(1-1/9) -\def\LogTwo #1% get log(2)=-2log(1-13/256)- 5log(1-1/9) with #1 digits precision -{% this #1 may be a count register, if desired - \romannumeral0\expandafter\LogTwoDoIt \expandafter - {\the\numexpr #1*150/143\expandafter}\expandafter % Nb Terms for 1/9 - {\the\numexpr #1*100/129\expandafter}\expandafter % Nb Terms for 13/256 - {\the\numexpr #1\relax }% -}% -\def\LogTwoDoIt #1#2#3% #1=nb of terms for 1/9, #2=nb of terms for 13/256, -{% #3=nb of digits for computations - \xinttrunc {#3} - {\xintAdd - {\xintMul {2}{\xintFxPtPowerSeries {1}{#2}{\coefflog}{\xa}{#3}}} - {\xintMul {5}{\xintFxPtPowerSeries {1}{#1}{\coefflog}{\xb}{#3}}}% - }% -}% - -\noindent $\log 2 \approx {}$\digitstt{\LogTwo {60}\dots}\endgraf -\noindent\phantom{$\log 2$}${}\approx{}$\digitstt{\printnumber{\LogTwo - {65}}\dots}\endgraf -\noindent\phantom{$\log 2$}${}\approx{}$\digitstt{\printnumber{\LogTwo - {70}}\dots}\endgraf - -Here is the code doing an exact evaluation of the partial sums. We have -added a |+1| to the number of digits for estimating the number of terms -to keep from the log series: we experimented that this gets exactly the -first |D| digits, for all values from |D=0| to |D=100|, except in one -case (|D=40|) where the last digit is wrong. For values of |D| -higher than |100| it is more efficient to use the code using -\csa{xintFxPtPowerSeries}. -\dverb|@ -\def\LogTwo #1% get log(2)=-2log(1-13/256)- 5log(1-1/9) -{% - \romannumeral0\expandafter\LogTwoDoIt \expandafter - {\the\numexpr (#1+1)*150/143\expandafter}\expandafter - {\the\numexpr (#1+1)*100/129\expandafter}\expandafter - {\the\numexpr #1\relax}% -}% -\def\LogTwoDoIt #1#2#3% -{% #3=nb of digits for truncating an EXACT partial sum - \xinttrunc {#3} - {\xintAdd - {\xintMul {2}{\xintPowerSeries {1}{#2}{\coefflog}{\xa}}} - {\xintMul {5}{\xintPowerSeries {1}{#1}{\coefflog}{\xb}}}% - }% -}%| - -Let us turn now to Pi, computed with the Machin formula. Again the numbers of -terms to keep in the two |arctg| series were roughly estimated, and some -experimentations showed that removing the last three digits was enough (at least -for |D=0-100| range). And the algorithm does print the correct digits when used -with |D=1000| (to be convinced of that one needs to run it for |D=1000| and -again, say for |D=1010|.) A theoretical analysis could help confirm that this -algorithm always gets better than |10^{-D}| precision, but again, strings of -zeroes or nines encountered in the decimal expansion may falsify the ending -digits, nines may be zeroes (and the last non-nine one should be increased) and -zeroes may be nine (and the last non-zero one should be decreased). - -\hypertarget{MachinCode}{} -\dverb|@ -% pi = 16 Arctg(1/5) - 4 Arctg(1/239) (John Machin's formula) -\def\coeffarctg #1{\the\numexpr\ifodd#1 -1\else1\fi\relax/% - \the\numexpr 2*#1+1\relax [0]}% -% the above computes (-1)^n/(2n+1). -\def\xa {1/25[0]}% 1/5^2, the [0] for (infinitesimally) faster parsing -\def\xb {1/57121[0]}% 1/239^2, the [0] for faster parsing -\def\Machin #1{% \Machin {\mycount} is allowed - \romannumeral0\expandafter\MachinA \expandafter - % number of terms for arctg(1/5): - {\the\numexpr (#1+3)*5/7\expandafter}\expandafter - % number of terms for arctg(1/239): - {\the\numexpr (#1+3)*10/45\expandafter}\expandafter - % do the computations with 3 additional digits: - {\the\numexpr #1+3\expandafter}\expandafter - % allow #1 to be a count register: - {\the\numexpr #1\relax }}% -\def\MachinA #1#2#3#4% -% #4: digits to keep after decimal point for final printing -% #3=#4+3: digits for evaluation of the necessary number of terms -% to be kept in the arctangent series, also used to truncate each -% individual summand. -{\xinttrunc {#4} % lowercase macro to match the initial \romannumeral0. - {\xintSub - {\xintMul {16/5}{\xintFxPtPowerSeries {0}{#1}{\coeffarctg}{\xa}{#3}}} - {\xintMul {4/239}{\xintFxPtPowerSeries {0}{#2}{\coeffarctg}{\xb}{#3}}}% - }}% -\[ \pi = \Machin {60}\dots \]| - -\def\coeffarctg #1{\the\numexpr\ifodd#1 -1\else1\fi\relax/% - \the\numexpr 2*#1+1\relax [0]}% -%\def\coeffarctg #1{\romannumeral0\xintmon{#1}/\the\numexpr 2*#1+1\relax }% -\def\xa {1/25[0]}% 1/5^2, the [0] for faster parsing -\def\xb {1/57121[0]}% 1/239^2, the [0] for faster parsing -\def\Machin #1{% #1 may be a count register, \Machin {\mycount} is allowed - \romannumeral0\expandafter\MachinA \expandafter - % number of terms for arctg(1/5): - {\the\numexpr (#1+3)*5/7\expandafter}\expandafter - % number of terms for arctg(1/239): - {\the\numexpr (#1+3)*10/45\expandafter}\expandafter - % do the computations with 3 additional digits: - {\the\numexpr #1+3\expandafter}\expandafter - % allow #1 to be a count register: - {\the\numexpr #1\relax }}% -\def\MachinA #1#2#3#4% -{\xinttrunc {#4} - {\xintSub - {\xintMul {16/5}{\xintFxPtPowerSeries {0}{#1}{\coeffarctg}{\xa}{#3}}} - {\xintMul{4/239}{\xintFxPtPowerSeries {0}{#2}{\coeffarctg}{\xb}{#3}}}% - }}% -\begin{framed} - \[ \pi = \Machin {60}\dots \] -\end{framed} -Here is a variant|\MachinBis|, -which evaluates the partial sums \emph{exactly} using -\csa{xintPowerSeries}, before their final truncation. No need for a -``|+3|'' then. -\dverb|@ -\def\MachinBis #1{% #1 may be a count register, -% the final result will be truncated to #1 digits post decimal point - \romannumeral0\expandafter\MachinBisA \expandafter - % number of terms for arctg(1/5): - {\the\numexpr #1*5/7\expandafter}\expandafter - % number of terms for arctg(1/239): - {\the\numexpr #1*10/45\expandafter}\expandafter - % allow #1 to be a count register: - {\the\numexpr #1\relax }}% -\def\MachinBisA #1#2#3% -{\xinttrunc {#3} % - {\xintSub - {\xintMul {16/5}{\xintPowerSeries {0}{#1}{\coeffarctg}{\xa}}} - {\xintMul{4/239}{\xintPowerSeries {0}{#2}{\coeffarctg}{\xb}}}% -}}%| - -\def\MachinBis #1{% #1 may be a count register, -% the final result will be truncated to #1 digits post decimal point - \romannumeral0\expandafter\MachinBisA \expandafter - % number of terms for arctg(1/5): - {\the\numexpr #1*5/7\expandafter}\expandafter - % number of terms for arctg(1/239): - {\the\numexpr #1*10/45\expandafter}\expandafter - % allow #1 to be a count register: - {\the\numexpr #1\relax }}% -\def\MachinBisA #1#2#3% -{\xinttrunc {#3} % - {\xintSub - {\xintMul {16/5}{\xintPowerSeries {0}{#1}{\coeffarctg}{\xa}}} - {\xintMul{4/239}{\xintPowerSeries {0}{#2}{\coeffarctg}{\xb}}}% -}}% - -Let us use this variant for a loop showing the build-up of digits: -\dverb|@ - \cnta 0 % previously declared \count register - \loop - \MachinBis{\cnta} \endgraf % Plain's \loop does not accept \par - \ifnum\cnta < 30 \advance\cnta 1 \repeat| - -\begin{multicols}{2} - \cnta 0 % previously declared \count register - \loop \noindent - \centeredline{\digitstt{\MachinBis{\cnta}}}% - \ifnum\cnta < 30 - \advance\cnta 1 \repeat -\end{multicols} - - -\hypertarget{Machin1000}{} -% -You want more digits and have some time? compile this copy of the -\hyperlink{MachinCode}{|\char 92 Machin|} with |etex| (or |pdftex|): -% -\dverb|@ -% Compile with e-TeX extensions enabled (etex, pdftex, ...) -\input xintfrac.sty -\input xintseries.sty -% pi = 16 Arctg(1/5) - 4 Arctg(1/239) (John Machin's formula) -\def\coeffarctg #1{\the\numexpr\ifodd#1 -1\else1\fi\relax/% - \the\numexpr 2*#1+1\relax [0]}% -\def\xa {1/25[0]}% -\def\xb {1/57121[0]}% -\def\Machin #1{% - \romannumeral0\expandafter\MachinA \expandafter - {\the\numexpr (#1+3)*5/7\expandafter}\expandafter - {\the\numexpr (#1+3)*10/45\expandafter}\expandafter - {\the\numexpr #1+3\expandafter}\expandafter - {\the\numexpr #1\relax }}% -\def\MachinA #1#2#3#4% -{\xinttrunc {#4} - {\xintSub - {\xintMul {16/5}{\xintFxPtPowerSeries {0}{#1}{\coeffarctg}{\xa}{#3}}} - {\xintMul {4/239}{\xintFxPtPowerSeries {0}{#2}{\coeffarctg}{\xb}{#3}}}% -}}% -\pdfresettimer -\oodef\Z {\Machin {1000}} -\odef\W {\the\pdfelapsedtime} -\message{\Z} -\message{computed in \xintRound {2}{\W/65536} seconds.} -\bye | - -This will log the first 1000 digits of $\pi$ after the decimal point. On my -laptop (a 2012 model) this took about @16@ seconds last time I tried. -\footnote{With \texttt{1.09i} and earlier \xintname releases, this used to be - \digitstt{42} seconds; the \texttt{1.09j} division is much faster with small - denominators as occurs here with \digitstt{\char92xa=1/25}, and I believe this - to be the main explanation for the speed gain.} As mentioned in the -introduction, the file \href{http://www.ctan.org/pkg/pi}{pi.tex} by \textsc{D. - Roegel} shows that orders of magnitude faster computations are possible within -\TeX{}, but recall our constraints of complete expandability and be merciful, -please. - - -\textbf{Why truncating rather than rounding?} One of our main competitors -on the market of scientific computing, a canadian product (not -encumbered with expandability constraints, and having barely ever heard -of \TeX{} ;-), prints numbers rounded in the last digit. Why didn't we -follow suit in the macros \csa{xintFxPtPowerSeries} and -\csa{xintFxPtPowerSeriesX}? To round at |D| digits, and excluding a -rewrite or cloning of the division algorithm which anyhow would add to -it some overhead in its final steps, \xintfracname needs to truncate at -|D+1|, then round. And rounding loses information! So, with more time -spent, we obtain a worst result than the one truncated at |D+1| (one -could imagine that additions and so on, done with only |D| digits, cost -less; true, but this is a negligeable effect per summand compared to the -additional cost for this term of having been truncated at |D+1| then -rounded). Rounding is the way to go when setting up algorithms to -evaluate functions destined to be composed one after the other: exact -algebraic operations with many summands and an |f| variable which is a -fraction are costly and create an even bigger fraction; replacing |f| -with a reasonable rounding, and rounding the result, is necessary to -allow arbitrary chaining. - -But, for the -computation of a single constant, we are really interested in the exact -decimal expansion, so we truncate and compute more terms until the -earlier result gets validated. Finally if we do want the rounding we can -always do it on a value computed with |D+1| truncation. - -% \clearpage - -\section{Commands of the \xintcfracname package} -\label{sec:cfrac} - -This package was first included in release |1.04| of the \xintname bundle. - - -\localtableofcontents - - -\subsection{Package overview} - -A \emph{simple} continued fraction has coefficients -|[c0,c1,...,cN]| (usually called partial quotients, but I really -dislike this entrenched terminology), where |c0| is a positive or -negative integer and the others are positive integers. As we will -see it is possible with \xintcfracname to specify the coefficient -function |c:n->cn|. Note that the index then starts at zero as -indicated. With the |amsmath| macro |\cfrac| one can display such a -continued fraction as -\[ c_0 + \cfrac{1}{c_1+\cfrac1{c_2+\cfrac1{c_3+\cfrac1{\ddots}}}}\] -Here is a concrete example: -\[ \xintFrac {208341/66317}=\xintCFrac {208341/66317}\] But the -difference with |amsmath|'s |\cfrac| is that this was input as -\centeredline{|\[ \xintFrac {208341/66317}=\xintCFrac - {208341/66317} \]|} The command \csbxint{CFrac} produces in two -expansion steps the whole thing with the many chained |\cfrac|'s and all -necessary braces, ready to be printed, in math mode. This is \LaTeX{} -only and with the |amsmath| package (we shall mention another method for -Plain \TeX{} users of |amstex|). - -A \emph{generalized} continued fraction has the same structure but -the numerators are not restricted to be ones, and numbers used in -the continued fraction may be arbitrary, also fractions, -irrationals, indeterminates. The \emph{centered} continued -fraction associated to a rational number is an -example: -\[ \xintFrac {915286/188421}=\xintGCFrac {\xintFtoCC {915286/188421}} -=\xintCFrac {915286/188421}\] - \centeredline{|\[ \xintFrac {915286/188421}=\xintGCFrac {\xintFtoCC - {915286/188421}} \]|} -The command \csbxint{GCFrac}, contrarily to -\csbxint{CFrac}, does not compute anything, it just typesets. Here, it is the -command \csbxint{FtoCC} which did the computation of -the centered continued fraction of |f|. Its output has the `inline format' -described in the next paragraph. In the display, we also used \csa{xintCFrac} -(code not shown), for comparison of the two types of continued fractions. - -A generalized continued fraction may be input `inline' as: -\centeredline{|a0+b0/a1+b1/a2+b2/...../a(n-1)+b(n-1)/an|}% -Fractions among the coefficients are allowed but they must be enclosed -within braces. Signed integers may be left without braces (but the |+| -signs are mandatory). Or, they may -be macros expanding (in two steps) to some number or fractional number. -\centeredline{|\xintGCFrac {1+-1/57+\xintPow {-3}{7}/\xintQuo {132}{25}}|} -\[ \xintFrac{\xintGCtoF {1+-1/57+\xintPow {-3}{7}/\xintQuo {132}{25}}}= - \xintGCFrac {1+-1/57+\xintPow {-3}{7}/\xintQuo {132}{25}}\] -The left hand side was obtained with the following code: -\centeredline{|\xintFrac{\xintGCtoF {1+-1/57+\xintPow {-3}{7}/\xintQuo - {132}{25}}}|} -It uses the macro \csbxint{GCtoF} to convert a generalized fraction from the -`inline format' to the fraction it evaluates to. - -A simple continued fraction is a special case of a generalized continued -fraction and may be input as such to macros expecting the `inline format', for -example |-7+1/6+1/19+1/1+1/33|. There is a simpler comma separated format: -\centeredline -{|\xintFrac{\xintCstoF{-7,6,19,1,33}}=\xintCFrac{\xintCstoF{-7,6,19,1,33}}|} -\[ -\xintFrac{\xintCstoF{-7,6,19,1,33}}=\xintCFrac{\xintCstoF{-7,6,19,1,33}}\] This -comma separated format may also be used with fractions among the coefficients: -in that case, computing with \csbxint{FtoCs} from the resulting |f| -its real coefficients will give a new comma separated list -with only integers. This list has no spaces: the spaces in the display below -arise from the math mode processing. -\centeredline{|\xintFrac{1084483/398959}=[\xintFtoCs{1084483/398959}]|} -\[\xintFrac{1084483/398959}=[\xintFtoCs{1084483/398959}]\] -If one prefers other separators, one can use \csbxint{FtoCx} whose first -argument will be the separator to be used. -\centeredline{|\xintFrac{2721/1001}=\xintFtoCx {+1/(}{2721/1001})\cdots)|} -\[\xintFrac{2721/1001}=\xintFtoCx {+1/(}{2721/1001})\cdots)\] -People using Plain \TeX{} and |amstex| can achieve the same effect as -|\xintCFrac| with: -|$$\xintFwOver{2721/1001}=\xintFtoCx {+\cfrac1\\ }{2721/1001}\endcfrac$$| - -Using \csa{xintFtoCx} with first argument an empty pair of braces |{}| will -return the list of the coefficients of the continued fraction of |f|, without -separator, and each one enclosed in a pair of group braces. This can then be -manipulated by the non-expandable macro \csbxint{AssignArray} or the expandable -ones \csbxint{Apply} and \csbxint{ListWithSep}. - -As a shortcut to using \csa{xintFtoCx} with separator |1+/|, there is -\csbxint{FtoGC}: -\centeredline{|2721/1001=\xintFtoGC {2721/1001}|}% -\centeredline{\digitstt{2721/1001=\xintFtoGC {2721/1001}}} -Let us compare in that case with the output of \csbxint{FtoCC}: -\centeredline{|2721/1001=\xintFtoCC {2721/1001}|}% -\centeredline{\digitstt{2721/1001=\xintFtoCC {2721/1001}}} - -The `|\printnumber|' macro which we use to print long numbers can also -be useful on long continued fractions. -\centeredline{|\printnumber{\xintFtoCC {35037018906350720204351049/%|}% -\centeredline{|244241737886197404558180}}|}% -\digitstt{\printnumber{\xintFtoCC {35037018906350720204351049/244241737886197404558180}}}. -If we apply \csbxint{GCtoF} to this generalized continued fraction, we -discover that the original fraction was reducible: -\centeredline{|\xintGCtoF - {143+1/2+...+-1/9}|\digitstt{=\xintGCtoF{143+1/2+1/5+-1/4+-1/4+-1/4+-1/3+1/2+1/2+1/6+-1/22+1/2+1/10+-1/5+-1/11+-1/3+1/4+-1/2+1/2+1/4+-1/2+1/23+1/3+1/8+-1/6+-1/9}}} - -\def\mymacro #1{$\xintFrac{#1}=[\xintFtoCs{#1}]$\vtop to 6pt{}} - -\begingroup -\catcode`^\active -\def^#1^{\hbox{\fontfamily{lmtt}\selectfont #1}}% - -When a generalized continued fraction is built with integers, and -numerators are only |1|'s or |-1|'s, the produced fraction is -irreducible. And if we compute it again with the last sub-fraction -omitted we get another irreducible fraction related to the bigger one by -a Bezout identity. Doing this here we get: -\centeredline{|\xintGCtoF {143+1/2+...+-1/6}|\digitstt{=\xintGCtoF{143+1/2+1/5+-1/4+-1/4+-1/4+-1/3+1/2+1/2+1/6+-1/22+1/2+1/10+-1/5+-1/11+-1/3+1/4+-1/2+1/2+1/4+-1/2+1/23+1/3+1/8+-1/6}}} -and indeed: -\[ \begin{vmatrix} - ^2897319801297630107^ & ^328124887710626729^\\ - ^20197107104701740^ & ^2287346221788023^ - \end{vmatrix} = \mbox{\digitstt{\xintiSub {\xintiMul {2897319801297630107}{2287346221788023}}{\xintiMul{20197107104701740}{328124887710626729}}}}\] - -\endgroup -More generally the various fractions obtained from the truncation of a -continued fraction to its initial terms are called the convergents. The -commands of \xintcfracname such as \csbxint{FtoCv}, \csbxint{FtoCCv}, -and others which compute such convergents, return them as a list of -braced items, with no separator. This list can then be treated either -with \csa{xint\-AssignArray}, or \csa{xintListWithSep}, or any other way -(but then, some \TeX{} programming knowledge will be necessary). Here -is an example: - -\noindent -\centeredline{|$$\xintFrac{915286/188421}\to \xintListWithSep {,}%|}% -\centeredline{|{\xintApply{\xintFrac}{\xintFtoCv{915286/188421}}}$$|} -\[ \xintFrac{915286/188421}\to \xintListWithSep {,} -{\xintApply\xintFrac{\xintFtoCv{915286/188421}}}\] -\centeredline{|$$\xintFrac{915286/188421}\to \xintListWithSep {,}%|}% -\centeredline{|{\xintApply{\xintFrac}{\xintFtoCCv{915286/188421}}}$$|} -\[ \xintFrac{915286/188421}\to \xintListWithSep {,} -{\xintApply\xintFrac{\xintFtoCCv{915286/188421}}}\] We thus see that the -`centered convergents' obtained with \csbxint{FtoCCv} are among the fuller list -of convergents as returned by \csbxint{FtoCv}. - -Here is a more complicated use of \csa{xintApply} -and \csa{xintListWithSep}. We first define a macro which will be applied to each -convergent:\centeredline{|\newcommand{\mymacro}[1]|% - |{$\xintFrac{#1}=[\xintFtoCs{#1}]$\vtop to 6pt{}}|}% -Next, we use the following code: -\centeredline{|$\xintFrac{49171/18089}\to{}$|}% -\centeredline{|\xintListWithSep {, - }{\xintApply{\mymacro}{\xintFtoCv{49171/18089}}}|} -It produces:\par -\noindent$ \xintFrac{49171/18089}\to {}$\xintListWithSep {, - }{\xintApply{\mymacro}{\xintFtoCv{49171/18089}}}. - - -\def\cn #1{\xintiPow {2}{#1}}% - -The macro \csbxint{CntoF} allows to specify the coefficients as -functions of the index. The values to which expand the -coefficient function do not have to be integers. \centeredline{|\def\cn - #1{\xintiPow {2}{#1}}% 2^n|}% - \centeredline{|\[\xintFrac{\xintCntoF {6}{\cn}}=\xintCFrac - [l]{\xintCntoF {6}{\cn}}\]|}% -\[\xintFrac{\xintCntoF {6}{\cn}}=\xintCFrac [l]{\xintCntoF - {6}{\cn}}\] -Notice the use of the optional argument |[l]| to \csa{xintCFrac}. Other -possibilities are |[r]| and (default) |[c]|. -\def\cn #1{\xintPow {2}{-#1}}% -\centeredline{|\def\cn #1{\xintPow {2}{-#1}}% 1/2^n|}% -\centeredline{% -|\[\xintFrac{\xintCntoF {6}{\cn}} = \xintGCFrac [r]{\xintCntoGC {6}{\cn}}|}% -\centeredline{| = [\xintFtoCs {\xintCntoF {6}{\cn}}]\]|}% -\[\xintFrac{\xintCntoF {6}{\cn}}=\xintGCFrac [r]{\xintCntoGC {6}{\cn}}= - [\xintFtoCs {\xintCntoF {6}{\cn}}]\] -We used \csbxint{CntoGC} as we wanted to display also the continued fraction and -not only the fraction returned by \csa{xintCntoF}. - -There are also \csbxint{GCntoF} and \csbxint{GCntoGC} which allow the same for -generalized fractions. The following initial portion of a generalized continued -fraction for $\pi$: -\def\an #1{\the\numexpr 2*#1+1\relax }% -\def\bn #1{\the\numexpr (#1+1)*(#1+1)\relax }% -\[ \xintFrac{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}} = - \cfrac{4}{\xintGCFrac{\xintGCntoGC {5}{\an}{\bn}}} = -\xintTrunc {10}{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}}\dots\] -was obtained with this code: -\dverb|@ -\def\an #1{\the\numexpr 2*#1+1\relax }% -\def\bn #1{\the\numexpr (#1+1)*(#1+1)\relax }% -\[ \xintFrac{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}} = - \cfrac{4}{\xintGCFrac{\xintGCntoGC {5}{\an}{\bn}}} = -\xintTrunc {10}{\xintDiv {4}{\xintGCntoF {5}{\an}{\bn}}}\dots\]| - -We see that the quality of approximation is not fantastic compared to the simple -continued fraction of $\pi$ with about as many terms: -\dverb|@ -\[ \xintFrac{\xintCstoF{3,7,15,1,292,1,1}}= - \xintGCFrac{3+1/7+1/15+1/1+1/292+1/1+1/1}= - \xintTrunc{10}{\xintCstoF{3,7,15,1,292,1,1}}\dots\]| -\[ \xintFrac{\xintCstoF{3,7,15,1,292,1,1}}= -\xintGCFrac{3+1/7+1/15+1/1+1/292+1/1+1/1}= -\xintTrunc{10}{\xintCstoF{3,7,15,1,292,1,1}}\dots\] - -\hypertarget{e-convergents}{To} -conclude this overview of most of the package functionalities, let us explore -the convergents of Euler's number $e$. -\dverb|@ -\def\cn #1{\the\numexpr\ifcase \numexpr #1+3-3*((#1+2)/3)\relax - 1\or1\or2*(#1/3)\fi\relax } -% produces the pattern 1,1,2,1,1,4,1,1,6,1,1,8,... which are the -% coefficients of the simple continued fraction of e-1. -\cnta 0 -\def\mymacro #1{\advance\cnta by 1 - \noindent - \hbox to 3em {\hfil\small\texttt{\the\cnta.} }% - $\xintTrunc {30}{\xintAdd {1[0]}{#1}}\dots= - \xintFrac{\xintAdd {1[0]}{#1}}$}% -\xintListWithSep{\vtop to 6pt{}\vbox to 12pt{}\par} - {\xintApply\mymacro{\xintiCstoCv{\xintCntoCs {35}{\cn}}}}| - -\smallskip The volume of computation is kept minimal by the following steps: -\begin{itemize} -\item a comma separated list of the first 36 coefficients is produced by - \csbxint{CntoCs}, -\item this is then given to \csbxint{iCstoCv} which produces the list of the - convergents (there is also \csbxint{CstoCv}, but our - coefficients being integers we used the infinitesimally - faster \csbxint{iCstoCv}), -\item then the whole list was converted into a sequence of one-line paragraphs, - each convergent becomes the argument to a macro printing it - together with its decimal expansion with 30 digits after the decimal point. -\item A count register |\cnta| was used to give a line count serving as a visual - aid: we could also have done that in an expandable way, but well, let's relax - from time to time\dots -\end{itemize} - - -\def\cn #1{\the\numexpr\ifcase \numexpr #1+3-3*((#1+2)/3)\relax - 1\or1\or2*(#1/3)\fi\relax } -% produces the pattern 1,1,2,1,1,4,1,1,6,1,1,8,... which are the -% coefficients of the simple continued fraction of e-1. -\cnta 0 -\def\mymacro #1{\advance\cnta by 1 - \noindent - \hbox to 3em {\hfil\small\digitstt{\the\cnta.} }% - $\xintTrunc {30}{\xintAdd {1[0]}{#1}}\dots= - \xintFrac{\xintAdd {1[0]}{#1}}$}% -\xintListWithSep{\vtop to 6pt{}\vbox to 12pt{}\par} - {\xintApply\mymacro{\xintiCstoCv{\xintCntoCs {35}{\cn}}}} - -% \def\testmacro #1{\xintTrunc {30}{\xintAdd {1[0]}{#1}}\xintAdd {1[0]}{#1}} -% \pdfresettimer -% \edef\z{\xintApply\testmacro{\xintiCstoCv{\xintCntoCs {35}{\cn}}}} -% (\the\pdfelapsedtime) - - -\smallskip The actual computation of the list of all 36 convergents accounts for -only 8\% of the total time (total time equal to about 5 hundredths of a second -in my testing, on my laptop): another 80\% is occupied with the computation of -the truncated decimal expansions (and the addition of 1 to everything as the -formula gives the continued fraction of $e-1$). One can with no problem compute -much bigger convergents. Let's get the 200th convergent. It turns out to -have the same first 268 digits after the decimal point as $e-1$. Higher -convergents get more and more digits in proportion to their index: the 500th -convergent already gets 799 digits correct! To allow speedy compilation of the -source of this document when the need arises, I limit here to the 200th -convergent (getting the 500th took about 1.2s on my laptop last time I tried, -and the 200th convergent is obtained ten times faster). -\dverb|@ -\oodef\z {\xintCntoF {199}{\cn}}% -\begingroup\parindent 0pt \leftskip 2.5cm -\indent\llap {Numerator = }{\printnumber{\xintNumerator\z}\par -\indent\llap {Denominator = }\printnumber{\xintDenominator\z}\par -\indent\llap {Expansion = }\printnumber{\xintTrunc{268}\z}\dots -\par\endgroup| - -\oodef\z {\xintCntoF {199}{\cn}}% - -\begingroup\parindent 0pt \leftskip 2.5cm -\indent\llap {Numerator = }\digitstt{\printnumber{\xintNumerator\z}}\par -\indent\llap {Denominator = }\digitstt{\printnumber{\xintDenominator\z}}\par -\indent\llap - {Expansion = }\digitstt{\printnumber{\xintTrunc{268}\z}\dots}\par\endgroup - -One can also use a centered continued fraction: we get more digits but there are -also more computations as the numerators may be either -$1$ or $-1$. - -\subsection{\csbh{xintCFrac}}\label{xintCFrac} - -\csa{xintCFrac}|{f}|\ntype{\Ff} is a math-mode only, \LaTeX{} with |amsmath| -only, macro which first computes then displays with the help of |\cfrac| the -simple continued fraction corresponding to the given fraction. It admits an -optional argument which may be |[l]|, |[r]| or (the default) |[c]| to specify -the location of the one's in the numerators of the sub-fractions. Each -coefficient is typeset using the \csbxint{Frac} macro from the \xintfracname -package. This macro is \fexpan dable in the sense that it prepares expandably -the whole expression with the multiple |\cfrac|'s, but it is not completely -expandable naturally. - -\subsection{\csbh{xintGCFrac}}\label{xintGCFrac} - -\csa{xintGCFrac}|{a+b/c+d/e+f/g+h/...}|\etype{f} uses similarly |\cfrac| to -typeset a -generalized continued fraction in inline format. It admits the same optional -argument as \csa{xintCFrac}. -\centeredline{|\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}\]|} -\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}\] -As can be seen this is typesetting macro, although it does proceed to the -evaluation of the coefficients themselves. See \csbxint{GCtoF} if you are -impatient to see this fraction computed. Numerators and denominators are made -arguments to the -\csbxint{Frac} macro. - -\subsection{\csbh{xintGCtoGCx}}\label{xintGCtoGCx} -%{\small New with release |1.05|.\par} - - -\csa{xintGCtoGCx}|{sepa}{sepb}{a+b/c+d/e+f/...+x/y}|\etype{nnf} returns the list -of the coefficients of the generalized continued fraction of |f|, each one -within a pair of braces, and separated with the help of |sepa| and |sepb|. Thus -\centeredline{|\xintGCtoGCx :;{1+2/3+4/5+6/7}| gives \xintGCtoGCx - :;{1+2/3+4/5+6/7}} Plain \TeX{}+|amstex| users may be interested in:\par -\dverb|@ -$$\xintGCtoGCx {+\cfrac}{\\}{a+b/...}\endcfrac$$ -$$\xintGCtoGCx {+\cfrac\xintFwOver}{\\\xintFwOver}{a+b/...}\endcfrac$$| - - -\subsection{\csbh{xintFtoCs}}\label{xintFtoCs} - -\csa{xintFtoCs}|{f}|\etype{\Ff} returns the comma separated list of the -coefficients of the simple continued fraction of |f|. -\centeredline{% - |\[ \xintSignedFrac{-5262046/89233} = [\xintFtoCs{-5262046/89233}]\]|}% -\[ \xintSignedFrac{-5262046/89233} = [\xintFtoCs{-5262046/89233}]\] - - -\subsection{\csbh{xintFtoCx}}\label{xintFtoCx} - -\csa{xintFtoCx}|{sep}{f}|\etype{n\Ff} returns the list of the coefficients of -the simple continued fraction of |f|, withing group braces and separated with -the help of |sep|. \centeredline{|$$\xintFtoCx {+\cfrac1\\ }{f}\endcfrac$$|} -will display the continued fraction in |\cfrac| format, with Plain \TeX{} and -|amstex|. - -\subsection{\csbh{xintFtoGC}}\label{xintFtoGC} - -\csa{xintFtoGC}|{f}|\etype{\Ff} does the same as \csa{xintFtoCx}|{+1/}{f}|. Its -output may thus be used in the package macros expecting such an `inline -format'. This continued fraction is a \emph{simple} one, not a -\emph{generalized} one, but as it is produced in the format used for -user input of generalized continued fractions, the macro was called -\csa{xintFtoGC} rather than \csa{xintFtoC} for example. -\centeredline{|566827/208524=\xintFtoGC {566827/208524}|}% -\centeredline{566827/208524=\xintFtoGC {566827/208524}} - -\subsection{\csbh{xintFtoCC}}\label{xintFtoCC} - -\csa{xintFtoCC}|{f}|\etype{\Ff} returns the `centered' continued fraction of -|f|, in `inline format'. \centeredline{|566827/208524=\xintFtoCC - {566827/208524}|}% -\centeredline{566827/208524=\xintFtoCC {566827/208524}} \centeredline{% - |\[\xintFrac{566827/208524} = \xintGCFrac{\xintFtoCC{566827/208524}}\]|}% -\[\xintFrac{566827/208524} = \xintGCFrac{\xintFtoCC{566827/208524}}\] - -\subsection{\csbh{xintFtoCv}}\label{xintFtoCv} - -\csa{xintFtoCv}|{f}|\etype{\Ff} returns the list of the (braced) convergents of -|f|, with no separator. To be treated with \csbxint{AssignArray} or -\csbxint{ListWithSep}. \centeredline{% - |\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCv{5211/3748}}}\]|}% -\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCv{5211/3748}}}\] - -\subsection{\csbh{xintFtoCCv}}\label{xintFtoCCv} - -\csa{xintFtoCCv}|{f}|\etype{\Ff} returns the list of the (braced) centered -convergents of |f|, with no separator. To be treated with \csbxint{AssignArray} -or \csbxint{ListWithSep}. \centeredline{% - |\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCCv{5211/3748}}}\]|}% -\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintFtoCCv{5211/3748}}}\] - -\subsection{\csbh{xintCstoF}}\label{xintCstoF} - -\csa{xintCstoF}|{a,b,c,d,...,z}|\etype{f} computes the fraction corresponding to -the coefficients, which may be fractions or even macros expanding to such -fractions. The final fraction may then be highly reducible. -\centeredline{|\[\xintGCFrac {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}|}% -\centeredline{|=\xintSignedFrac{\xintCstoF {-1,3,-5,7,-9,11,-13}}|}% -\centeredline{|=\xintSignedFrac{\xintGCtoF - {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}}\]|}% -\[\xintGCFrac {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}= -\xintSignedFrac{\xintCstoF {-1,3,-5,7,-9,11,-13}} -=\xintSignedFrac{\xintGCtoF {-1+1/3+1/-5+1/7+1/-9+1/11+1/-13}}\] -\centeredline{|\xintGCFrac{{1/2}+1/{1/3}+1/{1/4}+1/{1/5}}= |}% -\centeredline{| \xintFrac{\xintCstoF {1/2,1/3,1/4,1/5}}|}% -\[\xintGCFrac{{1/2}+1/{1/3}+1/{1/4}+1/{1/5}}= -\xintFrac{\xintCstoF {1/2,1/3,1/4,1/5}}\] A generalized continued fraction may -produce a reducible fraction (\csa{xintCstoF} tries its best not to accumulate -in a silly way superfluous factors but will not do simplifications which would -be obvious to a human, like simplification by 3 in the result above). - -\subsection{\csbh{xintCstoCv}}\label{xintCstoCv} - -\csa{xintCstoCv}|{a,b,c,d,...,z}|\etype{f} returns the list of the corresponding -convergents. It is allowed to use fractions as coefficients (the computed -convergents have then no reason to be the real convergents of the final -fraction). When the coefficients are integers, the convergents are irreducible -fractions, but otherwise it is not necessarily the case. -\centeredline{|\xintListWithSep:{\xintCstoCv{1,2,3,4,5,6}}|}% -\centeredline{\digitstt{\xintListWithSep:{\xintCstoCv{1,2,3,4,5,6}}}} -\centeredline{|\xintListWithSep:{\xintCstoCv{1,1/2,1/3,1/4,1/5,1/6}}|}% -\centeredline{\digitstt{\xintListWithSep:{\xintCstoCv{1,1/2,1/3,1/4,1/5,1/6}}}} -% j'ai retiré les [0] à partir de la version 1.09b, le 3 octobre 2013. -\centeredline{|\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintCstoCv|}% - \centeredline{|{\xintPow {-.3}{-5},7.3/4.57,\xintCstoF{3/4,9,-1/3}}}}\]|}% -\[\xintListWithSep{\to}{\xintApply\xintFrac{\xintCstoCv - {\xintPow {-.3}{-5},7.3/4.57,\xintCstoF{3/4,9,-1/3}}}}\] - - -\subsection{\csbh{xintCstoGC}}\label{xintCstoGC} - -\csa{xintCstoGC}|{a,b,..,z}|\etype{f} transforms a comma separated list (or -something expanding to such a list) into an `inline format' continued fraction -|{a}+1/{b}+1/...+1/{z}|. The coefficients are just copied and put within braces, -without expansion. The output can then be used in \csbxint{GCFrac} for example. -\centeredline{|\[\xintGCFrac {\xintCstoGC {-1,1/2,-1/3,1/4,-1/5}}|}% -\centeredline{|=\xintSignedFrac {\xintCstoF {-1,1/2,-1/3,1/4,-1/5}}\]|}% -\[\xintGCFrac {\xintCstoGC {-1,1/2,-1/3,1/4,-1/5}} = -\xintSignedFrac{\xintCstoF {-1,1/2,-1/3,1/4,-1/5}}\] - -\subsection{\csbh{xintGCtoF}}\label{xintGCtoF} - -\csa{xintGCtoF}|{a+b/c+d/e+f/g+......+v/w+x/y}|\etype{f} computes the fraction -defined by the inline generalized continued fraction. Coefficients may be -fractions but must then be put within braces. They can be macros. The plus signs -are mandatory. \dverb|@ -\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}} = -\xintFrac{\xintGCtoF {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}} = -\xintFrac{\xintIrr{\xintGCtoF - {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}}}\]| -\[\xintGCFrac {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}} = -\xintFrac{\xintGCtoF {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}} = -\xintFrac{\xintIrr{\xintGCtoF {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}}}}\] -\dverb|@ -\[ \xintGCFrac{{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}} = - \xintFrac{\xintGCtoF {{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}}} \]| -\[ \xintGCFrac{{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}} = - \xintFrac{\xintGCtoF {{1/2}+{2/3}/{4/5}+{1/2}/{1/5}+{3/2}/{5/3}}} \] -The macro tries its best not to accumulate superfluous factor in the -denominators, but doesn't reduce the fraction to irreducible form before -returning it and does not do simplifications which would be obvious to a human. - -\subsection{\csbh{xintGCtoCv}}\label{xintGCtoCv} - -\csa{xintGCtoCv}|{a+b/c+d/e+f/g+......+v/w+x/y}|\etype{f} returns the list of -the corresponding convergents. The coefficients may be fractions, but must then -be inside braces. Or they may be macros, too. - -The convergents will in the general case be reducible. To put them into -irreducible form, one needs one more step, for example it can be done -with |\xintApply\xintIrr|. -\dverb|@ -\[\xintListWithSep{,}{\xintApply\xintFrac - {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}\] -\[\xintListWithSep{,}{\xintApply\xintFrac{\xintApply\xintIrr - {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}}\]| -\[\xintListWithSep{,}{\xintApply\xintFrac - {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}\] -\[\xintListWithSep{,}{\xintApply\xintFrac{\xintApply\xintIrr - {\xintGCtoCv{3+{-2}/{7/2}+{3/4}/12+{-56}/3}}}}\] - -\subsection{\csbh{xintCntoF}}\label{xintCntoF} - -\def\macro #1{\the\numexpr 1+#1*#1\relax} - -\csa{xintCntoF}|{N}{\macro}|\etype{\numx f} computes the fraction |f| having coefficients -|c(j)=\macro{j}| for |j=0,1,...,N|. The |N| parameter is given to a |\numexpr|. -The values of the coefficients, as returned by |\macro| do not have to be -positive, nor integers, and it is thus not necessarily the case that the -original |c(j)| are the true coefficients of the final |f|. \centeredline{% - |\def\macro #1{\the\numexpr 1+#1*#1\relax}\xintCntoF {5}{\macro}|}% -\centeredline{\digitstt{\xintCntoF {5}{\macro}}} - -\subsection{\csbh{xintGCntoF}}\label{xintGCntoF} - -\def\coeffA #1{\the\numexpr #1+4-3*((#1+2)/3)\relax }% -\def\coeffB #1{\xintMON{#1}}% (-1)^n - -\csa{xintGCntoF}|{N}{\macroA}{\macroB}|\etype{\numx ff} returns the fraction |f| -corresponding to the inline generalized continued fraction -|a0+b0/a1+b1/a2+....+b(N-1)/aN|, with |a(j)=\macroA{j}| and |b(j)=\macroB{j}|. -The |N| parameter is given to a |\numexpr|. -\[\xintGCFrac{\xintGCntoGC {6}{\coeffA}{\coeffB}} -= \xintFrac{\xintGCntoF {6}{\coeffA}{\coeffB}}\] -There is also \csbxint{GCntoGC} to get the `inline format' continued -fraction. The previous display was obtained with: -\centeredline{|\def\coeffA #1{\the\numexpr #1+4-3*((#1+2)/3)\relax }%|}% -\centeredline{|\def\coeffB #1{\xintMON{#1}}% (-1)^n|}% -\centeredline{|\[\xintGCFrac{\xintGCntoGC {6}{\coeffA}{\coeffB}}|}% -\centeredline{| = \xintFrac{\xintGCntoF {6}{\coeffA}{\coeffB}}\]|} - - -\subsection{\csbh{xintCntoCs}}\label{xintCntoCs} - -\csa{xintCntoCs}|{N}{\macro}|\etype{\numx f} produces the comma separated list -of the corresponding coefficients, from |n=0| to |n=N|. The |N| is given to a -|\numexpr|. \centeredline{% - |\def\macro #1{\the\numexpr 1+#1*#1\relax}|}% -\centeredline{|\xintCntoCs {5}{\macro}|\digitstt{->\xintCntoCs {5}{\macro}}}% -\centeredline{|\[\xintFrac{\xintCntoF {5}{\macro}}=\xintCFrac{\xintCntoF - {5}{\macro}}\]|}% -\[ \xintFrac{\xintCntoF - {5}{\macro}}=\xintCFrac{\xintCntoF {5}{\macro}}\] - -\subsection{\csbh{xintCntoGC}}\label{xintCntoGC} - -\def\macro #1{\the\numexpr\ifodd#1 -1-#1\else1+#1\fi\relax/% - \the\numexpr 1+#1*#1\relax} -% -\csa{xintCntoGC}|{N}{\macro}|\etype{\numx f} evaluates the |c(j)=\macro{j}| from -|j=0| to |j=N| and returns a continued fraction written in inline format: -|{c(0)}+1/{c(1)}+1/...+1/{c(N)}|. The parameter |N| is given to a |\numexpr|. -The coefficients, after expansion, are, as shown, being enclosed in an added -pair of braces, they may thus be fractions. \centeredline{% - |\def\macro #1{\the\numexpr\ifodd#1 -1-#1\else1+#1\fi\relax/%|}% - \centeredline{|\the\numexpr 1+#1*#1\relax}|}% - \centeredline{|\edef\x{\xintCntoGC {5}{\macro}}\meaning\x|}% - \centeredline{\edef\x{\xintCntoGC {5}{\macro}}\digitstt{\meaning\x}}% - \centeredline{|\[\xintGCFrac{\xintCntoGC {5}{\macro}}\]|}% -\[\xintGCFrac{\xintCntoGC {5}{\macro}}\] - -\subsection{\csbh{xintGCntoGC}}\label{xintGCntoGC} - -\csa{xintGCntoGC}|{N}{\macroA}{\macroB}|\etype{\numx ff} evaluates the -coefficients and then returns the corresponding -|{a0}+{b0}/{a1}+{b1}/{a2}+...+{b(N-1)}/{aN}| inline generalized fraction. |N| is -givent to a |\numexpr|. As shown, the coefficients are enclosed into added pairs -of braces, and may thus be fractions. \dverb|@ \def\an #1{\the\numexpr - #1*#1*#1+1\relax}% -\def\bn #1{\the\numexpr \xintiiMON{#1}*(#1+1)\relax}% -$\xintGCntoGC {5}{\an}{\bn}}=\xintGCFrac {\xintGCntoGC {5}{\an}{\bn}} = -\displaystyle\xintFrac {\xintGCntoF {5}{\an}{\bn}}$\par| - -\def\an #1{\the\numexpr #1*#1*#1+1\relax}% -\def\bn #1{\the\numexpr \xintiiMON{#1}*(#1+1)\relax}% -$\xintGCntoGC {5}{\an}{\bn}=\xintGCFrac {\xintGCntoGC {5}{\an}{\bn}} - = \displaystyle\xintFrac {\xintGCntoF {5}{\an}{\bn}}$\par - - - -\subsection{\csbh{xintiCstoF}, \csbh{xintiGCtoF}, \csbh{xint\-iCstoCv}, \csbh{xintiGCtoCv}}\label{xintiCstoF} -\label{xintiGCtoF} -\label{xintiCstoCv} -\label{xintiGCtoCv} - -The same as the corresponding macros without the `i', but for -integer-only input. Infinitesimally faster; to notice the higher -efficiency one would need to use them with an input having (at least) -hundreds of coefficients. - - -\subsection{\csbh{xintGCtoGC}}\label{xintGCtoGC} - -\csa{xintGCtoGC}|{a+b/c+d/e+f/g+......+v/w+x/y}|\etype{f} expands (with the -usual meaning) each one of the coefficients and returns an inline continued -fraction of the same type, each expanded coefficient being enclosed withing -braces. \dverb|@ \edef\x {\xintGCtoGC {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac - {6}+\xintCstoF {2,-7,-5}/16}} \meaning\x| - -\edef\x {\xintGCtoGC - {1+\xintPow{1.5}{3}/{1/7}+{-3/5}/\xintFac {6}+\xintCstoF {2,-7,-5}/16}} -\digitstt{\meaning\x} - -To be honest I have, it seems, forgotten why I wrote this macro in the -first place. - -% will be used by the \lverb things - -\def\givesomestretch{% -\fontdimen2\font=0.33333\fontdimen6\font -\fontdimen3\font=0.16666\fontdimen6\font -\fontdimen4\font=0.11111\fontdimen6\font -}% -\def\MacroFont{\ttfamily\small\givesomestretch\hyphenchar\font45 - \baselineskip12pt\relax } - - -\ifnum\NoSourceCode=1 -\bigskip -\begin{framed} - \ttfamily\small\givesomestretch\hyphenchar\font45 This documentation - has been compiled without the source code. To produce the - documentation with the source code included, run "tex xint.dtx" to - generate xint.tex (if not already available), then thrice latex on - xint.tex and finally dvipdfmx on xint.dvi (ignore the dvipdfmx - warnings; see also - \autoref{sec:install}). -\end{framed} -\fi - -\makeatletter -\StopEventually{\end{document}\endinput} - -\def\storedlinecounts {} -\def\StoreCodelineNo #1{\edef\storedlinecounts{% - \unexpanded\expandafter{\storedlinecounts}% - {{#1}{\the\c@CodelineNo}}}\c@CodelineNo\z@ } - -\makeatother - -\newgeometry{hmarginratio=4:3,hscale=0.75} - - -\etocdepthtag.toc {implementation} - -\MakePercentIgnore -% -% \catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 -% \let</dtx>\relax -% \def<*xinttools>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 } -%</dtx> -%<*xinttools> -% \def\MARGEPAGENO{2.5em} -% \section {Package \xinttoolsnameimp implementation} -% \label{sec:toolsimp} -% -% Release |1.09g| splits off |xinttools.sty| from |xint.sty|. -% -% \localtableofcontents -% -% \subsection{Catcodes, \protect\eTeX{} and reload detection} -% -% The method for package identification and reload detection is copied verbatim -% from the packages by \textsc{Heiko Oberdiek} (with some modifications starting -% with release |1.09b|). -% -% The method for catcodes was also inspired by these packages, we proceed -% slightly differently. -% -% Starting with version |1.06| of the package, also |`| must be -% catcode-protected, because we replace everywhere in the code the -% twice-expansion done with |\expandafter| by the systematic use of -% |\romannumeral-`0|. -% -% Starting with version |1.06b| I decide that I suffer from an indigestion of @ -% signs, so I replace them all with underscores |_|, \`a la \LaTeX 3. -% -% Release |1.09b| is more economical: some macros are defined already in -% |xint.sty| (now |xinttools.sty|) and re-used in other modules. All catcode -% changes have been unified and \csa{XINT_storecatcodes} will be used by each -% module to redefine |\XINT_restorecatcodes_endinput| in case catcodes have -% changed in-between the loading of |xint.sty| (now |xinttools.sty|) and the -% module (not very probable but...). -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode95=11 % _ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \expandafter\let\expandafter\x\csname ver@xint.sty\endcsname - \expandafter - \ifx\csname PackageInfo\endcsname\relax - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \else - \def\y#1#2{\PackageInfo{#1}{#2}}% - \fi - \expandafter - \ifx\csname numexpr\endcsname\relax - \y{xinttools}{\numexpr not available, aborting input}% - \aftergroup\endinput - \else - \ifx\x\relax % plain-TeX, first loading - \else - \def\empty {}% - \ifx\x\empty % LaTeX, first loading, - % variable is initialized, but \ProvidesPackage not yet seen - \else - \y{xinttools}{I was already loaded, aborting input}% - \aftergroup\endinput - \fi - \fi - \fi - \def\ChangeCatcodesIfInputNotAborted - {% - \endgroup - \def\XINT_storecatcodes - {% takes care of all, to allow more economical code in modules - \catcode34=\the\catcode34 % " xintbinhex, and 1.09k xintexpr - \catcode63=\the\catcode63 % ? xintexpr - \catcode124=\the\catcode124 % | xintexpr - \catcode38=\the\catcode38 % & xintexpr - \catcode64=\the\catcode64 % @ xintexpr - \catcode33=\the\catcode33 % ! xintexpr - \catcode93=\the\catcode93 % ] -, xintfrac, xintseries, xintcfrac - \catcode91=\the\catcode91 % [ -, xintfrac, xintseries, xintcfrac - \catcode36=\the\catcode36 % $ xintgcd only - \catcode94=\the\catcode94 % ^ - \catcode96=\the\catcode96 % ` - \catcode47=\the\catcode47 % / - \catcode41=\the\catcode41 % ) - \catcode40=\the\catcode40 % ( - \catcode42=\the\catcode42 % * - \catcode43=\the\catcode43 % + - \catcode62=\the\catcode62 % > - \catcode60=\the\catcode60 % < - \catcode58=\the\catcode58 % : - \catcode46=\the\catcode46 % . - \catcode45=\the\catcode45 % - - \catcode44=\the\catcode44 % , - \catcode35=\the\catcode35 % # - \catcode95=\the\catcode95 % _ - \catcode125=\the\catcode125 % } - \catcode123=\the\catcode123 % { - \endlinechar=\the\endlinechar - \catcode13=\the\catcode13 % ^^M - \catcode32=\the\catcode32 % - \catcode61=\the\catcode61\relax % = - }% - \edef\XINT_restorecatcodes_endinput - {% - \XINT_storecatcodes\noexpand\endinput % - }% - \def\XINT_setcatcodes - {% - \catcode61=12 % = - \catcode32=10 % space - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode95=11 % _ (replaces @ everywhere, starting with 1.06b) - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=11 % : (made letter for error cs) - \catcode60=12 % < - \catcode62=12 % > - \catcode43=12 % + - \catcode42=12 % * - \catcode40=12 % ( - \catcode41=12 % ) - \catcode47=12 % / - \catcode96=12 % ` (for ubiquitous \romannumeral-`0 and some \catcode ) - \catcode94=11 % ^ - \catcode36=3 % $ - \catcode91=12 % [ - \catcode93=12 % ] - \catcode33=11 % ! - \catcode64=11 % @ - \catcode38=12 % & - \catcode124=12 % | - \catcode63=11 % ? - \catcode34=12 % " missing from v < 1.09k although needed in xintbinhex - }% - \XINT_setcatcodes - }% -\ChangeCatcodesIfInputNotAborted -\def\XINTsetupcatcodes {% for use by other modules - \edef\XINT_restorecatcodes_endinput - {% - \XINT_storecatcodes\noexpand\endinput % - }% - \XINT_setcatcodes -}% -% \end{macrocode} -% \subsection{Package identification} -% -% Inspired from \textsc{Heiko Oberdiek}'s packages. Modified in |1.09b| to allow -% re-use in the other modules. Also I assume now that if |\ProvidesPackage| -% exists it then does define |\ver@<pkgname>.sty|, code of |HO| for some reason -% escaping me (compatibility with LaTeX 2.09 or other things ??) seems to set -% extra precautions. -% -% |1.09c| uses e-\TeX{} |\ifdefined|. -% \begin{macrocode} -\ifdefined\ProvidesPackage - \let\XINT_providespackage\relax -\else - \def\XINT_providespackage #1#2[#3]% - {\immediate\write-1{Package: #2 #3}% - \expandafter\xdef\csname ver@#2.sty\endcsname{#3}}% -\fi -\XINT_providespackage -\ProvidesPackage {xinttools}% - [2014/02/05 v1.09ka Expandable and non-expandable utilities (jfB)]% -% \end{macrocode} -% \subsection{Token management, constants} -% \lverb|In 1.09e \xint_undef replaced everywhere by \xint_bye. -% Release 1.09h makes most everything \long.| -% \begin{macrocode} -\long\def\xint_gobble_ {}% -\long\def\xint_gobble_i #1{}% -\long\def\xint_gobble_ii #1#2{}% -\long\def\xint_gobble_iii #1#2#3{}% -\long\def\xint_gobble_iv #1#2#3#4{}% -\long\def\xint_gobble_v #1#2#3#4#5{}% -\long\def\xint_gobble_vi #1#2#3#4#5#6{}% -\long\def\xint_gobble_vii #1#2#3#4#5#6#7{}% -\long\def\xint_gobble_viii #1#2#3#4#5#6#7#8{}% -\long\def\xint_firstofone #1{#1}% -\xint_firstofone{\let\XINT_sptoken= } %<- space here! -\long\def\xint_firstoftwo #1#2{#1}% -\long\def\xint_secondoftwo #1#2{#2}% -\long\def\xint_firstoftwo_thenstop #1#2{ #1}% -\long\def\xint_secondoftwo_thenstop #1#2{ #2}% -\def\xint_minus_thenstop { -}% -\long\def\xint_gob_til_R #1\R {}% -\long\def\xint_gob_til_W #1\W {}% -\long\def\xint_gob_til_Z #1\Z {}% -\long\def\xint_bye #1\xint_bye {}% -\let\xint_relax\relax -\def\xint_brelax {\xint_relax }% -\long\def\xint_gob_til_xint_relax #1\xint_relax {}% -\long\def\xint_afterfi #1#2\fi {\fi #1}% -\chardef\xint_c_ 0 -\chardef\xint_c_i 1 % 1.09k did not have it, but needed in \xintSeq -\chardef\xint_c_viii 8 -\newtoks\XINT_toks -% \end{macrocode} -% \subsection{ \csh{xintodef}, \csh{xintgodef}, \csh{odef}} -% \lverb|1.09i. For use in \xintAssign. No parameter text! 1.09j uses \xint... -% rather than \XINT_.... \xintAssign [o] will use the preexisting \odef if there -% was one before xint' loading.| -% \begin{macrocode} -\def\xintodef #1{\expandafter\def\expandafter#1\expandafter }% -\ifdefined\odef\else\let\odef\xintodef\fi -\def\xintgodef {\global\xintodef }% -% \end{macrocode} -% \subsection{ \csh{xintoodef}, \csh{xintgoodef}, \csh{oodef}} -% \lverb|1.09i. Can be prefixed with \global. No parameter text. The alternative -% $\ -% $null \def\oodef #1#{\def\XINT_tmpa{#1}%$\ -% $null $quad $quad $quad \expandafter\expandafter\expandafter\expandafter$\ -% $null $quad $quad $quad \expandafter\expandafter\expandafter\def$\ -% $null $quad $quad $quad \expandafter\expandafter\expandafter\XINT_tmpa$\ -% $null $quad $quad $quad \expandafter\expandafter\expandafter }%$\ -% could not be prefixed by \global. Anyhow, macro parameter tokens would have to -% somehow not be seen by expanded stuff, except if designed for it. -% \xintAssign [oo] (etc...) uses the pre-existing \oodef if there was one. | -% \begin{macrocode} -\def\xintoodef #1{\expandafter\expandafter\expandafter\def - \expandafter\expandafter\expandafter#1% - \expandafter\expandafter\expandafter }% -\ifdefined\oodef\else\let\oodef\xintoodef\fi -\def\xintgoodef {\global\xintoodef }% -% \end{macrocode} -% \subsection{ \csh{xintfdef}, \csh{xintgfdef}, \csh{fdef}} -% \lverb|1.09i. No parameter text! | -% \begin{macrocode} -\def\xintfdef #1#2{\expandafter\def\expandafter#1\expandafter - {\romannumeral-`0#2}}% -\ifdefined\fdef\else\let\fdef\xintfdef\fi -\def\xintgfdef {\global\xintfdef }% should be \global\fdef if \fdef pre-exists? -% \end{macrocode} -% \subsection{ \csh{xintReverseOrder}} -% \lverb|\xintReverseOrder: does NOT expand its argument.| -% \begin{macrocode} -\def\xintReverseOrder {\romannumeral0\xintreverseorder }% -\long\def\xintreverseorder #1% -{% - \XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\long\def\XINT_rord_main #1#2#3#4#5#6#7#8#9% -{% - \xint_bye #9\XINT_rord_cleanup\xint_bye - \XINT_rord_main {#9#8#7#6#5#4#3#2#1}% -}% -\long\edef\XINT_rord_cleanup\xint_bye\XINT_rord_main #1#2\xint_relax -{% - \noexpand\expandafter\space\noexpand\xint_gob_til_xint_relax #1% -}% -% \end{macrocode} -% \subsection{\csh{xintRevWithBraces}} -% \lverb|New with 1.06. Makes the expansion of its argument and then reverses -% the resulting tokens or braced tokens, adding a pair of braces to each (thus, -% maintaining it when it was already there. -% -% As in some other places, 1.09e replaces \Z by \xint_bye, although here it is -% just for coherence of notation as \Z would be perfectly safe. The reason for -% \xint_relax, here and in other locations, is in case #1 expands to nothing, -% the \romannumeral-`0 must be stopped| -% \begin{macrocode} -\def\xintRevWithBraces {\romannumeral0\xintrevwithbraces }% -\def\xintRevWithBracesNoExpand {\romannumeral0\xintrevwithbracesnoexpand }% -\long\def\xintrevwithbraces #1% -{% - \expandafter\XINT_revwbr_loop\expandafter{\expandafter}% - \romannumeral-`0#1\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye -}% -\long\def\xintrevwithbracesnoexpand #1% -{% - \XINT_revwbr_loop {}% - #1\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye -}% -\long\def\XINT_revwbr_loop #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_xint_relax #9\XINT_revwbr_finish_a\xint_relax - \XINT_revwbr_loop {{#9}{#8}{#7}{#6}{#5}{#4}{#3}{#2}#1}% -}% -\long\def\XINT_revwbr_finish_a\xint_relax\XINT_revwbr_loop #1#2\xint_bye -{% - \XINT_revwbr_finish_b #2\R\R\R\R\R\R\R\Z #1% -}% -\def\XINT_revwbr_finish_b #1#2#3#4#5#6#7#8\Z -{% - \xint_gob_til_R - #1\XINT_revwbr_finish_c 8% - #2\XINT_revwbr_finish_c 7% - #3\XINT_revwbr_finish_c 6% - #4\XINT_revwbr_finish_c 5% - #5\XINT_revwbr_finish_c 4% - #6\XINT_revwbr_finish_c 3% - #7\XINT_revwbr_finish_c 2% - \R\XINT_revwbr_finish_c 1\Z -}% -\def\XINT_revwbr_finish_c #1#2\Z -{% - \expandafter\expandafter\expandafter - \space - \csname xint_gobble_\romannumeral #1\endcsname -}% -% \end{macrocode} -% \subsection{\csh{xintLength}} -% \lverb|\xintLength does NOT expand its argument.$\ -% 1.09g adds the missing \xintlength, which was previously called \XINT_length, -% and suppresses \XINT_Length$\ -% 1.06: improved code is roughly 20$% faster than the one from earlier -% versions. 1.09a, \xintnum inserted. 1.09e: \Z->\xint_bye as this is called -% from \xintNthElt, and there it was necessary not to use \Z. Later use of \Z -% and \W perfectly safe here.| -% \begin{macrocode} -\def\xintLength {\romannumeral0\xintlength }% -\long\def\xintlength #1% -{% - \XINT_length_loop - {0}#1\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye -}% -\long\def\XINT_length_loop #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_xint_relax #9\XINT_length_finish_a\xint_relax - \expandafter\XINT_length_loop\expandafter {\the\numexpr #1+8\relax}% -}% -\def\XINT_length_finish_a\xint_relax - \expandafter\XINT_length_loop\expandafter #1#2\xint_bye -{% - \XINT_length_finish_b #2\W\W\W\W\W\W\W\Z {#1}% -}% -\def\XINT_length_finish_b #1#2#3#4#5#6#7#8\Z -{% - \xint_gob_til_W - #1\XINT_length_finish_c 8% - #2\XINT_length_finish_c 7% - #3\XINT_length_finish_c 6% - #4\XINT_length_finish_c 5% - #5\XINT_length_finish_c 4% - #6\XINT_length_finish_c 3% - #7\XINT_length_finish_c 2% - \W\XINT_length_finish_c 1\Z -}% -\edef\XINT_length_finish_c #1#2\Z #3% - {\noexpand\expandafter\space\noexpand\the\numexpr #3-#1\relax}% -% \end{macrocode} -% \subsection{\csh{xintZapFirstSpaces}} -% \lverb|1.09f, written [2013/11/01].| -% \begin{macrocode} -\def\xintZapFirstSpaces {\romannumeral0\xintzapfirstspaces }% -% \end{macrocode} -% \lverb|defined via an \edef in order to inject space tokens inside.| -% \begin{macrocode} -\long\edef\xintzapfirstspaces #1% - {\noexpand\XINT_zapbsp_a \space #1\space\space\noexpand\xint_bye\xint_relax }% -\xint_firstofone {\long\def\XINT_zapbsp_a #1 } %<- space token here -{% -% \end{macrocode} -% \lverb|If the original #1 started with a space, here #1 will be in fact empty, -% so the effect will be to remove precisely one space from the original, because -% the first two space tokens are matched to the ones of the macro parameter -% text. If the original #1 did not start with a space then the #1 will be this -% original #1, with its added first space, up to the first <sp><sp> found. The -% added initial space will stop later the \romannumeral0. And in -% \xintZapLastSpaces we also carried along a space in order to be able to mix -% tne two codes in \xintZapSpaces. Testing for \emptiness of #1 is NOT done with -% an \if test because #1 may contain \if, \fi things (one could use a -% \detokenize method), and also because xint.sty has a style of its own for -% doing these things...| -% \begin{macrocode} - \XINT_zapbsp_again? #1\xint_bye\XINT_zapbsp_b {#1}% -% \end{macrocode} -% \lverb|The #1 above is thus either empty, or it starts with a (char 32) space -% token followed with a non (char 32) space token and at any rate #1 is -% protected from brace stripping. It is assumed that the initial input does not -% contain space tokens of other than 32 as character code.| -% \begin{macrocode} -}% -\long\def\XINT_zapbsp_again? #1{\xint_bye #1\XINT_zapbsp_again }% -% \end{macrocode} -% \lverb|In the "empty" situation above, here #1=\xint_bye, else #1 could be -% some brace things, but unbracing will anyhow not reveal any \xint_bye. When we -% do below \XINT_zapbsp_again we recall that we have stripped two spaces out of -% <sp><original #1>, so we have one <sp> less in #1, and when we loop we better -% not forget to re-insert one initial <sp>.| -% \begin{macrocode} -\edef\XINT_zapbsp_again\XINT_zapbsp_b #1{\noexpand\XINT_zapbsp_a\space }% -% \end{macrocode} -% \lverb|We now have now gotten rid of the initial spaces, but #1 perhaps extend -% only to some initial chunk which was delimited by <sp><sp>.| -% \begin{macrocode} -\long\def\XINT_zapbsp_b #1#2\xint_relax - {\XINT_zapbsp_end? #2\XINT_zapbsp_e\empty #2{#1}}% -% \end{macrocode} -% \lverb|If the initial chunk up to <sp><sp> (after stripping away the first -% spaces) was maximal, then #2 above is some spaces followed by \xint_bye, so in -% the \XINT_zapbsp_end? below it appears as \xint_bye, else the #1 below will -% not be nor give rise after brace removal to \xint_bye. And then the original -% \xint_bye in #2 will have the effect that all is swallowed and we continue -% with \XINT_zapbsp_e. If the chunk was maximal, then the #2 above contains as -% many space tokens as there were originally at the end.| -% \begin{macrocode} -\long\def\XINT_zapbsp_end? #1{\xint_bye #1\XINT_zapbsp_end }% -% \end{macrocode} -% \lverb|The #2 starts with a space which stops the \romannumeral. -% The #1 contains the same number of space tokens there was originally.| -% \begin{macrocode} -\long\def\XINT_zapbsp_end\XINT_zapbsp_e\empty #1\xint_bye #2{#2#1}% -% \end{macrocode} -% \lverb|& -% Here the initial chunk was not maximal. So we need to get a second piece -% all the way up to \xint_bye, we take this opportunity to remove the two -% initially added ending space tokens. We inserted an \empty to prevent brace -% removal. The \expandafter get rid of the \empty.| -% \begin{macrocode} -\xint_firstofone{\long\def\XINT_zapbsp_e #1 } \xint_bye - {\expandafter\XINT_zapbsp_f \expandafter{#1}}% -% \end{macrocode} -% \lverb|Let's not forget when we glue to reinsert the two intermediate space -% tokens. | -% \begin{macrocode} -\long\edef\XINT_zapbsp_f #1#2{#2\space\space #1}% -% \end{macrocode} -% \subsection{\csh{xintZapLastSpaces}} -% \lverb+1.09f, written [2013/11/01].+ -% \begin{macrocode} -\def\xintZapLastSpaces {\romannumeral0\xintzaplastspaces }% -% \end{macrocode} -% \lverb|Next macro is defined via an \edef for the space tokens.| -% \begin{macrocode} -\long\edef\xintzaplastspaces #1{\noexpand\XINT_zapesp_a {\space}\noexpand\empty - #1\space\space\noexpand\xint_bye \xint_relax}% -% \end{macrocode} -% \lverb|This creates a delimited macro with two space tokens:| -% \begin{macrocode} -\xint_firstofone {\long\def\XINT_zapesp_a #1#2 } %<- second space here - {\expandafter\XINT_zapesp_b\expandafter{#2}{#1}}% -% \end{macrocode} -% \lverb|The \empty from \xintzaplastspaces is to prevent brace removal in the -% #2 above. The \expandafter chain removes it.| -% \begin{macrocode} -\long\def\XINT_zapesp_b #1#2#3\xint_relax - {\XINT_zapesp_end? #3\XINT_zapesp_e {#2#1}\empty #3\xint_relax }% -% \end{macrocode} -% \lverb|& -% When we have reached the ending space tokens, #3 is a bunch of spaces followed -% by \xint_bye. So the #1 below will be \xint_bye. In all other cases #1 can not -% be \xint_bye nor can it give birth to it via brace stripping.| -% \begin{macrocode} -\long\def\XINT_zapesp_end? #1{\xint_bye #1\XINT_zapesp_end }% -% \end{macrocode} -% \lverb|& -% We are done. The #1 here has accumulated all the previous material. It started -% with a space token which stops the \romannumeral0. The reason for the space is -% the recycling of this code in \xintZapSpaces.| -% \begin{macrocode} -\long\def\XINT_zapesp_end\XINT_zapesp_e #1#2\xint_relax {#1}% -% \end{macrocode} -% \lverb|We haven't yet reached the end, so we need to re-inject two space -% tokens after what we have gotten so far. Then we loop. We might wonder why in -% \XINT_zapesp_b we scooped everything up to the end, rather than trying to test -% if the next thing was a bunch of spaces followed by \xint_bye\xint_relax. But -% how can we expandably examine what comes next? if we pick up something as -% undelimited parameter token we risk brace removal and we will never know about -% it so we cannot reinsert correctly; the only way is to gather a delimited -% macro parameter and be sure some token will be inside to forbid brace removal. -% I do not see (so far) any other way than scooping everything up to the end. -% Anyhow, 99$% of the use cases will NOT have <sp><sp> inside!.| -% \begin{macrocode} -\long\edef\XINT_zapesp_e #1{\noexpand \XINT_zapesp_a {#1\space\space}}% -% \end{macrocode} -% \subsection{\csh{xintZapSpaces}} -% \lverb+1.09f, written [2013/11/01].+ -% \begin{macrocode} -\def\xintZapSpaces {\romannumeral0\xintzapspaces }% -% \end{macrocode} -% \lverb|We start like \xintZapStartSpaces.| -% \begin{macrocode} -\long\edef\xintzapspaces #1% - {\noexpand\XINT_zapsp_a \space #1\space\space\noexpand\xint_bye\xint_relax}% -% \end{macrocode} -% \lverb|& -% Once the loop stripping the starting spaces is done, we plug into the -% \xintZapLast$-Spaces code via \XINT_zapesp_b. As our #1 will always have an -% initial space, this is why we arranged code of \xintZapLastSpaces to do the -% same.| -% \begin{macrocode} -\xint_firstofone {\long\def\XINT_zapsp_a #1 } %<- space token here -{% - \XINT_zapsp_again? #1\xint_bye\XINT_zapesp_b {#1}{}% -}% -\long\def\XINT_zapsp_again? #1{\xint_bye #1\XINT_zapsp_again }% -\long\edef\XINT_zapsp_again\XINT_zapesp_b #1#2{\noexpand\XINT_zapsp_a\space }% -% \end{macrocode} -% \subsection{\csh{xintZapSpacesB}} -% \lverb+1.09f, written [2013/11/01].+ -% \begin{macrocode} -\def\xintZapSpacesB {\romannumeral0\xintzapspacesb }% -\long\def\xintzapspacesb #1{\XINT_zapspb_one? #1\xint_relax\xint_relax - \xint_bye\xintzapspaces {#1}}% -\long\def\XINT_zapspb_one? #1#2% - {\xint_gob_til_xint_relax #1\XINT_zapspb_onlyspaces\xint_relax - \xint_gob_til_xint_relax #2\XINT_zapspb_bracedorone\xint_relax - \xint_bye {#1}}% -\def\XINT_zapspb_onlyspaces\xint_relax - \xint_gob_til_xint_relax\xint_relax\XINT_zapspb_bracedorone\xint_relax - \xint_bye #1\xint_bye\xintzapspaces #2{ }% -\long\def\XINT_zapspb_bracedorone\xint_relax - \xint_bye #1\xint_relax\xint_bye\xintzapspaces #2{ #1}% -% \end{macrocode} -% \subsection{\csh{xintCSVtoList}, \csh{xintCSVtoListNonStripped}} -% \lverb|& -% \xintCSVtoList transforms a,b,..,z into {a}{b}...{z}. The comma separated list -% may be a macro which is first expanded (protect the first item with a space if -% it is not to be expanded). First included in release 1.06. Here, use of \Z -% (and \R) perfectly safe. -% -% [2013/11/02]: Starting with 1.09f, automatically filters items through -% \xintZapSpacesB to strip off all spaces around commas, and spaces at the start -% and end of the list. The original is kept as \xintCSVtoListNonStripped, and is -% faster. But ... it doesn't strip spaces.| -% \begin{macrocode} -\def\xintCSVtoList {\romannumeral0\xintcsvtolist }% -\long\def\xintcsvtolist #1{\expandafter\xintApply - \expandafter\xintzapspacesb - \expandafter{\romannumeral0\xintcsvtolistnonstripped{#1}}}% -\def\xintCSVtoListNoExpand {\romannumeral0\xintcsvtolistnoexpand }% -\long\def\xintcsvtolistnoexpand #1{\expandafter\xintApply - \expandafter\xintzapspacesb - \expandafter{\romannumeral0\xintcsvtolistnonstrippednoexpand{#1}}}% -\def\xintCSVtoListNonStripped {\romannumeral0\xintcsvtolistnonstripped }% -\def\xintCSVtoListNonStrippedNoExpand - {\romannumeral0\xintcsvtolistnonstrippednoexpand }% -\long\def\xintcsvtolistnonstripped #1% -{% - \expandafter\XINT_csvtol_loop_a\expandafter - {\expandafter}\romannumeral-`0#1% - ,\xint_bye,\xint_bye,\xint_bye,\xint_bye - ,\xint_bye,\xint_bye,\xint_bye,\xint_bye,\Z -}% -\long\def\xintcsvtolistnonstrippednoexpand #1% -{% - \XINT_csvtol_loop_a - {}#1,\xint_bye,\xint_bye,\xint_bye,\xint_bye - ,\xint_bye,\xint_bye,\xint_bye,\xint_bye,\Z -}% -\long\def\XINT_csvtol_loop_a #1#2,#3,#4,#5,#6,#7,#8,#9,% -{% - \xint_bye #9\XINT_csvtol_finish_a\xint_bye - \XINT_csvtol_loop_b {#1}{{#2}{#3}{#4}{#5}{#6}{#7}{#8}{#9}}% -}% -\long\def\XINT_csvtol_loop_b #1#2{\XINT_csvtol_loop_a {#1#2}}% -\long\def\XINT_csvtol_finish_a\xint_bye\XINT_csvtol_loop_b #1#2#3\Z -{% - \XINT_csvtol_finish_b #3\R,\R,\R,\R,\R,\R,\R,\Z #2{#1}% -}% -\def\XINT_csvtol_finish_b #1,#2,#3,#4,#5,#6,#7,#8\Z -{% - \xint_gob_til_R - #1\XINT_csvtol_finish_c 8% - #2\XINT_csvtol_finish_c 7% - #3\XINT_csvtol_finish_c 6% - #4\XINT_csvtol_finish_c 5% - #5\XINT_csvtol_finish_c 4% - #6\XINT_csvtol_finish_c 3% - #7\XINT_csvtol_finish_c 2% - \R\XINT_csvtol_finish_c 1\Z -}% -\def\XINT_csvtol_finish_c #1#2\Z -{% - \csname XINT_csvtol_finish_d\romannumeral #1\endcsname -}% -\long\def\XINT_csvtol_finish_dviii #1#2#3#4#5#6#7#8#9{ #9}% -\long\def\XINT_csvtol_finish_dvii #1#2#3#4#5#6#7#8#9{ #9{#1}}% -\long\def\XINT_csvtol_finish_dvi #1#2#3#4#5#6#7#8#9{ #9{#1}{#2}}% -\long\def\XINT_csvtol_finish_dv #1#2#3#4#5#6#7#8#9{ #9{#1}{#2}{#3}}% -\long\def\XINT_csvtol_finish_div #1#2#3#4#5#6#7#8#9{ #9{#1}{#2}{#3}{#4}}% -\long\def\XINT_csvtol_finish_diii #1#2#3#4#5#6#7#8#9{ #9{#1}{#2}{#3}{#4}{#5}}% -\long\def\XINT_csvtol_finish_dii #1#2#3#4#5#6#7#8#9% - { #9{#1}{#2}{#3}{#4}{#5}{#6}}% -\long\def\XINT_csvtol_finish_di #1#2#3#4#5#6#7#8#9% - { #9{#1}{#2}{#3}{#4}{#5}{#6}{#7}}% -% \end{macrocode} -% \subsection{\csh{xintListWithSep}} -% \lverb|& -% \xintListWithSep {\sep}{{a}{b}...{z}} returns a \sep b \sep .... \sep z$\ -% Included in release 1.04. The 'sep' can be \par's: the macro -% xintlistwithsep etc... are all declared long. 'sep' does not have to be a -% single token. It is not expanded. The list may be a macro and it is expanded. -% 1.06 modifies the `feature' of returning sep if the list is empty: the output -% is now empty in that case. (sep was not used for a one element list, but -% strangely it was for a zero-element list). -% -% Use of \Z as delimiter was objectively an error, which I fix here in 1.09e, -% now the code uses \xint_bye.| -% \begin{macrocode} -\def\xintListWithSep {\romannumeral0\xintlistwithsep }% -\def\xintListWithSepNoExpand {\romannumeral0\xintlistwithsepnoexpand }% -\long\def\xintlistwithsep #1#2% - {\expandafter\XINT_lws\expandafter {\romannumeral-`0#2}{#1}}% -\long\def\XINT_lws #1#2{\XINT_lws_start {#2}#1\xint_bye }% -\long\def\xintlistwithsepnoexpand #1#2{\XINT_lws_start {#1}#2\xint_bye }% -\long\def\XINT_lws_start #1#2% -{% - \xint_bye #2\XINT_lws_dont\xint_bye - \XINT_lws_loop_a {#2}{#1}% -}% -\long\def\XINT_lws_dont\xint_bye\XINT_lws_loop_a #1#2{ }% -\long\def\XINT_lws_loop_a #1#2#3% -{% - \xint_bye #3\XINT_lws_end\xint_bye - \XINT_lws_loop_b {#1}{#2#3}{#2}% -}% -\long\def\XINT_lws_loop_b #1#2{\XINT_lws_loop_a {#1#2}}% -\long\def\XINT_lws_end\xint_bye\XINT_lws_loop_b #1#2#3{ #1}% -% \end{macrocode} -% \subsection{\csh{xintNthElt}} -% \lverb|& -% \xintNthElt {i}{{a}{b}...{z}} (or `tokens' abcd...z) returns the i th -% element (one pair of braces removed). The list is first expanded. -% First included in release 1.06. With 1.06a, a value of i = 0 (or negative) -% makes the macro return the length. This is different from \xintLen which is -% for numbers (checks sign) and different from \xintLength which does not first -% expand its argument. With 1.09b, only i=0 gives the length, negative values -% return the i th element from the end. 1.09c has some slightly less quick -% initial preparation (if #2 is very long, not good to have it twice), I wanted -% to respect the noexpand directive in all cases, and the alternative would be -% to define more macros. -% -% At some point I turned the \W's into \xint_relax's but forgot to modify -% accordingly \XINT_nthelt_finish. So in case the index is larger than the -% number of items the macro returned was an \xint_relax token rather than -% nothing. Fixed in 1.09e. I also take the opportunity of this fix to replace -% uses of \Z by \xint_bye. (and as a result I must do the change also in -% \XINT_length_loop and related macros). -% | -% \begin{macrocode} -\def\xintNthElt {\romannumeral0\xintnthelt }% -\def\xintNthEltNoExpand {\romannumeral0\xintntheltnoexpand }% -\def\xintnthelt #1% -{% - \expandafter\XINT_nthelt_a\expandafter {\the\numexpr #1}% -}% -\def\xintntheltnoexpand #1% -{% - \expandafter\XINT_ntheltnoexpand_a\expandafter {\the\numexpr #1}% -}% -\long\def\XINT_nthelt_a #1#2% -{% - \ifnum #1<0 - \xint_afterfi{\expandafter\XINT_nthelt_c\expandafter - {\romannumeral0\xintrevwithbraces {#2}}{-#1}}% - \else - \xint_afterfi{\expandafter\XINT_nthelt_c\expandafter - {\romannumeral-`0#2}{#1}}% - \fi -}% -\long\def\XINT_ntheltnoexpand_a #1#2% -{% - \ifnum #1<0 - \xint_afterfi{\expandafter\XINT_nthelt_c\expandafter - {\romannumeral0\xintrevwithbracesnoexpand {#2}}{-#1}}% - \else - \xint_afterfi{\expandafter\XINT_nthelt_c\expandafter - {#2}{#1}}% - \fi -}% -\long\def\XINT_nthelt_c #1#2% -{% - \ifnum #2>\xint_c_ - \expandafter\XINT_nthelt_loop_a - \else - \expandafter\XINT_length_loop - \fi {#2}#1\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye -}% -\def\XINT_nthelt_loop_a #1% -{% - \ifnum #1>\xint_c_viii - \expandafter\XINT_nthelt_loop_b - \else - \expandafter\XINT_nthelt_getit - \fi - {#1}% -}% -\long\def\XINT_nthelt_loop_b #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_xint_relax #9\XINT_nthelt_silentend\xint_relax - \expandafter\XINT_nthelt_loop_a\expandafter{\the\numexpr #1-8}% -}% -\def\XINT_nthelt_silentend #1\xint_bye { }% -\def\XINT_nthelt_getit #1% -{% - \expandafter\expandafter\expandafter\XINT_nthelt_finish - \csname xint_gobble_\romannumeral\numexpr#1-1\endcsname -}% -\long\edef\XINT_nthelt_finish #1#2\xint_bye - {\noexpand\xint_gob_til_xint_relax #1\noexpand\expandafter\space - \noexpand\xint_gobble_iii\xint_relax\space #1}% -% \end{macrocode} -% \subsection{\csh{xintApply}} -% \lverb|& -% \xintApply {\macro}{{a}{b}...{z}} returns {\macro{a}}...{\macro{b}} -% where each instance of \macro is ff-expanded. The list is first -% expanded and may thus be a macro. Introduced with release 1.04. -% -% Modified in 1.09e to not use \Z but rather \xint_bye.| -% \begin{macrocode} -\def\xintApply {\romannumeral0\xintapply }% -\def\xintApplyNoExpand {\romannumeral0\xintapplynoexpand }% -\long\def\xintapply #1#2% -{% - \expandafter\XINT_apply\expandafter {\romannumeral-`0#2}% - {#1}% -}% -\long\def\XINT_apply #1#2{\XINT_apply_loop_a {}{#2}#1\xint_bye }% -\long\def\xintapplynoexpand #1#2{\XINT_apply_loop_a {}{#1}#2\xint_bye }% -\long\def\XINT_apply_loop_a #1#2#3% -{% - \xint_bye #3\XINT_apply_end\xint_bye - \expandafter - \XINT_apply_loop_b - \expandafter {\romannumeral-`0#2{#3}}{#1}{#2}% -}% -\long\def\XINT_apply_loop_b #1#2{\XINT_apply_loop_a {#2{#1}}}% -\long\def\XINT_apply_end\xint_bye\expandafter\XINT_apply_loop_b - \expandafter #1#2#3{ #2}% -% \end{macrocode} -% \subsection{\csh{xintApplyUnbraced}} -% \lverb|& -% \xintApplyUnbraced {\macro}{{a}{b}...{z}} returns \macro{a}...\macro{z} -% where each instance of \macro is expanded using \romannumeral-`0. The second -% argument may be a macro as it is first expanded itself (fully). No braces -% are added: this allows for example a non-expandable \def in \macro, without -% having to do \gdef. The list is first expanded. Introduced with release 1.06b. -% Define \macro to start with a space if it is not expandable or its execution -% should be delayed only when all of \macro{a}...\macro{z} is ready. -% -% Modified in 1.09e to use \xint_bye rather than \Z.| -% \begin{macrocode} -\def\xintApplyUnbraced {\romannumeral0\xintapplyunbraced }% -\def\xintApplyUnbracedNoExpand {\romannumeral0\xintapplyunbracednoexpand }% -\long\def\xintapplyunbraced #1#2% -{% - \expandafter\XINT_applyunbr\expandafter {\romannumeral-`0#2}% - {#1}% -}% -\long\def\XINT_applyunbr #1#2{\XINT_applyunbr_loop_a {}{#2}#1\xint_bye }% -\long\def\xintapplyunbracednoexpand #1#2% - {\XINT_applyunbr_loop_a {}{#1}#2\xint_bye }% -\long\def\XINT_applyunbr_loop_a #1#2#3% -{% - \xint_bye #3\XINT_applyunbr_end\xint_bye - \expandafter\XINT_applyunbr_loop_b - \expandafter {\romannumeral-`0#2{#3}}{#1}{#2}% -}% -\long\def\XINT_applyunbr_loop_b #1#2{\XINT_applyunbr_loop_a {#2#1}}% -\long\def\XINT_applyunbr_end\xint_bye\expandafter\XINT_applyunbr_loop_b - \expandafter #1#2#3{ #2}% -% \end{macrocode} -% \subsection{\csh{xintSeq}} -% \lverb|1.09c. Without the optional argument puts stress on the input stack, -% should not be used to generated thousands of terms then. Here also, let's use -% \xint_bye rather than \Z as delimiter (1.09e; necessary change as #1 is used -% prior to being expanded, thus \Z might very well arise here as a macro).| -% \begin{macrocode} -\def\xintSeq {\romannumeral0\xintseq }% -\def\xintseq #1{\XINT_seq_chkopt #1\xint_bye }% -\def\XINT_seq_chkopt #1% -{% - \ifx [#1\expandafter\XINT_seq_opt - \else\expandafter\XINT_seq_noopt - \fi #1% -}% -\def\XINT_seq_noopt #1\xint_bye #2% -{% - \expandafter\XINT_seq\expandafter - {\the\numexpr#1\expandafter}\expandafter{\the\numexpr #2}% -}% -\def\XINT_seq #1#2% -{% - \ifcase\ifnum #1=#2 0\else\ifnum #2>#1 1\else -1\fi\fi\space - \expandafter\xint_firstoftwo_thenstop - \or - \expandafter\XINT_seq_p - \else - \expandafter\XINT_seq_n - \fi - {#2}{#1}% -}% -\def\XINT_seq_p #1#2% -{% - \ifnum #1>#2 - \expandafter\expandafter\expandafter\XINT_seq_p - \else - \expandafter\XINT_seq_e - \fi - \expandafter{\the\numexpr #1-\xint_c_i}{#2}{#1}% -}% -\def\XINT_seq_n #1#2% -{% - \ifnum #1<#2 - \expandafter\expandafter\expandafter\XINT_seq_n - \else - \expandafter\XINT_seq_e - \fi - \expandafter{\the\numexpr #1+\xint_c_i}{#2}{#1}% -}% -\def\XINT_seq_e #1#2#3{ }% -\def\XINT_seq_opt [\xint_bye #1]#2#3% -{% - \expandafter\XINT_seqo\expandafter - {\the\numexpr #2\expandafter}\expandafter - {\the\numexpr #3\expandafter}\expandafter - {\the\numexpr #1}% -}% -\def\XINT_seqo #1#2% -{% - \ifcase\ifnum #1=#2 0\else\ifnum #2>#1 1\else -1\fi\fi\space - \expandafter\XINT_seqo_a - \or - \expandafter\XINT_seqo_pa - \else - \expandafter\XINT_seqo_na - \fi - {#1}{#2}% -}% -\def\XINT_seqo_a #1#2#3{ {#1}}% -\def\XINT_seqo_o #1#2#3#4{ #4}% -\def\XINT_seqo_pa #1#2#3% -{% - \ifcase\ifnum #3=\xint_c_ 0\else\ifnum #3>\xint_c_ 1\else -1\fi\fi\space - \expandafter\XINT_seqo_o - \or - \expandafter\XINT_seqo_pb - \else - \xint_afterfi{\expandafter\space\xint_gobble_iv}% - \fi - {#1}{#2}{#3}{{#1}}% -}% -\def\XINT_seqo_pb #1#2#3% -{% - \expandafter\XINT_seqo_pc\expandafter{\the\numexpr #1+#3}{#2}{#3}% -}% -\def\XINT_seqo_pc #1#2% -{% - \ifnum #1>#2 - \expandafter\XINT_seqo_o - \else - \expandafter\XINT_seqo_pd - \fi - {#1}{#2}% -}% -\def\XINT_seqo_pd #1#2#3#4{\XINT_seqo_pb {#1}{#2}{#3}{#4{#1}}}% -\def\XINT_seqo_na #1#2#3% -{% - \ifcase\ifnum #3=\xint_c_ 0\else\ifnum #3>\xint_c_ 1\else -1\fi\fi\space - \expandafter\XINT_seqo_o - \or - \xint_afterfi{\expandafter\space\xint_gobble_iv}% - \else - \expandafter\XINT_seqo_nb - \fi - {#1}{#2}{#3}{{#1}}% -}% -\def\XINT_seqo_nb #1#2#3% -{% - \expandafter\XINT_seqo_nc\expandafter{\the\numexpr #1+#3}{#2}{#3}% -}% -\def\XINT_seqo_nc #1#2% -{% - \ifnum #1<#2 - \expandafter\XINT_seqo_o - \else - \expandafter\XINT_seqo_nd - \fi - {#1}{#2}% -}% -\def\XINT_seqo_nd #1#2#3#4{\XINT_seqo_nb {#1}{#2}{#3}{#4{#1}}}% -% \end{macrocode} -%\subsection{\csh{xintloop}, \csh{xintbreakloop}, \csh{xintbreakloopanddo}, -% \csh{xintloopskiptonext}} -% \lverb|1.09g [2013/11/22]. Made long with 1.09h.| -% \begin{macrocode} -\long\def\xintloop #1#2\repeat {#1#2\xintloop_again\fi\xint_gobble_i {#1#2}}% -\long\def\xintloop_again\fi\xint_gobble_i #1{\fi - #1\xintloop_again\fi\xint_gobble_i {#1}}% -\long\def\xintbreakloop #1\xintloop_again\fi\xint_gobble_i #2{}% -\long\def\xintbreakloopanddo #1#2\xintloop_again\fi\xint_gobble_i #3{#1}% -\long\def\xintloopskiptonext #1\xintloop_again\fi\xint_gobble_i #2{% - #2\xintloop_again\fi\xint_gobble_i {#2}}% -% \end{macrocode} -% \subsection{\csh{xintiloop}, \csh{xintiloopindex}, \csh{xintouteriloopindex}, -% \csh{xintbreakiloop}, \csh{xintbreakiloopanddo}, \csh{xintiloopskiptonext}, -% \csh{xintiloopskipandredo}} -% \lverb|1.09g [2013/11/22]. Made long with 1.09h.| -% \begin{macrocode} -\def\xintiloop [#1+#2]{% - \expandafter\xintiloop_a\the\numexpr #1\expandafter.\the\numexpr #2.}% -\long\def\xintiloop_a #1.#2.#3#4\repeat{% - #3#4\xintiloop_again\fi\xint_gobble_iii {#1}{#2}{#3#4}}% -\def\xintiloop_again\fi\xint_gobble_iii #1#2{% - \fi\expandafter\xintiloop_again_b\the\numexpr#1+#2.#2.}% -\long\def\xintiloop_again_b #1.#2.#3{% - #3\xintiloop_again\fi\xint_gobble_iii {#1}{#2}{#3}}% -\long\def\xintbreakiloop #1\xintiloop_again\fi\xint_gobble_iii #2#3#4{}% -\long\def\xintbreakiloopanddo - #1.#2\xintiloop_again\fi\xint_gobble_iii #3#4#5{#1}% -\long\def\xintiloopindex #1\xintiloop_again\fi\xint_gobble_iii #2% - {#2#1\xintiloop_again\fi\xint_gobble_iii {#2}}% -\long\def\xintouteriloopindex #1\xintiloop_again - #2\xintiloop_again\fi\xint_gobble_iii #3% - {#3#1\xintiloop_again #2\xintiloop_again\fi\xint_gobble_iii {#3}}% -\long\def\xintiloopskiptonext #1\xintiloop_again\fi\xint_gobble_iii #2#3{% - \expandafter\xintiloop_again_b \the\numexpr#2+#3.#3.}% -\long\def\xintiloopskipandredo #1\xintiloop_again\fi\xint_gobble_iii #2#3#4{% - #4\xintiloop_again\fi\xint_gobble_iii {#2}{#3}{#4}}% -% \end{macrocode} -% \subsection{\csh{XINT\_xflet}} -% \lverb|1.09e [2013/10/29]: we expand fully unbraced tokens and swallow arising -% space tokens until the dust settles. For treating cases -% {<blank>\x<blank>\y...}, with guaranteed expansion of the \x (which may itself -% give space tokens), a simpler approach is possible with doubled -% \romannumeral-`0, this is what I first did, but it had the feature that -% <sptoken><sptoken>\x would not expand the \x. At any rate, <sptoken>'s before -% the list terminator z were all correctly moved out of the way, hence the stuff -% was robust for use in (the then current versions of) \xintApplyInline and -% \xintFor. Although *two* space tokens would need devilishly prepared input, -% nevertheless I decided to also survive that, so here the method is a bit more -% complicated. But it simplifies things on the caller side.| -% \begin{macrocode} -\def\XINT_xflet #1% -{% - \def\XINT_xflet_macro {#1}\XINT_xflet_zapsp -}% -\def\XINT_xflet_zapsp -{% - \expandafter\futurelet\expandafter\XINT_token - \expandafter\XINT_xflet_sp?\romannumeral-`0% -}% -\def\XINT_xflet_sp? -{% - \ifx\XINT_token\XINT_sptoken - \expandafter\XINT_xflet_zapsp - \else\expandafter\XINT_xflet_zapspB - \fi -}% -\def\XINT_xflet_zapspB -{% - \expandafter\futurelet\expandafter\XINT_tokenB - \expandafter\XINT_xflet_spB?\romannumeral-`0% -}% -\def\XINT_xflet_spB? -{% - \ifx\XINT_tokenB\XINT_sptoken - \expandafter\XINT_xflet_zapspB - \else\expandafter\XINT_xflet_eq? - \fi -}% -\def\XINT_xflet_eq? -{% - \ifx\XINT_token\XINT_tokenB - \expandafter\XINT_xflet_macro - \else\expandafter\XINT_xflet_zapsp - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintApplyInline}} -% \lverb|& -% 1.09a: \xintApplyInline\macro{{a}{b}...{z}} has the same effect as executing -% \macro{a} and then applying again \xintApplyInline to the shortened list -% {{b}...{z}} until -% nothing is left. This is a non-expandable command which will result in -% quicker code than using -% \xintApplyUnbraced. It expands (fully) its second (list) argument -% first, which may thus be encapsulated in a macro. -% -% Release 1.09c has a new \xintApplyInline: the new version, while not -% expandable, is designed to survive when the applied macro closes a group, as -% is the case in alignemnts when it contains a $& or \\. It uses catcode 3 Z as -% list terminator. Don't use it among the list items. -% -% 1.09d: the bug which was discovered in \xintFor* regarding space tokens at the -% very end of the item list also was in \xintApplyInline. The new version will -% expand unbraced item elements and this is in fact convenient to simulate -% insertion of lists in others. -% -% 1.09e: the applied macro is allowed to be long, with items (or the first fixed -% arguments of he macro, passed together with it as #1 to \xintApplyInline) -% containing explicit \par's. (1.09g: some missing \long's added) -% -% 1.09f: terminator used to be z, now Z (still catcode 3). -%| -% \begin{macrocode} -\catcode`Z 3 -\long\def\xintApplyInline #1#2% -{% - \long\expandafter\def\expandafter\XINT_inline_macro - \expandafter ##\expandafter 1\expandafter {#1{##1}}% - \XINT_xflet\XINT_inline_b #2Z% this Z has catcode 3 -}% -\def\XINT_inline_b -{% - \ifx\XINT_token Z\expandafter\xint_gobble_i - \else\expandafter\XINT_inline_d\fi -}% -\long\def\XINT_inline_d #1% -{% - \long\def\XINT_item{{#1}}\XINT_xflet\XINT_inline_e -}% -\def\XINT_inline_e -{% - \ifx\XINT_token Z\expandafter\XINT_inline_w - \else\expandafter\XINT_inline_f\fi -}% -\def\XINT_inline_f -{% - \expandafter\XINT_inline_g\expandafter{\XINT_inline_macro {##1}}% -}% -\long\def\XINT_inline_g #1% -{% - \expandafter\XINT_inline_macro\XINT_item - \long\def\XINT_inline_macro ##1{#1}\XINT_inline_d -}% -\def\XINT_inline_w #1% -{% - \expandafter\XINT_inline_macro\XINT_item -}% -% \end{macrocode} -% \subsection{\csh{xintFor}, -% \csh{xintFor*}, \csh{xintBreakFor}, \csh{xintBreakForAndDo}} -% \lverb|1.09c [2013/10/09]: a new kind of loop which uses macro parameters -% #1, #2, #3, #4 rather than macros; while not expandable it survives executing -% code closing groups, like what happens in an alignment with the $& character. -% When inserted in a macro for later use, the # character must be doubled. -% -% The non-star variant works on a csv list, which it expands once, the -% star variant works on a token list, expanded fully. -% -% 1.09d: [2013/10/22] \xintFor* crashed when a space token was at the very end -% of the list. It is crucial in this code to not let the ending Z be picked up -% as a macro parameter without knowing in advance that it is its turn. So, we -% conscientiously clean out of the way space tokens, but also we ff-expand with -% \romannumeral-`0 (unbraced) items, a process which may create new space -% tokens, so it is iterated. As unbraced items are expanded, it is easy to -% simulate insertion of a list in another. -% Unbraced items consecutive to an even (non-zero) number of space tokens will -% not get expanded. -% -% 1.09e: [2013/10/29] does this better, no difference between an even or odd -% number of explicit consecutive space tokens. Normal situations anyhow only -% create at most one space token, but well. There was a feature in \xintFor (not -% \xintFor*) from 1.09c that it treated an empty list as a list with one, empty, -% item. This feature is kept in 1.09e, knowingly... Also, macros are made long, -% hence the iterated text may contain \par and also the looped over items. I -% thought about providing some macro expanding to the loop count, but as the -% \xintFor is not expandable anyhow, there is no loss of generality if the -% iterated commands do themselves the bookkeeping using a count or a LaTeX -% counter, and deal with nesting or other problems. I can't do *everything*! -% -% 1.09e adds \XINT_forever with \xintintegers, \xintdimensions, \xintrationals -% and \xintBreakFor, \xintBreakForAndDo, \xintifForFirst, \xintifForLast. On -% this occasion \xint_firstoftwo and \xint_secondoftwo are made long. -% -% 1.09f: rewrites large parts of \xintFor code in order to filter the comma -% separated list via \xintCSVtoList which gets rid of spaces. Compatibility -% with \XINT_forever, the necessity to prevent unwanted brace stripping, and -% shared code with \xintFor*, make this all a delicate balancing act. The #1 in -% \XINT_for_forever? has an initial space token which serves two purposes: -% preventing brace stripping, and stopping the expansion made by \xintcsvtolist. -% If the \XINT_forever branch is taken, the added space will not be a problem -% there. -% -% 1.09f rewrites (2013/11/03) the code which now allows all macro parameters -% from #1 to #9 in \xintFor, \xintFor*, and \XINT_forever. -% -% The 1.09f \xintFor and \xintFor* modified the value of \count 255 -% which was silly, 1.09g used \XINT_count, but requiring a \count only -% for that was also silly, 1.09h just uses \numexpr (all of that was only to -% get rid simply of a possibly space in #2...). -% -% 1.09ka [2014/02/05] corrects the following bug: \xintBreakFor and -% \xintBreakForAndDo could not be used in the last iteration.| -% \begin{macrocode} -\def\XINT_tmpa #1#2{\ifnum #2<#1 \xint_afterfi {{#########2}}\fi}% -\def\XINT_tmpb #1#2{\ifnum #1<#2 \xint_afterfi {{#########2}}\fi}% -\def\XINT_tmpc #1% -{% - \expandafter\edef \csname XINT_for_left#1\endcsname - {\xintApplyUnbraced {\XINT_tmpa #1}{123456789}}% - \expandafter\edef \csname XINT_for_right#1\endcsname - {\xintApplyUnbraced {\XINT_tmpb #1}{123456789}}% -}% -\xintApplyInline \XINT_tmpc {123456789}% -\long\def\xintBreakFor #1Z{}% -\long\def\xintBreakForAndDo #1#2Z{#1}% -\def\xintFor {\let\xintifForFirst\xint_firstoftwo - \futurelet\XINT_token\XINT_for_ifstar }% -\def\XINT_for_ifstar {\ifx\XINT_token*\expandafter\XINT_forx - \else\expandafter\XINT_for \fi }% -\catcode`U 3 % with numexpr -\catcode`V 3 % with xintfrac.sty (xint.sty not enough) -\catcode`D 3 % with dimexpr -% \def\XINT_flet #1% -% {% -% \def\XINT_flet_macro {#1}\XINT_flet_zapsp -% }% -\def\XINT_flet_zapsp -{% - \futurelet\XINT_token\XINT_flet_sp? -}% -\def\XINT_flet_sp? -{% - \ifx\XINT_token\XINT_sptoken - \xint_afterfi{\expandafter\XINT_flet_zapsp\romannumeral0}% - \else\expandafter\XINT_flet_macro - \fi -}% -\long\def\XINT_for #1#2in#3#4#5% -{% - \expandafter\XINT_toks\expandafter - {\expandafter\XINT_for_d\the\numexpr #2\relax {#5}}% - \def\XINT_flet_macro {\expandafter\XINT_for_forever?\space}% - \expandafter\XINT_flet_zapsp #3Z% -}% -\def\XINT_for_forever? #1Z% -{% - \ifx\XINT_token U\XINT_to_forever\fi - \ifx\XINT_token V\XINT_to_forever\fi - \ifx\XINT_token D\XINT_to_forever\fi - \expandafter\the\expandafter\XINT_toks\romannumeral0\xintcsvtolist {#1}Z% -}% -\def\XINT_to_forever\fi #1\xintcsvtolist #2{\fi \XINT_forever #2}% -\long\def\XINT_forx *#1#2in#3#4#5% -{% - \expandafter\XINT_toks\expandafter - {\expandafter\XINT_forx_d\the\numexpr #2\relax {#5}}% - \XINT_xflet\XINT_forx_forever? #3Z% -}% -\def\XINT_forx_forever? -{% - \ifx\XINT_token U\XINT_to_forxever\fi - \ifx\XINT_token V\XINT_to_forxever\fi - \ifx\XINT_token D\XINT_to_forxever\fi - \XINT_forx_empty? -}% -\def\XINT_to_forxever\fi #1\XINT_forx_empty? {\fi \XINT_forever }% -\catcode`U 11 -\catcode`D 11 -\catcode`V 11 -\def\XINT_forx_empty? -{% - \ifx\XINT_token Z\expandafter\xintBreakFor\fi - \the\XINT_toks -}% -\long\def\XINT_for_d #1#2#3% -{% - \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}% - \XINT_toks {{#3}}% - \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname - \the\XINT_toks \csname XINT_for_right#1\endcsname }% - \XINT_toks {\XINT_x\let\xintifForFirst\xint_secondoftwo\XINT_for_d #1{#2}}% - \futurelet\XINT_token\XINT_for_last? -}% -\long\def\XINT_forx_d #1#2#3% -{% - \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}% - \XINT_toks {{#3}}% - \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname - \the\XINT_toks \csname XINT_for_right#1\endcsname }% - \XINT_toks {\XINT_x\let\xintifForFirst\xint_secondoftwo\XINT_forx_d #1{#2}}% - \XINT_xflet\XINT_for_last? -}% -\def\XINT_for_last? -{% - \let\xintifForLast\xint_secondoftwo - \ifx\XINT_token Z\let\xintifForLast\xint_firstoftwo - \xint_afterfi{\xintBreakForAndDo{\XINT_x\xint_gobble_i Z}}\fi - \the\XINT_toks -}% -% \end{macrocode} -% \subsection{\csh{XINT\_forever}, \csh{xintintegers}, \csh{xintdimensions}, \csh{xintrationals}} -% \lverb|New with 1.09e. But this used inadvertently \xintiadd/\xintimul which -% have the unnecessary \xintnum overhead. Changed in 1.09f to use -% \xintiiadd/\xintiimul which do not have this overhead. Also 1.09f has -% \xintZapSpacesB which helps getting rid of spaces for the \xintrationals case -% (the other cases end up inside a \numexpr, or \dimexpr, so not necessary).| -% \begin{macrocode} -\catcode`U 3 -\catcode`D 3 -\catcode`V 3 -\let\xintegers U% -\let\xintintegers U% -\let\xintdimensions D% -\let\xintrationals V% -\def\XINT_forever #1% -{% - \expandafter\XINT_forever_a - \csname XINT_?expr_\ifx#1UU\else\ifx#1DD\else V\fi\fi a\expandafter\endcsname - \csname XINT_?expr_\ifx#1UU\else\ifx#1DD\else V\fi\fi i\expandafter\endcsname - \csname XINT_?expr_\ifx#1UU\else\ifx#1DD\else V\fi\fi \endcsname -}% -\catcode`U 11 -\catcode`D 11 -\catcode`V 11 -\def\XINT_?expr_Ua #1#2% - {\expandafter{\expandafter\numexpr\the\numexpr #1\expandafter\relax - \expandafter\relax\expandafter}% - \expandafter{\the\numexpr #2}}% -\def\XINT_?expr_Da #1#2% - {\expandafter{\expandafter\dimexpr\number\dimexpr #1\expandafter\relax - \expandafter s\expandafter p\expandafter\relax\expandafter}% - \expandafter{\number\dimexpr #2}}% -\catcode`Z 11 -\def\XINT_?expr_Va #1#2% -{% - \expandafter\XINT_?expr_Vb\expandafter - {\romannumeral-`0\xintrawwithzeros{\xintZapSpacesB{#2}}}% - {\romannumeral-`0\xintrawwithzeros{\xintZapSpacesB{#1}}}% -}% -\catcode`Z 3 -\def\XINT_?expr_Vb #1#2{\expandafter\XINT_?expr_Vc #2.#1.}% -\def\XINT_?expr_Vc #1/#2.#3/#4.% -{% - \xintifEq {#2}{#4}% - {\XINT_?expr_Vf {#3}{#1}{#2}}% - {\expandafter\XINT_?expr_Vd\expandafter - {\romannumeral0\xintiimul {#2}{#4}}% - {\romannumeral0\xintiimul {#1}{#4}}% - {\romannumeral0\xintiimul {#2}{#3}}% - }% -}% -\def\XINT_?expr_Vd #1#2#3{\expandafter\XINT_?expr_Ve\expandafter {#2}{#3}{#1}}% -\def\XINT_?expr_Ve #1#2{\expandafter\XINT_?expr_Vf\expandafter {#2}{#1}}% -\def\XINT_?expr_Vf #1#2#3{{#2/#3}{{0}{#1}{#2}{#3}}}% -\def\XINT_?expr_Ui {{\numexpr 1\relax}{1}}% -\def\XINT_?expr_Di {{\dimexpr 0pt\relax}{65536}}% -\def\XINT_?expr_Vi {{1/1}{0111}}% -\def\XINT_?expr_U #1#2% - {\expandafter{\expandafter\numexpr\the\numexpr #1+#2\relax\relax}{#2}}% -\def\XINT_?expr_D #1#2% - {\expandafter{\expandafter\dimexpr\the\numexpr #1+#2\relax sp\relax}{#2}}% -\def\XINT_?expr_V #1#2{\XINT_?expr_Vx #2}% -\def\XINT_?expr_Vx #1#2% -{% - \expandafter\XINT_?expr_Vy\expandafter - {\romannumeral0\xintiiadd {#1}{#2}}{#2}% -}% -\def\XINT_?expr_Vy #1#2#3#4% -{% - \expandafter{\romannumeral0\xintiiadd {#3}{#1}/#4}{{#1}{#2}{#3}{#4}}% -}% -\def\XINT_forever_a #1#2#3#4% -{% - \ifx #4[\expandafter\XINT_forever_opt_a - \else\expandafter\XINT_forever_b - \fi #1#2#3#4% -}% -\def\XINT_forever_b #1#2#3Z{\expandafter\XINT_forever_c\the\XINT_toks #2#3}% -\long\def\XINT_forever_c #1#2#3#4#5% - {\expandafter\XINT_forever_d\expandafter #2#4#5{#3}Z}% -\def\XINT_forever_opt_a #1#2#3[#4+#5]#6Z% -{% - \expandafter\expandafter\expandafter - \XINT_forever_opt_c\expandafter\the\expandafter\XINT_toks - \romannumeral-`0#1{#4}{#5}#3% -}% -\long\def\XINT_forever_opt_c #1#2#3#4#5#6{\XINT_forever_d #2{#4}{#5}#6{#3}Z}% -\long\def\XINT_forever_d #1#2#3#4#5% -{% - \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#5}% - \XINT_toks {{#2}}% - \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname - \the\XINT_toks \csname XINT_for_right#1\endcsname }% - \XINT_x - \let\xintifForFirst\xint_secondoftwo - \expandafter\XINT_forever_d\expandafter #1\romannumeral-`0#4{#2}{#3}#4{#5}% -}% -% \end{macrocode} -% \subsection{\csh{xintForpair}, \csh{xintForthree}, \csh{xintForfour}} -% \lverb|1.09c: I don't know yet if {a}{b} is better for the user or worse than -% (a,b). I prefer the former. I am not very motivated to deal with spaces in the -% (a,b) approach which is the one (currently) followed here. -% -% [2013/11/02] 1.09f: I may not have been very motivated in 1.09c, but since -% then I developped the \xintZapSpaces/\xintZapSpacesB tools (much to my -% satisfaction). Based on this, and better parameter texts, \xintForpair and its -% cousins now handle spaces very satisfactorily (this relies partly on the new -% \xintCSVtoList which makes use of \xintZapSpacesB). Does not share code with -% \xintFor anymore. -% -% [2013/11/03] 1.09f: \xintForpair extended to accept #1#2, #2#3 etc... up to -% #8#9, \xintForthree, #1#2#3 up to #7#8#9, \xintForfour id. | -% \begin{macrocode} -\catcode`j 3 -\long\def\xintForpair #1#2#3in#4#5#6% -{% - \let\xintifForFirst\xint_firstoftwo - \XINT_toks {\XINT_forpair_d #2{#6}}% - \expandafter\the\expandafter\XINT_toks #4jZ% -}% -\long\def\XINT_forpair_d #1#2#3(#4)#5% -{% - \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}% - \XINT_toks \expandafter{\romannumeral0\xintcsvtolist{ #4}}% - \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname - \the\XINT_toks \csname XINT_for_right\the\numexpr#1+1\endcsname}% - \let\xintifForLast\xint_secondoftwo - \ifx #5j\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi - {\let\xintifForLast\xint_firstoftwo - \xintBreakForAndDo {\XINT_x \xint_gobble_i Z}}% - \XINT_x - \let\xintifForFirst\xint_secondoftwo\XINT_forpair_d #1{#2}% -}% -\long\def\xintForthree #1#2#3in#4#5#6% -{% - \let\xintifForFirst\xint_firstoftwo - \XINT_toks {\XINT_forthree_d #2{#6}}% - \expandafter\the\expandafter\XINT_toks #4jZ% -}% -\long\def\XINT_forthree_d #1#2#3(#4)#5% -{% - \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}% - \XINT_toks \expandafter{\romannumeral0\xintcsvtolist{ #4}}% - \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname - \the\XINT_toks \csname XINT_for_right\the\numexpr#1+2\endcsname}% - \let\xintifForLast\xint_secondoftwo - \ifx #5j\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi - {\let\xintifForLast\xint_firstoftwo - \xintBreakForAndDo {\XINT_x \xint_gobble_i Z}}% - \XINT_x - \let\xintifForFirst\xint_secondoftwo\XINT_forthree_d #1{#2}% -}% -\long\def\xintForfour #1#2#3in#4#5#6% -{% - \let\xintifForFirst\xint_firstoftwo - \XINT_toks {\XINT_forfour_d #2{#6}}% - \expandafter\the\expandafter\XINT_toks #4jZ% -}% -\long\def\XINT_forfour_d #1#2#3(#4)#5% -{% - \long\def\XINT_y ##1##2##3##4##5##6##7##8##9{#2}% - \XINT_toks \expandafter{\romannumeral0\xintcsvtolist{ #4}}% - \long\edef\XINT_x {\noexpand\XINT_y \csname XINT_for_left#1\endcsname - \the\XINT_toks \csname XINT_for_right\the\numexpr#1+3\endcsname}% - \let\xintifForLast\xint_secondoftwo - \ifx #5j\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi - {\let\xintifForLast\xint_firstoftwo - \xintBreakForAndDo {\XINT_x \xint_gobble_i Z}}% - \XINT_x - \let\xintifForFirst\xint_secondoftwo\XINT_forfour_d #1{#2}% -}% -\catcode`Z 11 -\catcode`j 11 -% \end{macrocode} -% \subsection{\csh{xintAssign}, \csh{xintAssignArray}, \csh{xintDigitsOf}} -% \lverb|& -% \xintAssign {a}{b}..{z}\to\A\B...\Z,$\ -% \xintAssignArray {a}{b}..{z}\to\U -% -% version 1.01 corrects an oversight in 1.0 related to the value of -% \escapechar at the time of using \xintAssignArray or \xintRelaxArray -% These macros are non-expandable. -% -% In version 1.05a I suddenly see some incongruous \expandafter's in (what is -% called now) \XINT_assignarray_end_c, which I remove. -% -% Release 1.06 modifies the macros created by \xintAssignArray to feed their -% argument to a \numexpr. Release 1.06a detects an incredible typo in 1.01, (bad -% copy-paste from -% \xintRelaxArray) which caused \xintAssignArray to use #1 rather than the #2 as -% in the correct earlier 1.0 version!!! This went through undetected because -% \xint_arrayname, although weird, was still usable: the probability to -% overwrite something was almost zero. The bug got finally revealed doing -% \xintAssignArray {}{}{}\to\Stuff. -% -% With release 1.06b an empty argument (or expanding to empty) to -% \xintAssignArray is ok. -% -% 1.09h simplifies the coding of \xintAssignArray (no more _end_a, _end_b, -% etc...), and no use of a \count register anymore, and uses \xintiloop in -% \xintRelaxArray. Furthermore, macros are made long. -% -% 1.09i allows an optional parameter \xintAssign [oo] for example, then \oodef -% rather than \edef is used. Idem for \xintAssignArray. However in the latter -% case, the global variant is not available, one should use \globaldefs for -% that. -% -% 1.09j: I decide that the default behavior of \xintAssign should be to use -% \def, not \edef when assigning to a cs an item of the list. This is a -% breaking change but I don't think anybody on earth is using xint anyhow. -% Also use of the optional parameter was broken if it was [], [g], [e], [x] as -% the corresponding \XINT_... macros had not been defined (in the initial -% version I did not have the XINT_ prefix; then I added it in case \oodef was -% pre-existing and thus was not redefined by the package which instead had -% \XINT_oodef, now \xintoodef.)| -% \begin{macrocode} -\def\xintAssign{\def\XINT_flet_macro {\XINT_assign_fork}\XINT_flet_zapsp }% -\def\XINT_assign_fork -{% - \let\XINT_assign_def\def - \ifx\XINT_token[\expandafter\XINT_assign_opt - \else\expandafter\XINT_assign_a - \fi -}% -\def\XINT_assign_opt [#1]% -{% - \ifcsname #1def\endcsname - \expandafter\let\expandafter\XINT_assign_def \csname #1def\endcsname - \else - \expandafter\let\expandafter\XINT_assign_def \csname xint#1def\endcsname - \fi - \XINT_assign_a -}% -\long\def\XINT_assign_a #1\to -{% - \expandafter\XINT_assign_b\romannumeral-`0#1{}\to -}% -\long\def\XINT_assign_b #1% attention to the # at the beginning of next line -#{% - \def\xint_temp {#1}% - \ifx\empty\xint_temp - \expandafter\XINT_assign_c - \else - \expandafter\XINT_assign_d - \fi -}% -\long\def\XINT_assign_c #1#2\to #3% -{% - \XINT_assign_def #3{#1}% - \def\xint_temp {#2}% - \unless\ifx\empty\xint_temp\xint_afterfi{\XINT_assign_b #2\to }\fi -}% -\def\XINT_assign_d #1\to #2% normally #1 is {} here. -{% - \expandafter\XINT_assign_def\expandafter #2\expandafter{\xint_temp}% -}% -\def\xintRelaxArray #1% -{% - \edef\XINT_restoreescapechar {\escapechar\the\escapechar\relax}% - \escapechar -1 - \expandafter\def\expandafter\xint_arrayname\expandafter {\string #1}% - \XINT_restoreescapechar - \xintiloop [\csname\xint_arrayname 0\endcsname+-1] - \global - \expandafter\let\csname\xint_arrayname\xintiloopindex\endcsname\relax - \ifnum \xintiloopindex > \xint_c_ - \repeat - \global\expandafter\let\csname\xint_arrayname 00\endcsname\relax - \global\let #1\relax -}% -\def\xintAssignArray{\def\XINT_flet_macro {\XINT_assignarray_fork}% - \XINT_flet_zapsp }% -\def\XINT_assignarray_fork -{% - \let\XINT_assignarray_def\def - \ifx\XINT_token[\expandafter\XINT_assignarray_opt - \else\expandafter\XINT_assignarray - \fi -}% -\def\XINT_assignarray_opt [#1]% -{% - \ifcsname #1def\endcsname - \expandafter\let\expandafter\XINT_assignarray_def \csname #1def\endcsname - \else - \expandafter\let\expandafter\XINT_assignarray_def - \csname xint#1def\endcsname - \fi - \XINT_assignarray -}% -\long\def\XINT_assignarray #1\to #2% -{% - \edef\XINT_restoreescapechar {\escapechar\the\escapechar\relax }% - \escapechar -1 - \expandafter\def\expandafter\xint_arrayname\expandafter {\string #2}% - \XINT_restoreescapechar - \def\xint_itemcount {0}% - \expandafter\XINT_assignarray_loop \romannumeral-`0#1\xint_relax - \csname\xint_arrayname 00\expandafter\endcsname - \csname\xint_arrayname 0\expandafter\endcsname - \expandafter {\xint_arrayname}#2% -}% -\long\def\XINT_assignarray_loop #1% -{% - \def\xint_temp {#1}% - \ifx\xint_brelax\xint_temp - \expandafter\def\csname\xint_arrayname 0\expandafter\endcsname - \expandafter{\the\numexpr\xint_itemcount}% - \expandafter\expandafter\expandafter\XINT_assignarray_end - \else - \expandafter\def\expandafter\xint_itemcount\expandafter - {\the\numexpr\xint_itemcount+\xint_c_i}% - \expandafter\XINT_assignarray_def - \csname\xint_arrayname\xint_itemcount\expandafter\endcsname - \expandafter{\xint_temp }% - \expandafter\XINT_assignarray_loop - \fi -}% -\def\XINT_assignarray_end #1#2#3#4% -{% - \def #4##1% - {% - \romannumeral0\expandafter #1\expandafter{\the\numexpr ##1}% - }% - \def #1##1% - {% - \ifnum ##1<\xint_c_ - \xint_afterfi {\xintError:ArrayIndexIsNegative\space }% - \else - \xint_afterfi {% - \ifnum ##1>#2 - \xint_afterfi {\xintError:ArrayIndexBeyondLimit\space }% - \else\xint_afterfi - {\expandafter\expandafter\expandafter\space\csname #3##1\endcsname}% - \fi}% - \fi - }% -}% -\let\xintDigitsOf\xintAssignArray -\let\XINT_tmpa\relax \let\XINT_tmpb\relax \let\XINT_tmpc\relax -\XINT_restorecatcodes_endinput% -% \end{macrocode} -%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 -%\let</xinttools>\relax -%\def<*xint>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 } -%</xinttools> -%<*xint> -% -% \StoreCodelineNo {xinttools} -% -% \section{Package \xintnameimp implementation} -% \label{sec:xintimp} -% -% With release |1.09a| all macros doing arithmetic operations and a few more -% apply systematically |\xintnum| to their arguments; this adds a little -% overhead but this is more convenient for using count registers even with infix -% notation; also this is what |xintfrac.sty| did all along. Simplifies the -% discussion in the documentation too. -% -% \localtableofcontents -% -% \subsection{Catcodes, \protect\eTeX{} and reload detection} -% -% The code for reload detection is copied from \textsc{Heiko -% Oberdiek}'s packages, and adapted here to check for previous -% loading of the master \xintname package. -% -% The method for catcodes is slightly different, but still -% directly inspired by these packages. -% -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \def\space { }% - \let\z\endgroup - \expandafter\let\expandafter\x\csname ver@xint.sty\endcsname - \expandafter\let\expandafter\w\csname ver@xinttools.sty\endcsname - \expandafter - \ifx\csname PackageInfo\endcsname\relax - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \else - \def\y#1#2{\PackageInfo{#1}{#2}}% - \fi - \expandafter - \ifx\csname numexpr\endcsname\relax - \y{xint}{\numexpr not available, aborting input}% - \aftergroup\endinput - \else - \ifx\x\relax % plain-TeX, first loading of xint.sty - \ifx\w\relax % but xinttools.sty not yet loaded. - \y{xint}{now issuing \string\input\space xinttools.sty}% - \def\z{\endgroup\input xinttools.sty\relax}% - \fi - \else - \def\empty {}% - \ifx\x\empty % LaTeX, first loading, - % variable is initialized, but \ProvidesPackage not yet seen - \ifx\w\relax % xinttools.sty not yet loaded. - \y{xint}{now issuing \string\RequirePackage{xinttools}}% - \def\z{\endgroup\RequirePackage{xinttools}}% - \fi - \else - \y{xint}{I was already loaded, aborting input}% - \aftergroup\endinput - \fi - \fi - \fi -\z% -% \end{macrocode} -% \subsection{Confirmation of \xinttoolsnameimp loading} -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \ifdefined\PackageInfo - \def\y#1#2{\PackageInfo{#1}{#2}}% - \else - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \fi - \def\empty {}% - \expandafter\let\expandafter\w\csname ver@xinttools.sty\endcsname - \ifx\w\relax % Plain TeX, user gave a file name at the prompt - \y{xint}{Loading of package xinttools failed, aborting input}% - \aftergroup\endinput - \fi - \ifx\w\empty % LaTeX, user gave a file name at the prompt - \y{xint}{Loading of package xinttools failed, aborting input}% - \aftergroup\endinput - \fi -\endgroup% -% \end{macrocode} -% \subsection{Catcodes} -% \begin{macrocode} -\XINTsetupcatcodes% -% \end{macrocode} -% \subsection{Package identification} -% \begin{macrocode} -\XINT_providespackage -\ProvidesPackage{xint}% - [2014/02/05 v1.09ka Expandable operations on long numbers (jfB)]% -% \end{macrocode} -% \subsection{Token management, constants} -% \begin{macrocode} -\long\def\xint_firstofthree #1#2#3{#1}% -\long\def\xint_secondofthree #1#2#3{#2}% -\long\def\xint_thirdofthree #1#2#3{#3}% -\long\def\xint_firstofthree_thenstop #1#2#3{ #1}% 1.09i -\long\def\xint_secondofthree_thenstop #1#2#3{ #2}% -\long\def\xint_thirdofthree_thenstop #1#2#3{ #3}% -\def\xint_gob_til_zero #10{}% -\def\xint_gob_til_zeros_iii #1000{}% -\def\xint_gob_til_zeros_iv #10000{}% -\def\xint_gob_til_one #11{}% -\def\xint_gob_til_G #1G{}% -\def\xint_gob_til_minus #1-{}% -\def\xint_gob_til_relax #1\relax {}% -\def\xint_exchangetwo_keepbraces #1#2{{#2}{#1}}% -\def\xint_exchangetwo_keepbraces_thenstop #1#2{ {#2}{#1}}% -\def\xint_UDzerofork #10#2#3\krof {#2}% -\def\xint_UDsignfork #1-#2#3\krof {#2}% -\def\xint_UDwfork #1\W#2#3\krof {#2}% -\def\xint_UDzerosfork #100#2#3\krof {#2}% -\def\xint_UDonezerofork #110#2#3\krof {#2}% -\def\xint_UDzerominusfork #10-#2#3\krof {#2}% -\def\xint_UDsignsfork #1--#2#3\krof {#2}% -% \chardef\xint_c_ 0 % already done in xinttools -% \chardef\xint_c_i 1 % already done in xinttools -\chardef\xint_c_ii 2 -\chardef\xint_c_iii 3 -\chardef\xint_c_iv 4 -\chardef\xint_c_v 5 -% \chardef\xint_c_vi 6 % will be done in xintfrac -% \chardef\xinf_c_vii 7 % will be done in xintfrac -% \chardef\xint_c_viii 8 % already done in xinttools -\chardef\xint_c_ix 9 -\chardef\xint_c_x 10 -\chardef\xint_c_ii^v 32 % not used in xint, common to xintfrac and xintbinhex -\chardef\xint_c_ii^vi 64 -\mathchardef\xint_c_ixixixix 9999 -\mathchardef\xint_c_x^iv 10000 -\newcount\xint_c_x^viii \xint_c_x^viii 100000000 -% \end{macrocode} -% \subsection{\csh{xintRev}} -% \lverb|& -% \xintRev: expands fully its argument \romannumeral-`0, and checks the sign. -% However this last aspect does not appear like a very useful thing. And despite -% the fact that a special check is made for a sign, actually the input is not -% given to \xintnum, contrarily to \xintLen. This is all a bit incoherent. -% Should be fixed.| -% \begin{macrocode} -\def\xintRev {\romannumeral0\xintrev }% -\def\xintrev #1% -{% - \expandafter\XINT_rev_fork - \romannumeral-`0#1\xint_relax % empty #1 ok, \xint_relax stops expansion - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\def\XINT_rev_fork #1% -{% - \xint_UDsignfork - #1{\expandafter\xint_minus_thenstop\romannumeral0\XINT_rord_main {}}% - -{\XINT_rord_main {}#1}% - \krof -}% -% \end{macrocode} -% \subsection{\csh{xintLen}} -% \lverb|\xintLen is ONLY for (possibly long) integers. Gets extended to -% fractions by xintfrac.sty| -% \begin{macrocode} -\def\xintLen {\romannumeral0\xintlen }% -\def\xintlen #1% -{% - \expandafter\XINT_len_fork - \romannumeral0\xintnum{#1}\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye -}% -\def\XINT_Len #1% variant which does not expand via \xintnum. -{% - \romannumeral0\XINT_len_fork - #1\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\xint_bye -}% -\def\XINT_len_fork #1% -{% - \expandafter\XINT_length_loop - \xint_UDsignfork - #1{{0}}% - -{{0}#1}% - \krof -}% -% \end{macrocode} -% \subsection{\csh{XINT\_RQ}} -% \lverb|& -% cette macro renverse et ajoute le nombre minimal de zéros à -% la fin pour que la longueur soit alors multiple de 4$\ -% \romannumeral0\XINT_RQ {}<le truc à renverser>\R\R\R\R\R\R\R\R\Z$\ -% Attention, ceci n'est utilisé que pour des chaînes de chiffres, et donc le -% comportement avec des {..} ou autres espaces n'a fait l'objet d'aucune -% attention | -% \begin{macrocode} -\def\XINT_RQ #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_R #9\XINT_RQ_end_a\R\XINT_RQ {#9#8#7#6#5#4#3#2#1}% -}% -\def\XINT_RQ_end_a\R\XINT_RQ #1#2\Z -{% - \XINT_RQ_end_b #1\Z -}% -\def\XINT_RQ_end_b #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_R - #8\XINT_RQ_end_viii - #7\XINT_RQ_end_vii - #6\XINT_RQ_end_vi - #5\XINT_RQ_end_v - #4\XINT_RQ_end_iv - #3\XINT_RQ_end_iii - #2\XINT_RQ_end_ii - \R\XINT_RQ_end_i - \Z #2#3#4#5#6#7#8% -}% -\def\XINT_RQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}% -\def\XINT_RQ_end_vii #1\Z #2#3#4#5#6#7#8#9\Z { #8#9000}% -\def\XINT_RQ_end_vi #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#900}% -\def\XINT_RQ_end_v #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#90}% -\def\XINT_RQ_end_iv #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#9}% -\def\XINT_RQ_end_iii #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}% -\def\XINT_RQ_end_ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% -\def\XINT_RQ_end_i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% -\def\XINT_SQ #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_R #8\XINT_SQ_end_a\R\XINT_SQ {#8#7#6#5#4#3#2#1}% -}% -\def\XINT_SQ_end_a\R\XINT_SQ #1#2\Z -{% - \XINT_SQ_end_b #1\Z -}% -\def\XINT_SQ_end_b #1#2#3#4#5#6#7% -{% - \xint_gob_til_R - #7\XINT_SQ_end_vii - #6\XINT_SQ_end_vi - #5\XINT_SQ_end_v - #4\XINT_SQ_end_iv - #3\XINT_SQ_end_iii - #2\XINT_SQ_end_ii - \R\XINT_SQ_end_i - \Z #2#3#4#5#6#7% -}% -\def\XINT_SQ_end_vii #1\Z #2#3#4#5#6#7#8\Z { #8}% -\def\XINT_SQ_end_vi #1\Z #2#3#4#5#6#7#8\Z { #7#8000000}% -\def\XINT_SQ_end_v #1\Z #2#3#4#5#6#7#8\Z { #6#7#800000}% -\def\XINT_SQ_end_iv #1\Z #2#3#4#5#6#7#8\Z { #5#6#7#80000}% -\def\XINT_SQ_end_iii #1\Z #2#3#4#5#6#7#8\Z { #4#5#6#7#8000}% -\def\XINT_SQ_end_ii #1\Z #2#3#4#5#6#7#8\Z { #3#4#5#6#7#800}% -\def\XINT_SQ_end_i \Z #1#2#3#4#5#6#7\Z { #1#2#3#4#5#6#70}% -\def\XINT_OQ #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_R #9\XINT_OQ_end_a\R\XINT_OQ {#9#8#7#6#5#4#3#2#1}% -}% -\def\XINT_OQ_end_a\R\XINT_OQ #1#2\Z -{% - \XINT_OQ_end_b #1\Z -}% -\def\XINT_OQ_end_b #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_R - #8\XINT_OQ_end_viii - #7\XINT_OQ_end_vii - #6\XINT_OQ_end_vi - #5\XINT_OQ_end_v - #4\XINT_OQ_end_iv - #3\XINT_OQ_end_iii - #2\XINT_OQ_end_ii - \R\XINT_OQ_end_i - \Z #2#3#4#5#6#7#8% -}% -\def\XINT_OQ_end_viii #1\Z #2#3#4#5#6#7#8#9\Z { #9}% -\def\XINT_OQ_end_vii #1\Z #2#3#4#5#6#7#8#9\Z { #8#90000000}% -\def\XINT_OQ_end_vi #1\Z #2#3#4#5#6#7#8#9\Z { #7#8#9000000}% -\def\XINT_OQ_end_v #1\Z #2#3#4#5#6#7#8#9\Z { #6#7#8#900000}% -\def\XINT_OQ_end_iv #1\Z #2#3#4#5#6#7#8#9\Z { #5#6#7#8#90000}% -\def\XINT_OQ_end_iii #1\Z #2#3#4#5#6#7#8#9\Z { #4#5#6#7#8#9000}% -\def\XINT_OQ_end_ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% -\def\XINT_OQ_end_i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% -% \end{macrocode} -% \subsection{\csh{XINT\_cuz}} -% \begin{macrocode} -\edef\xint_cleanupzeros_andstop #1#2#3#4% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax -}% -\def\xint_cleanupzeros_nostop #1#2#3#4% -{% - \the\numexpr #1#2#3#4\relax -}% -\def\XINT_rev_andcuz #1% -{% - \expandafter\xint_cleanupzeros_andstop - \romannumeral0\XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -% \end{macrocode} -% \lverb|& -% routine CleanUpZeros. Utilisée en particulier par la -% soustraction.$\ -% INPUT: longueur **multiple de 4** (<-- ATTENTION)$\ -% OUTPUT: on a retiré tous les leading zéros, on n'est **plus* -% nécessairement de longueur 4n$\ -% Délimiteur pour _main: \W\W\W\W\W\W\W\Z avec SEPT \W| -% \begin{macrocode} -\def\XINT_cuz #1% -{% - \XINT_cuz_loop #1\W\W\W\W\W\W\W\Z% -}% -\def\XINT_cuz_loop #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_W #8\xint_cuz_end_a\W - \xint_gob_til_Z #8\xint_cuz_end_A\Z - \XINT_cuz_check_a {#1#2#3#4#5#6#7#8}% -}% -\def\xint_cuz_end_a #1\XINT_cuz_check_a #2% -{% - \xint_cuz_end_b #2% -}% -\edef\xint_cuz_end_b #1#2#3#4#5\Z -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4\relax -}% -\def\xint_cuz_end_A \Z\XINT_cuz_check_a #1{ 0}% -\def\XINT_cuz_check_a #1% -{% - \expandafter\XINT_cuz_check_b\the\numexpr #1\relax -}% -\def\XINT_cuz_check_b #1% -{% - \xint_gob_til_zero #1\xint_cuz_backtoloop 0\XINT_cuz_stop #1% -}% -\def\XINT_cuz_stop #1\W #2\Z{ #1}% -\def\xint_cuz_backtoloop 0\XINT_cuz_stop 0{\XINT_cuz_loop }% -% \end{macrocode} -% \subsection{\csh{xintIsOne}} -% \lverb|& -% Added in 1.03. Attention: \XINT_isOne does not do any expansion. Release 1.09a -% defines \xintIsOne which is more user-friendly. Will be modified if xintfrac -% is loaded. | -% \begin{macrocode} -\def\xintIsOne {\romannumeral0\xintisone }% -\def\xintisone #1{\expandafter\XINT_isone\romannumeral0\xintnum{#1}\W\Z }% -\def\XINT_isOne #1{\romannumeral0\XINT_isone #1\W\Z }% -\def\XINT_isone #1#2% -{% - \xint_gob_til_one #1\XINT_isone_b 1% - \expandafter\space\expandafter 0\xint_gob_til_Z #2% -}% -\def\XINT_isone_b #1\xint_gob_til_Z #2% -{% - \xint_gob_til_W #2\XINT_isone_yes \W - \expandafter\space\expandafter 0\xint_gob_til_Z -}% -\def\XINT_isone_yes #1\Z { 1}% -% \end{macrocode} -% \subsection{\csh{xintNum}} -% \lverb|& -% For example \xintNum {----+-+++---+----000000000000003}$\ -% 1.05 defines \xintiNum, which allows redefinition of \xintNum by xintfrac.sty -% Slightly modified in 1.06b (\R->\xint_relax) to avoid initial re-scan of -% input stack (while still allowing empty #1). In versions earlier than 1.09a -% it was entirely up to the user to apply \xintnum; starting with 1.09a -% arithmetic -% macros of xint.sty (like earlier already xintfrac.sty with its own \xintnum) -% make use of \xintnum. This allows arguments to -% be count registers, or even \numexpr arbitrary long expressions (with the -% trick of braces, see the user documentation).| -% \begin{macrocode} -\def\xintiNum {\romannumeral0\xintinum }% -\def\xintinum #1% -{% - \expandafter\XINT_num_loop - \romannumeral-`0#1\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z -}% -\let\xintNum\xintiNum \let\xintnum\xintinum -\def\XINT_num #1% -{% - \XINT_num_loop #1\xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z -}% -\def\XINT_num_loop #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_xint_relax #8\XINT_num_end\xint_relax - \XINT_num_NumEight #1#2#3#4#5#6#7#8% -}% -\edef\XINT_num_end\xint_relax\XINT_num_NumEight #1\xint_relax #2\Z -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1+0\relax -}% -\def\XINT_num_NumEight #1#2#3#4#5#6#7#8% -{% - \ifnum \numexpr #1#2#3#4#5#6#7#8+0= 0 - \xint_afterfi {\expandafter\XINT_num_keepsign_a - \the\numexpr #1#2#3#4#5#6#7#81\relax}% - \else - \xint_afterfi {\expandafter\XINT_num_finish - \the\numexpr #1#2#3#4#5#6#7#8\relax}% - \fi -}% -\def\XINT_num_keepsign_a #1% -{% - \xint_gob_til_one#1\XINT_num_gobacktoloop 1\XINT_num_keepsign_b -}% -\def\XINT_num_gobacktoloop 1\XINT_num_keepsign_b {\XINT_num_loop }% -\def\XINT_num_keepsign_b #1{\XINT_num_loop -}% -\def\XINT_num_finish #1\xint_relax #2\Z { #1}% -% \end{macrocode} -% \subsection{\csh{xintSgn}, \csh{xintiiSgn}, \csh{XINT\_Sgn}, \csh{XINT\_cntSgn}} -% \lverb|& -% Changed in 1.05. Earlier code was unnecessarily strange. 1.09a with \xintnum -% -% 1.09i defines \XINT_Sgn and \XINT_cntSgn (was \XINT__Sgn in 1.09i) for reasons -% of internal optimizations| -% \begin{macrocode} -\def\xintiiSgn {\romannumeral0\xintiisgn }% -\def\xintiisgn #1% -{% - \expandafter\XINT_sgn \romannumeral-`0#1\Z% -}% -\def\xintSgn {\romannumeral0\xintsgn }% -\def\xintsgn #1% -{% - \expandafter\XINT_sgn \romannumeral0\xintnum{#1}\Z% -}% -\def\XINT_sgn #1#2\Z -{% - \xint_UDzerominusfork - #1-{ 0}% - 0#1{ -1}% - 0-{ 1}% - \krof -}% -\def\XINT_Sgn #1#2\Z -{% - \xint_UDzerominusfork - #1-{0}% - 0#1{-1}% - 0-{1}% - \krof -}% -\def\XINT_cntSgn #1#2\Z -{% - \xint_UDzerominusfork - #1-\z@ - 0#1\m@ne - 0-\@ne - \krof -}% -% \end{macrocode} -% \subsection{\csh{xintBool}, \csh{xintToggle}} -% \lverb|1.09c| -% \begin{macrocode} -\def\xintBool #1{\romannumeral-`0% - \csname if#1\endcsname\expandafter1\else\expandafter0\fi }% -\def\xintToggle #1{\romannumeral-`0\iftoggle{#1}{1}{0}}% -% \end{macrocode} -% \subsection{\csh{xintSgnFork}} -% \lverb|Expandable three-way fork added in 1.07. The argument #1 must expand -% to -1,0 or 1. 1.09i has _afterstop, renamed _thenstop later, for efficiency.| -% \begin{macrocode} -\def\xintSgnFork {\romannumeral0\xintsgnfork }% -\def\xintsgnfork #1% -{% - \ifcase #1 \expandafter\xint_secondofthree_thenstop - \or\expandafter\xint_thirdofthree_thenstop - \else\expandafter\xint_firstofthree_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{XINT\_cntSgnFork}} -% \lverb|1.09i. Used internally, #1 must expand to \m@ne, \z@, or \@ne or -% equivalent. Does not insert a space token to stop a romannumeral0 expansion.| -% \begin{macrocode} -\def\XINT_cntSgnFork #1% -{% - \ifcase #1\expandafter\xint_secondofthree - \or\expandafter\xint_thirdofthree - \else\expandafter\xint_firstofthree - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintifSgn}} -% \lverb|Expandable three-way fork added in 1.09a. Branches expandably -% depending on whether <0, =0, >0. Choice of branch guaranteed in two steps. -% -% The use of \romannumeral0\xintsgn rather than \xintSgn is for matters related -% to the transformation of the ternary operator : in \xintNewExpr. I hope I have -% explained there the details because right now off hand I can't recall why. -% -% 1.09i has \xint_firstofthreeafterstop (now _thenstop) etc for faster -% expansion.| -% \begin{macrocode} -\def\xintifSgn {\romannumeral0\xintifsgn }% -\def\xintifsgn #1% -{% - \ifcase \romannumeral0\xintsgn{#1} - \expandafter\xint_secondofthree_thenstop - \or\expandafter\xint_thirdofthree_thenstop - \else\expandafter\xint_firstofthree_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintifZero}, \csh{xintifNotZero}} -% \lverb|& -% Expandable two-way fork added in 1.09a. Branches expandably depending on -% whether the argument is zero (branch A) or not (branch B). 1.09i restyling. By -% the way it appears (not thoroughly tested, though) that \if tests are faster -% than \ifnum tests. | -% \begin{macrocode} -\def\xintifZero {\romannumeral0\xintifzero }% -\def\xintifzero #1% -{% - \if0\xintSgn{#1}% - \expandafter\xint_firstoftwo_thenstop - \else - \expandafter\xint_secondoftwo_thenstop - \fi -}% -\def\xintifNotZero {\romannumeral0\xintifnotzero }% -\def\xintifnotzero #1% -{% - \if0\xintSgn{#1}% - \expandafter\xint_secondoftwo_thenstop - \else - \expandafter\xint_firstoftwo_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintifOne}} -% \lverb|added in 1.09i.| -% \begin{macrocode} -\def\xintifOne {\romannumeral0\xintifone }% -\def\xintifone #1% -{% - \if1\xintIsOne{#1}% - \expandafter\xint_firstoftwo_thenstop - \else - \expandafter\xint_secondoftwo_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintifTrueAelseB}, \csh{xint\-ifFalseAelseB}} -% \lverb|1.09i. Warning, \xintifTrueFalse, \xintifTrue deprecated, to be -% removed| -% \begin{macrocode} -\let\xintifTrueAelseB\xintifNotZero -\let\xintifFalseAelseB\xintifZero -\let\xintifTrue\xintifNotZero -\let\xintifTrueFalse\xintifNotZero -% \end{macrocode} -% \subsection{\csh{xintifCmp}} -% \lverb|& -% 1.09e -% \xintifCmp {n}{m}{if n<m}{if n=m}{if n>m}.| -% \begin{macrocode} -\def\xintifCmp {\romannumeral0\xintifcmp }% -\def\xintifcmp #1#2% -{% - \ifcase\xintCmp {#1}{#2} - \expandafter\xint_secondofthree_thenstop - \or\expandafter\xint_thirdofthree_thenstop - \else\expandafter\xint_firstofthree_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintifEq}} -% \lverb|& -% 1.09a -% \xintifEq {n}{m}{YES if n=m}{NO if n<>m}.| -% \begin{macrocode} -\def\xintifEq {\romannumeral0\xintifeq }% -\def\xintifeq #1#2% -{% - \if0\xintCmp{#1}{#2}% - \expandafter\xint_firstoftwo_thenstop - \else\expandafter\xint_secondoftwo_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintifGt}} -% \lverb|& -% 1.09a \xintifGt {n}{m}{YES if n>m}{NO if n<=m}.| -% \begin{macrocode} -\def\xintifGt {\romannumeral0\xintifgt }% -\def\xintifgt #1#2% -{% - \if1\xintCmp{#1}{#2}% - \expandafter\xint_firstoftwo_thenstop - \else\expandafter\xint_secondoftwo_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintifLt}} -% \lverb|& -% 1.09a \xintifLt {n}{m}{YES if n<m}{NO if n>=m}. Restyled in 1.09i| -% \begin{macrocode} -\def\xintifLt {\romannumeral0\xintiflt }% -\def\xintiflt #1#2% -{% - \ifnum\xintCmp{#1}{#2}<\xint_c_ - \expandafter\xint_firstoftwo_thenstop - \else \expandafter\xint_secondoftwo_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintifOdd}} -% \lverb|1.09e. Restyled in 1.09i.| -% \begin{macrocode} -\def\xintifOdd {\romannumeral0\xintifodd }% -\def\xintifodd #1% -{% - \if\xintOdd{#1}1% - \expandafter\xint_firstoftwo_thenstop - \else - \expandafter\xint_secondoftwo_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintOpp}} -% \lverb|\xintnum added in 1.09a| -% \begin{macrocode} -\def\xintiiOpp {\romannumeral0\xintiiopp }% -\def\xintiiopp #1% -{% - \expandafter\XINT_opp \romannumeral-`0#1% -}% -\def\xintiOpp {\romannumeral0\xintiopp }% -\def\xintiopp #1% -{% - \expandafter\XINT_opp \romannumeral0\xintnum{#1}% -}% -\let\xintOpp\xintiOpp \let\xintopp\xintiopp -\def\XINT_Opp #1{\romannumeral0\XINT_opp #1}% -\def\XINT_opp #1% -{% - \xint_UDzerominusfork - #1-{ 0}% zero - 0#1{ }% negative - 0-{ -#1}% positive - \krof -}% -% \end{macrocode} -% \subsection{\csh{xintAbs}} -% \lverb|Release 1.09a has now \xintiabs which does \xintnum (contrarily to some -% other i-macros, but similarly as \xintiAdd etc...) and this is -% inherited by DecSplit, by Sqr, and macros of xintgcd.sty.| -% \begin{macrocode} -\def\xintiiAbs {\romannumeral0\xintiiabs }% -\def\xintiiabs #1% -{% - \expandafter\XINT_abs \romannumeral-`0#1% -}% -\def\xintiAbs {\romannumeral0\xintiabs }% -\def\xintiabs #1% -{% - \expandafter\XINT_abs \romannumeral0\xintnum{#1}% -}% -\let\xintAbs\xintiAbs \let\xintabs\xintiabs -\def\XINT_Abs #1{\romannumeral0\XINT_abs #1}% -\def\XINT_abs #1% -{% - \xint_UDsignfork - #1{ }% - -{ #1}% - \krof -}% -% \end{macrocode} -% \lverb|& -% -----------------------------------------------------------------$\ -% -----------------------------------------------------------------$\ -% ARITHMETIC OPERATIONS: ADDITION, SUBTRACTION, SUMS, -% MULTIPLICATION, PRODUCTS, FACTORIAL, POWERS, EUCLIDEAN DIVISION. -% -% Release 1.03 re-organizes sub-routines to facilitate future developments: the -% diverse variants of addition, with diverse conditions on inputs and output are -% first listed; they will be used in multiplication, or in the summation, or in -% the power routines. I am aware that the commenting is close to non-existent, -% sorry about that. -% -% ADDITION I: \XINT_add_A -% -% INPUT:$\ -% \romannumeral0\XINT_add_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z$\ -% 1. <N1> et <N2> renversés $\ -% 2. de longueur 4n (avec des leading zéros éventuels)$\ -% 3. l'un des deux ne doit pas se terminer par 0000$\$relax -% [Donc on peut avoir 0000 comme input si l'autre est >0 et ne se termine pas en -% 0000 bien sûr]. On peut avoir l'un des deux vides. Mais alors l'autre ne doit -% être ni vide ni 0000. -% -% OUTPUT: la somme <N1>+<N2>, ordre normal, plus sur 4n, pas de leading zeros -% La procédure est plus rapide lorsque <N1> est le plus court des deux.$\ -% Nota bene: (30 avril 2013). J'ai une version qui est deux fois plus rapide sur -% des nombres d'environ 1000 chiffres chacun, et qui commence à être avantageuse -% pour des nombres d'au moins 200 chiffres. Cependant il serait vraiment -% compliqué d'en étendre l'utilisation aux emplois de l'addition dans les -% autres routines, comme celle de multiplication ou celle de division; et son -% implémentation ajouterait au minimum la mesure de la longueur des summands.| -% \begin{macrocode} -\def\XINT_add_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_add_az\W - \XINT_add_AB #1{#3#4#5#6}{#2}% -}% -\def\xint_add_az\W\XINT_add_AB #1#2% -{% - \XINT_add_AC_checkcarry #1% -}% -% \end{macrocode} -% \lverb|& -% ici #2 est prévu pour l'addition, mais attention il devra être renversé -% pour \numexpr. #3 = résultat partiel. #4 = chiffres qui restent. On vérifie si -% le deuxième nombre s'arrête.| -% \begin{macrocode} -\def\XINT_add_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint_gob_til_W #5\xint_add_bz\W - \XINT_add_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_add_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_add_ABEA\the\numexpr #1+10#5#4#3#2+#6.% -}% -\def\XINT_add_ABEA #1#2#3.#4% -{% - \XINT_add_A #2{#3#4}% -}% -% \end{macrocode} -% \lverb|& -% ici le deuxième nombre est fini -% #6 part à la poubelle, #2#3#4#5 est le #2 dans \XINT_add_AB -% on ne vérifie pas la retenue cette fois, mais les fois suivantes| -% \begin{macrocode} -\def\xint_add_bz\W\XINT_add_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_add_CC\the\numexpr #1+10#5#4#3#2.% -}% -\def\XINT_add_CC #1#2#3.#4% -{% - \XINT_add_AC_checkcarry #2{#3#4}% on va examiner et \'eliminer #2 -}% -% \end{macrocode} -% \lverb|& -% retenue plus chiffres qui restent de l'un des deux nombres. -% #2 = résultat partiel -% #3#4#5#6 = summand, avec plus significatif à droite| -% \begin{macrocode} -\def\XINT_add_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_add_AC_nocarry 0\XINT_add_C -}% -\def\xint_add_AC_nocarry 0\XINT_add_C #1#2\W\X\Y\Z -{% - \expandafter - \xint_cleanupzeros_andstop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% -}% -\def\XINT_add_C #1#2#3#4#5% -{% - \xint_gob_til_W #2\xint_add_cz\W - \XINT_add_CD {#5#4#3#2}{#1}% -}% -\def\XINT_add_CD #1% -{% - \expandafter\XINT_add_CC\the\numexpr 1+10#1.% -}% -\def\xint_add_cz\W\XINT_add_CD #1#2{ 1#2}% -% \end{macrocode} -% \lverb|Addition II: \XINT_addr_A.$\ -% INPUT: \romannumeral0\XINT_addr_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z -% -% Comme \XINT_add_A, la différence principale c'est qu'elle donne son résultat -% aussi *sur 4n*, renversé. De plus cette variante accepte que l'un ou même les -% deux inputs soient vides. Utilisé par la sommation et par la division (pour -% les quotients). Et aussi par la multiplication d'ailleurs.$\ -% INPUT: comme pour \XINT_add_A$\ -% 1. <N1> et <N2> renversés $\ -% 2. de longueur 4n (avec des leading zéros éventuels)$\ -% 3. l'un des deux ne doit pas se terminer par 0000$\ -% OUTPUT: la somme <N1>+<N2>, *aussi renversée* et *sur 4n*| -% \begin{macrocode} -\def\XINT_addr_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_addr_az\W - \XINT_addr_B #1{#3#4#5#6}{#2}% -}% -\def\xint_addr_az\W\XINT_addr_B #1#2% -{% - \XINT_addr_AC_checkcarry #1% -}% -\def\XINT_addr_B #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint_gob_til_W #5\xint_addr_bz\W - \XINT_addr_E #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_addr_E #1#2#3#4#5#6% -{% - \expandafter\XINT_addr_ABEA\the\numexpr #1+10#5#4#3#2+#6\relax -}% -\def\XINT_addr_ABEA #1#2#3#4#5#6#7% -{% - \XINT_addr_A #2{#7#6#5#4#3}% -}% -\def\xint_addr_bz\W\XINT_addr_E #1#2#3#4#5#6% -{% - \expandafter\XINT_addr_CC\the\numexpr #1+10#5#4#3#2\relax -}% -\def\XINT_addr_CC #1#2#3#4#5#6#7% -{% - \XINT_addr_AC_checkcarry #2{#7#6#5#4#3}% -}% -\def\XINT_addr_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_addr_AC_nocarry 0\XINT_addr_C -}% -\def\xint_addr_AC_nocarry 0\XINT_addr_C #1#2\W\X\Y\Z { #1#2}% -\def\XINT_addr_C #1#2#3#4#5% -{% - \xint_gob_til_W #2\xint_addr_cz\W - \XINT_addr_D {#5#4#3#2}{#1}% -}% -\def\XINT_addr_D #1% -{% - \expandafter\XINT_addr_CC\the\numexpr 1+10#1\relax -}% -\def\xint_addr_cz\W\XINT_addr_D #1#2{ #21000}% -% \end{macrocode} -% \lverb|ADDITION III, \XINT_addm_A$\ -% INPUT:\romannumeral0\XINT_addm_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z$\ -% 1. <N1> et <N2> renversés$\ -% 2. <N1> de longueur 4n ; <N2> non$\ -% 3. <N2> est *garanti au moins aussi long* que <N1>$\ -% OUTPUT: la somme <N1>+<N2>, ordre normal, pas sur 4n, leading zeros retirés. -% Utilisé par la multiplication.| -% \begin{macrocode} -\def\XINT_addm_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_addm_az\W - \XINT_addm_AB #1{#3#4#5#6}{#2}% -}% -\def\xint_addm_az\W\XINT_addm_AB #1#2% -{% - \XINT_addm_AC_checkcarry #1% -}% -\def\XINT_addm_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \XINT_addm_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_addm_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_addm_ABEA\the\numexpr #1+10#5#4#3#2+#6.% -}% -\def\XINT_addm_ABEA #1#2#3.#4% -{% - \XINT_addm_A #2{#3#4}% -}% -\def\XINT_addm_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_addm_AC_nocarry 0\XINT_addm_C -}% -\def\xint_addm_AC_nocarry 0\XINT_addm_C #1#2\W\X\Y\Z -{% - \expandafter - \xint_cleanupzeros_andstop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% -}% -\def\XINT_addm_C #1#2#3#4#5% -{% - \xint_gob_til_W - #5\xint_addm_cw - #4\xint_addm_cx - #3\xint_addm_cy - #2\xint_addm_cz - \W\XINT_addm_CD {#5#4#3#2}{#1}% -}% -\def\XINT_addm_CD #1% -{% - \expandafter\XINT_addm_CC\the\numexpr 1+10#1.% -}% -\def\XINT_addm_CC #1#2#3.#4% -{% - \XINT_addm_AC_checkcarry #2{#3#4}% -}% -\def\xint_addm_cw - #1\xint_addm_cx - #2\xint_addm_cy - #3\xint_addm_cz - \W\XINT_addm_CD -{% - \expandafter\XINT_addm_CDw\the\numexpr 1+#1#2#3.% -}% -\def\XINT_addm_CDw #1.#2#3\X\Y\Z -{% - \XINT_addm_end #1#3% -}% -\def\xint_addm_cx - #1\xint_addm_cy - #2\xint_addm_cz - \W\XINT_addm_CD -{% - \expandafter\XINT_addm_CDx\the\numexpr 1+#1#2.% -}% -\def\XINT_addm_CDx #1.#2#3\Y\Z -{% - \XINT_addm_end #1#3% -}% -\def\xint_addm_cy - #1\xint_addm_cz - \W\XINT_addm_CD -{% - \expandafter\XINT_addm_CDy\the\numexpr 1+#1.% -}% -\def\XINT_addm_CDy #1.#2#3\Z -{% - \XINT_addm_end #1#3% -}% -\def\xint_addm_cz\W\XINT_addm_CD #1#2#3{\XINT_addm_end #1#3}% -\edef\XINT_addm_end #1#2#3#4#5% - {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5\relax}% -% \end{macrocode} -% \lverb|ADDITION IV, variante \XINT_addp_A$\ -% INPUT: -% \romannumeral0\XINT_addp_A 0{}<N1>\W\X\Y\Z <N2>\W\X\Y\Z$\ -% 1. <N1> et <N2> renversés$\ -% 2. <N1> de longueur 4n ; <N2> non$\ -% 3. <N2> est *garanti au moins aussi long* que <N1>$\ -% OUTPUT: la somme <N1>+<N2>, dans l'ordre renversé, sur 4n, et en faisant -% attention de ne pas terminer en 0000. -% Utilisé par la multiplication servant pour le calcul des puissances.| -% \begin{macrocode} -\def\XINT_addp_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_addp_az\W - \XINT_addp_AB #1{#3#4#5#6}{#2}% -}% -\def\xint_addp_az\W\XINT_addp_AB #1#2% -{% - \XINT_addp_AC_checkcarry #1% -}% -\def\XINT_addp_AC_checkcarry #1% -{% - \xint_gob_til_zero #1\xint_addp_AC_nocarry 0\XINT_addp_C -}% -\def\xint_addp_AC_nocarry 0\XINT_addp_C -{% - \XINT_addp_F -}% -\def\XINT_addp_AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \XINT_addp_ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_addp_ABE #1#2#3#4#5#6% -{% - \expandafter\XINT_addp_ABEA\the\numexpr #1+10#5#4#3#2+#6\relax -}% -\def\XINT_addp_ABEA #1#2#3#4#5#6#7% -{% - \XINT_addp_A #2{#7#6#5#4#3}%<-- attention on met donc \`a droite -}% -\def\XINT_addp_C #1#2#3#4#5% -{% - \xint_gob_til_W - #5\xint_addp_cw - #4\xint_addp_cx - #3\xint_addp_cy - #2\xint_addp_cz - \W\XINT_addp_CD {#5#4#3#2}{#1}% -}% -\def\XINT_addp_CD #1% -{% - \expandafter\XINT_addp_CC\the\numexpr 1+10#1\relax -}% -\def\XINT_addp_CC #1#2#3#4#5#6#7% -{% - \XINT_addp_AC_checkcarry #2{#7#6#5#4#3}% -}% -\def\xint_addp_cw - #1\xint_addp_cx - #2\xint_addp_cy - #3\xint_addp_cz - \W\XINT_addp_CD -{% - \expandafter\XINT_addp_CDw\the\numexpr \xint_c_i+10#1#2#3\relax -}% -\def\XINT_addp_CDw #1#2#3#4#5#6% -{% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDw_zeros - 0000\XINT_addp_endDw #2#3#4#5% -}% -\def\XINT_addp_endDw_zeros 0000\XINT_addp_endDw 0000#1\X\Y\Z{ #1}% -\def\XINT_addp_endDw #1#2#3#4#5\X\Y\Z{ #5#4#3#2#1}% -\def\xint_addp_cx - #1\xint_addp_cy - #2\xint_addp_cz - \W\XINT_addp_CD -{% - \expandafter\XINT_addp_CDx\the\numexpr \xint_c_i+100#1#2\relax -}% -\def\XINT_addp_CDx #1#2#3#4#5#6% -{% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDx_zeros - 0000\XINT_addp_endDx #2#3#4#5% -}% -\def\XINT_addp_endDx_zeros 0000\XINT_addp_endDx 0000#1\Y\Z{ #1}% -\def\XINT_addp_endDx #1#2#3#4#5\Y\Z{ #5#4#3#2#1}% -\def\xint_addp_cy #1\xint_addp_cz\W\XINT_addp_CD -{% - \expandafter\XINT_addp_CDy\the\numexpr \xint_c_i+1000#1\relax -}% -\def\XINT_addp_CDy #1#2#3#4#5#6% -{% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_addp_endDy_zeros - 0000\XINT_addp_endDy #2#3#4#5% -}% -\def\XINT_addp_endDy_zeros 0000\XINT_addp_endDy 0000#1\Z{ #1}% -\def\XINT_addp_endDy #1#2#3#4#5\Z{ #5#4#3#2#1}% -\def\xint_addp_cz\W\XINT_addp_CD #1#2{ #21000}% -\def\XINT_addp_F #1#2#3#4#5% -{% - \xint_gob_til_W - #5\xint_addp_Gw - #4\xint_addp_Gx - #3\xint_addp_Gy - #2\xint_addp_Gz - \W\XINT_addp_G {#2#3#4#5}{#1}% -}% -\def\XINT_addp_G #1#2% -{% - \XINT_addp_F {#2#1}% -}% -\def\xint_addp_Gw - #1\xint_addp_Gx - #2\xint_addp_Gy - #3\xint_addp_Gz - \W\XINT_addp_G #4% -{% - \xint_gob_til_zeros_iv #3#2#10\XINT_addp_endGw_zeros - 0000\XINT_addp_endGw #3#2#10% -}% -\def\XINT_addp_endGw_zeros 0000\XINT_addp_endGw 0000#1\X\Y\Z{ #1}% -\def\XINT_addp_endGw #1#2#3#4#5\X\Y\Z{ #5#1#2#3#4}% -\def\xint_addp_Gx - #1\xint_addp_Gy - #2\xint_addp_Gz - \W\XINT_addp_G #3% -{% - \xint_gob_til_zeros_iv #2#100\XINT_addp_endGx_zeros - 0000\XINT_addp_endGx #2#100% -}% -\def\XINT_addp_endGx_zeros 0000\XINT_addp_endGx 0000#1\Y\Z{ #1}% -\def\XINT_addp_endGx #1#2#3#4#5\Y\Z{ #5#1#2#3#4}% -\def\xint_addp_Gy - #1\xint_addp_Gz - \W\XINT_addp_G #2% -{% - \xint_gob_til_zeros_iv #1000\XINT_addp_endGy_zeros - 0000\XINT_addp_endGy #1000% -}% -\def\XINT_addp_endGy_zeros 0000\XINT_addp_endGy 0000#1\Z{ #1}% -\def\XINT_addp_endGy #1#2#3#4#5\Z{ #5#1#2#3#4}% -\def\xint_addp_Gz\W\XINT_addp_G #1#2{ #2}% -% \end{macrocode} -% \subsection{\csh{xintAdd}} -% \lverb|Release 1.09a has \xintnum added into \xintiAdd.| -% \begin{macrocode} -\def\xintiiAdd {\romannumeral0\xintiiadd }% -\def\xintiiadd #1% -{% - \expandafter\xint_iiadd\expandafter{\romannumeral-`0#1}% -}% -\def\xint_iiadd #1#2% -{% - \expandafter\XINT_add_fork \romannumeral-`0#2\Z #1\Z -}% -\def\xintiAdd {\romannumeral0\xintiadd }% -\def\xintiadd #1% -{% - \expandafter\xint_add\expandafter{\romannumeral0\xintnum{#1}}% -}% -\def\xint_add #1#2% -{% - \expandafter\XINT_add_fork \romannumeral0\xintnum{#2}\Z #1\Z -}% -\let\xintAdd\xintiAdd \let\xintadd\xintiadd -\def\XINT_Add #1#2{\romannumeral0\XINT_add_fork #2\Z #1\Z }% -\def\XINT_add #1#2{\XINT_add_fork #2\Z #1\Z }% -% \end{macrocode} -% \lverb|ADDITION -% Ici #1#2 vient du *deuxième* argument de \xintAdd et #3#4 donc du *premier* -% [algo plus efficace lorsque le premier est plus long que le second]| -% \begin{macrocode} -\def\XINT_add_fork #1#2\Z #3#4\Z -{% - \xint_UDzerofork - #1\XINT_add_secondiszero - #3\XINT_add_firstiszero - 0 - {\xint_UDsignsfork - #1#3\XINT_add_minusminus % #1 = #3 = - - #1-\XINT_add_minusplus % #1 = - - #3-\XINT_add_plusminus % #3 = - - --\XINT_add_plusplus - \krof }% - \krof - {#2}{#4}#1#3% -}% -\def\XINT_add_secondiszero #1#2#3#4{ #4#2}% -\def\XINT_add_firstiszero #1#2#3#4{ #3#1}% -% \end{macrocode} -% \lverb|#1 vient du *deuxième* et #2 vient du *premier*| -% \begin{macrocode} -\def\XINT_add_minusminus #1#2#3#4% -{% - \expandafter\xint_minus_thenstop% - \romannumeral0\XINT_add_pre {#2}{#1}% -}% -\def\XINT_add_minusplus #1#2#3#4% -{% - \XINT_sub_pre {#4#2}{#1}% -}% -\def\XINT_add_plusminus #1#2#3#4% -{% - \XINT_sub_pre {#3#1}{#2}% -}% -\def\XINT_add_plusplus #1#2#3#4% -{% - \XINT_add_pre {#4#2}{#3#1}% -}% -\def\XINT_add_pre #1% -{% - \expandafter\XINT_add_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% -}% -\def\XINT_add_pre_b #1#2% -{% - \expandafter\XINT_add_A - \expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z -}% -% \end{macrocode} -% \subsection{\csh{xintSub}} -% \lverb|Release 1.09a has \xintnum added into \xintiSub.| -% \begin{macrocode} -\def\xintiiSub {\romannumeral0\xintiisub }% -\def\xintiisub #1% -{% - \expandafter\xint_iisub\expandafter{\romannumeral-`0#1}% -}% -\def\xint_iisub #1#2% -{% - \expandafter\XINT_sub_fork \romannumeral-`0#2\Z #1\Z -}% -\def\xintiSub {\romannumeral0\xintisub }% -\def\xintisub #1% -{% - \expandafter\xint_sub\expandafter{\romannumeral0\xintnum{#1}}% -}% -\def\xint_sub #1#2% -{% - \expandafter\XINT_sub_fork \romannumeral0\xintnum{#2}\Z #1\Z -}% -\def\XINT_Sub #1#2{\romannumeral0\XINT_sub_fork #2\Z #1\Z }% -\def\XINT_sub #1#2{\XINT_sub_fork #2\Z #1\Z }% -\let\xintSub\xintiSub \let\xintsub\xintisub -% \end{macrocode} -% \lverb|& -% SOUSTRACTION -% #3#4-#1#2: -% #3#4 vient du *premier* -% #1#2 vient du *second*| -% \begin{macrocode} -\def\XINT_sub_fork #1#2\Z #3#4\Z -{% - \xint_UDsignsfork - #1#3\XINT_sub_minusminus - #1-\XINT_sub_minusplus % attention, #3=0 possible - #3-\XINT_sub_plusminus % attention, #1=0 possible - --{\xint_UDzerofork - #1\XINT_sub_secondiszero - #3\XINT_sub_firstiszero - 0\XINT_sub_plusplus - \krof }% - \krof - {#2}{#4}#1#3% -}% -\def\XINT_sub_secondiszero #1#2#3#4{ #4#2}% -\def\XINT_sub_firstiszero #1#2#3#4{ -#3#1}% -\def\XINT_sub_plusplus #1#2#3#4% -{% - \XINT_sub_pre {#4#2}{#3#1}% -}% -\def\XINT_sub_minusminus #1#2#3#4% -{% - \XINT_sub_pre {#1}{#2}% -}% -\def\XINT_sub_minusplus #1#2#3#4% -{% - \xint_gob_til_zero #4\xint_sub_mp0\XINT_add_pre {#4#2}{#1}% -}% -\def\xint_sub_mp0\XINT_add_pre #1#2{ #2}% -\def\XINT_sub_plusminus #1#2#3#4% -{% - \xint_gob_til_zero #3\xint_sub_pm0\expandafter\xint_minus_thenstop% - \romannumeral0\XINT_add_pre {#2}{#3#1}% -}% -\def\xint_sub_pm #1\XINT_add_pre #2#3{ -#2}% -\def\XINT_sub_pre #1% -{% - \expandafter\XINT_sub_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% -}% -\def\XINT_sub_pre_b #1#2% -{% - \expandafter\XINT_sub_A - \expandafter1\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1 \W\X\Y\Z -}% -% \end{macrocode} -% \lverb|& -% \romannumeral0\XINT_sub_A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z$\ -% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS -% POUR QUE LEURS LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS -% AUCUN NE SE TERMINE EN 0000.$\ output: N2 - N1$\ -% Elle donne le résultat dans le **bon ordre**, avec le bon signe, -% et sans zéros superflus.| -% \begin{macrocode} -\def\XINT_sub_A #1#2#3\W\X\Y\Z #4#5#6#7% -{% - \xint_gob_til_W - #4\xint_sub_az - \W\XINT_sub_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z -}% -\def\XINT_sub_B #1#2#3#4#5#6#7% -{% - \xint_gob_til_W - #4\xint_sub_bz - \W\XINT_sub_onestep #1#2{#7#6#5#4}{#3}% -}% -% \end{macrocode} -% \lverb|& -% d'abord la branche principale -% #6 = 4 chiffres de N1, plus significatif en *premier*, -% #2#3#4#5 chiffres de N2, plus significatif en *dernier* -% On veut N2 - N1.| -% \begin{macrocode} -\def\XINT_sub_onestep #1#2#3#4#5#6% -{% - \expandafter\XINT_sub_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.% -}% -% \end{macrocode} -% \lverb|ON PRODUIT LE RÉSULTAT DANS LE BON ORDRE| -% \begin{macrocode} -\def\XINT_sub_backtoA #1#2#3.#4% -{% - \XINT_sub_A #2{#3#4}% -}% -\def\xint_sub_bz - \W\XINT_sub_onestep #1#2#3#4#5#6#7% -{% - \xint_UDzerofork - #1\XINT_sub_C % une retenue - 0\XINT_sub_D % pas de retenue - \krof - {#7}#2#3#4#5% -}% -\def\XINT_sub_D #1#2\W\X\Y\Z -{% - \expandafter - \xint_cleanupzeros_andstop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% -}% -\def\XINT_sub_C #1#2#3#4#5% -{% - \xint_gob_til_W - #2\xint_sub_cz - \W\XINT_sub_AC_onestep {#5#4#3#2}{#1}% -}% -\def\XINT_sub_AC_onestep #1% -{% - \expandafter\XINT_sub_backtoC\the\numexpr 11#1-\xint_c_i.% -}% -\def\XINT_sub_backtoC #1#2#3.#4% -{% - \XINT_sub_AC_checkcarry #2{#3#4}% la retenue va \^etre examin\'ee -}% -\def\XINT_sub_AC_checkcarry #1% -{% - \xint_gob_til_one #1\xint_sub_AC_nocarry 1\XINT_sub_C -}% -\def\xint_sub_AC_nocarry 1\XINT_sub_C #1#2\W\X\Y\Z -{% - \expandafter - \XINT_cuz_loop - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1\W\W\W\W\W\W\W\Z -}% -\def\xint_sub_cz\W\XINT_sub_AC_onestep #1% -{% - \XINT_cuz -}% -\def\xint_sub_az\W\XINT_sub_B #1#2#3#4#5#6#7% -{% - \xint_gob_til_W - #4\xint_sub_ez - \W\XINT_sub_Eenter #1{#3}#4#5#6#7% -}% -% \end{macrocode} -% \lverb|le premier nombre continue, le résultat sera < 0.| -% \begin{macrocode} -\def\XINT_sub_Eenter #1#2% -{% - \expandafter - \XINT_sub_E\expandafter1\expandafter{\expandafter}% - \romannumeral0% - \XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - \W\X\Y\Z #1% -}% -\def\XINT_sub_E #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_sub_F\W - \XINT_sub_Eonestep #1{#6#5#4#3}{#2}% -}% -\def\XINT_sub_Eonestep #1#2% -{% - \expandafter\XINT_sub_backtoE\the\numexpr 109999-#2+#1.% -}% -\def\XINT_sub_backtoE #1#2#3.#4% -{% - \XINT_sub_E #2{#3#4}% -}% -\def\xint_sub_F\W\XINT_sub_Eonestep #1#2#3#4% -{% - \xint_UDonezerofork - #4#1{\XINT_sub_Fdec 0}% soustraire 1. Et faire signe - - #1#4{\XINT_sub_Finc 1}% additionner 1. Et faire signe - - 10\XINT_sub_DD % terminer. Mais avec signe - - \krof - {#3}% -}% -\def\XINT_sub_DD {\expandafter\xint_minus_thenstop\romannumeral0\XINT_sub_D }% -\def\XINT_sub_Fdec #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_sub_Fdec_finish\W - \XINT_sub_Fdec_onestep #1{#6#5#4#3}{#2}% -}% -\def\XINT_sub_Fdec_onestep #1#2% -{% - \expandafter\XINT_sub_backtoFdec\the\numexpr 11#2+#1-\xint_c_i.% -}% -\def\XINT_sub_backtoFdec #1#2#3.#4% -{% - \XINT_sub_Fdec #2{#3#4}% -}% -\def\xint_sub_Fdec_finish\W\XINT_sub_Fdec_onestep #1#2% -{% - \expandafter\xint_minus_thenstop\romannumeral0\XINT_cuz -}% -\def\XINT_sub_Finc #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_sub_Finc_finish\W - \XINT_sub_Finc_onestep #1{#6#5#4#3}{#2}% -}% -\def\XINT_sub_Finc_onestep #1#2% -{% - \expandafter\XINT_sub_backtoFinc\the\numexpr 10#2+#1.% -}% -\def\XINT_sub_backtoFinc #1#2#3.#4% -{% - \XINT_sub_Finc #2{#3#4}% -}% -\def\xint_sub_Finc_finish\W\XINT_sub_Finc_onestep #1#2#3% -{% - \xint_UDzerofork - #1{\expandafter\expandafter\expandafter - \xint_minus_thenstop\xint_cleanupzeros_nostop}% - 0{ -1}% - \krof - #3% -}% -\def\xint_sub_ez\W\XINT_sub_Eenter #1% -{% - \xint_UDzerofork - #1\XINT_sub_K % il y a une retenue - 0\XINT_sub_L % pas de retenue - \krof -}% -\def\XINT_sub_L #1\W\X\Y\Z {\XINT_cuz_loop #1\W\W\W\W\W\W\W\Z }% -\def\XINT_sub_K #1% -{% - \expandafter - \XINT_sub_KK\expandafter1\expandafter{\expandafter}% - \romannumeral0% - \XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\def\XINT_sub_KK #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_sub_KK_finish\W - \XINT_sub_KK_onestep #1{#6#5#4#3}{#2}% -}% -\def\XINT_sub_KK_onestep #1#2% -{% - \expandafter\XINT_sub_backtoKK\the\numexpr 109999-#2+#1.% -}% -\def\XINT_sub_backtoKK #1#2#3.#4% -{% - \XINT_sub_KK #2{#3#4}% -}% -\def\xint_sub_KK_finish\W\XINT_sub_KK_onestep #1#2#3% -{% - \expandafter\xint_minus_thenstop - \romannumeral0\XINT_cuz_loop #3\W\W\W\W\W\W\W\Z -}% -% \end{macrocode} -% \subsection{\csh{xintCmp}} -% \lverb|Release 1.09a has \xintnum inserted into \xintCmp. Unnecessary -% \xintiCmp suppressed in 1.09f.| -% \begin{macrocode} -\def\xintCmp {\romannumeral0\xintcmp }% -\def\xintcmp #1% -{% - \expandafter\xint_cmp\expandafter{\romannumeral0\xintnum{#1}}% -}% -\def\xint_cmp #1#2% -{% - \expandafter\XINT_cmp_fork \romannumeral0\xintnum{#2}\Z #1\Z -}% -\def\XINT_Cmp #1#2{\romannumeral0\XINT_cmp_fork #2\Z #1\Z }% -% \end{macrocode} -% \lverb|& -% COMPARAISON $\ -% 1 si #3#4>#1#2, 0 si #3#4=#1#2, -1 si #3#4<#1#2$\ -% #3#4 vient du *premier*,$ -% #1#2 vient du *second*| -% \begin{macrocode} -\def\XINT_cmp_fork #1#2\Z #3#4\Z -{% - \xint_UDsignsfork - #1#3\XINT_cmp_minusminus - #1-\XINT_cmp_minusplus - #3-\XINT_cmp_plusminus - --{\xint_UDzerosfork - #1#3\XINT_cmp_zerozero - #10\XINT_cmp_zeroplus - #30\XINT_cmp_pluszero - 00\XINT_cmp_plusplus - \krof }% - \krof - {#2}{#4}#1#3% -}% -\def\XINT_cmp_minusplus #1#2#3#4{ 1}% -\def\XINT_cmp_plusminus #1#2#3#4{ -1}% -\def\XINT_cmp_zerozero #1#2#3#4{ 0}% -\def\XINT_cmp_zeroplus #1#2#3#4{ 1}% -\def\XINT_cmp_pluszero #1#2#3#4{ -1}% -\def\XINT_cmp_plusplus #1#2#3#4% -{% - \XINT_cmp_pre {#4#2}{#3#1}% -}% -\def\XINT_cmp_minusminus #1#2#3#4% -{% - \XINT_cmp_pre {#1}{#2}% -}% -\def\XINT_cmp_pre #1% -{% - \expandafter\XINT_cmp_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% -}% -\def\XINT_cmp_pre_b #1#2% -{% - \expandafter\XINT_cmp_A - \expandafter1\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z -}% -% \end{macrocode} -% \lverb|& -% COMPARAISON$\ -% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS -% POUR QUE LEUR LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS -% AUCUN NE SE TERMINE EN 0000. -% routine appelée via$\ -% \XINT_cmp_A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z$\ -% ATTENTION RENVOIE 1 SI N1 < N2, 0 si N1 = N2, -1 si N1 > N2| -% \begin{macrocode} -\def\XINT_cmp_A #1#2#3\W\X\Y\Z #4#5#6#7% -{% - \xint_gob_til_W #4\xint_cmp_az\W - \XINT_cmp_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z -}% -\def\XINT_cmp_B #1#2#3#4#5#6#7% -{% - \xint_gob_til_W#4\xint_cmp_bz\W - \XINT_cmp_onestep #1#2{#7#6#5#4}{#3}% -}% -\def\XINT_cmp_onestep #1#2#3#4#5#6% -{% - \expandafter\XINT_cmp_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.% -}% -\def\XINT_cmp_backtoA #1#2#3.#4% -{% - \XINT_cmp_A #2{#3#4}% -}% -\def\xint_cmp_bz\W\XINT_cmp_onestep #1\Z { 1}% -\def\xint_cmp_az\W\XINT_cmp_B #1#2#3#4#5#6#7% -{% - \xint_gob_til_W #4\xint_cmp_ez\W - \XINT_cmp_Eenter #1{#3}#4#5#6#7% -}% -\def\XINT_cmp_Eenter #1\Z { -1}% -\def\xint_cmp_ez\W\XINT_cmp_Eenter #1% -{% - \xint_UDzerofork - #1\XINT_cmp_K % il y a une retenue - 0\XINT_cmp_L % pas de retenue - \krof -}% -\def\XINT_cmp_K #1\Z { -1}% -\def\XINT_cmp_L #1{\XINT_OneIfPositive_main #1}% -\def\XINT_OneIfPositive #1% -{% - \XINT_OneIfPositive_main #1\W\X\Y\Z% -}% -\def\XINT_OneIfPositive_main #1#2#3#4% -{% - \xint_gob_til_Z #4\xint_OneIfPositive_terminated\Z - \XINT_OneIfPositive_onestep #1#2#3#4% -}% -\def\xint_OneIfPositive_terminated\Z\XINT_OneIfPositive_onestep\W\X\Y\Z { 0}% -\def\XINT_OneIfPositive_onestep #1#2#3#4% -{% - \expandafter\XINT_OneIfPositive_check\the\numexpr #1#2#3#4\relax -}% -\def\XINT_OneIfPositive_check #1% -{% - \xint_gob_til_zero #1\xint_OneIfPositive_backtomain 0% - \XINT_OneIfPositive_finish #1% -}% -\def\XINT_OneIfPositive_finish #1\W\X\Y\Z{ 1}% -\def\xint_OneIfPositive_backtomain 0\XINT_OneIfPositive_finish 0% - {\XINT_OneIfPositive_main }% -% \end{macrocode} -% \subsection{\csh{xintEq}, \csh{xintGt}, \csh{xintLt}} -% \lverb|1.09a.| -% \begin{macrocode} -\def\xintEq {\romannumeral0\xinteq }% -\def\xinteq #1#2{\xintifeq{#1}{#2}{1}{0}}% -\def\xintGt {\romannumeral0\xintgt }% -\def\xintgt #1#2{\xintifgt{#1}{#2}{1}{0}}% -\def\xintLt {\romannumeral0\xintlt }% -\def\xintlt #1#2{\xintiflt{#1}{#2}{1}{0}}% -% \end{macrocode} -% \subsection{\csh{xintIsZero}, \csh{xintIsNotZero}} -% \lverb|1.09a. restyled in 1.09i.| -% \begin{macrocode} -\def\xintIsZero {\romannumeral0\xintiszero }% -\def\xintiszero #1{\if0\xintSgn{#1}\xint_afterfi{ 1}\else\xint_afterfi{ 0}\fi}% -\def\xintIsNotZero {\romannumeral0\xintisnotzero }% -\def\xintisnotzero - #1{\if0\xintSgn{#1}\xint_afterfi{ 0}\else\xint_afterfi{ 1}\fi}% -% \end{macrocode} -% \subsection{\csh{xintIsTrue}, \csh{xintNot}, \csh{xintIsFalse}} -% \lverb|1.09c| -% \begin{macrocode} -\let\xintIsTrue\xintIsNotZero -\let\xintNot\xintIsZero -\let\xintIsFalse\xintIsZero -% \end{macrocode} -% \subsection{\csh{xintAND}, \csh{xintOR}, \csh{xintXOR}} -% \lverb|1.09a. Embarrasing bugs in \xintAND and \xintOR which inserted a space -% token corrected in 1.09i. \xintxor restyled with \if (faster) in 1.09i| -% \begin{macrocode} -\def\xintAND {\romannumeral0\xintand }% -\def\xintand #1#2{\if0\xintSgn{#1}\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo\fi - { 0}{\xintisnotzero{#2}}}% -\def\xintOR {\romannumeral0\xintor }% -\def\xintor #1#2{\if0\xintSgn{#1}\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo\fi - {\xintisnotzero{#2}}{ 1}}% -\def\xintXOR {\romannumeral0\xintxor }% -\def\xintxor #1#2{\if\xintIsZero{#1}\xintIsZero{#2}% - \xint_afterfi{ 0}\else\xint_afterfi{ 1}\fi }% -% \end{macrocode} -% \subsection{\csh{xintANDof}} -% \lverb|New with 1.09a. \xintANDof works also with an empty list.| -% \begin{macrocode} -\def\xintANDof {\romannumeral0\xintandof }% -\def\xintandof #1{\expandafter\XINT_andof_a\romannumeral-`0#1\relax }% -\def\XINT_andof_a #1{\expandafter\XINT_andof_b\romannumeral-`0#1\Z }% -\def\XINT_andof_b #1% - {\xint_gob_til_relax #1\XINT_andof_e\relax\XINT_andof_c #1}% -\def\XINT_andof_c #1\Z - {\xintifTrueAelseB {#1}{\XINT_andof_a}{\XINT_andof_no}}% -\def\XINT_andof_no #1\relax { 0}% -\def\XINT_andof_e #1\Z { 1}% -% \end{macrocode} -% \subsection{\csh{xintORof}} -% \lverb|New with 1.09a. Works also with an empty list.| -% \begin{macrocode} -\def\xintORof {\romannumeral0\xintorof }% -\def\xintorof #1{\expandafter\XINT_orof_a\romannumeral-`0#1\relax }% -\def\XINT_orof_a #1{\expandafter\XINT_orof_b\romannumeral-`0#1\Z }% -\def\XINT_orof_b #1% - {\xint_gob_til_relax #1\XINT_orof_e\relax\XINT_orof_c #1}% -\def\XINT_orof_c #1\Z - {\xintifTrueAelseB {#1}{\XINT_orof_yes}{\XINT_orof_a}}% -\def\XINT_orof_yes #1\relax { 1}% -\def\XINT_orof_e #1\Z { 0}% -% \end{macrocode} -% \subsection{\csh{xintXORof}} -% \lverb|New with 1.09a. Works with an empty list, too. \XINT_xorof_c more -% efficient in 1.09i| -% \begin{macrocode} -\def\xintXORof {\romannumeral0\xintxorof }% -\def\xintxorof #1{\expandafter\XINT_xorof_a\expandafter - 0\romannumeral-`0#1\relax }% -\def\XINT_xorof_a #1#2{\expandafter\XINT_xorof_b\romannumeral-`0#2\Z #1}% -\def\XINT_xorof_b #1% - {\xint_gob_til_relax #1\XINT_xorof_e\relax\XINT_xorof_c #1}% -\def\XINT_xorof_c #1\Z #2% - {\xintifTrueAelseB {#1}{\if #20\xint_afterfi{\XINT_xorof_a 1}% - \else\xint_afterfi{\XINT_xorof_a 0}\fi}% - {\XINT_xorof_a #2}% - }% -\def\XINT_xorof_e #1\Z #2{ #2}% -% \end{macrocode} -% \subsection{\csh{xintGeq}} -% \lverb|& -% Release 1.09a has \xintnum added into \xintGeq. Unused and useless \xintiGeq -% removed in 1.09e. -% PLUS GRAND OU ÉGAL -% attention compare les **valeurs absolues**| -% \begin{macrocode} -\def\xintGeq {\romannumeral0\xintgeq }% -\def\xintgeq #1% -{% - \expandafter\xint_geq\expandafter {\romannumeral0\xintnum{#1}}% -}% -\def\xint_geq #1#2% -{% - \expandafter\XINT_geq_fork \romannumeral0\xintnum{#2}\Z #1\Z -}% -\def\XINT_Geq #1#2{\romannumeral0\XINT_geq_fork #2\Z #1\Z }% -% \end{macrocode} -% \lverb|& -% PLUS GRAND OU ÉGAL -% ATTENTION, TESTE les VALEURS ABSOLUES| -% \begin{macrocode} -\def\XINT_geq_fork #1#2\Z #3#4\Z -{% - \xint_UDzerofork - #1\XINT_geq_secondiszero % |#1#2|=0 - #3\XINT_geq_firstiszero % |#1#2|>0 - 0{\xint_UDsignsfork - #1#3\XINT_geq_minusminus - #1-\XINT_geq_minusplus - #3-\XINT_geq_plusminus - --\XINT_geq_plusplus - \krof }% - \krof - {#2}{#4}#1#3% -}% -\def\XINT_geq_secondiszero #1#2#3#4{ 1}% -\def\XINT_geq_firstiszero #1#2#3#4{ 0}% -\def\XINT_geq_plusplus #1#2#3#4{\XINT_geq_pre {#4#2}{#3#1}}% -\def\XINT_geq_minusminus #1#2#3#4{\XINT_geq_pre {#2}{#1}}% -\def\XINT_geq_minusplus #1#2#3#4{\XINT_geq_pre {#4#2}{#1}}% -\def\XINT_geq_plusminus #1#2#3#4{\XINT_geq_pre {#2}{#3#1}}% -\def\XINT_geq_pre #1% -{% - \expandafter\XINT_geq_pre_b\expandafter - {\romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z }% -}% -\def\XINT_geq_pre_b #1#2% -{% - \expandafter\XINT_geq_A - \expandafter1\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1 \W\X\Y\Z -}% -% \end{macrocode} -% \lverb|& -% PLUS GRAND OU ÉGAL$\ -% N1 et N2 sont présentés à l'envers ET ON A RAJOUTÉ DES ZÉROS -% POUR QUE LEURS LONGUEURS À CHACUN SOIENT MULTIPLES DE 4, MAIS -% AUCUN NE SE TERMINE EN 0000$\ -% routine appelée via$\ -% \romannumeral0\XINT_geq_A 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z$\ -% ATTENTION RENVOIE 1 SI N1 < N2 ou N1 = N2 et 0 si N1 > N2| -% \begin{macrocode} -\def\XINT_geq_A #1#2#3\W\X\Y\Z #4#5#6#7% -{% - \xint_gob_til_W #4\xint_geq_az\W - \XINT_geq_B #1{#4#5#6#7}{#2}#3\W\X\Y\Z -}% -\def\XINT_geq_B #1#2#3#4#5#6#7% -{% - \xint_gob_til_W #4\xint_geq_bz\W - \XINT_geq_onestep #1#2{#7#6#5#4}{#3}% -}% -\def\XINT_geq_onestep #1#2#3#4#5#6% -{% - \expandafter\XINT_geq_backtoA\the\numexpr 11#5#4#3#2-#6+#1-\xint_c_i.% -}% -\def\XINT_geq_backtoA #1#2#3.#4% -{% - \XINT_geq_A #2{#3#4}% -}% -\def\xint_geq_bz\W\XINT_geq_onestep #1\W\X\Y\Z { 1}% -\def\xint_geq_az\W\XINT_geq_B #1#2#3#4#5#6#7% -{% - \xint_gob_til_W #4\xint_geq_ez\W - \XINT_geq_Eenter #1% -}% -\def\XINT_geq_Eenter #1\W\X\Y\Z { 0}% -\def\xint_geq_ez\W\XINT_geq_Eenter #1% -{% - \xint_UDzerofork - #1{ 0} % il y a une retenue - 0{ 1} % pas de retenue - \krof -}% -% \end{macrocode} -% \subsection{\csh{xintMax}} -% \lverb|& -% The rationale is that it is more efficient than using \xintCmp. -% 1.03 makes the code a tiny bit slower but easier to re-use for fractions. -% Note: actually since 1.08a code for fractions does not all reduce to these -% entry points, so perhaps I should revert the changes made in 1.03. Release -% 1.09a has \xintnum added into \xintiMax.| -% \begin{macrocode} -\def\xintiMax {\romannumeral0\xintimax }% -\def\xintimax #1% -{% - \expandafter\xint_max\expandafter {\romannumeral0\xintnum{#1}}% -}% -\let\xintMax\xintiMax \let\xintmax\xintimax -\def\xint_max #1#2% -{% - \expandafter\XINT_max_pre\expandafter {\romannumeral0\xintnum{#2}}{#1}% -}% -\def\XINT_max_pre #1#2{\XINT_max_fork #1\Z #2\Z {#2}{#1}}% -\def\XINT_Max #1#2{\romannumeral0\XINT_max_fork #2\Z #1\Z {#1}{#2}}% -% \end{macrocode} -% \lverb|& -% #3#4 vient du *premier*, -% #1#2 vient du *second*| -% \begin{macrocode} -\def\XINT_max_fork #1#2\Z #3#4\Z -{% - \xint_UDsignsfork - #1#3\XINT_max_minusminus % A < 0, B < 0 - #1-\XINT_max_minusplus % B < 0, A >= 0 - #3-\XINT_max_plusminus % A < 0, B >= 0 - --{\xint_UDzerosfork - #1#3\XINT_max_zerozero % A = B = 0 - #10\XINT_max_zeroplus % B = 0, A > 0 - #30\XINT_max_pluszero % A = 0, B > 0 - 00\XINT_max_plusplus % A, B > 0 - \krof }% - \krof - {#2}{#4}#1#3% -}% -% \end{macrocode} -% \lverb|& -% A = #4#2, B = #3#1| -% \begin{macrocode} -\def\XINT_max_zerozero #1#2#3#4{\xint_firstoftwo_thenstop }% -\def\XINT_max_zeroplus #1#2#3#4{\xint_firstoftwo_thenstop }% -\def\XINT_max_pluszero #1#2#3#4{\xint_secondoftwo_thenstop }% -\def\XINT_max_minusplus #1#2#3#4{\xint_firstoftwo_thenstop }% -\def\XINT_max_plusminus #1#2#3#4{\xint_secondoftwo_thenstop }% -\def\XINT_max_plusplus #1#2#3#4% -{% - \ifodd\XINT_Geq {#4#2}{#3#1} - \expandafter\xint_firstoftwo_thenstop - \else - \expandafter\xint_secondoftwo_thenstop - \fi -}% -% \end{macrocode} -% \lverb+#3=-, #4=-, #1 = |B| = -B, #2 = |A| = -A+ -% \begin{macrocode} -\def\XINT_max_minusminus #1#2#3#4% -{% - \ifodd\XINT_Geq {#1}{#2} - \expandafter\xint_firstoftwo_thenstop - \else - \expandafter\xint_secondoftwo_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintMaxof}} -% \lverb|New with 1.09a.| -% \begin{macrocode} -\def\xintiMaxof {\romannumeral0\xintimaxof }% -\def\xintimaxof #1{\expandafter\XINT_imaxof_a\romannumeral-`0#1\relax }% -\def\XINT_imaxof_a #1{\expandafter\XINT_imaxof_b\romannumeral0\xintnum{#1}\Z }% -\def\XINT_imaxof_b #1\Z #2% - {\expandafter\XINT_imaxof_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_imaxof_c #1% - {\xint_gob_til_relax #1\XINT_imaxof_e\relax\XINT_imaxof_d #1}% -\def\XINT_imaxof_d #1\Z - {\expandafter\XINT_imaxof_b\romannumeral0\xintimax {#1}}% -\def\XINT_imaxof_e #1\Z #2\Z { #2}% -\let\xintMaxof\xintiMaxof \let\xintmaxof\xintimaxof -% \end{macrocode} -% \subsection{\csh{xintMin}} -% \lverb|\xintnum added New with 1.09a.| -% \begin{macrocode} -\def\xintiMin {\romannumeral0\xintimin }% -\def\xintimin #1% -{% - \expandafter\xint_min\expandafter {\romannumeral0\xintnum{#1}}% -}% -\let\xintMin\xintiMin \let\xintmin\xintimin -\def\xint_min #1#2% -{% - \expandafter\XINT_min_pre\expandafter {\romannumeral0\xintnum{#2}}{#1}% -}% -\def\XINT_min_pre #1#2{\XINT_min_fork #1\Z #2\Z {#2}{#1}}% -\def\XINT_Min #1#2{\romannumeral0\XINT_min_fork #2\Z #1\Z {#1}{#2}}% -% \end{macrocode} -% \lverb|& -% #3#4 vient du *premier*, -% #1#2 vient du *second*| -% \begin{macrocode} -\def\XINT_min_fork #1#2\Z #3#4\Z -{% - \xint_UDsignsfork - #1#3\XINT_min_minusminus % A < 0, B < 0 - #1-\XINT_min_minusplus % B < 0, A >= 0 - #3-\XINT_min_plusminus % A < 0, B >= 0 - --{\xint_UDzerosfork - #1#3\XINT_min_zerozero % A = B = 0 - #10\XINT_min_zeroplus % B = 0, A > 0 - #30\XINT_min_pluszero % A = 0, B > 0 - 00\XINT_min_plusplus % A, B > 0 - \krof }% - \krof - {#2}{#4}#1#3% -}% -% \end{macrocode} -% \lverb|& -% A = #4#2, B = #3#1| -% \begin{macrocode} -\def\XINT_min_zerozero #1#2#3#4{\xint_firstoftwo_thenstop }% -\def\XINT_min_zeroplus #1#2#3#4{\xint_secondoftwo_thenstop }% -\def\XINT_min_pluszero #1#2#3#4{\xint_firstoftwo_thenstop }% -\def\XINT_min_minusplus #1#2#3#4{\xint_secondoftwo_thenstop }% -\def\XINT_min_plusminus #1#2#3#4{\xint_firstoftwo_thenstop }% -\def\XINT_min_plusplus #1#2#3#4% -{% - \ifodd\XINT_Geq {#4#2}{#3#1} - \expandafter\xint_secondoftwo_thenstop - \else - \expandafter\xint_firstoftwo_thenstop - \fi -}% -% \end{macrocode} -% \lverb+#3=-, #4=-, #1 = |B| = -B, #2 = |A| = -A+ -% \begin{macrocode} -\def\XINT_min_minusminus #1#2#3#4% -{% - \ifodd\XINT_Geq {#1}{#2} - \expandafter\xint_secondoftwo_thenstop - \else - \expandafter\xint_firstoftwo_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintMinof}} -% \lverb|1.09a| -% \begin{macrocode} -\def\xintiMinof {\romannumeral0\xintiminof }% -\def\xintiminof #1{\expandafter\XINT_iminof_a\romannumeral-`0#1\relax }% -\def\XINT_iminof_a #1{\expandafter\XINT_iminof_b\romannumeral0\xintnum{#1}\Z }% -\def\XINT_iminof_b #1\Z #2% - {\expandafter\XINT_iminof_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_iminof_c #1% - {\xint_gob_til_relax #1\XINT_iminof_e\relax\XINT_iminof_d #1}% -\def\XINT_iminof_d #1\Z - {\expandafter\XINT_iminof_b\romannumeral0\xintimin {#1}}% -\def\XINT_iminof_e #1\Z #2\Z { #2}% -\let\xintMinof\xintiMinof \let\xintminof\xintiminof -% \end{macrocode} -% \subsection{\csh{xintSum}} -% \lverb|& -% \xintSum {{a}{b}...{z}}$\ -% \xintSumExpr {a}{b}...{z}\relax$\ -% 1.03 (drastically) simplifies and makes the routines more efficient (for big -% computations). Also the way \xintSum and \xintSumExpr ...\relax are related. -% has been modified. Now \xintSumExpr \z \relax is accepted input when -% \z expands to a list of braced terms (prior only \xintSum {\z} or \xintSum \z -% was possible). -% -% 1.09a does NOT add the \xintnum overhead. 1.09h renames \xintiSum to -% \xintiiSum to correctly reflect this.| -% \begin{macrocode} -\def\xintiiSum {\romannumeral0\xintiisum }% -\def\xintiisum #1{\xintiisumexpr #1\relax }% -\def\xintiiSumExpr {\romannumeral0\xintiisumexpr }% -\def\xintiisumexpr {\expandafter\XINT_sumexpr\romannumeral-`0}% -\let\xintSum\xintiiSum \let\xintsum\xintiisum -\let\xintSumExpr\xintiiSumExpr \let\xintsumexpr\xintiisumexpr -\def\XINT_sumexpr {\XINT_sum_loop {0000}{0000}}% -\def\XINT_sum_loop #1#2#3% -{% - \expandafter\XINT_sum_checksign\romannumeral-`0#3\Z {#1}{#2}% -}% -\def\XINT_sum_checksign #1% -{% - \xint_gob_til_relax #1\XINT_sum_finished\relax - \xint_gob_til_zero #1\XINT_sum_skipzeroinput0% - \xint_UDsignfork - #1\XINT_sum_N - -{\XINT_sum_P #1}% - \krof -}% -\def\XINT_sum_finished #1\Z #2#3% -{% - \XINT_sub_A 1{}#3\W\X\Y\Z #2\W\X\Y\Z -}% -\def\XINT_sum_skipzeroinput #1\krof #2\Z {\XINT_sum_loop }% -\def\XINT_sum_P #1\Z #2% -{% - \expandafter\XINT_sum_loop\expandafter - {\romannumeral0\expandafter - \XINT_addr_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #2\W\X\Y\Z }% -}% -\def\XINT_sum_N #1\Z #2#3% -{% - \expandafter\XINT_sum_NN\expandafter - {\romannumeral0\expandafter - \XINT_addr_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #3\W\X\Y\Z }{#2}% -}% -\def\XINT_sum_NN #1#2{\XINT_sum_loop {#2}{#1}}% -% \end{macrocode} -% \subsection{\csh{xintMul}} -% \lverb|1.09a adds \xintnum| -% \begin{macrocode} -\def\xintiiMul {\romannumeral0\xintiimul }% -\def\xintiimul #1% -{% - \expandafter\xint_iimul\expandafter {\romannumeral-`0#1}% -}% -\def\xint_iimul #1#2% -{% - \expandafter\XINT_mul_fork \romannumeral-`0#2\Z #1\Z -}% -\def\xintiMul {\romannumeral0\xintimul }% -\def\xintimul #1% -{% - \expandafter\xint_mul\expandafter {\romannumeral0\xintnum{#1}}% -}% -\def\xint_mul #1#2% -{% - \expandafter\XINT_mul_fork \romannumeral0\xintnum{#2}\Z #1\Z -}% -\let\xintMul\xintiMul \let\xintmul\xintimul -\def\XINT_Mul #1#2{\romannumeral0\XINT_mul_fork #2\Z #1\Z }% -% \end{macrocode} -% \lverb|& -% MULTIPLICATION$\ -% Ici #1#2 = 2e input et #3#4 = 1er input $\ -% Release 1.03 adds some overhead to first compute and compare the -% lengths of the two inputs. The algorithm is asymmetrical and whether -% the first input is the longest or the shortest sometimes has a strong -% impact. 50 digits times 1000 digits used to be 5 times faster -% than 1000 digits times 50 digits. With the new code, the user input -% order does not matter as it is decided by the routine what is best. -% This is important for the extension to fractions, as there is no way -% then to generally control or guess the most frequent sizes of the -% inputs besides actually computing their lengths. | -% \begin{macrocode} -\def\XINT_mul_fork #1#2\Z #3#4\Z -{% - \xint_UDzerofork - #1\XINT_mul_zero - #3\XINT_mul_zero - 0{\xint_UDsignsfork - #1#3\XINT_mul_minusminus % #1 = #3 = - - #1-{\XINT_mul_minusplus #3}% % #1 = - - #3-{\XINT_mul_plusminus #1}% % #3 = - - --{\XINT_mul_plusplus #1#3}% - \krof }% - \krof - {#2}{#4}% -}% -\def\XINT_mul_zero #1#2{ 0}% -\def\XINT_mul_minusminus #1#2% -{% - \expandafter\XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#2}}% - {\romannumeral0\xintlength {#1}}{#1}{#2}% -}% -\def\XINT_mul_minusplus #1#2#3% -{% - \expandafter\xint_minus_thenstop\romannumeral0\expandafter - \XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#1#3}}% - {\romannumeral0\xintlength {#2}}{#2}{#1#3}% -}% -\def\XINT_mul_plusminus #1#2#3% -{% - \expandafter\xint_minus_thenstop\romannumeral0\expandafter - \XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#3}}% - {\romannumeral0\xintlength {#1#2}}{#1#2}{#3}% -}% -\def\XINT_mul_plusplus #1#2#3#4% -{% - \expandafter\XINT_mul_choice_a - \expandafter{\romannumeral0\xintlength {#2#4}}% - {\romannumeral0\xintlength {#1#3}}{#1#3}{#2#4}% -}% -\def\XINT_mul_choice_a #1#2% -{% - \expandafter\XINT_mul_choice_b\expandafter{#2}{#1}% -}% -\def\XINT_mul_choice_b #1#2% -{% - \ifnum #1<\xint_c_v - \expandafter\XINT_mul_choice_littlebyfirst - \else - \ifnum #2<\xint_c_v - \expandafter\expandafter\expandafter\XINT_mul_choice_littlebysecond - \else - \expandafter\expandafter\expandafter\XINT_mul_choice_compare - \fi - \fi - {#1}{#2}% -}% -\def\XINT_mul_choice_littlebyfirst #1#2#3#4% -{% - \expandafter\XINT_mul_M - \expandafter{\the\numexpr #3\expandafter}% - \romannumeral0\XINT_RQ {}#4\R\R\R\R\R\R\R\R\Z \Z\Z\Z\Z -}% -\def\XINT_mul_choice_littlebysecond #1#2#3#4% -{% - \expandafter\XINT_mul_M - \expandafter{\the\numexpr #4\expandafter}% - \romannumeral0\XINT_RQ {}#3\R\R\R\R\R\R\R\R\Z \Z\Z\Z\Z -}% -\def\XINT_mul_choice_compare #1#2% -{% - \ifnum #1>#2 - \expandafter \XINT_mul_choice_i - \else - \expandafter \XINT_mul_choice_ii - \fi - {#1}{#2}% -}% -\def\XINT_mul_choice_i #1#2% -{% - \ifnum #1<\numexpr\ifcase \numexpr (#2-\xint_c_iii)/\xint_c_iv\relax - \or 330\or 168\or 109\or 80\or 66\or 52\else 0\fi\relax - \expandafter\XINT_mul_choice_same - \else - \expandafter\XINT_mul_choice_permute - \fi -}% -\def\XINT_mul_choice_ii #1#2% -{% - \ifnum #2<\numexpr\ifcase \numexpr (#1-\xint_c_iii)/\xint_c_iv\relax - \or 330\or 168\or 109\or 80\or 66\or 52\else 0\fi\relax - \expandafter\XINT_mul_choice_permute - \else - \expandafter\XINT_mul_choice_same - \fi -}% -\def\XINT_mul_choice_same #1#2% -{% - \expandafter\XINT_mul_enter - \romannumeral0\XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #2\W\W\W\W -}% -\def\XINT_mul_choice_permute #1#2% -{% - \expandafter\XINT_mul_enter - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #1\W\W\W\W -}% -% \end{macrocode} -% \lverb|& -% Cette portion de routine d'addition se branche directement sur _addr_ -% lorsque -% le premier nombre est épuisé, ce qui est garanti arriver avant le second -% nombre. Elle produit son résultat toujours sur 4n, renversé. Ses deux inputs -% sont garantis sur 4n.| -% \begin{macrocode} -\def\XINT_mul_Ar #1#2#3#4#5#6% -{% - \xint_gob_til_Z #6\xint_mul_br\Z\XINT_mul_Br #1{#6#5#4#3}{#2}% -}% -\def\xint_mul_br\Z\XINT_mul_Br #1#2% -{% - \XINT_addr_AC_checkcarry #1% -}% -\def\XINT_mul_Br #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \expandafter\XINT_mul_ABEAr - \the\numexpr #1+10#2+#8#7#6#5.{#3}#4\W\X\Y\Z -}% -\def\XINT_mul_ABEAr #1#2#3#4#5#6.#7% -{% - \XINT_mul_Ar #2{#7#6#5#4#3}% -}% -% \end{macrocode} -% \lverb|& -% << Petite >> multiplication. -% mul_Mr renvoie le résultat *à l'envers*, sur *4n*$\ -% \romannumeral0\XINT_mul_Mr {<n>}<N>\Z\Z\Z\Z$\ -% Fait la multiplication de <N> par <n>, qui est < 10000. -% <N> est présenté *à l'envers*, sur *4n*. Lorsque <n> vaut 0, donne 0000.| -% \begin{macrocode} -\def\XINT_mul_Mr #1% -{% - \expandafter\XINT_mul_Mr_checkifzeroorone\expandafter{\the\numexpr #1}% -}% -\def\XINT_mul_Mr_checkifzeroorone #1% -{% - \ifcase #1 - \expandafter\XINT_mul_Mr_zero - \or - \expandafter\XINT_mul_Mr_one - \else - \expandafter\XINT_mul_Nr - \fi - {0000}{}{#1}% -}% -\def\XINT_mul_Mr_zero #1\Z\Z\Z\Z { 0000}% -\def\XINT_mul_Mr_one #1#2#3#4\Z\Z\Z\Z { #4}% -\def\XINT_mul_Nr #1#2#3#4#5#6#7% -{% - \xint_gob_til_Z #4\xint_mul_pr\Z\XINT_mul_Pr {#1}{#3}{#7#6#5#4}{#2}{#3}% -}% -\def\XINT_mul_Pr #1#2#3% -{% - \expandafter\XINT_mul_Lr\the\numexpr \xint_c_x^viii+#1+#2*#3\relax -}% -\def\XINT_mul_Lr 1#1#2#3#4#5#6#7#8#9% -{% - \XINT_mul_Nr {#1#2#3#4}{#9#8#7#6#5}% -}% -\def\xint_mul_pr\Z\XINT_mul_Pr #1#2#3#4#5% -{% - \xint_gob_til_zeros_iv #1\XINT_mul_Mr_end_nocarry 0000% - \XINT_mul_Mr_end_carry #1{#4}% -}% -\def\XINT_mul_Mr_end_nocarry 0000\XINT_mul_Mr_end_carry 0000#1{ #1}% -\def\XINT_mul_Mr_end_carry #1#2#3#4#5{ #5#4#3#2#1}% -% \end{macrocode} -% \lverb|& -% << Petite >> multiplication. -% renvoie le résultat *à l'endroit*, avec *nettoyage des leading zéros*.$\ -% \romannumeral0\XINT_mul_M {<n>}<N>\Z\Z\Z\Z$\ -% Fait la multiplication de <N> par <n>, qui est < 10000. -% <N> est présenté *à l'envers*, sur *4n*. | -% \begin{macrocode} -\def\XINT_mul_M #1% -{% - \expandafter\XINT_mul_M_checkifzeroorone\expandafter{\the\numexpr #1}% -}% -\def\XINT_mul_M_checkifzeroorone #1% -{% - \ifcase #1 - \expandafter\XINT_mul_M_zero - \or - \expandafter\XINT_mul_M_one - \else - \expandafter\XINT_mul_N - \fi - {0000}{}{#1}% -}% -\def\XINT_mul_M_zero #1\Z\Z\Z\Z { 0}% -\def\XINT_mul_M_one #1#2#3#4\Z\Z\Z\Z -{% - \expandafter\xint_cleanupzeros_andstop\romannumeral0\xintreverseorder{#4}% -}% -\def\XINT_mul_N #1#2#3#4#5#6#7% -{% - \xint_gob_til_Z #4\xint_mul_p\Z\XINT_mul_P {#1}{#3}{#7#6#5#4}{#2}{#3}% -}% -\def\XINT_mul_P #1#2#3% -{% - \expandafter\XINT_mul_L\the\numexpr \xint_c_x^viii+#1+#2*#3\relax -}% -\def\XINT_mul_L 1#1#2#3#4#5#6#7#8#9% -{% - \XINT_mul_N {#1#2#3#4}{#5#6#7#8#9}% -}% -\def\xint_mul_p\Z\XINT_mul_P #1#2#3#4#5% -{% - \XINT_mul_M_end #1#4% -}% -\edef\XINT_mul_M_end #1#2#3#4#5#6#7#8% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax -}% -% \end{macrocode} -% \lverb|& -% Routine de multiplication principale -% (attention délimiteurs modifiés pour 1.08)$\ -% Le résultat partiel est toujours maintenu avec significatif à -% droite et il a un nombre multiple de 4 de chiffres$\ -% \romannumeral0\XINT_mul_enter <N1>\Z\Z\Z\Z <N2>\W\W\W\W$\ -% avec <N1> *renversé*, *longueur 4n* (zéros éventuellement ajoutés -% au-delà du chiffre le plus significatif) -% et <N2> dans l'ordre *normal*, et pas forcément longueur 4n. -% pas de signes.$\ -% Pour 1.08: dans \XINT_mul_enter et les modifs de 1.03 -% qui filtrent les courts, on pourrait croire que le -% second opérande a au moins quatre chiffres; mais le problème c'est que -% ceci est appelé par \XINT_sqr. Et de plus \XINT_sqr est utilisé dans -% la nouvelle routine d'extraction de racine carrée: je ne veux pas -% rajouter l'overhead à \XINT_sqr de voir si a longueur est au moins 4. -% Dilemme donc. Il ne semble pas y avoir d'autres accès -% directs (celui de big fac n'est pas un problème). J'ai presque été -% tenté de faire du 5x4, mais si on veut maintenir les résultats -% intermédiaires sur 4n, il y a des complications. Par ailleurs, -% je modifie aussi un petit peu la façon de coder la suite, compte tenu -% du style que j'ai développé ultérieurement. Attention terminaison -% modifiée pour le deuxième opérande.| -% \begin{macrocode} -\def\XINT_mul_enter #1\Z\Z\Z\Z #2#3#4#5% -{% - \xint_gob_til_W #5\XINT_mul_exit_a\W - \XINT_mul_start {#2#3#4#5}#1\Z\Z\Z\Z -}% -\def\XINT_mul_exit_a\W\XINT_mul_start #1% -{% - \XINT_mul_exit_b #1% -}% -\def\XINT_mul_exit_b #1#2#3#4% -{% - \xint_gob_til_W - #2\XINT_mul_exit_ci - #3\XINT_mul_exit_cii - \W\XINT_mul_exit_ciii #1#2#3#4% -}% -\def\XINT_mul_exit_ciii #1\W #2\Z\Z\Z\Z \W\W\W -{% - \XINT_mul_M {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mul_exit_cii\W\XINT_mul_exit_ciii #1\W\W #2\Z\Z\Z\Z \W\W -{% - \XINT_mul_M {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mul_exit_ci\W\XINT_mul_exit_cii - \W\XINT_mul_exit_ciii #1\W\W\W #2\Z\Z\Z\Z \W -{% - \XINT_mul_M {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mul_start #1#2\Z\Z\Z\Z -{% - \expandafter\XINT_mul_main\expandafter - {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z}#2\Z\Z\Z\Z -}% -\def\XINT_mul_main #1#2\Z\Z\Z\Z #3#4#5#6% -{% - \xint_gob_til_W #6\XINT_mul_finish_a\W - \XINT_mul_compute {#3#4#5#6}{#1}#2\Z\Z\Z\Z -}% -\def\XINT_mul_compute #1#2#3\Z\Z\Z\Z -{% - \expandafter\XINT_mul_main\expandafter - {\romannumeral0\expandafter - \XINT_mul_Ar\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z - \W\X\Y\Z 0000#2\W\X\Y\Z }#3\Z\Z\Z\Z -}% -% \end{macrocode} -% \lverb|& -% Ici, le deuxième nombre se termine. Fin du calcul. On utilise la variante -% \XINT_addm_A de l'addition car on sait que le deuxième terme est au moins -% aussi long que le premier. Lorsque le multiplicateur avait longueur 4n, la -% dernière addition a fourni le résultat à l'envers, il faut donc encore le -% renverser. | -% \begin{macrocode} -\def\XINT_mul_finish_a\W\XINT_mul_compute #1% -{% - \XINT_mul_finish_b #1% -}% -\def\XINT_mul_finish_b #1#2#3#4% -{% - \xint_gob_til_W - #1\XINT_mul_finish_c - #2\XINT_mul_finish_ci - #3\XINT_mul_finish_cii - \W\XINT_mul_finish_ciii #1#2#3#4% -}% -\def\XINT_mul_finish_ciii #1\W #2#3\Z\Z\Z\Z \W\W\W -{% - \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 000#2\W\X\Y\Z -}% -\def\XINT_mul_finish_cii - \W\XINT_mul_finish_ciii #1\W\W #2#3\Z\Z\Z\Z \W\W -{% - \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 00#2\W\X\Y\Z -}% -\def\XINT_mul_finish_ci #1\XINT_mul_finish_ciii #2\W\W\W #3#4\Z\Z\Z\Z \W -{% - \expandafter\XINT_addm_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#2}#4\Z\Z\Z\Z \W\X\Y\Z 0#3\W\X\Y\Z -}% -\def\XINT_mul_finish_c #1\XINT_mul_finish_ciii \W\W\W\W #2#3\Z\Z\Z\Z -{% - \expandafter\xint_cleanupzeros_andstop\romannumeral0\xintreverseorder{#2}% -}% -% \end{macrocode} -% \lverb|& -% Variante de la Multiplication$\ -% \romannumeral0\XINT_mulr_enter <N1>\Z\Z\Z\Z <N2>\W\W\W\W $\ -% Ici <N1> est à l'envers sur 4n, et <N2> est à l'endroit, pas sur 4n, comme -% dans \XINT_mul_enter, mais le résultat est lui-même fourni *à l'envers*, sur -% *4n* (en faisant attention de ne pas avoir 0000 à la fin).$\ -% Utilisé par le calcul des puissances. J'ai modifié dans 1.08 sur le -% modèle de la nouvelle version de \XINT_mul_enter. Je pourrais économiser des -% macros et fusionner \XINT_mul_enter et \XINT_mulr_enter. Une autre fois.| -% \begin{macrocode} -\def\XINT_mulr_enter #1\Z\Z\Z\Z #2#3#4#5% -{% - \xint_gob_til_W #5\XINT_mulr_exit_a\W - \XINT_mulr_start {#2#3#4#5}#1\Z\Z\Z\Z -}% -\def\XINT_mulr_exit_a\W\XINT_mulr_start #1% -{% - \XINT_mulr_exit_b #1% -}% -\def\XINT_mulr_exit_b #1#2#3#4% -{% - \xint_gob_til_W - #2\XINT_mulr_exit_ci - #3\XINT_mulr_exit_cii - \W\XINT_mulr_exit_ciii #1#2#3#4% -}% -\def\XINT_mulr_exit_ciii #1\W #2\Z\Z\Z\Z \W\W\W -{% - \XINT_mul_Mr {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_exit_cii\W\XINT_mulr_exit_ciii #1\W\W #2\Z\Z\Z\Z \W\W -{% - \XINT_mul_Mr {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_exit_ci\W\XINT_mulr_exit_cii - \W\XINT_mulr_exit_ciii #1\W\W\W #2\Z\Z\Z\Z \W -{% - \XINT_mul_Mr {#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_start #1#2\Z\Z\Z\Z -{% - \expandafter\XINT_mulr_main\expandafter - {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_main #1#2\Z\Z\Z\Z #3#4#5#6% -{% - \xint_gob_til_W #6\XINT_mulr_finish_a\W - \XINT_mulr_compute {#3#4#5#6}{#1}#2\Z\Z\Z\Z -}% -\def\XINT_mulr_compute #1#2#3\Z\Z\Z\Z -{% - \expandafter\XINT_mulr_main\expandafter - {\romannumeral0\expandafter - \XINT_mul_Ar\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z - \W\X\Y\Z 0000#2\W\X\Y\Z }#3\Z\Z\Z\Z -}% -\def\XINT_mulr_finish_a\W\XINT_mulr_compute #1% -{% - \XINT_mulr_finish_b #1% -}% -\def\XINT_mulr_finish_b #1#2#3#4% -{% - \xint_gob_til_W - #1\XINT_mulr_finish_c - #2\XINT_mulr_finish_ci - #3\XINT_mulr_finish_cii - \W\XINT_mulr_finish_ciii #1#2#3#4% -}% -\def\XINT_mulr_finish_ciii #1\W #2#3\Z\Z\Z\Z \W\W\W -{% - \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 000#2\W\X\Y\Z -}% -\def\XINT_mulr_finish_cii - \W\XINT_mulr_finish_ciii #1\W\W #2#3\Z\Z\Z\Z \W\W -{% - \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z \W\X\Y\Z 00#2\W\X\Y\Z -}% -\def\XINT_mulr_finish_ci #1\XINT_mulr_finish_ciii #2\W\W\W #3#4\Z\Z\Z\Z \W -{% - \expandafter\XINT_addp_A\expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT_mul_Mr {#2}#4\Z\Z\Z\Z \W\X\Y\Z 0#3\W\X\Y\Z -}% -\def\XINT_mulr_finish_c #1\XINT_mulr_finish_ciii \W\W\W\W #2#3\Z\Z\Z\Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintSqr}} -% \begin{macrocode} -\def\xintiiSqr {\romannumeral0\xintiisqr }% -\def\xintiisqr #1% -{% - \expandafter\XINT_sqr\expandafter {\romannumeral0\xintiiabs{#1}}% -}% -\def\xintiSqr {\romannumeral0\xintisqr }% -\def\xintisqr #1% -{% - \expandafter\XINT_sqr\expandafter {\romannumeral0\xintiabs{#1}}% -}% -\let\xintSqr\xintiSqr \let\xintsqr\xintisqr -\def\XINT_sqr #1% -{% - \expandafter\XINT_mul_enter - \romannumeral0% - \XINT_RQ {}#1\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #1\W\W\W\W -}% -% \end{macrocode} -% \subsection{\csh{xintPrd}} -% \lverb|& -% \xintPrd {{a}...{z}}$\ -% \xintPrdExpr {a}...{z}\relax$\ -% Release 1.02 modified the product routine. The earlier version was faster in -% situations where each new term is bigger than the product of all previous -% terms, a situation which arises in the algorithm for computing powers. The -% 1.02 version was changed to be more efficient on big products, where the new -% term is small compared to what has been computed so far (the power algorithm -% now has its own product routine). -% -% Finally, the 1.03 version just simplifies everything as the multiplication now -% decides what is best, with the price of a little overhead. So the code has -% been dramatically reduced here. -% -% In 1.03 I also modify the way \xintPrd and \xintPrdExpr ...\relax are -% related. Now \xintPrdExpr \z \relax is accepted input when \z expands -% to a list of braced terms (prior only \xintPrd {\z} or \xintPrd \z was -% possible). -% -% In 1.06a I suddenly decide that \xintProductExpr was a silly name, and as the -% package is new and certainly not used, I decide I may just switch to -% \xintPrdExpr which I should have used from the beginning. -% -% 1.09a does NOT add the \xintnum overhead. 1.09h renames \xintiPrd to -% \xintiiPrd to correctly reflect this.| -% \begin{macrocode} -\def\xintiiPrd {\romannumeral0\xintiiprd }% -\def\xintiiprd #1{\xintiiprdexpr #1\relax }% -\let\xintPrd\xintiiPrd -\let\xintprd\xintiiprd -\def\xintiiPrdExpr {\romannumeral0\xintiiprdexpr }% -\def\xintiiprdexpr {\expandafter\XINT_prdexpr\romannumeral-`0}% -\let\xintPrdExpr\xintiiPrdExpr -\let\xintprdexpr\xintiiprdexpr -\def\XINT_prdexpr {\XINT_prod_loop_a 1\Z }% -\def\XINT_prod_loop_a #1\Z #2% - {\expandafter\XINT_prod_loop_b \romannumeral-`0#2\Z #1\Z \Z}% -\def\XINT_prod_loop_b #1% - {\xint_gob_til_relax #1\XINT_prod_finished\relax\XINT_prod_loop_c #1}% -\def\XINT_prod_loop_c - {\expandafter\XINT_prod_loop_a\romannumeral0\XINT_mul_fork }% -\def\XINT_prod_finished #1\Z #2\Z \Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintFac}} -% \lverb|& -% Modified with 1.02 and again in 1.03 for greater efficiency. I am -% tempted, -% here and elsewhere, to use \ifcase\XINT_Geq {#1}{1000000000} rather than -% \ifnum\xintLength {#1}>9 but for the time being I leave things as they stand. -% With release 1.05, rather than using \xintLength I opt finally for direct use -% of \numexpr (which will throw a suitable number too big message), and to raise -% the \xintError:$\ FactorialOfTooBigNumber for argument larger than 1000000 -% (rather than 1000000000). With 1.09a, \xintFac uses \xintnum. -% -% 1.09j for no special reason, I lower the maximal number from 999999 to 100000. -% Any how this computation would need more memory than TL2013 standard allows to -% TeX. And I don't even mention time... | -% \begin{macrocode} -\def\xintiFac {\romannumeral0\xintifac }% -\def\xintifac #1% -{% - \expandafter\XINT_fac_fork\expandafter{\the\numexpr #1}% -}% -\let\xintFac\xintiFac \let\xintfac\xintifac -\def\XINT_fac_fork #1% -{% - \ifcase\XINT_cntSgn #1\Z - \xint_afterfi{\expandafter\space\expandafter 1\xint_gobble_i }% - \or - \expandafter\XINT_fac_checklength - \else - \xint_afterfi{\expandafter\xintError:FactorialOfNegativeNumber - \expandafter\space\expandafter 1\xint_gobble_i }% - \fi - {#1}% -}% -\def\XINT_fac_checklength #1% -{% - \ifnum #1>100000 - \xint_afterfi{\expandafter\xintError:FactorialOfTooBigNumber - \expandafter\space\expandafter 1\xint_gobble_i }% - \else - \xint_afterfi{\ifnum #1>\xint_c_ixixixix - \expandafter\XINT_fac_big_loop - \else - \expandafter\XINT_fac_loop - \fi }% - \fi - {#1}% -}% -\def\XINT_fac_big_loop #1{\XINT_fac_big_loop_main {10000}{#1}{}}% -\def\XINT_fac_big_loop_main #1#2#3% -{% - \ifnum #1<#2 - \expandafter - \XINT_fac_big_loop_main - \expandafter - {\the\numexpr #1+1\expandafter }% - \else - \expandafter\XINT_fac_big_docomputation - \fi - {#2}{#3{#1}}% -}% -\def\XINT_fac_big_docomputation #1#2% -{% - \expandafter \XINT_fac_bigcompute_loop \expandafter - {\romannumeral0\XINT_fac_loop {9999}}#2\relax -}% -\def\XINT_fac_bigcompute_loop #1#2% -{% - \xint_gob_til_relax #2\XINT_fac_bigcompute_end\relax - \expandafter\XINT_fac_bigcompute_loop\expandafter - {\expandafter\XINT_mul_enter - \romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z - \Z\Z\Z\Z #1\W\W\W\W }% -}% -\def\XINT_fac_bigcompute_end #1#2#3#4#5% -{% - \XINT_fac_bigcompute_end_ #5% -}% -\def\XINT_fac_bigcompute_end_ #1\R #2\Z \W\X\Y\Z #3\W\X\Y\Z { #3}% -\def\XINT_fac_loop #1{\XINT_fac_loop_main 1{1000}{#1}}% -\def\XINT_fac_loop_main #1#2#3% -{% - \ifnum #3>#1 - \else - \expandafter\XINT_fac_loop_exit - \fi - \expandafter\XINT_fac_loop_main\expandafter - {\the\numexpr #1+1\expandafter }\expandafter - {\romannumeral0\XINT_mul_Mr {#1}#2\Z\Z\Z\Z }% - {#3}% -}% -\def\XINT_fac_loop_exit #1#2#3#4#5#6#7% -{% - \XINT_fac_loop_exit_ #6% -}% -\def\XINT_fac_loop_exit_ #1#2#3% -{% - \XINT_mul_M -}% -% \end{macrocode} -% \subsection{\csh{xintPow}} -% \lverb|1.02 modified the \XINT_posprod routine, the was renamed -% \XINT_pow_posprod and moved here, as it was well adapted for computing powers. -% Then 1.03 moved the special variants of multiplication (hence of addition) -% which were needed to earlier in this style file. -% -% Modified in 1.06, the exponent is given to a \numexpr rather than twice -% expanded. \xintnum added in 1.09a. -% -% \XINT_pow_posprod: Routine de produit servant pour le calcul des -% puissances. Chaque nouveau terme est plus grand que ce qui a déjà été calculé. -% Par conséquent on a intérêt à le conserver en second dans la routine de -% multiplication, donc le précédent calcul a intérêt à avoir été donné sur 4n, à -% l'envers. Il faut donc modifier la multiplication pour qu'elle fasse cela. Ce -% qui oblige à utiliser une version spéciale de l'addition également. -% -% 1.09j has reorganized the main loop, the described above \XINT_pow_posprod -% routine has been removed, intermediate multiplications are done -% immediately. Also, the maximal accepted exponent is now 100000 (no such -% restriction in \xintFloatPow, which accepts any exponent less than 2^31, and -% in \xintFloatPower which accepts long integers as exponent). -% -% 2^100000=9.990020930143845e30102 and multiplication of two numbers -% with 30000 digits would take hours on my laptop (seconds for 1000 digits).| -% \begin{macrocode} -\def\xintiiPow {\romannumeral0\xintiipow }% -\def\xintiipow #1% -{% - \expandafter\xint_pow\romannumeral-`0#1\Z% -}% -\def\xintiPow {\romannumeral0\xintipow }% -\def\xintipow #1% -{% - \expandafter\xint_pow\romannumeral0\xintnum{#1}\Z% -}% -\let\xintPow\xintiPow \let\xintpow\xintipow -\def\xint_pow #1#2\Z -{% - \xint_UDsignfork - #1\XINT_pow_Aneg - -\XINT_pow_Anonneg - \krof - #1{#2}% -}% -\def\XINT_pow_Aneg #1#2#3% -{% - \expandafter\XINT_pow_Aneg_\expandafter{\the\numexpr #3}{#2}% -}% -\def\XINT_pow_Aneg_ #1% -{% - \ifodd #1 - \expandafter\XINT_pow_Aneg_Bodd - \fi - \XINT_pow_Anonneg_ {#1}% -}% -\def\XINT_pow_Aneg_Bodd #1% -{% - \expandafter\XINT_opp\romannumeral0\XINT_pow_Anonneg_ -}% -% \end{macrocode} -% \lverb|B = #3, faire le xpxp. Modified with 1.06: use of \numexpr.| -% \begin{macrocode} -\def\XINT_pow_Anonneg #1#2#3% -{% - \expandafter\XINT_pow_Anonneg_\expandafter {\the\numexpr #3}{#1#2}% -}% -% \end{macrocode} -% \lverb+#1 = B, #2 = |A|+ -% \begin{macrocode} -\def\XINT_pow_Anonneg_ #1#2% -{% - \ifcase\XINT_Cmp {#2}{1} - \expandafter\XINT_pow_AisOne - \or - \expandafter\XINT_pow_AatleastTwo - \else - \expandafter\XINT_pow_AisZero - \fi - {#1}{#2}% -}% -\def\XINT_pow_AisOne #1#2{ 1}% -% \end{macrocode} -% \lverb|#1 = B| -% \begin{macrocode} -\def\XINT_pow_AisZero #1#2% -{% - \ifcase\XINT_cntSgn #1\Z - \xint_afterfi { 1}% - \or - \xint_afterfi { 0}% - \else - \xint_afterfi {\xintError:DivisionByZero\space 0}% - \fi -}% -\def\XINT_pow_AatleastTwo #1% -{% - \ifcase\XINT_cntSgn #1\Z - \expandafter\XINT_pow_BisZero - \or - \expandafter\XINT_pow_checkBsize - \else - \expandafter\XINT_pow_BisNegative - \fi - {#1}% -}% -\edef\XINT_pow_BisNegative #1#2% - {\noexpand\xintError:FractionRoundedToZero\space 0}% -\def\XINT_pow_BisZero #1#2{ 1}% -% \end{macrocode} -% \lverb|B = #1 > 0, A = #2 > 1. With 1.05, I replace \xintiLen{#1}>9 by -% direct use of \numexpr [to generate an error message if the exponent is too -% large] 1.06: \numexpr was already used above.| -% \begin{macrocode} -\def\XINT_pow_checkBsize #1% -{% - \ifnum #1>100000 - \expandafter\XINT_pow_BtooBig - \else - \expandafter\XINT_pow_loopI - \fi - {#1}% -}% -\edef\XINT_pow_BtooBig #1#2{\noexpand\xintError:ExponentTooBig\space 0}% -\def\XINT_pow_loopI #1% -{% - \ifnum #1=\xint_c_i\XINT_pow_Iend\fi - \ifodd #1 - \expandafter\XINT_pow_loopI_odd - \else - \expandafter\XINT_pow_loopI_even - \fi - {#1}% -}% -\edef\XINT_pow_Iend\fi #1\fi #2#3{\noexpand\fi\space #3}% -\def\XINT_pow_loopI_even #1#2% -{% - \expandafter\XINT_pow_loopI\expandafter - {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter - {\romannumeral0\xintiisqr {#2}}% -}% -\def\XINT_pow_loopI_odd #1#2% -{% - \expandafter\XINT_pow_loopI_odda\expandafter - {\romannumeral0\XINT_RQ {}#2\R\R\R\R\R\R\R\R\Z }{#1}{#2}% -}% -\def\XINT_pow_loopI_odda #1#2#3% -{% - \expandafter\XINT_pow_loopII\expandafter - {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter - {\romannumeral0\xintiisqr {#3}}{#1}% -}% -\def\XINT_pow_loopII #1% -{% - \ifnum #1 = \xint_c_i\XINT_pow_IIend\fi - \ifodd #1 - \expandafter\XINT_pow_loopII_odd - \else - \expandafter\XINT_pow_loopII_even - \fi - {#1}% -}% -\def\XINT_pow_loopII_even #1#2% -{% - \expandafter\XINT_pow_loopII\expandafter - {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter - {\romannumeral0\xintiisqr {#2}}% -}% -\def\XINT_pow_loopII_odd #1#2#3% -{% - \expandafter\XINT_pow_loopII_odda\expandafter - {\romannumeral0\XINT_mulr_enter #3\Z\Z\Z\Z #2\W\W\W\W}{#1}{#2}% -}% -\def\XINT_pow_loopII_odda #1#2#3% -{% - \expandafter\XINT_pow_loopII\expandafter - {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter - {\romannumeral0\xintiisqr {#3}}{#1}% -}% -\def\XINT_pow_IIend\fi #1\fi #2#3#4% -{% - \fi\XINT_mul_enter #4\Z\Z\Z\Z #3\W\W\W\W -}% -% \end{macrocode} -% \subsection{\csh{xintDivision}, \csh{xintQuo}, \csh{xintRem}} -% \lverb|The 1.09a release inserted the use of \xintnum. The \xintiiDivision -% etc... are the ones which do only \romannumeral-`0. -% -% January 5, 2014: Naturally, addition, subtraction, multiplication and division -% are the first things I did and since then I had left the division -% untouched. So in preparation of release 1.09j, I started revisiting the -% division, I did various minor improvements obtaining roughly -% 10$% efficiency gain. Then I decided I -% should deliberately impact the input save stack, with the hope to gain more -% speed from removing tokens and leaving them upstream. -% -% For this however I had to modify the underlying mathematical algorithm. The -% initial one is a bit unusual I guess, and, I trust, rather efficient, but it -% does not produce the quotient digits (in base 10000) one by one; at any given -% time it is possible that some correction will be made, which means it is not -% an appropriate algorithm for a TeX implementation which will abandon the -% quotient upstream. Thus I now have with 1.09j a new underlying mathematical -% algorithm, presumably much more standard. It is a bit complicated to implement -% expandably these things, but in the end I had regained the already mentioned -% 10$% efficiency and even more for -% small to medium sized inputs (up to 30$% perhaps). And in passing I did a -% special routine for divisors < 10000, which is 5 to 10 times faster still. -% -% But, I then tested a variant of my new implementation which again did not -% impact the input save stack and, for sizes of up to 200 digits, it is not much -% worse, indeed it is perhaps actually better than the one abandoning the -% quotient digits upstream (and in the end putting them in the correct order). -% So, finally, I re-incorporated the produced quotient digits within a tail -% recursion. Hence \xintDivision, like all other routines in xint (except -% \xintSeq without optional parameter) still does not impact the input save -% stack. One can have a produced quotient longer than 4x5000=20000 digits, and -% no need to worry about \xintTrunc, \xintRound, \xintFloat, \xintFloatSqrt, -% etc... and all other places using the division. -% -% However outputting to a file (which is basically the only thing one can do, -% multiplying out two 20000 digits numbers already takes hours, for 100000 it -% would be days if not weeks) 100000 digits is slow... the truncation routine -% will add 100000 zeros (circa) and then trim them four by four. Definitely I -% should do a routine XTrunc which will work by blocks of say 64, and -% furthermore, being destined to be used in and \edef or a \write, it could be -% much more efficient as it could simply be based on tail loop, which so far -% nothing in xint does because I want things to expand fully under -% \romannumeral-`0 (and don't imagine inserting chains of thousands of -% \expandafter's...) in order to be nestable. Inside \xintexpr such style of -% tail recursion leaving downstream things should definitely be implemented for -% the routines for which it is possible as things get expanded inside -% \csname..\endcsname. I don't do yet anything like this for 1.09j. | -% \begin{macrocode} -\def\xintiiQuo {\romannumeral0\xintiiquo }% -\def\xintiiRem {\romannumeral0\xintiirem }% -\def\xintiiquo {\expandafter\xint_firstoftwo_thenstop - \romannumeral0\xintiidivision }% -\def\xintiirem {\expandafter\xint_secondoftwo_thenstop - \romannumeral0\xintiidivision }% -\def\xintQuo {\romannumeral0\xintquo }% -\def\xintRem {\romannumeral0\xintrem }% -\def\xintquo {\expandafter\xint_firstoftwo_thenstop - \romannumeral0\xintdivision }% -\def\xintrem {\expandafter\xint_secondoftwo_thenstop - \romannumeral0\xintdivision }% -% \end{macrocode} -% \lverb|#1 = A, #2 = B. On calcule le quotient et le reste dans la division -% euclidienne de A par B.| -% \begin{macrocode} -\def\xintiiDivision {\romannumeral0\xintiidivision }% -\def\xintiidivision #1% -{% - \expandafter\xint_iidivision\expandafter {\romannumeral-`0#1}% -}% -\def\xint_iidivision #1#2% -{% - \expandafter\XINT_div_fork \romannumeral-`0#2\Z #1\Z -}% -\def\xintDivision {\romannumeral0\xintdivision }% -\def\xintdivision #1% -{% - \expandafter\xint_division\expandafter {\romannumeral0\xintnum{#1}}% -}% -\def\xint_division #1#2% -{% - \expandafter\XINT_div_fork \romannumeral0\xintnum{#2}\Z #1\Z -}% -% \end{macrocode} -% \lverb|#1#2 = 2e input = diviseur = B. -% #3#4 = 1er input = divisé = A.| -% \begin{macrocode} -\def\XINT_div_fork #1#2\Z #3#4\Z -{% - \xint_UDzerofork - #1\XINT_div_BisZero - #3\XINT_div_AisZero - 0{\xint_UDsignfork - #1\XINT_div_BisNegative % B < 0 - #3\XINT_div_AisNegative % A < 0, B > 0 - -\XINT_div_plusplus % B > 0, A > 0 - \krof }% - \krof - {#2}{#4}#1#3% #1#2=B, #3#4=A -}% -\edef\XINT_div_BisZero #1#2#3#4{\noexpand\xintError:DivisionByZero\space {0}{0}}% -\def\XINT_div_AisZero #1#2#3#4{ {0}{0}}% -% \end{macrocode} -% \lverb|& -% jusqu'à présent c'est facile.$\ -% minusplus signifie B < 0, A > 0$\ -% plusminus signifie B > 0, A < 0$\ -% Ici #3#1 correspond au diviseur B et #4#2 au divisé A. -% -% Cases with B<0 or especially A<0 are treated sub-optimally in terms of -% post-processing, things get reversed which could have been produced directly -% in the wanted order, but A,B>0 is given priority for optimization. I should -% revise the next few macros, definitely.| -% \begin{macrocode} -\def\XINT_div_plusplus #1#2#3#4{\XINT_div_prepare {#3#1}{#4#2}}% -% \end{macrocode} -% \lverb|B = #3#1 < 0, A non nul positif ou négatif| -% \begin{macrocode} -\def\XINT_div_BisNegative #1#2#3#4% -{% - \expandafter\XINT_div_BisNegative_b - \romannumeral0\XINT_div_fork #1\Z #4#2\Z -}% -\edef\XINT_div_BisNegative_b #1% -{% - \noexpand\expandafter\space\noexpand\expandafter - {\noexpand\romannumeral0\noexpand\XINT_opp #1}% -}% -% \end{macrocode} -% \lverb|B = #3#1 > 0, A =-#2< 0| -% \begin{macrocode} -\def\XINT_div_AisNegative #1#2#3#4% -{% - \expandafter\XINT_div_AisNegative_b - \romannumeral0\XINT_div_prepare {#3#1}{#2}{#3#1}% -}% -\def\XINT_div_AisNegative_b #1#2% -{% - \if0\XINT_Sgn #2\Z - \expandafter \XINT_div_AisNegative_Rzero - \else - \expandafter \XINT_div_AisNegative_Rpositive - \fi - {#1}{#2}% -}% -% \end{macrocode} -% \lverb|en #3 on a une copie de B (à l'endroit)| -% \begin{macrocode} -\edef\XINT_div_AisNegative_Rzero #1#2#3% -{% - \noexpand\expandafter\space\noexpand\expandafter - {\noexpand\romannumeral0\noexpand\XINT_opp #1}{0}% -}% -% \end{macrocode} -% \lverb!#1 = quotient, #2 = reste, #3 = diviseur initial (à l'endroit) -% remplace Reste par B - Reste, après avoir remplacé Q par -(Q+1) -% de sorte que la formule a = qb + r, 0<= r < |b| est valable! -% \begin{macrocode} -\def\XINT_div_AisNegative_Rpositive #1% -{% - \expandafter \XINT_div_AisNegative_Rpositive_b \expandafter - {\romannumeral0\xintiiopp{\xintInc {#1}}}% -}% -\def\XINT_div_AisNegative_Rpositive_b #1#2#3% -{% - \expandafter \xint_exchangetwo_keepbraces_thenstop \expandafter - {\romannumeral0\XINT_sub {#3}{#2}}{#1}% -}% -% \end{macrocode} -% \lverb|& -% Pour la suite A et B sont > 0. -% #1 = B. Pour le moment à l'endroit. -% Calcul du plus petit K = 4n >= longueur de B| -% \begin{macrocode} -\def\XINT_div_prepare #1% -{% - \expandafter \XINT_div_prepareB_aa \expandafter - {\romannumeral0\xintlength {#1}}{#1}% B > 0 ici -}% -\def\XINT_div_prepareB_aa #1% -{% - \ifnum #1=\xint_c_i - \expandafter\XINT_div_prepareB_onedigit - \else - \expandafter\XINT_div_prepareB_a - \fi - {#1}% -}% -\def\XINT_div_prepareB_a #1% -{% - \expandafter\XINT_div_prepareB_c\expandafter - {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}% -}% -% \end{macrocode} -% \lverb|B=1 and B=2 treated specially.| -% \begin{macrocode} -\def\XINT_div_prepareB_onedigit #1#2% -{% - \ifcase#2 - \or\expandafter\XINT_div_BisOne - \or\expandafter\XINT_div_BisTwo - \else\expandafter\XINT_div_prepareB_e - \fi {000}{0}{4}{#2}% -}% -\def\XINT_div_BisOne #1#2#3#4#5{ {#5}{0}}% -\def\XINT_div_BisTwo #1#2#3#4#5% -{% - \expandafter\expandafter\expandafter\XINT_div_BisTwo_a - \ifodd\xintiiLDg{#5} \expandafter1\else \expandafter0\fi {#5}% -}% -\edef\XINT_div_BisTwo_a #1#2% -{% - \noexpand\expandafter\space\noexpand\expandafter - {\noexpand\romannumeral0\noexpand\xinthalf {#2}}{#1}% -}% -% \end{macrocode} -% \lverb|#1 = K. 1.09j uses \csname, earlier versions did it with -% \ifcase.| -% \begin{macrocode} -\def\XINT_div_prepareB_c #1#2% -{% - \csname XINT_div_prepareB_d\romannumeral\numexpr#1-#2\endcsname - {#1}% -}% -\def\XINT_div_prepareB_d {\XINT_div_prepareB_e {}{0000}}% -\def\XINT_div_prepareB_di {\XINT_div_prepareB_e {0}{000}}% -\def\XINT_div_prepareB_dii {\XINT_div_prepareB_e {00}{00}}% -\def\XINT_div_prepareB_diii {\XINT_div_prepareB_e {000}{0}}% -\def\XINT_div_cleanR #10000.{{#1}}% -% \end{macrocode} -% \lverb|#1 = zéros à rajouter à B, #2=c [modifié dans 1.09j, ce sont maintenant -% des zéros explicites en nombre 4 - ancien c, et on utilisera -% \XINT_div_cleanR et non plus \XINT_dsh_checksignx pour nettoyer à la fin -% des zéros en excès dans le Reste; in all comments next, «c» stands now {0} or -% {00} or {000} or {0000} rather than a digit as in earlier versions], #3=K, #4 -% = B| -% \begin{macrocode} -\def\XINT_div_prepareB_e #1#2#3#4% -{% - \ifnum#3=\xint_c_iv\expandafter\XINT_div_prepareLittleB_f - \else\expandafter\XINT_div_prepareB_f - \fi - #4#1{#3}{#2}{#1}% -}% -% \end{macrocode} -% \lverb|x = #1#2#3#4 = 4 premiers chiffres de B. #1 est non nul. B is reversed. -% With 1.09j or latter x+1 and (x+1)/2 are pre-computed. Si K=4 on ne renverse -% pas B, et donc B=x dans la suite. De plus pour K=4 on ne travaille pas avec -% x+1 et (x+1)/2 mais avec x et x/2.| -% \begin{macrocode} -\def\XINT_div_prepareB_f #1#2#3#4#5#{% - \expandafter\XINT_div_prepareB_g - \the\numexpr #1#2#3#4+\xint_c_i\expandafter - .\the\numexpr (#1#2#3#4+\xint_c_i)/\xint_c_ii\expandafter - .\romannumeral0\xintreverseorder {#1#2#3#4#5}.{#1#2#3#4}% -}% -\def\XINT_div_prepareLittleB_f #1#{% - \expandafter\XINT_div_prepareB_g \the\numexpr #1/\xint_c_ii.{}.{}.{#1}% -}% -% \end{macrocode} -% \lverb|& -% #1 = x' = x+1= 1+quatre premiers chiffres de B, #2 = y = (x+1)/2 précalculé -% #3 = B préparé et maintenant renversé, #4=x, -% #5 = K, #6 = «c», #7= {} ou {0} ou {00} ou {000}, #8 = A initial -% On multiplie aussi A par 10^c. -> AK{x'yx}B«c». Par contre dans le -% cas little on a #1=y=(x/2), #2={}, #3={}, #4=x, donc cela donne -% ->AK{y{}x}{}«c», il n'y a pas de B.| -% \begin{macrocode} -\def\XINT_div_prepareB_g #1.#2.#3.#4#5#6#7#8% -{% - \XINT_div_prepareA_a {#8#7}{#5}{{#1}{#2}{#4}}{#3}{#6}% -}% -% \end{macrocode} -% \lverb|A, K, {x'yx}, B«c» | -% \begin{macrocode} -\def\XINT_div_prepareA_a #1% -{% - \expandafter\XINT_div_prepareA_b\expandafter - {\romannumeral0\xintlength {#1}}{#1}% -}% -% \end{macrocode} -% \lverb|L0, A, K, {x'yx}, B«c»| -% \begin{macrocode} -\def\XINT_div_prepareA_b #1% -{% - \expandafter\XINT_div_prepareA_c\expandafter - {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}% -}% -% \end{macrocode} -% \lverb|L, L0, A, K, {x'yx}, B, «c»| -% \begin{macrocode} -\def\XINT_div_prepareA_c #1#2% -{% - \csname XINT_div_prepareA_d\romannumeral\numexpr #1-#2\endcsname - {#1}% -}% -\def\XINT_div_prepareA_d {\XINT_div_prepareA_e {}}% -\def\XINT_div_prepareA_di {\XINT_div_prepareA_e {0}}% -\def\XINT_div_prepareA_dii {\XINT_div_prepareA_e {00}}% -\def\XINT_div_prepareA_diii {\XINT_div_prepareA_e {000}}% -% \end{macrocode} -% \lverb|#1#3 = A préparé, #2 = longueur de ce A préparé, #4=K, #5={x'yx}-> -% LKAx'yxB«c»| -% \begin{macrocode} -\def\XINT_div_prepareA_e #1#2#3#4#5% -{% - \XINT_div_start_a {#2}{#4}{#1#3}#5% -}% -% \end{macrocode} -% \lverb|L, K, A, x',y,x, B, «c» (avec y{}x{} au lieu de x'yxB dans la -% variante little)| -% \begin{macrocode} -\def\XINT_div_start_a #1#2% -{% - \ifnum #2=\xint_c_iv \expandafter\XINT_div_little_b - \else - \ifnum #1 < #2 - \expandafter\expandafter\expandafter\XINT_div_III_aa - \else - \expandafter\expandafter\expandafter\XINT_div_start_b - \fi - \fi - {#1}{#2}% -}% -% \end{macrocode} -% \lverb|L, K, A, x',y,x, B, «c».| -% \begin{macrocode} -\def\XINT_div_III_aa #1#2#3#4#5#6#7% -{% - \expandafter\expandafter\expandafter - \XINT_div_III_b\xint_cleanupzeros_nostop #3.{0000}% -}% -% \end{macrocode} -% \lverb|R.Q«c».| -% \begin{macrocode} -\def\XINT_div_III_b #1% -{% - \if0#1% - \expandafter\XINT_div_III_bRzero - \else - \expandafter\XINT_div_III_bRpos - \fi - #1% -}% -\def\XINT_div_III_bRzero 0.#1#2% -{% - \expandafter\space\expandafter - {\romannumeral0\XINT_cuz_loop #1\W\W\W\W\W\W\W\Z}{0}% -}% -\def\XINT_div_III_bRpos #1.#2#3% -{% - \expandafter\XINT_div_III_c \XINT_div_cleanR #1#3.{#2}% -}% -\def\XINT_div_III_c #1#2% -{% - \expandafter\space\expandafter - {\romannumeral0\XINT_cuz_loop #2\W\W\W\W\W\W\W\Z}{#1}% -}% -% \end{macrocode} -% \lverb|L, K, A, x',y,x, B, «c»->K.A.x{LK{x'y}x}B«c»| -% \begin{macrocode} -\def\XINT_div_start_b #1#2#3#4#5#6% -{% - \XINT_div_start_c {#2}.#3.{#6}{{#1}{#2}{{#4}{#5}}{#6}}% -}% -% \end{macrocode} -% \lverb|Kalpha.A.x{LK{x'y}x}, B, «c», au début #2=alpha est vide| -% \begin{macrocode} -\def\XINT_div_start_c #1#2.#3#4#5#6% -{% - \ifnum #1=\xint_c_iv\XINT_div_start_ca\fi - \expandafter\XINT_div_start_c\expandafter - {\the\numexpr #1-\xint_c_iv}#2#3#4#5#6.% -}% -\def\XINT_div_start_ca\fi\expandafter\XINT_div_start_c\expandafter - #1#2#3#4#5{\fi\XINT_div_start_d {#2#3#4#5}#2#3#4#5}% -% \end{macrocode} -% \lverb|#1=a, #2=alpha (de longueur K, à l'endroit).#3=reste de A.#4=x, -% #5={LK{x'y}x},#6=B,«c» -> a, x, alpha, B, {0000}, L, K, {x'y},x, -% alpha'=reste de A, B{}«c». Pour K=4 on a en fait B=x, faudra revoir après.| -% \begin{macrocode} -\def\XINT_div_start_d #1#2.#3.#4#5#6% -{% - \XINT_div_I_a {#1}{#4}{#2}{#6}{0000}#5{#3}{#6}{}% -}% -% \end{macrocode} -% \lverb|Ceci est le point de retour de la boucle principale. a, x, alpha, B, -% q0, L, K, {x'y}, x, alpha', BQ«c» | -% \begin{macrocode} -\def\XINT_div_I_a #1#2% -{% - \expandafter\XINT_div_I_b\the\numexpr #1/#2.{#1}{#2}% -}% -\def\XINT_div_I_b #1% -{% - \xint_gob_til_zero #1\XINT_div_I_czero 0\XINT_div_I_c #1% -}% -% \end{macrocode} -% \lverb|On intercepte quotient nul: #1=a, x, alpha, B, #5=q0, L, K, {x'y}, x, -% alpha', BQ«c» -> q{alpha} L, K, {x'y}, x, alpha', BQ«c»| -% \begin{macrocode} -\def\XINT_div_I_czero 0% - \XINT_div_I_c 0.#1#2#3#4#5{\XINT_div_I_g {#5}{#3}}% -\def\XINT_div_I_c #1.#2#3% -{% - \expandafter\XINT_div_I_da\the\numexpr #2-#1*#3.#1.% -}% -% \end{macrocode} -% \lverb|r.q.alpha, B, q0, L, K, {x'y}, x, alpha', BQ«c»| -% \begin{macrocode} -\def\XINT_div_I_da #1.% -{% - \ifnum #1>\xint_c_ix - \expandafter\XINT_div_I_dP - \else - \ifnum #1<\xint_c_ - \expandafter\expandafter\expandafter\XINT_div_I_dN - \else - \expandafter\expandafter\expandafter\XINT_div_I_db - \fi - \fi -}% -\def\XINT_div_I_dN #1.% -{% - \expandafter\XINT_div_I_dP\the\numexpr #1-\xint_c_i.% -}% -\def\XINT_div_I_db #1.#2#3% #1=q=un chiffre, #2=alpha, #3=B -{% - \expandafter\XINT_div_I_dc\expandafter - {\romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#2}}% - {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}% - #1{#2}{#3}% -}% -\def\XINT_div_I_dc #1#2% -{% - \if-#1% s'arranger pour que si négatif on ait renvoyé alpha=-. - \expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo\fi - {\expandafter\XINT_div_I_dP\the\numexpr #2-\xint_c_i.}% - {\XINT_div_I_e {#1}#2}% -}% -% \end{macrocode} -% \lverb|alpha,q,ancien alpha,B, q0->1nouveauq.alpha, L, K, {x'y},x, alpha', -% BQ«c»| -% \begin{macrocode} -\def\XINT_div_I_e #1#2#3#4#5% -{% - \expandafter\XINT_div_I_f \the\numexpr \xint_c_x^iv+#2+#5{#1}% -}% -% \end{macrocode} -% \lverb|q.alpha, B, q0, L, K, {x'y},x, alpha'BQ«c» (intercepter q=0?) -% -> 1nouveauq.nouvel alpha, L, K, {x'y}, x, alpha',BQ«c»| -% \begin{macrocode} -\def\XINT_div_I_dP #1.#2#3#4% -{% - \expandafter \XINT_div_I_f \the\numexpr \xint_c_x^iv+#1+#4\expandafter - {\romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#2}}% - {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}% -}% -% \end{macrocode} -% \lverb|1#1#2#3#4=nouveau q, nouvel alpha, L, K, {x'y},x,alpha', BQ«c»| -% \begin{macrocode} -\def\XINT_div_I_f 1#1#2#3#4{\XINT_div_I_g {#1#2#3#4}}% -% \end{macrocode} -% \lverb|#1=q,#2=nouvel alpha,#3=L, #4=K, #5={x'y}, #6=x, #7= alpha',#8=B, -% #9=Q«c» -> {x'y}alpha.alpha'.{{x'y}xKL}B{Qq}«c»| -% \begin{macrocode} -\def\XINT_div_I_g #1#2#3#4#5#6#7#8#9% -{% - \ifnum#3=#4 - \expandafter\XINT_div_III_ab - \else - \expandafter\XINT_div_I_h - \fi - {#5}#2.#7.{{#5}{#6}{#4}{#3}}{#8}{#9#1}% -}% -% \end{macrocode} -% \lverb|{x'y}alpha.alpha'.{{x'y}xKL}B{Qq}«c» -> R sans leading zeros.{Qq}«c»| -% \begin{macrocode} -\def\XINT_div_III_ab #1#2.#3.#4#5% -{% - \expandafter\XINT_div_III_b - \romannumeral0\XINT_cuz_loop #2#3\W\W\W\W\W\W\W\Z.% -}% -% \end{macrocode} -% \lverb|#1={x'y}alpha.#2#3#4#5#6=reste de A. -% #7={{x'y},x,K,L},#8=B,nouveauQ«c» devient {x'y},alpha sur K+4 chiffres.B, -% {{x'y},x,K,L}, #6= nouvel alpha',B,nouveauQ«c»| -% \begin{macrocode} -\def\XINT_div_I_h #1.#2#3#4#5#6.#7#8% -{% - \XINT_div_II_b #1#2#3#4#5.{#8}{#7}{#6}{#8}% -}% -% \end{macrocode} -% \lverb|{x'y}alpha.B, {{x'y},x,K,L}, nouveau alpha',B, Q«c» On -% intercepte la situation avec alpha débutant par 0000 qui est la seule qui -% pourrait donner un q1 nul. Donc q1 est non nul et la soustraction spéciale -% recevra un q1*B de longueur K ou K+4 et jamais 0000. Ensuite un q2 éventuel -% s'il est calculé est nécessairement non nul lui aussi. Comme dans la phase I -% on a aussi intercepté un q nul, la soustraction spéciale ne reçoit donc jamais -% un qB nul. Note: j'ai testé plusieurs fois que ma technique de gob_til_zeros -% est plus rapide que d'utiliser un \ifnum | -% \begin{macrocode} -\def\XINT_div_II_b #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_zeros_iv #2#3#4#5\XINT_div_II_skipc 0000% - \XINT_div_II_c #1{#2#3#4#5}{#6#7#8#9}% -}% -% \end{macrocode} -% \lverb|x'y{0000}{4chiffres}reste de alpha.#6=B,#7={{x'y},x,K,L}, alpha',B, -% Q«c» -> {x'y}x,K,L (à diminuer de 4), {alpha sur -% K}B{q1=0000}{alpha'}B,Q«c»| -% \begin{macrocode} -\def\XINT_div_II_skipc 0000\XINT_div_II_c #1#2#3#4#5.#6#7% -{% - \XINT_div_II_k #7{#4#5}{#6}{0000}% -}% -% \end{macrocode} -% \lverb|x'ya->1qx'yalpha.B, {{x'y},x,K,L}, nouveau alpha',B, Q«c»| -% \begin{macrocode} -\def\XINT_div_II_c #1#2#3#4% -{% - \expandafter\XINT_div_II_d\the\numexpr (#3#4+#2)/#1+\xint_c_ixixixix\relax - {#1}{#2}#3#4% -}% -% \end{macrocode} -% \lverb|1 suivi de q1 sur quatre chiffres, #5=x', #6=y, #7=alpha.#8=B, -% {{x'y},x,K,L}, alpha', B, Q«c» --> nouvel alpha.x',y,B,q1,{{x'y},x,K,L}, -% alpha', B, Q«c» | -% \begin{macrocode} -\def\XINT_div_II_d 1#1#2#3#4#5#6#7.#8% -{% - \expandafter\XINT_div_II_e - \romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#7}}% - {\romannumeral0\XINT_mul_Mr {#1#2#3#4}#8\Z\Z\Z\Z }.% - {#5}{#6}{#8}{#1#2#3#4}% -}% -% \end{macrocode} -% \lverb|alpha.x',y,B,q1, {{x'y},x,K,L}, alpha', B, Q«c»| -% \begin{macrocode} -\def\XINT_div_II_e #1#2#3#4% -{% - \xint_gob_til_zeros_iv #1#2#3#4\XINT_div_II_skipf 0000% - \XINT_div_II_f #1#2#3#4% -}% -% \end{macrocode} -% \lverb|0000alpha sur K chiffres.#2=x',#3=y,#4=B,#5=q1, #6={{x'y},x,K,L}, -% #7=alpha',BQ«c» -> {x'y}x,K,L (à diminuer de 4), -% {alpha sur K}B{q1}{alpha'}BQ«c»| -% \begin{macrocode} -\def\XINT_div_II_skipf 0000\XINT_div_II_f 0000#1.#2#3#4#5#6% -{% - \XINT_div_II_k #6{#1}{#4}{#5}% -}% -% \end{macrocode} -% \lverb|a1 (huit chiffres), alpha (sur K+4), x', y, B, q1, {{x'y},x,K,L}, -% alpha', B,Q«c»| -% \begin{macrocode} -\def\XINT_div_II_f #1#2#3#4#5#6#7#8#9.% -{% - \XINT_div_II_fa {#1#2#3#4#5#6#7#8}{#1#2#3#4#5#6#7#8#9}% -}% -\def\XINT_div_II_fa #1#2#3#4% -{% - \expandafter\XINT_div_II_g\expandafter - {\the\numexpr (#1+#4)/#3-\xint_c_i}{#2}% -}% -% \end{macrocode} -% \lverb|#1=q, #2=alpha (K+4), #3=B, #4=q1, {{x'y},x,K,L}, alpha', BQ«c» -% -> 1 puis nouveau q sur 4 chiffres, nouvel alpha sur K chiffres, -% B, {{x'y},x,K,L}, alpha',BQ«c» | -% \begin{macrocode} -\def\XINT_div_II_g #1#2#3#4% -{% - \expandafter \XINT_div_II_h - \the\numexpr #4+#1+\xint_c_x^iv\expandafter\expandafter\expandafter - {\expandafter\xint_gobble_iv - \romannumeral0\expandafter\XINT_div_sub_xpxp\expandafter - {\romannumeral0\xintreverseorder{#2}}% - {\romannumeral0\XINT_mul_Mr {#1}#3\Z\Z\Z\Z }}{#3}% -}% -% \end{macrocode} -% \lverb|1 puis nouveau q sur 4 chiffres, #5=nouvel alpha sur K chiffres, -% #6=B, #7={{x'y},x,K,L} avec L à ajuster, alpha', BQ«c» -% -> {x'y}x,K,L à diminuer de 4, {alpha}B{q}, alpha', BQ«c»| -% \begin{macrocode} -\def\XINT_div_II_h 1#1#2#3#4#5#6#7% -{% - \XINT_div_II_k #7{#5}{#6}{#1#2#3#4}% -}% -% \end{macrocode} -% \lverb|{x'y}x,K,L à diminuer de 4, alpha, B{q}alpha',BQ«c» -% ->nouveau L.K,x',y,x,alpha.B,q,alpha',B,Q«c» -% ->{LK{x'y}x},x,a,alpha.B,q,alpha',B,Q«c»| -% \begin{macrocode} -\def\XINT_div_II_k #1#2#3#4#5% -{% - \expandafter\XINT_div_II_l \the\numexpr #4-\xint_c_iv.{#3}#1{#2}#5.% -}% -\def\XINT_div_II_l #1.#2#3#4#5#6#7#8#9% -{% - \XINT_div_II_m {{#1}{#2}{{#3}{#4}}{#5}}{#5}{#6#7#8#9}#6#7#8#9% -}% -% \end{macrocode} -% \lverb|{LK{x'y}x},x,a,alpha.B{q}alpha'BQ -> a, x, alpha, B, q, -% L, K, {x'y}, x, alpha', BQ«c» | -% \begin{macrocode} -\def\XINT_div_II_m #1#2#3#4.#5#6% -{% - \XINT_div_I_a {#3}{#2}{#4}{#5}{#6}#1% -}% -% \end{macrocode} -% \lverb|L, K, A, y,{},x, {},«c»->A.{yx}L{}«c» Comme ici K=4, dans -% la phase I on n'a pas besoin de alpha, car a = alpha. De plus on a maintenu B -% dans l'ordre qui est donc la même chose que x. Par ailleurs la phase I est -% simplifiée, il s'agit simplement de la division euclidienne de a par x, et de -% plus on n'a à la faire qu'une unique fois et ensuite la phase II peut boucler -% sur elle-même au lieu de revenir en phase I, par conséquent il n'y a pas non -% plus de q0 ici. Enfin, le y est (x/2) pas ((x+1)/2) il n'y a pas de x'=x+1| -% \begin{macrocode} -\def\XINT_div_little_b #1#2#3#4#5#6#7% -{% - \XINT_div_little_c #3.{{#4}{#6}}{#1}% -}% -% \end{macrocode} -% \lverb|#1#2#3#4=a, #5=alpha'=reste de A.#6={yx}, #7=L, «c» -> a, -% y, x, L, alpha'=reste de A, «c».| -% \begin{macrocode} -\def\XINT_div_little_c #1#2#3#4#5.#6#7% -{% - \XINT_div_littleI_a {#1#2#3#4}#6{#7}{#5}% -}% -% \end{macrocode} -% \lverb|a, y, x, L, alpha',«c» On calcule ici (contrairement à la -% phase I générale) le vrai quotient euclidien de a par x=B, c'est donc un -% chiffre de 0 à 9. De plus on n'a à faire cela qu'une unique fois.| -% \begin{macrocode} -\def\XINT_div_littleI_a #1#2#3% -{% - \expandafter\XINT_div_littleI_b - \the\numexpr (#1+#2)/#3-\xint_c_i{#1}{#2}{#3}% -}% -% \end{macrocode} -% \lverb|On intercepte quotient nul: [est-ce vraiment utile? ou n'est-ce pas -% plutôt une perte de temps en moyenne? il faudrait tester] q=0#1=a, -% #2=y, x, L, alpha', «c» -> -% II_a avec L{alpha}alpha'.{yx}{0000}«c». Et en cas de quotient non nul on -% procède avec littleI_c avec #1=q, #2=a, #3=y, #4=x -> {nouvel alpha sur 4 -% chiffres}q{yx},L,alpha',«c».| -% \begin{macrocode} -\def\XINT_div_littleI_b #1% -{% - \xint_gob_til_zero #1\XINT_div_littleI_skip 0\XINT_div_littleI_c #1% -}% -\def\XINT_div_littleI_skip 0\XINT_div_littleI_c 0#1#2#3#4#5% - {\XINT_div_littleII_a {#4}{#1}#5.{{#2}{#3}}{0000}}% -\def\XINT_div_littleI_c #1#2#3#4% -{% - \expandafter\expandafter\expandafter\XINT_div_littleI_e - \expandafter\expandafter\expandafter - {\expandafter\xint_gobble_i\the\numexpr \xint_c_x^iv+#2-#1*#4}#1{{#3}{#4}}% -}% -% \end{macrocode} -% \lverb|#1=nouvel alpha sur 4 chiffres#2=q,#3={yx}, #4=L, #5=alpha',«c» -> -% L{alpha}alpha'.{yx}{000q}«c» point d'entrée de la boucle principale| -% \begin{macrocode} -\def\XINT_div_littleI_e #1#2#3#4#5% - {\XINT_div_littleII_a {#4}{#1}#5.{#3}{000#2}}% -% \end{macrocode} -% \lverb|L{alpha}alpha'.{yx}Q«c» et c'est là qu'on boucle| -% \begin{macrocode} -\def\XINT_div_littleII_a #1% -{% - \ifnum#1=\xint_c_iv - \expandafter\XINT_div_littleIII_ab - \else - \expandafter\XINT_div_littleII_b - \fi {#1}% -}% -% \end{macrocode} -% \lverb|L{alpha}alpha'.{yx}Q«c» -> (en fait #3 est vide normalement ici) R -% sans leading zeros.Q«c»| -% \begin{macrocode} -\def\XINT_div_littleIII_ab #1#2#3.#4% -{% - \expandafter\XINT_div_III_b\the\numexpr #2#3.% -}% -% \end{macrocode} -% \lverb|L{alpha}alpha'.{yx}Q«c». On diminue L de quatre, comme cela c'est -% fait.| -% \begin{macrocode} -\def\XINT_div_littleII_b #1% -{% - \expandafter\XINT_div_littleII_c\expandafter {\the\numexpr #1-\xint_c_iv}% -}% -% \end{macrocode} -% \lverb|{nouveauL}{alpha}alpha'.{yx}Q«c». On prélève 4 chiffres de alpha' -> -% {nouvel alpha sur huit chiffres}yx{nouveau L}{nouvel alpha'}Q«c». Regarder -% si l'ancien alpha était 0000 n'avancerait à rien car obligerait à refaire une -% chose comme la phase I, donc on ne perd pas de temps avec ça, on reste en -% permanence en phase II.| -% \begin{macrocode} -\def\XINT_div_littleII_c #1#2#3#4#5#6#7.#8% -{% - \XINT_div_littleII_d {#2#3#4#5#6}#8{#1}{#7}% -}% -\def\XINT_div_littleII_d #1#2#3% -{% - \expandafter\XINT_div_littleII_e\the\numexpr (#1+#2)/#3+\xint_c_ixixixix.% - {#1}{#2}{#3}% -}% -% \end{macrocode} -% \lverb|1 suivi de #1=q1 sur quatre chiffres.#2=alpha, #3=y, #4=x, -% L, alpha', Q«c» --> nouvel alpha sur 4.{q1}{yx},L,alpha', Q«c» | -% \begin{macrocode} -\def\XINT_div_littleII_e 1#1.#2#3#4% -{% - \expandafter\expandafter\expandafter\XINT_div_littleII_f - \expandafter\xint_gobble_i\the\numexpr \xint_c_x^iv+#2-#1*#4.% - {#1}{{#3}{#4}}% -}% -% \end{macrocode} -% \lverb|alpha.q,{yx},L,alpha',Q«c»->L{alpha}alpha'.{yx}{Qq}«c»| -% \begin{macrocode} -\def\XINT_div_littleII_f #1.#2#3#4#5#6% -{% - \XINT_div_littleII_a {#4}{#1}#5.{#3}{#6#2}% -}% -% \end{macrocode} -% \lverb|La soustraction spéciale. Dans 1.09j, elle fait A-qB, pour A (en fait -% alpha dans mes dénominations des commentaires du code) et qB chacun de -% longueur K ou K+4, avec K au moins huit multiple de quatre, qB a ses quatre -% chiffres significatifs (qui sont à droite) non nuls. Si A-qB<0 il suffit de -% renvoyer -, le résultat n'importe pas. On est sûr que qB est non nul. On le -% met dans cette version en premier pour tester plus facilement le cas avec qB -% de longueur K+4 et A de longueur seulement K. Lorsque la longueur de qB est -% inférieure ou égale à celle de A, on va jusqu'à la fin de A et donc c'est la -% retenue finale qui décide du cas négatif éventuel. Le résultat non négatif est -% toujours donc renvoyé avec la même longueur que A, et il est dans l'ordre. -% J'ai fait une implémentation des phases I et II en maintenant alpha toujours à -% l'envers afin d'éviter le reverse order systématique fait sur A (ou plutôt -% alpha), mais alors il fallait que la soustraction ici s'arrange pour repérer -% les huit chiffres les plus significatifs, au final ce n'était pas plus rapide, -% et même pénalisant pour de gros inputs. Dans les versions 1.09i et antérieures -% (en fait je pense qu'ici rien quasiment n'avait bougé depuis la première -% implémentation), la soustraction spéciale n'était pratiquée que dans des cas -% avec certainement A-qB positif ou nul. De plus on n'excluait pas q=0, donc il -% fallait aussi faire un éventuel reverseorder sur ce qui était encore non -% traité. Les cas avec q=0 sont maintenant interceptés en amont et comme A et qB -% ont toujours quasiment la même longueur on ne s'embarrasse pas de -% complications pour la fin.| -% \begin{macrocode} -\def\XINT_div_sub_xpxp #1#2% #1=alpha déjà renversé, #2 se développe en qB -{% - \expandafter\XINT_div_sub_xpxp_b #2\W\X\Y\Z #1\W\X\Y\Z -}% -\def\XINT_div_sub_xpxp_b -{% - \XINT_div_sub_A 1{}% -}% -\def\XINT_div_sub_A #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_div_sub_az\W - \XINT_div_sub_B #1{#3#4#5#6}{#2}% -}% -\def\XINT_div_sub_B #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint_gob_til_W #5\xint_div_sub_bz\W - \XINT_div_sub_onestep #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT_div_sub_onestep #1#2#3#4#5#6% -{% - \expandafter\XINT_div_sub_backtoA - \the\numexpr 11#6-#5#4#3#2+#1-\xint_c_i.% -}% -\def\XINT_div_sub_backtoA #1#2#3.#4% -{% - \XINT_div_sub_A #2{#3#4}% -}% -% \end{macrocode} -% \lverb|si on arrive en sub_bz c'est que qB était de longueur K+4 et A -% seulement de longueur K, le résultat est donc < 0, renvoyer juste -| -% \begin{macrocode} -\def\xint_div_sub_bz\W\XINT_div_sub_onestep #1\Z { -}% -% \end{macrocode} -% \lverb|si on arrive en sub_az c'est que qB était de longueur inférieure ou -% égale à celle de A, donc on continue jusqu'à la fin de A, et on vérifiera la -% retenue à la fin.| -% \begin{macrocode} -\def\xint_div_sub_az\W\XINT_div_sub_B #1#2{\XINT_div_sub_C #1}% -\def\XINT_div_sub_C #1#2#3#4#5#6% -{% - \xint_gob_til_W #3\xint_div_sub_cz\W - \XINT_div_sub_C_onestep #1{#6#5#4#3}{#2}% -}% -\def\XINT_div_sub_C_onestep #1#2% -{% - \expandafter\XINT_div_sub_backtoC \the\numexpr 11#2+#1-\xint_c_i.% -}% -\def\XINT_div_sub_backtoC #1#2#3.#4% -{% - \XINT_div_sub_C #2{#3#4}% -}% -% \end{macrocode} -% \lverb|une fois arrivé en sub_cz on teste la retenue pour voir si le résultat -% final est en fait négatif, dans ce cas on renvoie seulement -| -% \begin{macrocode} -\def\xint_div_sub_cz\W\XINT_div_sub_C_onestep #1#2% -{% - \if#10% retenue - \expandafter\xint_div_sub_neg - \else\expandafter\xint_div_sub_ok - \fi -}% -\def\xint_div_sub_neg #1{ -}% -\def\xint_div_sub_ok #1{ #1}% -% \end{macrocode} -% \lverb|& -% & -% -----------------------------------------------------------------$\ -% -----------------------------------------------------------------$\ -% DECIMAL OPERATIONS: FIRST DIGIT, LASTDIGIT, ODDNESS, -% MULTIPLICATION BY TEN, QUOTIENT BY TEN, QUOTIENT OR -% MULTIPLICATION BY POWER OF TEN, SPLIT OPERATION.| -% \subsection{\csh{xintFDg}} -% \lverb|& -% FIRST DIGIT. Code simplified in 1.05. -% And prepared for redefinition by xintfrac to parse through \xintNum. Version -% 1.09a inserts the \xintnum already here.| -% \begin{macrocode} -\def\xintiiFDg {\romannumeral0\xintiifdg }% -\def\xintiifdg #1% -{% - \expandafter\XINT_fdg \romannumeral-`0#1\W\Z -}% -\def\xintFDg {\romannumeral0\xintfdg }% -\def\xintfdg #1% -{% - \expandafter\XINT_fdg \romannumeral0\xintnum{#1}\W\Z -}% -\def\XINT_FDg #1{\romannumeral0\XINT_fdg #1\W\Z }% -\def\XINT_fdg #1#2#3\Z -{% - \xint_UDzerominusfork - #1-{ 0}% zero - 0#1{ #2}% negative - 0-{ #1}% positive - \krof -}% -% \end{macrocode} -% \subsection{\csh{xintLDg}} -% \lverb|& -% LAST DIGIT. Simplified in 1.05. And prepared for extension by xintfrac -% to parse through \xintNum. Release 1.09a adds the \xintnum already here, -% and this propagates to \xintOdd, etc... 1.09e The \xintiiLDg is for -% defining \xintiiOdd which is used once (currently) elsewhere .| -% \begin{macrocode} -\def\xintiiLDg {\romannumeral0\xintiildg }% -\def\xintiildg #1% -{% - \expandafter\XINT_ldg\expandafter {\romannumeral-`0#1}% -}% -\def\xintLDg {\romannumeral0\xintldg }% -\def\xintldg #1% -{% - \expandafter\XINT_ldg\expandafter {\romannumeral0\xintnum{#1}}% -}% -\def\XINT_LDg #1{\romannumeral0\XINT_ldg {#1}}% -\def\XINT_ldg #1% -{% - \expandafter\XINT_ldg_\romannumeral0\xintreverseorder {#1}\Z -}% -\def\XINT_ldg_ #1#2\Z{ #1}% -% \end{macrocode} -% \subsection{\csh{xintMON}, \csh{xintMMON}} -% \lverb|& -% MINUS ONE TO THE POWER N and (-1)^{N-1}| -% \begin{macrocode} -\def\xintiiMON {\romannumeral0\xintiimon }% -\def\xintiimon #1% -{% - \ifodd\xintiiLDg {#1} - \xint_afterfi{ -1}% - \else - \xint_afterfi{ 1}% - \fi -}% -\def\xintiiMMON {\romannumeral0\xintiimmon }% -\def\xintiimmon #1% -{% - \ifodd\xintiiLDg {#1} - \xint_afterfi{ 1}% - \else - \xint_afterfi{ -1}% - \fi -}% -\def\xintMON {\romannumeral0\xintmon }% -\def\xintmon #1% -{% - \ifodd\xintLDg {#1} - \xint_afterfi{ -1}% - \else - \xint_afterfi{ 1}% - \fi -}% -\def\xintMMON {\romannumeral0\xintmmon }% -\def\xintmmon #1% -{% - \ifodd\xintLDg {#1} - \xint_afterfi{ 1}% - \else - \xint_afterfi{ -1}% - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintOdd}} -% \lverb|1.05 has \xintiOdd, whereas \xintOdd parses through \xintNum. -% Inadvertently, 1.09a redefined \xintiLDg so \xintiOdd also parsed through -% \xintNum. Anyway, having a \xintOdd and a \xintiOdd was silly. Removed in -% 1.09f | -% \begin{macrocode} -\def\xintiiOdd {\romannumeral0\xintiiodd }% -\def\xintiiodd #1% -{% - \ifodd\xintiiLDg{#1} - \xint_afterfi{ 1}% - \else - \xint_afterfi{ 0}% - \fi -}% -\def\xintOdd {\romannumeral0\xintodd }% -\def\xintodd #1% -{% - \ifodd\xintLDg{#1} - \xint_afterfi{ 1}% - \else - \xint_afterfi{ 0}% - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintDSL}} -% \lverb|& -% DECIMAL SHIFT LEFT (=MULTIPLICATION PAR 10)| -% \begin{macrocode} -\def\xintDSL {\romannumeral0\xintdsl }% -\def\xintdsl #1% -{% - \expandafter\XINT_dsl \romannumeral-`0#1\Z -}% -\def\XINT_DSL #1{\romannumeral0\XINT_dsl #1\Z }% -\def\XINT_dsl #1% -{% - \xint_gob_til_zero #1\xint_dsl_zero 0\XINT_dsl_ #1% -}% -\def\xint_dsl_zero 0\XINT_dsl_ 0#1\Z { 0}% -\def\XINT_dsl_ #1\Z { #10}% -% \end{macrocode} -% \subsection{\csh{xintDSR}} -% \lverb|& -% DECIMAL SHIFT RIGHT (=DIVISION PAR 10). Release 1.06b which replaced all @'s -% by -% underscores left undefined the \xint_minus used in \XINT_dsr_b, and this bug -% was fixed only later in release 1.09b| -% \begin{macrocode} -\def\xintDSR {\romannumeral0\xintdsr }% -\def\xintdsr #1% -{% - \expandafter\XINT_dsr_a\expandafter {\romannumeral-`0#1}\W\Z -}% -\def\XINT_DSR #1{\romannumeral0\XINT_dsr_a {#1}\W\Z }% -\def\XINT_dsr_a -{% - \expandafter\XINT_dsr_b\romannumeral0\xintreverseorder -}% -\def\XINT_dsr_b #1#2#3\Z -{% - \xint_gob_til_W #2\xint_dsr_onedigit\W - \xint_gob_til_minus #2\xint_dsr_onedigit-% - \expandafter\XINT_dsr_removew - \romannumeral0\xintreverseorder {#2#3}% -}% -\def\xint_dsr_onedigit #1\xintreverseorder #2{ 0}% -\def\XINT_dsr_removew #1\W { }% -% \end{macrocode} -% \subsection{\csh{xintDSH}, \csh{xintDSHr}} -% \lverb+DECIMAL SHIFTS \xintDSH {x}{A}$\ -% si x <= 0, fait A -> A.10^(|x|). v1.03 corrige l'oversight pour A=0.$\ -% si x > 0, et A >=0, fait A -> quo(A,10^(x))$\ -% si x > 0, et A < 0, fait A -> -quo(-A,10^(x))$\ -% (donc pour x > 0 c'est comme DSR itéré x fois)$\ -% \xintDSHr donne le `reste' (si x<=0 donne zéro). -% -% Release 1.06 now feeds x to a \numexpr first. I will have to revise this code -% at some point.+ -% \begin{macrocode} -\def\xintDSHr {\romannumeral0\xintdshr }% -\def\xintdshr #1% -{% - \expandafter\XINT_dshr_checkxpositive \the\numexpr #1\relax\Z -}% -\def\XINT_dshr_checkxpositive #1% -{% - \xint_UDzerominusfork - 0#1\XINT_dshr_xzeroorneg - #1-\XINT_dshr_xzeroorneg - 0-\XINT_dshr_xpositive - \krof #1% -}% -\def\XINT_dshr_xzeroorneg #1\Z #2{ 0}% -\def\XINT_dshr_xpositive #1\Z -{% - \expandafter\xint_secondoftwo_thenstop\romannumeral0\xintdsx {#1}% -}% -\def\xintDSH {\romannumeral0\xintdsh }% -\def\xintdsh #1#2% -{% - \expandafter\xint_dsh\expandafter {\romannumeral-`0#2}{#1}% -}% -\def\xint_dsh #1#2% -{% - \expandafter\XINT_dsh_checksignx \the\numexpr #2\relax\Z {#1}% -}% -\def\XINT_dsh_checksignx #1% -{% - \xint_UDzerominusfork - #1-\XINT_dsh_xiszero - 0#1\XINT_dsx_xisNeg_checkA % on passe direct dans DSx - 0-{\XINT_dsh_xisPos #1}% - \krof -}% -\def\XINT_dsh_xiszero #1\Z #2{ #2}% -\def\XINT_dsh_xisPos #1\Z #2% -{% - \expandafter\xint_firstoftwo_thenstop - \romannumeral0\XINT_dsx_checksignA #2\Z {#1}% via DSx -}% -% \end{macrocode} -% \subsection{\csh{xintDSx}} -% \lverb+Je fais cette routine pour la version 1.01, après modification de -% \xintDecSplit. Dorénavant \xintDSx fera appel à \xintDecSplit et de même -% \xintDSH fera appel à \xintDSx. J'ai donc supprimé entièrement l'ancien code -% de \xintDSH et re-écrit entièrement celui de \xintDecSplit pour x positif. -% -% --> Attention le cas x=0 est traité dans la même catégorie que x > 0 <--$\ -% si x < 0, fait A -> A.10^(|x|)$\ -% si x >= 0, et A >=0, fait A -> {quo(A,10^(x))}{rem(A,10^(x))}$\ -% si x >= 0, et A < 0, d'abord on calcule {quo(-A,10^(x))}{rem(-A,10^(x))}$\ -% puis, si le premier n'est pas nul on lui donne le signe -$\ -% si le premier est nul on donne le signe - au second. -% -% On peut donc toujours reconstituer l'original A par 10^x Q \pm R -% où il faut prendre le signe plus si Q est positif ou nul et le signe moins si -% Q est strictement négatif. -% -% Release 1.06 has a faster and more compactly coded \XINT_dsx_zeroloop. -% Also, x is now given to a \numexpr. The earlier code should be then -% simplified, but I leave as is for the time being. -% -% Release 1.07 modified the coding of \XINT_dsx_zeroloop, to avoid impacting the -% input stack. Indeed the truncating, rounding, and conversion to float routines -% all use internally \XINT_dsx_zeroloop (via \XINT_dsx_addzerosnofuss), and they -% were thus roughly limited to generating N = 8 times the input save stack size -% digits. On TL2012 and TL2013, this means 40000 = 8x5000 digits. Although -% generating more than 40000 digits is more like a one shot thing, I wanted to -% open the possibility of outputting tens of thousands of digits to faile, thus -% I re-organized \XINT_dsx_zeroloop. -% -% January 5, 2014: but it is only with the new division implementation of 1.09j -% and also with its special \xintXTrunc routine that the possibility mentioned -% in the last paragraph has become a concrete one in terms of computation time.+ -% \begin{macrocode} -\def\xintDSx {\romannumeral0\xintdsx }% -\def\xintdsx #1#2% -{% - \expandafter\xint_dsx\expandafter {\romannumeral-`0#2}{#1}% -}% -\def\xint_dsx #1#2% -{% - \expandafter\XINT_dsx_checksignx \the\numexpr #2\relax\Z {#1}% -}% -\def\XINT_DSx #1#2{\romannumeral0\XINT_dsx_checksignx #1\Z {#2}}% -\def\XINT_dsx #1#2{\XINT_dsx_checksignx #1\Z {#2}}% -\def\XINT_dsx_checksignx #1% -{% - \xint_UDzerominusfork - #1-\XINT_dsx_xisZero - 0#1\XINT_dsx_xisNeg_checkA - 0-{\XINT_dsx_xisPos #1}% - \krof -}% -\def\XINT_dsx_xisZero #1\Z #2{ {#2}{0}}% attention comme x > 0 -\def\XINT_dsx_xisNeg_checkA #1\Z #2% -{% - \XINT_dsx_xisNeg_checkA_ #2\Z {#1}% -}% -\def\XINT_dsx_xisNeg_checkA_ #1#2\Z #3% -{% - \xint_gob_til_zero #1\XINT_dsx_xisNeg_Azero 0% - \XINT_dsx_xisNeg_checkx {#3}{#3}{}\Z {#1#2}% -}% -\def\XINT_dsx_xisNeg_Azero #1\Z #2{ 0}% -\def\XINT_dsx_xisNeg_checkx #1% -{% - \ifnum #1>1000000 - \xint_afterfi - {\xintError:TooBigDecimalShift - \expandafter\space\expandafter 0\xint_gobble_iv }% - \else - \expandafter \XINT_dsx_zeroloop - \fi -}% -\def\XINT_dsx_addzerosnofuss #1{\XINT_dsx_zeroloop {#1}{}\Z }% -\def\XINT_dsx_zeroloop #1#2% -{% - \ifnum #1<\xint_c_ix \XINT_dsx_exita\fi - \expandafter\XINT_dsx_zeroloop\expandafter - {\the\numexpr #1-\xint_c_viii}{#200000000}% -}% -\def\XINT_dsx_exita\fi\expandafter\XINT_dsx_zeroloop -{% - \fi\expandafter\XINT_dsx_exitb -}% -\def\XINT_dsx_exitb #1#2% -{% - \expandafter\expandafter\expandafter - \XINT_dsx_addzeros\csname xint_gobble_\romannumeral -#1\endcsname #2% -}% -\def\XINT_dsx_addzeros #1\Z #2{ #2#1}% -\def\XINT_dsx_xisPos #1\Z #2% -{% - \XINT_dsx_checksignA #2\Z {#1}% -}% -\def\XINT_dsx_checksignA #1% -{% - \xint_UDzerominusfork - #1-\XINT_dsx_AisZero - 0#1\XINT_dsx_AisNeg - 0-{\XINT_dsx_AisPos #1}% - \krof -}% -\def\XINT_dsx_AisZero #1\Z #2{ {0}{0}}% -\def\XINT_dsx_AisNeg #1\Z #2% -{% - \expandafter\XINT_dsx_AisNeg_dosplit_andcheckfirst - \romannumeral0\XINT_split_checksizex {#2}{#1}% -}% -\def\XINT_dsx_AisNeg_dosplit_andcheckfirst #1% -{% - \XINT_dsx_AisNeg_checkiffirstempty #1\Z -}% -\def\XINT_dsx_AisNeg_checkiffirstempty #1% -{% - \xint_gob_til_Z #1\XINT_dsx_AisNeg_finish_zero\Z - \XINT_dsx_AisNeg_finish_notzero #1% -}% -\def\XINT_dsx_AisNeg_finish_zero\Z - \XINT_dsx_AisNeg_finish_notzero\Z #1% -{% - \expandafter\XINT_dsx_end - \expandafter {\romannumeral0\XINT_num {-#1}}{0}% -}% -\def\XINT_dsx_AisNeg_finish_notzero #1\Z #2% -{% - \expandafter\XINT_dsx_end - \expandafter {\romannumeral0\XINT_num {#2}}{-#1}% -}% -\def\XINT_dsx_AisPos #1\Z #2% -{% - \expandafter\XINT_dsx_AisPos_finish - \romannumeral0\XINT_split_checksizex {#2}{#1}% -}% -\def\XINT_dsx_AisPos_finish #1#2% -{% - \expandafter\XINT_dsx_end - \expandafter {\romannumeral0\XINT_num {#2}}% - {\romannumeral0\XINT_num {#1}}% -}% -\edef\XINT_dsx_end #1#2% -{% - \noexpand\expandafter\space\noexpand\expandafter{#2}{#1}% -}% -% \end{macrocode} -% \subsection{\csh{xintDecSplit}, \csh{xintDecSplitL}, \csh{xintDecSplitR}} -% \lverb!DECIMAL SPLIT -% -% The macro \xintDecSplit {x}{A} first replaces A with |A| (*) -% This macro cuts the number into two pieces L and R. The concatenation LR -% always reproduces |A|, and R may be empty or have leading zeros. The -% position of the cut is specified by the first argument x. If x is zero or -% positive the cut location is x slots to the left of the right end of the -% number. If x becomes equal to or larger than the length of the number then L -% becomes empty. If x is negative the location of the cut is |x| slots to the -% right of the left end of the number. -% -% (*) warning: this may change in a future version. Only the behavior -% for A non-negative is guaranteed to remain the same. -% -% v1.05a: \XINT_split_checksizex does not compute the length anymore, rather the -% error will be from a \numexpr; but the limit of 999999999 does not make much -% sense. -% -% v1.06: Improvements in \XINT_split_fromleft_loop, \XINT_split_fromright_loop -% and related macros. More readable coding, speed gains. -% Also, I now feed immediately a \numexpr with x. Some simplifications should -% probably be made to the code, which is kept as is for the time being. -% -% 1.09e pays attention to the use of xintiabs which acquired in 1.09a the -% xintnum overhead. So xintiiabs rather without that overhead. -% ! -% \begin{macrocode} -\def\xintDecSplitL {\romannumeral0\xintdecsplitl }% -\def\xintDecSplitR {\romannumeral0\xintdecsplitr }% -\def\xintdecsplitl -{% - \expandafter\xint_firstoftwo_thenstop - \romannumeral0\xintdecsplit -}% -\def\xintdecsplitr -{% - \expandafter\xint_secondoftwo_thenstop - \romannumeral0\xintdecsplit -}% -\def\xintDecSplit {\romannumeral0\xintdecsplit }% -\def\xintdecsplit #1#2% -{% - \expandafter \xint_split \expandafter - {\romannumeral0\xintiiabs {#2}}{#1}% fait expansion de A -}% -\def\xint_split #1#2% -{% - \expandafter\XINT_split_checksizex\expandafter{\the\numexpr #2}{#1}% -}% -\def\XINT_split_checksizex #1% 999999999 is anyhow very big, could be reduced -{% - \ifnum\numexpr\XINT_Abs{#1}>999999999 - \xint_afterfi {\xintError:TooBigDecimalSplit\XINT_split_bigx }% - \else - \expandafter\XINT_split_xfork - \fi - #1\Z -}% -\def\XINT_split_bigx #1\Z #2% -{% - \ifcase\XINT_cntSgn #1\Z - \or \xint_afterfi { {}{#2}}% positive big x - \else - \xint_afterfi { {#2}{}}% negative big x - \fi -}% -\def\XINT_split_xfork #1% -{% - \xint_UDzerominusfork - #1-\XINT_split_zerosplit - 0#1\XINT_split_fromleft - 0-{\XINT_split_fromright #1}% - \krof -}% -\def\XINT_split_zerosplit #1\Z #2{ {#2}{}}% -\def\XINT_split_fromleft #1\Z #2% -{% - \XINT_split_fromleft_loop {#1}{}#2\W\W\W\W\W\W\W\W\Z -}% -\def\XINT_split_fromleft_loop #1% -{% - \ifnum #1<\xint_c_viii\XINT_split_fromleft_exita\fi - \expandafter\XINT_split_fromleft_loop_perhaps\expandafter - {\the\numexpr #1-\xint_c_viii\expandafter}\XINT_split_fromleft_eight -}% -\def\XINT_split_fromleft_eight #1#2#3#4#5#6#7#8#9{#9{#1#2#3#4#5#6#7#8#9}}% -\def\XINT_split_fromleft_loop_perhaps #1#2% -{% - \xint_gob_til_W #2\XINT_split_fromleft_toofar\W - \XINT_split_fromleft_loop {#1}% -}% -\def\XINT_split_fromleft_toofar\W\XINT_split_fromleft_loop #1#2#3\Z -{% - \XINT_split_fromleft_toofar_b #2\Z -}% -\def\XINT_split_fromleft_toofar_b #1\W #2\Z { {#1}{}}% -\def\XINT_split_fromleft_exita\fi - \expandafter\XINT_split_fromleft_loop_perhaps\expandafter #1#2% - {\fi \XINT_split_fromleft_exitb #1}% -\def\XINT_split_fromleft_exitb\the\numexpr #1-\xint_c_viii\expandafter -{% - \csname XINT_split_fromleft_endsplit_\romannumeral #1\endcsname -}% -\def\XINT_split_fromleft_endsplit_ #1#2\W #3\Z { {#1}{#2}}% -\def\XINT_split_fromleft_endsplit_i #1#2% - {\XINT_split_fromleft_checkiftoofar #2{#1#2}}% -\def\XINT_split_fromleft_endsplit_ii #1#2#3% - {\XINT_split_fromleft_checkiftoofar #3{#1#2#3}}% -\def\XINT_split_fromleft_endsplit_iii #1#2#3#4% - {\XINT_split_fromleft_checkiftoofar #4{#1#2#3#4}}% -\def\XINT_split_fromleft_endsplit_iv #1#2#3#4#5% - {\XINT_split_fromleft_checkiftoofar #5{#1#2#3#4#5}}% -\def\XINT_split_fromleft_endsplit_v #1#2#3#4#5#6% - {\XINT_split_fromleft_checkiftoofar #6{#1#2#3#4#5#6}}% -\def\XINT_split_fromleft_endsplit_vi #1#2#3#4#5#6#7% - {\XINT_split_fromleft_checkiftoofar #7{#1#2#3#4#5#6#7}}% -\def\XINT_split_fromleft_endsplit_vii #1#2#3#4#5#6#7#8% - {\XINT_split_fromleft_checkiftoofar #8{#1#2#3#4#5#6#7#8}}% -\def\XINT_split_fromleft_checkiftoofar #1#2#3\W #4\Z -{% - \xint_gob_til_W #1\XINT_split_fromleft_wenttoofar\W - \space {#2}{#3}% -}% -\def\XINT_split_fromleft_wenttoofar\W\space #1% -{% - \XINT_split_fromleft_wenttoofar_b #1\Z -}% -\def\XINT_split_fromleft_wenttoofar_b #1\W #2\Z { {#1}}% -\def\XINT_split_fromright #1\Z #2% -{% - \expandafter \XINT_split_fromright_a \expandafter - {\romannumeral0\xintreverseorder {#2}}{#1}{#2}% -}% -\def\XINT_split_fromright_a #1#2% -{% - \XINT_split_fromright_loop {#2}{}#1\W\W\W\W\W\W\W\W\Z -}% -\def\XINT_split_fromright_loop #1% -{% - \ifnum #1<\xint_c_viii\XINT_split_fromright_exita\fi - \expandafter\XINT_split_fromright_loop_perhaps\expandafter - {\the\numexpr #1-\xint_c_viii\expandafter }\XINT_split_fromright_eight -}% -\def\XINT_split_fromright_eight #1#2#3#4#5#6#7#8#9{#9{#9#8#7#6#5#4#3#2#1}}% -\def\XINT_split_fromright_loop_perhaps #1#2% -{% - \xint_gob_til_W #2\XINT_split_fromright_toofar\W - \XINT_split_fromright_loop {#1}% -}% -\def\XINT_split_fromright_toofar\W\XINT_split_fromright_loop #1#2#3\Z { {}}% -\def\XINT_split_fromright_exita\fi - \expandafter\XINT_split_fromright_loop_perhaps\expandafter #1#2% - {\fi \XINT_split_fromright_exitb #1}% -\def\XINT_split_fromright_exitb\the\numexpr #1-\xint_c_viii\expandafter -{% - \csname XINT_split_fromright_endsplit_\romannumeral #1\endcsname -}% -\edef\XINT_split_fromright_endsplit_ #1#2\W #3\Z #4% -{% - \noexpand\expandafter\space\noexpand\expandafter - {\noexpand\romannumeral0\noexpand\xintreverseorder {#2}}{#1}% -}% -\def\XINT_split_fromright_endsplit_i #1#2% - {\XINT_split_fromright_checkiftoofar #2{#2#1}}% -\def\XINT_split_fromright_endsplit_ii #1#2#3% - {\XINT_split_fromright_checkiftoofar #3{#3#2#1}}% -\def\XINT_split_fromright_endsplit_iii #1#2#3#4% - {\XINT_split_fromright_checkiftoofar #4{#4#3#2#1}}% -\def\XINT_split_fromright_endsplit_iv #1#2#3#4#5% - {\XINT_split_fromright_checkiftoofar #5{#5#4#3#2#1}}% -\def\XINT_split_fromright_endsplit_v #1#2#3#4#5#6% - {\XINT_split_fromright_checkiftoofar #6{#6#5#4#3#2#1}}% -\def\XINT_split_fromright_endsplit_vi #1#2#3#4#5#6#7% - {\XINT_split_fromright_checkiftoofar #7{#7#6#5#4#3#2#1}}% -\def\XINT_split_fromright_endsplit_vii #1#2#3#4#5#6#7#8% - {\XINT_split_fromright_checkiftoofar #8{#8#7#6#5#4#3#2#1}}% -\def\XINT_split_fromright_checkiftoofar #1% -{% - \xint_gob_til_W #1\XINT_split_fromright_wenttoofar\W - \XINT_split_fromright_endsplit_ -}% -\def\XINT_split_fromright_wenttoofar\W\XINT_split_fromright_endsplit_ #1\Z #2% - { {}{#2}}% -% \end{macrocode} -% \subsection{\csh{xintDouble}} -% \lverb|v1.08| -% \begin{macrocode} -\def\xintDouble {\romannumeral0\xintdouble }% -\def\xintdouble #1% -{% - \expandafter\XINT_dbl\romannumeral-`0#1% - \R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W -}% -\def\XINT_dbl #1% -{% - \xint_UDzerominusfork - #1-\XINT_dbl_zero - 0#1\XINT_dbl_neg - 0-{\XINT_dbl_pos #1}% - \krof -}% -\def\XINT_dbl_zero #1\Z \W\W\W\W\W\W\W { 0}% -\def\XINT_dbl_neg - {\expandafter\xint_minus_thenstop\romannumeral0\XINT_dbl_pos }% -\def\XINT_dbl_pos -{% - \expandafter\XINT_dbl_a \expandafter{\expandafter}\expandafter 0% - \romannumeral0\XINT_SQ {}% -}% -\def\XINT_dbl_a #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #9\XINT_dbl_end_a\W - \expandafter\XINT_dbl_b - \the\numexpr \xint_c_x^viii+#2+\xint_c_ii*#9#8#7#6#5#4#3\relax {#1}% -}% -\def\XINT_dbl_b 1#1#2#3#4#5#6#7#8#9% -{% - \XINT_dbl_a {#2#3#4#5#6#7#8#9}{#1}% -}% -\def\XINT_dbl_end_a #1+#2+#3\relax #4% -{% - \expandafter\XINT_dbl_end_b #2#4% -}% -\edef\XINT_dbl_end_b #1#2#3#4#5#6#7#8% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax -}% -% \end{macrocode} -% \subsection{\csh{xintHalf}} -% \lverb!v1.08! -% \begin{macrocode} -\def\xintHalf {\romannumeral0\xinthalf }% -\def\xinthalf #1% -{% - \expandafter\XINT_half\romannumeral-`0#1% - \R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W -}% -\def\XINT_half #1% -{% - \xint_UDzerominusfork - #1-\XINT_half_zero - 0#1\XINT_half_neg - 0-{\XINT_half_pos #1}% - \krof -}% -\def\XINT_half_zero #1\Z \W\W\W\W\W\W\W { 0}% -\def\XINT_half_neg {\expandafter\XINT_opp\romannumeral0\XINT_half_pos }% -\def\XINT_half_pos {\expandafter\XINT_half_a\romannumeral0\XINT_SQ {}}% -\def\XINT_half_a #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_W #8\XINT_half_dont\W - \expandafter\XINT_half_b - \the\numexpr \xint_c_x^viii+\xint_c_v*#7#6#5#4#3#2#1\relax #8% -}% -\edef\XINT_half_dont\W\expandafter\XINT_half_b - \the\numexpr \xint_c_x^viii+\xint_c_v*#1#2#3#4#5#6#7\relax \W\W\W\W\W\W\W -{% - \noexpand\expandafter\space - \noexpand\the\numexpr (#1#2#3#4#5#6#7+\xint_c_i)/\xint_c_ii-\xint_c_i \relax -}% -\def\XINT_half_b 1#1#2#3#4#5#6#7#8% -{% - \XINT_half_c {#2#3#4#5#6#7}{#1}% -}% -\def\XINT_half_c #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #3\XINT_half_end_a #2\W - \expandafter\XINT_half_d - \the\numexpr \xint_c_x^viii+\xint_c_v*#9#8#7#6#5#4#3+#2\relax {#1}% -}% -\def\XINT_half_d 1#1#2#3#4#5#6#7#8#9% -{% - \XINT_half_c {#2#3#4#5#6#7#8#9}{#1}% -}% -\def\XINT_half_end_a #1\W #2\relax #3% -{% - \xint_gob_til_zero #1\XINT_half_end_b 0\space #1#3% -}% -\edef\XINT_half_end_b 0\space 0#1#2#3#4#5#6#7% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7\relax -}% -% \end{macrocode} -% \subsection{\csh{xintDec}} -% \lverb!v1.08! -% \begin{macrocode} -\def\xintDec {\romannumeral0\xintdec }% -\def\xintdec #1% -{% - \expandafter\XINT_dec\romannumeral-`0#1% - \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W -}% -\def\XINT_dec #1% -{% - \xint_UDzerominusfork - #1-\XINT_dec_zero - 0#1\XINT_dec_neg - 0-{\XINT_dec_pos #1}% - \krof -}% -\def\XINT_dec_zero #1\W\W\W\W\W\W\W\W { -1}% -\def\XINT_dec_neg - {\expandafter\xint_minus_thenstop\romannumeral0\XINT_inc_pos }% -\def\XINT_dec_pos -{% - \expandafter\XINT_dec_a \expandafter{\expandafter}% - \romannumeral0\XINT_OQ {}% -}% -\def\XINT_dec_a #1#2#3#4#5#6#7#8#9% -{% - \expandafter\XINT_dec_b - \the\numexpr 11#9#8#7#6#5#4#3#2-\xint_c_i\relax {#1}% -}% -\def\XINT_dec_b 1#1% -{% - \xint_gob_til_one #1\XINT_dec_A 1\XINT_dec_c -}% -\def\XINT_dec_c #1#2#3#4#5#6#7#8#9{\XINT_dec_a {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_dec_A 1\XINT_dec_c #1#2#3#4#5#6#7#8#9% - {\XINT_dec_B {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_dec_B #1#2\W\W\W\W\W\W\W\W -{% - \expandafter\XINT_dec_cleanup - \romannumeral0\XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - #1% -}% -\edef\XINT_dec_cleanup #1#2#3#4#5#6#7#8% - {\noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax }% -% \end{macrocode} -% \subsection{\csh{xintInc}} -% \lverb!v1.08! -% \begin{macrocode} -\def\xintInc {\romannumeral0\xintinc }% -\def\xintinc #1% -{% - \expandafter\XINT_inc\romannumeral-`0#1% - \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W -}% -\def\XINT_inc #1% -{% - \xint_UDzerominusfork - #1-\XINT_inc_zero - 0#1\XINT_inc_neg - 0-{\XINT_inc_pos #1}% - \krof -}% -\def\XINT_inc_zero #1\W\W\W\W\W\W\W\W { 1}% -\def\XINT_inc_neg {\expandafter\XINT_opp\romannumeral0\XINT_dec_pos }% -\def\XINT_inc_pos -{% - \expandafter\XINT_inc_a \expandafter{\expandafter}% - \romannumeral0\XINT_OQ {}% -}% -\def\XINT_inc_a #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #9\XINT_inc_end\W - \expandafter\XINT_inc_b - \the\numexpr 10#9#8#7#6#5#4#3#2+\xint_c_i\relax {#1}% -}% -\def\XINT_inc_b 1#1% -{% - \xint_gob_til_zero #1\XINT_inc_A 0\XINT_inc_c -}% -\def\XINT_inc_c #1#2#3#4#5#6#7#8#9{\XINT_inc_a {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_inc_A 0\XINT_inc_c #1#2#3#4#5#6#7#8#9% - {\XINT_dec_B {#1#2#3#4#5#6#7#8#9}}% -\def\XINT_inc_end\W #1\relax #2{ 1#2}% -% \end{macrocode} -% \subsection{\csh{xintiSqrt}, \csh{xintiSquareRoot}} -% \lverb|v1.08. 1.09a uses \xintnum. -% -% Some overhead was added inadvertently in 1.09a to inner routines when -% \xintiquo and \xintidivision were also promoted to use \xintnum; release 1.09f -% thus uses \xintiiquo and \xintiidivision xhich avoid this \xintnum overhead. -% -% 1.09j replaced the previous long \ifcase from \XINT_sqrt_c by some nested -% \ifnum's.| -% \begin{macrocode} -\def\xintiSqrt {\romannumeral0\xintisqrt }% -\def\xintisqrt - {\expandafter\XINT_sqrt_post\romannumeral0\xintisquareroot }% -\def\XINT_sqrt_post #1#2{\XINT_dec_pos #1\R\R\R\R\R\R\R\R\Z - \W\W\W\W\W\W\W\W }% -\def\xintiSquareRoot {\romannumeral0\xintisquareroot }% -\def\xintisquareroot #1% - {\expandafter\XINT_sqrt_checkin\romannumeral0\xintnum{#1}\Z}% -\def\XINT_sqrt_checkin #1% -{% - \xint_UDzerominusfork - #1-\XINT_sqrt_iszero - 0#1\XINT_sqrt_isneg - 0-{\XINT_sqrt #1}% - \krof -}% -\def\XINT_sqrt_iszero #1\Z { 1.}% -\edef\XINT_sqrt_isneg #1\Z {\noexpand\xintError:RootOfNegative\space 1.}% -\def\XINT_sqrt #1\Z -{% - \expandafter\XINT_sqrt_start\expandafter - {\romannumeral0\xintlength {#1}}{#1}% -}% -\def\XINT_sqrt_start #1% -{% - \ifnum #1<\xint_c_x - \expandafter\XINT_sqrt_small_a - \else - \expandafter\XINT_sqrt_big_a - \fi - {#1}% -}% -\def\XINT_sqrt_small_a #1{\XINT_sqrt_a {#1}\XINT_sqrt_small_d }% -\def\XINT_sqrt_big_a #1{\XINT_sqrt_a {#1}\XINT_sqrt_big_d }% -\def\XINT_sqrt_a #1% -{% - \ifodd #1 - \expandafter\XINT_sqrt_bB - \else - \expandafter\XINT_sqrt_bA - \fi - {#1}% -}% -\def\XINT_sqrt_bA #1#2#3% -{% - \XINT_sqrt_bA_b #3\Z #2{#1}{#3}% -}% -\def\XINT_sqrt_bA_b #1#2#3\Z -{% - \XINT_sqrt_c {#1#2}% -}% -\def\XINT_sqrt_bB #1#2#3% -{% - \XINT_sqrt_bB_b #3\Z #2{#1}{#3}% -}% -\def\XINT_sqrt_bB_b #1#2\Z -{% - \XINT_sqrt_c #1% -}% -\def\XINT_sqrt_c #1#2% -{% - \expandafter #2\expandafter - {\the\numexpr\ifnum #1>\xint_c_iii - \ifnum #1>\xint_c_viii - \ifnum #1>15 \ifnum #1>24 \ifnum #1>35 - \ifnum #1>48 \ifnum #1>63 \ifnum #1>80 - 10\else 9\fi \else 8\fi \else 7\fi \else 6\fi - \else 5\fi \else 4\fi \else 3\fi \else 2\fi \relax }% -}% -\def\XINT_sqrt_small_d #1#2% -{% - \expandafter\XINT_sqrt_small_e\expandafter - {\the\numexpr #1\ifcase \numexpr #2/\xint_c_ii-\xint_c_i\relax - \or 0\or 00\or 000\or 0000\fi }% -}% -\def\XINT_sqrt_small_e #1#2% -{% - \expandafter\XINT_sqrt_small_f\expandafter {\the\numexpr #1*#1-#2}{#1}% -}% -\def\XINT_sqrt_small_f #1#2% -{% - \expandafter\XINT_sqrt_small_g\expandafter - {\the\numexpr ((#1+#2)/(\xint_c_ii*#2))-\xint_c_i}{#1}{#2}% -}% -\def\XINT_sqrt_small_g #1% -{% - \ifnum #1>\xint_c_ - \expandafter\XINT_sqrt_small_h - \else - \expandafter\XINT_sqrt_small_end - \fi - {#1}% -}% -\def\XINT_sqrt_small_h #1#2#3% -{% - \expandafter\XINT_sqrt_small_f\expandafter - {\the\numexpr #2-\xint_c_ii*#1*#3+#1*#1\expandafter}\expandafter - {\the\numexpr #3-#1}% -}% -\def\XINT_sqrt_small_end #1#2#3{ {#3}{#2}}% -\def\XINT_sqrt_big_d #1#2% -{% - \ifodd #2 - \expandafter\expandafter\expandafter\XINT_sqrt_big_eB - \else - \expandafter\expandafter\expandafter\XINT_sqrt_big_eA - \fi - \expandafter {\the\numexpr #2/\xint_c_ii }{#1}% -}% -\def\XINT_sqrt_big_eA #1#2#3% -{% - \XINT_sqrt_big_eA_a #3\Z {#2}{#1}{#3}% -}% -\def\XINT_sqrt_big_eA_a #1#2#3#4#5#6#7#8#9\Z -{% - \XINT_sqrt_big_eA_b {#1#2#3#4#5#6#7#8}% -}% -\def\XINT_sqrt_big_eA_b #1#2% -{% - \expandafter\XINT_sqrt_big_f - \romannumeral0\XINT_sqrt_small_e {#2000}{#1}{#1}% -}% -\def\XINT_sqrt_big_eB #1#2#3% -{% - \XINT_sqrt_big_eB_a #3\Z {#2}{#1}{#3}% -}% -\def\XINT_sqrt_big_eB_a #1#2#3#4#5#6#7#8#9% -{% - \XINT_sqrt_big_eB_b {#1#2#3#4#5#6#7#8#9}% -}% -\def\XINT_sqrt_big_eB_b #1#2\Z #3% -{% - \expandafter\XINT_sqrt_big_f - \romannumeral0\XINT_sqrt_small_e {#30000}{#1}{#1}% -}% -\def\XINT_sqrt_big_f #1#2#3#4% -{% - \expandafter\XINT_sqrt_big_f_a\expandafter - {\the\numexpr #2+#3\expandafter}\expandafter - {\romannumeral0\XINT_dsx_addzerosnofuss - {\numexpr #4-\xint_c_iv\relax}{#1}}{#4}% -}% -\def\XINT_sqrt_big_f_a #1#2#3#4% -{% - \expandafter\XINT_sqrt_big_g\expandafter - {\romannumeral0\xintiisub - {\XINT_dsx_addzerosnofuss - {\numexpr \xint_c_ii*#3-\xint_c_viii\relax}{#1}}{#4}}% - {#2}{#3}% -}% -\def\XINT_sqrt_big_g #1#2% -{% - \expandafter\XINT_sqrt_big_j - \romannumeral0\xintiidivision{#1}% - {\romannumeral0\XINT_dbl_pos #2\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W }{#2}% -}% -\def\XINT_sqrt_big_j #1% -{% - \if0\XINT_Sgn #1\Z - \expandafter \XINT_sqrt_big_end - \else \expandafter \XINT_sqrt_big_k - \fi {#1}% -}% -\def\XINT_sqrt_big_k #1#2#3% -{% - \expandafter\XINT_sqrt_big_l\expandafter - {\romannumeral0\xintiisub {#3}{#1}}% - {\romannumeral0\xintiiadd {#2}{\xintiiSqr {#1}}}% -}% -\def\XINT_sqrt_big_l #1#2% -{% - \expandafter\XINT_sqrt_big_g\expandafter - {#2}{#1}% -}% -\def\XINT_sqrt_big_end #1#2#3#4{ {#3}{#2}}% -% \end{macrocode} -% \subsection{\csh{xintIsTrue:csv}} -% \lverb|1.09c. For use by \xinttheboolexpr.(inside \csname, no need for a -% \romannumeral here). The macros may well be defined already here. I -% make no advertisement because I have inserted no space parsing in the -% :csv macros, as they will be used only with privately created comma -% separated lists, having no space naturally. Nevertheless they exist -% and can be used.| -% \begin{macrocode} -\def\xintIsTrue:csv #1{\expandafter\XINT_istrue:_a\romannumeral-`0#1,,^}% -\def\XINT_istrue:_a {\XINT_istrue:_b {}}% -\def\XINT_istrue:_b #1#2,% - {\expandafter\XINT_istrue:_c\romannumeral-`0#2,{#1}}% -\def\XINT_istrue:_c #1{\if #1,\expandafter\XINT_:_f - \else\expandafter\XINT_istrue:_d\fi #1}% -\def\XINT_istrue:_d #1,% - {\expandafter\XINT_istrue:_e\romannumeral0\xintisnotzero {#1},}% -\def\XINT_istrue:_e #1,#2{\XINT_istrue:_b {#2,#1}}% -\def\XINT_:_f ,#1#2^{\xint_gobble_i #1}% -% \end{macrocode} -% \subsection{\csh{xintANDof:csv}} -% \lverb|1.09a. For use by \xintexpr (inside \csname, no need for a -% \romannumeral here).| -% \begin{macrocode} -\def\xintANDof:csv #1{\expandafter\XINT_andof:_a\romannumeral-`0#1,,^}% -\def\XINT_andof:_a {\expandafter\XINT_andof:_b\romannumeral-`0}% -\def\XINT_andof:_b #1{\if #1,\expandafter\XINT_andof:_e - \else\expandafter\XINT_andof:_c\fi #1}% -\def\XINT_andof:_c #1,{\xintifTrueAelseB {#1}{\XINT_andof:_a}{\XINT_andof:_no}}% -\def\XINT_andof:_no #1^{0}% -\def\XINT_andof:_e #1^{1}% works with empty list -% \end{macrocode} -% \subsection{\csh{xintORof:csv}} -% \lverb|1.09a. For use by \xintexpr.| -% \begin{macrocode} -\def\xintORof:csv #1{\expandafter\XINT_orof:_a\romannumeral-`0#1,,^}% -\def\XINT_orof:_a {\expandafter\XINT_orof:_b\romannumeral-`0}% -\def\XINT_orof:_b #1{\if #1,\expandafter\XINT_orof:_e - \else\expandafter\XINT_orof:_c\fi #1}% -\def\XINT_orof:_c #1,{\xintifTrueAelseB{#1}{\XINT_orof:_yes}{\XINT_orof:_a}}% -\def\XINT_orof:_yes #1^{1}% -\def\XINT_orof:_e #1^{0}% works with empty list -% \end{macrocode} -% \subsection{\csh{xintXORof:csv}} -% \lverb|1.09a. For use by \xintexpr (inside a \csname..\endcsname).| -% \begin{macrocode} -\def\xintXORof:csv #1{\expandafter\XINT_xorof:_a\expandafter - 0\romannumeral-`0#1,,^}% -\def\XINT_xorof:_a #1#2,{\expandafter\XINT_xorof:_b\romannumeral-`0#2,#1}% -\def\XINT_xorof:_b #1{\if #1,\expandafter\XINT_:_e - \else\expandafter\XINT_xorof:_c\fi #1}% -\def\XINT_xorof:_c #1,#2% - {\xintifTrueAelseB {#1}{\if #20\xint_afterfi{\XINT_xorof:_a 1}% - \else\xint_afterfi{\XINT_xorof:_a 0}\fi}% - {\XINT_xorof:_a #2}% - }% -\def\XINT_:_e ,#1#2^{#1}% allows empty list -% \end{macrocode} -% \subsection{\csh{xintiMaxof:csv}} -% \lverb|1.09i. For use by \xintiiexpr.| -% \begin{macrocode} -\def\xintiMaxof:csv #1{\expandafter\XINT_imaxof:_b\romannumeral-`0#1,,}% -\def\XINT_imaxof:_b #1,#2,{\expandafter\XINT_imaxof:_c\romannumeral-`0#2,{#1},}% -\def\XINT_imaxof:_c #1{\if #1,\expandafter\XINT_of:_e - \else\expandafter\XINT_imaxof:_d\fi #1}% -\def\XINT_imaxof:_d #1,{\expandafter\XINT_imaxof:_b\romannumeral0\xintimax {#1}}% -\def\XINT_of:_e ,#1,{#1}% -\let\xintMaxof:csv\xintiMaxof:csv -% \end{macrocode} -% \subsection{\csh{xintiMinof:csv}} -% \lverb|1.09i. For use by \xintiiexpr.| -% \begin{macrocode} -\def\xintiMinof:csv #1{\expandafter\XINT_iminof:_b\romannumeral-`0#1,,}% -\def\XINT_iminof:_b #1,#2,{\expandafter\XINT_iminof:_c\romannumeral-`0#2,{#1},}% -\def\XINT_iminof:_c #1{\if #1,\expandafter\XINT_of:_e - \else\expandafter\XINT_iminof:_d\fi #1}% -\def\XINT_iminof:_d #1,{\expandafter\XINT_iminof:_b\romannumeral0\xintimin {#1}}% -\let\xintMinof:csv\xintiMinof:csv -% \end{macrocode} -% \subsection{\csh{xintiiSum:csv}} -% \lverb|1.09i. For use by \xintiiexpr.| -% \begin{macrocode} -\def\xintiiSum:csv #1{\expandafter\XINT_iisum:_a\romannumeral-`0#1,,^}% -\def\XINT_iisum:_a {\XINT_iisum:_b {0}}% -\def\XINT_iisum:_b #1#2,{\expandafter\XINT_iisum:_c\romannumeral-`0#2,{#1}}% -\def\XINT_iisum:_c #1{\if #1,\expandafter\XINT_:_e - \else\expandafter\XINT_iisum:_d\fi #1}% -\def\XINT_iisum:_d #1,#2{\expandafter\XINT_iisum:_b\expandafter - {\romannumeral0\xintiiadd {#2}{#1}}}% -\let\xintSum:csv\xintiiSum:csv -% \end{macrocode} -% \subsection{\csh{xintiiPrd:csv}} -% \lverb|1.09i. For use by \xintiiexpr.| -% \begin{macrocode} -\def\xintiiPrd:csv #1{\expandafter\XINT_iiprd:_a\romannumeral-`0#1,,^}% -\def\XINT_iiprd:_a {\XINT_iiprd:_b {1}}% -\def\XINT_iiprd:_b #1#2,{\expandafter\XINT_iiprd:_c\romannumeral-`0#2,{#1}}% -\def\XINT_iiprd:_c #1{\if #1,\expandafter\XINT_:_e - \else\expandafter\XINT_iiprd:_d\fi #1}% -\def\XINT_iiprd:_d #1,#2{\expandafter\XINT_iiprd:_b\expandafter - {\romannumeral0\xintiimul {#2}{#1}}}% -\let\xintPrd:csv\xintiiPrd:csv -\XINT_restorecatcodes_endinput% -% \end{macrocode} -%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 -%\let</xint>\relax -%\def<*xintbinhex>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 } -%</xint> -%<*xintbinhex> -% -% \StoreCodelineNo {xint} -% -% \section{Package \xintbinhexnameimp implementation} -% \label{sec:binheximp} -% -% The commenting is currently (\docdate) very sparse. -% -% \localtableofcontents -% \subsection{Catcodes, \protect\eTeX{} and reload detection} -% -% The code for reload detection is copied from \textsc{Heiko -% Oberdiek}'s packages, and adapted here to check for previous -% loading of the master \xintname package. -% -% The method for catcodes is slightly different, but still -% directly inspired by these packages. -% -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \def\space { }% - \let\z\endgroup - \expandafter\let\expandafter\x\csname ver@xintbinhex.sty\endcsname - \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname - \expandafter - \ifx\csname PackageInfo\endcsname\relax - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \else - \def\y#1#2{\PackageInfo{#1}{#2}}% - \fi - \expandafter - \ifx\csname numexpr\endcsname\relax - \y{xintbinhex}{\numexpr not available, aborting input}% - \aftergroup\endinput - \else - \ifx\x\relax % plain-TeX, first loading of xintbinhex.sty - \ifx\w\relax % but xint.sty not yet loaded. - \y{xintbinhex}{now issuing \string\input\space xint.sty}% - \def\z{\endgroup\input xint.sty\relax}% - \fi - \else - \def\empty {}% - \ifx\x\empty % LaTeX, first loading, - % variable is initialized, but \ProvidesPackage not yet seen - \ifx\w\relax % xint.sty not yet loaded. - \y{xintbinhex}{now issuing \string\RequirePackage{xint}}% - \def\z{\endgroup\RequirePackage{xint}}% - \fi - \else - \y{xintbinhex}{I was already loaded, aborting input}% - \aftergroup\endinput - \fi - \fi - \fi -\z% -% \end{macrocode} -% \subsection{Confirmation of \xintnameimp loading} -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \ifdefined\PackageInfo - \def\y#1#2{\PackageInfo{#1}{#2}}% - \else - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \fi - \def\empty {}% - \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname - \ifx\w\relax % Plain TeX, user gave a file name at the prompt - \y{xintbinhex}{Loading of package xint failed, aborting input}% - \aftergroup\endinput - \fi - \ifx\w\empty % LaTeX, user gave a file name at the prompt - \y{xintbinhex}{Loading of package xint failed, aborting input}% - \aftergroup\endinput - \fi -\endgroup% -% \end{macrocode} -% \subsection{Catcodes} -% \begin{macrocode} -\XINTsetupcatcodes% -% \end{macrocode} -% \subsection{Package identification} -% \begin{macrocode} -\XINT_providespackage -\ProvidesPackage{xintbinhex}% - [2014/02/05 v1.09ka Expandable binary and hexadecimal conversions (jfB)]% -% \end{macrocode} -% \subsection{Constants, etc...} -% \lverb!v1.08! -% \begin{macrocode} -\chardef\xint_c_xvi 16 -% \chardef\xint_c_ii^v 32 % already done in xint.sty -% \chardef\xint_c_ii^vi 64 % already done in xint.sty -\chardef\xint_c_ii^vii 128 -\mathchardef\xint_c_ii^viii 256 -\mathchardef\xint_c_ii^xii 4096 -\newcount\xint_c_ii^xv \xint_c_ii^xv 32768 -\newcount\xint_c_ii^xvi \xint_c_ii^xvi 65536 -\newcount\xint_c_x^v \xint_c_x^v 100000 -\newcount\xint_c_x^ix \xint_c_x^ix 1000000000 -\def\XINT_tmpa #1{% - \expandafter\edef\csname XINT_sdth_#1\endcsname - {\ifcase #1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or - 8\or 9\or A\or B\or C\or D\or E\or F\fi}}% -\xintApplyInline\XINT_tmpa - {{0}{1}{2}{3}{4}{5}{6}{7}{8}{9}{10}{11}{12}{13}{14}{15}}% -\def\XINT_tmpa #1{% - \expandafter\edef\csname XINT_sdtb_#1\endcsname - {\ifcase #1 - 0000\or 0001\or 0010\or 0011\or 0100\or 0101\or 0110\or 0111\or - 1000\or 1001\or 1010\or 1011\or 1100\or 1101\or 1110\or 1111\fi}}% -\xintApplyInline\XINT_tmpa - {{0}{1}{2}{3}{4}{5}{6}{7}{8}{9}{10}{11}{12}{13}{14}{15}}% -\let\XINT_tmpa\relax -\expandafter\def\csname XINT_sbtd_0000\endcsname {0}% -\expandafter\def\csname XINT_sbtd_0001\endcsname {1}% -\expandafter\def\csname XINT_sbtd_0010\endcsname {2}% -\expandafter\def\csname XINT_sbtd_0011\endcsname {3}% -\expandafter\def\csname XINT_sbtd_0100\endcsname {4}% -\expandafter\def\csname XINT_sbtd_0101\endcsname {5}% -\expandafter\def\csname XINT_sbtd_0110\endcsname {6}% -\expandafter\def\csname XINT_sbtd_0111\endcsname {7}% -\expandafter\def\csname XINT_sbtd_1000\endcsname {8}% -\expandafter\def\csname XINT_sbtd_1001\endcsname {9}% -\expandafter\def\csname XINT_sbtd_1010\endcsname {10}% -\expandafter\def\csname XINT_sbtd_1011\endcsname {11}% -\expandafter\def\csname XINT_sbtd_1100\endcsname {12}% -\expandafter\def\csname XINT_sbtd_1101\endcsname {13}% -\expandafter\def\csname XINT_sbtd_1110\endcsname {14}% -\expandafter\def\csname XINT_sbtd_1111\endcsname {15}% -\expandafter\let\csname XINT_sbth_0000\expandafter\endcsname - \csname XINT_sbtd_0000\endcsname -\expandafter\let\csname XINT_sbth_0001\expandafter\endcsname - \csname XINT_sbtd_0001\endcsname -\expandafter\let\csname XINT_sbth_0010\expandafter\endcsname - \csname XINT_sbtd_0010\endcsname -\expandafter\let\csname XINT_sbth_0011\expandafter\endcsname - \csname XINT_sbtd_0011\endcsname -\expandafter\let\csname XINT_sbth_0100\expandafter\endcsname - \csname XINT_sbtd_0100\endcsname -\expandafter\let\csname XINT_sbth_0101\expandafter\endcsname - \csname XINT_sbtd_0101\endcsname -\expandafter\let\csname XINT_sbth_0110\expandafter\endcsname - \csname XINT_sbtd_0110\endcsname -\expandafter\let\csname XINT_sbth_0111\expandafter\endcsname - \csname XINT_sbtd_0111\endcsname -\expandafter\let\csname XINT_sbth_1000\expandafter\endcsname - \csname XINT_sbtd_1000\endcsname -\expandafter\let\csname XINT_sbth_1001\expandafter\endcsname - \csname XINT_sbtd_1001\endcsname -\expandafter\def\csname XINT_sbth_1010\endcsname {A}% -\expandafter\def\csname XINT_sbth_1011\endcsname {B}% -\expandafter\def\csname XINT_sbth_1100\endcsname {C}% -\expandafter\def\csname XINT_sbth_1101\endcsname {D}% -\expandafter\def\csname XINT_sbth_1110\endcsname {E}% -\expandafter\def\csname XINT_sbth_1111\endcsname {F}% -\expandafter\def\csname XINT_shtb_0\endcsname {0000}% -\expandafter\def\csname XINT_shtb_1\endcsname {0001}% -\expandafter\def\csname XINT_shtb_2\endcsname {0010}% -\expandafter\def\csname XINT_shtb_3\endcsname {0011}% -\expandafter\def\csname XINT_shtb_4\endcsname {0100}% -\expandafter\def\csname XINT_shtb_5\endcsname {0101}% -\expandafter\def\csname XINT_shtb_6\endcsname {0110}% -\expandafter\def\csname XINT_shtb_7\endcsname {0111}% -\expandafter\def\csname XINT_shtb_8\endcsname {1000}% -\expandafter\def\csname XINT_shtb_9\endcsname {1001}% -\def\XINT_shtb_A {1010}% -\def\XINT_shtb_B {1011}% -\def\XINT_shtb_C {1100}% -\def\XINT_shtb_D {1101}% -\def\XINT_shtb_E {1110}% -\def\XINT_shtb_F {1111}% -\def\XINT_shtb_G {}% -\def\XINT_smallhex #1% -{% - \expandafter\XINT_smallhex_a\expandafter - {\the\numexpr (#1+\xint_c_viii)/\xint_c_xvi-\xint_c_i}{#1}% -}% -\def\XINT_smallhex_a #1#2% -{% - \csname XINT_sdth_#1\expandafter\expandafter\expandafter\endcsname - \csname XINT_sdth_\the\numexpr #2-\xint_c_xvi*#1\endcsname -}% -\def\XINT_smallbin #1% -{% - \expandafter\XINT_smallbin_a\expandafter - {\the\numexpr (#1+\xint_c_viii)/\xint_c_xvi-\xint_c_i}{#1}% -}% -\def\XINT_smallbin_a #1#2% -{% - \csname XINT_sdtb_#1\expandafter\expandafter\expandafter\endcsname - \csname XINT_sdtb_\the\numexpr #2-\xint_c_xvi*#1\endcsname -}% -% \end{macrocode} -% \subsection{\csh{xintDecToHex}, \csh{xintDecToBin}} -% \lverb!v1.08! -% \begin{macrocode} -\def\xintDecToHex {\romannumeral0\xintdectohex }% -\def\xintdectohex #1% - {\expandafter\XINT_dth_checkin\romannumeral-`0#1\W\W\W\W \T}% -\def\XINT_dth_checkin #1% -{% - \xint_UDsignfork - #1\XINT_dth_N - -{\XINT_dth_P #1}% - \krof -}% -\def\XINT_dth_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_dth_P }% -\def\XINT_dth_P {\expandafter\XINT_dth_III\romannumeral-`0\XINT_dtbh_I {0.}}% -\def\xintDecToBin {\romannumeral0\xintdectobin }% -\def\xintdectobin #1% - {\expandafter\XINT_dtb_checkin\romannumeral-`0#1\W\W\W\W \T }% -\def\XINT_dtb_checkin #1% -{% - \xint_UDsignfork - #1\XINT_dtb_N - -{\XINT_dtb_P #1}% - \krof -}% -\def\XINT_dtb_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_dtb_P }% -\def\XINT_dtb_P {\expandafter\XINT_dtb_III\romannumeral-`0\XINT_dtbh_I {0.}}% -\def\XINT_dtbh_I #1#2#3#4#5% -{% - \xint_gob_til_W #5\XINT_dtbh_II_a\W\XINT_dtbh_I_a {}{#2#3#4#5}#1\Z.% -}% -\def\XINT_dtbh_II_a\W\XINT_dtbh_I_a #1#2{\XINT_dtbh_II_b #2}% -\def\XINT_dtbh_II_b #1#2#3#4% -{% - \xint_gob_til_W - #1\XINT_dtbh_II_c - #2\XINT_dtbh_II_ci - #3\XINT_dtbh_II_cii - \W\XINT_dtbh_II_ciii #1#2#3#4% -}% -\def\XINT_dtbh_II_c \W\XINT_dtbh_II_ci - \W\XINT_dtbh_II_cii - \W\XINT_dtbh_II_ciii \W\W\W\W {{}}% -\def\XINT_dtbh_II_ci #1\XINT_dtbh_II_ciii #2\W\W\W - {\XINT_dtbh_II_d {}{#2}{0}}% -\def\XINT_dtbh_II_cii\W\XINT_dtbh_II_ciii #1#2\W\W - {\XINT_dtbh_II_d {}{#1#2}{00}}% -\def\XINT_dtbh_II_ciii #1#2#3\W - {\XINT_dtbh_II_d {}{#1#2#3}{000}}% -\def\XINT_dtbh_I_a #1#2#3.% -{% - \xint_gob_til_Z #3\XINT_dtbh_I_z\Z - \expandafter\XINT_dtbh_I_b\the\numexpr #2+#30000.{#1}% -}% -\def\XINT_dtbh_I_b #1.% -{% - \expandafter\XINT_dtbh_I_c\the\numexpr - (#1+\xint_c_ii^xv)/\xint_c_ii^xvi-\xint_c_i.#1.% -}% -\def\XINT_dtbh_I_c #1.#2.% -{% - \expandafter\XINT_dtbh_I_d\expandafter - {\the\numexpr #2-\xint_c_ii^xvi*#1}{#1}% -}% -\def\XINT_dtbh_I_d #1#2#3{\XINT_dtbh_I_a {#3#1.}{#2}}% -\def\XINT_dtbh_I_z\Z\expandafter\XINT_dtbh_I_b\the\numexpr #1+#2.% -{% - \ifnum #1=\xint_c_ \expandafter\XINT_dtbh_I_end_zb\fi - \XINT_dtbh_I_end_za {#1}% -}% -\def\XINT_dtbh_I_end_za #1#2{\XINT_dtbh_I {#2#1.}}% -\def\XINT_dtbh_I_end_zb\XINT_dtbh_I_end_za #1#2{\XINT_dtbh_I {#2}}% -\def\XINT_dtbh_II_d #1#2#3#4.% -{% - \xint_gob_til_Z #4\XINT_dtbh_II_z\Z - \expandafter\XINT_dtbh_II_e\the\numexpr #2+#4#3.{#1}{#3}% -}% -\def\XINT_dtbh_II_e #1.% -{% - \expandafter\XINT_dtbh_II_f\the\numexpr - (#1+\xint_c_ii^xv)/\xint_c_ii^xvi-\xint_c_i.#1.% -}% -\def\XINT_dtbh_II_f #1.#2.% -{% - \expandafter\XINT_dtbh_II_g\expandafter - {\the\numexpr #2-\xint_c_ii^xvi*#1}{#1}% -}% -\def\XINT_dtbh_II_g #1#2#3{\XINT_dtbh_II_d {#3#1.}{#2}}% -\def\XINT_dtbh_II_z\Z\expandafter\XINT_dtbh_II_e\the\numexpr #1+#2.% -{% - \ifnum #1=\xint_c_ \expandafter\XINT_dtbh_II_end_zb\fi - \XINT_dtbh_II_end_za {#1}% -}% -\def\XINT_dtbh_II_end_za #1#2#3{{}#2#1.\Z.}% -\def\XINT_dtbh_II_end_zb\XINT_dtbh_II_end_za #1#2#3{{}#2\Z.}% -\def\XINT_dth_III #1#2.% -{% - \xint_gob_til_Z #2\XINT_dth_end\Z - \expandafter\XINT_dth_III\expandafter - {\romannumeral-`0\XINT_dth_small #2.#1}% -}% -\def\XINT_dth_small #1.% -{% - \expandafter\XINT_smallhex\expandafter - {\the\numexpr (#1+\xint_c_ii^vii)/\xint_c_ii^viii-\xint_c_i\expandafter}% - \romannumeral-`0\expandafter\XINT_smallhex\expandafter - {\the\numexpr - #1-((#1+\xint_c_ii^vii)/\xint_c_ii^viii-\xint_c_i)*\xint_c_ii^viii}% -}% -\def\XINT_dth_end\Z\expandafter\XINT_dth_III\expandafter #1#2\T -{% - \XINT_dth_end_b #1% -}% -\def\XINT_dth_end_b #1.{\XINT_dth_end_c }% -\def\XINT_dth_end_c #1{\xint_gob_til_zero #1\XINT_dth_end_d 0\space #1}% -\def\XINT_dth_end_d 0\space 0#1% -{% - \xint_gob_til_zero #1\XINT_dth_end_e 0\space #1% -}% -\def\XINT_dth_end_e 0\space 0#1% -{% - \xint_gob_til_zero #1\XINT_dth_end_f 0\space #1% -}% -\def\XINT_dth_end_f 0\space 0{ }% -\def\XINT_dtb_III #1#2.% -{% - \xint_gob_til_Z #2\XINT_dtb_end\Z - \expandafter\XINT_dtb_III\expandafter - {\romannumeral-`0\XINT_dtb_small #2.#1}% -}% -\def\XINT_dtb_small #1.% -{% - \expandafter\XINT_smallbin\expandafter - {\the\numexpr (#1+\xint_c_ii^vii)/\xint_c_ii^viii-\xint_c_i\expandafter}% - \romannumeral-`0\expandafter\XINT_smallbin\expandafter - {\the\numexpr - #1-((#1+\xint_c_ii^vii)/\xint_c_ii^viii-\xint_c_i)*\xint_c_ii^viii}% -}% -\def\XINT_dtb_end\Z\expandafter\XINT_dtb_III\expandafter #1#2\T -{% - \XINT_dtb_end_b #1% -}% -\def\XINT_dtb_end_b #1.{\XINT_dtb_end_c }% -\def\XINT_dtb_end_c #1#2#3#4#5#6#7#8% -{% - \expandafter\XINT_dtb_end_d\the\numexpr #1#2#3#4#5#6#7#8\relax -}% -\edef\XINT_dtb_end_d #1#2#3#4#5#6#7#8#9% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8#9\relax -}% -% \end{macrocode} -% \subsection{\csh{xintHexToDec}} -% \lverb!v1.08! -% \begin{macrocode} -\def\xintHexToDec {\romannumeral0\xinthextodec }% -\def\xinthextodec #1% - {\expandafter\XINT_htd_checkin\romannumeral-`0#1\W\W\W\W \T }% -\def\XINT_htd_checkin #1% -{% - \xint_UDsignfork - #1\XINT_htd_neg - -{\XINT_htd_I {0000}#1}% - \krof -}% -\def\XINT_htd_neg {\expandafter\xint_minus_thenstop - \romannumeral0\XINT_htd_I {0000}}% -\def\XINT_htd_I #1#2#3#4#5% -{% - \xint_gob_til_W #5\XINT_htd_II_a\W - \XINT_htd_I_a {}{"#2#3#4#5}#1\Z\Z\Z\Z -}% -\def\XINT_htd_II_a \W\XINT_htd_I_a #1#2{\XINT_htd_II_b #2}% -\def\XINT_htd_II_b "#1#2#3#4% -{% - \xint_gob_til_W - #1\XINT_htd_II_c - #2\XINT_htd_II_ci - #3\XINT_htd_II_cii - \W\XINT_htd_II_ciii #1#2#3#4% -}% -\def\XINT_htd_II_c \W\XINT_htd_II_ci - \W\XINT_htd_II_cii - \W\XINT_htd_II_ciii \W\W\W\W #1\Z\Z\Z\Z\T -{% - \expandafter\xint_cleanupzeros_andstop - \romannumeral0\XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\def\XINT_htd_II_ci #1\XINT_htd_II_ciii - #2\W\W\W {\XINT_htd_II_d {}{"#2}{\xint_c_xvi}}% -\def\XINT_htd_II_cii\W\XINT_htd_II_ciii - #1#2\W\W {\XINT_htd_II_d {}{"#1#2}{\xint_c_ii^viii}}% -\def\XINT_htd_II_ciii #1#2#3\W {\XINT_htd_II_d {}{"#1#2#3}{\xint_c_ii^xii}}% -\def\XINT_htd_I_a #1#2#3#4#5#6% -{% - \xint_gob_til_Z #3\XINT_htd_I_end_a\Z - \expandafter\XINT_htd_I_b\the\numexpr - #2+\xint_c_ii^xvi*#6#5#4#3+\xint_c_x^ix\relax {#1}% -}% -\def\XINT_htd_I_b 1#1#2#3#4#5#6#7#8#9{\XINT_htd_I_c {#1#2#3#4#5}{#9#8#7#6}}% -\def\XINT_htd_I_c #1#2#3{\XINT_htd_I_a {#3#2}{#1}}% -\def\XINT_htd_I_end_a\Z\expandafter\XINT_htd_I_b\the\numexpr #1+#2\relax -{% - \expandafter\XINT_htd_I_end_b\the\numexpr \xint_c_x^v+#1\relax -}% -\def\XINT_htd_I_end_b 1#1#2#3#4#5% -{% - \xint_gob_til_zero #1\XINT_htd_I_end_bz0% - \XINT_htd_I_end_c #1#2#3#4#5% -}% -\def\XINT_htd_I_end_c #1#2#3#4#5#6{\XINT_htd_I {#6#5#4#3#2#1000}}% -\def\XINT_htd_I_end_bz0\XINT_htd_I_end_c 0#1#2#3#4% -{% - \xint_gob_til_zeros_iv #1#2#3#4\XINT_htd_I_end_bzz 0000% - \XINT_htd_I_end_D {#4#3#2#1}% -}% -\def\XINT_htd_I_end_D #1#2{\XINT_htd_I {#2#1}}% -\def\XINT_htd_I_end_bzz 0000\XINT_htd_I_end_D #1{\XINT_htd_I }% -\def\XINT_htd_II_d #1#2#3#4#5#6#7% -{% - \xint_gob_til_Z #4\XINT_htd_II_end_a\Z - \expandafter\XINT_htd_II_e\the\numexpr - #2+#3*#7#6#5#4+\xint_c_x^viii\relax {#1}{#3}% -}% -\def\XINT_htd_II_e 1#1#2#3#4#5#6#7#8{\XINT_htd_II_f {#1#2#3#4}{#5#6#7#8}}% -\def\XINT_htd_II_f #1#2#3{\XINT_htd_II_d {#2#3}{#1}}% -\def\XINT_htd_II_end_a\Z\expandafter\XINT_htd_II_e - \the\numexpr #1+#2\relax #3#4\T -{% - \XINT_htd_II_end_b #1#3% -}% -\edef\XINT_htd_II_end_b #1#2#3#4#5#6#7#8% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8\relax -}% -% \end{macrocode} -% \subsection{\csh{xintBinToDec}} -% \lverb!v1.08! -% \begin{macrocode} -\def\xintBinToDec {\romannumeral0\xintbintodec }% -\def\xintbintodec #1{\expandafter\XINT_btd_checkin - \romannumeral-`0#1\W\W\W\W\W\W\W\W \T }% -\def\XINT_btd_checkin #1% -{% - \xint_UDsignfork - #1\XINT_btd_neg - -{\XINT_btd_I {000000}#1}% - \krof -}% -\def\XINT_btd_neg {\expandafter\xint_minus_thenstop - \romannumeral0\XINT_btd_I {000000}}% -\def\XINT_btd_I #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #9\XINT_btd_II_a {#2#3#4#5#6#7#8#9}\W - \XINT_btd_I_a {}{\csname XINT_sbtd_#2#3#4#5\endcsname*\xint_c_xvi+% - \csname XINT_sbtd_#6#7#8#9\endcsname}% - #1\Z\Z\Z\Z\Z\Z -}% -\def\XINT_btd_II_a #1\W\XINT_btd_I_a #2#3{\XINT_btd_II_b #1}% -\def\XINT_btd_II_b #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_W - #1\XINT_btd_II_c - #2\XINT_btd_II_ci - #3\XINT_btd_II_cii - #4\XINT_btd_II_ciii - #5\XINT_btd_II_civ - #6\XINT_btd_II_cv - #7\XINT_btd_II_cvi - \W\XINT_btd_II_cvii #1#2#3#4#5#6#7#8% -}% -\def\XINT_btd_II_c #1\XINT_btd_II_cvii \W\W\W\W\W\W\W\W #2\Z\Z\Z\Z\Z\Z\T -{% - \expandafter\XINT_btd_II_c_end - \romannumeral0\XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\edef\XINT_btd_II_c_end #1#2#3#4#5#6% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6\relax -}% -\def\XINT_btd_II_ci #1\XINT_btd_II_cvii #2\W\W\W\W\W\W\W - {\XINT_btd_II_d {}{#2}{\xint_c_ii }}% -\def\XINT_btd_II_cii #1\XINT_btd_II_cvii #2\W\W\W\W\W\W - {\XINT_btd_II_d {}{\csname XINT_sbtd_00#2\endcsname }{\xint_c_iv }}% -\def\XINT_btd_II_ciii #1\XINT_btd_II_cvii #2\W\W\W\W\W - {\XINT_btd_II_d {}{\csname XINT_sbtd_0#2\endcsname }{\xint_c_viii }}% -\def\XINT_btd_II_civ #1\XINT_btd_II_cvii #2\W\W\W\W - {\XINT_btd_II_d {}{\csname XINT_sbtd_#2\endcsname}{\xint_c_xvi }}% -\def\XINT_btd_II_cv #1\XINT_btd_II_cvii #2#3#4#5#6\W\W\W -{% - \XINT_btd_II_d {}{\csname XINT_sbtd_#2#3#4#5\endcsname*\xint_c_ii+% - #6}{\xint_c_ii^v }% -}% -\def\XINT_btd_II_cvi #1\XINT_btd_II_cvii #2#3#4#5#6#7\W\W -{% - \XINT_btd_II_d {}{\csname XINT_sbtd_#2#3#4#5\endcsname*\xint_c_iv+% - \csname XINT_sbtd_00#6#7\endcsname}{\xint_c_ii^vi }% -}% -\def\XINT_btd_II_cvii #1#2#3#4#5#6#7\W -{% - \XINT_btd_II_d {}{\csname XINT_sbtd_#1#2#3#4\endcsname*\xint_c_viii+% - \csname XINT_sbtd_0#5#6#7\endcsname}{\xint_c_ii^vii }% -}% -\def\XINT_btd_II_d #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_Z #4\XINT_btd_II_end_a\Z - \expandafter\XINT_btd_II_e\the\numexpr - #2+(\xint_c_x^ix+#3*#9#8#7#6#5#4)\relax {#1}{#3}% -}% -\def\XINT_btd_II_e 1#1#2#3#4#5#6#7#8#9{\XINT_btd_II_f {#1#2#3}{#4#5#6#7#8#9}}% -\def\XINT_btd_II_f #1#2#3{\XINT_btd_II_d {#2#3}{#1}}% -\def\XINT_btd_II_end_a\Z\expandafter\XINT_btd_II_e - \the\numexpr #1+(#2\relax #3#4\T -{% - \XINT_btd_II_end_b #1#3% -}% -\edef\XINT_btd_II_end_b #1#2#3#4#5#6#7#8#9% -{% - \noexpand\expandafter\space\noexpand\the\numexpr #1#2#3#4#5#6#7#8#9\relax -}% -\def\XINT_btd_I_a #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_Z #3\XINT_btd_I_end_a\Z - \expandafter\XINT_btd_I_b\the\numexpr - #2+\xint_c_ii^viii*#8#7#6#5#4#3+\xint_c_x^ix\relax {#1}% -}% -\def\XINT_btd_I_b 1#1#2#3#4#5#6#7#8#9{\XINT_btd_I_c {#1#2#3}{#9#8#7#6#5#4}}% -\def\XINT_btd_I_c #1#2#3{\XINT_btd_I_a {#3#2}{#1}}% -\def\XINT_btd_I_end_a\Z\expandafter\XINT_btd_I_b - \the\numexpr #1+\xint_c_ii^viii #2\relax -{% - \expandafter\XINT_btd_I_end_b\the\numexpr 1000+#1\relax -}% -\def\XINT_btd_I_end_b 1#1#2#3% -{% - \xint_gob_til_zeros_iii #1#2#3\XINT_btd_I_end_bz 000% - \XINT_btd_I_end_c #1#2#3% -}% -\def\XINT_btd_I_end_c #1#2#3#4{\XINT_btd_I {#4#3#2#1000}}% -\def\XINT_btd_I_end_bz 000\XINT_btd_I_end_c 000{\XINT_btd_I }% -% \end{macrocode} -% \subsection{\csh{xintBinToHex}} -% \lverb!v1.08! -% \begin{macrocode} -\def\xintBinToHex {\romannumeral0\xintbintohex }% -\def\xintbintohex #1% -{% - \expandafter\XINT_bth_checkin - \romannumeral0\expandafter\XINT_num_loop - \romannumeral-`0#1\xint_relax\xint_relax - \xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z - \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W -}% -\def\XINT_bth_checkin #1% -{% - \xint_UDsignfork - #1\XINT_bth_N - -{\XINT_bth_P #1}% - \krof -}% -\def\XINT_bth_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_bth_P }% -\def\XINT_bth_P {\expandafter\XINT_bth_I\expandafter{\expandafter}% - \romannumeral0\XINT_OQ {}}% -\def\XINT_bth_I #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #9\XINT_bth_end_a\W - \expandafter\expandafter\expandafter - \XINT_bth_I - \expandafter\expandafter\expandafter - {\csname XINT_sbth_#9#8#7#6\expandafter\expandafter\expandafter\endcsname - \csname XINT_sbth_#5#4#3#2\endcsname #1}% -}% -\def\XINT_bth_end_a\W \expandafter\expandafter\expandafter - \XINT_bth_I \expandafter\expandafter\expandafter #1% -{% - \XINT_bth_end_b #1% -}% -\def\XINT_bth_end_b #1\endcsname #2\endcsname #3% -{% - \xint_gob_til_zero #3\XINT_bth_end_z 0\space #3% -}% -\def\XINT_bth_end_z0\space 0{ }% -% \end{macrocode} -% \subsection{\csh{xintHexToBin}} -% \lverb!v1.08! -% \begin{macrocode} -\def\xintHexToBin {\romannumeral0\xinthextobin }% -\def\xinthextobin #1% -{% - \expandafter\XINT_htb_checkin\romannumeral-`0#1GGGGGGGG\T -}% -\def\XINT_htb_checkin #1% -{% - \xint_UDsignfork - #1\XINT_htb_N - -{\XINT_htb_P #1}% - \krof -}% -\def\XINT_htb_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_htb_P }% -\def\XINT_htb_P {\XINT_htb_I_a {}}% -\def\XINT_htb_I_a #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_G #9\XINT_htb_II_a G% - \expandafter\expandafter\expandafter - \XINT_htb_I_b - \expandafter\expandafter\expandafter - {\csname XINT_shtb_#2\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#3\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#4\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#5\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#6\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#7\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#8\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#9\endcsname }{#1}% -}% -\def\XINT_htb_I_b #1#2{\XINT_htb_I_a {#2#1}}% -\def\XINT_htb_II_a G\expandafter\expandafter\expandafter\XINT_htb_I_b -{% - \expandafter\expandafter\expandafter \XINT_htb_II_b -}% -\def\XINT_htb_II_b #1#2#3\T -{% - \XINT_num_loop #2#1% - \xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z -}% -% \end{macrocode} -% \subsection{\csh{xintCHexToBin}} -% \lverb!v1.08! -% \begin{macrocode} -\def\xintCHexToBin {\romannumeral0\xintchextobin }% -\def\xintchextobin #1% -{% - \expandafter\XINT_chtb_checkin\romannumeral-`0#1% - \R\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W\W -}% -\def\XINT_chtb_checkin #1% -{% - \xint_UDsignfork - #1\XINT_chtb_N - -{\XINT_chtb_P #1}% - \krof -}% -\def\XINT_chtb_N {\expandafter\xint_minus_thenstop\romannumeral0\XINT_chtb_P }% -\def\XINT_chtb_P {\expandafter\XINT_chtb_I\expandafter{\expandafter}% - \romannumeral0\XINT_OQ {}}% -\def\XINT_chtb_I #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_W #9\XINT_chtb_end_a\W - \expandafter\expandafter\expandafter - \XINT_chtb_I - \expandafter\expandafter\expandafter - {\csname XINT_shtb_#9\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#8\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#7\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#6\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#5\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#4\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#3\expandafter\expandafter\expandafter\endcsname - \csname XINT_shtb_#2\endcsname - #1}% -}% -\def\XINT_chtb_end_a\W\expandafter\expandafter\expandafter - \XINT_chtb_I\expandafter\expandafter\expandafter #1% -{% - \XINT_chtb_end_b #1% - \xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z -}% -\def\XINT_chtb_end_b #1\W#2\W#3\W#4\W#5\W#6\W#7\W#8\W\endcsname -{% - \XINT_num_loop -}% -\XINT_restorecatcodes_endinput% -% \end{macrocode} -%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 -%\let</xintbinhex>\relax -%\def<*xintgcd>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 } -%</xintbinhex> -%<*xintgcd> -% -% \StoreCodelineNo {xintbinhex} -% -% \section{Package \xintgcdnameimp implementation} -% \label{sec:gcdimp} -% -% The commenting is currently (\docdate) very sparse. Release |1.09h| has -% modified a bit the |\xintTypesetEuclideAlgorithm| and -% |\xintTypesetBezoutAlgorithm| layout with respect to line indentation in -% particular. And they use the \xinttoolsnameimp |\xintloop| rather than the -% Plain \TeX{} or \LaTeX{}'s |\loop|. -% -% \localtableofcontents -% \subsection{Catcodes, \protect\eTeX{} and reload detection} -% -% The code for reload detection is copied from \textsc{Heiko -% Oberdiek}'s packages, and adapted here to check for previous -% loading of the master \xintname package. -% -% The method for catcodes is slightly different, but still -% directly inspired by these packages. -% -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \def\space { }% - \let\z\endgroup - \expandafter\let\expandafter\x\csname ver@xintgcd.sty\endcsname - \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname - \expandafter - \ifx\csname PackageInfo\endcsname\relax - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \else - \def\y#1#2{\PackageInfo{#1}{#2}}% - \fi - \expandafter - \ifx\csname numexpr\endcsname\relax - \y{xintgcd}{\numexpr not available, aborting input}% - \aftergroup\endinput - \else - \ifx\x\relax % plain-TeX, first loading of xintgcd.sty - \ifx\w\relax % but xint.sty not yet loaded. - \y{xintgcd}{now issuing \string\input\space xint.sty}% - \def\z{\endgroup\input xint.sty\relax}% - \fi - \else - \def\empty {}% - \ifx\x\empty % LaTeX, first loading, - % variable is initialized, but \ProvidesPackage not yet seen - \ifx\w\relax % xint.sty not yet loaded. - \y{xintgcd}{now issuing \string\RequirePackage{xint}}% - \def\z{\endgroup\RequirePackage{xint}}% - \fi - \else - \y{xintgcd}{I was already loaded, aborting input}% - \aftergroup\endinput - \fi - \fi - \fi -\z% -% \end{macrocode} -% \subsection{Confirmation of \xintnameimp loading} -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \ifdefined\PackageInfo - \def\y#1#2{\PackageInfo{#1}{#2}}% - \else - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \fi - \def\empty {}% - \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname - \ifx\w\relax % Plain TeX, user gave a file name at the prompt - \y{xintgcd}{Loading of package xint failed, aborting input}% - \aftergroup\endinput - \fi - \ifx\w\empty % LaTeX, user gave a file name at the prompt - \y{xintgcd}{Loading of package xint failed, aborting input}% - \aftergroup\endinput - \fi -\endgroup% -% \end{macrocode} -% \subsection{Catcodes} -% \begin{macrocode} -\XINTsetupcatcodes% -% \end{macrocode} -% \subsection{Package identification} -% \begin{macrocode} -\XINT_providespackage -\ProvidesPackage{xintgcd}% - [2014/02/05 v1.09ka Euclide algorithm with xint package (jfB)]% -% \end{macrocode} -% \subsection{\csh{xintGCD}} -% The macros of |1.09a| benefits from the |\xintnum| which has been inserted -% inside |\xintiabs| in \xintname; -% this is a little overhead but is more convenient for the -% user and also makes it easier to use into |\xintexpr|essions. -% \begin{macrocode} -\def\xintGCD {\romannumeral0\xintgcd }% -\def\xintgcd #1% -{% - \expandafter\XINT_gcd\expandafter{\romannumeral0\xintiabs {#1}}% -}% -\def\XINT_gcd #1#2% -{% - \expandafter\XINT_gcd_fork\romannumeral0\xintiabs {#2}\Z #1\Z -}% -% \end{macrocode} -% \lverb|& -% Ici #3#4=A, #1#2=B| -% \begin{macrocode} -\def\XINT_gcd_fork #1#2\Z #3#4\Z -{% - \xint_UDzerofork - #1\XINT_gcd_BisZero - #3\XINT_gcd_AisZero - 0\XINT_gcd_loop - \krof - {#1#2}{#3#4}% -}% -\def\XINT_gcd_AisZero #1#2{ #1}% -\def\XINT_gcd_BisZero #1#2{ #2}% -\def\XINT_gcd_CheckRem #1#2\Z -{% - \xint_gob_til_zero #1\xint_gcd_end0\XINT_gcd_loop {#1#2}% -}% -\def\xint_gcd_end0\XINT_gcd_loop #1#2{ #2}% -% \end{macrocode} -% \lverb|#1=B, #2=A| -% \begin{macrocode} -\def\XINT_gcd_loop #1#2% -{% - \expandafter\expandafter\expandafter - \XINT_gcd_CheckRem - \expandafter\xint_secondoftwo - \romannumeral0\XINT_div_prepare {#1}{#2}\Z - {#1}% -}% -% \end{macrocode} -% \subsection{\csh{xintGCDof}} -% \lverb|New with 1.09a. I also tried an optimization (not working two by two) -% which I thought was clever but -% it seemed to be less efficient ...| -% \begin{macrocode} -\def\xintGCDof {\romannumeral0\xintgcdof }% -\def\xintgcdof #1{\expandafter\XINT_gcdof_a\romannumeral-`0#1\relax }% -\def\XINT_gcdof_a #1{\expandafter\XINT_gcdof_b\romannumeral-`0#1\Z }% -\def\XINT_gcdof_b #1\Z #2{\expandafter\XINT_gcdof_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_gcdof_c #1{\xint_gob_til_relax #1\XINT_gcdof_e\relax\XINT_gcdof_d #1}% -\def\XINT_gcdof_d #1\Z {\expandafter\XINT_gcdof_b\romannumeral0\xintgcd {#1}}% -\def\XINT_gcdof_e #1\Z #2\Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintLCM}} -% \lverb|New with 1.09a. Inadvertent use of \xintiQuo which was promoted at the -% same time to add the \xintnum overhead. So with 1.09f \xintiiQuo without the -% overhead.| -% \begin{macrocode} -\def\xintLCM {\romannumeral0\xintlcm}% -\def\xintlcm #1% -{% - \expandafter\XINT_lcm\expandafter{\romannumeral0\xintiabs {#1}}% -}% -\def\XINT_lcm #1#2% -{% - \expandafter\XINT_lcm_fork\romannumeral0\xintiabs {#2}\Z #1\Z -}% -\def\XINT_lcm_fork #1#2\Z #3#4\Z -{% - \xint_UDzerofork - #1\XINT_lcm_BisZero - #3\XINT_lcm_AisZero - 0\expandafter - \krof - \XINT_lcm_notzero\expandafter{\romannumeral0\XINT_gcd_loop {#1#2}{#3#4}}% - {#1#2}{#3#4}% -}% -\def\XINT_lcm_AisZero #1#2#3#4#5{ 0}% -\def\XINT_lcm_BisZero #1#2#3#4#5{ 0}% -\def\XINT_lcm_notzero #1#2#3{\xintiimul {#2}{\xintiiQuo{#3}{#1}}}% -% \end{macrocode} -% \subsection{\csh{xintLCMof}} -% \lverb|New with 1.09a| -% \begin{macrocode} -\def\xintLCMof {\romannumeral0\xintlcmof }% -\def\xintlcmof #1{\expandafter\XINT_lcmof_a\romannumeral-`0#1\relax }% -\def\XINT_lcmof_a #1{\expandafter\XINT_lcmof_b\romannumeral-`0#1\Z }% -\def\XINT_lcmof_b #1\Z #2{\expandafter\XINT_lcmof_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_lcmof_c #1{\xint_gob_til_relax #1\XINT_lcmof_e\relax\XINT_lcmof_d #1}% -\def\XINT_lcmof_d #1\Z {\expandafter\XINT_lcmof_b\romannumeral0\xintlcm {#1}}% -\def\XINT_lcmof_e #1\Z #2\Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintBezout}} -% \lverb|1.09a inserts use of \xintnum| -% \begin{macrocode} -\def\xintBezout {\romannumeral0\xintbezout }% -\def\xintbezout #1% -{% - \expandafter\xint_bezout\expandafter {\romannumeral0\xintnum{#1}}% -}% -\def\xint_bezout #1#2% -{% - \expandafter\XINT_bezout_fork \romannumeral0\xintnum{#2}\Z #1\Z -}% -% \end{macrocode} -% \lverb|#3#4 = A, #1#2=B| -% \begin{macrocode} -\def\XINT_bezout_fork #1#2\Z #3#4\Z -{% - \xint_UDzerosfork - #1#3\XINT_bezout_botharezero - #10\XINT_bezout_secondiszero - #30\XINT_bezout_firstiszero - 00{\xint_UDsignsfork - #1#3\XINT_bezout_minusminus % A < 0, B < 0 - #1-\XINT_bezout_minusplus % A > 0, B < 0 - #3-\XINT_bezout_plusminus % A < 0, B > 0 - --\XINT_bezout_plusplus % A > 0, B > 0 - \krof }% - \krof - {#2}{#4}#1#3{#3#4}{#1#2}% #1#2=B, #3#4=A -}% -\edef\XINT_bezout_botharezero #1#2#3#4#5#6% -{% - \noexpand\xintError:NoBezoutForZeros - \space {0}{0}{0}{0}{0}% -}% -% \end{macrocode} -% \lverb|& -% attention première entrée doit être ici (-1)^n donc 1$\ -% #4#2 = 0 = A, B = #3#1| -% \begin{macrocode} -\def\XINT_bezout_firstiszero #1#2#3#4#5#6% -{% - \xint_UDsignfork - #3{ {0}{#3#1}{0}{1}{#1}}% - -{ {0}{#3#1}{0}{-1}{#1}}% - \krof -}% -% \end{macrocode} -% \lverb|#4#2 = A, B = #3#1 = 0| -% \begin{macrocode} -\def\XINT_bezout_secondiszero #1#2#3#4#5#6% -{% - \xint_UDsignfork - #4{ {#4#2}{0}{-1}{0}{#2}}% - -{ {#4#2}{0}{1}{0}{#2}}% - \krof -}% -% \end{macrocode} -% \lverb|#4#2= A < 0, #3#1 = B < 0| -% \begin{macrocode} -\def\XINT_bezout_minusminus #1#2#3#4% -{% - \expandafter\XINT_bezout_mm_post - \romannumeral0\XINT_bezout_loop_a 1{#1}{#2}1001% -}% -\def\XINT_bezout_mm_post #1#2% -{% - \expandafter\XINT_bezout_mm_postb\expandafter - {\romannumeral0\xintiiopp{#2}}{\romannumeral0\xintiiopp{#1}}% -}% -\def\XINT_bezout_mm_postb #1#2% -{% - \expandafter\XINT_bezout_mm_postc\expandafter {#2}{#1}% -}% -\edef\XINT_bezout_mm_postc #1#2#3#4#5% -{% - \space {#4}{#5}{#1}{#2}{#3}% -}% -% \end{macrocode} -% \lverb|minusplus #4#2= A > 0, B < 0| -% \begin{macrocode} -\def\XINT_bezout_minusplus #1#2#3#4% -{% - \expandafter\XINT_bezout_mp_post - \romannumeral0\XINT_bezout_loop_a 1{#1}{#4#2}1001% -}% -\def\XINT_bezout_mp_post #1#2% -{% - \expandafter\XINT_bezout_mp_postb\expandafter - {\romannumeral0\xintiiopp {#2}}{#1}% -}% -\edef\XINT_bezout_mp_postb #1#2#3#4#5% -{% - \space {#4}{#5}{#2}{#1}{#3}% -}% -% \end{macrocode} -% \lverb|plusminus A < 0, B > 0| -% \begin{macrocode} -\def\XINT_bezout_plusminus #1#2#3#4% -{% - \expandafter\XINT_bezout_pm_post - \romannumeral0\XINT_bezout_loop_a 1{#3#1}{#2}1001% -}% -\def\XINT_bezout_pm_post #1% -{% - \expandafter \XINT_bezout_pm_postb \expandafter - {\romannumeral0\xintiiopp{#1}}% -}% -\edef\XINT_bezout_pm_postb #1#2#3#4#5% -{% - \space {#4}{#5}{#1}{#2}{#3}% -}% -% \end{macrocode} -% \lverb|plusplus| -% \begin{macrocode} -\def\XINT_bezout_plusplus #1#2#3#4% -{% - \expandafter\XINT_bezout_pp_post - \romannumeral0\XINT_bezout_loop_a 1{#3#1}{#4#2}1001% -}% -% \end{macrocode} -% \lverb|la parité (-1)^N est en #1, et on la jette ici.| -% \begin{macrocode} -\edef\XINT_bezout_pp_post #1#2#3#4#5% -{% - \space {#4}{#5}{#1}{#2}{#3}% -}% -% \end{macrocode} -% \lverb|& -% n = 0: 1BAalpha(0)beta(0)alpha(-1)beta(-1)$\ -% n général: -% {(-1)^n}{r(n-1)}{r(n-2)}{alpha(n-1)}{beta(n-1)}{alpha(n-2)}{beta(n-2)}$\ -% #2 = B, #3 = A| -% \begin{macrocode} -\def\XINT_bezout_loop_a #1#2#3% -{% - \expandafter\XINT_bezout_loop_b - \expandafter{\the\numexpr -#1\expandafter }% - \romannumeral0\XINT_div_prepare {#2}{#3}{#2}% -}% -% \end{macrocode} -% \lverb|& -% Le q(n) a ici une existence éphémère, dans le version Bezout Algorithm -% il faudra le conserver. On voudra à la fin -% {{q(n)}{r(n)}{alpha(n)}{beta(n)}}. -% De plus ce n'est plus (-1)^n que l'on veut mais n. (ou dans un autre ordre)$\ -% {-(-1)^n}{q(n)}{r(n)}{r(n-1)}{alpha(n-1)}{beta(n-1)}{alpha(n-2)}{beta(n-2)}| -% \begin{macrocode} -\def\XINT_bezout_loop_b #1#2#3#4#5#6#7#8% -{% - \expandafter \XINT_bezout_loop_c \expandafter - {\romannumeral0\xintiiadd{\XINT_Mul{#5}{#2}}{#7}}% - {\romannumeral0\xintiiadd{\XINT_Mul{#6}{#2}}{#8}}% - {#1}{#3}{#4}{#5}{#6}% -}% -% \end{macrocode} -% \lverb|{alpha(n)}{->beta(n)}{-(-1)^n}{r(n)}{r(n-1)}{alpha(n-1)}{beta(n-1)}| -% \begin{macrocode} -\def\XINT_bezout_loop_c #1#2% -{% - \expandafter \XINT_bezout_loop_d \expandafter - {#2}{#1}% -}% -% \end{macrocode} -% \lverb|{beta(n)}{alpha(n)}{(-1)^(n+1)}{r(n)}{r(n-1)}{alpha(n-1)}{beta(n-1)}| -% \begin{macrocode} -\def\XINT_bezout_loop_d #1#2#3#4#5% -{% - \XINT_bezout_loop_e #4\Z {#3}{#5}{#2}{#1}% -}% -% \end{macrocode} -% \lverb|r(n)\Z {(-1)^(n+1)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}| -% \begin{macrocode} -\def\XINT_bezout_loop_e #1#2\Z -{% - \xint_gob_til_zero #1\xint_bezout_loop_exit0\XINT_bezout_loop_f - {#1#2}% -}% -% \end{macrocode} -% \lverb|{r(n)}{(-1)^(n+1)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}| -% \begin{macrocode} -\def\XINT_bezout_loop_f #1#2% -{% - \XINT_bezout_loop_a {#2}{#1}% -}% -% \end{macrocode} -% \lverb|{(-1)^(n+1)}{r(n)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)} -% et itération| -% \begin{macrocode} -\def\xint_bezout_loop_exit0\XINT_bezout_loop_f #1#2% -{% - \ifcase #2 - \or \expandafter\XINT_bezout_exiteven - \else\expandafter\XINT_bezout_exitodd - \fi -}% -\edef\XINT_bezout_exiteven #1#2#3#4#5% -{% - \space {#5}{#4}{#1}% -}% -\edef\XINT_bezout_exitodd #1#2#3#4#5% -{% - \space {-#5}{-#4}{#1}% -}% -% \end{macrocode} -% \subsection{\csh{xintEuclideAlgorithm}} -% \lverb|& -% Pour Euclide: -% {N}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0}$\ -% u<2n> = u<2n+3>u<2n+2> + u<2n+4> à la n ième étape| -% \begin{macrocode} -\def\xintEuclideAlgorithm {\romannumeral0\xinteuclidealgorithm }% -\def\xinteuclidealgorithm #1% -{% - \expandafter \XINT_euc \expandafter{\romannumeral0\xintiabs {#1}}% -}% -\def\XINT_euc #1#2% -{% - \expandafter\XINT_euc_fork \romannumeral0\xintiabs {#2}\Z #1\Z -}% -% \end{macrocode} -% \lverb|Ici #3#4=A, #1#2=B| -% \begin{macrocode} -\def\XINT_euc_fork #1#2\Z #3#4\Z -{% - \xint_UDzerofork - #1\XINT_euc_BisZero - #3\XINT_euc_AisZero - 0\XINT_euc_a - \krof - {0}{#1#2}{#3#4}{{#3#4}{#1#2}}{}\Z -}% -% \end{macrocode} -% \lverb|& -% Le {} pour protéger {{A}{B}} si on s'arrête après une étape (B divise -% A). -% On va renvoyer:$\ -% {N}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0}| -% \begin{macrocode} -\def\XINT_euc_AisZero #1#2#3#4#5#6{ {1}{0}{#2}{#2}{0}{0}}% -\def\XINT_euc_BisZero #1#2#3#4#5#6{ {1}{0}{#3}{#3}{0}{0}}% -% \end{macrocode} -% \lverb|& -% {n}{rn}{an}{{qn}{rn}}...{{A}{B}}{}\Z$\ -% a(n) = r(n-1). Pour n=0 on a juste {0}{B}{A}{{A}{B}}{}\Z$\ -% \XINT_div_prepare {u}{v} divise v par u| -% \begin{macrocode} -\def\XINT_euc_a #1#2#3% -{% - \expandafter\XINT_euc_b - \expandafter {\the\numexpr #1+1\expandafter }% - \romannumeral0\XINT_div_prepare {#2}{#3}{#2}% -}% -% \end{macrocode} -% \lverb|{n+1}{q(n+1)}{r(n+1)}{rn}{{qn}{rn}}...| -% \begin{macrocode} -\def\XINT_euc_b #1#2#3#4% -{% - \XINT_euc_c #3\Z {#1}{#3}{#4}{{#2}{#3}}% -}% -% \end{macrocode} -% \lverb|r(n+1)\Z {n+1}{r(n+1)}{r(n)}{{q(n+1)}{r(n+1)}}{{qn}{rn}}...$\ -% Test si r(n+1) est nul.| -% \begin{macrocode} -\def\XINT_euc_c #1#2\Z -{% - \xint_gob_til_zero #1\xint_euc_end0\XINT_euc_a -}% -% \end{macrocode} -% \lverb|& -% {n+1}{r(n+1)}{r(n)}{{q(n+1)}{r(n+1)}}...{}\Z -% Ici r(n+1) = 0. On arrête on se prépare à inverser -% {n+1}{0}{r(n)}{{q(n+1)}{r(n+1)}}.....{{q1}{r1}}{{A}{B}}{}\Z$\ -% On veut renvoyer: {N=n+1}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0}| -% \begin{macrocode} -\def\xint_euc_end0\XINT_euc_a #1#2#3#4\Z% -{% - \expandafter\xint_euc_end_ - \romannumeral0% - \XINT_rord_main {}#4{{#1}{#3}}% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -\edef\xint_euc_end_ #1#2#3% -{% - \space {#1}{#3}{#2}% -}% -% \end{macrocode} -% \subsection{\csh{xintBezoutAlgorithm}} -% \lverb|& -% Pour Bezout: objectif, renvoyer$\ -% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1}$\ -% {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}$\ -% alpha0=1, beta0=0, alpha(-1)=0, beta(-1)=1| -% \begin{macrocode} -\def\xintBezoutAlgorithm {\romannumeral0\xintbezoutalgorithm }% -\def\xintbezoutalgorithm #1% -{% - \expandafter \XINT_bezalg \expandafter{\romannumeral0\xintiabs {#1}}% -}% -\def\XINT_bezalg #1#2% -{% - \expandafter\XINT_bezalg_fork \romannumeral0\xintiabs {#2}\Z #1\Z -}% -% \end{macrocode} -% \lverb|Ici #3#4=A, #1#2=B| -% \begin{macrocode} -\def\XINT_bezalg_fork #1#2\Z #3#4\Z -{% - \xint_UDzerofork - #1\XINT_bezalg_BisZero - #3\XINT_bezalg_AisZero - 0\XINT_bezalg_a - \krof - 0{#1#2}{#3#4}1001{{#3#4}{#1#2}}{}\Z -}% -\def\XINT_bezalg_AisZero #1#2#3\Z{ {1}{0}{0}{1}{#2}{#2}{1}{0}{0}{0}{0}{1}}% -\def\XINT_bezalg_BisZero #1#2#3#4\Z{ {1}{0}{0}{1}{#3}{#3}{1}{0}{0}{0}{0}{1}}% -% \end{macrocode} -% \lverb|& -% pour préparer l'étape n+1 il faut -% {n}{r(n)}{r(n-1)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}& -% {{q(n)}{r(n)}{alpha(n)}{beta(n)}}... -% division de #3 par #2| -% \begin{macrocode} -\def\XINT_bezalg_a #1#2#3% -{% - \expandafter\XINT_bezalg_b - \expandafter {\the\numexpr #1+1\expandafter }% - \romannumeral0\XINT_div_prepare {#2}{#3}{#2}% -}% -% \end{macrocode} -% \lverb|& -% {n+1}{q(n+1)}{r(n+1)}{r(n)}{alpha(n)}{beta(n)}{alpha(n-1)}{beta(n-1)}...| -% \begin{macrocode} -\def\XINT_bezalg_b #1#2#3#4#5#6#7#8% -{% - \expandafter\XINT_bezalg_c\expandafter - {\romannumeral0\xintiiadd {\xintiiMul {#6}{#2}}{#8}}% - {\romannumeral0\xintiiadd {\xintiiMul {#5}{#2}}{#7}}% - {#1}{#2}{#3}{#4}{#5}{#6}% -}% -% \end{macrocode} -% \lverb|& -% {beta(n+1)}{alpha(n+1)}{n+1}{q(n+1)}{r(n+1)}{r(n)}{alpha(n)}{beta(n}}| -% \begin{macrocode} -\def\XINT_bezalg_c #1#2#3#4#5#6% -{% - \expandafter\XINT_bezalg_d\expandafter {#2}{#3}{#4}{#5}{#6}{#1}% -}% -% \end{macrocode} -% \lverb|{alpha(n+1)}{n+1}{q(n+1)}{r(n+1)}{r(n)}{beta(n+1)}| -% \begin{macrocode} -\def\XINT_bezalg_d #1#2#3#4#5#6#7#8% -{% - \XINT_bezalg_e #4\Z {#2}{#4}{#5}{#1}{#6}{#7}{#8}{{#3}{#4}{#1}{#6}}% -}% -% \end{macrocode} -% \lverb|r(n+1)\Z {n+1}{r(n+1)}{r(n)}{alpha(n+1)}{beta(n+1)}$\ -% {alpha(n)}{beta(n)}{q,r,alpha,beta(n+1)}$\ -% Test si r(n+1) est nul.| -% \begin{macrocode} -\def\XINT_bezalg_e #1#2\Z -{% - \xint_gob_til_zero #1\xint_bezalg_end0\XINT_bezalg_a -}% -% \end{macrocode} -% \lverb|& -% Ici r(n+1) = 0. On arrête on se prépare à inverser.$\ -% {n+1}{r(n+1)}{r(n)}{alpha(n+1)}{beta(n+1)}{alpha(n)}{beta(n)}$\ -% {q,r,alpha,beta(n+1)}...{{A}{B}}{}\Z$\ -% On veut renvoyer$\ -% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1}$\ -% {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}| -% \begin{macrocode} -\def\xint_bezalg_end0\XINT_bezalg_a #1#2#3#4#5#6#7#8\Z -{% - \expandafter\xint_bezalg_end_ - \romannumeral0% - \XINT_rord_main {}#8{{#1}{#3}}% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax -}% -% \end{macrocode} -% \lverb|& -% {N}{D}{A}{B}{q1}{r1}{alpha1=q1}{beta1=1}{q2}{r2}{alpha2}{beta2}$\ -% ....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}$\ -% On veut renvoyer$\ -% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1}$\ -% {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}| -% \begin{macrocode} -\edef\xint_bezalg_end_ #1#2#3#4% -{% - \space {#1}{#3}{0}{1}{#2}{#4}{1}{0}% -}% -% \end{macrocode} -% \subsection{\csh{xintTypesetEuclideAlgorithm}} -% \lverb|& -% TYPESETTING -% -% Organisation: -% -% {N}{A}{D}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0}$\ -% \U1 = N = nombre d'étapes, \U3 = PGCD, \U2 = A, \U4=B -% q1 = \U5, q2 = \U7 --> qn = \U<2n+3>, rn = \U<2n+4> -% bn = rn. B = r0. A=r(-1) -% -% r(n-2) = q(n)r(n-1)+r(n) (n e étape) -% -% \U{2n} = \U{2n+3} \times \U{2n+2} + \U{2n+4}, n e étape. -% (avec n entre 1 et N) -% -% 1.09h uses \xintloop, and \par rather than \endgraf; and \par rather than -% \hfill\break| -% \begin{macrocode} -\def\xintTypesetEuclideAlgorithm #1#2% -{% l'algo remplace #1 et #2 par |#1| et |#2| - \par - \begingroup - \xintAssignArray\xintEuclideAlgorithm {#1}{#2}\to\U - \edef\A{\U2}\edef\B{\U4}\edef\N{\U1}% - \setbox 0 \vbox{\halign {$##$\cr \A\cr \B \cr}}% - \count 255 1 - \xintloop - \indent\hbox to \wd 0 {\hfil$\U{\numexpr 2*\count255\relax}$}% - ${} = \U{\numexpr 2*\count255 + 3\relax} - \times \U{\numexpr 2*\count255 + 2\relax} - + \U{\numexpr 2*\count255 + 4\relax}$% - \ifnum \count255 < \N - \par - \advance \count255 1 - \repeat - \endgroup -}% -% \end{macrocode} -% \subsection{\csh{xintTypesetBezoutAlgorithm}} -% \lverb|& -% Pour Bezout on a: -% {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1}$\ -% {q2}{r2}{alpha2}{beta2}....{qN}{rN=0}{alphaN=A/D}{betaN=B/D}% -% Donc 4N+8 termes: -% U1 = N, U2= A, U5=D, U6=B, q1 = U9, qn = U{4n+5}, n au moins 1$\ -% rn = U{4n+6}, n au moins -1$\ -% alpha(n) = U{4n+7}, n au moins -1$\ -% beta(n) = U{4n+8}, n au moins -1 -% -% 1.09h uses \xintloop, and \par rather than \endgraf; and no more \parindent0pt -% | -% \begin{macrocode} -\def\xintTypesetBezoutAlgorithm #1#2% -{% - \par - \begingroup - \xintAssignArray\xintBezoutAlgorithm {#1}{#2}\to\BEZ - \edef\A{\BEZ2}\edef\B{\BEZ6}\edef\N{\BEZ1}% A = |#1|, B = |#2| - \setbox 0 \vbox{\halign {$##$\cr \A\cr \B \cr}}% - \count255 1 - \xintloop - \indent\hbox to \wd 0 {\hfil$\BEZ{4*\count255 - 2}$}% - ${} = \BEZ{4*\count255 + 5} - \times \BEZ{4*\count255 + 2} - + \BEZ{4*\count255 + 6}$\hfill\break - \hbox to \wd 0 {\hfil$\BEZ{4*\count255 +7}$}% - ${} = \BEZ{4*\count255 + 5} - \times \BEZ{4*\count255 + 3} - + \BEZ{4*\count255 - 1}$\hfill\break - \hbox to \wd 0 {\hfil$\BEZ{4*\count255 +8}$}% - ${} = \BEZ{4*\count255 + 5} - \times \BEZ{4*\count255 + 4} - + \BEZ{4*\count255 }$ - \par - \ifnum \count255 < \N - \advance \count255 1 - \repeat - \edef\U{\BEZ{4*\N + 4}}% - \edef\V{\BEZ{4*\N + 3}}% - \edef\D{\BEZ5}% - \ifodd\N - $\U\times\A - \V\times \B = -\D$% - \else - $\U\times\A - \V\times\B = \D$% - \fi - \par - \endgroup -}% -% \end{macrocode} -% \subsection{\csh{xintGCDof:csv}} -% \lverb|1.09a. For use by \xintexpr.| -% \begin{macrocode} -\def\xintGCDof:csv #1{\expandafter\XINT_gcdof:_b\romannumeral-`0#1,,}% -\def\XINT_gcdof:_b #1,#2,{\expandafter\XINT_gcdof:_c\romannumeral-`0#2,{#1},}% -\def\XINT_gcdof:_c #1{\if #1,\expandafter\XINT_of:_e - \else\expandafter\XINT_gcdof:_d\fi #1}% -\def\XINT_gcdof:_d #1,{\expandafter\XINT_gcdof:_b\romannumeral0\xintgcd {#1}}% -% \end{macrocode} -% \subsection{\csh{xintLCMof:csv}} -% \lverb|1.09a. For use by \xintexpr.| -% \begin{macrocode} -\def\xintLCMof:csv #1{\expandafter\XINT_lcmof:_a\romannumeral-`0#1,,}% -\def\XINT_lcmof:_a #1,#2,{\expandafter\XINT_lcmof:_c\romannumeral-`0#2,{#1},}% -\def\XINT_lcmof:_c #1{\if#1,\expandafter\XINT_of:_e - \else\expandafter\XINT_lcmof:_d\fi #1}% -\def\XINT_lcmof:_d #1,{\expandafter\XINT_lcmof:_a\romannumeral0\xintlcm {#1}}% -\XINT_restorecatcodes_endinput% -% \end{macrocode} -%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 -%\let</xintgcd>\relax -%\def<*xintfrac>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 } -%</xintgcd> -%<*xintfrac> -% -% \StoreCodelineNo {xintgcd} -% -% \section{Package \xintfracnameimp implementation} -% \label{sec:fracimp} -% -% The commenting is currently (\docdate) very sparse. -% -% \localtableofcontents -% \subsection{Catcodes, \protect\eTeX{} and reload detection} -% -% The code for reload detection is copied from \textsc{Heiko -% Oberdiek}'s packages, and adapted here to check for previous -% loading of the master \xintname package. -% -% The method for catcodes is slightly different, but still -% directly inspired by these packages. -% -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \def\space { }% - \let\z\endgroup - \expandafter\let\expandafter\x\csname ver@xintfrac.sty\endcsname - \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname - \expandafter - \ifx\csname PackageInfo\endcsname\relax - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \else - \def\y#1#2{\PackageInfo{#1}{#2}}% - \fi - \expandafter - \ifx\csname numexpr\endcsname\relax - \y{xintfrac}{\numexpr not available, aborting input}% - \aftergroup\endinput - \else - \ifx\x\relax % plain-TeX, first loading of xintfrac.sty - \ifx\w\relax % but xint.sty not yet loaded. - \y{xintfrac}{now issuing \string\input\space xint.sty}% - \def\z{\endgroup\input xint.sty\relax}% - \fi - \else - \def\empty {}% - \ifx\x\empty % LaTeX, first loading, - % variable is initialized, but \ProvidesPackage not yet seen - \ifx\w\relax % xint.sty not yet loaded. - \y{xintfrac}{now issuing \string\RequirePackage{xint}}% - \def\z{\endgroup\RequirePackage{xint}}% - \fi - \else - \y{xintfrac}{I was already loaded, aborting input}% - \aftergroup\endinput - \fi - \fi - \fi -\z% -% \end{macrocode} -% \subsection{Confirmation of \xintnameimp loading} -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \ifdefined\PackageInfo - \def\y#1#2{\PackageInfo{#1}{#2}}% - \else - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \fi - \def\empty {}% - \expandafter\let\expandafter\w\csname ver@xint.sty\endcsname - \ifx\w\relax % Plain TeX, user gave a file name at the prompt - \y{xintfrac}{Loading of package xint failed, aborting input}% - \aftergroup\endinput - \fi - \ifx\w\empty % LaTeX, user gave a file name at the prompt - \y{xintfrac}{Loading of package xint failed, aborting input}% - \aftergroup\endinput - \fi -\endgroup% -% \end{macrocode} -% \subsection{Catcodes} -% \begin{macrocode} -\XINTsetupcatcodes% -% \end{macrocode} -% \subsection{Package identification} -% \begin{macrocode} -\XINT_providespackage -\ProvidesPackage{xintfrac}% - [2014/02/05 v1.09ka Expandable operations on fractions (jfB)]% -\chardef\xint_c_vi 6 -\chardef\xint_c_vii 7 -\chardef\xint_c_xviii 18 -% \end{macrocode} -% \subsection{\csh{xintLen}} -% \begin{macrocode} -\def\xintLen {\romannumeral0\xintlen }% -\def\xintlen #1% -{% - \expandafter\XINT_flen\romannumeral0\XINT_infrac {#1}% -}% -\def\XINT_flen #1#2#3% -{% - \expandafter\space - \the\numexpr -1+\XINT_Abs {#1}+\XINT_Len {#2}+\XINT_Len {#3}\relax -}% -% \end{macrocode} -% \subsection{\csh{XINT\_lenrord\_loop}} -% \begin{macrocode} -\def\XINT_lenrord_loop #1#2#3#4#5#6#7#8#9% -{% faire \romannumeral-`0\XINT_lenrord_loop 0{}#1\Z\W\W\W\W\W\W\W\Z - \xint_gob_til_W #9\XINT_lenrord_W\W - \expandafter\XINT_lenrord_loop\expandafter - {\the\numexpr #1+7}{#9#8#7#6#5#4#3#2}% -}% -\def\XINT_lenrord_W\W\expandafter\XINT_lenrord_loop\expandafter #1#2#3\Z -{% - \expandafter\XINT_lenrord_X\expandafter {#1}#2\Z -}% -\def\XINT_lenrord_X #1#2\Z -{% - \XINT_lenrord_Y #2\R\R\R\R\R\R\T {#1}% -}% -\def\XINT_lenrord_Y #1#2#3#4#5#6#7#8\T -{% - \xint_gob_til_W - #7\XINT_lenrord_Z \xint_c_viii - #6\XINT_lenrord_Z \xint_c_vii - #5\XINT_lenrord_Z \xint_c_vi - #4\XINT_lenrord_Z \xint_c_v - #3\XINT_lenrord_Z \xint_c_iv - #2\XINT_lenrord_Z \xint_c_iii - \W\XINT_lenrord_Z \xint_c_ii \Z -}% -\def\XINT_lenrord_Z #1#2\Z #3% retourne: {longueur}renverse\Z -{% - \expandafter{\the\numexpr #3-#1\relax}% -}% -% \end{macrocode} -% \subsection{\csh{XINT\_outfrac}} -% \lverb|& -% 1.06a version now outputs 0/1[0] and not 0[0] in case of zero. More generally -% all macros have been checked in xintfrac, xintseries, xintcfrac, to make sure -% the output format for fractions was always A/B[n]. (except \xintIrr, -% \xintJrr, \xintRawWithZeros) -% -% The problem with statements like those in the previous paragraph is that it is -% hard to maintain consistencies across relases. | -% \begin{macrocode} -\def\XINT_outfrac #1#2#3% -{% - \ifcase\XINT_cntSgn #3\Z - \expandafter \XINT_outfrac_divisionbyzero - \or - \expandafter \XINT_outfrac_P - \else - \expandafter \XINT_outfrac_N - \fi - {#2}{#3}[#1]% -}% -\def\XINT_outfrac_divisionbyzero #1#2{\xintError:DivisionByZero\space #1/0}% -\edef\XINT_outfrac_P #1#2% -{% - \noexpand\if0\noexpand\XINT_Sgn #1\noexpand\Z - \noexpand\expandafter\noexpand\XINT_outfrac_Zero - \noexpand\fi - \space #1/#2% -}% -\def\XINT_outfrac_Zero #1[#2]{ 0/1[0]}% -\def\XINT_outfrac_N #1#2% -{% - \expandafter\XINT_outfrac_N_a\expandafter - {\romannumeral0\XINT_opp #2}{\romannumeral0\XINT_opp #1}% -}% -\def\XINT_outfrac_N_a #1#2% -{% - \expandafter\XINT_outfrac_P\expandafter {#2}{#1}% -}% -% \end{macrocode} -% \subsection{\csh{XINT\_inFrac}} -% \lverb|Extended in 1.07 to accept scientific notation on input. With lowercase -% e only. The \xintexpr parser does accept uppercase E also.| -% \begin{macrocode} -\def\XINT_inFrac {\romannumeral0\XINT_infrac }% -\def\XINT_infrac #1% -{% - \expandafter\XINT_infrac_ \romannumeral-`0#1[\W]\Z\T -}% -\def\XINT_infrac_ #1[#2#3]#4\Z -{% - \xint_UDwfork - #2\XINT_infrac_A - \W\XINT_infrac_B - \krof - #1[#2#3]#4% -}% -\def\XINT_infrac_A #1[\W]\T -{% - \XINT_frac #1/\W\Z -}% -\def\XINT_infrac_B #1% -{% - \xint_gob_til_zero #1\XINT_infrac_Zero0\XINT_infrac_BB #1% -}% -\def\XINT_infrac_BB #1[\W]\T {\XINT_infrac_BC #1/\W\Z }% -\def\XINT_infrac_BC #1/#2#3\Z -{% - \xint_UDwfork - #2\XINT_infrac_BCa - \W{\expandafter\XINT_infrac_BCb \romannumeral-`0#2}% - \krof - #3\Z #1\Z -}% -\def\XINT_infrac_BCa \Z #1[#2]#3\Z { {#2}{#1}{1}}% -\def\XINT_infrac_BCb #1[#2]/\W\Z #3\Z { {#2}{#3}{#1}}% -\def\XINT_infrac_Zero #1\T { {0}{0}{1}}% -% \end{macrocode} -% \subsection{\csh{XINT\_frac}} -% \lverb|Extended in 1.07 to recognize and accept scientific notation both at -% the numerator and (possible) denominator. Only a lowercase e will do here, but -% uppercase E is possible within an \xintexpr..\relax | -% \begin{macrocode} -\def\XINT_frac #1/#2#3\Z -{% - \xint_UDwfork - #2\XINT_frac_A - \W{\expandafter\XINT_frac_U \romannumeral-`0#2}% - \krof - #3e\W\Z #1e\W\Z -}% -\def\XINT_frac_U #1e#2#3\Z -{% - \xint_UDwfork - #2\XINT_frac_Ua - \W{\XINT_frac_Ub #2}% - \krof - #3\Z #1\Z -}% -\def\XINT_frac_Ua \Z #1/\W\Z {\XINT_frac_B #1.\W\Z {0}}% -\def\XINT_frac_Ub #1/\W e\W\Z #2\Z {\XINT_frac_B #2.\W\Z {#1}}% -\def\XINT_frac_B #1.#2#3\Z -{% - \xint_UDwfork - #2\XINT_frac_Ba - \W{\XINT_frac_Bb #2}% - \krof - #3\Z #1\Z -}% -\def\XINT_frac_Ba \Z #1\Z {\XINT_frac_T {0}{#1}}% -\def\XINT_frac_Bb #1.\W\Z #2\Z -{% - \expandafter \XINT_frac_T \expandafter - {\romannumeral0\xintlength {#1}}{#2#1}% -}% -\def\XINT_frac_A e\W\Z {\XINT_frac_T {0}{1}{0}}% -\def\XINT_frac_T #1#2#3#4e#5#6\Z -{% - \xint_UDwfork - #5\XINT_frac_Ta - \W{\XINT_frac_Tb #5}% - \krof - #6\Z #4\Z {#1}{#2}{#3}% -}% -\def\XINT_frac_Ta \Z #1\Z {\XINT_frac_C #1.\W\Z {0}}% -\def\XINT_frac_Tb #1e\W\Z #2\Z {\XINT_frac_C #2.\W\Z {#1}}% -\def\XINT_frac_C #1.#2#3\Z -{% - \xint_UDwfork - #2\XINT_frac_Ca - \W{\XINT_frac_Cb #2}% - \krof - #3\Z #1\Z -}% -\def\XINT_frac_Ca \Z #1\Z {\XINT_frac_D {0}{#1}}% -\def\XINT_frac_Cb #1.\W\Z #2\Z -{% - \expandafter\XINT_frac_D\expandafter - {\romannumeral0\xintlength {#1}}{#2#1}% -}% -\def\XINT_frac_D #1#2#3#4#5#6% -{% - \expandafter \XINT_frac_E \expandafter - {\the\numexpr -#1+#3+#4-#6\expandafter}\expandafter - {\romannumeral0\XINT_num_loop #2% - \xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z }% - {\romannumeral0\XINT_num_loop #5% - \xint_relax\xint_relax\xint_relax\xint_relax - \xint_relax\xint_relax\xint_relax\xint_relax\Z }% -}% -\def\XINT_frac_E #1#2#3% -{% - \expandafter \XINT_frac_F #3\Z {#2}{#1}% -}% -\def\XINT_frac_F #1% -{% - \xint_UDzerominusfork - #1-\XINT_frac_Gdivisionbyzero - 0#1\XINT_frac_Gneg - 0-{\XINT_frac_Gpos #1}% - \krof -}% -\edef\XINT_frac_Gdivisionbyzero #1\Z #2#3% -{% - \noexpand\xintError:DivisionByZero\space {0}{#2}{0}% -}% -\def\XINT_frac_Gneg #1\Z #2#3% -{% - \expandafter\XINT_frac_H \expandafter{\romannumeral0\XINT_opp #2}{#3}{#1}% -}% -\def\XINT_frac_H #1#2{ {#2}{#1}}% -\def\XINT_frac_Gpos #1\Z #2#3{ {#3}{#2}{#1}}% -% \end{macrocode} -% \subsection{\csh{XINT\_factortens}, \csh{XINT\_cuz\_cnt}} -% \begin{macrocode} -\def\XINT_factortens #1% -{% - \expandafter\XINT_cuz_cnt_loop\expandafter - {\expandafter}\romannumeral0\XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - \R\R\R\R\R\R\R\R\Z -}% -\def\XINT_cuz_cnt #1% -{% - \XINT_cuz_cnt_loop {}#1\R\R\R\R\R\R\R\R\Z -}% -\def\XINT_cuz_cnt_loop #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_R #9\XINT_cuz_cnt_toofara \R - \expandafter\XINT_cuz_cnt_checka\expandafter - {\the\numexpr #1+8\relax}{#2#3#4#5#6#7#8#9}% -}% -\def\XINT_cuz_cnt_toofara\R - \expandafter\XINT_cuz_cnt_checka\expandafter #1#2% -{% - \XINT_cuz_cnt_toofarb {#1}#2% -}% -\def\XINT_cuz_cnt_toofarb #1#2\Z {\XINT_cuz_cnt_toofarc #2\Z {#1}}% -\def\XINT_cuz_cnt_toofarc #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_R #2\XINT_cuz_cnt_toofard 7% - #3\XINT_cuz_cnt_toofard 6% - #4\XINT_cuz_cnt_toofard 5% - #5\XINT_cuz_cnt_toofard 4% - #6\XINT_cuz_cnt_toofard 3% - #7\XINT_cuz_cnt_toofard 2% - #8\XINT_cuz_cnt_toofard 1% - \Z #1#2#3#4#5#6#7#8% -}% -\def\XINT_cuz_cnt_toofard #1#2\Z #3\R #4\Z #5% -{% - \expandafter\XINT_cuz_cnt_toofare - \the\numexpr #3\relax \R\R\R\R\R\R\R\R\Z - {\the\numexpr #5-#1\relax}\R\Z -}% -\def\XINT_cuz_cnt_toofare #1#2#3#4#5#6#7#8% -{% - \xint_gob_til_R #2\XINT_cuz_cnt_stopc 1% - #3\XINT_cuz_cnt_stopc 2% - #4\XINT_cuz_cnt_stopc 3% - #5\XINT_cuz_cnt_stopc 4% - #6\XINT_cuz_cnt_stopc 5% - #7\XINT_cuz_cnt_stopc 6% - #8\XINT_cuz_cnt_stopc 7% - \Z #1#2#3#4#5#6#7#8% -}% -\def\XINT_cuz_cnt_checka #1#2% -{% - \expandafter\XINT_cuz_cnt_checkb\the\numexpr #2\relax \Z {#1}% -}% -\def\XINT_cuz_cnt_checkb #1% -{% - \xint_gob_til_zero #1\expandafter\XINT_cuz_cnt_loop\xint_gob_til_Z - 0\XINT_cuz_cnt_stopa #1% -}% -\def\XINT_cuz_cnt_stopa #1\Z -{% - \XINT_cuz_cnt_stopb #1\R\R\R\R\R\R\R\R\Z % -}% -\def\XINT_cuz_cnt_stopb #1#2#3#4#5#6#7#8#9% -{% - \xint_gob_til_R #2\XINT_cuz_cnt_stopc 1% - #3\XINT_cuz_cnt_stopc 2% - #4\XINT_cuz_cnt_stopc 3% - #5\XINT_cuz_cnt_stopc 4% - #6\XINT_cuz_cnt_stopc 5% - #7\XINT_cuz_cnt_stopc 6% - #8\XINT_cuz_cnt_stopc 7% - #9\XINT_cuz_cnt_stopc 8% - \Z #1#2#3#4#5#6#7#8#9% -}% -\def\XINT_cuz_cnt_stopc #1#2\Z #3\R #4\Z #5% -{% - \expandafter\XINT_cuz_cnt_stopd\expandafter - {\the\numexpr #5-#1}#3% -}% -\def\XINT_cuz_cnt_stopd #1#2\R #3\Z -{% - \expandafter\space\expandafter - {\romannumeral0\XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax }{#1}% -}% -% \end{macrocode} -% \subsection{\csh{xintRaw}} -% \lverb|& -% 1.07: this macro simply prints in a user readable form the fraction after its -% initial scanning. Useful when put inside braces in an \xintexpr, when the -% input is not yet in the A/B[n] form.| -% \begin{macrocode} -\def\xintRaw {\romannumeral0\xintraw }% -\def\xintraw -{% - \expandafter\XINT_raw\romannumeral0\XINT_infrac -}% -\def\XINT_raw #1#2#3{ #2/#3[#1]}% -% \end{macrocode} -% \subsection{\csh{xintPRaw}} -% \lverb|& -% 1.09b: these [n]'s and especially the possible /1 are truly annoying at -% times.| -% \begin{macrocode} -\def\xintPRaw {\romannumeral0\xintpraw }% -\def\xintpraw -{% - \expandafter\XINT_praw\romannumeral0\XINT_infrac -}% -\def\XINT_praw #1% -{% - \ifnum #1=\xint_c_ \expandafter\XINT_praw_a\fi \XINT_praw_A {#1}% -}% -\def\XINT_praw_A #1#2#3% -{% - \if\XINT_isOne{#3}1\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi { #2[#1]}{ #2/#3[#1]}% -}% -\def\XINT_praw_a\XINT_praw_A #1#2#3% -{% - \if\XINT_isOne{#3}1\expandafter\xint_firstoftwo - \else\expandafter\xint_secondoftwo - \fi { #2}{ #2/#3}% -}% -% \end{macrocode} -% \subsection{\csh{xintRawWithZeros}} -% \lverb|& -% This was called \xintRaw in versions earlier than 1.07| -% \begin{macrocode} -\def\xintRawWithZeros {\romannumeral0\xintrawwithzeros }% -\def\xintrawwithzeros -{% - \expandafter\XINT_rawz\romannumeral0\XINT_infrac -}% -\def\XINT_rawz #1% -{% - \ifcase\XINT_cntSgn #1\Z - \expandafter\XINT_rawz_Ba - \or - \expandafter\XINT_rawz_A - \else - \expandafter\XINT_rawz_Ba - \fi - {#1}% -}% -\def\XINT_rawz_A #1#2#3{\xint_dsh {#2}{-#1}/#3}% -\def\XINT_rawz_Ba #1#2#3{\expandafter\XINT_rawz_Bb - \expandafter{\romannumeral0\xint_dsh {#3}{#1}}{#2}}% -\def\XINT_rawz_Bb #1#2{ #2/#1}% -% \end{macrocode} -% \subsection{\csh{xintFloor}} -% \lverb|1.09a| -% \begin{macrocode} -\def\xintFloor {\romannumeral0\xintfloor }% -\def\xintfloor #1{\expandafter\XINT_floor - \romannumeral0\xintrawwithzeros {#1}.}% -\def\XINT_floor #1/#2.{\xintiiquo {#1}{#2}}% -% \end{macrocode} -% \subsection{\csh{xintCeil}} -% \lverb|1.09a| -% \begin{macrocode} -\def\xintCeil {\romannumeral0\xintceil }% -\def\xintceil #1{\xintiiopp {\xintFloor {\xintOpp{#1}}}}% -% \end{macrocode} -% \subsection{\csh{xintNumerator}} -% \begin{macrocode} -\def\xintNumerator {\romannumeral0\xintnumerator }% -\def\xintnumerator -{% - \expandafter\XINT_numer\romannumeral0\XINT_infrac -}% -\def\XINT_numer #1% -{% - \ifcase\XINT_cntSgn #1\Z - \expandafter\XINT_numer_B - \or - \expandafter\XINT_numer_A - \else - \expandafter\XINT_numer_B - \fi - {#1}% -}% -\def\XINT_numer_A #1#2#3{\xint_dsh {#2}{-#1}}% -\def\XINT_numer_B #1#2#3{ #2}% -% \end{macrocode} -% \subsection{\csh{xintDenominator}} -% \begin{macrocode} -\def\xintDenominator {\romannumeral0\xintdenominator }% -\def\xintdenominator -{% - \expandafter\XINT_denom\romannumeral0\XINT_infrac -}% -\def\XINT_denom #1% -{% - \ifcase\XINT_cntSgn #1\Z - \expandafter\XINT_denom_B - \or - \expandafter\XINT_denom_A - \else - \expandafter\XINT_denom_B - \fi - {#1}% -}% -\def\XINT_denom_A #1#2#3{ #3}% -\def\XINT_denom_B #1#2#3{\xint_dsh {#3}{#1}}% -% \end{macrocode} -% \subsection{\csh{xintFrac}} -% \begin{macrocode} -\def\xintFrac {\romannumeral0\xintfrac }% -\def\xintfrac #1% -{% - \expandafter\XINT_fracfrac_A\romannumeral0\XINT_infrac {#1}% -}% -\def\XINT_fracfrac_A #1{\XINT_fracfrac_B #1\Z }% -\catcode`^=7 -\def\XINT_fracfrac_B #1#2\Z -{% - \xint_gob_til_zero #1\XINT_fracfrac_C 0\XINT_fracfrac_D {10^{#1#2}}% -}% -\def\XINT_fracfrac_C 0\XINT_fracfrac_D #1#2#3% -{% - \if1\XINT_isOne {#3}% - \xint_afterfi {\expandafter\xint_firstoftwo_thenstop\xint_gobble_ii }% - \fi - \space - \frac {#2}{#3}% -}% -\def\XINT_fracfrac_D #1#2#3% -{% - \if1\XINT_isOne {#3}\XINT_fracfrac_E\fi - \space - \frac {#2}{#3}#1% -}% -\def\XINT_fracfrac_E \fi\space\frac #1#2{\fi \space #1\cdot }% -% \end{macrocode} -% \subsection{\csh{xintSignedFrac}} -% \begin{macrocode} -\def\xintSignedFrac {\romannumeral0\xintsignedfrac }% -\def\xintsignedfrac #1% -{% - \expandafter\XINT_sgnfrac_a\romannumeral0\XINT_infrac {#1}% -}% -\def\XINT_sgnfrac_a #1#2% -{% - \XINT_sgnfrac_b #2\Z {#1}% -}% -\def\XINT_sgnfrac_b #1% -{% - \xint_UDsignfork - #1\XINT_sgnfrac_N - -{\XINT_sgnfrac_P #1}% - \krof -}% -\def\XINT_sgnfrac_P #1\Z #2% -{% - \XINT_fracfrac_A {#2}{#1}% -}% -\def\XINT_sgnfrac_N -{% - \expandafter\xint_minus_thenstop\romannumeral0\XINT_sgnfrac_P -}% -% \end{macrocode} -% \subsection{\csh{xintFwOver}} -% \begin{macrocode} -\def\xintFwOver {\romannumeral0\xintfwover }% -\def\xintfwover #1% -{% - \expandafter\XINT_fwover_A\romannumeral0\XINT_infrac {#1}% -}% -\def\XINT_fwover_A #1{\XINT_fwover_B #1\Z }% -\def\XINT_fwover_B #1#2\Z -{% - \xint_gob_til_zero #1\XINT_fwover_C 0\XINT_fwover_D {10^{#1#2}}% -}% -\catcode`^=11 -\def\XINT_fwover_C #1#2#3#4#5% -{% - \if0\XINT_isOne {#5}\xint_afterfi { {#4\over #5}}% - \else\xint_afterfi { #4}% - \fi -}% -\def\XINT_fwover_D #1#2#3% -{% - \if0\XINT_isOne {#3}\xint_afterfi { {#2\over #3}}% - \else\xint_afterfi { #2\cdot }% - \fi - #1% -}% -% \end{macrocode} -% \subsection{\csh{xintSignedFwOver}} -% \begin{macrocode} -\def\xintSignedFwOver {\romannumeral0\xintsignedfwover }% -\def\xintsignedfwover #1% -{% - \expandafter\XINT_sgnfwover_a\romannumeral0\XINT_infrac {#1}% -}% -\def\XINT_sgnfwover_a #1#2% -{% - \XINT_sgnfwover_b #2\Z {#1}% -}% -\def\XINT_sgnfwover_b #1% -{% - \xint_UDsignfork - #1\XINT_sgnfwover_N - -{\XINT_sgnfwover_P #1}% - \krof -}% -\def\XINT_sgnfwover_P #1\Z #2% -{% - \XINT_fwover_A {#2}{#1}% -}% -\def\XINT_sgnfwover_N -{% - \expandafter\xint_minus_thenstop\romannumeral0\XINT_sgnfwover_P -}% -% \end{macrocode} -% \subsection{\csh{xintREZ}} -% \begin{macrocode} -\def\xintREZ {\romannumeral0\xintrez }% -\def\xintrez -{% - \expandafter\XINT_rez_A\romannumeral0\XINT_infrac -}% -\def\XINT_rez_A #1#2% -{% - \XINT_rez_AB #2\Z {#1}% -}% -\def\XINT_rez_AB #1% -{% - \xint_UDzerominusfork - #1-\XINT_rez_zero - 0#1\XINT_rez_neg - 0-{\XINT_rez_B #1}% - \krof -}% -\def\XINT_rez_zero #1\Z #2#3{ 0/1[0]}% -\def\XINT_rez_neg {\expandafter\xint_minus_thenstop\romannumeral0\XINT_rez_B }% -\def\XINT_rez_B #1\Z -{% - \expandafter\XINT_rez_C\romannumeral0\XINT_factortens {#1}% -}% -\def\XINT_rez_C #1#2#3#4% -{% - \expandafter\XINT_rez_D\romannumeral0\XINT_factortens {#4}{#3}{#2}{#1}% -}% -\def\XINT_rez_D #1#2#3#4#5% -{% - \expandafter\XINT_rez_E\expandafter - {\the\numexpr #3+#4-#2}{#1}{#5}% -}% -\def\XINT_rez_E #1#2#3{ #3/#2[#1]}% -% \end{macrocode} -% \subsection{\csh{xintE}} -% \lverb|1.07: The fraction is the first argument contrarily to \xintTrunc and -% \xintRound. -% -% \xintfE (1.07) and \xintiE (1.09i) are for \xintexpr and cousins. It is quite -% annoying that \numexpr does not know how to deal correctly with a minus sign - -% as prefix: \numexpr -(1)\relax is illegal! (one can do \numexpr 0-(1)\relax). -% -% the 1.07 \xintE puts directly its second argument in a \numexpr. The \xintfE -% first uses \xintNum on it, this is necessary for use in \xintexpr. (but -% one cannot use directly infix notation in the second argument of \xintfE) -% -% 1.09i also adds \xintFloatE and modifies \XINTinFloatfE, although currently -% the latter is only used from \xintfloatexpr hence always with \XINTdigits, it -% comes equipped with its first argument withing brackets as the other -% \XINTinFloat... macros. | -% \begin{macrocode} -\def\xintE {\romannumeral0\xinte }% -\def\xinte #1% -{% - \expandafter\XINT_e \romannumeral0\XINT_infrac {#1}% -}% -\def\XINT_e #1#2#3#4% -{% - \expandafter\XINT_e_end\expandafter{\the\numexpr #1+#4}{#2}{#3}% -}% -\def\XINT_e_end #1#2#3{ #2/#3[#1]}% -\def\xintfE {\romannumeral0\xintfe }% -\def\xintfe #1% -{% - \expandafter\XINT_fe \romannumeral0\XINT_infrac {#1}% -}% -\def\XINT_fe #1#2#3#4% -{% - \expandafter\XINT_e_end\expandafter{\the\numexpr #1+\xintNum{#4}}{#2}{#3}% -}% -\def\xintFloatE {\romannumeral0\xintfloate }% -\def\xintfloate #1{\XINT_floate_chkopt #1\Z }% -\def\XINT_floate_chkopt #1% -{% - \ifx [#1\expandafter\XINT_floate_opt - \else\expandafter\XINT_floate_noopt - \fi #1% -}% -\def\XINT_floate_noopt #1\Z -{% - \expandafter\XINT_floate_a\expandafter\XINTdigits - \romannumeral0\XINT_infrac {#1}% -}% -\def\XINT_floate_opt [\Z #1]#2% -{% - \expandafter\XINT_floate_a\expandafter - {\the\numexpr #1\expandafter}\romannumeral0\XINT_infrac {#2}% -}% -\def\XINT_floate_a #1#2#3#4#5% -{% - \expandafter\expandafter\expandafter\XINT_float_a - \expandafter\xint_exchangetwo_keepbraces\expandafter - {\the\numexpr #2+#5}{#1}{#3}{#4}\XINT_float_Q -}% -\def\XINTinFloatfE {\romannumeral0\XINTinfloatfe }% -\def\XINTinfloatfe [#1]#2% -{% - \expandafter\XINT_infloatfe_a\expandafter - {\the\numexpr #1\expandafter}\romannumeral0\XINT_infrac {#2}% -}% -\def\XINT_infloatfe_a #1#2#3#4#5% -{% - \expandafter\expandafter\expandafter\XINT_infloat_a - \expandafter\xint_exchangetwo_keepbraces\expandafter - {\the\numexpr #2+\xintNum{#5}}{#1}{#3}{#4}\XINT_infloat_Q -}% -\def\xintiE {\romannumeral0\xintie }% for \xintiiexpr only -\def\xintie #1% -{% - \expandafter\XINT_ie \romannumeral0\XINT_infrac {#1}% allows 3.123e3 -}% -\def\XINT_ie #1#2#3#4% assumes #3=1 and uses \xint_dsh with its \numexpr -{% - \xint_dsh {#2}{0-(#1+#4)}% could have \xintNum{#4} for a bit more general -}% -% \end{macrocode} -% \subsection{\csh{xintIrr}} -% \lverb|& -% 1.04 fixes a buggy \xintIrr {0}. -% 1.05 modifies the initial parsing and post-processing to use \xintrawwithzeros -% and to -% more quickly deal with an input denominator equal to 1. 1.08 version does -% not remove a /1 denominator.| -% \begin{macrocode} -\def\xintIrr {\romannumeral0\xintirr }% -\def\xintirr #1% -{% - \expandafter\XINT_irr_start\romannumeral0\xintrawwithzeros {#1}\Z -}% -\def\XINT_irr_start #1#2/#3\Z -{% - \if0\XINT_isOne {#3}% - \xint_afterfi - {\xint_UDsignfork - #1\XINT_irr_negative - -{\XINT_irr_nonneg #1}% - \krof}% - \else - \xint_afterfi{\XINT_irr_denomisone #1}% - \fi - #2\Z {#3}% -}% -\def\XINT_irr_denomisone #1\Z #2{ #1/1}% changed in 1.08 -\def\XINT_irr_negative #1\Z #2{\XINT_irr_D #1\Z #2\Z \xint_minus_thenstop}% -\def\XINT_irr_nonneg #1\Z #2{\XINT_irr_D #1\Z #2\Z \space}% -\def\XINT_irr_D #1#2\Z #3#4\Z -{% - \xint_UDzerosfork - #3#1\XINT_irr_indeterminate - #30\XINT_irr_divisionbyzero - #10\XINT_irr_zero - 00\XINT_irr_loop_a - \krof - {#3#4}{#1#2}{#3#4}{#1#2}% -}% -\def\XINT_irr_indeterminate #1#2#3#4#5{\xintError:NaN\space 0/0}% -\def\XINT_irr_divisionbyzero #1#2#3#4#5{\xintError:DivisionByZero #5#2/0}% -\def\XINT_irr_zero #1#2#3#4#5{ 0/1}% changed in 1.08 -\def\XINT_irr_loop_a #1#2% -{% - \expandafter\XINT_irr_loop_d - \romannumeral0\XINT_div_prepare {#1}{#2}{#1}% -}% -\def\XINT_irr_loop_d #1#2% -{% - \XINT_irr_loop_e #2\Z -}% -\def\XINT_irr_loop_e #1#2\Z -{% - \xint_gob_til_zero #1\xint_irr_loop_exit0\XINT_irr_loop_a {#1#2}% -}% -\def\xint_irr_loop_exit0\XINT_irr_loop_a #1#2#3#4% -{% - \expandafter\XINT_irr_loop_exitb\expandafter - {\romannumeral0\xintiiquo {#3}{#2}}% - {\romannumeral0\xintiiquo {#4}{#2}}% -}% -\def\XINT_irr_loop_exitb #1#2% -{% - \expandafter\XINT_irr_finish\expandafter {#2}{#1}% -}% -\def\XINT_irr_finish #1#2#3{#3#1/#2}% changed in 1.08 -% \end{macrocode} -% \subsection{\csh{xintNum}} -% \lverb|& -% This extension of the xint original xintNum is added in 1.05, as a -% synonym to -% \xintIrr, but raising an error when the input does not evaluate to an integer. -% Usable with not too much overhead on integer input as \xintIrr -% checks quickly for a denominator equal to 1 (which will be put there by the -% \XINT_infrac called by \xintrawwithzeros). This way, macros such as \xintQuo -% can be -% modified with minimal overhead to accept fractional input as long as it -% evaluates to an integer. | -% \begin{macrocode} -\def\xintNum {\romannumeral0\xintnum }% -\def\xintnum #1{\expandafter\XINT_intcheck\romannumeral0\xintirr {#1}\Z }% -\edef\XINT_intcheck #1/#2\Z -{% - \noexpand\if 0\noexpand\XINT_isOne {#2}\noexpand\xintError:NotAnInteger - \noexpand\fi\space #1% -}% -% \end{macrocode} -% \subsection{\csh{xintifInt}} -% \lverb|1.09e. xintfrac.sty only.| -% \begin{macrocode} -\def\xintifInt {\romannumeral0\xintifint }% -\def\xintifint #1{\expandafter\XINT_ifint\romannumeral0\xintirr {#1}\Z }% -\def\XINT_ifint #1/#2\Z -{% - \if\XINT_isOne {#2}1% - \expandafter\xint_firstoftwo_thenstop - \else - \expandafter\xint_secondoftwo_thenstop - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintJrr}} -% \lverb|& -% Modified similarly as \xintIrr in release 1.05. 1.08 version does -% not remove a /1 denominator.| -% \begin{macrocode} -\def\xintJrr {\romannumeral0\xintjrr }% -\def\xintjrr #1% -{% - \expandafter\XINT_jrr_start\romannumeral0\xintrawwithzeros {#1}\Z -}% -\def\XINT_jrr_start #1#2/#3\Z -{% - \if0\XINT_isOne {#3}\xint_afterfi - {\xint_UDsignfork - #1\XINT_jrr_negative - -{\XINT_jrr_nonneg #1}% - \krof}% - \else - \xint_afterfi{\XINT_jrr_denomisone #1}% - \fi - #2\Z {#3}% -}% -\def\XINT_jrr_denomisone #1\Z #2{ #1/1}% changed in 1.08 -\def\XINT_jrr_negative #1\Z #2{\XINT_jrr_D #1\Z #2\Z \xint_minus_thenstop }% -\def\XINT_jrr_nonneg #1\Z #2{\XINT_jrr_D #1\Z #2\Z \space}% -\def\XINT_jrr_D #1#2\Z #3#4\Z -{% - \xint_UDzerosfork - #3#1\XINT_jrr_indeterminate - #30\XINT_jrr_divisionbyzero - #10\XINT_jrr_zero - 00\XINT_jrr_loop_a - \krof - {#3#4}{#1#2}1001% -}% -\def\XINT_jrr_indeterminate #1#2#3#4#5#6#7{\xintError:NaN\space 0/0}% -\def\XINT_jrr_divisionbyzero #1#2#3#4#5#6#7{\xintError:DivisionByZero #7#2/0}% -\def\XINT_jrr_zero #1#2#3#4#5#6#7{ 0/1}% changed in 1.08 -\def\XINT_jrr_loop_a #1#2% -{% - \expandafter\XINT_jrr_loop_b - \romannumeral0\XINT_div_prepare {#1}{#2}{#1}% -}% -\def\XINT_jrr_loop_b #1#2#3#4#5#6#7% -{% - \expandafter \XINT_jrr_loop_c \expandafter - {\romannumeral0\xintiiadd{\XINT_Mul{#4}{#1}}{#6}}% - {\romannumeral0\xintiiadd{\XINT_Mul{#5}{#1}}{#7}}% - {#2}{#3}{#4}{#5}% -}% -\def\XINT_jrr_loop_c #1#2% -{% - \expandafter \XINT_jrr_loop_d \expandafter{#2}{#1}% -}% -\def\XINT_jrr_loop_d #1#2#3#4% -{% - \XINT_jrr_loop_e #3\Z {#4}{#2}{#1}% -}% -\def\XINT_jrr_loop_e #1#2\Z -{% - \xint_gob_til_zero #1\xint_jrr_loop_exit0\XINT_jrr_loop_a {#1#2}% -}% -\def\xint_jrr_loop_exit0\XINT_jrr_loop_a #1#2#3#4#5#6% -{% - \XINT_irr_finish {#3}{#4}% -}% -% \end{macrocode} -% \subsection{\csh{xintTFrac}} -% \lverb|1.09i, for frac in \xintexpr. And \xintFrac is already assigned. T for -% truncation. However, potentially not very efficient with numbers in scientific -% notations, with big exponents. Will have to think it again some day. I -% hesitated how to call the macro. Same convention as in maple, but some people -% reserve fractional part to x - floor(x). Also, not clear if I had to make it -% negative (or zero) if x < 0, or rather always positive. There should be in -% fact such a thing for each rounding function, trunc, round, floor, ceil. | -% \begin{macrocode} -\def\xintTFrac {\romannumeral0\xinttfrac }% -\def\xinttfrac #1% - {\expandafter\XINT_tfrac_fork\romannumeral0\xintrawwithzeros {#1}\Z }% -\def\XINT_tfrac_fork #1% -{% - \xint_UDzerominusfork - #1-\XINT_tfrac_zero - 0#1\XINT_tfrac_N - 0-{\XINT_tfrac_P #1}% - \krof -}% -\def\XINT_tfrac_zero #1\Z { 0/1[0]}% -\def\XINT_tfrac_N {\expandafter\XINT_opp\romannumeral0\XINT_tfrac_P }% -\def\XINT_tfrac_P #1/#2\Z -{% - \expandafter\XINT_rez_AB\romannumeral0\xintiirem{#1}{#2}\Z {0}{#2}% -}% -% \end{macrocode} -% \subsection{\csh{XINTinFloatFrac}} -% \lverb|1.09i, for frac in \xintfloatexpr. This version computes -% exactly from the input the fractional part and then only converts it -% into a float with the asked-for number of digits. I will have to think -% it again some day, certainly. | -% \begin{macrocode} -\def\XINTinFloatFrac {\romannumeral0\XINTinfloatfrac }% -\def\XINTinfloatfrac [#1]#2% -{% - \expandafter\XINT_infloatfrac_a\expandafter - {\romannumeral0\xinttfrac{#2}}{#1}% -}% -\def\XINT_infloatfrac_a #1#2{\XINTinFloat [#2]{#1}}% -% \end{macrocode} -% \subsection{\csh{xintTrunc}, \csh{xintiTrunc}} -% \lverb|& -% Modified in 1.06 to give the first argument to a \numexpr. -% -% 1.09f fixes the overhead added in 1.09a to some inner routines when \xintiquo -% was redefined to use \xintnum. Now uses \xintiiquo, rather. -% -% 1.09j: minor improvements, \XINT_trunc_E was very strange and defined two -% never occuring branches; also, optimizes the call to the division routine, and -% the zero loops.| -% \begin{macrocode} -\def\xintTrunc {\romannumeral0\xinttrunc }% -\def\xintiTrunc {\romannumeral0\xintitrunc }% -\def\xinttrunc #1% -{% - \expandafter\XINT_trunc\expandafter {\the\numexpr #1}% -}% -\def\XINT_trunc #1#2% -{% - \expandafter\XINT_trunc_G - \romannumeral0\expandafter\XINT_trunc_A - \romannumeral0\XINT_infrac {#2}{#1}{#1}% -}% -\def\xintitrunc #1% -{% - \expandafter\XINT_itrunc\expandafter {\the\numexpr #1}% -}% -\def\XINT_itrunc #1#2% -{% - \expandafter\XINT_itrunc_G - \romannumeral0\expandafter\XINT_trunc_A - \romannumeral0\XINT_infrac {#2}{#1}{#1}% -}% -\def\XINT_trunc_A #1#2#3#4% -{% - \expandafter\XINT_trunc_checkifzero - \expandafter{\the\numexpr #1+#4}#2\Z {#3}% -}% -\def\XINT_trunc_checkifzero #1#2#3\Z -{% - \xint_gob_til_zero #2\XINT_trunc_iszero0\XINT_trunc_B {#1}{#2#3}% -}% -\def\XINT_trunc_iszero0\XINT_trunc_B #1#2#3{ 0\Z 0}% -\def\XINT_trunc_B #1% -{% - \ifcase\XINT_cntSgn #1\Z - \expandafter\XINT_trunc_D - \or - \expandafter\XINT_trunc_D - \else - \expandafter\XINT_trunc_C - \fi - {#1}% -}% -\def\XINT_trunc_C #1#2#3% -{% - \expandafter\XINT_trunc_CE\expandafter - {\romannumeral0\XINT_dsx_zeroloop {-#1}{}\Z {#3}}{#2}% -}% -\def\XINT_trunc_CE #1#2{\XINT_trunc_E #2.{#1}}% -\def\XINT_trunc_D #1#2% -{% - \expandafter\XINT_trunc_E - \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z {#2}.% -}% -\def\XINT_trunc_E #1% -{% - \xint_UDsignfork - #1\XINT_trunc_Fneg - -{\XINT_trunc_Fpos #1}% - \krof -}% -\def\XINT_trunc_Fneg #1.#2{\expandafter\xint_firstoftwo_thenstop - \romannumeral0\XINT_div_prepare {#2}{#1}\Z \xint_minus_thenstop}% -\def\XINT_trunc_Fpos #1.#2{\expandafter\xint_firstoftwo_thenstop - \romannumeral0\XINT_div_prepare {#2}{#1}\Z \space }% -\def\XINT_itrunc_G #1#2\Z #3#4% -{% - \xint_gob_til_zero #1\XINT_trunc_zero 0#3#1#2% -}% -\def\XINT_trunc_zero 0#1#20{ 0}% -\def\XINT_trunc_G #1\Z #2#3% -{% - \xint_gob_til_zero #2\XINT_trunc_zero 0% - \expandafter\XINT_trunc_H\expandafter - {\the\numexpr\romannumeral0\xintlength {#1}-#3}{#3}{#1}#2% -}% -\def\XINT_trunc_H #1#2% -{% - \ifnum #1 > \xint_c_ - \xint_afterfi {\XINT_trunc_Ha {#2}}% - \else - \xint_afterfi {\XINT_trunc_Hb {-#1}}% -0,--1,--2, .... - \fi -}% -\def\XINT_trunc_Ha -{% - \expandafter\XINT_trunc_Haa\romannumeral0\xintdecsplit -}% -\def\XINT_trunc_Haa #1#2#3% -{% - #3#1.#2% -}% -\def\XINT_trunc_Hb #1#2#3% -{% - \expandafter #3\expandafter0\expandafter.% - \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z {}#2% #1=-0 autorisé ! -}% -% \end{macrocode} -% \subsection{\csh{xintRound}, \csh{xintiRound}} -% \lverb|Modified in 1.06 to give the first argument to a \numexpr.| -% \begin{macrocode} -\def\xintRound {\romannumeral0\xintround }% -\def\xintiRound {\romannumeral0\xintiround }% -\def\xintround #1% -{% - \expandafter\XINT_round\expandafter {\the\numexpr #1}% -}% -\def\XINT_round -{% - \expandafter\XINT_trunc_G\romannumeral0\XINT_round_A -}% -\def\xintiround #1% -{% - \expandafter\XINT_iround\expandafter {\the\numexpr #1}% -}% -\def\XINT_iround -{% - \expandafter\XINT_itrunc_G\romannumeral0\XINT_round_A -}% -\def\XINT_round_A #1#2% -{% - \expandafter\XINT_round_B - \romannumeral0\expandafter\XINT_trunc_A - \romannumeral0\XINT_infrac {#2}{\the\numexpr #1+1\relax}{#1}% -}% -\def\XINT_round_B #1\Z -{% - \expandafter\XINT_round_C - \romannumeral0\XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax - \Z -}% -\def\XINT_round_C #1% -{% - \ifnum #1<5 - \expandafter\XINT_round_Daa - \else - \expandafter\XINT_round_Dba - \fi -}% -\def\XINT_round_Daa #1% -{% - \xint_gob_til_Z #1\XINT_round_Daz\Z \XINT_round_Da #1% -}% -\def\XINT_round_Daz\Z \XINT_round_Da \Z { 0\Z }% -\def\XINT_round_Da #1\Z -{% - \XINT_rord_main {}#1% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax \Z -}% -\def\XINT_round_Dba #1% -{% - \xint_gob_til_Z #1\XINT_round_Dbz\Z \XINT_round_Db #1% -}% -\def\XINT_round_Dbz\Z \XINT_round_Db \Z { 1\Z }% -\def\XINT_round_Db #1\Z -{% - \XINT_addm_A 0{}1000\W\X\Y\Z #1000\W\X\Y\Z \Z -}% -% \end{macrocode} -% \subsection{\csh{xintXTrunc}} -% \lverb|1.09j [2014/01/06] This is completely expandable but not f-expandable. -% Designed be used inside an \edef or a \write, if one is interested in getting -% tens of thousands of digits from the decimal expansion of some fraction... it -% is not worth using it rather than \xintTrunc if for less than *hundreds* of -% digits. For efficiency it clones part of the preparatory division macros, as -% the same denominator will be used again and again. The D parameter which says -% how many digits to keep after decimal mark must be at least 1 (and it is -% forcefully set to such a value if found negative or zero, to avoid an eternal -% loop). -% -% For reasons of efficiency I try to use the shortest possible denominator, so -% if the fraction is A/B[N], I want to use B. For N at least zero, just -% immediately replace A by A.10^N. The first division then may be a little -% longish but the next ones will be fast (if B is not too big). For N<0, this is -% a bit more complicated. I thought somewhat about this, and I would need a -% rather complicated approach going through a long division algorithm, forcing -% me to essentially clone the actual division with some differences; a side -% thing is that as this would use blocks of four digits I would have a hard time -% allowing a non-multiple of four number of post decimal mark digits. -% -% Thus, for N<0, another method is followed. First the euclidean division -% A/B=Q+R/B is done. The number of digits of Q is M. If |N|\leq D, we launch -% inside a \csname the routine for obtaining D-|N| next digits (this may impact -% TeX's memory if D is very big), call them T. We then need to position the -% decimal mark D slots from the right of QT, which has length M+D-|N|, hence |N| -% slots from the right of Q. We thus avoid having to work will the T, as D may -% be very very big (\xintXTrunc's only goal is to make it possible to learn by -% hearts decimal expansions with thousands of digits). We can use the -% \xintDecSplit for that on Q . Computing the length M of Q was a more or less -% unavoidable step. If |N|>D, the \csname step is skipped we need to remove the -% D-|N| last digits from Q, etc.. we compare D-|N| with the length M of Q etc... -% (well in this last, very uncommon, branch, I stopped trying to optimize thinsg -% and I even do an \xintnum to ensure a 0 if something comes out empty from -% \xintDecSplit).| -% \begin{macrocode} -\def\xintXTrunc #1#2% -{% - \expandafter\XINT_xtrunc_a\expandafter - {\the\numexpr #1\expandafter}\romannumeral0\xintraw {#2}% -}% -\def\XINT_xtrunc_a #1% -{% - \expandafter\XINT_xtrunc_b\expandafter - {\the\numexpr\ifnum#1<\xint_c_i \xint_c_i-\fi #1}% -}% -\def\XINT_xtrunc_b #1% -{% - \expandafter\XINT_xtrunc_c\expandafter - {\the\numexpr (#1+\xint_c_ii^v)/\xint_c_ii^vi-\xint_c_i}{#1}% -}% -\def\XINT_xtrunc_c #1#2% -{% - \expandafter\XINT_xtrunc_d\expandafter - {\the\numexpr #2-\xint_c_ii^vi*#1}{#1}{#2}% -}% -\def\XINT_xtrunc_d #1#2#3#4/#5[#6]% -{% - \XINT_xtrunc_e #4.{#6}{#5}{#3}{#2}{#1}% -}% -% #1=numerator.#2=N,#3=B,#4=D,#5=Blocs,#6=extra -\def\XINT_xtrunc_e #1% -{% - \xint_UDzerominusfork - #1-\XINT_xtrunc_zero - 0#1\XINT_xtrunc_N - 0-{\XINT_xtrunc_P #1}% - \krof -}% -\def\XINT_xtrunc_zero .#1#2#3#4#5% -{% - 0.\romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter - {\the\numexpr #5}{}\Z {}% - \xintiloop [#4+-1] - \ifnum \xintiloopindex>\xint_c_ - 0000000000000000000000000000000000000000000000000000000000000000% - \repeat -}% -\def\XINT_xtrunc_N {-\XINT_xtrunc_P }% -\def\XINT_xtrunc_P #1.#2% -{% - \ifnum #2<\xint_c_ - \expandafter\XINT_xtrunc_negN_Q - \else - \expandafter\XINT_xtrunc_Q - \fi {#2}{#1}.% -}% -\def\XINT_xtrunc_negN_Q #1#2.#3#4#5#6% -{% - \expandafter\XINT_xtrunc_negN_R - \romannumeral0\XINT_div_prepare {#3}{#2}{#3}{#1}{#4}% -}% -% #1=Q, #2=R, #3=B, #4=N<0, #5=D -\def\XINT_xtrunc_negN_R #1#2#3#4#5% -{% - \expandafter\XINT_xtrunc_negN_S\expandafter - {\the\numexpr -#4}{#5}{#2}{#3}{#1}% -}% -\def\XINT_xtrunc_negN_S #1#2% -{% - \expandafter\XINT_xtrunc_negN_T\expandafter - {\the\numexpr #2-#1}{#1}{#2}% -}% -\def\XINT_xtrunc_negN_T #1% -{% - \ifnum \xint_c_<#1 - \expandafter\XINT_xtrunc_negNA - \else - \expandafter\XINT_xtrunc_negNW - \fi {#1}% -}% -% #1=D-|N|>0, #2=|N|, #3=D, #4=R, #5=B, #6=Q -\def\XINT_xtrunc_unlock #10.{ }% -\def\XINT_xtrunc_negNA #1#2#3#4#5#6% -{% - \expandafter\XINT_xtrunc_negNB\expandafter - {\romannumeral0\expandafter\expandafter\expandafter - \XINT_xtrunc_unlock\expandafter\string - \csname\XINT_xtrunc_b {#1}#4/#5[0]\expandafter\endcsname - \expandafter}\expandafter - {\the\numexpr\xintLength{#6}-#2}{#6}% -}% -\def\XINT_xtrunc_negNB #1#2#3{\XINT_xtrunc_negNC {#2}{#3}#1}% -\def\XINT_xtrunc_negNC #1% -{% - \ifnum \xint_c_ < #1 - \expandafter\XINT_xtrunc_negNDa - \else - \expandafter\XINT_xtrunc_negNE - \fi {#1}% -}% -\def\XINT_xtrunc_negNDa #1#2% -{% - \expandafter\XINT_xtrunc_negNDb% - \romannumeral0\XINT_split_fromleft_loop {#1}{}#2\W\W\W\W\W\W\W\W\Z -}% -\def\XINT_xtrunc_negNDb #1#2{#1.#2}% -\def\XINT_xtrunc_negNE #1#2% -{% - 0.\romannumeral0\XINT_dsx_zeroloop {-#1}{}\Z {}#2% -}% -% #1=D-|N|<=0, #2=|N|, #3=D, #4=R, #5=B, #6=Q -\def\XINT_xtrunc_negNW #1#2#3#4#5#6% -{% - \expandafter\XINT_xtrunc_negNX\expandafter - {\romannumeral0\xintnum{\xintDecSplitL {-#1}{#6}}}{#3}% -}% -\def\XINT_xtrunc_negNX #1#2% -{% - \expandafter\XINT_xtrunc_negNC\expandafter - {\the\numexpr\xintLength {#1}-#2}{#1}% -}% -\def\XINT_xtrunc_Q #1% -{% - \expandafter\XINT_xtrunc_prepare_I - \romannumeral0\XINT_dsx_zeroloop {#1}{}\Z -}% -\def\XINT_xtrunc_prepare_I #1.#2#3% -{% - \expandafter\XINT_xtrunc_prepareB_aa\expandafter - {\romannumeral0\xintlength {#2}}{#2}{#1}% -}% -\def\XINT_xtrunc_prepareB_aa #1% -{% - \ifnum #1=\xint_c_i - \expandafter\XINT_xtrunc_prepareB_onedigit - \else - \expandafter\XINT_xtrunc_prepareB_PaBa - \fi - {#1}% -}% -\def\XINT_xtrunc_prepareB_onedigit #1#2% -{% - \ifcase#2 - \or\expandafter\XINT_xtrunc_BisOne - \or\expandafter\XINT_xtrunc_BisTwo - \else\expandafter\XINT_xtrunc_prepareB_PaBe - \fi {000}{0}{4}{#2}% -}% -\def\XINT_xtrunc_BisOne #1#2#3#4#5#6#7% -{% - #5.\romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter - {\the\numexpr #7}{}\Z {}% - \xintiloop [#6+-1] - \ifnum \xintiloopindex>\xint_c_ - 0000000000000000000000000000000000000000000000000000000000000000% - \repeat -}% -\def\XINT_xtrunc_BisTwo #1#2#3#4#5#6#7% -{% - \xintHalf {#5}.\ifodd\xintiiLDg{#5} 5\else 0\fi - \romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter - {\the\numexpr #7-\xint_c_i}{}\Z {}% - \xintiloop [#6+-1] - \ifnum \xintiloopindex>\xint_c_ - 0000000000000000000000000000000000000000000000000000000000000000% - \repeat -}% -\def\XINT_xtrunc_prepareB_PaBa #1#2% -{% - \expandafter\XINT_xtrunc_Pa\expandafter - {\romannumeral0\XINT_xtrunc_prepareB_a {#1}{#2}}% -}% -\def\XINT_xtrunc_prepareB_a #1% -{% - \expandafter\XINT_xtrunc_prepareB_c\expandafter - {\the\numexpr \xint_c_iv*((#1+\xint_c_i)/\xint_c_iv)}{#1}% -}% -\def\XINT_xtrunc_prepareB_c #1#2% -{% - \csname XINT_xtrunc_prepareB_d\romannumeral\numexpr#1-#2\endcsname - {#1}% -}% -\def\XINT_xtrunc_prepareB_d {\XINT_xtrunc_prepareB_e {}{0000}}% -\def\XINT_xtrunc_prepareB_di {\XINT_xtrunc_prepareB_e {0}{000}}% -\def\XINT_xtrunc_prepareB_dii {\XINT_xtrunc_prepareB_e {00}{00}}% -\def\XINT_xtrunc_prepareB_diii {\XINT_xtrunc_prepareB_e {000}{0}}% -\def\XINT_xtrunc_prepareB_PaBe #1#2#3#4% -{% - \expandafter\XINT_xtrunc_Pa\expandafter - {\romannumeral0\XINT_xtrunc_prepareB_e {#1}{#2}{#3}{#4}}% -}% -\def\XINT_xtrunc_prepareB_e #1#2#3#4% -{% - \ifnum#3=\xint_c_iv\expandafter\XINT_xtrunc_prepareLittleB_f - \else\expandafter\XINT_xtrunc_prepareB_f - \fi - #4#1{#3}{#2}{#1}% -}% -\def\XINT_xtrunc_prepareB_f #1#2#3#4#5#{% - \expandafter\space - \expandafter\XINT_div_prepareB_g - \the\numexpr #1#2#3#4+\xint_c_i\expandafter - .\the\numexpr (#1#2#3#4+\xint_c_i)/\xint_c_ii\expandafter - .\romannumeral0\xintreverseorder {#1#2#3#4#5}.{#1#2#3#4}% -}% -\def\XINT_xtrunc_prepareLittleB_f #1#{% - \expandafter\space\expandafter - \XINT_div_prepareB_g \the\numexpr #1/\xint_c_ii.{}.{}.{#1}% -}% -\def\XINT_xtrunc_Pa #1#2% -{% - \expandafter\XINT_xtrunc_Pb\romannumeral0#1{#2}{#1}% -}% -\def\XINT_xtrunc_Pb #1#2#3#4{#1.\XINT_xtrunc_A {#4}{#2}{#3}}% -\def\XINT_xtrunc_A #1% -{% - \unless\ifnum #1>\xint_c_ \XINT_xtrunc_transition\fi - \expandafter\XINT_xtrunc_B\expandafter{\the\numexpr #1-\xint_c_i}% -}% -\def\XINT_xtrunc_B #1#2#3% -{% - \expandafter\XINT_xtrunc_D\romannumeral0#3% - {#20000000000000000000000000000000000000000000000000000000000000000}% - {#1}{#3}% -}% -\def\XINT_xtrunc_D #1#2#3% -{% - \romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter - {\the\numexpr \xint_c_ii^vi-\xintLength{#1}}{}\Z {}#1% - \XINT_xtrunc_A {#3}{#2}% -}% -\def\XINT_xtrunc_transition\fi - \expandafter\XINT_xtrunc_B\expandafter #1#2#3#4% -{% - \fi - \ifnum #4=\xint_c_ \XINT_xtrunc_abort\fi - \expandafter\XINT_xtrunc_x\expandafter - {\romannumeral0\XINT_dsx_zeroloop {#4}{}\Z {#2}}{#3}{#4}% -}% -\def\XINT_xtrunc_x #1#2% -{% - \expandafter\XINT_xtrunc_y\romannumeral0#2{#1}% -}% -\def\XINT_xtrunc_y #1#2#3% -{% - \romannumeral0\expandafter\XINT_dsx_zeroloop\expandafter - {\the\numexpr #3-\xintLength{#1}}{}\Z {}#1% -}% -\def\XINT_xtrunc_abort\fi\expandafter\XINT_xtrunc_x\expandafter #1#2#3{\fi}% -% \end{macrocode} -% \subsection{\csh{xintDigits}} -% \lverb|& -% The mathchardef used to be called \XINT_digits, but for reasons originating in -% \xintNewExpr, release 1.09a uses \XINTdigits without underscore.| -% \begin{macrocode} -\mathchardef\XINTdigits 16 -\def\xintDigits #1#2% - {\afterassignment \xint_gobble_i \mathchardef\XINTdigits=}% -\def\xinttheDigits {\number\XINTdigits }% -% \end{macrocode} -% \subsection{\csh{xintFloat}} -% \lverb|1.07. Completely re-written in 1.08a, with spectacular speed -% gains. The earlier version was seriously silly when dealing with -% inputs having a big power of ten. Again some modifications in 1.08b -% for a better treatment of cases with long explicit numerators or -% denominators. -% -% Here again some inner macros used the \xintiquo with extra \xintnum overhead -% in 1.09a, 1.09f reinstalled use of \xintiiquo without this overhead.| -% \begin{macrocode} -\def\xintFloat {\romannumeral0\xintfloat }% -\def\xintfloat #1{\XINT_float_chkopt #1\Z }% -\def\XINT_float_chkopt #1% -{% - \ifx [#1\expandafter\XINT_float_opt - \else\expandafter\XINT_float_noopt - \fi #1% -}% -\def\XINT_float_noopt #1\Z -{% - \expandafter\XINT_float_a\expandafter\XINTdigits - \romannumeral0\XINT_infrac {#1}\XINT_float_Q -}% -\def\XINT_float_opt [\Z #1]#2% -{% - \expandafter\XINT_float_a\expandafter - {\the\numexpr #1\expandafter}% - \romannumeral0\XINT_infrac {#2}\XINT_float_Q -}% -\def\XINT_float_a #1#2#3% #1=P, #2=n, #3=A, #4=B -{% - \XINT_float_fork #3\Z {#1}{#2}% #1 = precision, #2=n -}% -\def\XINT_float_fork #1% -{% - \xint_UDzerominusfork - #1-\XINT_float_zero - 0#1\XINT_float_J - 0-{\XINT_float_K #1}% - \krof -}% -\def\XINT_float_zero #1\Z #2#3#4#5{ 0.e0}% -\def\XINT_float_J {\expandafter\xint_minus_thenstop\romannumeral0\XINT_float_K }% -\def\XINT_float_K #1\Z #2% #1=A, #2=P, #3=n, #4=B -{% - \expandafter\XINT_float_L\expandafter - {\the\numexpr\xintLength{#1}\expandafter}\expandafter - {\the\numexpr #2+\xint_c_ii}{#1}{#2}% -}% -\def\XINT_float_L #1#2% -{% - \ifnum #1>#2 - \expandafter\XINT_float_Ma - \else - \expandafter\XINT_float_Mc - \fi {#1}{#2}% -}% -\def\XINT_float_Ma #1#2#3% -{% - \expandafter\XINT_float_Mb\expandafter - {\the\numexpr #1-#2\expandafter\expandafter\expandafter}% - \expandafter\expandafter\expandafter - {\expandafter\xint_firstoftwo - \romannumeral0\XINT_split_fromleft_loop {#2}{}#3\W\W\W\W\W\W\W\W\Z - }{#2}% -}% -\def\XINT_float_Mb #1#2#3#4#5#6% #2=A', #3=P+2, #4=P, #5=n, #6=B -{% - \expandafter\XINT_float_N\expandafter - {\the\numexpr\xintLength{#6}\expandafter}\expandafter - {\the\numexpr #3\expandafter}\expandafter - {\the\numexpr #1+#5}% - {#6}{#3}{#2}{#4}% -}% long de B, P+2, n', B, |A'|=P+2, A', P -\def\XINT_float_Mc #1#2#3#4#5#6% -{% - \expandafter\XINT_float_N\expandafter - {\romannumeral0\xintlength{#6}}{#2}{#5}{#6}{#1}{#3}{#4}% -}% long de B, P+2, n, B, |A|, A, P -\def\XINT_float_N #1#2% -{% - \ifnum #1>#2 - \expandafter\XINT_float_O - \else - \expandafter\XINT_float_P - \fi {#1}{#2}% -}% -\def\XINT_float_O #1#2#3#4% -{% - \expandafter\XINT_float_P\expandafter - {\the\numexpr #2\expandafter}\expandafter - {\the\numexpr #2\expandafter}\expandafter - {\the\numexpr #3-#1+#2\expandafter\expandafter\expandafter}% - \expandafter\expandafter\expandafter - {\expandafter\xint_firstoftwo - \romannumeral0\XINT_split_fromleft_loop {#2}{}#4\W\W\W\W\W\W\W\W\Z - }% -}% |B|,P+2,n,B,|A|,A,P -\def\XINT_float_P #1#2#3#4#5#6#7#8% -{% - \expandafter #8\expandafter {\the\numexpr #1-#5+#2-\xint_c_i}% - {#6}{#4}{#7}{#3}% -}% |B|-|A|+P+1,A,B,P,n -\def\XINT_float_Q #1% -{% - \ifnum #1<\xint_c_ - \expandafter\XINT_float_Ri - \else - \expandafter\XINT_float_Rii - \fi {#1}% -}% -\def\XINT_float_Ri #1#2#3% -{% - \expandafter\XINT_float_Sa - \romannumeral0\xintiiquo {#2}% - {\XINT_dsx_addzerosnofuss {-#1}{#3}}\Z {#1}% -}% -\def\XINT_float_Rii #1#2#3% -{% - \expandafter\XINT_float_Sa - \romannumeral0\xintiiquo - {\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}\Z {#1}% -}% -\def\XINT_float_Sa #1% -{% - \if #19% - \xint_afterfi {\XINT_float_Sb\XINT_float_Wb }% - \else - \xint_afterfi {\XINT_float_Sb\XINT_float_Wa }% - \fi #1% -}% -\def\XINT_float_Sb #1#2\Z #3#4% -{% - \expandafter\XINT_float_T\expandafter - {\the\numexpr #4+\xint_c_i\expandafter}% - \romannumeral-`0\XINT_lenrord_loop 0{}#2\Z\W\W\W\W\W\W\W\Z #1{#3}{#4}% -}% -\def\XINT_float_T #1#2#3% -{% - \ifnum #2>#1 - \xint_afterfi{\XINT_float_U\XINT_float_Xb}% - \else - \xint_afterfi{\XINT_float_U\XINT_float_Xa #3}% - \fi -}% -\def\XINT_float_U #1#2% -{% - \ifnum #2<\xint_c_v - \expandafter\XINT_float_Va - \else - \expandafter\XINT_float_Vb - \fi #1% -}% -\def\XINT_float_Va #1#2\Z #3% -{% - \expandafter#1% - \romannumeral0\expandafter\XINT_float_Wa - \romannumeral0\XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax \Z -}% -\def\XINT_float_Vb #1#2\Z #3% -{% - \expandafter #1% - \romannumeral0\expandafter #3% - \romannumeral0\XINT_addm_A 0{}1000\W\X\Y\Z #2000\W\X\Y\Z \Z -}% -\def\XINT_float_Wa #1{ #1.}% -\def\XINT_float_Wb #1#2% - {\if #11\xint_afterfi{ 10.}\else\xint_afterfi{ #1.#2}\fi }% -\def\XINT_float_Xa #1\Z #2#3#4% -{% - \expandafter\XINT_float_Y\expandafter - {\the\numexpr #3+#4-#2}{#1}% -}% -\def\XINT_float_Xb #1\Z #2#3#4% -{% - \expandafter\XINT_float_Y\expandafter - {\the\numexpr #3+#4+\xint_c_i-#2}{#1}% -}% -\def\XINT_float_Y #1#2{ #2e#1}% -% \end{macrocode} -% \subsection{\csh{XINTinFloat}} -% \lverb|1.07. Completely rewritten in 1.08a for immensely greater efficiency -% when the power of ten is big: previous version had some very serious -% bottlenecks arising from the creation of long strings of zeros, which made -% things such as 2^999999 completely impossible, but now even 2^999999999 with -% 24 significant digits is no problem! Again (slightly) improved in 1.08b. -% -% I decide in 1.09a not to use anymore \romannumeral`-0 mais \romannumeral0 also -% in the float routines, for consistency of style. -% -% Here again some inner macros used the \xintiquo with extra \xintnum overhead -% in 1.09a, 1.09f fixed that to use \xintiiquo for example. -% -% 1.09i added a stupid bug to \XINT_infloat_zero when it changed 0[0] to a silly -% 0/1[0], breaking in particular \xintFloatAdd when one of the argument is zero -% :((( -% -% 1.09j fixes this. Besides, for notational coherence \XINT_inFloat and -% \XINT_infloat have been renamed respectively \XINTinFloat and \XINTinfloat in -% release 1.09j.| -% \begin{macrocode} -\def\XINTinFloat {\romannumeral0\XINTinfloat }% -\def\XINTinfloat [#1]#2% -{% - \expandafter\XINT_infloat_a\expandafter - {\the\numexpr #1\expandafter}% - \romannumeral0\XINT_infrac {#2}\XINT_infloat_Q -}% -\def\XINT_infloat_a #1#2#3% #1=P, #2=n, #3=A, #4=B -{% - \XINT_infloat_fork #3\Z {#1}{#2}% #1 = precision, #2=n -}% -\def\XINT_infloat_fork #1% -{% - \xint_UDzerominusfork - #1-\XINT_infloat_zero - 0#1\XINT_infloat_J - 0-{\XINT_float_K #1}% - \krof -}% -\def\XINT_infloat_zero #1\Z #2#3#4#5{ 0[0]}% -% the 0[0] was stupidly changed to 0/1[0] in 1.09i, with the result that the -% Float addition would crash when an operand was zero -\def\XINT_infloat_J {\expandafter-\romannumeral0\XINT_float_K }% -\def\XINT_infloat_Q #1% -{% - \ifnum #1<\xint_c_ - \expandafter\XINT_infloat_Ri - \else - \expandafter\XINT_infloat_Rii - \fi {#1}% -}% -\def\XINT_infloat_Ri #1#2#3% -{% - \expandafter\XINT_infloat_S\expandafter - {\romannumeral0\xintiiquo {#2}% - {\XINT_dsx_addzerosnofuss {-#1}{#3}}}{#1}% -}% -\def\XINT_infloat_Rii #1#2#3% -{% - \expandafter\XINT_infloat_S\expandafter - {\romannumeral0\xintiiquo - {\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}}{#1}% -}% -\def\XINT_infloat_S #1#2#3% -{% - \expandafter\XINT_infloat_T\expandafter - {\the\numexpr #3+\xint_c_i\expandafter}% - \romannumeral-`0\XINT_lenrord_loop 0{}#1\Z\W\W\W\W\W\W\W\Z - {#2}% -}% -\def\XINT_infloat_T #1#2#3% -{% - \ifnum #2>#1 - \xint_afterfi{\XINT_infloat_U\XINT_infloat_Wb}% - \else - \xint_afterfi{\XINT_infloat_U\XINT_infloat_Wa #3}% - \fi -}% -\def\XINT_infloat_U #1#2% -{% - \ifnum #2<\xint_c_v - \expandafter\XINT_infloat_Va - \else - \expandafter\XINT_infloat_Vb - \fi #1% -}% -\def\XINT_infloat_Va #1#2\Z -{% - \expandafter#1% - \romannumeral0\XINT_rord_main {}#2% - \xint_relax - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_bye\xint_bye\xint_bye\xint_bye - \xint_relax \Z -}% -\def\XINT_infloat_Vb #1#2\Z -{% - \expandafter #1% - \romannumeral0\XINT_addm_A 0{}1000\W\X\Y\Z #2000\W\X\Y\Z \Z -}% -\def\XINT_infloat_Wa #1\Z #2#3% -{% - \expandafter\XINT_infloat_X\expandafter - {\the\numexpr #3+\xint_c_i-#2}{#1}% -}% -\def\XINT_infloat_Wb #1\Z #2#3% -{% - \expandafter\XINT_infloat_X\expandafter - {\the\numexpr #3+\xint_c_ii-#2}{#1}% -}% -\def\XINT_infloat_X #1#2{ #2[#1]}% -% \end{macrocode} -% \subsection{\csh{xintAdd}} -% \begin{macrocode} -\def\xintAdd {\romannumeral0\xintadd }% -\def\xintadd #1% -{% - \expandafter\xint_fadd\expandafter {\romannumeral0\XINT_infrac {#1}}% -}% -\def\xint_fadd #1#2{\expandafter\XINT_fadd_A\romannumeral0\XINT_infrac{#2}#1}% -\def\XINT_fadd_A #1#2#3#4% -{% - \ifnum #4 > #1 - \xint_afterfi {\XINT_fadd_B {#1}}% - \else - \xint_afterfi {\XINT_fadd_B {#4}}% - \fi - {#1}{#4}{#2}{#3}% -}% -\def\XINT_fadd_B #1#2#3#4#5#6#7% -{% - \expandafter\XINT_fadd_C\expandafter - {\romannumeral0\xintiimul {#7}{#5}}% - {\romannumeral0\xintiiadd - {\romannumeral0\xintiimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% - {\romannumeral0\xintiimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}% - }% - {#1}% -}% -\def\XINT_fadd_C #1#2#3% -{% - \expandafter\XINT_fadd_D\expandafter {#2}{#3}{#1}% -}% -\def\XINT_fadd_D #1#2{\XINT_outfrac {#2}{#1}}% -% \end{macrocode} -% \subsection{\csh{xintSub}} -% \begin{macrocode} -\def\xintSub {\romannumeral0\xintsub }% -\def\xintsub #1% -{% - \expandafter\xint_fsub\expandafter {\romannumeral0\XINT_infrac {#1}}% -}% -\def\xint_fsub #1#2% - {\expandafter\XINT_fsub_A\romannumeral0\XINT_infrac {#2}#1}% -\def\XINT_fsub_A #1#2#3#4% -{% - \ifnum #4 > #1 - \xint_afterfi {\XINT_fsub_B {#1}}% - \else - \xint_afterfi {\XINT_fsub_B {#4}}% - \fi - {#1}{#4}{#2}{#3}% -}% -\def\XINT_fsub_B #1#2#3#4#5#6#7% -{% - \expandafter\XINT_fsub_C\expandafter - {\romannumeral0\xintiimul {#7}{#5}}% - {\romannumeral0\xintiisub - {\romannumeral0\xintiimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% - {\romannumeral0\xintiimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}% - }% - {#1}% -}% -\def\XINT_fsub_C #1#2#3% -{% - \expandafter\XINT_fsub_D\expandafter {#2}{#3}{#1}% -}% -\def\XINT_fsub_D #1#2{\XINT_outfrac {#2}{#1}}% -% \end{macrocode} -% \subsection{\csh{xintSum}} -% \begin{macrocode} -\def\xintSum {\romannumeral0\xintsum }% -\def\xintsum #1{\xintsumexpr #1\relax }% -\def\xintSumExpr {\romannumeral0\xintsumexpr }% -\def\xintsumexpr {\expandafter\XINT_fsumexpr\romannumeral-`0}% -\def\XINT_fsumexpr {\XINT_fsum_loop_a {0/1[0]}}% -\def\XINT_fsum_loop_a #1#2% -{% - \expandafter\XINT_fsum_loop_b \romannumeral-`0#2\Z {#1}% -}% -\def\XINT_fsum_loop_b #1% -{% - \xint_gob_til_relax #1\XINT_fsum_finished\relax - \XINT_fsum_loop_c #1% -}% -\def\XINT_fsum_loop_c #1\Z #2% -{% - \expandafter\XINT_fsum_loop_a\expandafter{\romannumeral0\xintadd {#2}{#1}}% -}% -\def\XINT_fsum_finished #1\Z #2{ #2}% -% \end{macrocode} -% \subsection{\csh{xintMul}} -% \begin{macrocode} -\def\xintMul {\romannumeral0\xintmul }% -\def\xintmul #1% -{% - \expandafter\xint_fmul\expandafter {\romannumeral0\XINT_infrac {#1}}% -}% -\def\xint_fmul #1#2% - {\expandafter\XINT_fmul_A\romannumeral0\XINT_infrac {#2}#1}% -\def\XINT_fmul_A #1#2#3#4#5#6% -{% - \expandafter\XINT_fmul_B - \expandafter{\the\numexpr #1+#4\expandafter}% - \expandafter{\romannumeral0\xintiimul {#6}{#3}}% - {\romannumeral0\xintiimul {#5}{#2}}% -}% -\def\XINT_fmul_B #1#2#3% -{% - \expandafter \XINT_fmul_C \expandafter{#3}{#1}{#2}% -}% -\def\XINT_fmul_C #1#2{\XINT_outfrac {#2}{#1}}% -% \end{macrocode} -% \subsection{\csh{xintSqr}} -% \begin{macrocode} -\def\xintSqr {\romannumeral0\xintsqr }% -\def\xintsqr #1% -{% - \expandafter\xint_fsqr\expandafter{\romannumeral0\XINT_infrac {#1}}% -}% -\def\xint_fsqr #1{\XINT_fmul_A #1#1}% -% \end{macrocode} -% \subsection{\csh{xintPow}} -% \lverb|& -% Modified in 1.06 to give the exponent to a \numexpr. -% -% With 1.07 and for use within the \xintexpr parser, we must allow -% fractions (which are integers in disguise) as input to the exponent, so we -% must have a variant which uses \xintNum and not only \numexpr -% for normalizing the input. Hence the \xintfPow here. -% -% 1.08b: well actually I -% think that with xintfrac.sty loaded the exponent should always be allowed to -% be a fraction giving an integer. So I do as for \xintFac, and remove here the -% duplicated. Then \xintexpr can use the \xintPow as defined here.| -% \begin{macrocode} -\def\xintPow {\romannumeral0\xintpow }% -\def\xintpow #1% -{% - \expandafter\xint_fpow\expandafter {\romannumeral0\XINT_infrac {#1}}% -}% -\def\xint_fpow #1#2% -{% - \expandafter\XINT_fpow_fork\the\numexpr \xintNum{#2}\relax\Z #1% -}% -\def\XINT_fpow_fork #1#2\Z -{% - \xint_UDzerominusfork - #1-\XINT_fpow_zero - 0#1\XINT_fpow_neg - 0-{\XINT_fpow_pos #1}% - \krof - {#2}% -}% -\def\XINT_fpow_zero #1#2#3#4{ 1/1[0]}% -\def\XINT_fpow_pos #1#2#3#4#5% -{% - \expandafter\XINT_fpow_pos_A\expandafter - {\the\numexpr #1#2*#3\expandafter}\expandafter - {\romannumeral0\xintiipow {#5}{#1#2}}% - {\romannumeral0\xintiipow {#4}{#1#2}}% -}% -\def\XINT_fpow_neg #1#2#3#4% -{% - \expandafter\XINT_fpow_pos_A\expandafter - {\the\numexpr -#1*#2\expandafter}\expandafter - {\romannumeral0\xintiipow {#3}{#1}}% - {\romannumeral0\xintiipow {#4}{#1}}% -}% -\def\XINT_fpow_pos_A #1#2#3% -{% - \expandafter\XINT_fpow_pos_B\expandafter {#3}{#1}{#2}% -}% -\def\XINT_fpow_pos_B #1#2{\XINT_outfrac {#2}{#1}}% -% \end{macrocode} -% \subsection{\csh{xintFac}} -% \lverb|1.07: to be used by the \xintexpr scanner which needs to be able to -% apply \xintFac -% to a fraction which is an integer in disguise; so we use \xintNum and not only -% \numexpr. Je modifie cela dans 1.08b, au lieu d'avoir un \xintfFac -% spécialement pour \xintexpr, tout simplement j'étends \xintFac comme les -% autres macros, pour qu'elle utilise \xintNum. | -% \begin{macrocode} -\def\xintFac {\romannumeral0\xintfac }% -\def\xintfac #1% -{% - \expandafter\XINT_fac_fork\expandafter{\the\numexpr \xintNum{#1}}% -}% -% \end{macrocode} -% \subsection{\csh{xintPrd}} -% \begin{macrocode} -\def\xintPrd {\romannumeral0\xintprd }% -\def\xintprd #1{\xintprdexpr #1\relax }% -\def\xintPrdExpr {\romannumeral0\xintprdexpr }% -\def\xintprdexpr {\expandafter\XINT_fprdexpr \romannumeral-`0}% -\def\XINT_fprdexpr {\XINT_fprod_loop_a {1/1[0]}}% -\def\XINT_fprod_loop_a #1#2% -{% - \expandafter\XINT_fprod_loop_b \romannumeral-`0#2\Z {#1}% -}% -\def\XINT_fprod_loop_b #1% -{% - \xint_gob_til_relax #1\XINT_fprod_finished\relax - \XINT_fprod_loop_c #1% -}% -\def\XINT_fprod_loop_c #1\Z #2% -{% - \expandafter\XINT_fprod_loop_a\expandafter{\romannumeral0\xintmul {#1}{#2}}% -}% -\def\XINT_fprod_finished #1\Z #2{ #2}% -% \end{macrocode} -% \subsection{\csh{xintDiv}} -% \begin{macrocode} -\def\xintDiv {\romannumeral0\xintdiv }% -\def\xintdiv #1% -{% - \expandafter\xint_fdiv\expandafter {\romannumeral0\XINT_infrac {#1}}% -}% -\def\xint_fdiv #1#2% - {\expandafter\XINT_fdiv_A\romannumeral0\XINT_infrac {#2}#1}% -\def\XINT_fdiv_A #1#2#3#4#5#6% -{% - \expandafter\XINT_fdiv_B - \expandafter{\the\numexpr #4-#1\expandafter}% - \expandafter{\romannumeral0\xintiimul {#2}{#6}}% - {\romannumeral0\xintiimul {#3}{#5}}% -}% -\def\XINT_fdiv_B #1#2#3% -{% - \expandafter\XINT_fdiv_C - \expandafter{#3}{#1}{#2}% -}% -\def\XINT_fdiv_C #1#2{\XINT_outfrac {#2}{#1}}% -% \end{macrocode} -% \subsection{\csh{xintIsOne}} -% \lverb|& -% New with 1.09a. Could be more efficient. For fractions with big powers of -% tens, it is better to use \xintCmp{f}{1}. Restyled in 1.09i.| -% \begin{macrocode} -\def\xintIsOne {\romannumeral0\xintisone }% -\def\xintisone #1{\expandafter\XINT_fracisone - \romannumeral0\xintrawwithzeros{#1}\Z }% -\def\XINT_fracisone #1/#2\Z - {\if0\XINT_Cmp {#1}{#2}\xint_afterfi{ 1}\else\xint_afterfi{ 0}\fi}% -% \end{macrocode} -% \subsection{\csh{xintGeq}} -% \lverb|& -% Rewritten completely in 1.08a to be less dumb when comparing fractions having -% big powers of tens.| -% \begin{macrocode} -\def\xintGeq {\romannumeral0\xintgeq }% -\def\xintgeq #1% -{% - \expandafter\xint_fgeq\expandafter {\romannumeral0\xintabs {#1}}% -}% -\def\xint_fgeq #1#2% -{% - \expandafter\XINT_fgeq_A \romannumeral0\xintabs {#2}#1% -}% -\def\XINT_fgeq_A #1% -{% - \xint_gob_til_zero #1\XINT_fgeq_Zii 0% - \XINT_fgeq_B #1% -}% -\def\XINT_fgeq_Zii 0\XINT_fgeq_B #1[#2]#3[#4]{ 1}% -\def\XINT_fgeq_B #1/#2[#3]#4#5/#6[#7]% -{% - \xint_gob_til_zero #4\XINT_fgeq_Zi 0% - \expandafter\XINT_fgeq_C\expandafter - {\the\numexpr #7-#3\expandafter}\expandafter - {\romannumeral0\xintiimul {#4#5}{#2}}% - {\romannumeral0\xintiimul {#6}{#1}}% -}% -\def\XINT_fgeq_Zi 0#1#2#3#4#5#6#7{ 0}% -\def\XINT_fgeq_C #1#2#3% -{% - \expandafter\XINT_fgeq_D\expandafter - {#3}{#1}{#2}% -}% -\def\XINT_fgeq_D #1#2#3% -{% - \expandafter\XINT_cntSgnFork\romannumeral-`0\expandafter\XINT_cntSgn - \the\numexpr #2+\xintLength{#3}-\xintLength{#1}\relax\Z - { 0}{\XINT_fgeq_E #2\Z {#3}{#1}}{ 1}% -}% -\def\XINT_fgeq_E #1% -{% - \xint_UDsignfork - #1\XINT_fgeq_Fd - -{\XINT_fgeq_Fn #1}% - \krof -}% -\def\XINT_fgeq_Fd #1\Z #2#3% -{% - \expandafter\XINT_fgeq_Fe\expandafter - {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#3}}{#2}% -}% -\def\XINT_fgeq_Fe #1#2{\XINT_geq_pre {#2}{#1}}% -\def\XINT_fgeq_Fn #1\Z #2#3% -{% - \expandafter\XINT_geq_pre\expandafter - {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}% -}% -% \end{macrocode} -% \subsection{\csh{xintMax}} -% \lverb|& -% Rewritten completely in 1.08a.| -% \begin{macrocode} -\def\xintMax {\romannumeral0\xintmax }% -\def\xintmax #1% -{% - \expandafter\xint_fmax\expandafter {\romannumeral0\xintraw {#1}}% -}% -\def\xint_fmax #1#2% -{% - \expandafter\XINT_fmax_A\romannumeral0\xintraw {#2}#1% -}% -\def\XINT_fmax_A #1#2/#3[#4]#5#6/#7[#8]% -{% - \xint_UDsignsfork - #1#5\XINT_fmax_minusminus - -#5\XINT_fmax_firstneg - #1-\XINT_fmax_secondneg - --\XINT_fmax_nonneg_a - \krof - #1#5{#2/#3[#4]}{#6/#7[#8]}% -}% -\def\XINT_fmax_minusminus --% - {\expandafter\xint_minus_thenstop\romannumeral0\XINT_fmin_nonneg_b }% -\def\XINT_fmax_firstneg #1-#2#3{ #1#2}% -\def\XINT_fmax_secondneg -#1#2#3{ #1#3}% -\def\XINT_fmax_nonneg_a #1#2#3#4% -{% - \XINT_fmax_nonneg_b {#1#3}{#2#4}% -}% -\def\XINT_fmax_nonneg_b #1#2% -{% - \if0\romannumeral0\XINT_fgeq_A #1#2% - \xint_afterfi{ #1}% - \else \xint_afterfi{ #2}% - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintMaxof}} -% \begin{macrocode} -\def\xintMaxof {\romannumeral0\xintmaxof }% -\def\xintmaxof #1{\expandafter\XINT_maxof_a\romannumeral-`0#1\relax }% -\def\XINT_maxof_a #1{\expandafter\XINT_maxof_b\romannumeral0\xintraw{#1}\Z }% -\def\XINT_maxof_b #1\Z #2% - {\expandafter\XINT_maxof_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_maxof_c #1% - {\xint_gob_til_relax #1\XINT_maxof_e\relax\XINT_maxof_d #1}% -\def\XINT_maxof_d #1\Z - {\expandafter\XINT_maxof_b\romannumeral0\xintmax {#1}}% -\def\XINT_maxof_e #1\Z #2\Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintMin}} -% \lverb|& -% Rewritten completely in 1.08a.| -% \begin{macrocode} -\def\xintMin {\romannumeral0\xintmin }% -\def\xintmin #1% -{% - \expandafter\xint_fmin\expandafter {\romannumeral0\xintraw {#1}}% -}% -\def\xint_fmin #1#2% -{% - \expandafter\XINT_fmin_A\romannumeral0\xintraw {#2}#1% -}% -\def\XINT_fmin_A #1#2/#3[#4]#5#6/#7[#8]% -{% - \xint_UDsignsfork - #1#5\XINT_fmin_minusminus - -#5\XINT_fmin_firstneg - #1-\XINT_fmin_secondneg - --\XINT_fmin_nonneg_a - \krof - #1#5{#2/#3[#4]}{#6/#7[#8]}% -}% -\def\XINT_fmin_minusminus --% - {\expandafter\xint_minus_thenstop\romannumeral0\XINT_fmax_nonneg_b }% -\def\XINT_fmin_firstneg #1-#2#3{ -#3}% -\def\XINT_fmin_secondneg -#1#2#3{ -#2}% -\def\XINT_fmin_nonneg_a #1#2#3#4% -{% - \XINT_fmin_nonneg_b {#1#3}{#2#4}% -}% -\def\XINT_fmin_nonneg_b #1#2% -{% - \if0\romannumeral0\XINT_fgeq_A #1#2% - \xint_afterfi{ #2}% - \else \xint_afterfi{ #1}% - \fi -}% -% \end{macrocode} -% \subsection{\csh{xintMinof}} -% \begin{macrocode} -\def\xintMinof {\romannumeral0\xintminof }% -\def\xintminof #1{\expandafter\XINT_minof_a\romannumeral-`0#1\relax }% -\def\XINT_minof_a #1{\expandafter\XINT_minof_b\romannumeral0\xintraw{#1}\Z }% -\def\XINT_minof_b #1\Z #2% - {\expandafter\XINT_minof_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_minof_c #1% - {\xint_gob_til_relax #1\XINT_minof_e\relax\XINT_minof_d #1}% -\def\XINT_minof_d #1\Z - {\expandafter\XINT_minof_b\romannumeral0\xintmin {#1}}% -\def\XINT_minof_e #1\Z #2\Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintCmp}} -% \lverb|& -% Rewritten completely in 1.08a to be less dumb when comparing fractions having -% big powers of tens. Incredibly, it seems that 1.08b introduced a bug in -% delimited arguments making the macro just non-functional when one of the input -% was zero! I -% did not detect this until working on release 1.09a, somehow I had not tested -% that -% \xintCmp just did NOT work! I must have done some last minute change... | -% \begin{macrocode} -\def\xintCmp {\romannumeral0\xintcmp }% -\def\xintcmp #1% -{% - \expandafter\xint_fcmp\expandafter {\romannumeral0\xintraw {#1}}% -}% -\def\xint_fcmp #1#2% -{% - \expandafter\XINT_fcmp_A\romannumeral0\xintraw {#2}#1% -}% -\def\XINT_fcmp_A #1#2/#3[#4]#5#6/#7[#8]% -{% - \xint_UDsignsfork - #1#5\XINT_fcmp_minusminus - -#5\XINT_fcmp_firstneg - #1-\XINT_fcmp_secondneg - --\XINT_fcmp_nonneg_a - \krof - #1#5{#2/#3[#4]}{#6/#7[#8]}% -}% -\def\XINT_fcmp_minusminus --#1#2{\XINT_fcmp_B #2#1}% -\def\XINT_fcmp_firstneg #1-#2#3{ -1}% -\def\XINT_fcmp_secondneg -#1#2#3{ 1}% -\def\XINT_fcmp_nonneg_a #1#2% -{% - \xint_UDzerosfork - #1#2\XINT_fcmp_zerozero - 0#2\XINT_fcmp_firstzero - #10\XINT_fcmp_secondzero - 00\XINT_fcmp_pos - \krof - #1#2% -}% -\def\XINT_fcmp_zerozero #1#2#3#4{ 0}% 1.08b had some [ and ] here!!! -\def\XINT_fcmp_firstzero #1#2#3#4{ -1}% incredibly I never saw that until -\def\XINT_fcmp_secondzero #1#2#3#4{ 1}% preparing 1.09a. -\def\XINT_fcmp_pos #1#2#3#4% -{% - \XINT_fcmp_B #1#3#2#4% -}% -\def\XINT_fcmp_B #1/#2[#3]#4/#5[#6]% -{% - \expandafter\XINT_fcmp_C\expandafter - {\the\numexpr #6-#3\expandafter}\expandafter - {\romannumeral0\xintiimul {#4}{#2}}% - {\romannumeral0\xintiimul {#5}{#1}}% -}% -\def\XINT_fcmp_C #1#2#3% -{% - \expandafter\XINT_fcmp_D\expandafter - {#3}{#1}{#2}% -}% -\def\XINT_fcmp_D #1#2#3% -{% - \expandafter\XINT_cntSgnFork\romannumeral-`0\expandafter\XINT_cntSgn - \the\numexpr #2+\xintLength{#3}-\xintLength{#1}\relax\Z - { -1}{\XINT_fcmp_E #2\Z {#3}{#1}}{ 1}% -}% -\def\XINT_fcmp_E #1% -{% - \xint_UDsignfork - #1\XINT_fcmp_Fd - -{\XINT_fcmp_Fn #1}% - \krof -}% -\def\XINT_fcmp_Fd #1\Z #2#3% -{% - \expandafter\XINT_fcmp_Fe\expandafter - {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#3}}{#2}% -}% -\def\XINT_fcmp_Fe #1#2{\XINT_cmp_pre {#2}{#1}}% -\def\XINT_fcmp_Fn #1\Z #2#3% -{% - \expandafter\XINT_cmp_pre\expandafter - {\romannumeral0\XINT_dsx_addzerosnofuss {#1}{#2}}{#3}% -}% -% \end{macrocode} -% \subsection{\csh{xintAbs}} -% \lverb|Simplified in 1.09i. (original macro was written before \xintRaw)| -% \begin{macrocode} -\def\xintAbs {\romannumeral0\xintabs }% -\def\xintabs #1{\expandafter\XINT_abs\romannumeral0\xintraw {#1}}% -% \end{macrocode} -% \subsection{\csh{xintOpp}} -% \lverb|caution that -#1 would not be ok if #1 has [n] -% stuff. Simplified in 1.09i. (original macro was written before \xintRaw)| -% \begin{macrocode} -\def\xintOpp {\romannumeral0\xintopp }% -\def\xintopp #1{\expandafter\XINT_opp\romannumeral0\xintraw {#1}}% -% \end{macrocode} -% \subsection{\csh{xintSgn}} -% \lverb|Simplified in 1.09i. (original macro was written before \xintRaw)| -% \begin{macrocode} -\def\xintSgn {\romannumeral0\xintsgn }% -\def\xintsgn #1{\expandafter\XINT_sgn\romannumeral0\xintraw {#1}\Z }% -% \end{macrocode} -% \subsection{\csh{xintFloatAdd}, \csh{XINTinFloatAdd}} -% \lverb|1.07; 1.09ka improves a bit the efficieny of the coding of -% \XINT_FL_Add_d.| -% \begin{macrocode} -\def\xintFloatAdd {\romannumeral0\xintfloatadd }% -\def\xintfloatadd #1{\XINT_fladd_chkopt \xintfloat #1\Z }% -\def\XINTinFloatAdd {\romannumeral0\XINTinfloatadd }% -\def\XINTinfloatadd #1{\XINT_fladd_chkopt \XINTinfloat #1\Z }% -\def\XINT_fladd_chkopt #1#2% -{% - \ifx [#2\expandafter\XINT_fladd_opt - \else\expandafter\XINT_fladd_noopt - \fi #1#2% -}% -\def\XINT_fladd_noopt #1#2\Z #3% -{% - #1[\XINTdigits]{\XINT_FL_Add {\XINTdigits+\xint_c_ii}{#2}{#3}}% -}% -\def\XINT_fladd_opt #1[\Z #2]#3#4% -{% - #1[#2]{\XINT_FL_Add {#2+\xint_c_ii}{#3}{#4}}% -}% -\def\XINT_FL_Add #1#2% -{% - \expandafter\XINT_FL_Add_a\expandafter{\the\numexpr #1\expandafter}% - \expandafter{\romannumeral0\XINTinfloat [#1]{#2}}% -}% -\def\XINT_FL_Add_a #1#2#3% -{% - \expandafter\XINT_FL_Add_b\romannumeral0\XINTinfloat [#1]{#3}#2{#1}% -}% -\def\XINT_FL_Add_b #1% -{% - \xint_gob_til_zero #1\XINT_FL_Add_zero 0\XINT_FL_Add_c #1% -}% -\def\XINT_FL_Add_c #1[#2]#3% -{% - \xint_gob_til_zero #3\XINT_FL_Add_zerobis 0\XINT_FL_Add_d #1[#2]#3% -}% -\def\XINT_FL_Add_d #1[#2]#3[#4]#5% -{% - \ifnum \numexpr #2-#4-#5>\xint_c_i - \expandafter \xint_secondofthree_thenstop - \else - \ifnum \numexpr #4-#2-#5>\xint_c_i - \expandafter\expandafter\expandafter\xint_thirdofthree_thenstop - \fi - \fi - \xintadd {#1[#2]}{#3[#4]}% -}% -\def\XINT_FL_Add_zero 0\XINT_FL_Add_c 0[0]#1[#2]#3{#1[#2]}% -\def\XINT_FL_Add_zerobis 0\XINT_FL_Add_d #1[#2]0[0]#3{#1[#2]}% -% \end{macrocode} -% \subsection{\csh{xintFloatSub}, \csh{XINTinFloatSub}} -% \lverb|1.07| -% \begin{macrocode} -\def\xintFloatSub {\romannumeral0\xintfloatsub }% -\def\xintfloatsub #1{\XINT_flsub_chkopt \xintfloat #1\Z }% -\def\XINTinFloatSub {\romannumeral0\XINTinfloatsub }% -\def\XINTinfloatsub #1{\XINT_flsub_chkopt \XINTinfloat #1\Z }% -\def\XINT_flsub_chkopt #1#2% -{% - \ifx [#2\expandafter\XINT_flsub_opt - \else\expandafter\XINT_flsub_noopt - \fi #1#2% -}% -\def\XINT_flsub_noopt #1#2\Z #3% -{% - #1[\XINTdigits]{\XINT_FL_Add {\XINTdigits+\xint_c_ii}{#2}{\xintOpp{#3}}}% -}% -\def\XINT_flsub_opt #1[\Z #2]#3#4% -{% - #1[#2]{\XINT_FL_Add {#2+\xint_c_ii}{#3}{\xintOpp{#4}}}% -}% -% \end{macrocode} -% \subsection{\csh{xintFloatMul}, \csh{XINTinFloatMul}} -% \lverb|1.07| -% \begin{macrocode} -\def\xintFloatMul {\romannumeral0\xintfloatmul}% -\def\xintfloatmul #1{\XINT_flmul_chkopt \xintfloat #1\Z }% -\def\XINTinFloatMul {\romannumeral0\XINTinfloatmul }% -\def\XINTinfloatmul #1{\XINT_flmul_chkopt \XINTinfloat #1\Z }% -\def\XINT_flmul_chkopt #1#2% -{% - \ifx [#2\expandafter\XINT_flmul_opt - \else\expandafter\XINT_flmul_noopt - \fi #1#2% -}% -\def\XINT_flmul_noopt #1#2\Z #3% -{% - #1[\XINTdigits]{\XINT_FL_Mul {\XINTdigits+\xint_c_ii}{#2}{#3}}% -}% -\def\XINT_flmul_opt #1[\Z #2]#3#4% -{% - #1[#2]{\XINT_FL_Mul {#2+\xint_c_ii}{#3}{#4}}% -}% -\def\XINT_FL_Mul #1#2% -{% - \expandafter\XINT_FL_Mul_a\expandafter{\the\numexpr #1\expandafter}% - \expandafter{\romannumeral0\XINTinfloat [#1]{#2}}% -}% -\def\XINT_FL_Mul_a #1#2#3% -{% - \expandafter\XINT_FL_Mul_b\romannumeral0\XINTinfloat [#1]{#3}#2% -}% -\def\XINT_FL_Mul_b #1[#2]#3[#4]{\xintE{\xintiiMul {#1}{#3}}{#2+#4}}% -% \end{macrocode} -% \subsection{\csh{xintFloatDiv}, \csh{XINTinFloatDiv}} -% \lverb|1.07| -% \begin{macrocode} -\def\xintFloatDiv {\romannumeral0\xintfloatdiv}% -\def\xintfloatdiv #1{\XINT_fldiv_chkopt \xintfloat #1\Z }% -\def\XINTinFloatDiv {\romannumeral0\XINTinfloatdiv }% -\def\XINTinfloatdiv #1{\XINT_fldiv_chkopt \XINTinfloat #1\Z }% -\def\XINT_fldiv_chkopt #1#2% -{% - \ifx [#2\expandafter\XINT_fldiv_opt - \else\expandafter\XINT_fldiv_noopt - \fi #1#2% -}% -\def\XINT_fldiv_noopt #1#2\Z #3% -{% - #1[\XINTdigits]{\XINT_FL_Div {\XINTdigits+\xint_c_ii}{#2}{#3}}% -}% -\def\XINT_fldiv_opt #1[\Z #2]#3#4% -{% - #1[#2]{\XINT_FL_Div {#2+\xint_c_ii}{#3}{#4}}% -}% -\def\XINT_FL_Div #1#2% -{% - \expandafter\XINT_FL_Div_a\expandafter{\the\numexpr #1\expandafter}% - \expandafter{\romannumeral0\XINTinfloat [#1]{#2}}% -}% -\def\XINT_FL_Div_a #1#2#3% -{% - \expandafter\XINT_FL_Div_b\romannumeral0\XINTinfloat [#1]{#3}#2% -}% -\def\XINT_FL_Div_b #1[#2]#3[#4]{\xintE{#3/#1}{#4-#2}}% -% \end{macrocode} -% \subsection{\csh{XINTinFloatSum}} -% \lverb|1.09a: quick write-up, for use by \xintfloatexpr, will need to be -% thought through again. Renamed (and slightly modified) in 1.09h. Should be -% extended for optional precision. Should be rewritten for optimization. | -% \begin{macrocode} -\def\XINTinFloatSum {\romannumeral0\XINTinfloatsum }% -\def\XINTinfloatsum #1{\expandafter\XINT_floatsum_a\romannumeral-`0#1\relax }% -\def\XINT_floatsum_a #1{\expandafter\XINT_floatsum_b - \romannumeral0\XINTinfloat[\XINTdigits]{#1}\Z }% -\def\XINT_floatsum_b #1\Z #2% - {\expandafter\XINT_floatsum_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_floatsum_c #1% - {\xint_gob_til_relax #1\XINT_floatsum_e\relax\XINT_floatsum_d #1}% -\def\XINT_floatsum_d #1\Z - {\expandafter\XINT_floatsum_b\romannumeral0\XINTinfloatadd {#1}}% -\def\XINT_floatsum_e #1\Z #2\Z { #2}% -% \end{macrocode} -% \subsection{\csh{XINTinFloatPrd}} -% \lverb|1.09a: quick write-up, for use by \xintfloatexpr, will need to be -% thought through again. Renamed (and slightly modified) in 1.09h. Should be -% extended for optional precision. Should be rewritten for optimization. | -% \begin{macrocode} -\def\XINTinFloatPrd {\romannumeral0\XINTinfloatprd }% -\def\XINTinfloatprd #1{\expandafter\XINT_floatprd_a\romannumeral-`0#1\relax }% -\def\XINT_floatprd_a #1{\expandafter\XINT_floatprd_b - \romannumeral0\XINTinfloat[\XINTdigits]{#1}\Z }% -\def\XINT_floatprd_b #1\Z #2% - {\expandafter\XINT_floatprd_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_floatprd_c #1% - {\xint_gob_til_relax #1\XINT_floatprd_e\relax\XINT_floatprd_d #1}% -\def\XINT_floatprd_d #1\Z - {\expandafter\XINT_floatprd_b\romannumeral0\XINTinfloatmul {#1}}% -\def\XINT_floatprd_e #1\Z #2\Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintFloatPow}, \csh{XINTinFloatPow}} -% \lverb|1.07. Release 1.09j has re-organized the core loop, and -% \XINT_flpow_prd sub-routine has been removed.| -% \begin{macrocode} -\def\xintFloatPow {\romannumeral0\xintfloatpow}% -\def\xintfloatpow #1{\XINT_flpow_chkopt \xintfloat #1\Z }% -\def\XINTinFloatPow {\romannumeral0\XINTinfloatpow }% -\def\XINTinfloatpow #1{\XINT_flpow_chkopt \XINTinfloat #1\Z }% -\def\XINT_flpow_chkopt #1#2% -{% - \ifx [#2\expandafter\XINT_flpow_opt - \else\expandafter\XINT_flpow_noopt - \fi - #1#2% -}% -\def\XINT_flpow_noopt #1#2\Z #3% -{% - \expandafter\XINT_flpow_checkB_start\expandafter - {\the\numexpr #3\expandafter}\expandafter - {\the\numexpr \XINTdigits}{#2}{#1[\XINTdigits]}% -}% -\def\XINT_flpow_opt #1[\Z #2]#3#4% -{% - \expandafter\XINT_flpow_checkB_start\expandafter - {\the\numexpr #4\expandafter}\expandafter - {\the\numexpr #2}{#3}{#1[#2]}% -}% -\def\XINT_flpow_checkB_start #1{\XINT_flpow_checkB_a #1\Z }% -\def\XINT_flpow_checkB_a #1% -{% - \xint_UDzerominusfork - #1-\XINT_flpow_BisZero - 0#1{\XINT_flpow_checkB_b 1}% - 0-{\XINT_flpow_checkB_b 0#1}% - \krof -}% -\def\XINT_flpow_BisZero \Z #1#2#3{#3{1/1[0]}}% -\def\XINT_flpow_checkB_b #1#2\Z #3% -{% - \expandafter\XINT_flpow_checkB_c \expandafter - {\romannumeral0\xintlength{#2}}{#3}{#2}#1% -}% -\def\XINT_flpow_checkB_c #1#2% -{% - \expandafter\XINT_flpow_checkB_d \expandafter - {\the\numexpr \expandafter\xintLength\expandafter - {\the\numexpr #1*20/\xint_c_iii }+#1+#2+\xint_c_i }% -}% -\def\XINT_flpow_checkB_d #1#2#3#4% -{% - \expandafter \XINT_flpow_a - \romannumeral0\XINTinfloat [#1]{#4}{#1}{#2}#3% -}% -\def\XINT_flpow_a #1% -{% - \xint_UDzerominusfork - #1-\XINT_flpow_zero - 0#1{\XINT_flpow_b 1}% - 0-{\XINT_flpow_b 0#1}% - \krof -}% -\def\XINT_flpow_b #1#2[#3]#4#5% -{% - \XINT_flpow_loopI {#5}{#2[#3]}{\romannumeral0\XINTinfloatmul [#4]}% - {#1*\ifodd #5 1\else 0\fi}% -}% -\def\XINT_flpow_zero [#1]#2#3#4#5% -% xint is not equipped to signal infinity, the 2^31 will provoke -% deliberately a number too big and arithmetic overflow in \XINT_float_Xb -{% - \if #41\xint_afterfi {\xintError:DivisionByZero #5{1[2147483648]}}% - \else \xint_afterfi {#5{0[0]}}\fi -}% -\def\XINT_flpow_loopI #1% -{% - \ifnum #1=\xint_c_i\XINT_flpow_ItoIII\fi - \ifodd #1 - \expandafter\XINT_flpow_loopI_odd - \else - \expandafter\XINT_flpow_loopI_even - \fi - {#1}% -}% -\def\XINT_flpow_ItoIII\fi #1\fi #2#3#4#5% -{% - \fi\expandafter\XINT_flpow_III\the\numexpr #5\relax #3% -}% -\def\XINT_flpow_loopI_even #1#2#3% -{% - \expandafter\XINT_flpow_loopI\expandafter - {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter - {#3{#2}{#2}}{#3}% -}% -\def\XINT_flpow_loopI_odd #1#2#3% -{% - \expandafter\XINT_flpow_loopII\expandafter - {\the\numexpr #1/\xint_c_ii-\xint_c_i\expandafter}\expandafter - {#3{#2}{#2}}{#3}{#2}% -}% -\def\XINT_flpow_loopII #1% -{% - \ifnum #1 = \xint_c_i\XINT_flpow_IItoIII\fi - \ifodd #1 - \expandafter\XINT_flpow_loopII_odd - \else - \expandafter\XINT_flpow_loopII_even - \fi - {#1}% -}% -\def\XINT_flpow_loopII_even #1#2#3% -{% - \expandafter\XINT_flpow_loopII\expandafter - {\the\numexpr #1/\xint_c_ii\expandafter}\expandafter - {#3{#2}{#2}}{#3}% -}% -\def\XINT_flpow_loopII_odd #1#2#3#4% -{% - \expandafter\XINT_flpow_loopII_odda\expandafter - {#3{#2}{#4}}{#1}{#2}{#3}% -}% -\def\XINT_flpow_loopII_odda #1#2#3#4% -{% - \expandafter\XINT_flpow_loopII\expandafter - {\the\numexpr #2/\xint_c_ii-\xint_c_i\expandafter}\expandafter - {#4{#3}{#3}}{#4}{#1}% -}% -\def\XINT_flpow_IItoIII\fi #1\fi #2#3#4#5#6% -{% - \fi\expandafter\XINT_flpow_III\the\numexpr #6\expandafter\relax - #4{#3}{#5}% -}% -\def\XINT_flpow_III #1#2[#3]#4% -{% - \expandafter\XINT_flpow_IIIend\expandafter - {\the\numexpr\if #41-\fi#3\expandafter}% - \xint_UDzerofork - #4{{#2}}% - 0{{1/#2}}% - \krof #1% -}% -\def\XINT_flpow_IIIend #1#2#3#4% -{% - \xint_UDzerofork - #3{#4{#2[#1]}}% - 0{#4{-#2[#1]}}% - \krof -}% -% \end{macrocode} -% \subsection{\csh{xintFloatPower}, \csh{XINTinFloatPower}} -% \lverb|1.07. The core loop has been re-organized in 1.09j for some slight -% efficiency gain. | -% \begin{macrocode} -\def\xintFloatPower {\romannumeral0\xintfloatpower}% -\def\xintfloatpower #1{\XINT_flpower_chkopt \xintfloat #1\Z }% -\def\XINTinFloatPower {\romannumeral0\XINTinfloatpower}% -\def\XINTinfloatpower #1{\XINT_flpower_chkopt \XINTinfloat #1\Z }% -\def\XINT_flpower_chkopt #1#2% -{% - \ifx [#2\expandafter\XINT_flpower_opt - \else\expandafter\XINT_flpower_noopt - \fi - #1#2% -}% -\def\XINT_flpower_noopt #1#2\Z #3% -{% - \expandafter\XINT_flpower_checkB_start\expandafter - {\the\numexpr \XINTdigits\expandafter}\expandafter - {\romannumeral0\xintnum{#3}}{#2}{#1[\XINTdigits]}% -}% -\def\XINT_flpower_opt #1[\Z #2]#3#4% -{% - \expandafter\XINT_flpower_checkB_start\expandafter - {\the\numexpr #2\expandafter}\expandafter - {\romannumeral0\xintnum{#4}}{#3}{#1[#2]}% -}% -\def\XINT_flpower_checkB_start #1#2{\XINT_flpower_checkB_a #2\Z {#1}}% -\def\XINT_flpower_checkB_a #1% -{% - \xint_UDzerominusfork - #1-\XINT_flpower_BisZero - 0#1{\XINT_flpower_checkB_b 1}% - 0-{\XINT_flpower_checkB_b 0#1}% - \krof -}% -\def\XINT_flpower_BisZero \Z #1#2#3{#3{1/1[0]}}% -\def\XINT_flpower_checkB_b #1#2\Z #3% -{% - \expandafter\XINT_flpower_checkB_c \expandafter - {\romannumeral0\xintlength{#2}}{#3}{#2}#1% -}% -\def\XINT_flpower_checkB_c #1#2% -{% - \expandafter\XINT_flpower_checkB_d \expandafter - {\the\numexpr \expandafter\xintLength\expandafter - {\the\numexpr #1*20/\xint_c_iii }+#1+#2+\xint_c_i }% -}% -\def\XINT_flpower_checkB_d #1#2#3#4% -{% - \expandafter \XINT_flpower_a - \romannumeral0\XINTinfloat [#1]{#4}{#1}{#2}#3% -}% -\def\XINT_flpower_a #1% -{% - \xint_UDzerominusfork - #1-\XINT_flpow_zero - 0#1{\XINT_flpower_b 1}% - 0-{\XINT_flpower_b 0#1}% - \krof -}% -\def\XINT_flpower_b #1#2[#3]#4#5% -{% - \XINT_flpower_loopI {#5}{#2[#3]}{\romannumeral0\XINTinfloatmul [#4]}% - {#1*\xintiiOdd {#5}}% -}% -\def\XINT_flpower_loopI #1% -{% - \if1\XINT_isOne {#1}\XINT_flpower_ItoIII\fi - \if1\xintiiOdd{#1}% - \expandafter\expandafter\expandafter\XINT_flpower_loopI_odd - \else - \expandafter\expandafter\expandafter\XINT_flpower_loopI_even - \fi - \expandafter {\romannumeral0\xinthalf{#1}}% -}% -\def\XINT_flpower_ItoIII\fi #1\fi\expandafter #2#3#4#5% -{% - \fi\expandafter\XINT_flpow_III \the\numexpr #5\relax #3% -}% -\def\XINT_flpower_loopI_even #1#2#3% -{% - \expandafter\XINT_flpower_toI\expandafter {#3{#2}{#2}}{#1}{#3}% -}% -\def\XINT_flpower_loopI_odd #1#2#3% -{% - \expandafter\XINT_flpower_toII\expandafter {#3{#2}{#2}}{#1}{#3}{#2}% -}% -\def\XINT_flpower_toI #1#2{\XINT_flpower_loopI {#2}{#1}}% -\def\XINT_flpower_toII #1#2{\XINT_flpower_loopII {#2}{#1}}% -\def\XINT_flpower_loopII #1% -{% - \if1\XINT_isOne {#1}\XINT_flpower_IItoIII\fi - \if1\xintiiOdd{#1}% - \expandafter\expandafter\expandafter\XINT_flpower_loopII_odd - \else - \expandafter\expandafter\expandafter\XINT_flpower_loopII_even - \fi - \expandafter {\romannumeral0\xinthalf{#1}}% -}% -\def\XINT_flpower_loopII_even #1#2#3% -{% - \expandafter\XINT_flpower_toII\expandafter - {#3{#2}{#2}}{#1}{#3}% -}% -\def\XINT_flpower_loopII_odd #1#2#3#4% -{% - \expandafter\XINT_flpower_loopII_odda\expandafter - {#3{#2}{#4}}{#2}{#3}{#1}% -}% -\def\XINT_flpower_loopII_odda #1#2#3#4% -{% - \expandafter\XINT_flpower_toII\expandafter - {#3{#2}{#2}}{#4}{#3}{#1}% -}% -\def\XINT_flpower_IItoIII\fi #1\fi\expandafter #2#3#4#5#6% -{% - \fi\expandafter\XINT_flpow_III\the\numexpr #6\expandafter\relax - #4{#3}{#5}% -}% -% \end{macrocode} -% \subsection{\csh{xintFloatSqrt}, \csh{XINTinFloatSqrt}} -% \lverb|1.08| -% \begin{macrocode} -\def\xintFloatSqrt {\romannumeral0\xintfloatsqrt }% -\def\xintfloatsqrt #1{\XINT_flsqrt_chkopt \xintfloat #1\Z }% -\def\XINTinFloatSqrt {\romannumeral0\XINTinfloatsqrt }% -\def\XINTinfloatsqrt #1{\XINT_flsqrt_chkopt \XINTinfloat #1\Z }% -\def\XINT_flsqrt_chkopt #1#2% -{% - \ifx [#2\expandafter\XINT_flsqrt_opt - \else\expandafter\XINT_flsqrt_noopt - \fi #1#2% -}% -\def\XINT_flsqrt_noopt #1#2\Z -{% - #1[\XINTdigits]{\XINT_FL_sqrt \XINTdigits {#2}}% -}% -\def\XINT_flsqrt_opt #1[\Z #2]#3% -{% - #1[#2]{\XINT_FL_sqrt {#2}{#3}}% -}% -\def\XINT_FL_sqrt #1% -{% - \ifnum\numexpr #1<\xint_c_xviii - \xint_afterfi {\XINT_FL_sqrt_a\xint_c_xviii}% - \else - \xint_afterfi {\XINT_FL_sqrt_a {#1+\xint_c_i}}% - \fi -}% -\def\XINT_FL_sqrt_a #1#2% -{% - \expandafter\XINT_FL_sqrt_checkifzeroorneg - \romannumeral0\XINTinfloat [#1]{#2}% -}% -\def\XINT_FL_sqrt_checkifzeroorneg #1% -{% - \xint_UDzerominusfork - #1-\XINT_FL_sqrt_iszero - 0#1\XINT_FL_sqrt_isneg - 0-{\XINT_FL_sqrt_b #1}% - \krof -}% -\def\XINT_FL_sqrt_iszero #1[#2]{0[0]}% -\def\XINT_FL_sqrt_isneg #1[#2]{\xintError:RootOfNegative 0[0]}% -\def\XINT_FL_sqrt_b #1[#2]% -{% - \ifodd #2 - \xint_afterfi{\XINT_FL_sqrt_c 01}% - \else - \xint_afterfi{\XINT_FL_sqrt_c {}0}% - \fi - {#1}{#2}% -}% -\def\XINT_FL_sqrt_c #1#2#3#4% -{% - \expandafter\XINT_flsqrt\expandafter {\the\numexpr #4-#2}{#3#1}% -}% -\def\XINT_flsqrt #1#2% -{% - \expandafter\XINT_sqrt_a - \expandafter{\romannumeral0\xintlength {#2}}\XINT_flsqrt_big_d {#2}{#1}% -}% -\def\XINT_flsqrt_big_d #1#2% -{% - \ifodd #2 - \expandafter\expandafter\expandafter\XINT_flsqrt_big_eB - \else - \expandafter\expandafter\expandafter\XINT_flsqrt_big_eA - \fi - \expandafter {\the\numexpr (#2-\xint_c_i)/\xint_c_ii }{#1}% -}% -\def\XINT_flsqrt_big_eA #1#2#3% -{% - \XINT_flsqrt_big_eA_a #3\Z {#2}{#1}{#3}% -}% -\def\XINT_flsqrt_big_eA_a #1#2#3#4#5#6#7#8#9\Z -{% - \XINT_flsqrt_big_eA_b {#1#2#3#4#5#6#7#8}% -}% -\def\XINT_flsqrt_big_eA_b #1#2% -{% - \expandafter\XINT_flsqrt_big_f - \romannumeral0\XINT_flsqrt_small_e {#2001}{#1}% -}% -\def\XINT_flsqrt_big_eB #1#2#3% -{% - \XINT_flsqrt_big_eB_a #3\Z {#2}{#1}{#3}% -}% -\def\XINT_flsqrt_big_eB_a #1#2#3#4#5#6#7#8#9% -{% - \XINT_flsqrt_big_eB_b {#1#2#3#4#5#6#7#8#9}% -}% -\def\XINT_flsqrt_big_eB_b #1#2\Z #3% -{% - \expandafter\XINT_flsqrt_big_f - \romannumeral0\XINT_flsqrt_small_e {#30001}{#1}% -}% -\def\XINT_flsqrt_small_e #1#2% -{% - \expandafter\XINT_flsqrt_small_f\expandafter - {\the\numexpr #1*#1-#2-\xint_c_i}{#1}% -}% -\def\XINT_flsqrt_small_f #1#2% -{% - \expandafter\XINT_flsqrt_small_g\expandafter - {\the\numexpr (#1+#2)/(2*#2)-\xint_c_i }{#1}{#2}% -}% -\def\XINT_flsqrt_small_g #1% -{% - \ifnum #1>\xint_c_ - \expandafter\XINT_flsqrt_small_h - \else - \expandafter\XINT_flsqrt_small_end - \fi - {#1}% -}% -\def\XINT_flsqrt_small_h #1#2#3% -{% - \expandafter\XINT_flsqrt_small_f\expandafter - {\the\numexpr #2-\xint_c_ii*#1*#3+#1*#1\expandafter}\expandafter - {\the\numexpr #3-#1}% -}% -\def\XINT_flsqrt_small_end #1#2#3% -{% - \expandafter\space\expandafter - {\the\numexpr \xint_c_i+#3*\xint_c_x^iv- - (#2*\xint_c_x^iv+#3)/(\xint_c_ii*#3)}% -}% -\def\XINT_flsqrt_big_f #1% -{% - \expandafter\XINT_flsqrt_big_fa\expandafter - {\romannumeral0\xintiisqr {#1}}{#1}% -}% -\def\XINT_flsqrt_big_fa #1#2#3#4% -{% - \expandafter\XINT_flsqrt_big_fb\expandafter - {\romannumeral0\XINT_dsx_addzerosnofuss - {\numexpr #3-\xint_c_viii\relax}{#2}}% - {\romannumeral0\xintiisub - {\XINT_dsx_addzerosnofuss - {\numexpr \xint_c_ii*(#3-\xint_c_viii)\relax}{#1}}{#4}}% - {#3}% -}% -\def\XINT_flsqrt_big_fb #1#2% -{% - \expandafter\XINT_flsqrt_big_g\expandafter {#2}{#1}% -}% -\def\XINT_flsqrt_big_g #1#2% -{% - \expandafter\XINT_flsqrt_big_j - \romannumeral0\xintiidivision - {#1}{\romannumeral0\XINT_dbl_pos #2\R\R\R\R\R\R\R\Z \W\W\W\W\W\W\W }{#2}% -}% -\def\XINT_flsqrt_big_j #1% -{% - \if0\XINT_Sgn #1\Z - \expandafter \XINT_flsqrt_big_end_a - \else \expandafter \XINT_flsqrt_big_k - \fi {#1}% -}% -\def\XINT_flsqrt_big_k #1#2#3% -{% - \expandafter\XINT_flsqrt_big_l\expandafter - {\romannumeral0\XINT_sub_pre {#3}{#1}}% - {\romannumeral0\xintiiadd {#2}{\romannumeral0\XINT_sqr {#1}}}% -}% -\def\XINT_flsqrt_big_l #1#2% -{% - \expandafter\XINT_flsqrt_big_g\expandafter - {#2}{#1}% -}% -\def\XINT_flsqrt_big_end_a #1#2#3#4#5% -{% - \expandafter\XINT_flsqrt_big_end_b\expandafter - {\the\numexpr -#4+#5/\xint_c_ii\expandafter}\expandafter - {\romannumeral0\xintiisub - {\XINT_dsx_addzerosnofuss {#4}{#3}}% - {\xintHalf{\xintiiQuo{\XINT_dsx_addzerosnofuss {#4}{#2}}{#3}}}}% -}% -\def\XINT_flsqrt_big_end_b #1#2{#2[#1]}% -% \end{macrocode} -% \subsection{\csh{XINTinFloatMaxof}} -% \lverb|1.09a, for use by \xintNewFloatExpr. Name changed in 1.09h| -% \begin{macrocode} -\def\XINTinFloatMaxof {\romannumeral0\XINTinfloatmaxof }% -\def\XINTinfloatmaxof #1{\expandafter\XINT_flmaxof_a\romannumeral-`0#1\relax }% -\def\XINT_flmaxof_a #1{\expandafter\XINT_flmaxof_b - \romannumeral0\XINTinfloat [\XINTdigits]{#1}\Z }% -\def\XINT_flmaxof_b #1\Z #2% - {\expandafter\XINT_flmaxof_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_flmaxof_c #1% - {\xint_gob_til_relax #1\XINT_flmaxof_e\relax\XINT_flmaxof_d #1}% -\def\XINT_flmaxof_d #1\Z - {\expandafter\XINT_flmaxof_b\romannumeral0\xintmax - {\XINTinFloat [\XINTdigits]{#1}}}% -\def\XINT_flmaxof_e #1\Z #2\Z { #2}% -% \end{macrocode} -% \subsection{\csh{XINTinFloatMinof}} -% \lverb|1.09a, for use by \xintNewFloatExpr. Name changed in 1.09h| -% \begin{macrocode} -\def\XINTinFloatMinof {\romannumeral0\XINTinfloatminof }% -\def\XINTinfloatminof #1{\expandafter\XINT_flminof_a\romannumeral-`0#1\relax }% -\def\XINT_flminof_a #1{\expandafter\XINT_flminof_b - \romannumeral0\XINTinfloat [\XINTdigits]{#1}\Z }% -\def\XINT_flminof_b #1\Z #2% - {\expandafter\XINT_flminof_c\romannumeral-`0#2\Z {#1}\Z}% -\def\XINT_flminof_c #1% - {\xint_gob_til_relax #1\XINT_flminof_e\relax\XINT_flminof_d #1}% -\def\XINT_flminof_d #1\Z - {\expandafter\XINT_flminof_b\romannumeral0\xintmin - {\XINTinFloat [\XINTdigits]{#1}}}% -\def\XINT_flminof_e #1\Z #2\Z { #2}% -% \end{macrocode} -% \subsection{\csh{xintRound:csv}} -% \lverb|1.09a. For use by \xinttheiexpr.| -% \begin{macrocode} -\def\xintRound:csv #1{\expandafter\XINT_round:_a\romannumeral-`0#1,,^}% -\def\XINT_round:_a {\XINT_round:_b {}}% -\def\XINT_round:_b #1#2,% - {\expandafter\XINT_round:_c\romannumeral-`0#2,{#1}}% -\def\XINT_round:_c #1{\if #1,\expandafter\XINT_:_f - \else\expandafter\XINT_round:_d\fi #1}% -\def\XINT_round:_d #1,% - {\expandafter\XINT_round:_e\romannumeral0\xintiround 0{#1},}% -\def\XINT_round:_e #1,#2{\XINT_round:_b {#2,#1}}% -% \end{macrocode} -% \subsection{\csh{xintFloat:csv}} -% \lverb|1.09a. For use by \xintthefloatexpr.| -% \begin{macrocode} -\def\xintFloat:csv #1{\expandafter\XINT_float:_a\romannumeral-`0#1,,^}% -\def\XINT_float:_a {\XINT_float:_b {}}% -\def\XINT_float:_b #1#2,% - {\expandafter\XINT_float:_c\romannumeral-`0#2,{#1}}% -\def\XINT_float:_c #1{\if #1,\expandafter\XINT_:_f - \else\expandafter\XINT_float:_d\fi #1}% -\def\XINT_float:_d #1,% - {\expandafter\XINT_float:_e\romannumeral0\xintfloat {#1},}% -\def\XINT_float:_e #1,#2{\XINT_float:_b {#2,#1}}% -% \end{macrocode} -% \subsection{\csh{xintSum:csv}} -% \lverb|1.09a. For use by \xintexpr.| -% \begin{macrocode} -\def\xintSum:csv #1{\expandafter\XINT_sum:_a\romannumeral-`0#1,,^}% -\def\XINT_sum:_a {\XINT_sum:_b {0/1[0]}}% -\def\XINT_sum:_b #1#2,{\expandafter\XINT_sum:_c\romannumeral-`0#2,{#1}}% -\def\XINT_sum:_c #1{\if #1,\expandafter\XINT_:_e - \else\expandafter\XINT_sum:_d\fi #1}% -\def\XINT_sum:_d #1,#2{\expandafter\XINT_sum:_b\expandafter - {\romannumeral0\xintadd {#2}{#1}}}% -% \end{macrocode} -% \subsection{\csh{xintPrd:csv}} -% \lverb|1.09a. For use by \xintexpr.| -% \begin{macrocode} -\def\xintPrd:csv #1{\expandafter\XINT_prd:_a\romannumeral-`0#1,,^}% -\def\XINT_prd:_a {\XINT_prd:_b {1/1[0]}}% -\def\XINT_prd:_b #1#2,{\expandafter\XINT_prd:_c\romannumeral-`0#2,{#1}}% -\def\XINT_prd:_c #1{\if #1,\expandafter\XINT_:_e - \else\expandafter\XINT_prd:_d\fi #1}% -\def\XINT_prd:_d #1,#2{\expandafter\XINT_prd:_b\expandafter - {\romannumeral0\xintmul {#2}{#1}}}% -% \end{macrocode} -% \subsection{\csh{xintMaxof:csv}} -% \lverb|1.09a. For use by \xintexpr. Even with only one -% argument, there does not seem to be really a motive for using \xintraw?| -% \begin{macrocode} -\def\xintMaxof:csv #1{\expandafter\XINT_maxof:_b\romannumeral-`0#1,,}% -\def\XINT_maxof:_b #1,#2,{\expandafter\XINT_maxof:_c\romannumeral-`0#2,{#1},}% -\def\XINT_maxof:_c #1{\if #1,\expandafter\XINT_of:_e - \else\expandafter\XINT_maxof:_d\fi #1}% -\def\XINT_maxof:_d #1,{\expandafter\XINT_maxof:_b\romannumeral0\xintmax {#1}}% -% \end{macrocode} -% \subsection{\csh{xintMinof:csv}} -% \lverb|1.09a. For use by \xintexpr.| -% \begin{macrocode} -\def\xintMinof:csv #1{\expandafter\XINT_minof:_b\romannumeral-`0#1,,}% -\def\XINT_minof:_b #1,#2,{\expandafter\XINT_minof:_c\romannumeral-`0#2,{#1},}% -\def\XINT_minof:_c #1{\if #1,\expandafter\XINT_of:_e - \else\expandafter\XINT_minof:_d\fi #1}% -\def\XINT_minof:_d #1,{\expandafter\XINT_minof:_b\romannumeral0\xintmin {#1}}% -% \end{macrocode} -% \subsection{\csh{XINTinFloatMinof:csv}} -% \lverb|1.09a. For use by \xintfloatexpr. Name changed in 1.09h| -% \begin{macrocode} -\def\XINTinFloatMinof:csv #1{\expandafter\XINT_flminof:_a\romannumeral-`0#1,,}% -\def\XINT_flminof:_a #1,{\expandafter\XINT_flminof:_b - \romannumeral0\XINTinfloat [\XINTdigits]{#1},}% -\def\XINT_flminof:_b #1,#2,% - {\expandafter\XINT_flminof:_c\romannumeral-`0#2,{#1},}% -\def\XINT_flminof:_c #1{\if #1,\expandafter\XINT_of:_e - \else\expandafter\XINT_flminof:_d\fi #1}% -\def\XINT_flminof:_d #1,% - {\expandafter\XINT_flminof:_b\romannumeral0\xintmin - {\XINTinFloat [\XINTdigits]{#1}}}% -% \end{macrocode} -% \subsection{\csh{XINTinFloatMaxof:csv}} -% \lverb|1.09a. For use by \xintfloatexpr. Name changed in 1.09h| -% \begin{macrocode} -\def\XINTinFloatMaxof:csv #1{\expandafter\XINT_flmaxof:_a\romannumeral-`0#1,,}% -\def\XINT_flmaxof:_a #1,{\expandafter\XINT_flmaxof:_b - \romannumeral0\XINTinfloat [\XINTdigits]{#1},}% -\def\XINT_flmaxof:_b #1,#2,% - {\expandafter\XINT_flmaxof:_c\romannumeral-`0#2,{#1},}% -\def\XINT_flmaxof:_c #1{\if #1,\expandafter\XINT_of:_e - \else\expandafter\XINT_flmaxof:_d\fi #1}% -\def\XINT_flmaxof:_d #1,% - {\expandafter\XINT_flmaxof:_b\romannumeral0\xintmax - {\XINTinFloat [\XINTdigits]{#1}}}% -% \end{macrocode} -% \subsection{\csh{XINTinFloatSum:csv}} -% \lverb|1.09a. For use by \xintfloatexpr. Renamed in 1.09h| -% \begin{macrocode} -\def\XINTinFloatSum:csv #1{\expandafter\XINT_floatsum:_a\romannumeral-`0#1,,^}% -\def\XINT_floatsum:_a {\XINT_floatsum:_b {0[0]}}% -\def\XINT_floatsum:_b #1#2,% - {\expandafter\XINT_floatsum:_c\romannumeral-`0#2,{#1}}% -\def\XINT_floatsum:_c #1{\if #1,\expandafter\XINT_:_e - \else\expandafter\XINT_floatsum:_d\fi #1}% -\def\XINT_floatsum:_d #1,#2{\expandafter\XINT_floatsum:_b\expandafter - {\romannumeral0\XINTinfloatadd {#2}{#1}}}% -% \end{macrocode} -% \subsection{\csh{XINTinFloatPrd:csv}} -% \lverb|1.09a. For use by \xintfloatexpr. Renamed in 1.09h| -% \begin{macrocode} -\def\XINTinFloatPred:csv #1{\expandafter\XINT_floatprd:_a\romannumeral-`0#1,,^}% -\def\XINT_floatprd:_a {\XINT_floatprd:_b {1[0]}}% -\def\XINT_floatprd:_b #1#2,% - {\expandafter\XINT_floatprd:_c\romannumeral-`0#2,{#1}}% -\def\XINT_floatprd:_c #1{\if #1,\expandafter\XINT_:_e - \else\expandafter\XINT_floatprd:_d\fi #1}% -\def\XINT_floatprd:_d #1,#2{\expandafter\XINT_floatprd:_b\expandafter - {\romannumeral0\XINTinfloatmul {#2}{#1}}}% -\XINT_restorecatcodes_endinput% -% \end{macrocode} -%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 -%\let</xintfrac>\relax -%\def<*xintseries>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 } -%</xintfrac> -%<*xintseries> -% -% \StoreCodelineNo {xintfrac} -% -% \section{Package \xintseriesnameimp implementation} -% \label{sec:seriesimp} -% -% The commenting is currently (\docdate) very sparse. -% -% \localtableofcontents -% \subsection{Catcodes, \protect\eTeX{} and reload detection} -% -% The code for reload detection is copied from \textsc{Heiko -% Oberdiek}'s packages, and adapted here to check for previous -% loading of the \xintfracname package. -% -% The method for catcodes is slightly different, but still -% directly inspired by these packages. -% -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \def\space { }% - \let\z\endgroup - \expandafter\let\expandafter\x\csname ver@xintseries.sty\endcsname - \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname - \expandafter - \ifx\csname PackageInfo\endcsname\relax - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \else - \def\y#1#2{\PackageInfo{#1}{#2}}% - \fi - \expandafter - \ifx\csname numexpr\endcsname\relax - \y{xintseries}{\numexpr not available, aborting input}% - \aftergroup\endinput - \else - \ifx\x\relax % plain-TeX, first loading of xintseries.sty - \ifx\w\relax % but xintfrac.sty not yet loaded. - \y{xintseries}{now issuing \string\input\space xintfrac.sty}% - \def\z{\endgroup\input xintfrac.sty\relax}% - \fi - \else - \def\empty {}% - \ifx\x\empty % LaTeX, first loading, - % variable is initialized, but \ProvidesPackage not yet seen - \ifx\w\relax % xintfrac.sty not yet loaded. - \y{xintseries}{now issuing \string\RequirePackage{xintfrac}}% - \def\z{\endgroup\RequirePackage{xintfrac}}% - \fi - \else - \y{xintseries}{I was already loaded, aborting input}% - \aftergroup\endinput - \fi - \fi - \fi -\z% -% \end{macrocode} -% \subsection{Confirmation of \xintfracnameimp loading} -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \ifdefined\PackageInfo - \def\y#1#2{\PackageInfo{#1}{#2}}% - \else - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \fi - \def\empty {}% - \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname - \ifx\w\relax % Plain TeX, user gave a file name at the prompt - \y{xintseries}{Loading of package xintfrac failed, aborting input}% - \aftergroup\endinput - \fi - \ifx\w\empty % LaTeX, user gave a file name at the prompt - \y{xintseries}{Loading of package xintfrac failed, aborting input}% - \aftergroup\endinput - \fi -\endgroup% -% \end{macrocode} -% \subsection{Catcodes} -% \begin{macrocode} -\XINTsetupcatcodes% -% \end{macrocode} -% \subsection{Package identification} -% \begin{macrocode} -\XINT_providespackage -\ProvidesPackage{xintseries}% - [2014/02/05 v1.09ka Expandable partial sums with xint package (jfB)]% -% \end{macrocode} -% \subsection{\csh{xintSeries}} -% \lverb|& -% Modified in 1.06 to give the indices first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that. -% 1.08a adds the forgotten optimization following that previous change.| -% \begin{macrocode} -\def\xintSeries {\romannumeral0\xintseries }% -\def\xintseries #1#2% -{% - \expandafter\XINT_series\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}% -}% -\def\XINT_series #1#2#3% -{% - \ifnum #2<#1 - \xint_afterfi { 0/1[0]}% - \else - \xint_afterfi {\XINT_series_loop {#1}{0}{#2}{#3}}% - \fi -}% -\def\XINT_series_loop #1#2#3#4% -{% - \ifnum #3>#1 \else \XINT_series_exit \fi - \expandafter\XINT_series_loop\expandafter - {\the\numexpr #1+1\expandafter }\expandafter - {\romannumeral0\xintadd {#2}{#4{#1}}}% - {#3}{#4}% -}% -\def\XINT_series_exit \fi #1#2#3#4#5#6#7#8% -{% - \fi\xint_gobble_ii #6% -}% -% \end{macrocode} -% \subsection{\csh{xintiSeries}} -% \lverb|& -% Modified in 1.06 to give the indices first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that. -% 1.08a adds the forgotten optimization following that previous change.| -% \begin{macrocode} -\def\xintiSeries {\romannumeral0\xintiseries }% -\def\xintiseries #1#2% -{% - \expandafter\XINT_iseries\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}% -}% -\def\XINT_iseries #1#2#3% -{% - \ifnum #2<#1 - \xint_afterfi { 0}% - \else - \xint_afterfi {\XINT_iseries_loop {#1}{0}{#2}{#3}}% - \fi -}% -\def\XINT_iseries_loop #1#2#3#4% -{% - \ifnum #3>#1 \else \XINT_iseries_exit \fi - \expandafter\XINT_iseries_loop\expandafter - {\the\numexpr #1+1\expandafter }\expandafter - {\romannumeral0\xintiiadd {#2}{#4{#1}}}% - {#3}{#4}% -}% -\def\XINT_iseries_exit \fi #1#2#3#4#5#6#7#8% -{% - \fi\xint_gobble_ii #6% -}% -% \end{macrocode} -% \subsection{\csh{xintPowerSeries}} -% \lverb|& -% The 1.03 version was very lame and created a build-up of denominators. -% The Horner scheme for polynomial evaluation is used in 1.04, this -% cures the denominator problem and drastically improves the efficiency -% of the macro. -% Modified in 1.06 to give the indices first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that. -% 1.08a adds the forgotten optimization following that previous change.| -% \begin{macrocode} -\def\xintPowerSeries {\romannumeral0\xintpowerseries }% -\def\xintpowerseries #1#2% -{% - \expandafter\XINT_powseries\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}% -}% -\def\XINT_powseries #1#2#3#4% -{% - \ifnum #2<#1 - \xint_afterfi { 0/1[0]}% - \else - \xint_afterfi - {\XINT_powseries_loop_i {#3{#2}}{#1}{#2}{#3}{#4}}% - \fi -}% -\def\XINT_powseries_loop_i #1#2#3#4#5% -{% - \ifnum #3>#2 \else\XINT_powseries_exit_i\fi - \expandafter\XINT_powseries_loop_ii\expandafter - {\the\numexpr #3-1\expandafter}\expandafter - {\romannumeral0\xintmul {#1}{#5}}{#2}{#4}{#5}% -}% -\def\XINT_powseries_loop_ii #1#2#3#4% -{% - \expandafter\XINT_powseries_loop_i\expandafter - {\romannumeral0\xintadd {#4{#1}}{#2}}{#3}{#1}{#4}% -}% -\def\XINT_powseries_exit_i\fi #1#2#3#4#5#6#7#8#9% -{% - \fi \XINT_powseries_exit_ii #6{#7}% -}% -\def\XINT_powseries_exit_ii #1#2#3#4#5#6% -{% - \xintmul{\xintPow {#5}{#6}}{#4}% -}% -% \end{macrocode} -% \subsection{\csh{xintPowerSeriesX}} -% \lverb|& -% Same as \xintPowerSeries except for the initial expansion of the x parameter. -% Modified in 1.06 to give the indices first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that. -% 1.08a adds the forgotten optimization following that previous change.| -% \begin{macrocode} -\def\xintPowerSeriesX {\romannumeral0\xintpowerseriesx }% -\def\xintpowerseriesx #1#2% -{% - \expandafter\XINT_powseriesx\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}% -}% -\def\XINT_powseriesx #1#2#3#4% -{% - \ifnum #2<#1 - \xint_afterfi { 0/1[0]}% - \else - \xint_afterfi - {\expandafter\XINT_powseriesx_pre\expandafter - {\romannumeral-`0#4}{#1}{#2}{#3}% - }% - \fi -}% -\def\XINT_powseriesx_pre #1#2#3#4% -{% - \XINT_powseries_loop_i {#4{#3}}{#2}{#3}{#4}{#1}% -}% -% \end{macrocode} -% \subsection{\csh{xintRationalSeries}} -% \lverb|& -% This computes F(a)+...+F(b) on the basis of the value of F(a) and the -% ratios F(n)/F(n-1). As in \xintPowerSeries we use an iterative scheme which -% has the great advantage to avoid denominator build-up. This makes exact -% computations possible with exponential type series, which would be completely -% inaccessible to \xintSeries. -% #1=a, #2=b, #3=F(a), #4=ratio function -% Modified in 1.06 to give the indices first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that. -% 1.08a adds the forgotten optimization following that previous change.| -% \begin{macrocode} -\def\xintRationalSeries {\romannumeral0\xintratseries }% -\def\xintratseries #1#2% -{% - \expandafter\XINT_ratseries\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}% -}% -\def\XINT_ratseries #1#2#3#4% -{% - \ifnum #2<#1 - \xint_afterfi { 0/1[0]}% - \else - \xint_afterfi - {\XINT_ratseries_loop {#2}{1}{#1}{#4}{#3}}% - \fi -}% -\def\XINT_ratseries_loop #1#2#3#4% -{% - \ifnum #1>#3 \else\XINT_ratseries_exit_i\fi - \expandafter\XINT_ratseries_loop\expandafter - {\the\numexpr #1-1\expandafter}\expandafter - {\romannumeral0\xintadd {1}{\xintMul {#2}{#4{#1}}}}{#3}{#4}% -}% -\def\XINT_ratseries_exit_i\fi #1#2#3#4#5#6#7#8% -{% - \fi \XINT_ratseries_exit_ii #6% -}% -\def\XINT_ratseries_exit_ii #1#2#3#4#5% -{% - \XINT_ratseries_exit_iii #5% -}% -\def\XINT_ratseries_exit_iii #1#2#3#4% -{% - \xintmul{#2}{#4}% -}% -% \end{macrocode} -% \subsection{\csh{xintRationalSeriesX}} -% \lverb|& -% a,b,initial,ratiofunction,x$\ -% This computes F(a,x)+...+F(b,x) on the basis of the value of F(a,x) and the -% ratios F(n,x)/F(n-1,x). The argument x is first expanded and it is the value -% resulting from this which is used then throughout. The initial term F(a,x) -% must be defined as one-parameter macro which will be given x. -% Modified in 1.06 to give the indices first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that. -% 1.08a adds the forgotten optimization following that previous change.| -% \begin{macrocode} -\def\xintRationalSeriesX {\romannumeral0\xintratseriesx }% -\def\xintratseriesx #1#2% -{% - \expandafter\XINT_ratseriesx\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}% -}% -\def\XINT_ratseriesx #1#2#3#4#5% -{% - \ifnum #2<#1 - \xint_afterfi { 0/1[0]}% - \else - \xint_afterfi - {\expandafter\XINT_ratseriesx_pre\expandafter - {\romannumeral-`0#5}{#2}{#1}{#4}{#3}% - }% - \fi -}% -\def\XINT_ratseriesx_pre #1#2#3#4#5% -{% - \XINT_ratseries_loop {#2}{1}{#3}{#4{#1}}{#5{#1}}% -}% -% \end{macrocode} -% \subsection{\csh{xintFxPtPowerSeries}} -% \lverb|& -% I am not two happy with this piece of code. Will make it more economical -% another day. -% Modified in 1.06 to give the indices first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that. -% 1.08a: forgot last time some optimization from the change to \numexpr.| -% \begin{macrocode} -\def\xintFxPtPowerSeries {\romannumeral0\xintfxptpowerseries }% -\def\xintfxptpowerseries #1#2% -{% - \expandafter\XINT_fppowseries\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}% -}% -\def\XINT_fppowseries #1#2#3#4#5% -{% - \ifnum #2<#1 - \xint_afterfi { 0}% - \else - \xint_afterfi - {\expandafter\XINT_fppowseries_loop_pre\expandafter - {\romannumeral0\xinttrunc {#5}{\xintPow {#4}{#1}}}% - {#1}{#4}{#2}{#3}{#5}% - }% - \fi -}% -\def\XINT_fppowseries_loop_pre #1#2#3#4#5#6% -{% - \ifnum #4>#2 \else\XINT_fppowseries_dont_i \fi - \expandafter\XINT_fppowseries_loop_i\expandafter - {\the\numexpr #2+\xint_c_i\expandafter}\expandafter - {\romannumeral0\xintitrunc {#6}{\xintMul {#5{#2}}{#1}}}% - {#1}{#3}{#4}{#5}{#6}% -}% -\def\XINT_fppowseries_dont_i \fi\expandafter\XINT_fppowseries_loop_i - {\fi \expandafter\XINT_fppowseries_dont_ii }% -\def\XINT_fppowseries_dont_ii #1#2#3#4#5#6#7{\xinttrunc {#7}{#2[-#7]}}% -\def\XINT_fppowseries_loop_i #1#2#3#4#5#6#7% -{% - \ifnum #5>#1 \else \XINT_fppowseries_exit_i \fi - \expandafter\XINT_fppowseries_loop_ii\expandafter - {\romannumeral0\xinttrunc {#7}{\xintMul {#3}{#4}}}% - {#1}{#4}{#2}{#5}{#6}{#7}% -}% -\def\XINT_fppowseries_loop_ii #1#2#3#4#5#6#7% -{% - \expandafter\XINT_fppowseries_loop_i\expandafter - {\the\numexpr #2+\xint_c_i\expandafter}\expandafter - {\romannumeral0\xintiiadd {#4}{\xintiTrunc {#7}{\xintMul {#6{#2}}{#1}}}}% - {#1}{#3}{#5}{#6}{#7}% -}% -\def\XINT_fppowseries_exit_i\fi\expandafter\XINT_fppowseries_loop_ii - {\fi \expandafter\XINT_fppowseries_exit_ii }% -\def\XINT_fppowseries_exit_ii #1#2#3#4#5#6#7% -{% - \xinttrunc {#7} - {\xintiiadd {#4}{\xintiTrunc {#7}{\xintMul {#6{#2}}{#1}}}[-#7]}% -}% -% \end{macrocode} -% \subsection{\csh{xintFxPtPowerSeriesX}} -% \lverb|& -% a,b,coeff,x,D$\ -% Modified in 1.06 to give the indices first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that. -% 1.08a adds the forgotten optimization following that previous change.| -% \begin{macrocode} -\def\xintFxPtPowerSeriesX {\romannumeral0\xintfxptpowerseriesx }% -\def\xintfxptpowerseriesx #1#2% -{% - \expandafter\XINT_fppowseriesx\expandafter - {\the\numexpr #1\expandafter}\expandafter{\the\numexpr #2}% -}% -\def\XINT_fppowseriesx #1#2#3#4#5% -{% - \ifnum #2<#1 - \xint_afterfi { 0}% - \else - \xint_afterfi - {\expandafter \XINT_fppowseriesx_pre \expandafter - {\romannumeral-`0#4}{#1}{#2}{#3}{#5}% - }% - \fi -}% -\def\XINT_fppowseriesx_pre #1#2#3#4#5% -{% - \expandafter\XINT_fppowseries_loop_pre\expandafter - {\romannumeral0\xinttrunc {#5}{\xintPow {#1}{#2}}}% - {#2}{#1}{#3}{#4}{#5}% -}% -% \end{macrocode} -% \subsection{\csh{xintFloatPowerSeries}} -% \lverb|1.08a. I still have to re-visit \xintFxPtPowerSeries; temporarily I -% just adapted the code to the case of floats.| -% \begin{macrocode} -\def\xintFloatPowerSeries {\romannumeral0\xintfloatpowerseries }% -\def\xintfloatpowerseries #1{\XINT_flpowseries_chkopt #1\Z }% -\def\XINT_flpowseries_chkopt #1% -{% - \ifx [#1\expandafter\XINT_flpowseries_opt - \else\expandafter\XINT_flpowseries_noopt - \fi - #1% -}% -\def\XINT_flpowseries_noopt #1\Z #2% -{% - \expandafter\XINT_flpowseries\expandafter - {\the\numexpr #1\expandafter}\expandafter - {\the\numexpr #2}\XINTdigits -}% -\def\XINT_flpowseries_opt [\Z #1]#2#3% -{% - \expandafter\XINT_flpowseries\expandafter - {\the\numexpr #2\expandafter}\expandafter - {\the\numexpr #3\expandafter}{\the\numexpr #1}% -}% -\def\XINT_flpowseries #1#2#3#4#5% -{% - \ifnum #2<#1 - \xint_afterfi { 0.e0}% - \else - \xint_afterfi - {\expandafter\XINT_flpowseries_loop_pre\expandafter - {\romannumeral0\XINTinfloatpow [#3]{#5}{#1}}% - {#1}{#5}{#2}{#4}{#3}% - }% - \fi -}% -\def\XINT_flpowseries_loop_pre #1#2#3#4#5#6% -{% - \ifnum #4>#2 \else\XINT_flpowseries_dont_i \fi - \expandafter\XINT_flpowseries_loop_i\expandafter - {\the\numexpr #2+\xint_c_i\expandafter}\expandafter - {\romannumeral0\XINTinfloatmul [#6]{#5{#2}}{#1}}% - {#1}{#3}{#4}{#5}{#6}% -}% -\def\XINT_flpowseries_dont_i \fi\expandafter\XINT_flpowseries_loop_i - {\fi \expandafter\XINT_flpowseries_dont_ii }% -\def\XINT_flpowseries_dont_ii #1#2#3#4#5#6#7{\xintfloat [#7]{#2}}% -\def\XINT_flpowseries_loop_i #1#2#3#4#5#6#7% -{% - \ifnum #5>#1 \else \XINT_flpowseries_exit_i \fi - \expandafter\XINT_flpowseries_loop_ii\expandafter - {\romannumeral0\XINTinfloatmul [#7]{#3}{#4}}% - {#1}{#4}{#2}{#5}{#6}{#7}% -}% -\def\XINT_flpowseries_loop_ii #1#2#3#4#5#6#7% -{% - \expandafter\XINT_flpowseries_loop_i\expandafter - {\the\numexpr #2+\xint_c_i\expandafter}\expandafter - {\romannumeral0\XINTinfloatadd [#7]{#4}% - {\XINTinfloatmul [#7]{#6{#2}}{#1}}}% - {#1}{#3}{#5}{#6}{#7}% -}% -\def\XINT_flpowseries_exit_i\fi\expandafter\XINT_flpowseries_loop_ii - {\fi \expandafter\XINT_flpowseries_exit_ii }% -\def\XINT_flpowseries_exit_ii #1#2#3#4#5#6#7% -{% - \xintfloatadd [#7]{#4}{\XINTinfloatmul [#7]{#6{#2}}{#1}}% -}% -% \end{macrocode} -% \subsection{\csh{xintFloatPowerSeriesX}} -% \lverb|1.08a| -% \begin{macrocode} -\def\xintFloatPowerSeriesX {\romannumeral0\xintfloatpowerseriesx }% -\def\xintfloatpowerseriesx #1{\XINT_flpowseriesx_chkopt #1\Z }% -\def\XINT_flpowseriesx_chkopt #1% -{% - \ifx [#1\expandafter\XINT_flpowseriesx_opt - \else\expandafter\XINT_flpowseriesx_noopt - \fi - #1% -}% -\def\XINT_flpowseriesx_noopt #1\Z #2% -{% - \expandafter\XINT_flpowseriesx\expandafter - {\the\numexpr #1\expandafter}\expandafter - {\the\numexpr #2}\XINTdigits -}% -\def\XINT_flpowseriesx_opt [\Z #1]#2#3% -{% - \expandafter\XINT_flpowseriesx\expandafter - {\the\numexpr #2\expandafter}\expandafter - {\the\numexpr #3\expandafter}{\the\numexpr #1}% -}% -\def\XINT_flpowseriesx #1#2#3#4#5% -{% - \ifnum #2<#1 - \xint_afterfi { 0.e0}% - \else - \xint_afterfi - {\expandafter \XINT_flpowseriesx_pre \expandafter - {\romannumeral-`0#5}{#1}{#2}{#4}{#3}% - }% - \fi -}% -\def\XINT_flpowseriesx_pre #1#2#3#4#5% -{% - \expandafter\XINT_flpowseries_loop_pre\expandafter - {\romannumeral0\XINTinfloatpow [#5]{#1}{#2}}% - {#2}{#1}{#3}{#4}{#5}% -}% -\XINT_restorecatcodes_endinput% -% \end{macrocode} -%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 -%\let</xintseries>\relax -%\def<*xintcfrac>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 } -%</xintseries> -%<*xintcfrac> -% -% \StoreCodelineNo {xintseries} -% -% \section{Package \xintcfracnameimp implementation} -% \label{sec:cfracimp} -% -% The commenting is currently (\docdate) very sparse. -% -% \localtableofcontents -% \subsection{Catcodes, \protect\eTeX{} and reload detection} -% -% The code for reload detection is copied from \textsc{Heiko -% Oberdiek}'s packages, and adapted here to check for previous -% loading of the \xintfracname package. -% -% The method for catcodes is slightly different, but still -% directly inspired by these packages. -% -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \def\space { }% - \let\z\endgroup - \expandafter\let\expandafter\x\csname ver@xintcfrac.sty\endcsname - \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname - \expandafter - \ifx\csname PackageInfo\endcsname\relax - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \else - \def\y#1#2{\PackageInfo{#1}{#2}}% - \fi - \expandafter - \ifx\csname numexpr\endcsname\relax - \y{xintcfrac}{\numexpr not available, aborting input}% - \aftergroup\endinput - \else - \ifx\x\relax % plain-TeX, first loading of xintcfrac.sty - \ifx\w\relax % but xintfrac.sty not yet loaded. - \y{xintcfrac}{now issuing \string\input\space xintfrac.sty}% - \def\z{\endgroup\input xintfrac.sty\relax}% - \fi - \else - \def\empty {}% - \ifx\x\empty % LaTeX, first loading, - % variable is initialized, but \ProvidesPackage not yet seen - \ifx\w\relax % xintfrac.sty not yet loaded. - \y{xintcfrac}{now issuing \string\RequirePackage{xintfrac}}% - \def\z{\endgroup\RequirePackage{xintfrac}}% - \fi - \else - \y{xintcfrac}{I was already loaded, aborting input}% - \aftergroup\endinput - \fi - \fi - \fi -\z% -% \end{macrocode} -% \subsection{Confirmation of \xintfracnameimp loading} -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \ifdefined\PackageInfo - \def\y#1#2{\PackageInfo{#1}{#2}}% - \else - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \fi - \def\empty {}% - \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname - \ifx\w\relax % Plain TeX, user gave a file name at the prompt - \y{xintcfrac}{Loading of package xintfrac failed, aborting input}% - \aftergroup\endinput - \fi - \ifx\w\empty % LaTeX, user gave a file name at the prompt - \y{xintcfrac}{Loading of package xintfrac failed, aborting input}% - \aftergroup\endinput - \fi -\endgroup% -% \end{macrocode} -% \subsection{Catcodes} -% \begin{macrocode} -\XINTsetupcatcodes% -% \end{macrocode} -% \subsection{Package identification} -% \begin{macrocode} -\XINT_providespackage -\ProvidesPackage{xintcfrac}% - [2014/02/05 v1.09ka Expandable continued fractions with xint package (jfB)]% -% \end{macrocode} -% \subsection{\csh{xintCFrac}} -% \begin{macrocode} -\def\xintCFrac {\romannumeral0\xintcfrac }% -\def\xintcfrac #1% -{% - \XINT_cfrac_opt_a #1\Z -}% -\def\XINT_cfrac_opt_a #1% -{% - \ifx[#1\XINT_cfrac_opt_b\fi \XINT_cfrac_noopt #1% -}% -\def\XINT_cfrac_noopt #1\Z -{% - \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z - \relax\relax -}% -\def\XINT_cfrac_opt_b\fi\XINT_cfrac_noopt [\Z #1]% -{% - \fi\csname XINT_cfrac_opt#1\endcsname -}% -\def\XINT_cfrac_optl #1% -{% - \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z - \relax\hfill -}% -\def\XINT_cfrac_optc #1% -{% - \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z - \relax\relax -}% -\def\XINT_cfrac_optr #1% -{% - \expandafter\XINT_cfrac_A\romannumeral0\xintrawwithzeros {#1}\Z - \hfill\relax -}% -\def\XINT_cfrac_A #1/#2\Z -{% - \expandafter\XINT_cfrac_B\romannumeral0\xintiidivision {#1}{#2}{#2}% -}% -\def\XINT_cfrac_B #1#2% -{% - \XINT_cfrac_C #2\Z {#1}% -}% -\def\XINT_cfrac_C #1% -{% - \xint_gob_til_zero #1\XINT_cfrac_integer 0\XINT_cfrac_D #1% -}% -\def\XINT_cfrac_integer 0\XINT_cfrac_D 0#1\Z #2#3#4#5{ #2}% -\def\XINT_cfrac_D #1\Z #2#3{\XINT_cfrac_loop_a {#1}{#3}{#1}{{#2}}}% -\def\XINT_cfrac_loop_a -{% - \expandafter\XINT_cfrac_loop_d\romannumeral0\XINT_div_prepare -}% -\def\XINT_cfrac_loop_d #1#2% -{% - \XINT_cfrac_loop_e #2.{#1}% -}% -\def\XINT_cfrac_loop_e #1% -{% - \xint_gob_til_zero #1\xint_cfrac_loop_exit0\XINT_cfrac_loop_f #1% -}% -\def\XINT_cfrac_loop_f #1.#2#3#4% -{% - \XINT_cfrac_loop_a {#1}{#3}{#1}{{#2}#4}% -}% -\def\xint_cfrac_loop_exit0\XINT_cfrac_loop_f #1.#2#3#4#5#6% - {\XINT_cfrac_T #5#6{#2}#4\Z }% -\def\XINT_cfrac_T #1#2#3#4% -{% - \xint_gob_til_Z #4\XINT_cfrac_end\Z\XINT_cfrac_T #1#2{#4+\cfrac{#11#2}{#3}}% -}% -\def\XINT_cfrac_end\Z\XINT_cfrac_T #1#2#3% -{% - \XINT_cfrac_end_b #3% -}% -\def\XINT_cfrac_end_b \Z+\cfrac#1#2{ #2}% -% \end{macrocode} -% \subsection{\csh{xintGCFrac}} -% \begin{macrocode} -\def\xintGCFrac {\romannumeral0\xintgcfrac }% -\def\xintgcfrac #1{\XINT_gcfrac_opt_a #1\Z }% -\def\XINT_gcfrac_opt_a #1% -{% - \ifx[#1\XINT_gcfrac_opt_b\fi \XINT_gcfrac_noopt #1% -}% -\def\XINT_gcfrac_noopt #1\Z -{% - \XINT_gcfrac #1+\W/\relax\relax -}% -\def\XINT_gcfrac_opt_b\fi\XINT_gcfrac_noopt [\Z #1]% -{% - \fi\csname XINT_gcfrac_opt#1\endcsname -}% -\def\XINT_gcfrac_optl #1% -{% - \XINT_gcfrac #1+\W/\relax\hfill -}% -\def\XINT_gcfrac_optc #1% -{% - \XINT_gcfrac #1+\W/\relax\relax -}% -\def\XINT_gcfrac_optr #1% -{% - \XINT_gcfrac #1+\W/\hfill\relax -}% -\def\XINT_gcfrac -{% - \expandafter\XINT_gcfrac_enter\romannumeral-`0% -}% -\def\XINT_gcfrac_enter {\XINT_gcfrac_loop {}}% -\def\XINT_gcfrac_loop #1#2+#3/% -{% - \xint_gob_til_W #3\XINT_gcfrac_endloop\W - \XINT_gcfrac_loop {{#3}{#2}#1}% -}% -\def\XINT_gcfrac_endloop\W\XINT_gcfrac_loop #1#2#3% -{% - \XINT_gcfrac_T #2#3#1\Z\Z -}% -\def\XINT_gcfrac_T #1#2#3#4{\XINT_gcfrac_U #1#2{\xintFrac{#4}}}% -\def\XINT_gcfrac_U #1#2#3#4#5% -{% - \xint_gob_til_Z #5\XINT_gcfrac_end\Z\XINT_gcfrac_U - #1#2{\xintFrac{#5}% - \ifcase\xintSgn{#4} - +\or+\else-\fi - \cfrac{#1\xintFrac{\xintAbs{#4}}#2}{#3}}% -}% -\def\XINT_gcfrac_end\Z\XINT_gcfrac_U #1#2#3% -{% - \XINT_gcfrac_end_b #3% -}% -\def\XINT_gcfrac_end_b #1\cfrac#2#3{ #3}% -% \end{macrocode} -% \subsection{\csh{xintGCtoGCx}} -% \begin{macrocode} -\def\xintGCtoGCx {\romannumeral0\xintgctogcx }% -\def\xintgctogcx #1#2#3% -{% - \expandafter\XINT_gctgcx_start\expandafter {\romannumeral-`0#3}{#1}{#2}% -}% -\def\XINT_gctgcx_start #1#2#3{\XINT_gctgcx_loop_a {}{#2}{#3}#1+\W/}% -\def\XINT_gctgcx_loop_a #1#2#3#4+#5/% -{% - \xint_gob_til_W #5\XINT_gctgcx_end\W - \XINT_gctgcx_loop_b {#1{#4}}{#2{#5}#3}{#2}{#3}% -}% -\def\XINT_gctgcx_loop_b #1#2% -{% - \XINT_gctgcx_loop_a {#1#2}% -}% -\def\XINT_gctgcx_end\W\XINT_gctgcx_loop_b #1#2#3#4{ #1}% -% \end{macrocode} -% \subsection{\csh{xintFtoCs}} -% \begin{macrocode} -\def\xintFtoCs {\romannumeral0\xintftocs }% -\def\xintftocs #1% -{% - \expandafter\XINT_ftc_A\romannumeral0\xintrawwithzeros {#1}\Z -}% -\def\XINT_ftc_A #1/#2\Z -{% - \expandafter\XINT_ftc_B\romannumeral0\xintiidivision {#1}{#2}{#2}% -}% -\def\XINT_ftc_B #1#2% -{% - \XINT_ftc_C #2.{#1}% -}% -\def\XINT_ftc_C #1% -{% - \xint_gob_til_zero #1\XINT_ftc_integer 0\XINT_ftc_D #1% -}% -\def\XINT_ftc_integer 0\XINT_ftc_D 0#1.#2#3{ #2}% -\def\XINT_ftc_D #1.#2#3{\XINT_ftc_loop_a {#1}{#3}{#1}{#2,}}% -\def\XINT_ftc_loop_a -{% - \expandafter\XINT_ftc_loop_d\romannumeral0\XINT_div_prepare -}% -\def\XINT_ftc_loop_d #1#2% -{% - \XINT_ftc_loop_e #2.{#1}% -}% -\def\XINT_ftc_loop_e #1% -{% - \xint_gob_til_zero #1\xint_ftc_loop_exit0\XINT_ftc_loop_f #1% -}% -\def\XINT_ftc_loop_f #1.#2#3#4% -{% - \XINT_ftc_loop_a {#1}{#3}{#1}{#4#2,}% -}% -\def\xint_ftc_loop_exit0\XINT_ftc_loop_f #1.#2#3#4{ #4#2}% -% \end{macrocode} -% \subsection{\csh{xintFtoCx}} -% \begin{macrocode} -\def\xintFtoCx {\romannumeral0\xintftocx }% -\def\xintftocx #1#2% -{% - \expandafter\XINT_ftcx_A\romannumeral0\xintrawwithzeros {#2}\Z {#1}% -}% -\def\XINT_ftcx_A #1/#2\Z -{% - \expandafter\XINT_ftcx_B\romannumeral0\xintiidivision {#1}{#2}{#2}% -}% -\def\XINT_ftcx_B #1#2% -{% - \XINT_ftcx_C #2.{#1}% -}% -\def\XINT_ftcx_C #1% -{% - \xint_gob_til_zero #1\XINT_ftcx_integer 0\XINT_ftcx_D #1% -}% -\def\XINT_ftcx_integer 0\XINT_ftcx_D 0#1.#2#3#4{ #2}% -\def\XINT_ftcx_D #1.#2#3#4{\XINT_ftcx_loop_a {#1}{#3}{#1}{#2#4}{#4}}% -\def\XINT_ftcx_loop_a -{% - \expandafter\XINT_ftcx_loop_d\romannumeral0\XINT_div_prepare -}% -\def\XINT_ftcx_loop_d #1#2% -{% - \XINT_ftcx_loop_e #2.{#1}% -}% -\def\XINT_ftcx_loop_e #1% -{% - \xint_gob_til_zero #1\xint_ftcx_loop_exit0\XINT_ftcx_loop_f #1% -}% -\def\XINT_ftcx_loop_f #1.#2#3#4#5% -{% - \XINT_ftcx_loop_a {#1}{#3}{#1}{#4{#2}#5}{#5}% -}% -\def\xint_ftcx_loop_exit0\XINT_ftcx_loop_f #1.#2#3#4#5{ #4{#2}}% -% \end{macrocode} -% \subsection{\csh{xintFtoGC}} -% \begin{macrocode} -\def\xintFtoGC {\romannumeral0\xintftogc }% -\def\xintftogc {\xintftocx {+1/}}% -% \end{macrocode} -% \subsection{\csh{xintFtoCC}} -% \begin{macrocode} -\def\xintFtoCC {\romannumeral0\xintftocc }% -\def\xintftocc #1% -{% - \expandafter\XINT_ftcc_A\expandafter {\romannumeral0\xintrawwithzeros {#1}}% -}% -\def\XINT_ftcc_A #1% -{% - \expandafter\XINT_ftcc_B - \romannumeral0\xintrawwithzeros {\xintAdd {1/2[0]}{#1[0]}}\Z {#1[0]}% -}% -\def\XINT_ftcc_B #1/#2\Z -{% - \expandafter\XINT_ftcc_C\expandafter {\romannumeral0\xintiiquo {#1}{#2}}% -}% -\def\XINT_ftcc_C #1#2% -{% - \expandafter\XINT_ftcc_D\romannumeral0\xintsub {#2}{#1}\Z {#1}% -}% -\def\XINT_ftcc_D #1% -{% - \xint_UDzerominusfork - #1-\XINT_ftcc_integer - 0#1\XINT_ftcc_En - 0-{\XINT_ftcc_Ep #1}% - \krof -}% -\def\XINT_ftcc_Ep #1\Z #2% -{% - \expandafter\XINT_ftcc_loop_a\expandafter - {\romannumeral0\xintdiv {1[0]}{#1}}{#2+1/}% -}% -\def\XINT_ftcc_En #1\Z #2% -{% - \expandafter\XINT_ftcc_loop_a\expandafter - {\romannumeral0\xintdiv {1[0]}{#1}}{#2+-1/}% -}% -\def\XINT_ftcc_integer #1\Z #2{ #2}% -\def\XINT_ftcc_loop_a #1% -{% - \expandafter\XINT_ftcc_loop_b - \romannumeral0\xintrawwithzeros {\xintAdd {1/2[0]}{#1}}\Z {#1}% -}% -\def\XINT_ftcc_loop_b #1/#2\Z -{% - \expandafter\XINT_ftcc_loop_c\expandafter - {\romannumeral0\xintiiquo {#1}{#2}}% -}% -\def\XINT_ftcc_loop_c #1#2% -{% - \expandafter\XINT_ftcc_loop_d - \romannumeral0\xintsub {#2}{#1[0]}\Z {#1}% -}% -\def\XINT_ftcc_loop_d #1% -{% - \xint_UDzerominusfork - #1-\XINT_ftcc_end - 0#1\XINT_ftcc_loop_N - 0-{\XINT_ftcc_loop_P #1}% - \krof -}% -\def\XINT_ftcc_end #1\Z #2#3{ #3#2}% -\def\XINT_ftcc_loop_P #1\Z #2#3% -{% - \expandafter\XINT_ftcc_loop_a\expandafter - {\romannumeral0\xintdiv {1[0]}{#1}}{#3#2+1/}% -}% -\def\XINT_ftcc_loop_N #1\Z #2#3% -{% - \expandafter\XINT_ftcc_loop_a\expandafter - {\romannumeral0\xintdiv {1[0]}{#1}}{#3#2+-1/}% -}% -% \end{macrocode} -% \subsection{\csh{xintFtoCv}} -% \begin{macrocode} -\def\xintFtoCv {\romannumeral0\xintftocv }% -\def\xintftocv #1% -{% - \xinticstocv {\xintFtoCs {#1}}% -}% -% \end{macrocode} -% \subsection{\csh{xintFtoCCv}} -% \begin{macrocode} -\def\xintFtoCCv {\romannumeral0\xintftoccv }% -\def\xintftoccv #1% -{% - \xintigctocv {\xintFtoCC {#1}}% -}% -% \end{macrocode} -% \subsection{\csh{xintCstoF}} -% \begin{macrocode} -\def\xintCstoF {\romannumeral0\xintcstof }% -\def\xintcstof #1% -{% - \expandafter\XINT_cstf_prep \romannumeral-`0#1,\W,% -}% -\def\XINT_cstf_prep -{% - \XINT_cstf_loop_a 1001% -}% -\def\XINT_cstf_loop_a #1#2#3#4#5,% -{% - \xint_gob_til_W #5\XINT_cstf_end\W - \expandafter\XINT_cstf_loop_b - \romannumeral0\xintrawwithzeros {#5}.{#1}{#2}{#3}{#4}% -}% -\def\XINT_cstf_loop_b #1/#2.#3#4#5#6% -{% - \expandafter\XINT_cstf_loop_c\expandafter - {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% - {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% -}% -\def\XINT_cstf_loop_c #1#2% -{% - \expandafter\XINT_cstf_loop_d\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_cstf_loop_d #1#2% -{% - \expandafter\XINT_cstf_loop_e\expandafter {\expandafter{#2}#1}% -}% -\def\XINT_cstf_loop_e #1#2% -{% - \expandafter\XINT_cstf_loop_a\expandafter{#2}#1% -}% -\def\XINT_cstf_end #1.#2#3#4#5{\xintrawwithzeros {#2/#3}}% 1.09b removes [0] -% \end{macrocode} -% \subsection{\csh{xintiCstoF}} -% \begin{macrocode} -\def\xintiCstoF {\romannumeral0\xinticstof }% -\def\xinticstof #1% -{% - \expandafter\XINT_icstf_prep \romannumeral-`0#1,\W,% -}% -\def\XINT_icstf_prep -{% - \XINT_icstf_loop_a 1001% -}% -\def\XINT_icstf_loop_a #1#2#3#4#5,% -{% - \xint_gob_til_W #5\XINT_icstf_end\W - \expandafter - \XINT_icstf_loop_b \romannumeral-`0#5.{#1}{#2}{#3}{#4}% -}% -\def\XINT_icstf_loop_b #1.#2#3#4#5% -{% - \expandafter\XINT_icstf_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% - {#2}{#3}% -}% -\def\XINT_icstf_loop_c #1#2% -{% - \expandafter\XINT_icstf_loop_a\expandafter {#2}{#1}% -}% -\def\XINT_icstf_end#1.#2#3#4#5{\xintrawwithzeros {#2/#3}}% 1.09b removes [0] -% \end{macrocode} -% \subsection{\csh{xintGCtoF}} -% \begin{macrocode} -\def\xintGCtoF {\romannumeral0\xintgctof }% -\def\xintgctof #1% -{% - \expandafter\XINT_gctf_prep \romannumeral-`0#1+\W/% -}% -\def\XINT_gctf_prep -{% - \XINT_gctf_loop_a 1001% -}% -\def\XINT_gctf_loop_a #1#2#3#4#5+% -{% - \expandafter\XINT_gctf_loop_b - \romannumeral0\xintrawwithzeros {#5}.{#1}{#2}{#3}{#4}% -}% -\def\XINT_gctf_loop_b #1/#2.#3#4#5#6% -{% - \expandafter\XINT_gctf_loop_c\expandafter - {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% - {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% -}% -\def\XINT_gctf_loop_c #1#2% -{% - \expandafter\XINT_gctf_loop_d\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_gctf_loop_d #1#2% -{% - \expandafter\XINT_gctf_loop_e\expandafter {\expandafter{#2}#1}% -}% -\def\XINT_gctf_loop_e #1#2% -{% - \expandafter\XINT_gctf_loop_f\expandafter {\expandafter{#2}#1}% -}% -\def\XINT_gctf_loop_f #1#2/% -{% - \xint_gob_til_W #2\XINT_gctf_end\W - \expandafter\XINT_gctf_loop_g - \romannumeral0\xintrawwithzeros {#2}.#1% -}% -\def\XINT_gctf_loop_g #1/#2.#3#4#5#6% -{% - \expandafter\XINT_gctf_loop_h\expandafter - {\romannumeral0\XINT_mul_fork #1\Z #6\Z }% - {\romannumeral0\XINT_mul_fork #1\Z #5\Z }% - {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% - {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% -}% -\def\XINT_gctf_loop_h #1#2% -{% - \expandafter\XINT_gctf_loop_i\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_gctf_loop_i #1#2% -{% - \expandafter\XINT_gctf_loop_j\expandafter {\expandafter{#2}#1}% -}% -\def\XINT_gctf_loop_j #1#2% -{% - \expandafter\XINT_gctf_loop_a\expandafter {#2}#1% -}% -\def\XINT_gctf_end #1.#2#3#4#5{\xintrawwithzeros {#2/#3}}% 1.09b removes [0] -% \end{macrocode} -% \subsection{\csh{xintiGCtoF}} -% \begin{macrocode} -\def\xintiGCtoF {\romannumeral0\xintigctof }% -\def\xintigctof #1% -{% - \expandafter\XINT_igctf_prep \romannumeral-`0#1+\W/% -}% -\def\XINT_igctf_prep -{% - \XINT_igctf_loop_a 1001% -}% -\def\XINT_igctf_loop_a #1#2#3#4#5+% -{% - \expandafter\XINT_igctf_loop_b - \romannumeral-`0#5.{#1}{#2}{#3}{#4}% -}% -\def\XINT_igctf_loop_b #1.#2#3#4#5% -{% - \expandafter\XINT_igctf_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% - {#2}{#3}% -}% -\def\XINT_igctf_loop_c #1#2% -{% - \expandafter\XINT_igctf_loop_f\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_igctf_loop_f #1#2#3#4/% -{% - \xint_gob_til_W #4\XINT_igctf_end\W - \expandafter\XINT_igctf_loop_g - \romannumeral-`0#4.{#2}{#3}#1% -}% -\def\XINT_igctf_loop_g #1.#2#3% -{% - \expandafter\XINT_igctf_loop_h\expandafter - {\romannumeral0\XINT_mul_fork #1\Z #3\Z }% - {\romannumeral0\XINT_mul_fork #1\Z #2\Z }% -}% -\def\XINT_igctf_loop_h #1#2% -{% - \expandafter\XINT_igctf_loop_i\expandafter {#2}{#1}% -}% -\def\XINT_igctf_loop_i #1#2#3#4% -{% - \XINT_igctf_loop_a {#3}{#4}{#1}{#2}% -}% -\def\XINT_igctf_end #1.#2#3#4#5{\xintrawwithzeros {#4/#5}}% 1.09b removes [0] -% \end{macrocode} -% \subsection{\csh{xintCstoCv}} -% \begin{macrocode} -\def\xintCstoCv {\romannumeral0\xintcstocv }% -\def\xintcstocv #1% -{% - \expandafter\XINT_cstcv_prep \romannumeral-`0#1,\W,% -}% -\def\XINT_cstcv_prep -{% - \XINT_cstcv_loop_a {}1001% -}% -\def\XINT_cstcv_loop_a #1#2#3#4#5#6,% -{% - \xint_gob_til_W #6\XINT_cstcv_end\W - \expandafter\XINT_cstcv_loop_b - \romannumeral0\xintrawwithzeros {#6}.{#2}{#3}{#4}{#5}{#1}% -}% -\def\XINT_cstcv_loop_b #1/#2.#3#4#5#6% -{% - \expandafter\XINT_cstcv_loop_c\expandafter - {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% - {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% -}% -\def\XINT_cstcv_loop_c #1#2% -{% - \expandafter\XINT_cstcv_loop_d\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_cstcv_loop_d #1#2% -{% - \expandafter\XINT_cstcv_loop_e\expandafter {\expandafter{#2}#1}% -}% -\def\XINT_cstcv_loop_e #1#2% -{% - \expandafter\XINT_cstcv_loop_f\expandafter{#2}#1% -}% -\def\XINT_cstcv_loop_f #1#2#3#4#5% -{% - \expandafter\XINT_cstcv_loop_g\expandafter - {\romannumeral0\xintrawwithzeros {#1/#2}}{#5}{#1}{#2}{#3}{#4}% -}% -\def\XINT_cstcv_loop_g #1#2{\XINT_cstcv_loop_a {#2{#1}}}% 1.09b removes [0] -\def\XINT_cstcv_end #1.#2#3#4#5#6{ #6}% -% \end{macrocode} -% \subsection{\csh{xintiCstoCv}} -% \begin{macrocode} -\def\xintiCstoCv {\romannumeral0\xinticstocv }% -\def\xinticstocv #1% -{% - \expandafter\XINT_icstcv_prep \romannumeral-`0#1,\W,% -}% -\def\XINT_icstcv_prep -{% - \XINT_icstcv_loop_a {}1001% -}% -\def\XINT_icstcv_loop_a #1#2#3#4#5#6,% -{% - \xint_gob_til_W #6\XINT_icstcv_end\W - \expandafter - \XINT_icstcv_loop_b \romannumeral-`0#6.{#2}{#3}{#4}{#5}{#1}% -}% -\def\XINT_icstcv_loop_b #1.#2#3#4#5% -{% - \expandafter\XINT_icstcv_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% - {{#2}{#3}}% -}% -\def\XINT_icstcv_loop_c #1#2% -{% - \expandafter\XINT_icstcv_loop_d\expandafter {#2}{#1}% -}% -\def\XINT_icstcv_loop_d #1#2% -{% - \expandafter\XINT_icstcv_loop_e\expandafter - {\romannumeral0\xintrawwithzeros {#1/#2}}{{#1}{#2}}% -}% -\def\XINT_icstcv_loop_e #1#2#3#4{\XINT_icstcv_loop_a {#4{#1}}#2#3}% -\def\XINT_icstcv_end #1.#2#3#4#5#6{ #6}% 1.09b removes [0] -% \end{macrocode} -% \subsection{\csh{xintGCtoCv}} -% \begin{macrocode} -\def\xintGCtoCv {\romannumeral0\xintgctocv }% -\def\xintgctocv #1% -{% - \expandafter\XINT_gctcv_prep \romannumeral-`0#1+\W/% -}% -\def\XINT_gctcv_prep -{% - \XINT_gctcv_loop_a {}1001% -}% -\def\XINT_gctcv_loop_a #1#2#3#4#5#6+% -{% - \expandafter\XINT_gctcv_loop_b - \romannumeral0\xintrawwithzeros {#6}.{#2}{#3}{#4}{#5}{#1}% -}% -\def\XINT_gctcv_loop_b #1/#2.#3#4#5#6% -{% - \expandafter\XINT_gctcv_loop_c\expandafter - {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% - {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#6}}{\XINT_Mul {#1}{#4}}}% - {\romannumeral0\xintiiadd {\XINT_Mul {#2}{#5}}{\XINT_Mul {#1}{#3}}}% -}% -\def\XINT_gctcv_loop_c #1#2% -{% - \expandafter\XINT_gctcv_loop_d\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_gctcv_loop_d #1#2% -{% - \expandafter\XINT_gctcv_loop_e\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_gctcv_loop_e #1#2% -{% - \expandafter\XINT_gctcv_loop_f\expandafter {#2}#1% -}% -\def\XINT_gctcv_loop_f #1#2% -{% - \expandafter\XINT_gctcv_loop_g\expandafter - {\romannumeral0\xintrawwithzeros {#1/#2}}{{#1}{#2}}% -}% -\def\XINT_gctcv_loop_g #1#2#3#4% -{% - \XINT_gctcv_loop_h {#4{#1}}{#2#3}% 1.09b removes [0] -}% -\def\XINT_gctcv_loop_h #1#2#3/% -{% - \xint_gob_til_W #3\XINT_gctcv_end\W - \expandafter\XINT_gctcv_loop_i - \romannumeral0\xintrawwithzeros {#3}.#2{#1}% -}% -\def\XINT_gctcv_loop_i #1/#2.#3#4#5#6% -{% - \expandafter\XINT_gctcv_loop_j\expandafter - {\romannumeral0\XINT_mul_fork #1\Z #6\Z }% - {\romannumeral0\XINT_mul_fork #1\Z #5\Z }% - {\romannumeral0\XINT_mul_fork #2\Z #4\Z }% - {\romannumeral0\XINT_mul_fork #2\Z #3\Z }% -}% -\def\XINT_gctcv_loop_j #1#2% -{% - \expandafter\XINT_gctcv_loop_k\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_gctcv_loop_k #1#2% -{% - \expandafter\XINT_gctcv_loop_l\expandafter {\expandafter{#2}#1}% -}% -\def\XINT_gctcv_loop_l #1#2% -{% - \expandafter\XINT_gctcv_loop_m\expandafter {\expandafter{#2}#1}% -}% -\def\XINT_gctcv_loop_m #1#2{\XINT_gctcv_loop_a {#2}#1}% -\def\XINT_gctcv_end #1.#2#3#4#5#6{ #6}% -% \end{macrocode} -% \subsection{\csh{xintiGCtoCv}} -% \begin{macrocode} -\def\xintiGCtoCv {\romannumeral0\xintigctocv }% -\def\xintigctocv #1% -{% - \expandafter\XINT_igctcv_prep \romannumeral-`0#1+\W/% -}% -\def\XINT_igctcv_prep -{% - \XINT_igctcv_loop_a {}1001% -}% -\def\XINT_igctcv_loop_a #1#2#3#4#5#6+% -{% - \expandafter\XINT_igctcv_loop_b - \romannumeral-`0#6.{#2}{#3}{#4}{#5}{#1}% -}% -\def\XINT_igctcv_loop_b #1.#2#3#4#5% -{% - \expandafter\XINT_igctcv_loop_c\expandafter - {\romannumeral0\xintiiadd {#5}{\XINT_Mul {#1}{#3}}}% - {\romannumeral0\xintiiadd {#4}{\XINT_Mul {#1}{#2}}}% - {{#2}{#3}}% -}% -\def\XINT_igctcv_loop_c #1#2% -{% - \expandafter\XINT_igctcv_loop_f\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_igctcv_loop_f #1#2#3#4/% -{% - \xint_gob_til_W #4\XINT_igctcv_end_a\W - \expandafter\XINT_igctcv_loop_g - \romannumeral-`0#4.#1#2{#3}% -}% -\def\XINT_igctcv_loop_g #1.#2#3#4#5% -{% - \expandafter\XINT_igctcv_loop_h\expandafter - {\romannumeral0\XINT_mul_fork #1\Z #5\Z }% - {\romannumeral0\XINT_mul_fork #1\Z #4\Z }% - {{#2}{#3}}% -}% -\def\XINT_igctcv_loop_h #1#2% -{% - \expandafter\XINT_igctcv_loop_i\expandafter {\expandafter{#2}{#1}}% -}% -\def\XINT_igctcv_loop_i #1#2{\XINT_igctcv_loop_k #2{#2#1}}% -\def\XINT_igctcv_loop_k #1#2% -{% - \expandafter\XINT_igctcv_loop_l\expandafter - {\romannumeral0\xintrawwithzeros {#1/#2}}% -}% -\def\XINT_igctcv_loop_l #1#2#3{\XINT_igctcv_loop_a {#3{#1}}#2}%1.09i removes [0] -\def\XINT_igctcv_end_a #1.#2#3#4#5% -{% - \expandafter\XINT_igctcv_end_b\expandafter - {\romannumeral0\xintrawwithzeros {#2/#3}}% -}% -\def\XINT_igctcv_end_b #1#2{ #2{#1}}% 1.09b removes [0] -% \end{macrocode} -% \subsection{\csh{xintCntoF}} -% \lverb|& -% Modified in 1.06 to give the N first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that.| -% \begin{macrocode} -\def\xintCntoF {\romannumeral0\xintcntof }% -\def\xintcntof #1% -{% - \expandafter\XINT_cntf\expandafter {\the\numexpr #1}% -}% -\def\XINT_cntf #1#2% -{% - \ifnum #1>\xint_c_ - \xint_afterfi {\expandafter\XINT_cntf_loop\expandafter - {\the\numexpr #1-1\expandafter}\expandafter - {\romannumeral-`0#2{#1}}{#2}}% - \else - \xint_afterfi - {\ifnum #1=\xint_c_ - \xint_afterfi {\expandafter\space \romannumeral-`0#2{0}}% - \else \xint_afterfi { 0/1[0]}% - \fi}% - \fi -}% -\def\XINT_cntf_loop #1#2#3% -{% - \ifnum #1>\xint_c_ \else \XINT_cntf_exit \fi - \expandafter\XINT_cntf_loop\expandafter - {\the\numexpr #1-1\expandafter }\expandafter - {\romannumeral0\xintadd {\xintDiv {1[0]}{#2}}{#3{#1}}}% - {#3}% -}% -\def\XINT_cntf_exit \fi - \expandafter\XINT_cntf_loop\expandafter - #1\expandafter #2#3% -{% - \fi\xint_gobble_ii #2% -}% -% \end{macrocode} -% \subsection{\csh{xintGCntoF}} -% \lverb|& -% Modified in 1.06 to give the N first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that.| -% \begin{macrocode} -\def\xintGCntoF {\romannumeral0\xintgcntof }% -\def\xintgcntof #1% -{% - \expandafter\XINT_gcntf\expandafter {\the\numexpr #1}% -}% -\def\XINT_gcntf #1#2#3% -{% - \ifnum #1>\xint_c_ - \xint_afterfi {\expandafter\XINT_gcntf_loop\expandafter - {\the\numexpr #1-1\expandafter}\expandafter - {\romannumeral-`0#2{#1}}{#2}{#3}}% - \else - \xint_afterfi - {\ifnum #1=\xint_c_ - \xint_afterfi {\expandafter\space\romannumeral-`0#2{0}}% - \else \xint_afterfi { 0/1[0]}% - \fi}% - \fi -}% -\def\XINT_gcntf_loop #1#2#3#4% -{% - \ifnum #1>\xint_c_ \else \XINT_gcntf_exit \fi - \expandafter\XINT_gcntf_loop\expandafter - {\the\numexpr #1-1\expandafter }\expandafter - {\romannumeral0\xintadd {\xintDiv {#4{#1}}{#2}}{#3{#1}}}% - {#3}{#4}% -}% -\def\XINT_gcntf_exit \fi - \expandafter\XINT_gcntf_loop\expandafter - #1\expandafter #2#3#4% -{% - \fi\xint_gobble_ii #2% -}% -% \end{macrocode} -% \subsection{\csh{xintCntoCs}} -% \lverb|& -% Modified in 1.06 to give the N first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that.| -% \begin{macrocode} -\def\xintCntoCs {\romannumeral0\xintcntocs }% -\def\xintcntocs #1% -{% - \expandafter\XINT_cntcs\expandafter {\the\numexpr #1}% -}% -\def\XINT_cntcs #1#2% -{% - \ifnum #1<0 - \xint_afterfi { }% 1.09i: a 0/1[0] was strangely here, removed - \else - \xint_afterfi {\expandafter\XINT_cntcs_loop\expandafter - {\the\numexpr #1-1\expandafter}\expandafter - {\expandafter{\romannumeral-`0#2{#1}}}{#2}}% - \fi -}% -\def\XINT_cntcs_loop #1#2#3% -{% - \ifnum #1>-1 \else \XINT_cntcs_exit \fi - \expandafter\XINT_cntcs_loop\expandafter - {\the\numexpr #1-1\expandafter }\expandafter - {\expandafter{\romannumeral-`0#3{#1}},#2}{#3}% -}% -\def\XINT_cntcs_exit \fi - \expandafter\XINT_cntcs_loop\expandafter - #1\expandafter #2#3% -{% - \fi\XINT_cntcs_exit_b #2% -}% -\def\XINT_cntcs_exit_b #1,{ }% -% \end{macrocode} -% \subsection{\csh{xintCntoGC}} -% \lverb|& -% Modified in 1.06 to give the N first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that.| -% \begin{macrocode} -\def\xintCntoGC {\romannumeral0\xintcntogc }% -\def\xintcntogc #1% -{% - \expandafter\XINT_cntgc\expandafter {\the\numexpr #1}% -}% -\def\XINT_cntgc #1#2% -{% - \ifnum #1<0 - \xint_afterfi { }% 1.09i there was as strange 0/1[0] here, removed - \else - \xint_afterfi {\expandafter\XINT_cntgc_loop\expandafter - {\the\numexpr #1-1\expandafter}\expandafter - {\expandafter{\romannumeral-`0#2{#1}}}{#2}}% - \fi -}% -\def\XINT_cntgc_loop #1#2#3% -{% - \ifnum #1>-1 \else \XINT_cntgc_exit \fi - \expandafter\XINT_cntgc_loop\expandafter - {\the\numexpr #1-1\expandafter }\expandafter - {\expandafter{\romannumeral-`0#3{#1}}+1/#2}{#3}% -}% -\def\XINT_cntgc_exit \fi - \expandafter\XINT_cntgc_loop\expandafter - #1\expandafter #2#3% -{% - \fi\XINT_cntgc_exit_b #2% -}% -\def\XINT_cntgc_exit_b #1+1/{ }% -% \end{macrocode} -% \subsection{\csh{xintGCntoGC}} -% \lverb|& -% Modified in 1.06 to give the N first to a \numexpr rather than expanding -% twice. I just use \the\numexpr and maintain the previous code after that.| -% \begin{macrocode} -\def\xintGCntoGC {\romannumeral0\xintgcntogc }% -\def\xintgcntogc #1% -{% - \expandafter\XINT_gcntgc\expandafter {\the\numexpr #1}% -}% -\def\XINT_gcntgc #1#2#3% -{% - \ifnum #1<0 - \xint_afterfi { }% 1.09i now returns nothing - \else - \xint_afterfi {\expandafter\XINT_gcntgc_loop\expandafter - {\the\numexpr #1-1\expandafter}\expandafter - {\expandafter{\romannumeral-`0#2{#1}}}{#2}{#3}}% - \fi -}% -\def\XINT_gcntgc_loop #1#2#3#4% -{% - \ifnum #1>-1 \else \XINT_gcntgc_exit \fi - \expandafter\XINT_gcntgc_loop_b\expandafter - {\expandafter{\romannumeral-`0#4{#1}}/#2}{#3{#1}}{#1}{#3}{#4}% -}% -\def\XINT_gcntgc_loop_b #1#2#3% -{% - \expandafter\XINT_gcntgc_loop\expandafter - {\the\numexpr #3-1\expandafter}\expandafter - {\expandafter{\romannumeral-`0#2}+#1}% -}% -\def\XINT_gcntgc_exit \fi - \expandafter\XINT_gcntgc_loop_b\expandafter #1#2#3#4#5% -{% - \fi\XINT_gcntgc_exit_b #1% -}% -\def\XINT_gcntgc_exit_b #1/{ }% -% \end{macrocode} -% \subsection{\csh{xintCstoGC}} -% \begin{macrocode} -\def\xintCstoGC {\romannumeral0\xintcstogc }% -\def\xintcstogc #1% -{% - \expandafter\XINT_cstc_prep \romannumeral-`0#1,\W,% -}% -\def\XINT_cstc_prep #1,{\XINT_cstc_loop_a {{#1}}}% -\def\XINT_cstc_loop_a #1#2,% -{% - \xint_gob_til_W #2\XINT_cstc_end\W - \XINT_cstc_loop_b {#1}{#2}% -}% -\def\XINT_cstc_loop_b #1#2{\XINT_cstc_loop_a {#1+1/{#2}}}% -\def\XINT_cstc_end\W\XINT_cstc_loop_b #1#2{ #1}% -% \end{macrocode} -% \subsection{\csh{xintGCtoGC}} -% \begin{macrocode} -\def\xintGCtoGC {\romannumeral0\xintgctogc }% -\def\xintgctogc #1% -{% - \expandafter\XINT_gctgc_start \romannumeral-`0#1+\W/% -}% -\def\XINT_gctgc_start {\XINT_gctgc_loop_a {}}% -\def\XINT_gctgc_loop_a #1#2+#3/% -{% - \xint_gob_til_W #3\XINT_gctgc_end\W - \expandafter\XINT_gctgc_loop_b\expandafter - {\romannumeral-`0#2}{#3}{#1}% -}% -\def\XINT_gctgc_loop_b #1#2% -{% - \expandafter\XINT_gctgc_loop_c\expandafter - {\romannumeral-`0#2}{#1}% -}% -\def\XINT_gctgc_loop_c #1#2#3% -{% - \XINT_gctgc_loop_a {#3{#2}+{#1}/}% -}% -\def\XINT_gctgc_end\W\expandafter\XINT_gctgc_loop_b -{% - \expandafter\XINT_gctgc_end_b -}% -\def\XINT_gctgc_end_b #1#2#3{ #3{#1}}% -\XINT_restorecatcodes_endinput% -% \end{macrocode} -%\catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 -%\let</xintcfrac>\relax -%\def<*xintexpr>{\catcode`\<=12 \catcode`\>=12 \catcode`\*=12 \catcode`\/=12 } -%</xintcfrac> -%<*xintexpr> -% -% \StoreCodelineNo {xintcfrac} -% -% \section{Package \xintexprnameimp implementation} -% \label{sec:exprimp} -% -% The first version was released in June 2013. I was greatly helped in this task -% of writing an expandable parser of infix operations by the comments provided -% in |l3fp-parse.dtx| (in its version as available in April-May 2013). One will -% recognize in particular the idea of the `until' macros; I have not looked into -% the actual |l3fp| code beyond the very useful comments provided in its -% documentation. -% -% A main worry was that my data has no a priori bound on its size; to keep the -% code reasonably efficient, I experimented with a technique of storing and -% retrieving data expandably as \emph{names} of control sequences. Intermediate -% computation results are stored as control sequences |\.=a/b[n]|. -% -% Another peculiarity is that the input is allowed to contain (but only where -% the scanner looks for a number or fraction) material within braces |{...}|. -% This will be expanded completely and must give an integer, decimal number or -% fraction (not in scientific notation). Conversely any explict fraction -% |A/B[n]| \emph{with the brackets} or macro expanding to such a thing -% \textbf{must} be enclosed within such braces: square brackets are not -% acceptable by the expression parser. -% -% These two things are a bit \emph{experimental} and perhaps I will opt for -% another approach at a later stage. To circumvent the potential hash-table -% impact of the |\.=a/b[n]| I have provided the macro creators |\xintNewExpr| -% and |\xintNewFloatExpr|. -% -% Roughly speaking, the parser mechanism is as follows: at any given time the -% last found ``operator'' has its associated |until| macro awaiting some news -% from the token flow; first |getnext| expands forward in the hope to construct -% some number, which may come from a parenthesized sub-expression, from some -% braced material, or from a digit by digit scan. After this number has been -% formed the next operator is looked for by the |getop| macro. Once |getop| has -% finished its job, |until| is presented with three tokens: the first one is the -% precedence level of the new found operator (which may be an end of expression -% marker), the second is the operator character token (earlier versions had here -% already some macro name, but in order to keep as much common code to expr and -% floatexpr common as possible, this was modied) of the new found operator, and -% the third one is the newly found number (which was encountered just before the -% new operator). -% -% The |until| macro of the earlier operator examines the precedence level of the -% new found one, and either executes the earlier operator (in the case of a -% binary operation, with the found number and a previously stored one) or it -% delays execution, giving the hand to the |until| macro of the operator having -% been found of higher precedence. -% -% A minus sign acting as prefix gets converted into a (unary) operator -% inheriting the precedence level of the previous operator. -% -% Once the end of the expression is found (it has to be marked by a |\relax|) -% the final result is output as four tokens: the first one a catcode 11 -% exclamation mark, the second one an error generating macro, the third one a -% printing macro and the fourth is |\.=a/b[n]|. The prefix |\xintthe| makes the -% output printable by killing the first two tokens. -% -% Version |1.08b| |[2013/06/14]| corrected a problem originating in the attempt -% to attribute a special rôle to braces: expansion could be stopped by space -% tokens, as various macros tried to expand without grabbing what came next. -% They now have a doubled |\romannumeral-`0|. -% -% Version |1.09a| |[2013/09/24]| has a better mechanism regarding |\xintthe|, -% more commenting and better organization of the code, and most importantly it -% implements functions, comparison operators, logic operators, conditionals. The -% code was reorganized and expansion proceeds a bit differently in order to have -% the |_getnext| and |_getop| codes entirely shared by |\xintexpr| and -% |\xintfloatexpr|. |\xintNewExpr| was rewritten in order to work with the -% standard macro parameter character |#|, to be catcode protected and to also -% allow comma separated expressions. -% -% Version |1.09c| |[2013/10/09]| added the |bool| and |togl| operators, -% |\xintboolexpr|, and |\xintNewNumExpr|, |\xintNewBoolExpr|. The code for -% |\xintNewExpr| is shared with |float|, |num|, and |bool|-expressions. Also the -% precedence level of the postfix operators |!|, |?| and |:| has been made lower -% than the one of functions. -% -% Version |1.09i| |[2013/12/18]| unpacks count and dimen registers and control -% squences, with tacit multiplication. It has also made small improvements. -% (speed gains in macro expansions in quite a few places.) -% -% Also, |1.09i| implements |\xintiiexpr|, |\xinttheiiexpr|. New function |frac|. -% And encapsulation in |\csname..\endcsname| is done with |.=| as first tokens, -% so unpacking with |\string| can be done in a completely escape char agnostic -% way. -% -% Version |1.09j| |[2014/01/09]| extends the tacit multiplication to the case of -% a sub |\xintexpr|-essions. Also, it now |\xint_protect|s the result of the -% |\xintexpr| full expansions, thus, an |\xintexpr| without |\xintthe| prefix -% can be used not only as the first item within an ``|\fdef|'' as previously but -% also now anywhere within an |\edef|. Five tokens are used to pack the -% computation result rather than the possibly hundreds or thousands of digits of -% an |\xintthe| unlocked result. I deliberately omit a second |\xint_protect| -% which, however would be necessary if some macro |\.=digits/digits[digits]| had -% acquired some expandable meaning elsewhere. But this seems not that probable, -% and adding the protection would mean impacting everything only to allow some -% crazy user which has loaded something else than xint to do an |\edef|... the -% |\xintexpr| computations are otherwise in no way affected if such control -% sequences have a meaning. -% -% Version |1.09k| |[2014/01/21]| does tacit multiplication also for an opening -% parenthesis encountered during the scanning of a number, or at a time when the -% parser expects an infix operator. -% -% And it adds to the syntax recognition of hexadecimal numbers starting with a -% |"|, and having possibly a fractional part (except in |\xintiiexpr|, -% naturally). -% -% \localtableofcontents -% \subsection{Catcodes, \protect\eTeX{} and reload detection} -% -% The code for reload detection is copied from \textsc{Heiko -% Oberdiek}'s packages, and adapted here to check for previous -% loading of the \xintfracname package. -% -% The method for catcodes is slightly different, but still -% directly inspired by these packages. -% -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \def\space { }% - \let\z\endgroup - \expandafter\let\expandafter\x\csname ver@xintexpr.sty\endcsname - \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname - \expandafter - \ifx\csname PackageInfo\endcsname\relax - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \else - \def\y#1#2{\PackageInfo{#1}{#2}}% - \fi - \expandafter - \ifx\csname numexpr\endcsname\relax - \y{xintexpr}{\numexpr not available, aborting input}% - \aftergroup\endinput - \else - \ifx\x\relax % plain-TeX, first loading of xintexpr.sty - \ifx\w\relax % but xintfrac.sty not yet loaded. - \y{xintexpr}{now issuing \string\input\space xintfrac.sty}% - \def\z{\endgroup\input xintfrac.sty\relax}% - \fi - \else - \def\empty {}% - \ifx\x\empty % LaTeX, first loading, - % variable is initialized, but \ProvidesPackage not yet seen - \ifx\w\relax % xintfrac.sty not yet loaded. - \y{xintexpr}{now issuing \string\RequirePackage{xintfrac}}% - \def\z{\endgroup\RequirePackage{xintfrac}}% - \fi - \else - \y{xintexpr}{I was already loaded, aborting input}% - \aftergroup\endinput - \fi - \fi - \fi -\z% -% \end{macrocode} -% \subsection{Confirmation of \xintfracnameimp loading} -% \begin{macrocode} -\begingroup\catcode61\catcode48\catcode32=10\relax% - \catcode13=5 % ^^M - \endlinechar=13 % - \catcode123=1 % { - \catcode125=2 % } - \catcode64=11 % @ - \catcode35=6 % # - \catcode44=12 % , - \catcode45=12 % - - \catcode46=12 % . - \catcode58=12 % : - \ifdefined\PackageInfo - \def\y#1#2{\PackageInfo{#1}{#2}}% - \else - \def\y#1#2{\immediate\write-1{Package #1 Info: #2.}}% - \fi - \def\empty {}% - \expandafter\let\expandafter\w\csname ver@xintfrac.sty\endcsname - \ifx\w\relax % Plain TeX, user gave a file name at the prompt - \y{xintexpr}{Loading of package xintfrac failed, aborting input}% - \aftergroup\endinput - \fi - \ifx\w\empty % LaTeX, user gave a file name at the prompt - \y{xintexpr}{Loading of package xintfrac failed, aborting input}% - \aftergroup\endinput - \fi -\endgroup% -% \end{macrocode} -% \subsection{Catcodes} -% \begin{macrocode} -\XINTsetupcatcodes% -% \end{macrocode} -% \subsection{Package identification} -% \begin{macrocode} -\XINT_providespackage -\ProvidesPackage{xintexpr}% - [2014/02/05 v1.09k Expandable expression parser (jfB)]% -% \end{macrocode} -% \subsection{Encapsulation in pseudo cs names, helper macros} -% \lverb|1.09i uses .= for encapsulation, thus allowing \escapechar to be -% anything (all previous releases were with ., so \escapechar 46 was forbidden). -% Besides, the \edef definition has \space already expanded, perhaps this will -% compensate a tiny bit the time penalty of `.=' viz `.' in unlocking... well -% not really, I guess. (for no special reason 1.09k uses some \expandafter's -% rather than \edef+\noexpand's for the definition of \XINT_expr_lock)| -% \begin{macrocode} -\def\xint_gob_til_! #1!{}% nota bene: this ! has catcode 11 -\expandafter\def\expandafter -\XINT_expr_lock\expandafter#\expandafter1\expandafter !\expandafter - {\expandafter\expandafter\space\csname .=#1\endcsname }% -\def\XINT_expr_unlock {\expandafter\XINT_expr_unlock_a\string }% -\def\XINT_expr_unlock_a #1.={}% -\def\XINT_expr_unexpectedtoken {\xintError:ignored }% -\def\XINT_newexpr_setprefix #1>{\noexpand\romannumeral-`0}% -\def\xint_UDxintrelaxfork #1\xint_relax #2#3\krof {#2}% -% \end{macrocode} -% \subsection{\csh{xintexpr}, \csh{xinttheexpr}, \csh{xintthe}, ...} -% \lverb|\xintthe is defined with a parameter, I guess I wanted to make sure no -% stray space tokens could cause a problem. -% -% With 1.09i, \xintiexpr replaces -% \xintnumexpr which is kept for compatibility but will be removed at some -% point. Should perhaps issue a warning, but well, people can also read the -% documentation. Also 1.09i removes \xinttheeval. -% -% 1.09i has re-organized the material here. -% -% 1.09j modifies the mechanism of \XINT_expr_usethe and -% \XINT_expr_print, etc... in order for \xintexpr-essions to be usable -% within \edef'initions. I hesitated quite a bit with adding -% \xint_protect in front of the \.=digits macros, which will in -% 99.99999$% of use cases supposed all have \relax meaning; and it is a -% bit of a pain, really, it is quite a pain to add these extra tokens -% only for \edef contexts and for situations which will never occur... -% well no damn'it let's *NOT* add this extra \xint_protect. Just one -% before the printing macro (which can not be \protected, else \xintthe -% could not work).| -% \begin{macrocode} -\def\xint_protect {\noexpand\xint_protect\noexpand }% 1.09j -\def\XINT_expr_done {!\XINT_expr_usethe\xint_protect\XINT_expr_print }% -\let\XINT_iiexpr_done \XINT_expr_done -\def\XINT_iexpr_done {!\XINT_expr_usethe\xint_protect\XINT_iexpr_print }% -\def\XINT_flexpr_done {!\XINT_expr_usethe\xint_protect\XINT_flexpr_print }% -\def\XINT_boolexpr_done {!\XINT_expr_usethe\xint_protect\XINT_boolexpr_print }% -\protected\def\XINT_expr_usethe #1#2#3% modified in 1.09j - {\xintError:missing_xintthe!\show#3missing xintthe (see log)!}% -\def\xintthe #1{\romannumeral-`0\expandafter\xint_gobble_iii\romannumeral-`0#1}% -\let\XINT_expr_print \XINT_expr_unlock -\def\XINT_iexpr_print #1{\xintRound:csv {\XINT_expr_unlock #1}}% -\def\XINT_flexpr_print #1{\xintFloat:csv {\XINT_expr_unlock #1}}% -\def\XINT_boolexpr_print #1{\xintIsTrue:csv{\XINT_expr_unlock #1}}% -\def\xintexpr {\romannumeral0\xinteval }% -\def\xintfloatexpr {\romannumeral0\xintfloateval }% -\def\xintiiexpr {\romannumeral0\xintiieval }% -\def\xinteval - {\expandafter\XINT_expr_until_end_a \romannumeral-`0\XINT_expr_getnext }% -\def\xintfloateval - {\expandafter\XINT_flexpr_until_end_a\romannumeral-`0\XINT_expr_getnext }% -\def\xintiieval - {\expandafter\XINT_iiexpr_until_end_a\romannumeral-`0\XINT_expr_getnext }% -\def\xinttheexpr - {\romannumeral-`0\expandafter\xint_gobble_iii\romannumeral0\xinteval }% -\def\xintthefloatexpr - {\romannumeral-`0\expandafter\xint_gobble_iii\romannumeral0\xintfloateval }% -\def\xinttheiiexpr - {\romannumeral-`0\expandafter\xint_gobble_iii\romannumeral0\xintiieval }% -\def\xintiexpr {\romannumeral0\expandafter\expandafter\expandafter - \XINT_iexpr_done \expandafter\xint_gobble_iv\romannumeral0\xinteval }% -\def\xinttheiexpr {\romannumeral-`0\expandafter\expandafter\expandafter - \XINT_iexpr_print\expandafter\xint_gobble_iv\romannumeral0\xinteval }% -\def\xintboolexpr {\romannumeral0\expandafter\expandafter\expandafter - \XINT_boolexpr_done \expandafter\xint_gobble_iv\romannumeral0\xinteval }% -\def\xinttheboolexpr {\romannumeral-`0\expandafter\expandafter\expandafter - \XINT_boolexpr_print\expandafter\xint_gobble_iv\romannumeral0\xinteval }% -\let\xintnumexpr \xintiexpr % deprecated -\let\xintthenumexpr\xinttheiexpr % deprecated -% \end{macrocode} -% \subsection{\csh{xintifboolexpr}, \csh{xintifboolfloatexpr}, csh{xintifbooliiexpr}} -% \lverb|1.09c. Does not work with comma separated expressions. I could -% make use \xintORof:csv (or AND, or XOR) to allow it, but don't know it the -% overhead is worth it. -% -% 1.09i adds \xintifbooliiexpr | -% \begin{macrocode} -\def\xintifboolexpr #1% - {\romannumeral0\xintifnotzero {\xinttheexpr #1\relax}}% -\def\xintifboolfloatexpr #1% - {\romannumeral0\xintifnotzero {\xintthefloatexpr #1\relax}}% -\def\xintifbooliiexpr #1% - {\romannumeral0\xintifnotzero {\xinttheiiexpr #1\relax}}% -% \end{macrocode} -% \subsection{\csh{XINT\_get\_next}: looking for a number} -% \lverb|June 14: 1.08b adds a second \romannumeral-`0 to \XINT_expr_getnext in -% an attempt to solve a problem with space tokens stopping the \romannumeral and -% thus preventing expansion of the following token. For example: 1+ \the\cnta -% caused a problem, as `\the' was not expanded. I did not define -% \XINT_expr_getnext as a macro with parameter (which would have cured -% preventively this), precisely to try to recognize brace pairs. The second -% \romannumeral-`0 is added for the same reason in other places. -% -% The get-next scans forward to find a number: after expansion of what comes -% next, an opening parenthesis signals a parenthesized sub-expression, a ! with -% catcode 11 signals there was there an \xintexpr.. \relax sub-expression (now -% evaluated), a minus is a prefix operator, a plus is silently ignored, a digit -% or decimal point signals to start gathering a number, braced material {...} is -% allowed and will be directly fed into a \csname..\endcsname for complete -% expansion which must delivers a (fractional) number, possibly ending in [n]; -% explicit square brackets must be enclosed into such braces. Once a number -% issues from the previous procedures, it is a locked into a -% \csname...\endcsname, and the flow then proceeds with \XINT_expr_getop which -% will scan for an infix or postfix operator following the number. -% -% A special r\^ole is played by underscores _ for use with \xintNewExpr -% to input macro parameters. -% -% Release 1.09a implements functions; the idea is that a letter (actually, -% anything not otherwise recognized!) triggers the function name gatherer, the -% comma is promoted to a binary operator of priority intermediate between -% parentheses and infix operators. The code had some other revisions in order -% for all the _getnext and _getop macros to now be shared by \xintexpr and -% \xintfloatexpr. -% -% 1.09i now allows direct insertion of \count's, \dimen's and \skip's which will -% be unpacked using \number. -% -% 1.09i speeds up a bit the recognition of a braced thing: the case of a single -% braced control sequence makes a third expansion mandatory, let's do it -% immediately and not wait. So macros got shuffled and modified a bit. -% -% \XINT_expr_unpackvariable does not insert a [0] for compatibility with -% \xintiiexpr. A [0] would have made a bit faster \xintexpr macros when dealing -% with an unpacked count control sequence, as without it the \xintnum will be -% used in the parsing by xintfrac macros when the number is used. But [0] is not -% accepted by most macros ultimately called by \xintiiexpr.| -% \begin{macrocode} -\def\XINT_expr_getnext -{% - \expandafter\XINT_expr_getnext_checkforbraced_a - \romannumeral-`0\romannumeral-`0% -}% -\def\XINT_expr_getnext_checkforbraced_a #1% was done later in <1.09i -{% - \expandafter\XINT_expr_getnext_checkforbraced_b\expandafter - {\romannumeral-`0#1}% -}% -\def\XINT_expr_getnext_checkforbraced_b #1% -{% - \XINT_expr_getnext_checkforbraced_c #1\xint_relax\Z {#1}% -}% -\def\XINT_expr_getnext_checkforbraced_c #1#2% -{% - \xint_UDxintrelaxfork - #1\XINT_expr_getnext_wasemptyorspace - #2\XINT_expr_getnext_gotonetoken_wehope - \xint_relax\XINT_expr_getnext_gotbracedstuff - \krof -}% doubly braced things are not acceptable, will cause errors. -\def\XINT_expr_getnext_wasemptyorspace #1{\XINT_expr_getnext }% -\def\XINT_expr_getnext_gotbracedstuff #1\xint_relax\Z #2% -{% - \expandafter\XINT_expr_getop\csname .=#2\endcsname -}% -\def\XINT_expr_getnext_gotonetoken_wehope\Z #1% -{% screens out sub-expressions and \count or \dimen registers/variables - \xint_gob_til_! #1\XINT_expr_subexpr !% recall this ! has catcode 11 - \ifcat\relax#1% \count or \numexpr etc... token or count, dimen, skip cs - \expandafter\XINT_expr_countdimenetc_fork - \else - \expandafter\expandafter\expandafter - \XINT_expr_getnext_onetoken_fork\expandafter\string - \fi - #1% -}% -\def\XINT_expr_subexpr !#1\fi !{\expandafter\XINT_expr_getop\xint_gobble_iii }% -\def\XINT_expr_countdimenetc_fork #1% -{% - \ifx\count#1\else\ifx#1\dimen\else\ifx#1\numexpr\else\ifx#1\dimexpr\else - \ifx\skip#1\else\ifx\glueexpr#1\else - \XINT_expr_unpackvariable - \fi\fi\fi\fi\fi\fi - \expandafter\XINT_expr_getnext\number #1% -}% -\def\XINT_expr_unpackvariable\fi\fi\fi\fi\fi\fi\expandafter\XINT_expr_getnext - \number #1{\fi\fi\fi\fi\fi\fi - \expandafter\XINT_expr_getop\csname .=\number#1\endcsname }% -% \end{macrocode} -% \lverb|1.09a: In order to have this code shared by \xintexpr and -% \xintfloatexpr, I have moved to the until macros the responsability to choose -% expr or floatexpr, hence here, the opening parenthesis for example can not be -% triggered directly as it would not know in which context it works. Hence the -% \xint_c_xviii ({}. And also the mechanism of \xintNewExpr has been modified to -% allow use of #. -% -% 1.09i also has \xintiiexpr. | -% \begin{macrocode} -\begingroup -\lccode`*=`# -\lowercase{\endgroup -\def\XINT_expr_sixwayfork #1(-.+*#2#3\krof {#2}% -\def\XINT_expr_getnext_onetoken_fork #1% -{% The * is in truth catcode 12 #. For (hacking) use with \xintNewExpr. - \XINT_expr_sixwayfork - #1-.+*{\xint_c_xviii ({}}% back to until for oparen triggering - (#1.+*{-}% - (-#1+*{\XINT_expr_scandec_II .}% - (-.#1*{\XINT_expr_getnext }% - (-.+#1{\XINT_expr_scandec_II }% - (-.+*{\XINT_expr_scan_dec_or_func #1}% - \krof -}}% -% \end{macrocode} -% \subsection{\csh{XINT\_expr\_scan\_dec\_or\_func}: collecting an integer or -% decimal number or hexa-decimal number or function name} -% \lverb|\XINT_expr_scanfunc_b rewritten in 1.09i. And 1.09k adds hexadecimal -% numbers to the syntax, with " as prefix, and possibly a fractional part. -% Naturally to postfix with an E in scientific notation, one would need to -% surround the hexadecimal number in parentheses to avoid ambiguities; or -% rather, just use a lowercase e. By the way, if I allowed only lowercase e for -% scientific notation I could possibly fuse together the scanning in the dec and -% hexa cases; up to some loss of syntax control in the dec case.| -% \begin{macrocode} -\def\XINT_expr_scan_dec_or_func #1% this #1 has necessarily here catcode 12 -{% - \ifnum \xint_c_ix<1#1 - \expandafter\XINT_expr_scandec_I - \else - \if #1"\expandafter\expandafter\expandafter\XINT_expr_scanhex_I - \else % We assume we are dealing with a function name!! - \expandafter\expandafter\expandafter\XINT_expr_scanfunc - \fi - \fi #1% -}% -\def\XINT_expr_scanfunc -{% - \expandafter\XINT_expr_func\romannumeral-`0\XINT_expr_scanfunc_c -}% -\def\XINT_expr_scanfunc_c #1% -{% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanfunc_a\romannumeral-`0\romannumeral-`0% -}% -\def\XINT_expr_scanfunc_a #1% please no braced things here! -{% - \ifcat #1\relax % missing opening parenthesis, probably - \expandafter\XINT_expr_scanfunc_panic - \else - \xint_afterfi{\expandafter\XINT_expr_scanfunc_b \string #1}% - \fi -}% -\def\xint_UDparenfork #1()#2#3\krof {#2}% -\def\XINT_expr_scanfunc_b #1% -{% - \xint_UDparenfork - #1){(}% and then \XINT_expr_func - (#1{(}% and then \XINT_expr_func (this is for bool/toggle names) - (){\XINT_expr_scanfunc_c #1}% - \krof -}% -\def\XINT_expr_scanfunc_panic {\xintError:bigtroubleahead(0\relax }% -\def\XINT_expr_func #1(% common to expr and flexpr and iiexpr -{% - \xint_c_xviii @{#1}% functions have the highest priority. -}% -% \end{macrocode} -% \lverb|Scanning for a number of fraction. Once gathered, lock it and do -% _getop. 1.09i modifies \XINT_expr_scanintpart_a (splits _aa) and also -% \XINT_expr_scanfracpart_a in -% order for the tacit multiplication of \count's and \dimen's to be compatible -% with escape-char=a digit. -% -% 1.09j further extends for recognition of an \xint..expr and then insertion -% of a * (which is done in \XINT_expr_getop_a).| -% \begin{macrocode} -\def\XINT_expr_scandec_I -{% - \expandafter\XINT_expr_getop\romannumeral-`0\expandafter - \XINT_expr_lock\romannumeral-`0\XINT_expr_scanintpart_b -}% -\def\XINT_expr_scandec_II -{% - \expandafter\XINT_expr_getop\romannumeral-`0\expandafter - \XINT_expr_lock\romannumeral-`0\XINT_expr_scanfracpart_b -}% -\def\XINT_expr_scanintpart_a #1% Please no braced material: 123{FORBIDDEN} -{% careful that ! has catcode letter here - \ifcat #1\relax\else - \ifx !#1\else - \expandafter\expandafter\expandafter - \xint_thirdofthree - \fi\fi - \xint_firstoftwo !% this stops the scan - {\expandafter\XINT_expr_scanintpart_aa\string }#1% -}% -\def\XINT_expr_scanintpart_aa #1% -{% - \ifnum \xint_c_ix<1#1 - \expandafter\XINT_expr_scanintpart_b - \else - \if .#1% - \expandafter\expandafter\expandafter - \XINT_expr_scandec_transition - \else % gather what we got so far, leave catcode 12 #1 in stream - \expandafter\expandafter\expandafter !% ! of catcode 11, space needed - \fi - \fi - #1% -}% -\def\XINT_expr_scanintpart_b #1% -{% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanintpart_a\romannumeral-`0\romannumeral-`0% -}% -\def\XINT_expr_scandec_transition .% -{% - \expandafter.\romannumeral-`0\expandafter - \XINT_expr_scanfracpart_a\romannumeral-`0\romannumeral-`0% -}% -\def\XINT_expr_scanfracpart_a #1% -{% - \ifcat #1\relax\else - \ifx !#1\else - \expandafter\expandafter\expandafter - \xint_thirdofthree - \fi\fi - \xint_firstoftwo !% this stops the scan - {\expandafter\XINT_expr_scanfracpart_aa\string }#1% -}% -\def\XINT_expr_scanfracpart_aa #1% -{% - \ifnum \xint_c_ix<1#1 - \expandafter\XINT_expr_scanfracpart_b - \else - \expandafter !% - \fi - #1% -}% -\def\XINT_expr_scanfracpart_b #1% -{% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanfracpart_a\romannumeral-`0\romannumeral-`0% -}% -% \end{macrocode} -% \lverb|1.09k [2014/01/21]: added scanning for an hexadecimal number, possibly -% with a "hexa-decimal" part, only with uppercase ABCDEF (xintbinhex.sty works -% with ABCDEF, as tex itself requires uppercase letters after ", thus at least I -% feel comfortable with not bothering allowing abcdef... which would be possible -% but would complicate things; although perhaps there could be some use for -% lowercase. If needed, can be implemented, but I will probably long be dead -% when an archivist droid will be the first around circa 2500 AD to read these -% lines). -% -% For compatibility with \xintiiexpr, the [] thing is incorporated only if there -% the parser encounters a . indicating a fractional part (this fractional part -% may be empty). Thus for (infinitesimally) faster further processing by -% \xintexpr, "ABC.+ etc... is better than "ABC+ etc... on the other hand the -% initial processing with a . followed by an empty fractional part adds its bit -% of overhead... The . is not allowed in \xintiiexpr, as it will provoke -% insertion of [0] which is incompatible with it.| -% \begin{macrocode} -\def\XINT_expr_scanhex_I #1% -{% - \expandafter\XINT_expr_getop\romannumeral-`0\expandafter - \XINT_expr_lock\expandafter\XINT_expr_inhex - \romannumeral-`0\XINT_expr_scanhexI_a -}% -\def\XINT_expr_inhex #1.#2#3;% expanded inside \csname..\endcsname -{% - \if#2I\xintHexToDec{#1}% - \else - \xintiiMul{\xintiiPow{625}{\xintLength{#3}}}{\xintHexToDec{#1#3}}% - [\the\numexpr-4*\xintLength{#3}]% - \fi -}% -\def\XINT_expr_scanhexI_a #1% -{% - \ifcat #1\relax\else - \ifx !#1\else - \expandafter\expandafter\expandafter - \xint_thirdofthree - \fi\fi - \xint_firstoftwo {.I;!}% - {\expandafter\XINT_expr_scanhexI_aa\string }#1% -}% -\def\XINT_expr_scanhexI_aa #1% -{% - \if\ifnum`#1>`/ - \ifnum`#1>`9 - \ifnum`#1>`@ - \ifnum`#1>`F - 0\else1\fi\else0\fi\else1\fi\else0\fi 1% - \expandafter\XINT_expr_scanhexI_b - \else - \if .#1% - \expandafter\xint_firstoftwo - \else % gather what we got so far, leave catcode 12 #1 in stream - \expandafter\xint_secondoftwo - \fi - {\expandafter\XINT_expr_scanhex_transition}% - {\xint_afterfi {.I;!}}% - \fi - #1% -}% -\def\XINT_expr_scanhexI_b #1% -{% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanhexI_a\romannumeral-`0\romannumeral-`0% -}% -\def\XINT_expr_scanhex_transition .% -{% - \expandafter.\expandafter.\romannumeral-`0\expandafter - \XINT_expr_scanhexII_a\romannumeral-`0\romannumeral-`0% -}% -\def\XINT_expr_scanhexII_a #1% -{% - \ifcat #1\relax\else - \ifx !#1\else - \expandafter\expandafter\expandafter - \xint_thirdofthree - \fi\fi - \xint_firstoftwo {;!}% this stops the scan - {\expandafter\XINT_expr_scanhexII_aa\string }#1% -}% -\def\XINT_expr_scanhexII_aa #1% -{% - \if\ifnum`#1>`/ - \ifnum`#1>`9 - \ifnum`#1>`@ - \ifnum`#1>`F - 0\else1\fi\else0\fi\else1\fi\else0\fi 1% - \expandafter\XINT_expr_scanhexII_b - \else - \xint_afterfi {;!}% - \fi - #1% -}% -\def\XINT_expr_scanhexII_b #1% -{% - \expandafter #1\romannumeral-`0\expandafter - \XINT_expr_scanhexII_a\romannumeral-`0\romannumeral-`0% -}% -% \end{macrocode} -% \subsection{\csh{XINT\_expr\_getop}: looking for an operator} -% \lverb|June 14 (1.08b): I add here a second \romannumeral-`0, because -% \XINT_expr_getnext and others try to expand the next token -% but without grabbing it. -% -% This finds the next infix operator or closing parenthesis or postfix -% exclamation mark ! -% or expression end. It then leaves in the token flow -% <precedence> <operator> <locked number>. The <precedence> is generally -% a character command which thus stops expansion and gives back control to an -% \XINT_expr_until_<op> command; or it is the minus sign which will be -% converted by a suitable \XINT_expr_checkifprefix_<p> into an operator -% with a given inherited precedence. Earlier releases than 1.09c used tricks for -% the postfix !, ?, :, with <precedence> being in fact a macro to act -% immediately, and then re-activate \XINT_expr_getop. -% -% In versions earlier than 1.09a the <operator> was already made in to a control -% sequence; but now it is a left as a token and will be (generally) converted by -% the until macro which knows if it is in a \xintexpr or an \xintfloatexpr. (or -% an \xintiiexpr, since 1.09i) -% -% 1.09i allows \count's, \dimen's, \skip's with tacit multiplication. -% -% 1.09j extends the mechanism of tacit multiplication to the case of a sub -% xintexpression in its various variants. Careful that our ! has catcode 11 so -% \ifx! would be a disaster... -% -% 1.09k extends tacit multiplication to the case of an encountered opening -% parenthesis. -% -% | -% \begin{macrocode} -\def\XINT_expr_getop #1% this #1 is the current locked computed value -{% full expansion of next token, first swallowing a possible space - \expandafter\XINT_expr_getop_a\expandafter #1% - \romannumeral-`0\romannumeral-`0% -}% -\def\XINT_expr_getop_a #1#2% -{% if a control sequence is found, must be either \relax or register|variable - \ifcat #2\relax\expandafter\xint_firstoftwo - \else \expandafter\xint_secondoftwo - \fi - {\ifx #2\relax\expandafter\xint_firstofthree - \else\expandafter\xint_secondofthree % tacit multiplication - \fi }% - {\ifx !#2\expandafter\xint_secondofthree % tacit multiplication - \else % 1.09k adds tacit multiplication in front of ( - \if (#2\expandafter\expandafter\expandafter\xint_secondofthree - \else - \expandafter\expandafter\expandafter\xint_thirdofthree - \fi - \fi }% - {\XINT_expr_foundend #1}% - {\XINT_expr_foundop *#1#2}% - {\XINT_expr_foundop #2#1}% -}% -\def\XINT_expr_foundend {\xint_c_ \relax }% \relax is a place holder here. -\def\XINT_expr_foundop #1% then becomes <prec> <op> and is followed by <\.=f> -{% 1.09a: no control sequence \XINT_expr_op_#1, code common to expr/flexpr - \ifcsname XINT_expr_precedence_#1\endcsname - \expandafter\xint_afterfi\expandafter - {\csname XINT_expr_precedence_#1\endcsname #1}% - \else - \XINT_expr_unexpectedtoken - \expandafter\XINT_expr_getop - \fi -}% -% \end{macrocode} -% \subsection{Parentheses} -% \lverb|1.09a removes some doubling of \romannumeral-`\0 from 1.08b -% which served no useful purpose here (I think...). | -% \begin{macrocode} -\def\XINT_tmpa #1#2#3#4#5% -{% - \def#1##1% - {% - \xint_UDsignfork - ##1{\expandafter#1\romannumeral-`0#3}% - -{#2##1}% - \krof - }% - \def#2##1##2% - {% - \ifcase ##1\expandafter #4% - \or\xint_afterfi{% - \XINT_expr_extra_closing_paren - \expandafter #1\romannumeral-`0\XINT_expr_getop - }% - \else - \xint_afterfi{\expandafter#1\romannumeral-`0\csname XINT_#5_op_##2\endcsname }% - \fi - }% -}% -\xintFor #1 in {expr,flexpr,iiexpr} \do {% -\expandafter\XINT_tmpa - \csname XINT_#1_until_end_a\expandafter\endcsname - \csname XINT_#1_until_end_b\expandafter\endcsname - \csname XINT_#1_op_-vi\expandafter\endcsname - \csname XINT_#1_done\endcsname - {#1}% -}% -\def\XINT_expr_extra_closing_paren {\xintError:removed }% -\def\XINT_tmpa #1#2#3#4#5#6% -{% - \def #1{\expandafter #3\romannumeral-`0\XINT_expr_getnext }% - \let #2#1% - \def #3##1{\xint_UDsignfork - ##1{\expandafter #3\romannumeral-`0#5}% - -{#4##1}% - \krof }% - \def #4##1##2% - {% - \ifcase ##1\expandafter \XINT_expr_missing_cparen - \or \expandafter \XINT_expr_getop - \else \xint_afterfi - {\expandafter #3\romannumeral-`0\csname XINT_#6_op_##2\endcsname }% - \fi - }% -}% -\xintFor #1 in {expr,flexpr,iiexpr} \do {% -\expandafter\XINT_tmpa - \csname XINT_#1_op_(\expandafter\endcsname - \csname XINT_#1_oparen\expandafter\endcsname - \csname XINT_#1_until_)_a\expandafter\endcsname - \csname XINT_#1_until_)_b\expandafter\endcsname - \csname XINT_#1_op_-vi\endcsname - {#1}% -}% -\def\XINT_expr_missing_cparen {\xintError:inserted \xint_c_ \XINT_expr_done }% -\expandafter\let\csname XINT_expr_precedence_)\endcsname \xint_c_i -\expandafter\let\csname XINT_flexpr_precedence_)\endcsname \xint_c_i -\expandafter\let\csname XINT_iiexpr_precedence_)\endcsname \xint_c_i -\expandafter\let\csname XINT_expr_op_)\endcsname \XINT_expr_getop -\expandafter\let\csname XINT_flexpr_op_)\endcsname\XINT_expr_getop -\expandafter\let\csname XINT_iiexpr_op_)\endcsname\XINT_expr_getop -% \end{macrocode} -% \subsection{The \csh{XINT\_expr\_until\_<op>} macros for boolean operators, -% comparison operators, arithmetic operators, scientfic notation.} -% \lverb|Extended in 1.09a with comparison and boolean operators. -% 1.09i adds \xintiiexpr and incorporates optional part [\XINTdigits] for a tiny -% bit faster float operations now already equipped with their optional -% argument|. -% \begin{macrocode} -\def\XINT_tmpb #1#2#3#4#5#6%#7% -{% - \expandafter\XINT_tmpc - \csname XINT_#1_op_#3\expandafter\endcsname - \csname XINT_#1_until_#3_a\expandafter\endcsname - \csname XINT_#1_until_#3_b\expandafter\endcsname - \csname XINT_#1_op_-#5\expandafter\endcsname - \csname xint_c_#4\expandafter\endcsname - \csname #2#6\expandafter\endcsname - \csname XINT_expr_precedence_#3\endcsname {#1}%{#7}% -}% -\def\XINT_tmpc #1#2#3#4#5#6#7#8#9% -{% - \def #1##1% \XINT_expr_op_<op> - {% keep value, get next number and operator, then do until - \expandafter #2\expandafter ##1% - \romannumeral-`0\expandafter\XINT_expr_getnext - }% - \def #2##1##2% \XINT_expr_until_<op>_a - {\xint_UDsignfork - ##2{\expandafter #2\expandafter ##1\romannumeral-`0#4}% - -{#3##1##2}% - \krof }% - \def #3##1##2##3##4% \XINT_expr_until_<op>_b - {% either execute next operation now, or first do next (possibly unary) - \ifnum ##2>#5% - \xint_afterfi {\expandafter #2\expandafter ##1\romannumeral-`0% - \csname XINT_#8_op_##3\endcsname {##4}}% - \else - \xint_afterfi - {\expandafter ##2\expandafter ##3% - \csname .=#6#9{\XINT_expr_unlock ##1}{\XINT_expr_unlock ##4}\endcsname }% - \fi - }% - \let #7#5% -}% -\def\XINT_tmpa #1{\XINT_tmpb {expr}{xint}#1{}}% -\xintApplyInline {\XINT_tmpa }{% - {|{iii}{vi}{OR}}% - {&{iv}{vi}{AND}}% - {<{v}{vi}{Lt}}% - {>{v}{vi}{Gt}}% - {={v}{vi}{Eq}}% - {+{vi}{vi}{Add}}% - {-{vi}{vi}{Sub}}% - {*{vii}{vii}{Mul}}% - {/{vii}{vii}{Div}}% - {^{viii}{viii}{Pow}}% - {e{ix}{ix}{fE}}% - {E{ix}{ix}{fE}}% -}% -\def\XINT_tmpa #1{\XINT_tmpb {flexpr}{xint}#1{}}% -\xintApplyInline {\XINT_tmpa }{% - {|{iii}{vi}{OR}}% - {&{iv}{vi}{AND}}% - {<{v}{vi}{Lt}}% - {>{v}{vi}{Gt}}% - {={v}{vi}{Eq}}% -}% -\def\XINT_tmpa #1{\XINT_tmpb {flexpr}{XINTinFloat}#1{[\XINTdigits]}}% -\xintApplyInline {\XINT_tmpa }{% - {+{vi}{vi}{Add}}% - {-{vi}{vi}{Sub}}% - {*{vii}{vii}{Mul}}% - {/{vii}{vii}{Div}}% - {^{viii}{viii}{Power}}% - {e{ix}{ix}{fE}}% - {E{ix}{ix}{fE}}% -}% -\def\XINT_tmpa #1{\XINT_tmpb {iiexpr}{xint}#1{}}% -\xintApplyInline {\XINT_tmpa }{% - {|{iii}{vi}{OR}}% - {&{iv}{vi}{AND}}% - {<{v}{vi}{Lt}}% - {>{v}{vi}{Gt}}% - {={v}{vi}{Eq}}% - {+{vi}{vi}{iiAdd}}% - {-{vi}{vi}{iiSub}}% - {*{vii}{vii}{iiMul}}% - {/{vii}{vii}{iiQuo}}% - {^{viii}{viii}{iiPow}}% - {e{ix}{ix}{iE}}% - {E{ix}{ix}{iE}}% -}% -% \end{macrocode} -% \subsection{The comma as binary operator} -% \lverb|New with 1.09a.| -% \begin{macrocode} -\def\XINT_tmpa #1#2#3#4#5#6% -{% - \def #1##1% \XINT_expr_op_,_a - {% - \expandafter #2\expandafter ##1\romannumeral-`0\XINT_expr_getnext - }% - \def #2##1##2% \XINT_expr_until_,_a - {\xint_UDsignfork - ##2{\expandafter #2\expandafter ##1\romannumeral-`0#4}% - -{#3##1##2}% - \krof }% - \def #3##1##2##3##4% \XINT_expr_until_,_b - {% - \ifnum ##2>\xint_c_ii - \xint_afterfi {\expandafter #2\expandafter ##1\romannumeral-`0% - \csname XINT_#6_op_##3\endcsname {##4}}% - \else - \xint_afterfi - {\expandafter ##2\expandafter ##3% - \csname .=\XINT_expr_unlock ##1,\XINT_expr_unlock ##4\endcsname }% - \fi - }% - \let #5\xint_c_ii -}% -\xintFor #1 in {expr,flexpr,iiexpr} \do {% -\expandafter\XINT_tmpa - \csname XINT_#1_op_,\expandafter\endcsname - \csname XINT_#1_until_,_a\expandafter\endcsname - \csname XINT_#1_until_,_b\expandafter\endcsname - \csname XINT_#1_op_-vi\expandafter\endcsname - \csname XINT_expr_precedence_,\endcsname {#1}% -}% -% \end{macrocode} -% \subsection{\csh{XINT\_expr\_op\_-<level>}: minus as prefix inherits its -% precedence level} -% \lverb|1.09i: \xintiiexpr must use \xintiiOpp (or at least \xintiOpp, but that -% would be a waste; however impacts round and trunc as I allow them).| -% \begin{macrocode} -\def\XINT_tmpa #1#2#3% -{% - \expandafter\XINT_tmpb - \csname XINT_#1_op_-#3\expandafter\endcsname - \csname XINT_#1_until_-#3_a\expandafter\endcsname - \csname XINT_#1_until_-#3_b\expandafter\endcsname - \csname xint_c_#3\endcsname {#1}#2% -}% -\def\XINT_tmpb #1#2#3#4#5#6% -{% - \def #1% \XINT_expr_op_-<level> - {% get next number+operator then switch to _until macro - \expandafter #2\romannumeral-`0\XINT_expr_getnext - }% - \def #2##1% \XINT_expr_until_-<l>_a - {\xint_UDsignfork - ##1{\expandafter #2\romannumeral-`0#1}% - -{#3##1}% - \krof }% - \def #3##1##2##3% \XINT_expr_until_-<l>_b - {% _until tests precedence level with next op, executes now or postpones - \ifnum ##1>#4% - \xint_afterfi {\expandafter #2\romannumeral-`0% - \csname XINT_#5_op_##2\endcsname {##3}}% - \else - \xint_afterfi {\expandafter ##1\expandafter ##2% - \csname .=#6{\XINT_expr_unlock ##3}\endcsname }% - \fi - }% -}% -\xintApplyInline{\XINT_tmpa {expr}\xintOpp}{{vi}{vii}{viii}{ix}}% -\xintApplyInline{\XINT_tmpa {flexpr}\xintOpp}{{vi}{vii}{viii}{ix}}% -\xintApplyInline{\XINT_tmpa {iiexpr}\xintiiOpp}{{vi}{vii}{viii}{ix}}% -% \end{macrocode} -% \subsection{? as two-way conditional} -% \lverb|New with 1.09a. Modified in 1.09c to have less precedence than -% functions. Code is cleaner as it does not play tricks with _precedence. There -% is no associated until macro, because action is immediate once activated (only -% a previously scanned function can delay activation).| -% \begin{macrocode} -\let\XINT_expr_precedence_? \xint_c_x -\def \XINT_expr_op_? #1#2#3% -{% - \xintifZero{\XINT_expr_unlock #1}% - {\XINT_expr_getnext #3}% - {\XINT_expr_getnext #2}% -}% -\let\XINT_flexpr_op_?\XINT_expr_op_? -\let\XINT_iiexpr_op_?\XINT_expr_op_? -% \end{macrocode} -% \subsection{: as three-way conditional} -% \lverb|New with 1.09a. Modified in 1.09c to have less precedence than -% functions. | -% \begin{macrocode} -\let\XINT_expr_precedence_: \xint_c_x -\def \XINT_expr_op_: #1#2#3#4% -{% - \xintifSgn {\XINT_expr_unlock #1}% - {\XINT_expr_getnext #2}% - {\XINT_expr_getnext #3}% - {\XINT_expr_getnext #4}% -}% -\let\XINT_flexpr_op_:\XINT_expr_op_: -\let\XINT_iiexpr_op_:\XINT_expr_op_: -% \end{macrocode} -% \subsection{! as postfix factorial operator} -% \lverb|The factorial is currently the exact one, there is no float version. -% Starting with 1.09c, it has lower priority than functions, it is not executed -% immediately anymore. The code is cleaner and does not abuse _precedence, but -% does assign it a true level. There is no until macro, because the factorial -% acts on what precedes it.| -% \begin{macrocode} -\let\XINT_expr_precedence_! \xint_c_x -\def\XINT_expr_op_! #1{\expandafter\XINT_expr_getop - \csname .=\xintFac{\XINT_expr_unlock #1}\endcsname }% -\let\XINT_flexpr_op_!\XINT_expr_op_! -\def\XINT_iiexpr_op_! #1{\expandafter\XINT_expr_getop - \csname .=\xintiFac{\XINT_expr_unlock #1}\endcsname }% -% \end{macrocode} -% \subsection{Functions} -% \lverb|New with 1.09a. Names of ..Float..:csv macros have been changed in -% 1.09h | -% \begin{macrocode} -\def\XINT_tmpa #1#2#3#4{% - \def #1##1% - {% - \ifcsname XINT_expr_onlitteral_##1\endcsname - \expandafter\XINT_expr_funcoflitteral - \else - \expandafter #2% - \fi {##1}% - }% - \def #2##1% - {% - \ifcsname XINT_#4_func_##1\endcsname - \xint_afterfi - {\expandafter\expandafter\csname XINT_#4_func_##1\endcsname}% - \else \csname xintError:unknown `##1\string'\endcsname - \xint_afterfi{\expandafter\XINT_expr_func_unknown}% - \fi - \romannumeral-`0#3% - }% -}% -\xintFor #1 in {expr,flexpr,iiexpr} \do {% - \expandafter\XINT_tmpa - \csname XINT_#1_op_@\expandafter\endcsname - \csname XINT_#1_op_@@\expandafter\endcsname - \csname XINT_#1_oparen\endcsname {#1}% -}% -\def\XINT_expr_funcoflitteral #1% -{% - \expandafter\expandafter\csname XINT_expr_onlitteral_#1\endcsname - \romannumeral-`0\XINT_expr_scanfunc -}% -\def\XINT_expr_onlitteral_bool #1#2#3{\expandafter\XINT_expr_getop - \csname .=\xintBool{#3}\endcsname }% -\def\XINT_expr_onlitteral_togl #1#2#3{\expandafter\XINT_expr_getop - \csname .=\xintToggle{#3}\endcsname }% -\def\XINT_expr_func_unknown #1#2#3% 1.09i removes [0], because \xintiiexpr - {\expandafter #1\expandafter #2\csname .=0\endcsname }% -\def\XINT_expr_func_reduce #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintIrr {\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_reduce\XINT_expr_func_reduce -% \XINT_iiexpr_func_reduce not defined -\def\XINT_expr_func_frac #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintTFrac {\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_flexpr_func_frac #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\XINTinFloatFrac [\XINTdigits]{\XINT_expr_unlock #3}\endcsname -}% -% \XINT_iiexpr_func_frac not defined -\def\XINT_expr_func_sqr #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintSqr {\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_flexpr_func_sqr #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\XINTinFloatMul [\XINTdigits]% - {\XINT_expr_unlock #3}{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_iiexpr_func_sqr #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintiiSqr {\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_expr_func_abs #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintAbs {\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_abs\XINT_expr_func_abs -\def\XINT_iiexpr_func_abs #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintiiAbs {\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_expr_func_sgn #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintSgn {\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_sgn\XINT_expr_func_sgn -\def\XINT_iiexpr_func_sgn #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintiiSgn {\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_expr_func_floor #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintFloor {\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_floor\XINT_expr_func_floor -\let\XINT_iiexpr_func_floor\XINT_expr_func_floor -\def\XINT_expr_func_ceil #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintCeil {\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_ceil\XINT_expr_func_ceil -\let\XINT_iiexpr_func_ceil\XINT_expr_func_ceil -\def\XINT_expr_twoargs #1,#2,{{#1}{#2}}% -\def\XINT_expr_func_quo #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\expandafter\expandafter\xintQuo - \expandafter\XINT_expr_twoargs - \romannumeral-`0\XINT_expr_unlock #3,\endcsname -}% -\let\XINT_flexpr_func_quo\XINT_expr_func_quo -\def\XINT_iiexpr_func_quo #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\expandafter\expandafter\xintiiQuo - \expandafter\XINT_expr_twoargs - \romannumeral-`0\XINT_expr_unlock #3,\endcsname -}% -\def\XINT_expr_func_rem #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\expandafter\expandafter\xintRem - \expandafter\XINT_expr_twoargs - \romannumeral-`0\XINT_expr_unlock #3,\endcsname -}% -\let\XINT_flexpr_func_rem\XINT_expr_func_rem -\def\XINT_iiexpr_func_rem #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\expandafter\expandafter\xintiiRem - \expandafter\XINT_expr_twoargs - \romannumeral-`0\XINT_expr_unlock #3,\endcsname -}% -\def\XINT_expr_oneortwo #1#2#3,#4,#5.% -{% - \if\relax#5\relax\expandafter\xint_firstoftwo\else - \expandafter\xint_secondoftwo\fi - {#1{0}}{#2{\xintNum {#4}}}{#3}% -}% -\def\XINT_expr_func_round #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\XINT_expr_oneortwo - \expandafter\xintiRound\expandafter\xintRound - \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname -}% -\let\XINT_flexpr_func_round\XINT_expr_func_round -\def\XINT_iiexpr_oneortwo #1#2,#3,#4.% -{% - \if\relax#4\relax\expandafter\xint_firstoftwo\else - \expandafter\xint_secondoftwo\fi - {#1{0}}{#1{#3}}{#2}% -}% -\def\XINT_iiexpr_func_round #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\XINT_iiexpr_oneortwo\expandafter\xintiRound - \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname -}% -\def\XINT_expr_func_trunc #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\XINT_expr_oneortwo - \expandafter\xintiTrunc\expandafter\xintTrunc - \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname -}% -\let\XINT_flexpr_func_trunc\XINT_expr_func_trunc -\def\XINT_iiexpr_func_trunc #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\XINT_iiexpr_oneortwo\expandafter\xintiTrunc - \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname -}% -\def\XINT_expr_argandopt #1,#2,#3.% -{% - \if\relax#3\relax\expandafter\xint_firstoftwo\else - \expandafter\xint_secondoftwo\fi - {[\XINTdigits]}{[\xintNum {#2}]}{#1}% -}% -\def\XINT_expr_func_float #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\XINTinFloat - \romannumeral-`0\expandafter\XINT_expr_argandopt - \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname -}% -\let\XINT_flexpr_func_float\XINT_expr_func_float -% \XINT_iiexpr_func_float not defined -\def\XINT_expr_func_sqrt #1#2#3% -{% - \expandafter #1\expandafter #2\csname .=% - \expandafter\XINTinFloatSqrt - \romannumeral-`0\expandafter\XINT_expr_argandopt - \romannumeral-`0\XINT_expr_unlock #3,,.\endcsname -}% -\let\XINT_flexpr_func_sqrt\XINT_expr_func_sqrt -\def\XINT_iiexpr_func_sqrt #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintiSqrt {\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_expr_func_gcd #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintGCDof:csv{\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_gcd\XINT_expr_func_gcd -\let\XINT_iiexpr_func_gcd\XINT_expr_func_gcd -\def\XINT_expr_func_lcm #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintLCMof:csv{\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_lcm\XINT_expr_func_lcm -\let\XINT_iiexpr_func_lcm\XINT_expr_func_lcm -\def\XINT_expr_func_max #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintMaxof:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_iiexpr_func_max #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintiMaxof:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_flexpr_func_max #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\XINTinFloatMaxof:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_expr_func_min #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintMinof:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_iiexpr_func_min #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintiMinof:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_flexpr_func_min #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\XINTinFloatMinof:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_expr_func_sum #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintSum:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_flexpr_func_sum #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\XINTinFloatSum:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_iiexpr_func_sum #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintiiSum:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_expr_func_prd #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintPrd:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_flexpr_func_prd #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\XINTinFloatPrd:csv{\XINT_expr_unlock #3}\endcsname -}% -\def\XINT_iiexpr_func_prd #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintiiPrd:csv{\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_expr_func_add\XINT_expr_func_sum -\let\XINT_expr_func_mul\XINT_expr_func_prd -\let\XINT_flexpr_func_add\XINT_flexpr_func_sum -\let\XINT_flexpr_func_mul\XINT_flexpr_func_prd -\let\XINT_iiexpr_func_add\XINT_iiexpr_func_sum -\let\XINT_iiexpr_func_mul\XINT_iiexpr_func_prd -\def\XINT_expr_func_? #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintIsNotZero {\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_? \XINT_expr_func_? -\let\XINT_iiexpr_func_? \XINT_expr_func_? -\def\XINT_expr_func_! #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintIsZero {\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_! \XINT_expr_func_! -\let\XINT_iiexpr_func_! \XINT_expr_func_! -\def\XINT_expr_func_not #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintIsZero {\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_not \XINT_expr_func_not -\let\XINT_iiexpr_func_not \XINT_expr_func_not -\def\XINT_expr_func_all #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintANDof:csv{\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_all\XINT_expr_func_all -\let\XINT_iiexpr_func_all\XINT_expr_func_all -\def\XINT_expr_func_any #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintORof:csv{\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_any\XINT_expr_func_any -\let\XINT_iiexpr_func_any\XINT_expr_func_any -\def\XINT_expr_func_xor #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\xintXORof:csv{\XINT_expr_unlock #3}\endcsname -}% -\let\XINT_flexpr_func_xor\XINT_expr_func_xor -\let\XINT_iiexpr_func_xor\XINT_expr_func_xor -\def\xintifNotZero:: #1,#2,#3,{\xintifNotZero{#1}{#2}{#3}}% -\def\XINT_expr_func_if #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\expandafter\xintifNotZero:: - \romannumeral-`0\XINT_expr_unlock #3,\endcsname -}% -\let\XINT_flexpr_func_if\XINT_expr_func_if -\let\XINT_iiexpr_func_if\XINT_expr_func_if -\def\xintifSgn:: #1,#2,#3,#4,{\xintifSgn{#1}{#2}{#3}{#4}}% -\def\XINT_expr_func_ifsgn #1#2#3% -{% - \expandafter #1\expandafter #2\csname - .=\expandafter\xintifSgn:: - \romannumeral-`0\XINT_expr_unlock #3,\endcsname -}% -\let\XINT_flexpr_func_ifsgn\XINT_expr_func_ifsgn -\let\XINT_iiexpr_func_ifsgn\XINT_expr_func_ifsgn -% \end{macrocode} -% \subsection{\csh{xintNewExpr}, \csh{xintNewFloatExpr}\dots} -% \lverb|& -% Rewritten in 1.09a. Now, the parameters of the formula are entered in the -% usual way by the user, with # not _. And _ is assigned to make macros -% not expand. This way, : is freed, as we now need it for the ternary operator. -% (on numeric data; if use with macro parameters, should be coded with the -% functionn ifsgn , rather) -% -% Code unified in 1.09c, and \xintNewNumExpr, \xintNewBoolExpr added. 1.09i -% renames \xintNewNumExpr to \xintNewIExpr, and defines \xintNewIIExpr.| -% \begin{macrocode} -\def\XINT_newexpr_print #1{\ifnum\xintNthElt{0}{#1}>1 - \expandafter\xint_firstoftwo - \else - \expandafter\xint_secondoftwo - \fi - {_xintListWithSep,{#1}}{\xint_firstofone#1}}% -\xintForpair #1#2 in {(fl,Float),(i,iRound0),(bool,IsTrue)}\do {% - \expandafter\def\csname XINT_new#1expr_print\endcsname - ##1{\ifnum\xintNthElt{0}{##1}>1 - \expandafter\xint_firstoftwo - \else - \expandafter\xint_secondoftwo - \fi - {_xintListWithSep,{\xintApply{_xint#2}{##1}}} - {_xint#2##1}}}% -\toks0 {}% -\xintFor #1 in {Bool,Toggle,Floor,Ceil,iRound,Round,iTrunc,Trunc,TFrac,% - Lt,Gt,Eq,AND,OR,IsNotZero,IsZero,ifNotZero,ifSgn,% - Irr,Num,Abs,Sgn,Opp,Quo,Rem,Add,Sub,Mul,Sqr,Div,Pow,Fac,fE,iSqrt,% - iiAdd,iiSub,iiMul,iiSqr,iiPow,iiQuo,iiRem,iiSgn,iiAbs,iiOpp,iE}\do - {\toks0 - \expandafter{\the\toks0\expandafter\def\csname xint#1\endcsname {_xint#1}}}% -\xintFor #1 in {,Sqrt,Add,Sub,Mul,Div,Power,fE,Frac}\do - {\toks0 - \expandafter{\the\toks0\expandafter\def\csname XINTinFloat#1\endcsname - {_XINTinFloat#1}}}% -\xintFor #1 in {GCDof,LCMof,Maxof,Minof,ANDof,ORof,XORof,Sum,Prd,% - iMaxof,iMinof,iiSum,iiPrd}\do - {\toks0 - \expandafter{\the\toks0\expandafter\def\csname xint#1:csv\endcsname - ####1{_xint#1{\xintCSVtoListNonStripped {####1}}}}}% -\xintFor #1 in {Maxof,Minof,Sum,Prd}\do - {\toks0 - \expandafter{\the\toks0\expandafter\def\csname XINTinFloat#1:csv\endcsname - ####1{_XINTinFloat#1{\xintCSVtoListNonStripped {####1}}}}}% -\expandafter\def\expandafter\XINT_expr_protect\expandafter{\the\toks0 - \def\XINTdigits {_XINTdigits}% - \def\XINT_expr_print ##1{\expandafter\XINT_newexpr_print\expandafter - {\romannumeral0\xintcsvtolistnonstripped{\XINT_expr_unlock ##1}}}% - \def\XINT_flexpr_print ##1{\expandafter\XINT_newflexpr_print\expandafter - {\romannumeral0\xintcsvtolistnonstripped{\XINT_expr_unlock ##1}}}% - \def\XINT_iexpr_print ##1{\expandafter\XINT_newiexpr_print\expandafter - {\romannumeral0\xintcsvtolistnonstripped{\XINT_expr_unlock ##1}}}% - \def\XINT_boolexpr_print ##1{\expandafter\XINT_newboolexpr_print\expandafter - {\romannumeral0\xintcsvtolistnonstripped{\XINT_expr_unlock ##1}}}% -}% -\toks0 {}% -\def\xintNewExpr {\xint_NewExpr\xinttheexpr }% -\def\xintNewFloatExpr {\xint_NewExpr\xintthefloatexpr }% -\def\xintNewIExpr {\xint_NewExpr\xinttheiexpr }% -\let\xintNewNumExpr\xintNewIExpr -\def\xintNewIIExpr {\xint_NewExpr\xinttheiiexpr }% -\def\xintNewBoolExpr {\xint_NewExpr\xinttheboolexpr }% -% \end{macrocode} -% \lverb|1.09i has added \escapechar 92, as \meaning is used in \XINT_NewExpr, -% and a non existent escape-char would be a problem with \scantokens. Also -% \catcode32 is set to 10 in \xintexprSafeCatcodes for being extra-safe.| -% \begin{macrocode} -\def\xint_NewExpr #1#2[#3]% -{% - \begingroup - \ifcase #3\relax - \toks0 {\xdef #2}% - \or \toks0 {\xdef #2##1}% - \or \toks0 {\xdef #2##1##2}% - \or \toks0 {\xdef #2##1##2##3}% - \or \toks0 {\xdef #2##1##2##3##4}% - \or \toks0 {\xdef #2##1##2##3##4##5}% - \or \toks0 {\xdef #2##1##2##3##4##5##6}% - \or \toks0 {\xdef #2##1##2##3##4##5##6##7}% - \or \toks0 {\xdef #2##1##2##3##4##5##6##7##8}% - \or \toks0 {\xdef #2##1##2##3##4##5##6##7##8##9}% - \fi - \xintexprSafeCatcodes - \escapechar92 - \XINT_NewExpr #1% -}% -\catcode`* 13 -\def\XINT_NewExpr #1#2% -{% - \def\XINT_tmpa ##1##2##3##4##5##6##7##8##9{#2}% - \XINT_expr_protect - \lccode`*=`_ \lowercase {\def*}{!noexpand!}% - \catcode`_ 13 \catcode`: 11 %\endlinechar -1 %not sure why I had that, \par? - \everyeof {\noexpand }% - \edef\XINT_tmpb ##1##2##3##4##5##6##7##8##9% - {\scantokens - \expandafter{\romannumeral-`0#1% - \XINT_tmpa {####1}{####2}{####3}% - {####4}{####5}{####6}% - {####7}{####8}{####9}% - \relax}}% - \lccode`*=`\$ \lowercase {\def*}{####}% - \catcode`\$ 13 \catcode`! 0 \catcode`_ 11 % - \the\toks0 - {\scantokens\expandafter{\expandafter - \XINT_newexpr_setprefix\meaning\XINT_tmpb}}% - \endgroup -}% -\let\xintexprRestoreCatcodes\empty -\def\xintexprSafeCatcodes -{% for end user. - \edef\xintexprRestoreCatcodes {% - \catcode34=\the\catcode34 % " - \catcode63=\the\catcode63 % ? - \catcode124=\the\catcode124 % | - \catcode38=\the\catcode38 % & - \catcode33=\the\catcode33 % ! - \catcode93=\the\catcode93 % ] - \catcode91=\the\catcode91 % [ - \catcode94=\the\catcode94 % ^ - \catcode95=\the\catcode95 % _ - \catcode47=\the\catcode47 % / - \catcode41=\the\catcode41 % ) - \catcode40=\the\catcode40 % ( - \catcode42=\the\catcode42 % * - \catcode43=\the\catcode43 % + - \catcode62=\the\catcode62 % > - \catcode60=\the\catcode60 % < - \catcode58=\the\catcode58 % : - \catcode46=\the\catcode46 % . - \catcode45=\the\catcode45 % - - \catcode44=\the\catcode44 % , - \catcode61=\the\catcode61 % = - \catcode32=\the\catcode32\relax % space - }% it's hard to know where to stop... - \catcode34=12 % " - \catcode63=12 % ? - \catcode124=12 % | - \catcode38=4 % & - \catcode33=12 % ! - \catcode93=12 % ] - \catcode91=12 % [ - \catcode94=7 % ^ - \catcode95=8 % _ - \catcode47=12 % / - \catcode41=12 % ) - \catcode40=12 % ( - \catcode42=12 % * - \catcode43=12 % + - \catcode62=12 % > - \catcode60=12 % < - \catcode58=12 % : - \catcode46=12 % . - \catcode45=12 % - - \catcode44=12 % , - \catcode61=12 % = - \catcode32=10 % space -}% -\let\XINT_tmpa\relax \let\XINT_tmpb\relax \let\XINT_tmpc\relax -\XINT_restorecatcodes_endinput% -% \end{macrocode} -% \DeleteShortVerb{\|} -% \MakePercentComment -%</xintexpr> -%<*dtx> -\StoreCodelineNo {xintexpr} - -\def\mymacro #1{\mymacroaux #1} -\def\mymacroaux #1#2{\strut \texttt{#1:}& \digitstt{ #2.}\tabularnewline } -\indent -\begin{tabular}[t]{r@{}r} -\xintApplyInline\mymacro\storedlinecounts -\end{tabular} -\def\mymacroaux #1#2{#2}% -% -\parbox[t]{10cm}{Total number of code lines: - \digitstt{\xintiiSum{\xintApply\mymacro\storedlinecounts}}. Each - package starts - with circa \digitstt{80} lines dealing with catcodes, package identification - and reloading management, also for Plain \TeX\strut. Version - \texttt{\xintversion} of \texttt{\xintdate}.\par} - - -\CharacterTable - {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z - Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z - Digits \0\1\2\3\4\5\6\7\8\9 - Exclamation \! Double quote \" Hash (number) \# - Dollar \$ Percent \% Ampersand \& - Acute accent \' Left paren \( Right paren \) - Asterisk \* Plus \+ Comma \, - Minus \- Point \. Solidus \/ - Colon \: Semicolon \; Less than \< - Equals \= Greater than \> Question mark \? - Commercial at \@ Left bracket \[ Backslash \\ - Right bracket \] Circumflex \^ Underscore \_ - Grave accent \` Left brace \{ Vertical bar \| - Right brace \} Tilde \~} -\CheckSum {21378} -\makeatletter\check@checksum\makeatother -\Finale -%% End of file xint.dtx diff --git a/Master/texmf-dist/doc/generic/xint/xint.pdf b/Master/texmf-dist/doc/generic/xint/xint.pdf Binary files differindex 4b2f17f993c..70e5dccdff3 100644 --- a/Master/texmf-dist/doc/generic/xint/xint.pdf +++ b/Master/texmf-dist/doc/generic/xint/xint.pdf |