diff options
Diffstat (limited to 'Master')
| -rw-r--r-- | Master/texmf-dist/doc/generic/xint/README | 76 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/generic/xint/xint.pdf | bin | 249177 -> 348223 bytes | |||
| -rw-r--r-- | Master/texmf-dist/source/generic/xint/xint.dtx | 5493 | ||||
| -rw-r--r-- | Master/texmf-dist/source/generic/xint/xint.ins | 9 | ||||
| -rw-r--r-- | Master/texmf-dist/tex/generic/xint/xint.sty | 1751 | ||||
| -rw-r--r-- | Master/texmf-dist/tex/generic/xint/xintfrac.sty | 1107 | ||||
| -rw-r--r-- | Master/texmf-dist/tex/generic/xint/xintgcd.sty | 62 | ||||
| -rw-r--r-- | Master/texmf-dist/tex/generic/xint/xintseries.sty | 349 |
8 files changed, 6397 insertions, 2450 deletions
diff --git a/Master/texmf-dist/doc/generic/xint/README b/Master/texmf-dist/doc/generic/xint/README index 988f53e845d..c313073378d 100644 --- a/Master/texmf-dist/doc/generic/xint/README +++ b/Master/texmf-dist/doc/generic/xint/README @@ -1,12 +1,17 @@ %---------------------------------------------------------------- % The xint bundle -% components: -% (base) xint: Expandable operations on long numbers -% xintgcd: Euclidean algorithm with xint package +% (base) xint.sty Expandable operations on long numbers +% xintgcd.sty Euclidean algorithm with xint package +% xintfrac.sty Expandable operations on fractions +% xintseries.sty Expandable partial sums with xint package % Copyright (C) 2013 by Jean-Francois Burnol -% Version: v1.02 (2013/04/05) +% Version: v1.03 (2013/04/14) %---------------------------------------------------------------- +This work consists of the source file xint.dtx and of its derived +files xint.sty, xintgcd.sty, xintfrac.sty, xintseries.sty, xint.ins +and the documentation xint.pdf (or xint.dvi). + 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 @@ -17,38 +22,39 @@ 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 xint.sty, xintgcd.sty, xint.ins 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'. - Installation and Usage: - ======================= - - Run tex or latex on xint.dtx. - - This will extract the style files xint.sty and xintgcd.sty (and - xint.ins). Files with the same names and in the same repertory - will be overwritten. The tex (not latex) run will stop with the - complaint that it does not understand \NeedsTeXFormat, but the - style files will already have been extracted by that time. - - Alternatively, run tex or latex on xint.ins if available. - - To get xint.pdf run pdflatex thrice on xint.dtx - - xint.sty, xintgcd.sty -> TDS:tex/generic/xint/ - xint.dtx -> TDS:source/generic/xint/ - xint.pdf -> TDS:doc/generic/xint/ - - It may well be necessary to then refresh the TeX installation - filename database. - - Usage with LaTeX: \usepackage{xint} - \usepackage{xintgcd} - - Usage with TeX: \input xint.sty\relax - \input xintgcd.sty\relax +Installation and Usage: +======================= + +Run tex or latex on xint.dtx. + +This will extract the style files xint.sty, xintgcd.sty, xintfrac.sty, +xintseries.sty (and xint.ins). Files with the same names and in the +same repertory will be overwritten. The tex (not latex) run will stop +with the complaint that it does not understand \NeedsTeXFormat, but the +style files will already have been extracted by that time. + +Alternatively, run tex or latex on xint.ins if available. + +To get xint.pdf run pdflatex thrice on xint.dtx + + xint.sty, xintgcd.sty, xintfrac.sty, xintseries.sty + -> TDS:tex/generic/xint/ + xint.dtx -> TDS:source/generic/xint/ + xint.pdf -> TDS:doc/generic/xint/ + +It may be necessary to then refresh the TeX installation filename +database. + +Usage with LaTeX: \usepackage{xint} + \usepackage{xintgcd} % (loads xint) + \usepackage{xintfrac} % (loads xint) + \usepackage{xintseries} % (loads xintfrac) + +Usage with TeX: \input xint.sty\relax + \input xintgcd.sty\relax % (loads xint) + \input xintfrac.sty\relax % (loads xint) + \input xintseries.sty\relax % (loads xintfrac) diff --git a/Master/texmf-dist/doc/generic/xint/xint.pdf b/Master/texmf-dist/doc/generic/xint/xint.pdf Binary files differindex 3390c97a79c..b9a92b732d1 100644 --- a/Master/texmf-dist/doc/generic/xint/xint.pdf +++ b/Master/texmf-dist/doc/generic/xint/xint.pdf diff --git a/Master/texmf-dist/source/generic/xint/xint.dtx b/Master/texmf-dist/source/generic/xint/xint.dtx index 7e8e6bb2c34..0fd44b0b4f1 100644 --- a/Master/texmf-dist/source/generic/xint/xint.dtx +++ b/Master/texmf-dist/source/generic/xint/xint.dtx @@ -1,20 +1,24 @@ % -*- coding: iso-latin-1; -*- -% This file: xint.dtx (1.02, 2013/04/05) +% This file: xint.dtx (1.03, 2013/04/14) %% %%---------------------------------------------------------------- -%% The xint bundle (version 1.02 of April 5, 2013) +%% The xint bundle (version 1.03 of April 14, 2013) %<xint>%% xint: Expandable operations on long numbers -%<xintgcd>%% xintgcd: Euclidean algorithm with xint package +%<xintgcd>%% xintgcd: Euclidean algorithm with xint package +%<xintfrac>%% xintfrac: Expandable operations on fractions +%<xintseries>%% xintseries: Expandable partial sums with xint package %% Copyright (C) 2013 by Jean-Francois Burnol %%---------------------------------------------------------------- %% % Style files in the bundle: -% (base) xint.sty Expandable operations on long numbers -% xintgcd.sty Euclidean algorithm with xint package +% (base) xint.sty Expandable operations on long numbers +% xintgcd.sty Euclidean algorithm with xint package +% xintfrac.sty Expandable operations on fractions +% xintseries.sty Expandable partial sums with xint package % -% This work consists of the source file xint.dtx and of its derived -% files xint.sty, xintgcd.sty, xint.ins and the documentation xint.pdf -% (or xint.dvi). +% This work consists of the source file xint.dtx and of its derived files +% xint.sty, xintgcd.sty, xintfrac.sty, xintseries.sty, xint.ins and the +% documentation xint.pdf (or xint.dvi). % % This work may be distributed and/or modified under the % conditions of the LaTeX Project Public License, either @@ -34,33 +38,38 @@ % % Run tex or latex on xint.dtx. % -% This will extract the style files xint.sty and xintgcd.sty (and -% xint.ins). Files with the same names and in the same repertory -% will be overwritten. The tex (not latex) run will stop with the -% complaint that it does not understand \NeedsTeXFormat, but the +% This will extract the style files xint.sty, xintgcd.sty, xintfrac.sty, +% xintseries.sty (and xint.ins). Files with the same names and in the +% same repertory will be overwritten. The tex (not latex) run will stop +% with the complaint that it does not understand \NeedsTeXFormat, but the % style files will already have been extracted by that time. % % Alternatively, run tex or latex on xint.ins if available. % % To get xint.pdf run pdflatex thrice on xint.dtx % -% xint.sty, xintgcd.sty -> TDS:tex/generic/xint/ -% xint.dtx -> TDS:source/generic/xint/ -% xint.pdf -> TDS:doc/generic/xint/ +% xint.sty, xintgcd.sty, xintfrac.sty, xintseries.sty +% -> TDS:tex/generic/xint/ +% xint.dtx -> TDS:source/generic/xint/ +% xint.pdf -> TDS:doc/generic/xint/ % -% It may well be necessary to then refresh the TeX installation -% filename database. +% It may be necessary to then refresh the TeX installation filename +% database. % % Usage with LaTeX: \usepackage{xint} -% \usepackage{xintgcd} +% \usepackage{xintgcd} % (loads xint) +% \usepackage{xintfrac} % (loads xint) +% \usepackage{xintseries} % (loads xintfrac) % -% Usage with TeX: \input xint.sty\relax -% \input xintgcd.sty\relax +% Usage with TeX: \input xint.sty\relax +% \input xintgcd.sty\relax % (loads xint) +% \input xintfrac.sty\relax % (loads xint) +% \input xintseries.sty\relax % (loads xintfrac) % %<*none> -\def\pkgversion{1.02} -\def\pkgdate{2013/04/05} -\def\lasttimestamp{Time-stamp: <05-04-2013 16:49:16 CEST JF>} +\def\pkgversion{1.03} +\def\pkgdate{2013/04/14} +\def\lasttimestamp{Time-stamp: <15-04-2013 10:58:29 CEST jfb>} \def\striptimestamp #1 <#2 #3 #4 #5>{#2 at #3 #4} \def\getdocdate #1 <#2-#3-#4 #5>{#4/#3/#2} \edef\docdate{\expandafter\getdocdate\lasttimestamp} @@ -72,7 +81,9 @@ \file{xint.ins}{\from{xint.dtx}{ins}} \usepreamble\defaultpreamble \file{xint.sty}{\from{xint.dtx}{xint}} -\file{xintgcd.sty}{\from{xint.dtx}{xintgcd}}} +\file{xintgcd.sty}{\from{xint.dtx}{xintgcd}} +\file{xintfrac.sty}{\from{xint.dtx}{xintfrac}} +\file{xintseries.sty}{\from{xint.dtx}{xintseries}}} \endgroup \iffalse %</none> @@ -80,7 +91,8 @@ %----------- to .ins file ---------------------------------------- %% %% This is a generated file. Run tex or latex on this file to -%% extract xint.sty and xintgcd.sty from xint.dtx +%% extract xint.sty, xintgcd.sty, xintfrac.sty and xintseries.sty +%% from xint.dtx %% %% See xint.dtx for the statements of copyright and conditions of %% distribution and/or modification of this work. @@ -89,26 +101,63 @@ \askforoverwritefalse \generate{\usepreamble\defaultpreamble \file{xint.sty}{\from{xint.dtx}{xint}} -\file{xintgcd.sty}{\from{xint.dtx}{xintgcd}}} +\file{xintgcd.sty}{\from{xint.dtx}{xintgcd}} +\file{xintfrac.sty}{\from{xint.dtx}{xintfrac}} +\file{xintseries.sty}{\from{xint.dtx}{xintseries}}} \endbatchfile %----------- end of .ins file ------------------------------------ %</ins> %<*none> \fi \NeedsTeXFormat{LaTeX2e} -\ProvidesFile{xint.dtx} - [bundle source and documentation (\dtxtimestamp)] +\ProvidesFile{xint.dtx}[bundle source and documentation (\dtxtimestamp)] + \documentclass[a4paper,11pt,abstract]{scrdoc} \pagestyle{headings} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} + +%---- GEOMETRY WILL BE CHANGED FOR SOURCE CODE SECTIONS + \usepackage[hscale=0.66,vscale=0.75]{geometry} -\usepackage{xint} +%---- WE NEED OUR BEAUTIFUL SELVES + +%% checking that dependencies are all-right +%\usepackage{xint} +%\usepackage{xintgcd} +%\usepackage{xintfrac} + +\usepackage{xintseries} \usepackage{xintgcd} -\usepackage{txfonts} +%---- CHANGING TOCDEPTH MIDWAY THROUGH THE MAIN TOC + +\usepackage{etoc} +\makeatletter + +\def\toctransition {% + \addtocontents {toc}{\protect\newtocdepth {1}}% + % \setcounter{tocdepth}{1}% à cause des bookmarks de hyperref, + % \def\etocaftertitlehook {\c@tocdepth 2 }% pour les local tocs + % non finalement puisque je laisse tocdepth à 2 globalement plus besoin + \let\newtocdepth\@gobble % mais ne pas oublier ça + \etocmulticolstyle [1]{\subsection *{Contents}}% + \def\@pnumwidth{2em}% attention ce n'est pas une longueur. + % fait pour problème de overfull box au niveau des numéros de + % page dans les local tocs des sections implémentations +} +% qu'est-ce qu'il faut pas faire ! +% (à cause de la gestion des bookmarks par hyperref) +% je ne veux pas le sous-sections de la partie implémentation dans les +% bookmarks, ou peut-être si en fait je les veux aussi? ce n'est pas gênant +% dans les bookmarks. +\def\newtocdepth #1{\c@tocdepth #1 } % ainsi on modifie localement seulement +\makeatother + +%--- TXFONTS, AND TXTT MADE SMALLER AND ALLOWING HYPHENATION +\usepackage{txfonts} % malheureusement, comme j'utilise des diacritiques dans mes % parties commentées, imprimées verbatim, je ne pourrai pas % utiliser dvipdfmx qui a un problème avec txtt @@ -184,6 +233,8 @@ pdfkeywords={Expansion, arithmetic, TeX},% pdfstartview=FitH,% pdfpagemode=UseOutlines} + +%---- OUR CLEVER PRIVATE LITTLE MACRO FOR CENTERED LINES \makeatletter % 7 mars 2013 % This macro allows to conveniently center a line inside a paragraph and still @@ -200,32 +251,41 @@ pdfpagemode=UseOutlines} \let\next=} \def\@centeredline {\hbox to \linewidth \bgroup \hss \bgroup \aftergroup\centeredline@ } - \makeatother +%---- ALLOWING COMMENTS INSIDE VERBATIM BLOCKS \makeatletter \let\original@check@percent\check@percent \let\check@percent\relax +\makeatother + +%---- A MORE FLEXIBLE \verb +\makeatletter % le \verb de doc.sty est très chiant car il a retiré % \verbatim@font pour mettre un \ttfamily hard-coded % à la place. % % Par ailleurs j'en ai marre des erreurs dues au fait que mes % paragraphes reformatés dans emacs passent à la ligne au milieu -% d'un \verb. Je décide donc d'annuler l'effet du \dospecials sur -% les espaces dans la source . Et donc je retire le -% \verb@eol@error et il n'y a donc plus lieu d'un comportement -% différent pour l'impression des blancs, donné par la version étoilée. +% d'un \verb. Je décide donc d'annuler l'effet du \dospecials sur les +% espaces dans la source. Et donc je retire le \verb@eol@error et +% il n'y a donc plus lieu d'un comportement différent pour +% l'impression des blancs, donné par la version étoilée. % % Et il n'y avait donc pas de \obeylines puisque la fin de ligne % devenait un message d'erreur dans \verb@eol@error % % De plus je retire le \do@noligs qui me gêne plutôt qu'autre chose, -% surtout maintenant que les espaces ne sont pas \ +% surtout maintenant que les espaces ne sont pas des control spaces +% attention au signe - par contre % -\def\verb {\relax \ifmmode \hbox \else \leavevmode \null \fi +\def\noligminus {\kern \z@ \char`\-} +\begingroup\catcode`\-\active +\gdef\verb {\relax \ifmmode \hbox \else \leavevmode \null \fi \bgroup \verbatim@font - \let \do \@makeother \dospecials \catcode32 10 \@ifstar {\@sverb }{\@sverb }} + \let \do \@makeother \dospecials \catcode`\-\active + \let-\noligminus \catcode32 10 \@ifstar {\@sverb }{\@sverb }} +\endgroup % ça c'est pour mes petits morceaux de code: \def\verbatim@font {\ttfamily } \def\MacroFont{\ttfamily\baselineskip12pt\relax} @@ -239,11 +299,16 @@ pdfpagemode=UseOutlines} \makeatother % Note: il n'y a plus de \hyphenchar-1 dans le \DeclareFontFamily de t1txtt +% ATTENTION CEPENDANT À CE QUI SE PASSE EN CAS DE CHANGEMENT DE TAILLE \DeclareRobustCommand\csa[1]{{\ttfamily\char`\\#1}} -\DeclareRobustCommand\csb[1]{{\color{blue}\ttfamily\char`\\#1}} -\newcommand\ch[1]{\texorpdfstring{\csa{#1}}{\textbackslash #1}} -\newcommand\chb[1]{\texorpdfstring{\csb{#1}}{\textbackslash #1}} + +\DeclareRobustCommand\csb[1]{\hyperref[#1]{\color{blue}\ttfamily\char`\\#1}} + +\DeclareRobustCommand\csbnolk[1]{{\color{blue}\ttfamily\char`\\#1}} + +\newcommand\csh[1]{\texorpdfstring{\csa{#1}}{\textbackslash #1}} +\newcommand\csbh[1]{\texorpdfstring{\csbnolk{#1}}{\textbackslash #1}} \newcommand\xintname {\texorpdfstring {{\color{joli}\ttfamily\bfseries xint}} @@ -252,15 +317,33 @@ pdfpagemode=UseOutlines} \newcommand\xintgcdname{\texorpdfstring {{\color{joli}\ttfamily\bfseries xintgcd}} {xintgcd}\xspace} + +\newcommand\xintfracname{\texorpdfstring + {{\color{joli}\ttfamily\bfseries xintfrac}} + {xintfrac}\xspace} +\newcommand\xintseriesname{\texorpdfstring + {{\color{joli}\ttfamily\bfseries xintseries}} + {xintseries}\xspace} \frenchspacing \renewcommand\familydefault\sfdefault +%---- WE WANT TO SEE ALL THOSE NUMBERS +\def\allownumbersplit #1% +{% + \ifx #1\relax \else #1\hskip 0pt plus 1pt\relax + \expandafter\allownumbersplit\fi +}% +\def\printnumber #1% +{\expandafter\expandafter\expandafter + \allownumbersplit #1\relax }% Expands twice before printing. + \begin{document} \thispagestyle{empty} \rmfamily \begin{center} - {\normalfont\Large The \xintname bundle: \xintname and \xintgcdname}\\ + {\normalfont\Large The \xintname bundle: \xintname, \xintgcdname, + \xintfracname, and \xintseriesname.\par}% \textsc{Jean-François Burnol}\par \footnotesize \ttfamily jfbu (at) free (dot) fr\\ @@ -277,9 +360,16 @@ pdfpagemode=UseOutlines} sign. The \xintgcdname package provides implementations of the - Euclidean algorithm and of its typesetting. - - The packages may be used with Plain and with \LaTeX. + Euclidean algorithm and of its typesetting. The \xintfracname + package extends the scope of \xintname to fractional numbers of + arbitrary sizes ; \xintseriesname provides some basic + functionality based on the \xintname and \xintfracname packages + for computing in an expandable manner partial sums of series and + power series with fractional coefficients. + + The packages may be used with Plain and with \LaTeX. All macros + dealing with computations work purely by expansion, and may thus + be used almost everywhere in \TeX{}. \end{abstract} @@ -295,14 +385,14 @@ pdfpagemode=UseOutlines} \section{Origins of this package} The package |bigintcalc| by \textsc{Heiko Oberdiek} already -provides expandable arithmetic operations on ``big numbers'', +provides expandable arithmetic operations on ``big integers'', exceeding the \TeX{} limits (of &2^{31}-1&), so why another 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 +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 @@ -312,19 +402,18 @@ 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} macro, they worked -one digit at a time, having previously stored digit arithmetic in -(many) macros. +one digit at a time, having previously stored carry-arithmetic in +1200 macros. I noticed that the |bigintcalc| package used the \csa{numexpr} \eTeX{} primitive when 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 1200 macros -storing pre-computed arithmetic with 2 or 3 digits. 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). +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. @@ -333,12 +422,14 @@ The present package is the result of this initial questioning. \end{framed} I have aimed at speed wherever I could, and to the extent that I -could guess what was more efficient for \TeX{}. After a while -though I did opt for more readable coding style in those parts of -the code which were not at the heart of repeatedly used loops. In -particular I started using \csa{ifnum} and \csa{ifcase} constructs -which I had completely avoided so far, working only with macro -expansions. +could guess what was more efficient for \TeX{}. + +% After a while +% though I did opt for more readable coding style in those parts of +% the code which were not at the heart of repeatedly used loops. In +% particular I started using \csa{ifnum} and \csa{ifcase} constructs +% which I had completely avoided so far, working only with macro +% expansions. I wrote a version of addition which does \csa{numexpr} operations eight digits at a time, but its additional overhead made it a bit slower @@ -348,27 +439,69 @@ taking a noticeable time, so I have chosen to retain the addition routine which was most efficient for numbers having a few dozens to a few hundreds digits. -This implementation is thus a \TeX nical thing, quite different -from what one would do in a structured programming language like -|C|, although the underlying algorithms are just the standard -steps applied to hand computations (nothing fancy like -Fast Fourier Transform...). +% This implementation is thus a \TeX nical thing, quite different +% from what one would do in a structured programming language like +% |C|, although the underlying algorithms are just the standard +% steps applied to hand computations (nothing fancy like +% Fast Fourier Transform...). + +By the way, I used the word `speed', and yes \xintname enjoys +working `fast and efficiently' (within many quotes...) with 200 +digits numbers, but surely any program in |C| using the |CPU| and +pointers to the memory for arithmetic operations on arrays of +numbers would do computations thousands of times faster (or more, +I don't know) than what \TeX{} can achieve when manipulating strings of +ASCII representations of digits via a game on up to nine +parameters per macro! And, besides, the underlying +algorithms used by \xintname are just the standard hand +computation methods, nothing fancy like Fast Fourier Transform. + + +\begin{framed} + Even within \TeX{} it is possible to set up arithmetic + operations working orders of magnitude faster than what + \xintname achieves,\footnotemark[2]\ but this does (I guess) + require the capacity to do assignments to memory storage. The + arithmetic implemented by \xintname does not do any assignment + and works by pure macro expansion: this has a toll on + speed.\footnotemark[3]\ Nevertheless, numbers with less than + thirty digits are quite ``small'' from the point of view of the + package, and a great many operations on such numbers can be done + in a document without real noticeable impact on the compilation + time. +\end{framed} + +To see \xintname in action on the traditional computations of +$\pi$ and $\log 2$, jump to the +\hyperref[xintFxPtPowerSeries]{\color{blue}{\csa{xintFxPtPowerSeries} + documentation}}. + +\footnotetext[2]{this is well demonstrated by the + \href{http://www.ctan.org/pkg/pi}{\color{niceone}pi computing file} by + \textsc{D. Roegel} from 1996. As will be seen at the end of this + manual, the \textsc{Machin formula} to compute $\pi$ can also be + implemented (in a completely expandable way) with the help of + \xintfracname and \xintseriesname: on my laptop it computes 200 digits + in less than one second, but this is much slower than what pi.tex + achieves.} + +\footnotetext[3]{not to mention the impact on coding style; after + having now completed the main work on these packages, the author + feels he cannot do anything in \TeX{} but expansion-only compatible + macros... } + +\setcounter{footnote}{3} -By the way, yes \xintname enjoys working fast and efficiently with -200 digits numbers, but surely any program (even poorly written) -in |C| using the |CPU| for arithmetic operations on arrays of -numbers (not digits!!!) will work thousands of times faster (or -more, I don't know) than what can be achieved using \TeX{} to -manipulate strings of ASCII representations of digits! \section{Expansions} -Except otherwise stated all macros are completely expandable. For -example, with the following code snippet within |myfile.tex| +Except for the assignments or typesetting macros, the bundle +macros are constructed to work in expansion-only context. For +example, with the following code snippet within |myfile.tex|: \begin{verbatim} \newwrite\outfile \immediate\openout\outfile \jobname-out\relax -\immediate\write\outfile {\xintQuo{\xintPow{2}{1000}}{\xintFac{100}}} +\immediate\write\outfile {\xintQuo{\xintiPow{2}{1000}}{\xintFac{100}}} % \immediate\closeout\outfile \end{verbatim} the tex run creates a file |myfile-out.tex| @@ -376,36 +509,38 @@ containing the decimal representation of the integer quotient &2^{1000}/100!&. Such macros can also be used inside a |\csname...\endcsname|, and of course in an |\edef|. -\edef\x{\xintQuo{\xintPow {2}{1000}}{\xintFac{100}}} +\edef\x{\xintQuo{\xintiPow {2}{1000}}{\xintFac{100}}} \edef\y{\xintLen{\x}} -\def\allownumbersplit #1% - {\ifx #1\relax \else #1\hskip 0pt plus 1pt - \expandafter\allownumbersplit\fi}% Furthermore the package macros give their final results in two expansion steps. They twice expand their arguments so that they can be arbitrarily chained. Hence \centeredline{% - |\xintLen{\xintQuo{\xintPow{2}{1000}}{\xintFac{100}}}|} expands + |\xintLen{\xintQuo{\xintiPow{2}{1000}}{\xintFac{100}}}|} expands in two steps and tells us that &[2^{1000}/100!]& has {\y} digits. This is not so many, let us print them here: -{\expandafter\expandafter\expandafter\allownumbersplit - \xintQuo{\xintPow {2}{1000}}{\xintFac{100}}\relax}. 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 this -little macro (not provided by the package): -\begin{verbatim} -\def\allownumbersplit #1{\ifx #1\relax \else #1\hskip 0pt plus 1pt - \expandafter\allownumbersplit\fi}% -\end{verbatim} -To provoke the double expansion first, it is used as in: +\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): \begin{verbatim} -\expandafter\expandafter\expandafter\allownumbersplit - \xintQuo{\xintPow {2}{1000}}{\xintFac{100}}\relax +\def\allownumbersplit #1% +{% + \ifx #1\relax \else #1\hskip 0pt plus 1pt\relax + \expandafter\allownumbersplit\fi +}% +\def\printnumber #1% +{\expandafter\expandafter\expandafter\allownumbersplit #1\relax }% +% Expands twice before printing. \end{verbatim} -Or, the computation can be done inside an \csa{edef} and then only one -\csa{expandafter} will be enough before |\allownumbersplit|. - -Remarks on the double expansion of arguments: +which is used for example as |\printnumber {\xintQuo{\xintiPow +{2}{1000}}{\xintFac{100}}}|. Or, the computation can be done inside +an \csa{edef}: |\edef\mynumber {\|\texttt{xint\-Quo}|{\xintiPow +{2}{1000}}{\xintFac{100}}}| followed by |\printnumber\mynumber|. The macro +is not part of the package and would need additional thinking for more +general use. + +Important points, to be noted, related to the double expansion of arguments: \begin{enumerate} \item When I say that the macros expand twice their arguments, this means that they expand the first token seen (for each @@ -423,20 +558,21 @@ Remarks on the double expansion of arguments: \item Unfortunately, after |\def\x {12}|, one can not use just {\color{blue}|-\x|} as input to one of the package macros: the rules above explain that the twice expansion will act only on the minus sign, - hence do nothing. The only way is to use the \csb{xintOpp} - macro, which replaces a number with its opposite. Example: |\xintAdd - {\xintOpp\x}{\x}|\,=\,{\xintAdd {\xintOpp\x}{\x}}. + hence do nothing. The only way is to use the \csb{xintiOpp} + macro, which replaces a number with its opposite. Example: |\xintiAdd + {\xint|\-|iOpp\x}{\x}|\,=\,{\xintiAdd {\xintiOpp\x}{\x}}. \def\x {12}% \item With the definition \centeredline{% - |\def\AplusBC #1#2#3{\xintAdd {#1}{\xintMul {#2}{#3}}}|} one + |\def\AplusBC #1#2#3{\xintiAdd {#1}{\xintiMul {#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 a result {|\xintAdd + by the macro expanding to its definition. As a result {|\xintiAdd {\AplusBC {1}{2}{3}}{4}|} would then miserably fail. The solution is to use the \emph{lowercase} form of - \csa{xintAdd}: \centeredline {|\def\AplusBC - #1#2#3{|{\color{blue}|\romannumeral0\xintadd |}|{#1}{\xintMul {#2}{#3}}}|}% + \csa{xintiAdd}: \smallskip\centeredline {|\def\AplusBC + #1#2#3{|{\color{blue}|\romannumeral0\xintiadd |}|{#1}{\xintiMul {#2}{#3}}}|} + and then \csa{AplusBC} will share the same properties as do the other \xintname `primitive' macros. % ENFIN DÉBARRASSÉ DES TRÈS TRÈS TRÈS CHIANTS EOL ERROR DE \verb !!! @@ -447,64 +583,75 @@ Remarks on the double expansion of arguments: precisely to facilitate building-up higher level macros based on them. \end{enumerate} -\section {Inputs} +\section {Inputs (integers)} + +\begin{framed} + \TeX{}'s count registers cannot be directly used but must be + prefixed by |\the| or |\number|. The same for \csa{numexpr} + expressions. +\end{framed} -After a twice expansion of the arguments, the ensuing numbers have -to be strings of digits with one (and not more) optional minus -sign (and not a plus sign). The first digit is not -zero if there are more than one digit. And |-0| is not legal -input. Syntax such as -|\xintMul\A\B| is accepted and equivalent to |\xintMul {\A}{\B}|. -Or course |\xintAdd\xintMul\A\B\C| does not work, the product -operation must be put within braces: +Each one of the package macros first does a double expansion of its +arguments, and it expects the ensuing numbers to be strings of digits +with one (and not more) optional minus sign (and not a plus +sign).\footnote{these conditions are relaxed (\emph{only} for the + extended macros) when \xintfracname is loaded: the number, even zero, + may start with many minus signs; but plus signs are still forbidden.} +The first digit is not zero if there are more than one digit. And |-0| +is not legal input. Syntax such as |\xintMul\A\B| is accepted and +equivalent\footnote{see however near the end of + \hyperref[sec:outputs]{\color{niceone}this later section} for the + important difference when used in contexts where \TeX{} expects a + number, such as following an \csa{ifcase} or an \csa{ifnum}.} to +|\xintMul {\A}{\B}|. Or course |\xintAdd\xintMul\A\B\C| does not work, +the product operation must be put within braces: |\xintAdd{\xintMul\A\B}\C|. It would be nice to have a functional form |\add(x,\mul(y,z))| but this is not provided by the package. Arguments must be either within braces or a single control sequence. -For the division (but not for addition, subtraction, or -multiplication), the two inputs must have at most -&2^{31}-9=&{\xintSub{\xintPow {2}{31}}{9}} digits. - -I guess anyhow that this is way way way beyond what is possible in terms -of memory in any implementation of \TeX{}. But if the situation did -arise nevertheless of such a gigantic input, an arithmetic overflow -would occur (after some long time I guess) as \xintname first computes -the lengths of the inputs by using \csa{numexpr} with successive -additions of the number |8| to itself until the whole input has been -parsed \footnote{It is the macro \csa{xintLen} (used by - \csa{xintDivision}) which will trigger an arithmetic overflow if it is - called with an input of more than {\xintSub{\xintPow {2}{31}}{9}} - digits. I thought it was not worthwile adding to the code of - \csa{xintLen} a safeguard against the arithmetic overflow in a - \csa{numexpr}: this check would have some general impact on speed, - whereas the situation can not realistically occur (or even not at all, - I admit not having double-checked the intrinsinc \TeX{} memory - limitations).} (this initial step is only for the division algorithm, -the three other arithmetic operations remain unaware of the sizes of -their inputs, although they do experience them in a sense, as they -initially reverse the order of digits of at least one of the inputs, -which means they have to scan it entirely). +For the multiplication and the division (but not for addition and +subtraction), the inputs must have each at most +&2^{31}-9=&{\xintiSub{\xintiPow {2}{31}}{9}} digits.\footnote{when + \xintfracname is loaded, this restriction on the length of the + numbers becomes a general one.} + +I guess anyhow that this is way way way beyond what is possible in +terms of memory in any implementation of \TeX{}. But if the +situation did arise nevertheless of such a gigantic input, an +arithmetic overflow would occur (after some long time I guess) +during the computation by \xintname of the lengths of the inputs, +as this computation uses \csa{numexpr} for successive additions of +the number |8| to itself until the whole input has been parsed. +\footnote{it is the macro \csa{xintLen} (used by the + multiplication and the division algorithms) which will trigger + an arithmetic overflow if it is called with an input of more + than {\xintiSub{\xintiPow {2}{31}}{9}} digits. I decided it wasn't + worth it to add to the code of \csa{xintLen} a safeguard against + this potential arithmetic overflow: it would have some general + impact on speed, whereas the situation can not realistically + occur (or even not at all, I admit not having double-checked the + intrinsinc \TeX{} memory limitations).} Also: the factorial function \csa{xintFac} will refuse to (start...) compute |N!| if |N| $\geq$ 1000000000, and the power function -|\xintPow {A}{B}|, when the absolute value \verb+|A|+ is at +|\xintiPow {A}{B}|, when the absolute value \verb+|A|+ is at least two, will refuse to start the computation if |B| $\geq$ 1000000000 (the minimal outcome is &2^{1000000000}& which has 301029996 digits...). -In those latter cases, no arithmetic overflow will happen, but rather, -copied from package |bigintcalc|, undefined control sequences with -names indicating the source of the problem are inserted in the -token stream and will appear in the log file in \TeX{} `undefined -macro' error messages. This will not stop the -computation, which (most of the time) will output a zero. +In those latter cases, no arithmetic overflow will happen, but the +compilation log will report an ``undefined control sequence +error'', where the name of the control sequence indicates the +source of the error (this method is copied from package +|bigintcalc|). Errors of this type do not stop the computation, +which (generally) will output a zero. No check is done on the format of the inputs after the initial twice expansion. Often, but not always, something starting with a -|0| will be assumed to be zero (throwing the rest away, or -sometimes not which then will lead to errors). Plus signs are not -accepted and will cause errors. +|0| will be assumed to be zero (throwing or not what comes after +the zero away). Plus signs are not accepted and will cause errors. +Spaces should be avoided. The sole exception is the macro \csb{xintNum} which accepts numbers starting with an arbitrary long sequence of plus signs, minus signs, @@ -517,16 +664,87 @@ within the initial signs. An empty string is also acceptable input: \csa{xintNum} expands twice its argument, and obtains its final result in two expansion steps. -\begin{framed} - \TeX{}'s count registers cannot be directly used but must be - prefixed by |\the| or |\number|. The same for \csa{numexpr} - expressions. -\end{framed} - -\section{Outputs} - -The output, when it consists of a single number, is always in the -normalized form described in the previous section. Some macros +\section{Inputs (fractions)} + +When package \xintfracname is loaded, there is a wider range of +input formats to most macros (some, such as \csb{xintQuo} which +computes the quotient in an euclidean division, remain +``integer-only'', and the previous section applies). + +\edef\z {\xintAdd + {367.8920280/---278.289287}{-109.2882/270.12898}} + +Here is a typical computation: \centeredline{|\xintAdd + {367.8920280/---278.289287}{-109.2882/270.12898}|}% + \centeredline{\texttt{=\z}}% + \centeredline{\texttt{=\xintIrr\z{} (irreducible)}}% + \centeredline{\texttt{=\xintTrunc {50}{\z}\dots}} Signs (on + input) may thus be either at the numerator or denominator, or at + both (chains of |-| are ok, but still no |+| sign). +An optional decimal point is + authorized, both in the numerator and the denominator. It is + also licit to use |\A/\B| as input if each of |\A| and |\B| + expands in at most two steps to a ``decimal number'' as + examplified above by the numerators and denominators. 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. + + Lastly, input such as |16000/289072[17]| (or |3[-4]|) is + accepted and represents, respectively |(16000/289072)10^{17}| + and |3|\raisebox{.5ex}{|.|}|10^{-4}|. It is 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\strut{} may then have a decimal + point.\vadjust{\vskip-\dp\strutbox + \hbox{\smash{\color{niceone}\llap{\strut\small + IMPORTANT!\ $\Bigg\{$\ }}}\vskip\dp\strutbox } + And, for this format, + ONLY the numerator may carry a UNIQUE minus sign (and no + superfluous zeros). + + This format with a power of ten represented by a number within + square brackets is used by the \xintfracname macros for output. + It is allowed for user input but the parsing is minimal and it + is very important to follow the above rules. This reduced + flexibility, compared to the format without the square brackets, + allows chaining package macros without too much speed impact, as + they always output computation results in the |A/B[n]| form (or + |A[n]|). \footnote{see however the \csb{xintFrac} and + \csb{xintFwOver} macros for print only, inside math mode.} + +All computations done by \xintfracname on fractions are exact. +Even when the inputs contains decimal points, it does not make +the package switch to a (currently non-existent) `floating-point' +mode: the inputs are converted into an exact internal representation. + + +Generally speaking, there should be no spaces in the inputs +(although most would be harmless, most of the time; the devil +being in the details, it is best to just not take chances with +these spaces). + +\edef\z {\xintSub {\xintMul {2.3}{\xintPow {5.6}{3}}} {17728/189.5}} + +It would certainly be nice to be able to input directly expressions such as +|2.3*5.6^3-17728/189.5|, but this is not possible. One must use, for +example: + \centeredline{|\xintSub {\xintMul {2.3}{\xintPow + {5.6}{3}}} {17728/189.5}|} or, an option in this case is: +\centeredline{|\xintAdd {\xintPrd {{2.3}{5.6}{5.6}{5.6}}}{-17728/189.5}|}% +\centeredline{\texttt{=\z =\xintIrr\z =\xintTrunc {15}\z\dots}} + + +\section{Outputs (integers)}\label{sec:outputs} + +The output of macros of the \xintname package, when it consists of +a single integer number, is in the normalized form previously +described.\footnote{see the next section for the modifications + brought by loading the \xintfracname package.} + +Some macros have an output consisting of more than one number, each one is then within braces. For example \csb{xintDivision} gives first the quotient and then the remainder, each of them within braces. This @@ -538,8 +756,7 @@ direct access. \def\n{\string{N\string}} \def\x{\string{x\string}} -The macro \csb{xintDecSplit}\x\n\footnote{Its behavior has been - modified in bundle version |1.01|, check its documentation.} cuts its second +The macro \csb{xintDecSplit}\x\n\ cuts its second argument |N| at a location specified by its first argument |x|, and returns the two pieces one after the other, each within braces. Depending on the value of |x| and the length of |N|, the first, or the second, output of @@ -565,22 +782,121 @@ With |\def\A{1}|: % \ifcase \xintSgn\A 0\or OK\else ERROR\fi\ % \ifcase \xintSgn{\A} 0\or OK\else ERROR\fi +\section{Outputs (fractions)} + +When the package \xintfracname is loaded, the routines +\csb{xintAdd}, \csb{xintSub}, \csb{xintMul}, \csb{xintPow}, +\csb{xintSum}, \csb{xintPrd} are modified to allow fractions on +input,\footnote{of course, the power function does not accept a + fractional exponent. Or rather, does not expect, and errors will + result if one is provided.}\,\footnote{macros \csb{xintiAdd}, + \csb{xintiSub}, \csb{xintiMul}, \csb{xintiPow}, \csb{xintiSum}, + \csb{xintiPrd} are the original ones dealing only with integers. + They are available as synonyms, also when \xintfracname is not + loaded. }\,\footnote{also \csb{xintCmp}, \csb{xintSgn}, + \csb{xintOpp}, \csb{xintAbs}, \csb{xintMax}, \csb{xintMin} are + extended to fractions and have their integer-only initial + synonyms.} and always produce on output a fractional number +|f=A/B[n]| where |A| and |B| are integers, with |B| positive, and +|n| is a signed ``small'' 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 zeros (\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).\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|.} + +As the present document loads the \xintfracname package, most +examples with integers will use the \csb{xintiAdd}, +\csb{xintiSub}, \csb{xintiMul}, \csb{xintiPow}, \csb{xintiSum}, +\csb{xintiPrd}, macros which are the original un-modified +integer-only versions. This is mandatory in particular when using +their ouput as input to integer-only macros such as \csb{xintQuo}. + + +The macro \csb{xintREZ} (remove zeros) puts all powers of ten into +the |[n]|, and removes the |B| if it is less than |1|. The macro +\csb{xintIrr} transforms |f| into its unique irreducible +representative |C/D|, and prints |C| if |D=1|. + +The macro \csb{xintTrunc}|{N}{f}| prints\footnote{`prints' does + not mean that this macro is only for outputting; to the contrary + it is recommended to use it in intermediate results when doing + things such as computing $\sum_{n=1}^{1000} \frac1n$, else the + numbers manipulated by \xintname will be as big as $1000!$. + Besides the package does not provide any `printing' facility; + such facilities are necessary as \TeX{} by default will print a + long number on a single line extending beyond the page limits. + The \csa{printnumber} macro used in this documentation is just + one way to deal with this problem (some other method should be + used to guarantee 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|. Of course a negative |f| is no problem, +needless to say. When the original fraction is negative and its +truncation has only zeros, it is printed as |-0.0...0|, with |N| +zeros following the decimal point: +\centeredline{|\xintTrunc {5}{\xintPow {-13}{-9}}=|\texttt{\xintTrunc {5}{\xintPow {-13}{-9}}}}% +\centeredline{|\xintTrunc {20}{\xintPow {-13}{-9}}=|\texttt{\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}}}=|% +\texttt{\xintTrunc {10}{\xintSum {{1/2}{1/3}{1/5}{-31/30}}}}} + +The output of \csb{xintTrunc} may of course serve as input to the other +macros. And this is almost necessary when summing hundreds of +terms of a series with fractional coefficients, as the exact +rational number quickly becomes quite big (when doing the sum from +|n=|1 to |n=|1000 of |1/n|, the raw denominator is &1000!&, which +has 2568 digits) ; but for less than fifty terms with small +denominators it is often possible to work with the exact +value without too much toll on the compilation time. + +The macro \csb{xintiTrunc}|{N}{f}| is like \csa{xintTrunc}|{N}{f}| +followed by multiplication by |10^N|. Thus, it ouputs an integer +in a format acceptable by the integer-only macros. This is also +convenient when computing partial sums of series: it is a bit +faster to sum with \csb{xintiSeries} the integers produced by +\csa{xintiTrunc}|{N}| than it is to use the general +\csb{xintSeries} on the decimal numbers produced by +\csa{xintTrunc}|{N}|. These latter macros belong to the \xintseriesname +package. + +Needless to say when using \csa{xintTrunc} or \csa{xintiTrunc} on +intermediate computations the ending digits of the final result +are, pending further analysis, only indications of those of the +fraction an exact computation would have produced. + +\edef\z {\xintPow {1.01}{100}} + +To get the integer part of the decimal expansion of |f|, use +|\xintiTrunc{0}{f}|: \centeredline{|\xintiTrunc {0}{\xintPow + {1.01}{100}}=|\texttt{\xintiTrunc {0}\z}}% +\centeredline{|\xintTrunc {10}{\xintPow + {1.01}{100}}=|\texttt{\xintTrunc {10}\z}} \section{Assignments} \xintAssign\xintBezout{357}{323}\to\tmpA\tmpB\tmpU\tmpV\tmpD -You might not need to maintain at all times complete +It might not be necessary to maintain at all times complete expandability. For example why not allow oneself the two definitions |\edef\A {\xintQuo{100}{3}}| and |\edef\B {\xintRem {100}{3}}|. A special syntax is provided to make these things more efficient, as the package provides \csa{xintDivision} which computes both quotient and remainder at the same time: - \centeredline{\csb{xintAssign}\csa{xintDivision}|{100}{3}|\csb{to}|\A\B|} + \centeredline{\csb{xintAssign}\csa{xintDivision}|{100}{3}|\csbnolk{to}|\A\B|} \centeredline{\csb{xintAssign}\csa{xintDivision}% -|{\xintPow {2}{1000}}{\xintFac{100}}|\csb{to}|\A\B|} gives -\xintAssign\xintDivision{\xintPow {2}{1000}}{\xintFac{100}}\to\A\B +|{\xintiPow {2}{1000}}{\xintFac{100}}|\csbnolk{to}|\A\B|} gives +\xintAssign\xintDivision{\xintiPow {2}{1000}}{\xintFac{100}}\to\A\B |\meaning\A|\texttt{: \expandafter\allownumbersplit\meaning\A\relax} and |\meaning\B|\texttt{: \expandafter\allownumbersplit\meaning\B\relax}. @@ -588,22 +904,22 @@ expandability. For example why not allow oneself the two definitions Another example (which uses a macro from the \xintgcdname package): \centeredline{\csb{xintAssign}\csa{xintBezout}|{357}{323}|% - \csb{to}|\A\B\U\V\D|} is equivalent to setting |\A| to + \csbnolk{to}|\A\B\U\V\D|} is equivalent to setting |\A| to \texttt{\tmpA}, |\B| to \texttt{\tmpB}, |\U| to \texttt{\tmpU}, |\V| to \texttt{\tmpV}, and |\D| to \texttt{\tmpD}. And indeed (\tmpU)$\times$\tmpA-(\tmpV)$\times$\tmpB= - \xintSub{\xintMul\tmpU\tmpA}{\xintMul\tmpV\tmpB} + \xintiSub{\xintiMul\tmpU\tmpA}{\xintiMul\tmpV\tmpB} is a Bezout Identity. \xintAssign\xintBezout{3570902836026}{200467139463}\to\tmpA\tmpB\tmpU\tmpV\tmpD \centeredline{\csb{xintAssign}\csa{xintBezout}|{3570902836026}{200467139463}|% - \csb{to}|\A\B\U\V\D|} gives then |\U|\texttt{: + \csbnolk{to}|\A\B\U\V\D|} gives then |\U|\texttt{: \expandafter\allownumbersplit\meaning\tmpU\relax}, |\V|\texttt{: \expandafter\allownumbersplit\meaning\tmpV\relax} and |\D=|\texttt{\tmpD}. When one does not know in advance the number of tokens, one can use \csa{xintAssignArray} or its synonym \csa{xintDigitsOf}: - \centeredline{\csb{xintDigitsOf}\csa{xintPow}|{2}{100}|\csb{to}\csa{Out}} + \centeredline{\csb{xintDigitsOf}\csa{xintiPow}|{2}{100}|\csbnolk{to}\csa{Out}} This defines \csa{Out} to be macro with one parameter, \csa{Out}|{0}| gives the size |N| of the array and \csa{Out}|{n}|, for |n| from |1| to |N| then gives the |n|th @@ -615,16 +931,16 @@ expandability. For example why not allow oneself the two definitions \newcount\cnta \newcount\cntb \begingroup -\xintDigitsOf\xintPow{2}{100}\to\Out +\xintDigitsOf\xintiPow{2}{100}\to\Out \cnta = 1 \cntb = 0 \loop -\advance \cntb \xintSqr{\Out{\the\cnta}} +\advance \cntb \xintiSqr{\Out{\the\cnta}} \ifnum \cnta < \Out{0} \advance\cnta 1 \repeat -|2^{100}| (=\xintPow {2}{100}) has \Out{0} digits and the sum of +|2^{100}| (=\xintiPow {2}{100}) has \Out{0} digits and the sum of their squares is \the\cntb. These digits are, from the least to the most significant: \cnta = \Out{0} \loop \Out{\the\cnta}\ifnum \cnta > 1 \advance\cnta -1 , \repeat. @@ -634,16 +950,16 @@ the most significant: \cnta = \Out{0} \newcount\cnta \newcount\cntb \begingroup -\xintDigitsOf\xintPow{2}{100}\to\Out +\xintDigitsOf\xintiPow{2}{100}\to\Out \cnta = 1 \cntb = 0 \loop -\advance \cntb \xintSqr{\Out{\the\cnta}} +\advance \cntb \xintiSqr{\Out{\the\cnta}} \ifnum \cnta < \Out{0} \advance\cnta 1 \repeat -&2^{100}& (=\xintPow {2}{100}) has \Out{0} digits and the sum of +&2^{100}& (=\xintiPow {2}{100}) has \Out{0} digits and the sum of their squares is \the\cntb. These digits are, from the least to the most significant: \cnta = \Out{0} \loop \Out{\the\cnta}\ifnum \cnta > 1 \advance\cnta -1 , \repeat. @@ -666,18 +982,18 @@ 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: -\centeredline{\csb{xintSum}|{\xintPow{2}{100}}=|\texttt{% - \xintSum{\xintPow{2}{100}}}} Indeed, \csa{xintSum} is usually +\centeredline{\csb{xintiSum}|{\xintiPow{2}{100}}=|\texttt{% + \xintiSum{\xintiPow{2}{100}}}} Indeed, \csa{xintiSum} is usually used as in \centeredline{% - \csb{xintSum}|{{123}{-345}{\xintFac{7}}{\xintOpp{\xintRem{3347}{591}}}}=|\texttt{% - \xintSum{{123}{-345}{\xintFac{7}}{\xintOpp{\xintRem{3347}{591}}}}}} + \csb{xintiSum}|{{123}{-345}{\xintFac{7}}{\xintiOpp{\xintRem{3347}{591}}}}=|\texttt{% + \xintiSum{{123}{-345}{\xintFac{7}}{\xintiOpp{\xintRem{3347}{591}}}}}} but in the example above each digit of &2^{100}& is treated as would have been a summand enclosed within braces, due to the rules of \TeX{} for parsing macro arguments. Note that |{-\xintRem{3347}{591}}| is not a valid input, because the double expansion will apply only to the minus sign and leave -unaffected the |\xintRem|. So we used \csa{xintOpp} which replaces +unaffected the |\xintRem|. So we used \csa{xint}\-|iOpp| which replaces a number with its opposite. As a last example of use of \csa{xintAssignArray} here is one line @@ -722,14 +1038,15 @@ errors: \section{Package namespace} -Inner macros of the \xintname and \xintgcdname packages all begin -either with |\XINT@| or with |\xint@|. The package public commands -all start with |\xint|. The major forms have their initials -capitalized, and lowercase forms, prefixed with |\romannumeral0|, -allow definitions of further macros expanding in two steps to -their full expansion (and can thus be chained with the `primitive' -\xintname macros). Some other control sequence names are used -only as delimiters, and left undefined. +Inner macros of \xintname, \xintgcdname, \xintfracname, and +\xintseriesname all begin either with |\XINT@| or with |\xint@|. +The package public commands all start with |\xint|. The major +forms have their initials capitalized, and lowercase forms, +prefixed with |\romannumeral0|, allow definitions of further +macros expanding in two steps to their full expansion (and can +thus be chained with the `primitive' \xintname macros). Some other +control sequence names are used only as delimiters, and left +undefined. The |\xintReverseOrder|\marg{tokens} macro uses |\xint@UNDEF| and |\xint@undef| as dummy tokens and can be used on arbitrary token @@ -737,15 +1054,19 @@ strings not containing these control sequence names. Anything within braces is treated as one unit: one level of exterior braces is removed and the contents are not reverted. -\clearpage +% \clearpage \section{Loading and usage} \begin{verbatim} Usage with LaTeX: \usepackage{xint} - \usepackage{xintgcd} - - Usage with TeX: \input xint.sty\relax - \input xintgcd.sty\relax + \usepackage{xintgcd} % (loads xint) + \usepackage{xintfrac} % (loads xint) + \usepackage{xintseries} % (loads xintfrac) + + Usage with TeX: \input xint.sty\relax + \input xintgcd.sty\relax % (loads xint) + \input xintfrac.sty\relax % (loads xint) + \input xintseries.sty\relax % (loads xintfrac) \end{verbatim} We have added, directly copied from packages by \textsc{Heiko @@ -755,9 +1076,10 @@ executable |tex| can not be used, |etex| or |pdftex| (version |1.40| or later) or ..., must be invoked. -Furthermore, the package \xintgcdname will check for previous -loading of \xintname, and will try to load it if this was not -already done. +Furthermore, the packages \xintgcdname and \xintfracname will +check for previous loading of \xintname, and will try to load it +if this was not already done. And package \xintseriesname loads +\xintfracname. Also inspired from the \textsc{Heiko Oberdiek} packages we have included a complete catcode protection mecanism. The packages may @@ -771,9 +1093,9 @@ This is for the loading of the packages. For the actual use of the macros, note that when feeding them with negative numbers the minus sign must have category code other, as is standard. -\xintname presupposes that the usual \csa{space} and -\csa{empty} macros are pre-defined, which is the case in Plain -\TeX{} as well as in \LaTeX. +The components of the \xintname bundle presuppose that the usual +\csa{space} and \csa{empty} macros are pre-defined, which is the case in +Plain \TeX{} as well as in \LaTeX. Lastly, the macros \csa{xintRelaxArray} (of \xintname) and \csa{xintTypesetEuclideAlgorithm} and @@ -788,22 +1110,23 @@ compatible. \csa{xintTypesetBezoutAlgorithm} also uses the \begin{verbatim} Run tex or latex on xint.dtx. - This will extract the style files xint.sty and xintgcd.sty (and - xint.ins). Files with the same names and in the same repertory - will be overwritten. The tex (not latex) run will stop with the - complaint that it does not understand \NeedsTeXFormat, but the + This will extract the style files xint.sty, xintgcd.sty, xintfrac.sty, + xintseries.sty (and xint.ins). Files with the same names and in the + same repertory will be overwritten. The tex (not latex) run will stop + with the complaint that it does not understand \NeedsTeXFormat, but the style files will already have been extracted by that time. Alternatively, run tex or latex on xint.ins if available. To get xint.pdf run pdflatex thrice on xint.dtx - xint.sty, xintgcd.sty -> TDS:tex/generic/xint/ - xint.dtx -> TDS:source/generic/xint/ - xint.pdf -> TDS:doc/generic/xint/ + xint.sty, xintgcd.sty, xintfrac.sty, xintseries.sty + -> TDS:tex/generic/xint/ + xint.dtx -> TDS:source/generic/xint/ + xint.pdf -> TDS:doc/generic/xint/ - It may well be necessary to then refresh the TeX installation - filename database. +It may be necessary to then refresh the TeX installation filename +database. \end{verbatim} @@ -814,13 +1137,36 @@ compatible. \csa{xintTypesetBezoutAlgorithm} also uses the \def\x{\string{x\string}} \n{} (resp. \m{} or \x) stands for a normalised number within braces as -described in the documentation, or for a control sequence expanding in -at most two steps to such a number (without the braces!), or for a -control sequence within braces expanding in at most two steps to such a -number, of for material within braces which expands in two expansion of -the first token to such a number. - -\subsection{\chb{xintRev}} +described in the documentation, or for a control sequence expanding in at most +two steps to such a number (without the braces!), or for a control sequence +within braces expanding in at most two steps to such a number, of for material +within braces which expands to such a number after two expansions of the first +token. + +Some of these macros are extended by \xintfracname to accept fractions on input, +and to output a fraction (except for those which output |1|, |0| or |-1|). This +will be mentioned and the original macro \csa{xintAbc} remains then available +under the name \csa{xintiAbc}. + +The integer-only macros are more efficient on integers, even for simple things +such as determining the sign of a number, as there is always some overhead due +to parsing the fraction format on input; however except if one does really a lot +of computations, there is no need in general to employ the integer-only +variants, apart from one mandatory context:\vadjust{\vskip-\dp\strutbox + \hbox{\smash{\color{niceone}\llap{\strut\small + IMPORTANT!\ $\Bigg\{$\ }}}\vskip\dp\strutbox } +when they are inside\strut{} other +integer-only macros. For example |\xintQuo {\xintMul {2}{3}}{2}| will generate +an error when \xintfracname is loaded, because |\xintMul {2}{3}| outputs +|6/1[0]| which |\xintQuo| will not understand. So |\xintQuo {\xintiMul + {2}{3}}{2}| is mandatory. And, when one has something which one knows to be an +integer such as |\xintMul {1/2}{12}|, one can use either |\xintIrr {\xintMul + {1/2}{12}}| or |\xintiTrunc {0}{\xintMul {1/2}{12}}| to produce it in the +format which will be understood by integer-only macros. + + + +\subsection{\csbh{xintRev}} \label{xintRev} \csa{xintRev\n} will revert the order of the digits of the number, keeping the optional sign. Leading zeros @@ -829,44 +1175,50 @@ resulting from the operation are not removed (see the \centeredline{|\xintRev{-123000}|\texttt{=\xintRev{-123000}}} \centeredline{|\xintNum{\xintRev{-123000}}|\texttt{=\xintNum{\xintRev{-123000}}}} -\subsection{\chb{xintReverseOrder}} +\subsection{\csbh{xintReverseOrder}}\label{xintReverseOrder} \csa{xintReverseOrder}\marg{token\_list} does not do any expansion of its argument and just reverses the order of the tokens. Brace pairs encountered are removed once and the enclosed material does not get reverted. -\centeredline{|\xintReverseOrder{\xintDigitsOf\xintPow {2}{100}\to\Stuff}|} +\centeredline{|\xintReverseOrder{\xintDigitsOf\xintiPow {2}{100}\to\Stuff}|} \centeredline{gives: \ttfamily \expandafter\expandafter\expandafter\detokenize \expandafter\expandafter\expandafter{% -\xintReverseOrder{\xintDigitsOf\xintPow {2}{100}\to\Stuff}}} +\xintReverseOrder{\xintDigitsOf\xintiPow {2}{100}\to\Stuff}}} -\subsection{\chb{xintNum}} +\subsection{\csbh{xintNum}}\label{xintNum} \csa{xintNum\n} removes chains of plus or minus signs, followed by zeros. \centeredline{|\xintNum{+---++----+--000000000367941789479}|\texttt {=\xintNum{+---++----+--000000000367941789479}}} -\subsection{\chb{xintLen}} +\subsection{\csbh{xintLen}}\label{xintiLen} \csa{xintLen\n} returns the length of the number, not counting the sign. \centeredline{|\xintLen{-12345678901234567890123456789}|\texttt -{=\xintLen{-12345678901234567890123456789}}} + {=\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 |N/1[0]| so the +minus one means that the extended \csa{xintLen} behaves the same +as the original for integers). The whole thing should sum up to +less than circa &2^{31}&. -\subsection{\chb{xintLength}} +\subsection{\csbh{xintLength}}\label{xintLength} \csa{xintLength}\marg{token\_list} does not do any expansion of its argument and just counts how many tokens there are. Things enclosed in braces count as one. -\centeredline{|\xintLength {\xintPow {2}{100}}=|\texttt{\xintLength - {\xintPow{2}{100}}}} -\centeredline{${}\neq{}$|\xintLen {\xintPow {2}{100}}=|\texttt{\xintLen - {\xintPow{2}{100}}}} +\centeredline{|\xintLength {\xintiPow {2}{100}}=|\texttt{\xintLength + {\xintiPow{2}{100}}}} +\centeredline{${}\neq{}$|\xintLen {\xintiPow {2}{100}}=|\texttt{\xintLen + {\xintiPow{2}{100}}}} -\subsection{\chb{xintAssign}} +\subsection{\csbh{xintAssign}}\label{xintAssign} \csa{xintAssign}\meta{braced things}\csa{to}% \meta{as many cs as they are things} defines (without checking if @@ -877,7 +1229,7 @@ the left of \csa{to} enclosed within braces. Important: a double expansion is applied first to the material extending up to \csa{to}. -\xintAssign\xintPow {7}{13}\to\SevenToThePowerThirteen +\xintAssign\xintiPow {7}{13}\to\SevenToThePowerThirteen \xintAssign\xintDivision{1000000000000}{133333333}\to\Q\R As a special exception, if after this initial double expansion a @@ -887,7 +1239,7 @@ defined to be the complete expansion of the material between \csa{xintAssign} and \csa{to}. \centeredline{|\xintAssign\xintDivision{1000000000000}{133333333}\to\Q\R|} \centeredline{|\meaning\Q: |\texttt{\meaning\Q}, |\meaning\R: - |\texttt{\meaning\R}} \centeredline{|\xintAssign\xintPow + |\texttt{\meaning\R}} \centeredline{|\xintAssign\xintiPow {7}{13}\to\SevenToThePowerThirteen|} \centeredline{|\SevenToThePowerThirteen=|\texttt{\SevenToThePowerThirteen}} @@ -895,7 +1247,7 @@ Of course this macro and its cousins completely break usage in pure expansion contexts, as assignments are made via the \csa{edef} primitive. -\subsection{\chb{xintAssignArray}} +\subsection{\csbh{xintAssignArray}}\label{xintAssignArray} \xintAssignArray\xintBezout {1000}{113}\to\Bez @@ -913,187 +1265,197 @@ so that the successive elements are \csa{myArray}|{1}|, \dots, |\Bez{5}| to \texttt{\Bez5}: (\Bez3)${}\times{}$\Bez1${}-{}$(\Bez4)${}\times{}$\Bez2${}={}$\Bez5. -\subsection{\chb{xintRelaxArray}} +\subsection{\csbh{xintRelaxArray}}\label{xintRelaxArray} \csa{xintRelaxArray}\csa{myArray} sets to \csa{relax} all macros which were defined by the previous \csa{xintAssignArray} with \csa{myArray} as array name. -\subsection{\chb{xintDigitsOf}} +\subsection{\csbh{xintDigitsOf}}\label{xintDigitsOf} This is a synonym for \csa{xintAssignArray}, to be used to define an array giving all the digits of a given number. -\begingroup\xintDigitsOf\xintPow {7}{500}\to\digits -\centeredline{|\xintDigitsOf\xintPow {7}{500}\to\digits|} +\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{\chb{xintSgn}} +\subsection{\csbh{xintSgn}}\label{xintiSgn} \csa{xintSgn\n} returns 1 if the number is positive, 0 if it is -zero and -1 if it is negative. +zero and -1 if it is negative. Extended by \xintfracname to fractions. -\subsection{\chb{xintOpp}} +\subsection{\csbh{xintOpp}}\label{xintiOpp} \csa{xintOpp\n} returns the opposite |-N| of the number |N|. +Extended by \xintfracname to fractions. + -\subsection{\chb{xintAbs}} +\subsection{\csbh{xintAbs}}\label{xintiAbs} -\csa{xintAbs\n} returns the absolute value of the number. +\csa{xintAbs\n} returns the absolute value of the number. Extended +by \xintfracname to fractions. -\subsection{\chb{xintAdd}} +\subsection{\csbh{xintAdd}}\label{xintiAdd} -\csa{xintAdd\n\m} returns the sum of the two numbers. It is more -efficient to have the longer of the two be the first argument. +\csa{xintAdd\n\m} returns the sum of the two numbers. Extended by +\xintfracname to fractions. -\subsection{\chb{xintSub}} +\subsection{\csbh{xintSub}}\label{xintiSub} -\csa{xintSub\n\m} returns the difference |N-M|. +\csa{xintSub\n\m} returns the difference |N-M|. Extended by +\xintfracname to fractions. -\subsection{\chb{xintCmp}} +\subsection{\csbh{xintCmp}}\label{xintiCmp} \csa{xintCmp\n\m} returns 1 if |N>M|, 0 if |N=M|, and -1 if |N<M|. +Extended by \xintfracname to fractions. -\subsection{\chb{xintGeq}} +\subsection{\csbh{xintGeq}}\label{xintGeq} \csa{xintGeq\n\m} returns 1 if the 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. -\subsection{\chb{xintMax}} +\subsection{\csbh{xintMax}}\label{xintiMax} \csa{xintMax\n\m} 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). +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}=|\texttt{\xintiMax{-5}{-6}}. Extended by \xintfracname to fractions. -\subsection{\chb{xintMin}} +\subsection{\csbh{xintMin}}\label{xintiMin} \csa{xintMin\n\m} 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). +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}=|\texttt{\xintiMin{-5}{-6}}. Extended by \xintfracname to fractions. -\subsection{\chb{xintSum}} +\subsection{\csbh{xintSum}}\label{xintiSum} \csa{xintSum}\marg{braced things} after expanding its argument twice expects to find a sequence of tokens (or braced material). Each is twice-expanded, and the sum of all these numbers is returned. \centeredline{% - \csa{xintSum}|{{123}{-98763450}{\xintFac{7}}{\xintMul{3347}{591}}}=|\texttt{% - \xintSum{{123}{-98763450}{\xintFac{7}}{\xintMul{3347}{591}}}}} -\centeredline{\csa{xintSum}|{1234567890}=|\texttt{% - \xintSum{1234567890}}} -An empty sum is no error and returns zero: |\xintSum -{}=|\texttt{\xintSum {}}. A sum with only one -term returns that number: |\xintSum {{-1234}}=|\texttt{\xintSum - {{-1234}}}. Attention that |\xintSum {-1234}| is not legal input -and will may the \TeX{} run fail. On the other hand |\xintSum -{1234}=|\texttt{\xintSum{1234}}. - -\subsection{\chb{xintSumExpr}} + \csa{xintiSum}|{{123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}}=|\texttt{% + \xintiSum{{123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}}}} +\centeredline{\csa{xintiSum}|{1234567890}=|\texttt{% + \xintiSum{1234567890}}} +An empty sum is no error and returns zero: |\xintiSum +{}=|\texttt{\xintiSum {}}. A sum with only one +term returns that number: |\xintiSum {{-1234}}=|\texttt{\xintiSum + {{-1234}}}. Attention that |\xintiSum {-1234}| is not legal input +and will make the \TeX{} run fail. On the other hand |\xintiSum +{1234}=|\texttt{\xintiSum{1234}}. Extended by \xintfracname +to fractions. + +\subsection{\csbh{xintSumExpr}}\label{xintiSumExpr} \csa{xintSum}\meta{braced things}\csa{relax} is to what \csa{xintSum} reduces after its initial double expansion of its argument. \centeredline{% - \csa{xintSumExpr}| {123}{-98763450}|% - |{\xintFac{7}}{\xintMul{3347}{591}}\relax=|\texttt{% - \xintSumExpr {123}{-98763450}{\xintFac{7}}{\xintMul{3347}{591}}\relax}} + \csa{xintiSumExpr}| {123}{-98763450}|% + |{\xintFac{7}}{\xintiMul{3347}{591}}\relax=|\texttt{% + \xintiSumExpr + {123}{-98763450}{\xintFac{7}}{\xintiMul{3347}{591}}\relax}} -\subsection{\chb{xintMul}} +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. -\csa{xintMul\n\m} returns the product of the two numbers. The order of the -numbers influences the efficiency of the computation: -\begin{enumerate} -\item if the shortest number has at most 4 digits, then it is (always) more - efficient if it comes as the second argument, -\item if both numbers have at most 50 digits, then it is generally a bit - more efficient to have the longest one be the first argument and the shortest - one be the second argument, -\item if one of the number has more than 250 digits, it is always advantageous - if the shortest number is the first argument (except if it has only 4 digits - or less!). For example, 50 digits $\times$ 1000 digits is five times faster - than 1000 digits $\times$ 50 digits, -\item if both numbers have less than 250 digits, then it is advantageous to have - the shortest one be the first argument, as long as it is not too short. The - limit depends on the size of the longer number; roughly, when the longer - number has 100 digits, this limit on the shorter one is already of 12 digits, - and the longer the long number, the lower the limit for the shorter. -\end{enumerate} -So when both numbers have at most 50 digits, put the longer one first and the -shorter one second; when both numbers have at least 50 digits, put the shorter -one first and the longer one second. The gain will be substantial if the long -number is very long. +Extended by \xintfracname to fractions. -When one of the number has at most 4 digits, always make it second. But if the -shortest has 5 digits or more, it is advantageous to have it in first -position when the longer number has 250 digits or more. +\subsection{\csbh{xintMul}}\label{xintiMul} +{\small Modified in bundle version |1.03|.\par} +\csa{xintMul\n\m} 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. -\subsection{\chb{xintSqr}} +\subsection{\csbh{xintSqr}}\label{xintiSqr} -\csa{xintSqr\n} returns the square. +\csa{xintSqr\n} returns the square. Extended by \xintfracname to fractions. -\subsection{\chb{xintPrd}} +\subsection{\csbh{xintPrd}}\label{xintiPrd} \csa{xintPrd}\marg{braced things} after expanding its argument twice expects to find a sequence of tokens (or braced material). Each is twice-expanded, and the product of all these numbers is returned. \centeredline{% - \csa{xintPrd}|{{-9876}{\xintFac{7}}{\xintMul{3347}{591}}}=|% + \csa{xintiPrd}|{{-9876}{\xintFac{7}}{\xintiMul{3347}{591}}}=|% \texttt{% - \xintPrd{{-9876}{\xintFac{7}}{\xintMul{3347}{591}}}}} -\centeredline{\csa{xintPrd}|{123456789123456789}=|\texttt{% - \xintPrd{123456789123456789}}} An empty product is no error -and returns 1: |\xintPrd {}=|\texttt{\xintPrd {}}. A product -reduced to a single term returns this number: |\xintPrd -{{-1234}}=|\texttt{\xintPrd {{-1234}}}. Attention that |\xintPrd + \xintiPrd{{-9876}{\xintFac{7}}{\xintiMul{3347}{591}}}}} +\centeredline{\csa{xintiPrd}|{123456789123456789}=|\texttt{% + \xintiPrd{123456789123456789}}} An empty product is no error +and returns 1: |\xintiPrd {}=|\texttt{\xintiPrd {}}. A product +reduced to a single term returns this number: |\xintiPrd +{{-1234}}=|\texttt{\xintiPrd {{-1234}}}. Attention that |\xintiPrd {-1234}| is not legal input and will make the \TeX{} compilation -fail. On the other hand |\xintPrd {1234}=|\texttt{\xintPrd +fail. On the other hand |\xintiPrd {1234}=|\texttt{\xintiPrd {1234}}. \centeredline{&2^{200}3^{100}7^{100}&} -\centeredline{=|\xintPrd {{\xintPow {2}{200}}{\xintPow - {3}{100}}{\xintPow {7}{100}}}|} +\centeredline{=|\xintiPrd {{\xintiPow {2}{200}}{\xintiPow + {3}{100}}{\xintiPow {7}{100}}}|} =\expandafter\expandafter\expandafter\allownumbersplit - \xintPrd {{\xintPow {2}{200}}{\xintPow {3}{100}}{\xintPow + \xintiPrd {{\xintiPow {2}{200}}{\xintiPow {3}{100}}{\xintiPow {7}{100}}}\relax -\centeredline{=|\xintPow {\xintMul {\xintPow {42}{9}}{43008}}{10}|} +\centeredline{=|\xintiPow {\xintiMul {\xintiPow {42}{9}}{43008}}{10}|} +Extended by \xintfracname to fractions. -% \expandafter\expandafter\expandafter\allownumbersplit -% \xintPow {\xintMul {\xintPow {42}{9}}{43008}}{10}\relax +% \printnumber{% +% \xintPow {\xintMul {\xintPow {42}{9}}{43008}}{10}} -\subsection{\chb{xintProductExpr}} +\subsection{\csbh{xintProductExpr}}\label{xintiProductExpr} \csa{xintProductExpr}\meta{braced things}\csa{relax} is to what \csa{xintPrd} reduces after its initial double expansion of its argument. -\centeredline{\csa{xintProductExpr}| 123456789123456789\relax=|\texttt{% - \xintProductExpr 123456789123456789\relax}} +\centeredline{\csa{xintiProductExpr}| 123456789123456789\relax=|\texttt{% + \xintiProductExpr 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. -\subsection{\chb{xintFac}} +Extended by \xintfracname to fractions. + +\subsection{\csbh{xintFac}}\label{xintFac} \csa{xintFac\n} returns the factorial. It is an error if the argument is negative or at least &10^9&. It is not recommended to launch the computation of things such as &100000!&, if you need -your computer for other tasks. 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{\chb{xintPow}} +your computer for other tasks. + +% 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{xintPow}}\label{xintiPow} \csa{xintPow\n\m} returns |N^M|. When |M| is zero, this is 1. Some cases (|N| zero and |M| negative, \verb+|N|>1+ and |M| negative, \verb+|N|>1+ and |M| at least &10^9&) make \xintname throw errors. -\subsection{\chb{xintDivision}} +Extended by \xintfracname to fractions. Of course, negative +exponents do not then cause errors anymore. + + +\subsection{\csbh{xintDivision}}\label{xintDivision} \csa{xintDivision\n\m} returns |{quotient Q}{remainder R}|. This is euclidean division: |N = QM + R|, |0|${}\leq{}$\verb+R < |M|+. So the @@ -1101,36 +1463,49 @@ 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 of course an error (even if |N| vanishes) and returns |{0}{0}|. -\subsection{\chb{xintQuo}} +This macro is integer only and not to be confused with the \xintfracname macro +\csb{xintDiv} which divides one fraction by another. -\csa{xintQuo\n\m} returns the quotient from the euclidean division. +\subsection{\csbh{xintQuo}}\label{xintQuo} -\subsection{\chb{xintRem}} +\csa{xintQuo\n\m} 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). + +\subsection{\csbh{xintRem}}\label{xintRem} \csa{xintRem\n\m} returns the remainder from the euclidean division. -\subsection{\chb{xintFDg}} +\subsection{\csbh{xintFDg}}\label{xintFDg} \csa{xintFDg\n} returns the first digit (most significant) of the decimal expansion. -\subsection{\chb{xintLDg}} +\subsection{\csbh{xintLDg}}\label{xintLDg} \csa{xintLDg\n} returns the least significant digit. When the number is positive, this is the same as the remainder in the euclidean division by ten. -\subsection{\chb{xintOdd}} +\subsection{\csbh{xintMON}, \csbh{xintMMON}}\label{xintMON}\label{xintMMON} +{\small New in bundle version |1.03|.\par} + +\csa{xintMON\n} returns |(-1)^N| and \csa{xintMMON\n} returns +|(-1)^{N-1}|. \centeredline{|\xintMON {-280914019374101929}=|\texttt{\xintMON + {280914019374101929}}, |\xintMMON +{-280914019374101929}=|\texttt{\xintMMON {280914019374101929}}} + +\subsection{\csbh{xintOdd}}\label{xintOdd} \csa{xintOdd\n} is 1 if the number is odd and 0 otherwise. -\subsection{\chb{xintDSL}} +\subsection{\csbh{xintDSL}}\label{xintDSL} \csa{xintDSL\n} is decimal shift left, \emph{i.e.} multiplication by ten. -\subsection{\chb{xintDSR}} +\subsection{\csbh{xintDSR}}\label{xintDSR} \csa{xintDSR\n} is decimal shift right, \emph{i.e.} it removes the last digit (keeping the sign). For a positive number, this is the @@ -1139,7 +1514,7 @@ course, done in a more efficient manner than via the general division algorithm). For |N| from |-9| to |-1|, the macro returns |0|. -\subsection{\chb{xintDSH}} +\subsection{\csbh{xintDSH}}\label{xintDSH} \csa{xintDSH\x\n} is parametrized decimal shift. When |x| is negative, it is like iterating \csa{xintDSL} \verb+|x|+ times @@ -1148,7 +1523,7 @@ it is like iterating \csa{DSR} |x| times (and is more efficient of course), and for a non-negative |N| this is thus the same as the quotient from the euclidean division by |10^x|. -\subsection{\chb{xintDSHr}, \chb{xintDSx}} +\subsection{\csbh{xintDSHr}, \csbh{xintDSx}}\label{xintDSHr}\label{xintDSx} {\small New in bundle version |1.01|.\par} \csa{xintDSHr\x\n} expects |x| to be zero or positive and it returns @@ -1211,9 +1586,9 @@ simultaneously. |\xintDSHr {9}{-123004321}=|\texttt{\xintDSHr {9}{-123004321}}\\ \end{flushleft} -\subsection{\chb{xintDecSplit}} +\subsection{\csbh{xintDecSplit}}\label{xintDecSplit} -{\small Modified in bundle version |1.01|!\par} +{\small This has been modified in bundle version |1.01|.\par} \csa{xintDecSplit\x\n} cuts the number into two pieces (each one within a pair of enclosing braces). First the sign if present is \emph{removed}. @@ -1222,16 +1597,14 @@ 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 zeros 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 +second piece the remaining digits (\emph{empty} if &|x|& equals or exceeds the length of |N|). Leading zeros 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. -{\small This macro is for use in future components of the \xintname - bundle. Its behavior for |N| non-negative is final and will not +{\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|. It is recommended to use the macro only for non-negative - |N| until the definitive version is released.\par} + negative |N|.\par} \xintAssign\xintDecSplit {0}{-123004321}\to\L\R @@ -1256,12 +1629,12 @@ of the first and second piece. \centeredline{|\xintAssign\xintDecSplit {-15}{-12300004321}\to\L\R|} |\meaning\L: |\texttt{\meaning\L}, |\meaning\R: |\texttt{\meaning\R.} -\subsection{\chb{xintDecSplitL}} +\subsection{\csbh{xintDecSplitL}}\label{xintDecSplitL} \csa{xintDecSplitL\x\n} returns the first piece after the action of \csa{xintDecSplit}. -\subsection{\chb{xintDecSplitR}} +\subsection{\csbh{xintDecSplitR}}\label{xintDecSplitR} \csa{xintDecSplitR\x\n} returns the second piece after the action of \csa{xintDecSplit}. @@ -1270,7 +1643,7 @@ of \csa{xintDecSplit}. \section{Commands of the \xintgcdname package} -\subsection{\chb{xintGCD}} +\subsection{\csbh{xintGCD}}\label{xintGCD} \csa{xintGCD\n\m} computes the greatest common divisor. It is positive, except when both |N| and |M| vanish, in which case the macro @@ -1279,7 +1652,7 @@ returns zero. \centeredline{|\xintGCD{123456789012345}{9876543210321}=|\texttt {\xintGCD{123456789012345}{9876543210321}}} -\subsection{\chb{xintBezout}} +\subsection{\csbh{xintBezout}}\label{xintBezout} \xintAssign{{\xintBezout {10000}{1113}}}\to\X \xintAssign {\xintBezout {10000}{1113}}\to\A\B\U\V\D @@ -1305,7 +1678,7 @@ second, |D| is the GCD, and \texttt{UA - VB = D}. |\D: |\texttt{\D }. -\subsection{\chb{xintEuclideAlgorithm}} +\subsection{\csbh{xintEuclideAlgorithm}}\label{xintEuclideAlgorithm} \xintAssign {{\xintEuclideAlgorithm {10000}{1113}}}\to\X @@ -1340,7 +1713,7 @@ 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{\chb{xintBezoutAlgorithm}} +\subsection{\csbh{xintBezoutAlgorithm}}\label{xintBezoutAlgorithm} \catcode`\& 4 @@ -1360,7 +1733,9 @@ The first token is the number of steps, the second is |N|, then 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{\chb{xintTypesetEuclideAlgorithm}} +\catcode`\& 13 + +\subsection{\csbh{xintTypesetEuclideAlgorithm}}\label{xintTypesetEuclideAlgorithm} This macro is just an example of how to organize the data returned by \csa{xintEuclideAlgorithm}. Copy the source code to a new macro @@ -1370,7 +1745,7 @@ and modify it to what is needed. {123456789012345}{9876543210321} -\subsection{\chb{xintTypesetBezoutAlgorithm}} +\subsection{\csbh{xintTypesetBezoutAlgorithm}}\label{xintTypesetBezoutAlgorithm} This macro is just an example of how to organize the data returned by \csa{xintBezoutAlgorithm}. Copy the source code to a new macro @@ -1378,16 +1753,587 @@ and modify it to what is needed. \centeredline{|\xintTypesetBezoutAlgorithm {10000}{1113}|} \xintTypesetBezoutAlgorithm {10000}{1113} +\section{Commands of the \xintfracname package} + +The general rule of the bundle that each macro first double-expands each one of +its arguments applies. + +\subsection{\csbh{xintLen}}\label{xintLen} + +The original macro is extended to accept a fraction on input. +\centeredline {|\xintLen {201710/298219}=|\texttt{\xintLen {201710/298219}}, +|\xintLen {1234/1}=|\texttt{\xintLen {1234/1}}, |\xintLen {1234}=|\texttt{\xintLen {1234}}} + +\subsection{\csbh{xintNumerator}}\label{xintNumerator} + +This returns the numerator corresponding to the internal representation of the +fraction:\footnote{recall that the |[]| construct excludes presence of a decimal + point.} \centeredline{|\xintNumerator + {178000/25600000[17]}=|\texttt{\xintNumerator {178000/25600000[17]}}}% +\centeredline{|\xintNumerator {178.000/25600000}=|\texttt{\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 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]}=|\texttt{\xintDenominator {178000/25600000[17]}}}% +\centeredline{|\xintDenominator {178.000/25600000}=|\texttt{\xintDenominator + {178.000/25600000}}} As shown by the examples, no simplification of the +input is done. The denominator looks wrong in the second 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{xintFrac}}\label{xintFrac} + +This is a \LaTeX{} only command, 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| and the denominator is +omitted when it is one, the number is then separated from the power of ten by a +|\cdot|. |$\xintFrac {178.000/25600000}$| gives $\xintFrac {178.000/25600000}$, +|$\xintFrac {178.000/1}$| gives $\xintFrac {178.000/1}$ and |$\xintFrac +{3.5/5.7}$| gives $\xintFrac {3.5/5.7}$. As shown by the examples, no +simplification of the input is done (apart from removing the decimal points and +moving the sign to the numerator). + + +\subsection{\csbh{xintFwOver}}\label{xintFwOver} + +This does the same as \csa{xintFrac} 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 $\xintFwOver +{178.000/1}$ and |$\xintFwOver {3.5/5.7}$| gives $\xintFwOver {3.5/5.7}$. + +\subsection{\csbh{xintSum}, \csbh{xintSumExpr}}\label{xintSum}\label{xintSumExpr} + +The originals are extended to accept fractions on input. Their outputs will now +always be in the form |A/B[n]| or |A[n]| and thus cannot be used inside +integer-only macros. +The originals are preserved as \csa{xintiSum} and \csa{xintiSumExpr}. + + +\subsection{\csbh{xintPrd}, \csbh{xintProductExpr}}\label{xintPrd}\label{xintProductExpr} + +The originals are extended to accept fractions on input. Their outputs will now +always be in the form |A/B[n]| or |A[n]| and thus cannot be used inside +integer-only macros. +The originals are preserved as \csa{xintiPrd} and \csa{xintiPrdExpr}. + + +\subsection{\csbh{xintREZ}}\label{xintREZ} + +This command normalizes a fraction by removing the powers of ten in its +numerator and denominator: |\xintREZ {178000/25600000[17]}=|\texttt{\xintREZ + {178000/25600000[17]}}. As shown by the example, it does not otherwise +simplify the fraction. + +\subsection{\csbh{xintIrr}}\label{xintIrr} + +This puts the fraction into its unique irreducible form: \centeredline{|\xintIrr + {178.256/256.178}=|% + \texttt{\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. + +To avoid some overhead in the parsing by |\xintFrac| of the output of +|\xintIrr|, add a |[0]|: |\xintFrac {\xintIrr {178.256/256.178}[0]}|. This +advice is only for \csa{xintIrr} (or \csa{xintJrr}) as these macros do not have +the |[n]| systematically present in the outputs of the other macros, |[n]| whose +rôle is also to signal that the format can be parsed in a minimal way, as it is +not arbitrary user-input but beautiful package crafted output... + + +\subsection{\csbh{xintJrr}}\label{xintJrr} + +This also puts the fraction into its unique irreducible form: +\centeredline{|\xintJrr {178.256/256.178}=|% + \texttt{\xintJrr {178.256/256.178}}}% +This is faster than \csa{xintIrr} for fractions having a substantial common +factor in the numerator and the denominator, as here: +|\xintJrr {\xintiMul{\xintFac {15}}{\xintFac + {15}}/\xintiMul{\xintFac{10}}{\xintFac{30}}}=|\texttt{% + \xintJrr {\xintiMul{\xintFac {15}}{\xintFac + {15}}/\xintiMul{\xintFac{10}}{\xintFac{30}}}}. But to notice the +difference one would need computations with much bigger numbers than in this +example. + + +\subsection{\csbh{xintTrunc}}\label{xintTrunc} + +\csa{xintTrunc}|{N}{f}| outputs the start of the decimal expansion of the +fraction |f|, with |N| digits after the decimal point. The argument |N| should +be non-negative. When |N=0|, the integer part of |f| results, with an ending +decimal point. Only when |f| evaluates to zero does \csa{xintTrunc} not print +the decimal point. When |f| is not zero, the sign is maintained in the output, +also when the digits are all zero. \centeredline{|\xintTrunc + {20}{-803.2028/20905.298}=|\texttt{\xintTrunc {20}{-803.2028/20905.298}}} The +digits printed are exact up to and including the last one. + +\subsection{\csbh{xintiTrunc}}\label{xintiTrunc} + +\csa{xintiTrunc}|{N}{f}| outputs the integer equal to |10^N| times what +\csa{xintTrunc}|{N}{f}| would return. \centeredline{|\xintiTrunc + {20}{-803.2028/20905.298}=|\texttt{\xintiTrunc {20}{-803.2028/20905.298}}} The +difference between \csa{xintTrunc}|{0}{f}| and \csa{xintiTrunc}|{0}{f}| is that +the former cannot be used inside integer-only macros, whereas the latter +removes the decimal point, and never returns |-0| (and of course removes +all superfluous leading zeros.) + +\subsection{\csbh{xintMul}}\label{xintMul} + +The original macro is extended to accept fractions on input. Its output will now +always be in the form |A/B[n]| or |A[n]| and thus cannot be used inside +integer-only macros. +The original is preserved as \csa{xintiMul}. + +\subsection{\csbh{xintSqr}}\label{xintSqr} + +The original macro is extended to accept a fraction on input. Its output will now +always be in the form |A/B[n]| or |A[n]| and thus cannot be used inside +integer-only macros. +The original is preserved as \csa{xintiSqr}. + +\subsection{\csbh{xintPow}}\label{xintPow} + +The original macro is extended to accept a fraction on input (the exponent must +be a signed integer of course). Its output will now always be in the form +|A/B[n]| or |A[n]| and thus cannot be used directly inside integer-only macros. +The original is preserved as \csa{xintiPow}. + +\subsection{\csbh{xintDiv}}\label{xintDiv} + +\csa{xintDiv}|{f}{g}| 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]| or |A[n]| and cannot be used directly inside +integer-only macros. + +\subsection{\csbh{xintAdd}}\label{xintAdd} +The original macro is extended to accept fractions on input. Its output will now +always be in the form |A/B[n]| or |A[n]| and thus cannot be used directly inside +integer-only macros. +The original is preserved as \csa{xintiAdd}. +\subsection{\csbh{xintSub}}\label{xintSub} + +The original macro is extended to accept fractions on input. Its output will now +always be in the form |A/B[n]| or |A[n]| and thus cannot be used directly inside +integer-only macros. +The original is preserved as \csa{xintiSub}. + +\subsection{\csbh{xintCmp}}\label{xintCmp} + +The macro is extended to fractions. +The original, which skips the overhead of +the fraction format parsing, is preserved as \csa{xintiCmp}. + +\subsection{\csbh{xintMax}}\label{xintMax} + +The macro is extended to fractions. The re-defined version cannot be used +directly inside integer-only macros anymore. The original is preserved as +\csa{xintiMax}. + +\subsection{\csbh{xintMin}}\label{xintMin} + +The macro is extended to fractions. The re-defined version cannot be used +directly inside integer-only macros anymore. The original is preserved as +\csa{xintiMin}. + +\subsection{\csbh{xintSgn}}\label{xintSgn} + +The macro is extended to fractions. The original, which skips the overhead of +the fraction format parsing, is preserved as \csa{xintiSgn}. + +\subsection{\csbh{xintOpp}}\label{xintOpp} + +The macro is extended to fractions. The re-defined version cannot be used +directly inside integer-only macros anymore. The original is preserved as +\csa{xintiOpp}. + +\subsection{\csbh{xintAbs}}\label{xintAbs} + +The macro is extended to fractions. The re-defined version cannot be used +directly inside integer-only macros anymore. The original is preserved as +\csa{xintiAbs}. + + +\section{Commands of the \xintseriesname package} + +There will be some exceptions to the general rule that +each macro first double-expands each one of its arguments. + +\subsection{\csbh{xintSeries}}\label{xintSeries} + +\def\coeff #1{\romannumeral0\xintmon{#1}/#1.5} % (-1)^n/(n+1/2) +\edef\w {\xintSeries {0}{50}{\coeff}} +\edef\z {\xintJrr {\w}[0]} + +\csa{xintSeries}|{A}{B}{\coeff}| evaluates the sum of all values of the |\coeff +{n}| from |n=A| to and including |n=B|. The initial and final indices must +(after double-expansion) obey the \TeX{} constraint of being explicit numbers of +values at most |2^31-1| (these conditions are not checked by the macro). The +|\coeff| macro (which, as argument to \csa{xintSeries} is double-expanded only +at the time of computing the successive |\coeff {n}|) should be defined as a +one-parameter command, accepting on input a number (not a count register) and +needing at most two expansions to compute its final result. +\begin{verbatim} +\def\coeff #1{\romannumeral0\xintmon{#1}/#1.5} % (-1)^n/(n+1/2) +\edef\w {\xintSeries {0}{50}{\coeff}} +\edef\z {\xintJrr {\w}[0]} +% \xintJrr as a big common factor is suspected. +\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} = \xintFrac\z \] +\end{verbatim} +\vspace*{-\baselineskip} +\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} = \xintFrac\z \] +For info, before action by |\xintJrr| the inner representation of the result +has a denominator of |\xintLen {\xintDenominator\w}=|\xintLen +{\xintDenominator\w} digits. + +\subsection{\csbh{xintiSeries}}\label{xintiSeries} + +\def\coeff #1{\romannumeral0\xintitrunc {40}{\xintMON{#1}/#1.5}} + +\csa{xintiSeries}|{A}{B}{\coeff}| evaluates the sum of all values of the |\coeff +{n}| from |n=A| to and including |n=B|. The initial and final indices must +(after double-expansion) obey the \TeX{} constraint of being explicit numbers of +values at most |2^31-1| (these conditions are not checked by the macro). The +|\coeff| macro (which, as argument to \csa{xintSeries} is double-expanded only +at the time of computing the successive |\coeff {n}|) should be defined as a +one-parameter command, accepting on input a number (not a count register) and +needing at most two expansions to compute its final result, \emph{which must be + an integer}, in the format understood by the package integer-only +\csa{xintiAdd}. +\begin{verbatim} +\def\coeff #1{\romannumeral0\xintitrunc {40}{\xintMON{#1}/#1.5}} +% (-1)^n/(n+1/2), with 40 digits post decimal point, as an integer +\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx + \xintTrunc {40}{\xintiSeries {0}{50}{\coeff}[-40]}\dots\] +\end{verbatim} +\vspace*{-\baselineskip} +\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} \approx + \xintTrunc {40}{\xintiSeries {0}{50}{\coeff}[-40]}\dots\] +We should have cut out some of the final digits, rather than print all 40 of +them. For comparison the decimal expansion of the exact result is: +\[ \sum_{n=0}^{n=50} \frac{(-1)^n}{n+\frac12} = \xintTrunc {40}{\z}\dots\] + +\subsection{\csbh{xintPowerSeries}}\label{xintPowerSeries} + +\csa{xintPowerSeries}|{A}{B}{\coeff}{x}| evaluates the sum of +|\coeff{n}|\raisebox{.5ex}{|.|}|x^n| from |n=A| up to and including |n=B|. The +initial and final indices must (after double-expansion) obey the \TeX{} +constraint of being explicit numbers of values at most |2^31-1| (these +conditions are not checked by the macro). The |\coeff| macro (which, as argument +to \csa{xintPowerSeries} is double-expanded only at the time of computing the +successive |\coeff{n}|) should be defined as a one-parameter command, accepting +on input a number (not a count register) and needing at most two expansions to +compute its final result. + +The |x| can be either a fraction directly input or a macro expanding in at most +two steps to such a fraction. It is actually more efficient for the various +macro expansions done by \TeX{} to encapsulate the fraction |x| in a macro (say +|\x|), if it has big numerators and denominators, as the less tokens there are, +the faster it goes, and some amount of shuffling around of the data given as the +fourth parameter to \csa{xintPowerSeries} is done internally, repeatedly. And, +for greater efficiency |x| should be a fraction in |A/B[n]| format. + +Note though that this macro computes the \emph{exact} result, which may quickly +become a very big (possibly highly reducible) fraction. +\begin{verbatim} +\def\geom #1{1[0]} % the geometric series +\def\x {5/17[0]} +\[ \sum_{n=0}^{n=20} \Bigl(\frac 5{17}\Bigr)^n + =\xintFrac{\xintIrr{\xintPowerSeries {0}{20}{\geom}{\x}}} + =\xintFrac{\xintDiv{\xintiSub{\xintiPow {17}{21}}{\xintiPow{5}{21}}} + {\xintiSub{\xintiPow {17}{21}}{\xintiMul {5}{\xintiPow {17}{20}}}}}\] +% a parser for arbitrary algebraic expressions with the +,-,/,*,and ^ +% operations would dearly appreciated here ; but implementing a +% completely expandable one would quite a lot of work. +\end{verbatim} +\def\geom #1{1[0]} % the geometric series +\def\x {5/17[0]} % +\[ \sum_{n=0}^{n=20} \Bigl(\frac 5{17}\Bigr)^n + =\xintFrac{\xintIrr{\xintPowerSeries {0}{20}{\geom}{\x}}} + =\xintFrac{\xintDiv{\xintiSub{\xintiPow {17}{21}}{\xintiPow{5}{21}}} + {\xintiSub{\xintiPow {17}{21}}{\xintiMul {5}{\xintiPow {17}{20}}}}}\] +Check it by hand\dots you can guess the common factor by looking at the last two +digits of the two denominators! + +% checking: + +% \xintIrr {69091933912531895722624092/5757661159377657976885341} + +% \xintIrr {48770776879770870268819212/4064231406647572522401601} + +% gives 12 in both cases. Hourrah! + +\begin{verbatim} +\def\coefflog #1{1/#1[0]}% 1/n +\def\x {1/2[0]}% +\[ \log 2 \approx \sum_{n=1}^{20} \frac1n \frac1{2^n} + = \xintFrac {\xintIrr {\xintPowerSeries {1}{20}{\coefflog}{\x}}}\] +\end{verbatim} +\def\coefflog #1{1/#1[0]} % 1/n +\def\x {1/2[0]}% +\[ \log 2 \approx \sum_{n=1}^{20} \frac1n \frac1{2^n} + = \xintFrac {\xintIrr {\xintPowerSeries + {1}{20}{\coefflog}{\x}}}\] + + +\begin{verbatim} +\def\coeffarctg #1{1/\the\numexpr\xintMON{#1}*(2*#1+1)\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) +\def\x {1/25[0]}% 1/5^2 +\[\mathrm{Arctg}(\frac15)\approx + \frac15\sum_{n=0}^{15} \frac{(-1)^n}{(2n+1)25^n} += \xintFrac{\xintIrr {\xintDiv + {\xintPowerSeries {0}{15}{\coeffarctg}{\x}}{5}}}\] +\end{verbatim} +\def\coeffarctg #1{1/\the\numexpr \xintMON{#1}*(2*#1+1)\relax }% (-1)^n/(2n+1) +\def\x {1/25[0]}% 1/5^2 +\[\mathrm{Arctg}(\frac15)\approx + \frac15\sum_{n=0}^{15} \frac{(-1)^n}{(2n+1)25^n} += \xintFrac{\xintIrr {\xintDiv + {\xintPowerSeries {0}{15}{\coeffarctg}{\x}}{5}}}\] + + +\subsection{\csbh{xintFxPtPowerSeries}}\label{xintFxPtPowerSeries} + +\csa{xintFxPtPowerSeries}|{A}{B}{\coeff}{x}{D}| computes the sum of +|\coeff{n}|\raisebox{.5ex}{|.|}|x^n| from |n=A| to |n=B| with each term of the +series being truncated to |D| digits after the decimal point. + +More precisely the first power |x^A| is computed exactly, then truncated. Then +each successive power is obtained from the previous one by multiplication by the +exact original value of |x|, then truncating. And +|\coeff{n}|\raisebox{.5ex}{|.|}|x^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 $\sum c_n \frac {x^n}{n!}$ to avoid having to compute the factorial +from scratch at each coefficient, the same way \csa{xintFxPtPowerSeries} +does not compute |x^n| from scratch at each |n|. For the next package +release (perhaps). + + +\def\coeffexp #1{1/\xintFac {#1}[0]}% [0] for faster parsing +\def\x {-1/2[0]}% +\def\ApproxExp #1#2{\xintFxPtPowerSeries {0}{#1}{\coeffexp}{\x}{#2}}% +\newcount\cnta + +\noindent\begin{minipage}{0.3\linewidth} +\centeredline{$e^{-\frac12}\approx{}$}% +\cnta 0 +\loop +$\ApproxExp {\the\cnta}{20}$\\ +\ifnum\cnta<19 +\advance\cnta 1 +\repeat\par +\end{minipage} +\hfil +\begin{minipage}{0.65\linewidth} +\ttfamily\hyphenchar\font-1 +\begin{verbatim} +\def\coeffexp #1{1/\xintFac {#1}[0]}% +\def\x {-1/2[0]}% [0] for faster parsing +\def\ApproxExp #1#2{\xintFxPtPowerSeries + {0}{#1}{\coeffexp}{\x}{#2}}% +\centeredline{$e^{-\frac12}\approx{}$}% +\cnta 0 % previously declared \count register +\loop +$\ApproxExp {\the\cnta}{20}$\\ +% truncates 20 digits after decimal point +\ifnum\cnta<19 +\advance\cnta 1 +\repeat\par +% One should **not** trust the final digits, +% independently of how many terms we compute, +% as errors from the initial terms will never +% disappear! and their cumulative value can +% make the last digit(s) wrong (especially when +% it is a 0 or a 9). We can see it is the case +% here via the computation with more digits: +\end{verbatim} +\end{minipage}%\medskip +\centeredline{|\xintFxPtPowerSeries {0}{30}{\coeffexp}{\x}{25}=|% +\texttt{\hyphenchar\font45 \xintFxPtPowerSeries {0}{30}{\coeffexp}{\x}{25}}} + +\catcode`\& 4 + +\edef\z{\xintIrr {\xintPowerSeries {0}{19}{\coeffexp}{\x}}} + +It is no difficulty for \xintfracname to compute exactly, with the +help of \csa{xintPowerSeries} and \csa{xintIrr}, the nineteenth partial +sum, and to then give the (start of the) exact decimal expansion (again +we see that the last digit from a `truncated' computation was wrong): +\[\halign {\hfil#&$#$&$#$\hfil\cr +|\xintPowerSeries {0}{19}{\coeffexp}{\x}| +&{}={}& \displaystyle\xintFrac{\z}\cr +\vphantom{\vrule height 15pt depth 3pt width 0pt }&{}={}& + \xintTrunc {25}{\z}\dots\cr }\] +Thus, one should always +estimate a priori how many ending digits are not reliable (and secretly +re-do the computation with at least five more digits...). + +\catcode`\& 13 + +\begin{verbatim} +\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% this #1 may be a count register, if desired +% get log(2)=-2log(1-13/256)- 5log(1-1/9) with #1 digits precision +{% + \romannumeral0\expandafter\LogTwoDoIt \expandafter + % Nb Terms for 1/9: + {\the\numexpr (#1+5)*150/143\expandafter}\expandafter + % Nb Terms for 13/256: + {\the\numexpr (#1+5)*100/129\expandafter}\expandafter + {\the\numexpr #1+5\expandafter}\expandafter{\the\numexpr #1\relax }% +}% +\def\LogTwoDoIt #1#2#3#4% +% #1=nb of terms for 1/9, #2=nb of terms for 13/256, +{% #3=nb of digits for computations, #4 for printing + \xinttrunc {#4} + {\xintAdd + {\xintMul {2}{\xintFxPtPowerSeries {1}{#2}{\coefflog}{\xa}{#3}}} + {\xintMul {5}{\xintFxPtPowerSeries {1}{#1}{\coefflog}{\xb}{#3}}}% + }% +}% +\[ \log 2 = \LogTwo {60}\dots\] +\end{verbatim} +\vspace*{-\baselineskip} +\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+5)*150/143\expandafter}\expandafter % Nb Terms for 1/9 + {\the\numexpr (#1+5)*100/129\expandafter}\expandafter % Nb Terms for 13/256 + {\the\numexpr #1+5\expandafter}\expandafter{\the\numexpr #1\relax }% +}% +\def\LogTwoDoIt #1#2#3#4% #1=nb of terms for 1/9, #2=nb of terms for 13/256, +{% #3=nb of digits for computations, #4 for printing + \xinttrunc {#4} + {\xintAdd + {\xintMul {2}{\xintFxPtPowerSeries {1}{#2}{\coefflog}{\xa}{#3}}} + {\xintMul {5}{\xintFxPtPowerSeries {1}{#1}{\coefflog}{\xb}{#3}}}% + }% +}% +\[ \log 2 = \LogTwo {60}\dots\] + +\begin{verbatim} +% pi = 16 Arctg(1/5) - 4 Arctg(1/239) (John Machin's formula) +\def\coeffarctg #1{\romannumeral0\xintmon{#1}/% + \the\numexpr 2*#1+1\relax [0]}% +% the above computes (-1)^n/(2n+1). +% Recall the xint macro \xintMON which does '(-1)^N' but ATTENTION: It +% is MANDATORY that \coeffarctg #1 gives the final numerato in two +% expansion steps (the denominator is then identified as what follows +% after the slash and will be subjected to its own additional two +% expansion steps). If we had written \xintMON {#1} then this would not +% have been the case, because one expansion step is used by the +% expansion of \coeffarctg to its definition. Most of the time not +% respecting these guidelines provokes errors on compilation, but here, +% as I discovered making the mistake myself, if we had written \xintMON +% {#1} the computation would have silently proceeded to a WRONG final +% value! So please follow the package's author instructions. +% Alternative: +% \def\coeffarctg #1{1/\the\numexpr\xintMON{#1}*(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{% \Machin {\mycount} is allowed + \romannumeral0\expandafter\MachinA \expandafter + % number of terms for arctg(1/5): + {\the\numexpr (#1+4)*5/7\expandafter}\expandafter + % number of terms for arctg(1/239): + {\the\numexpr (#1+4)*10/45\expandafter}\expandafter + % do the computations with 4 additional digits: + {\the\numexpr #1+4\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 printing +% #3=#4+4: digits for computing intermediate results +{\xinttrunc {#4} % lowercase! produces the space to stop \romannumeral + {\xintiSub % does the final subtraction exactly with integers + {\xintiTrunc {#3} % produces an integer for \xintiSub + {\xintMul {16/5}{\xintFxPtPowerSeries {0}{#1}{\coeffarctg}{\xa}{#3}}}} + {\xintiTrunc {#3} % above and below the main stuff + {\xintMul{4/239}{\xintFxPtPowerSeries {0}{#2}{\coeffarctg}{\xb}{#3}}}}% + [-#3]}} % this [-n] is ok as it follows an *integer* +\[ \pi = \Machin {60}\dots \] +\end{verbatim} +\vspace*{-\baselineskip} +\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+4)*5/7\expandafter}\expandafter + % number of terms for arctg(1/239): + {\the\numexpr (#1+4)*10/45\expandafter}\expandafter + % do the computations with 4 additional digits: + {\the\numexpr #1+4\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 printing +% #3=#4+4: digits for computing intermediate results +{\xinttrunc {#4} + {\xintiSub % does the final subtraction exactly with integers + {\xintiTrunc {#3} % produces an integer for \xintiSub + {\xintMul {16/5}{\xintFxPtPowerSeries {0}{#1}{\coeffarctg}{\xa}{#3}}}} + {\xintiTrunc {#3} % above and below the main stuff + {\xintMul{4/239}{\xintFxPtPowerSeries {0}{#2}{\coeffarctg}{\xb}{#3}}}}% + [-#3]}} % this [-n] is ok as it follows an *integer* +\[ \pi = \Machin {60}\dots \] + +You want more digits? (and have some time?) Copy this code to a Plain +\TeX{} or \LaTeX{} document loading \xintseriesname, and compile: +\begin{verbatim} +\newwrite\outfile +\immediate\openout\outfile \jobname-out\relax +\immediate\write\outfile {\Machin {1000}} +\immediate\closeout\outfile +\end{verbatim} +This will create a file with the correct first 1000 digits of $\pi$ +after the decimal point. On my laptop this took about 44 seconds last +time I tried (and for 200 digits it is less than 1 second). As mentioned +in the introduction, the file +\href{http://www.ctan.org/pkg/pi}{\color{niceone}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. + + + +\catcode`\& 4 \makeatletter \let\check@percent\original@check@percent \StopEventually{\check@checksum\end{document}\endinput} \makeatother +\newgeometry{hmarginratio=4:3,hscale=0.75} + \def\MacroFont{\ttfamily\small\baselineskip12pt\relax} +\toctransition + \MakePercentIgnore % % \catcode`\<=0 \catcode`\>=11 \catcode`\*=11 \catcode`\/=11 @@ -1402,6 +2348,7 @@ and modify it to what is needed. % sparse. Some comments may be left-overs from previous versions % of the macro, with parameters in another order for example. % +% \localtableofcontents % \subsection{Catcodes, \eTeX{} detection, reload detection} % % The method for package identification and reload detection is @@ -1524,7 +2471,7 @@ and modify it to what is needed. \fi \expandafter\x\csname ver@xint.sty\endcsname \ProvidesPackage{xint}% - [2013/04/05 v1.02 Expandable operations on long numbers (jfB)]% + [2013/04/14 v1.03 Expandable operations on long numbers (jfB)]% % \end{macrocode} % \subsection{Token management macros} % \begin{macrocode} @@ -1537,12 +2484,14 @@ and modify it to what is needed. \def\xint@gobble@six #1#2#3#4#5#6{}% \def\xint@gobble@seven #1#2#3#4#5#6#7{}% \def\xint@gobble@eight #1#2#3#4#5#6#7#8{}% +\def\xint@firstoftwo #1#2{#1}% \def\xint@secondoftwo #1#2{#2}% \def\xint@firstoftwo@andstop #1#2{ #1}% \def\xint@secondoftwo@andstop #1#2{ #2}% \def\xint@exchangetwo@keepbraces #1#2{{#2}{#1}}% \def\xint@exchangetwo@keepbraces@andstop #1#2{ {#2}{#1}}% \def\xint@xpxp@andstop {\expandafter\expandafter\expandafter\space }% +\def\xint@minus@andstop { -}% \def\xint@r #1\R {}% \def\xint@w #1\W {}% \def\xint@z #1\Z {}% @@ -1554,13 +2503,14 @@ and modify it to what is needed. \def\xint@bracedundef {\xint@undef }% \def\xint@UDzerofork #10\dummy #2#3\xint@UDkrof {#2}% \def\xint@UDsignfork #1-\dummy #2#3\xint@UDkrof {#2}% +\def\xint@UDwfork #1\W\dummy #2#3\xint@UDkrof {#2}% \def\xint@UDzerosfork #100\dummy #2#3\xint@UDkrof {#2}% \def\xint@UDonezerofork #110\dummy #2#3\xint@UDkrof {#2}% \def\xint@UDzerominusfork #10-\dummy #2#3\xint@UDkrof {#2}% \def\xint@UDsignsfork #1--\dummy #2#3\xint@UDkrof {#2}% \def\xint@afterfi #1#2\fi {\fi #1}% % \end{macrocode} -% \subsection{\ch{xintRev}, \ch{xintReverseOrder}} +% \subsection{\csh{xintRev}, \csh{xintReverseOrder}} % \begin{verbatim} % \xintRev: fait la double expansion, vérifie le signe % \xintReverseOrder: ne fait PAS la double expansion, ne regarde @@ -1590,11 +2540,7 @@ and modify it to what is needed. }% \def\XINT@rev@negative #1#2\Z {% - \expandafter - \space - \expandafter - -% - \romannumeral0\XINT@rev {#2}% + \expandafter \space \expandafter -\romannumeral0\XINT@rev {#2}% }% \def\XINT@rev@nonnegative #1\Z {% @@ -1622,11 +2568,14 @@ and modify it to what is needed. \def\XINT@strip@undef #1\xint@undef {}% \def\XINT@strip@UNDEF #1\xint@UNDEF {}% % \end{macrocode} -% \subsection{\ch{XINT@RQ}} +% \subsection{\csh{XINT@RQ}} % \begin{verbatim} % 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 % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -1660,7 +2609,7 @@ and modify it to what is needed. \def\XINT@RQ@end@ii #1\Z #2#3#4#5#6#7#8#9\Z { #3#4#5#6#7#8#900}% \def\XINT@RQ@end@i \Z #1#2#3#4#5#6#7#8\Z { #1#2#3#4#5#6#7#80}% % \end{macrocode} -% \subsection{\ch{XINT@cuz}} +% \subsection{\csh{XINT@cuz}} % \begin{macrocode} \def\xint@cleanupzeros@andstop #1#2#3#4% {\expandafter @@ -1723,7 +2672,21 @@ and modify it to what is needed. \def\XINT@cuz@Stop #1\W #2\Z{ #1}% \def\xint@cuz@backtoloop 0\XINT@cuz@Stop 0{\XINT@cuz@loop }% % \end{macrocode} -% \subsection{\ch{xintNum}} +% \subsection{\csh{XINT@isOne}} +% Added in |1.03|. +% \begin{macrocode} +\def\XINT@isOne #1{\romannumeral0\XINT@isone #1\W\Z }% +\def\XINT@isone #1#2% +{% + \xint@one #1\XINT@isone@b 1\expandafter\space\expandafter 0\xint@z #2% +}% +\def\XINT@isone@b #1\xint@z #2% +{% + \xint@w #2\XINT@isone@yes\W\expandafter\space\expandafter 0\xint@z +}% +\def\XINT@isone@yes #1\Z{ 1}% +% \end{macrocode} +% \subsection{\csh{xintNum}} % \begin{verbatim} % For example \xintNum {----+-+++---+----000000000000003} % \end{verbatim} @@ -1764,19 +2727,20 @@ and modify it to what is needed. \def\XINT@num@keepsign@b #1{\XINT@num@loop -}% \def\XINT@num@finish #1\R #2\Z { #1}% % \end{macrocode} -% \subsection{\ch{xintLen}, \ch{xintLength}} +% \subsection{\csh{xintLen}, \csh{xintLength}} % \begin{verbatim} % \xintLen -> fait la double expansion, ne compte PAS le signe % \xintLength -> ne fait PAS la double expansion, compte le signe % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\xintLen {\romannumeral0\xintlen }% -\def\xintlen #1% +\def\xintiLen {\romannumeral0\xintilen }% +\def\xintilen #1% {% \expandafter\expandafter\expandafter \XINT@length@fork #1\R\R\R\R\R\R\R\R\Z }% +\let\xintLen\xintiLen \let\xintlen\xintilen \def\XINT@Len #1{\romannumeral0\XINT@length@fork #1\R\R\R\R\R\R\R\R\Z }% \def\XINT@length@fork #1% {% @@ -1826,7 +2790,7 @@ and modify it to what is needed. \def\XINT@length@end@i #1\XINT@length@end@viii #2% {\expandafter\space\the\numexpr #2-1\relax}% % \end{macrocode} -% \subsection{\ch{xintAssign}, \ch{xintAssignArray}, \ch{xintDigitsOf}} +% \subsection{\csh{xintAssign}, \csh{xintAssignArray}, \csh{xintDigitsOf}} % \begin{verbatim} % \xintAssign {a}{b}..{z}\to\A\B...\Z, % \xintAssignArray {a}{b}..{z}\to\U @@ -1946,14 +2910,15 @@ and modify it to what is needed. }% \let\xintDigitsOf\xintAssignArray % \end{macrocode} -% \subsection{\ch{xintSgn}} +% \subsection{\csh{xintSgn}} % \begin{macrocode} -\def\xintSgn {\romannumeral0\xintsgn }% -\def\xintsgn #1% +\def\xintiSgn {\romannumeral0\xintisgn }% +\def\xintisgn #1% {% \expandafter\expandafter\expandafter \XINT@sgn #1\Z% }% +\let\xintSgn\xintiSgn \let\xintsgn\xintisgn \def\XINT@Sgn #1{\romannumeral0\XINT@sgn #1\Z }% \def\XINT@sgn #1% {% @@ -1966,14 +2931,15 @@ and modify it to what is needed. \xint@z }% % \end{macrocode} -% \subsection{\ch{xintOpp}} +% \subsection{\csh{xintOpp}} % \begin{macrocode} -\def\xintOpp {\romannumeral0\xintopp }% -\def\xintopp #1% +\def\xintiOpp {\romannumeral0\xintiopp }% +\def\xintiopp #1% {% \expandafter\expandafter\expandafter \XINT@opp #1% }% +\let\xintOpp\xintiOpp \let\xintopp\xintiopp \def\XINT@Opp #1{\romannumeral0\XINT@opp #1}% \def\XINT@opp #1% {% @@ -1985,15 +2951,16 @@ and modify it to what is needed. \xint@UDkrof }% % \end{macrocode} -% \subsection{\ch{xintAbs}} +% \subsection{\csh{xintAbs}} % \begin{macrocode} -\def\xintAbs {\romannumeral0\xintabs }% -\def\xintabs #1% +\def\xintiAbs {\romannumeral0\xintiabs }% +\def\xintiabs #1% {% \expandafter\expandafter\expandafter \XINT@abs #1% }% -\def\XINT@Abs {\romannumeral0\XINT@abs }% +\let\xintAbs\xintiAbs \let\xintabs\xintiabs +\def\XINT@Abs #1{\romannumeral0\XINT@abs #1}% \def\XINT@abs #1% {% \xint@UDsignfork @@ -2007,103 +2974,25 @@ and modify it to what is needed. %----------------------------------------------------------------- % ARITHMETIC OPERATIONS: ADDITION, SUBTRACTION, SUMS, % MULTIPLICATION, PRODUCTS, FACTORIAL, POWERS, EUCLIDEAN DIVISION. -% \end{verbatim} -% \vspace*{-2\baselineskip} -% \subsection{\ch{xintAdd}} -% \begin{macrocode} -\def\xintAdd {\romannumeral0\xintadd }% -\def\xintadd #1% -{% - \expandafter\expandafter\expandafter - \xint@add - \expandafter\expandafter\expandafter - {#1}% -}% -\def\xint@add #1#2% -{% - \expandafter\expandafter\expandafter - \XINT@add@fork #2\Z #1\Z -}% -\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} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% 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] -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} -\def\XINT@add@fork #1#2\Z #3#4\Z -{% - \xint@UDzerofork - #1\dummy \XINT@add@secondiszero - #3\dummy \XINT@add@firstiszero - 0\dummy - {\xint@UDsignsfork - #1#3\dummy \XINT@add@minusminus % #1 = #3 = - - #1-\dummy \XINT@add@minusplus % #1 = - - #3-\dummy \XINT@add@plusminus % #3 = - - --\dummy \XINT@add@plusplus - \xint@UDkrof }% - \xint@UDkrof - {#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} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% #1 vient du *deuxième* et #2 vient du *premier* -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} -\def\XINT@add@minusminus #1#2#3#4% -{% - \expandafter\space\expandafter-% - \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\expandafter{% - \romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z - }% -}% -\def\XINT@add@@pre #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} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% ADDITION \XINT@add@A +% +% 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. +% +% ADDITION +% I: \XINT@add@A % INPUT: -% \romannumeral0\XINT@add@A <N1>\W\X\Y\Z <N2>\W\X\Y\Z -% avec: N1 et N2 sur **4n**, et **renversés**, et le plus long ne -% doit pas se terminer par 0000. [Donc on peut avoir 0000 comme -% input si l'autre est >0 et ne se termine pas en 0000 bien sûr]. -% OUTPUT: -% La somme N1+N2, *PAS* sur 4n, dans l'ordre *normal*, et *sans -% leading zeros* -% La procédure est plus rapide lorsque la longueur de N2 est -% supérieure à celle de N1 +% \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 +% [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>, order normal, plus sur 4n, pas de leading zeros +% La procédure est plus rapide lorsque la longueur de <N2> est supérieure à +% celle de <N1> % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -2113,13 +3002,6 @@ and modify it to what is needed. #3\xint@add@az \W\XINT@add@AB #1{#3#4#5#6}{#2}% }% -% \end{macrocode} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% 1er nombre fini. -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} \def\xint@add@az\W\XINT@add@AB #1#2% {% \XINT@add@AC@checkcarry #1% @@ -2128,7 +3010,8 @@ and modify it to what is needed. % \vspace*{-.5\baselineskip} % \begin{verbatim} % ici #2 est prévu pour l'addition, mais attention il devra être renversé pour -% \numexpr. #3 = résultat partiel. #4 = chiffres qui restent +% \numexpr. #3 = résultat partiel. #4 = chiffres qui restent. On vérifie si le +% deuxième nombre s'arrête. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -2200,16 +3083,421 @@ and modify it to what is needed. }% \def\xint@add@cz\W\XINT@add@CD #1#2{ 1#2}% % \end{macrocode} -% \subsection{\ch{xintSub}} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% 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. +% 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* +% \end{verbatim} +% \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\xintSub {\romannumeral0\xintsub }% -\def\xintsub #1% +\def\XINT@addr@A #1#2#3#4#5#6% +{% + \xint@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@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@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@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} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% 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. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@addm@A #1#2#3#4#5#6% +{% + \xint@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\relax.% +}% +\def\XINT@addm@ABEA #1#2#3.#4% +{% + \XINT@addm@A #2{#3#4}% +}% +\def\XINT@addm@AC@checkcarry #1% +{% + \xint@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@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + #1% +}% +\def\XINT@addm@C #1#2#3#4#5% +{% + \xint@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\relax.% +}% +\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\relax.% +}% +\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\relax.% +}% +\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\relax.% +}% +\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}% +\def\XINT@addm@end #1#2#3#4#5% +{\expandafter\space\the\numexpr #1#2#3#4#5\relax +}% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% 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. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@addp@A #1#2#3#4#5#6% +{% + \xint@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@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@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 1+10#1#2#3\relax +}% +\def\XINT@addp@CDw #1#2#3#4#5#6% +{% + \xint@quatrezeros #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 1+100#1#2\relax +}% +\def\XINT@addp@CDx #1#2#3#4#5#6% +{% + \xint@quatrezeros #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 1+1000#1\relax +}% +\def\XINT@addp@CDy #1#2#3#4#5#6% +{% + \xint@quatrezeros #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@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@quatrezeros #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@quatrezeros #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@quatrezeros #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}} +% \begin{macrocode} +\def\xintiAdd {\romannumeral0\xintiadd }% +\def\xintiadd #1% +{% + \expandafter\expandafter\expandafter + \xint@add + \expandafter\expandafter\expandafter + {#1}% +}% +\let\xintAdd\xintiAdd \let\xintadd\xintiadd +\def\xint@add #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@add@fork #2\Z #1\Z +}% +\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} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% 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] +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@add@fork #1#2\Z #3#4\Z +{% + \xint@UDzerofork + #1\dummy \XINT@add@secondiszero + #3\dummy \XINT@add@firstiszero + 0\dummy + {\xint@UDsignsfork + #1#3\dummy \XINT@add@minusminus % #1 = #3 = - + #1-\dummy \XINT@add@minusplus % #1 = - + #3-\dummy \XINT@add@plusminus % #3 = - + --\dummy \XINT@add@plusplus + \xint@UDkrof }% + \xint@UDkrof + {#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} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% #1 vient du *deuxième* et #2 vient du *premier* +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\XINT@add@minusminus #1#2#3#4% +{% + \expandafter\space\expandafter-% + \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\expandafter{% + \romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z + }% +}% +\def\XINT@add@@pre #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}} +% \begin{macrocode} +\def\xintiSub {\romannumeral0\xintisub }% +\def\xintisub #1% {% \expandafter\expandafter\expandafter \xint@sub \expandafter\expandafter\expandafter {#1}% }% +\let\xintSub\xintiSub \let\xintsub\xintisub \def\xint@sub #1#2% {% \expandafter\expandafter\expandafter @@ -2278,7 +3566,7 @@ and modify it to what is needed. % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% \romannumeral0\XINT@subA 1{}<N1>\W\X\Y\Z<N2>\W\X\Y\Z +% \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 @@ -2498,7 +3786,7 @@ and modify it to what is needed. }% \def\XINT@sub@KK@onestep #1#2% {\expandafter - \XINT@sub@backtoKK\the\numexpr 110000-#2+#1-1\relax.% + \XINT@sub@backtoKK\the\numexpr 109999-#2+#1\relax.% }% \def\XINT@sub@backtoKK #1#2#3.#4% {% @@ -2510,16 +3798,17 @@ and modify it to what is needed. 0\XINT@cuz@loop #3\W\W\W\W\W\W\W\Z }% % \end{macrocode} -% \subsection{\ch{xintCmp}} +% \subsection{\csh{xintCmp}} % \begin{macrocode} -\def\xintCmp {\romannumeral0\xintcmp }% -\def\xintcmp #1% +\def\xintiCmp {\romannumeral0\xinticmp }% +\def\xinticmp #1% {% \expandafter\expandafter\expandafter \xint@cmp \expandafter\expandafter\expandafter {#1}% }% +\let\xintCmp\xintiCmp \let\xintcmp\xinticmp \def\xint@cmp #1#2% {% \expandafter\expandafter\expandafter @@ -2652,7 +3941,7 @@ and modify it to what is needed. \def\xint@OneIfPositive@backtomain 0\XINT@OneIfPositive@finish 0% {\XINT@OneIfPositive@main }% % \end{macrocode} -% \subsection{\ch{xintGeq}} +% \subsection{\csh{xintGeq}} % \begin{verbatim} % PLUS GRAND OU ÉGAL % attention compare les **valeurs absolues** @@ -2766,22 +4055,30 @@ and modify it to what is needed. \xint@UDkrof }% % \end{macrocode} -% \subsection{\ch{xintMax}} +% \subsection{\csh{xintMax}} +% \begin{verbatim} +% 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. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\xintMax {\romannumeral0\xintmax }% -\def\xintmax #1% +\def\xintiMax {\romannumeral0\xintimax }% +\def\xintimax #1% {% \expandafter\expandafter\expandafter \xint@max \expandafter\expandafter\expandafter {#1}% }% +\let\xintMax\xintiMax \let\xintmax\xintimax \def\xint@max #1#2% {% \expandafter\expandafter\expandafter - \XINT@max@fork #2\Z #1\Z + \XINT@max@pre + \expandafter\expandafter\expandafter {#2}{#1}% }% -\def\XINT@Max #1#2{\romannumeral0\XINT@max@fork #2\Z #1\Z }% +\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} % \vspace*{-.5\baselineskip} % \begin{verbatim} @@ -2812,17 +4109,17 @@ and modify it to what is needed. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\XINT@max@zerozero #1#2#3#4{ 0}% -\def\XINT@max@zeroplus #1#2#3#4{ #4#2}% -\def\XINT@max@pluszero #1#2#3#4{ #3#1}% -\def\XINT@max@minusplus #1#2#3#4{ #4#2}% -\def\XINT@max@plusminus #1#2#3#4{ #3#1}% -\def\XINT@max@plusplus #1#2#3#4% +\def\XINT@max@zerozero #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@max@zeroplus #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@max@pluszero #1#2#3#4{\xint@secondoftwo@andstop }% +\def\XINT@max@minusplus #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@max@plusminus #1#2#3#4{\xint@secondoftwo@andstop }% +\def\XINT@max@plusplus #1#2#3#4% {% \ifodd\XINT@Geq {#4#2}{#3#1} - \xint@afterfi { #4#2}% + \expandafter\xint@firstoftwo@andstop \else - \xint@afterfi { #3#1}% + \expandafter\xint@secondoftwo@andstop \fi }% % \end{macrocode} @@ -2835,28 +4132,31 @@ and modify it to what is needed. \def\XINT@max@minusminus #1#2#3#4% {% \ifodd\XINT@Geq {#1}{#2} - \xint@afterfi { -#2}% + \expandafter\xint@firstoftwo@andstop \else - \xint@afterfi { -#1}% + \expandafter\xint@secondoftwo@andstop \fi }% % \end{macrocode} -% \subsection{\ch{xintMin}} +% \subsection{\csh{xintMin}} % \begin{macrocode} -\def\xintMin {\romannumeral0\xintmin }% -\def\xintmin #1% +\def\xintiMin {\romannumeral0\xintimin }% +\def\xintimin #1% {% \expandafter\expandafter\expandafter \xint@min \expandafter\expandafter\expandafter {#1}% }% +\let\xintMin\xintiMin \let\xintmin\xintimin \def\xint@min #1#2% {% \expandafter\expandafter\expandafter - \XINT@min@fork #2\Z #1\Z + \XINT@min@pre + \expandafter\expandafter\expandafter {#2}{#1}% }% -\def\XINT@Min #1#2{\romannumeral0\XINT@min@fork #2\Z #1\Z }% +\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} % \vspace*{-.5\baselineskip} % \begin{verbatim} @@ -2887,17 +4187,17 @@ and modify it to what is needed. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\XINT@min@zerozero #1#2#3#4{ 0}% -\def\XINT@min@zeroplus #1#2#3#4{ 0}% -\def\XINT@min@pluszero #1#2#3#4{ 0}% -\def\XINT@min@minusplus #1#2#3#4{ #3#1}% -\def\XINT@min@plusminus #1#2#3#4{ #4#2}% +\def\XINT@min@zerozero #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@min@zeroplus #1#2#3#4{\xint@secondoftwo@andstop }% +\def\XINT@min@pluszero #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@min@minusplus #1#2#3#4{\xint@secondoftwo@andstop }% +\def\XINT@min@plusminus #1#2#3#4{\xint@firstoftwo@andstop }% \def\XINT@min@plusplus #1#2#3#4% {% \ifodd\XINT@Geq {#4#2}{#3#1} - \xint@afterfi { #3#1}% + \expandafter\xint@secondoftwo@andstop \else - \xint@afterfi { #4#2}% + \expandafter\xint@firstoftwo@andstop \fi }% % \end{macrocode} @@ -2910,161 +4210,82 @@ and modify it to what is needed. \def\XINT@min@minusminus #1#2#3#4% {% \ifodd\XINT@Geq {#1}{#2} - \xint@afterfi { -#1}% + \expandafter\xint@secondoftwo@andstop \else - \xint@afterfi { -#2}% + \expandafter\xint@firstoftwo@andstop \fi }% % \end{macrocode} -% \subsection{\ch{xintSum}, \ch{xintSumExpr}} +% \subsection{\csh{xintSum}, \csh{xintSumExpr}} % \begin{verbatim} % \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). % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\XINT@psum #1% -{% - \romannumeral0\XINT@psum@checkifemptysum #1\Z -}% -\def\XINT@psum@checkifemptysum #1% -{% - \xint@relax #1\XINT@psum@returnzero\relax \XINT@psum@RQfirst #1% -}% -\def\XINT@psum@returnzero #1\Z { 0}% -\def\XINT@psum@RQfirst #1\Z +\def\xintiSum {\romannumeral0\xintisum }% +\def\xintisum #1{\xintisumexpr #1\relax }% +\def\xintiSumExpr {\romannumeral0\xintisumexpr }% +\def\xintisumexpr {% - \expandafter\XINT@psum@loop\expandafter - {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z}% avant: #1\Z + \expandafter\expandafter\expandafter\XINT@sumexpr }% -\def\XINT@psum@loop #1#2% -{% - \xint@relax #2\XINT@psum@end\relax - \expandafter - \XINT@psum@loop\expandafter - {\romannumeral0\expandafter\XINT@sum@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 }% -}% -\def\XINT@psum@end\relax\expandafter - \XINT@psum@loop\expandafter #1% - {\XINT@psum@end@ #1}% -\def\XINT@psum@end@ #1\W\X\Y\Z #2\W\X\Y\Z -{% - \expandafter - \xint@cleanupzeros@andstop\romannumeral0\XINT@rev {#2}% -}% -\def\xintSumExpr {\romannumeral0\xintsumexpr }% -\def\xintSum {\romannumeral0\xintsum }% -\def\xintsum #1% +\let\xintSum\xintiSum \let\xintsum\xintisum +\let\xintSumExpr\xintiSumExpr \let\xintsumexpr\xintisumexpr +\def\XINT@sumexpr {\XINT@sum@loop {0000}{0000}}% +\def\XINT@sum@loop #1#2#3% {% + \xint@relax #3\XINT@sum@finished\relax \expandafter\expandafter\expandafter - \xintsumexpr #1\relax -}% -\def\xintsumexpr #1% -{% - \expandafter\expandafter\expandafter - \XINT@sum@checkifempty #1\Z {\XINT@psum }{\XINT@psum }% -}% -\def\XINT@sum@checkifempty #1% -{% - \xint@relax #1\XINT@sum@returnzero\relax - \XINT@sum@checksign #1% + \XINT@sum@checksign #3\Z {#1}{#2}% }% -\def\XINT@sum@returnzero #1\Z #2#3{ 0}% \def\XINT@sum@checksign #1% {% \xint@zero #1\XINT@sum@skipzeroinput0% \xint@UDsignfork - #1\dummy \XINT@sum@pushneg - -\dummy \XINT@sum@pushpos + #1\dummy \XINT@sum@N + -\dummy {\XINT@sum@P #1}% \xint@UDkrof - #1% -}% -\def\XINT@sum@skipzeroinput #1\xint@UDkrof #2\Z #3#4% -{% - \XINT@sum@xpxpnext {#3}{#4}% -}% -\def\XINT@sum@pushpos #1#2\Z #3#4% -{% - \XINT@sum@xpxpnext {#3{#1#2}}{#4}% -}% -\def\XINT@sum@pushneg #1#2\Z #3#4% -{% - \XINT@sum@xpxpnext {#3}{#4{#2}}% -}% -\def\XINT@sum@xpxpnext #1#2#3% -{% - \expandafter\expandafter\expandafter - \XINT@sum@checkiffinished #3\Z {#1}{#2}% -}% -\def\XINT@sum@checkiffinished #1% -{% - \xint@relax #1\XINT@sum@end\relax - \XINT@sum@checksign #1% -}% -\def\XINT@sum@end\relax\XINT@sum@checksign\relax #1\Z #2#3% - {\xintsub{#2\relax}{#3\relax}}% -\def\XINT@sum@A #1#2#3#4#5#6% -{% - \xint@w - #3\xint@sum@az - \W\XINT@sum@B #1{#3#4#5#6}{#2}% -}% -\def\xint@sum@az\W\XINT@sum@B #1#2% -{% - \XINT@sum@AC@checkcarry #1% -}% -\def\XINT@sum@B #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint@w - #5\xint@sum@bz - \W\XINT@sum@E #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z }% -\def\XINT@sum@E #1#2#3#4#5#6% -{\expandafter - \XINT@sum@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax -}% -\def\XINT@sum@ABEA #1#2#3#4#5#6#7% +\def\XINT@sum@skipzeroinput #1\xint@UDkrof #2\Z {\XINT@sum@loop }% +\def\XINT@sum@P #1\Z #2% {% - \XINT@sum@A #2{#7#6#5#4#3}% -}% -\def\xint@sum@bz\W\XINT@sum@E #1#2#3#4#5#6% -{\expandafter - \XINT@sum@CC\the\numexpr #1+10#5#4#3#2\relax + \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@CC #1#2#3#4#5#6#7% +\def\XINT@sum@N #1\Z #2#3% {% - \XINT@sum@AC@checkcarry #2{#7#6#5#4#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@AC@checkcarry #1% +\def\XINT@sum@NN #1#2{\XINT@sum@loop {#2}{#1}}% +\def\XINT@sum@finished #1\Z #2#3% {% - \xint@zero #1\xint@sum@AC@nocarry 0\XINT@sum@C -}% -\def\xint@sum@AC@nocarry 0\XINT@sum@C #1#2\W\X\Y\Z { #1#2}% -\def\XINT@sum@C #1#2#3#4#5% -{% - \xint@w - #2\xint@sum@cz - \W\XINT@sum@D {#5#4#3#2}{#1}% + \XINT@sub@A 1{}#3\W\X\Y\Z #2\W\X\Y\Z }% -\def\XINT@sum@D #1% -{\expandafter - \XINT@sum@CC\the\numexpr 1+10#1\relax -}% -\def\xint@sum@cz\W\XINT@sum@D #1#2{ #21000}% % \end{macrocode} -% \subsection{\ch{xintMul}} +% \subsection{\csh{xintMul}} % \begin{macrocode} -\def\xintMul {\romannumeral0\xintmul }% -\def\xintmul #1% +\def\xintiMul {\romannumeral0\xintimul }% +\def\xintimul #1% {% \expandafter\expandafter\expandafter \xint@mul \expandafter\expandafter\expandafter {#1}% }% +\let\xintMul\xintiMul \let\xintmul\xintimul \def\xint@mul #1#2% {\expandafter\expandafter\expandafter \XINT@mul@fork #2\Z #1\Z @@ -3074,21 +4295,16 @@ and modify it to what is needed. % \vspace*{-.5\baselineskip} % \begin{verbatim} % MULTIPLICATION -% Ici #1#2 = 2e input et #3#4 = 1er input -% Quelques précisions apportées à l'occasion de 1.02 (qui modifie le -% fonctionnement de \xintPrd pour tenir compte des règles ci-dessous, et aussi -% améliore au passage très légèrement la vitesse de calcul de la factorielle): -% La multiplication est plus rapide sous les conditions suivantes: -% - si le nombre le plus court a au plus 4 chiffres, il DOIT être en 2e -% - si les deux ont au plus 50 chiffres, le plus court en *second* -% - si les deux ont au moins 50 chiffres, le plus court en *premier* -% - si le plus long a au moins 250 chiffres, mettre ce plus long -% toujours en *second* (sauf si l'autre a au plus 4 chiffres) -% - si le plus long a au moins 100 chiffres, en second si le premier a au -% moins 12 chiffres. -% La règle générale est donc à peu près: pour les 'gros calculs' mettre le -% plus court en premier (SAUF pour multiplication par un nombre < 10000), et -% pour les 'petits calculs' le plus long en premier. +% 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. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -3100,159 +4316,147 @@ and modify it to what is needed. 0\dummy {\xint@UDsignsfork #1#3\dummy \XINT@mul@minusminus % #1 = #3 = - - #1-\dummy \XINT@mul@minusplus % #1 = - - #3-\dummy \XINT@mul@plusminus % #3 = - - --\dummy \XINT@mul@plusplus + #1-\dummy {\XINT@mul@minusplus #3}% % #1 = - + #3-\dummy {\XINT@mul@plusminus #1}% % #3 = - + --\dummy {\XINT@mul@plusplus #1#3}% \xint@UDkrof }% \xint@UDkrof - {#2}{#4}#1#3% + {#2}{#4}% }% -\def\XINT@mul@zero #1#2#3#4{ 0}% -% \end{macrocode} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% Dans ce qui suit #3#1 vient du #1#2 initial correspondant au -% ** 2e ** input. -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} -\def\XINT@mul@minusminus #1#2#3#4% +\def\XINT@mul@zero #1#2{ 0}% +\def\XINT@mul@minusminus #1#2% {% - \expandafter - \XINT@mul@enter\romannumeral0% - \XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z + \expandafter\XINT@mul@choice@a + \expandafter{\romannumeral0\XINT@length {#2}}% + {\romannumeral0\XINT@length {#1}}{#1}{#2}% }% -\def\XINT@mul@minusplus #1#2#3#4% +\def\XINT@mul@minusplus #1#2#3% {% - \expandafter\space\expandafter-% - \romannumeral0\expandafter - \XINT@mul@enter\romannumeral0% - \XINT@RQ {}#4#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z + \expandafter\space\expandafter-\romannumeral0\expandafter + \XINT@mul@choice@a + \expandafter{\romannumeral0\XINT@length {#1#3}}% + {\romannumeral0\XINT@length {#2}}{#2}{#1#3}% }% -\def\XINT@mul@plusminus #1#2#3#4% +\def\XINT@mul@plusminus #1#2#3% {% - \expandafter\space\expandafter-% - \romannumeral0\expandafter - \XINT@mul@enter\romannumeral0% - \XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #3#1\W\X\Y\Z + \expandafter\space\expandafter-\romannumeral0\expandafter + \XINT@mul@choice@a + \expandafter{\romannumeral0\XINT@length {#3}}% + {\romannumeral0\XINT@length {#1#2}}{#1#2}{#3}% }% -% \end{macrocode} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% Ici #3#1 correspond au **2e input** -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} \def\XINT@mul@plusplus #1#2#3#4% {% - \expandafter - \XINT@mul@enter\romannumeral0% - \XINT@RQ {}#4#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #3#1\W\X\Y\Z -}% -\def\XINT@mul@add@A #1#2#3#4#5#6% -{% - \xint@w - #3\xint@mul@add@az - \W\XINT@mul@add@AB #1{#3#4#5#6}{#2}% -}% -\def\xint@mul@add@az\W\XINT@mul@add@AB #1#2% -{% - \XINT@mul@add@AC@checkcarry #1% -}% -\def\XINT@mul@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \XINT@mul@add@ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT@mul@add@ABE #1#2#3#4#5#6% -{\expandafter - \XINT@mul@add@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax.% + \expandafter\XINT@mul@choice@a + \expandafter{\romannumeral0\XINT@length {#2#4}}% + {\romannumeral0\XINT@length {#1#3}}{#1#3}{#2#4}% }% -\def\XINT@mul@add@ABEA #1#2#3.#4% +\def\XINT@mul@choice@a #1#2% {% - \XINT@mul@add@A #2{#3#4}% + \expandafter\XINT@mul@choice@b \expandafter{#2}{#1}% }% -\def\XINT@mul@add@AC@checkcarry #1% +\def\XINT@mul@choice@b #1#2% {% - \xint@zero #1\xint@mul@add@AC@nocarry 0\XINT@mul@add@C + \ifnum #1<5 + \expandafter\XINT@mul@choice@littlebyfirst + \else + \ifnum #2<5 + \expandafter\expandafter\expandafter + \XINT@mul@choice@littlebysecond + \else + \expandafter\expandafter\expandafter + \XINT@mul@choice@compare + \fi + \fi + {#1}{#2}% }% -\def\xint@mul@add@AC@nocarry 0\XINT@mul@add@C #1#2\W\X\Y\Z +\def\XINT@mul@choice@littlebyfirst #1#2#3#4% {% - \expandafter - \xint@cleanupzeros@andstop - \romannumeral0% - \XINT@rord@main {}#2% - \xint@UNDEF - \xint@undef\xint@undef\xint@undef\xint@undef - \xint@undef\xint@undef\xint@undef\xint@undef - \xint@UNDEF - #1% + \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@add@C #1#2#3#4#5% +\def\XINT@mul@choice@littlebysecond #1#2#3#4% {% - \xint@w - #5\xint@mul@add@cw - #4\xint@mul@add@cx - #3\xint@mul@add@cy - #2\xint@mul@add@cz - \W\XINT@mul@add@CD {#5#4#3#2}{#1}% + \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@add@CD #1% -{\expandafter - \XINT@mul@add@CC\the\numexpr 1+10#1\relax.% -}% -\def\XINT@mul@add@CC #1#2#3.#4% +\def\XINT@mul@choice@compare #1#2% {% - \XINT@mul@add@AC@checkcarry #2{#3#4}% -}% -\def\xint@mul@add@cw - #1\xint@mul@add@cx - #2\xint@mul@add@cy - #3\xint@mul@add@cz - \W\XINT@mul@add@CD -{\expandafter - \XINT@mul@add@CDw\the\numexpr 1+#1#2#3\relax.% -}% -\def\XINT@mul@add@CDw #1.#2#3\X\Y\Z -{% - \XINT@mul@add@end #1#3% -}% -\def\xint@mul@add@cx - #1\xint@mul@add@cy - #2\xint@mul@add@cz - \W\XINT@mul@add@CD -{\expandafter - \XINT@mul@add@CDx\the\numexpr 1+#1#2\relax.% + \ifnum #1>#2 + \expandafter \XINT@mul@choice@i + \else + \expandafter \XINT@mul@choice@ii + \fi + {#1}{#2}% }% -\def\XINT@mul@add@CDx #1.#2#3\Y\Z +\def\XINT@mul@choice@i #1#2% +{% + \ifcase \numexpr (#2-3)/4\relax + \or \xint@afterfi {\ifnum #1<330 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<168 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<109 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<80 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<66 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<52 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \else \expandafter \XINT@mul@choice@permute + \fi +}% +\def\XINT@mul@choice@ii #1#2% +{% + \ifcase \numexpr (#1-3)/4\relax + \or \xint@afterfi {\ifnum #2<330 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<168 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<109 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<80 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<66 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<52 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \else \expandafter \XINT@mul@choice@same + \fi +}% +\def\XINT@mul@choice@same #1#2% {% - \XINT@mul@add@end #1#3% -}% -\def\xint@mul@add@cy - #1\xint@mul@add@cz - \W\XINT@mul@add@CD -{\expandafter - \XINT@mul@add@CDy\the\numexpr 1+#1\relax.% + \expandafter + \XINT@mul@enter\romannumeral0% + \XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #2\W\X\Y\Z }% -\def\XINT@mul@add@CDy #1.#2#3\Z +\def\XINT@mul@choice@permute #1#2% {% - \XINT@mul@add@end #1#3% -}% -\def\xint@mul@add@cz\W\XINT@mul@add@CD #1#2#3{\XINT@mul@add@end #1#3}% -\def\XINT@mul@add@end #1#2#3#4#5% -{\expandafter\space - \the\numexpr #1#2#3#4#5\relax + \expandafter + \XINT@mul@enter\romannumeral0% + \XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1\W\X\Y\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% 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. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} \def\XINT@mul@Ar #1#2#3#4#5#6% {% \xint@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@sum@AC@checkcarry #1% + \XINT@addr@AC@checkcarry #1% }% \def\XINT@mul@Br #1#2#3#4\W\X\Y\Z #5#6#7#8% {\expandafter @@ -3265,7 +4469,11 @@ and modify it to what is needed. % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% Mr renvoie le résultat ***à l'envers***, sur ***4n chiffres*** +% << 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. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -3307,6 +4515,17 @@ and modify it to what is needed. }% \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} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% << 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*. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} \def\XINT@mul@M #1% {\expandafter \XINT@mul@M@checkifzeroorone @@ -3324,7 +4543,8 @@ and modify it to what is needed. {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 {% +\def\XINT@mul@M@one #1#2#3#4\Z\Z\Z\Z +{% \expandafter \xint@cleanupzeros@andstop \romannumeral0\XINT@rev{#4}% @@ -3357,10 +4577,9 @@ and modify it to what is needed. % 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>\W\X\Y\Z <N2>\W\X\Y\Z -% avec N1: *renversé*, *longueur 4n* (zéros éventuellement ajoutés +% 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, -% et N2 est *non nul*. +% et <N2> dans l'ordre *normal*, et pas forcément longueur 4n. % pas de signes % \end{verbatim} % \vspace*{-1.5\baselineskip} @@ -3414,9 +4633,20 @@ and modify it to what is needed. \XINT@mul@main \expandafter {\romannumeral0\expandafter \XINT@mul@Ar \expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z \W\X\Y\Z 0000#1\W\X\Y\Z - }#3\W\X\Y\Z + \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z + \W\X\Y\Z 0000#1\W\X\Y\Z }#3\W\X\Y\Z }% +% \end{macrocode} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% 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. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} \def\xint@mul@mainw #1\xint@mul@mainx #2\xint@mul@mainy @@ -3424,7 +4654,7 @@ and modify it to what is needed. \W\XINT@mul@compute #4#5#6\W\X\Y\Z \X\Y\Z {% \expandafter - \XINT@mul@add@A \expandafter0\expandafter{\expandafter}% + \XINT@addm@A \expandafter0\expandafter{\expandafter}% \romannumeral0% \XINT@mul@Mr {#3#2#1}#6\Z\Z\Z\Z \W\X\Y\Z 000#4\W\X\Y\Z @@ -3435,7 +4665,7 @@ and modify it to what is needed. \W\XINT@mul@compute #3#4#5\W\X\Y\Z \Y\Z {% \expandafter - \XINT@mul@add@A \expandafter0\expandafter{\expandafter}% + \XINT@addm@A \expandafter0\expandafter{\expandafter}% \romannumeral0% \XINT@mul@Mr {#2#1}#5\Z\Z\Z\Z \W\X\Y\Z 00#3\W\X\Y\Z @@ -3445,7 +4675,7 @@ and modify it to what is needed. \W\XINT@mul@compute #2#3#4\W\X\Y\Z \Z {% \expandafter - \XINT@mul@add@A \expandafter0\expandafter{\expandafter}% + \XINT@addm@A \expandafter0\expandafter{\expandafter}% \romannumeral0% \XINT@mul@Mr {#1}#4\Z\Z\Z\Z \W\X\Y\Z 0#2\W\X\Y\Z @@ -3456,16 +4686,112 @@ and modify it to what is needed. \xint@cleanupzeros@andstop\romannumeral0\XINT@rev{#1}% }% % \end{macrocode} -% \subsection{\ch{xintSqr}} +% \vspace*{-.5\baselineskip} +% \begin{verbatim} +% Variante de la Multiplication +% \romannumeral0\XINT@mulr@enter <N1>\W\X\Y\Z <N2>\W\X\Y\Z +% 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 et aussi par la division. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\xintSqr {\romannumeral0\xintsqr }% -\def\xintsqr #1% +\def\XINT@mulr@enter #1\W\X\Y\Z #2#3#4#5% +{% + \xint@w + #5\xint@mulr@enterw + #4\xint@mulr@enterx + #3\xint@mulr@entery + #2\xint@mulr@enterz + \W\XINT@mulr@start {#2#3#4#5}#1\W\X\Y\Z +}% +\def\xint@mulr@enterw + #1\xint@mulr@enterx + #2\xint@mulr@entery + #3\xint@mulr@enterz + \W\XINT@mulr@start #4#5\W\X\Y\Z \X\Y\Z +{% + \XINT@mul@Mr {#3#2#1}#5\Z\Z\Z\Z +}% +\def\xint@mulr@enterx + #1\xint@mulr@entery + #2\xint@mulr@enterz + \W\XINT@mulr@start #3#4\W\X\Y\Z \Y\Z +{% + \XINT@mul@Mr {#2#1}#4\Z\Z\Z\Z +}% +\def\xint@mulr@entery + #1\xint@mulr@enterz + \W\XINT@mulr@start #2#3\W\X\Y\Z \Z +{% + \XINT@mul@Mr {#1}#3\Z\Z\Z\Z +}% +\def\XINT@mulr@start #1#2\W\X\Y\Z +{\expandafter + \XINT@mulr@main \expandafter + {\romannumeral0\XINT@mul@Mr {#1}#2\Z\Z\Z\Z }#2\W\X\Y\Z +}% +\def\XINT@mulr@main #1#2\W\X\Y\Z #3#4#5#6% +{% + \xint@w + #6\xint@mulr@mainw + #5\xint@mulr@mainx + #4\xint@mulr@mainy + #3\xint@mulr@mainz + \W\XINT@mulr@compute {#1}{#3#4#5#6}#2\W\X\Y\Z +}% +\def\XINT@mulr@compute #1#2#3\W\X\Y\Z +{\expandafter + \XINT@mulr@main \expandafter + {\romannumeral0\expandafter + \XINT@mul@Ar \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z \W\X\Y\Z 0000#1\W\X\Y\Z + }#3\W\X\Y\Z +}% +\def\xint@mulr@mainw + #1\xint@mulr@mainx + #2\xint@mulr@mainy + #3\xint@mulr@mainz + \W\XINT@mulr@compute #4#5#6\W\X\Y\Z \X\Y\Z +{\expandafter + \XINT@addp@A + \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@mul@Mr {#3#2#1}#6\Z\Z\Z\Z + \W\X\Y\Z 000#4\W\X\Y\Z +}% +\def\xint@mulr@mainx + #1\xint@mulr@mainy + #2\xint@mulr@mainz + \W\XINT@mulr@compute #3#4#5\W\X\Y\Z \Y\Z +{\expandafter + \XINT@addp@A + \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@mul@Mr {#2#1}#5\Z\Z\Z\Z + \W\X\Y\Z 00#3\W\X\Y\Z +}% +\def\xint@mulr@mainy + #1\xint@mulr@mainz + \W\XINT@mulr@compute #2#3#4\W\X\Y\Z \Z +{\expandafter + \XINT@addp@A + \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@mul@Mr {#1}#4\Z\Z\Z\Z + \W\X\Y\Z 0#2\W\X\Y\Z +}% +\def\xint@mulr@mainz\W\XINT@mulr@compute #1#2#3\W\X\Y\Z { #1}% +% \end{macrocode} +% \subsection{\csh{xintSqr}} +% \begin{macrocode} +\def\xintiSqr {\romannumeral0\xintisqr }% +\def\xintisqr #1% {% \expandafter\expandafter\expandafter \XINT@sqr \expandafter\expandafter\expandafter - {\xintAbs{#1}}% fait l'expansion de #1 et se d\'ebarrasse du signe + {\xintiAbs{#1}}% fait l'expansion de #1 et se d\'ebarrasse du signe }% +\let\xintSqr\xintiSqr \let\xintsqr\xintisqr \def\XINT@sqr #1% {\expandafter \XINT@mul@enter @@ -3474,165 +4800,57 @@ and modify it to what is needed. \W\X\Y\Z #1\W\X\Y\Z }% % \end{macrocode} -% \subsection{\ch{xintPrd}, \ch{xintProductExpr}} +% \subsection{\csh{xintPrd}, \csh{xintProductExpr}} % \begin{verbatim} % \xintPrd {{a}...{z}} % \xintProductExpr {a}...{z}\relax -% Release 1.02 modifies \XINT@posprod. The new version takes into account that -% multiplication on long numbers is more efficient with the shorter one first; -% the earlier version was on the premise that it was more efficient to give -% the longer number first and used a special version of multiplication to -% produce its output in reversed order to serve as next first number. This was -% optimal for numbers of at most 50 digits, but the bad choice for long -% numbers. As \xintPrd should be fast when it is used to produce long numbers -% (producing short numbers means few factors means computation does not take -% much time anyhow), I revert this choice here. On the other hand this was the -% correct choice (and tested so) for use in the \xintPow recursion which each -% times multiplies by something even bigger than what has been obtained so -% far. So the original version is renamed and moved to serve for \xintPow. -% The factorial receives separate special revision. +% 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 \xintProductExpr ...\relax are +% related. Now \xintProductExpr \z \relax is accepted input when \z expands +% to a list of braced terms (prior only \xintPrd {\z} or \xintPrd \z was +% possible). % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\XINT@posprod #1% -{% - \XINT@pprod@checkifempty #1\Z -}% -\def\XINT@pprod@checkifempty #1% -{% - \xint@relax #1\XINT@pprod@emptyproduct\relax - \XINT@pprod@first #1% -}% -\def\XINT@pprod@emptyproduct #1\Z { 1}% -\def\XINT@pprod@first #1\Z -{% - \XINT@pprod@getnext {#1}% -}% -\def\XINT@pprod@getnext #1#2% -{% - \XINT@pprod@checkiffinished #2\Z {#1}% -}% -\def\XINT@pprod@checkiffinished #1% -{% - \xint@relax #1\XINT@pprod@end\relax - \XINT@pprod@RQnew #1% -}% -\def\XINT@pprod@RQnew #1\Z -{% - \expandafter\XINT@pprod@compute - \expandafter - {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z }% -}% -\def\XINT@pprod@compute #1#2% +\def\xintiPrd {\romannumeral0\xintiprd }% +\def\xintiprd #1{\xintiproductexpr #1\relax }% +\let\xintPrd\xintiPrd +\let\xintprd\xintiprd +\def\xintiProductExpr {\romannumeral0\xintiproductexpr }% +\def\xintiproductexpr {% - \expandafter - \XINT@pprod@getnext - \expandafter - {\romannumeral0\XINT@mul@enter #1\W\X\Y\Z #2\W\X\Y\Z }% -}% -\def\XINT@pprod@end\relax\XINT@pprod@RQnew #1\Z #2{ #2}% -\def\xintProductExpr {\romannumeral0\xintproductexpr }% -\def\xintPrd {\romannumeral0\xintprd }% -\def\xintprd #1% -{% - \expandafter\expandafter\expandafter - \xintproductexpr #1\relax -}% -\def\xintproductexpr #1% -{% - \expandafter\expandafter\expandafter - \XINT@prod@checkifempty #1\Z -}% -\def\XINT@prod@checkifempty #1% -{% - \xint@relax #1\XINT@prod@emptyproduct\relax - \XINT@prod@checkfirstsign #1% -}% -\def\XINT@prod@emptyproduct #1\Z { 1}% -\def\XINT@prod@checkfirstsign #1% -{% - \xint@zero #1\XINT@prod@returnzero0% - \xint@UDsignfork - #1\dummy \XINT@prod@firstisneg - -\dummy \XINT@prod@firstispos - \xint@UDkrof - #1% -}% -\def\XINT@prod@returnzero #1\relax { 0}% -\def\XINT@prod@firstisneg #1#2\Z -{% - \XINT@prod@xpxpnext 0{#2}% -}% -\def\XINT@prod@firstispos #1\Z -{% - \XINT@prod@xpxpnext 1{#1}% + \expandafter\expandafter\expandafter\XINT@productexpr }% -\def\XINT@prod@xpxpnext #1#2#3% +\let\xintProductExpr\xintiProductExpr +\let\xintproductexpr\xintiproductexpr +\def\XINT@productexpr {\XINT@prod@loop {1}}% +\def\XINT@prod@loop #1#2% {% - \expandafter\expandafter\expandafter - \XINT@prod@checkiffinished #3\Z {#2}#1% + \xint@relax #2\XINT@prod@finished\relax + \expandafter\XINT@prod@loop\expandafter + {\romannumeral0\xintimul {#2}{#1}}% }% -\def\XINT@prod@checkiffinished #1% +\def\XINT@prod@finished #1#2#3#4#5% {% - \xint@relax #1\XINT@prod@end\relax - \XINT@prod@checksign #1% -}% -\def\XINT@prod@checksign #1% -{% - \xint@zero #1\XINT@prod@returnzero0% - \xint@UDsignfork - #1\dummy \XINT@prod@neg@RQnew - -\dummy \XINT@prod@pos@RQnew - \xint@UDkrof - #1% -}% -\def\XINT@prod@pos@RQnew #1\Z -{% - \expandafter - \XINT@prod@pos - \expandafter - {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z }% -}% -\def\XINT@prod@pos #1#2#3% -{% - \expandafter - \XINT@prod@xpxpnext - \expandafter - #3% - \expandafter - {\romannumeral0\XINT@mul@enter #1\W\X\Y\Z #2\W\X\Y\Z }% -}% -\def\XINT@prod@neg@RQnew #1#2\Z -{% - \expandafter - \XINT@prod@neg - \expandafter - {\romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z }% -}% -\def\XINT@prod@neg #1#2#3% -{% - \expandafter - \XINT@prod@xpxpnext - \expandafter - {\the\numexpr 1-#3\expandafter}% - \expandafter - {\romannumeral0\XINT@mul@enter #1\W\X\Y\Z #2\W\X\Y\Z }% -}% -\def\XINT@prod@end\relax\XINT@prod@checksign #1\Z #2#3% -{% - \xint@prod@cleanupzeros #3#2% -}% -\def\xint@prod@cleanupzeros #1#2#3#4#5% -{% - \expandafter\space\the\numexpr (2*#1-1)*#2#3#4#5\relax + \XINT@prod@finished@ #5% }% +\def\XINT@prod@finished@ #1#2#3#4#5{ #5}% % \end{macrocode} -% \subsection{\ch{xintFac}} +% \subsection{\csh{xintFac}} % \begin{verbatim} -% Modified a bit with 1.02, following changes to \XINT@posprod -% I am tempted, here and elsewhere, to use \ifcase\XINT@Geq {#1}{1000000000} -% rather than \ifnum\XINT@Length {#1}>9 but for the time being I leave things -% as they stand. +% 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\XINT@Length {#1}>9 but for the time being I leave things as they stand. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -3647,7 +4865,7 @@ and modify it to what is needed. \def\XINT@Fac {\romannumeral0\XINT@fac@fork }% \def\XINT@fac@fork #1% {% - \ifcase\xintSgn {#1} + \ifcase\xintiSgn {#1} \xint@afterfi{\expandafter\space\expandafter 1\xint@gobble }% \or \expandafter\XINT@fac@checklength @@ -3686,74 +4904,60 @@ and modify it to what is needed. }% \def\XINT@fac@big@docomputation #1#2% {% - \expandafter - \XINT@pprod@getnext - \expandafter + \expandafter \XINT@fac@bigcompute@loop \expandafter {\romannumeral0\XINT@fac@loop {9999}}#2\relax }% -\def\XINT@fac@loop #1{\XINT@fac@loop@main 1{#1}{}}% -\def\XINT@fac@loop@main #1#2#3% +\def\XINT@fac@bigcompute@loop #1#2% {% - \ifnum #1<#2 - \expandafter - \XINT@fac@loop@main - \expandafter - {\the\numexpr #1+1\expandafter }% - \else - \expandafter\XINT@fac@docomputation - \fi - {#2}{#3{#1}}% + \xint@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 + \W\X\Y\Z #1\W\X\Y\Z }% }% -\def\XINT@fac@docomputation #1#2% +\def\XINT@fac@bigcompute@end #1#2#3#4#5% {% - \XINT@fprod@getnext {1000}#2\relax + \XINT@fac@bigcompute@end@ #5% }% -\def\XINT@fprod@getnext #1#2% +\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% {% - \XINT@fprod@checkiffinished #2\Z {#1}% + \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@fprod@checkiffinished #1% +\def\XINT@fac@loop@exit #1#2#3#4#5#6#7% {% - \xint@relax #1\XINT@fprod@end\relax - \XINT@fprod@compute #1% + \XINT@fac@loop@exit@ #6% }% -\def\XINT@fprod@compute #1\Z #2% +\def\XINT@fac@loop@exit@ #1#2#3% {% - \expandafter - \XINT@fprod@getnext - \expandafter - {\romannumeral0\XINT@mul@Mr {#1}#2\Z\Z\Z\Z }% -}% -\def\XINT@fprod@end\relax\XINT@fprod@compute #1\Z #2% -{% - \expandafter - \xint@cleanupzeros@andstop - \romannumeral0\XINT@rev {#2}% + \XINT@mul@M }% % \end{macrocode} -% \subsection{\ch{xintPow}} +% \subsection{\csh{xintPow}} % \begin{verbatim} -% 1.02 this is the same as in earlier versions, but I had to move here the -% special routine of Product as it was previously done, with its accompanying -% special multiplication and addition (to maintain the intermediate results in a -% special reversed form), now that I have modified the \XINT@posprod routine. +% 1.02 modified the \XINT@posprod routine, and this meant that the original +% version was moved here and renamed to \XINT@pow@posprod, as it was well +% adapted for computing powers. Then I moved in 1.03 the special variants of +% multiplication (hence of addition) which were needed to earlier in this file. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\xintPow {\romannumeral0\xintpow }% -\def\xintpow #1% +\def\xintiPow {\romannumeral0\xintipow }% +\def\xintipow #1% {% \expandafter\expandafter\expandafter \xint@pow #1\Z% }% -% \end{macrocode} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% #1#2 = A -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} +\let\xintPow\xintiPow \let\xintpow\xintipow \def\xint@pow #1#2\Z {% \xint@UDsignfork @@ -3859,7 +5063,7 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@pow@checkBlength #1#2% {% - \ifnum\xintLen{#1} >9 + \ifnum\xintiLen{#1} >9 \expandafter\XINT@pow@BtooBig \else \expandafter\XINT@pow@loop @@ -3880,7 +5084,7 @@ and modify it to what is needed. \xint@afterfi{\expandafter\XINT@pow@loop@a \expandafter{\the\numexpr 2*(#1/2)-#1\expandafter }% b mod 2 \expandafter{\the\numexpr #1-#1/2\expandafter }% [b/2] - \expandafter{\romannumeral0\xintsqr{#2}}}% + \expandafter{\romannumeral0\xintisqr{#2}}}% \fi {{#2}}% }% @@ -3900,7 +5104,12 @@ and modify it to what is needed. % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% Routine de produit servant pour le calcul des puissances. +% 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. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -3930,10 +5139,8 @@ and modify it to what is needed. }% \def\XINT@pow@pprod@compute #1\Z #2% {% - \expandafter - \XINT@pow@pprod@getnext - \expandafter - {\romannumeral0\XINT@pow@mul@enter #2\W\X\Y\Z #1\W\X\Y\Z}% + \expandafter \XINT@pow@pprod@getnext \expandafter + {\romannumeral0\XINT@mulr@enter #2\W\X\Y\Z #1\W\X\Y\Z}% }% \def\XINT@pow@pprod@end\relax\XINT@pow@pprod@compute #1\Z #2% {% @@ -3942,246 +5149,7 @@ and modify it to what is needed. \romannumeral0\XINT@rev {#2}% }% % \end{macrocode} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% Multiplication spéciale pour emploi par le Produit servant pour le calcul -% des Puissances (sic) -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} -\def\XINT@pow@mul@enter #1\W\X\Y\Z #2#3#4#5% -{% - \xint@w - #5\xint@pow@mul@enterw - #4\xint@pow@mul@enterx - #3\xint@pow@mul@entery - #2\xint@pow@mul@enterz - \W\XINT@pow@mul@start {#2#3#4#5}#1\W\X\Y\Z -}% -\def\xint@pow@mul@enterw - #1\xint@pow@mul@enterx - #2\xint@pow@mul@entery - #3\xint@pow@mul@enterz - \W\XINT@pow@mul@start #4#5\W\X\Y\Z \X\Y\Z -{% - \XINT@mul@Mr {#3#2#1}#5\Z\Z\Z\Z -}% -\def\xint@pow@mul@enterx - #1\xint@pow@mul@entery - #2\xint@pow@mul@enterz - \W\XINT@pow@mul@start #3#4\W\X\Y\Z \Y\Z -{% - \XINT@mul@Mr {#2#1}#4\Z\Z\Z\Z -}% -\def\xint@pow@mul@entery - #1\xint@pow@mul@enterz - \W\XINT@pow@mul@start #2#3\W\X\Y\Z \Z -{% - \XINT@mul@Mr {#1}#3\Z\Z\Z\Z -}% -\def\XINT@pow@mul@start #1#2\W\X\Y\Z -{\expandafter - \XINT@pow@mul@main \expandafter - {\romannumeral0% - \XINT@mul@Mr {#1}#2\Z\Z\Z\Z - }#2\W\X\Y\Z -}% -\def\XINT@pow@mul@main #1#2\W\X\Y\Z #3#4#5#6% -{% - \xint@w - #6\xint@pow@mul@mainw - #5\xint@pow@mul@mainx - #4\xint@pow@mul@mainy - #3\xint@pow@mul@mainz - \W\XINT@pow@mul@compute {#1}{#3#4#5#6}#2\W\X\Y\Z -}% -\def\XINT@pow@mul@compute #1#2#3\W\X\Y\Z -{\expandafter - \XINT@pow@mul@main \expandafter - {\romannumeral0\expandafter - \XINT@mul@Ar \expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z \W\X\Y\Z 0000#1\W\X\Y\Z - }#3\W\X\Y\Z -}% -\def\xint@pow@mul@mainw - #1\xint@pow@mul@mainx - #2\xint@pow@mul@mainy - #3\xint@pow@mul@mainz - \W\XINT@pow@mul@compute #4#5#6\W\X\Y\Z \X\Y\Z -{% - \expandafter - \XINT@pow@add@A \expandafter0\expandafter{\expandafter}% - \romannumeral0% - \XINT@mul@Mr {#3#2#1}#6\Z\Z\Z\Z - \W\X\Y\Z 000#4\W\X\Y\Z -}% -\def\xint@pow@mul@mainx - #1\xint@pow@mul@mainy - #2\xint@pow@mul@mainz - \W\XINT@pow@mul@compute #3#4#5\W\X\Y\Z \Y\Z -{% - \expandafter - \XINT@pow@add@A \expandafter0\expandafter{\expandafter}% - \romannumeral0% - \XINT@mul@Mr {#2#1}#5\Z\Z\Z\Z - \W\X\Y\Z 00#3\W\X\Y\Z -}% -\def\xint@pow@mul@mainy - #1\xint@pow@mul@mainz - \W\XINT@pow@mul@compute #2#3#4\W\X\Y\Z \Z -{% - \expandafter - \XINT@pow@add@A \expandafter0\expandafter{\expandafter}% - \romannumeral0% - \XINT@mul@Mr {#1}#4\Z\Z\Z\Z - \W\X\Y\Z 0#2\W\X\Y\Z -}% -\def\xint@pow@mul@mainz\W\XINT@pow@mul@compute #1#2#3\W\X\Y\Z -{ #1}% -% \end{macrocode} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% ADDITION spéciale pour emploi dans la routine de Multiplication utilisée -% dans la routine de Produit servant pour le calcul des puissances (sic). -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} -\def\XINT@pow@add@A #1#2#3#4#5#6% -{% - \xint@w - #3\xint@pow@add@az - \W\XINT@pow@add@AB #1{#3#4#5#6}{#2}% -}% -\def\xint@pow@add@az\W\XINT@pow@add@AB #1#2% -{% - \XINT@pow@add@AC@checkcarry #1% -}% -\def\XINT@pow@add@AC@checkcarry #1% -{% - \xint@zero #1\xint@pow@add@AC@nocarry 0\XINT@pow@add@C -}% -\def\xint@pow@add@AC@nocarry 0\XINT@pow@add@C -{% - \XINT@pow@add@F -}% -\def\XINT@pow@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \XINT@pow@add@ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT@pow@add@ABE #1#2#3#4#5#6% -{\expandafter - \XINT@pow@add@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax -}% -\def\XINT@pow@add@ABEA #1#2#3#4#5#6#7% -{% - \XINT@pow@add@A #2{#7#6#5#4#3}%<-- attention on met donc \`a droite -}% -\def\XINT@pow@add@C #1#2#3#4#5% -{% - \xint@w - #5\xint@pow@add@cw - #4\xint@pow@add@cx - #3\xint@pow@add@cy - #2\xint@pow@add@cz - \W\XINT@pow@add@CD {#5#4#3#2}{#1}% -}% -\def\XINT@pow@add@CD #1% -{\expandafter - \XINT@pow@add@CC\the\numexpr 1+10#1\relax -}% -\def\XINT@pow@add@CC #1#2#3#4#5#6#7% -{% - \XINT@pow@add@AC@checkcarry #2{#7#6#5#4#3}% -}% -\def\xint@pow@add@cw - #1\xint@pow@add@cx - #2\xint@pow@add@cy - #3\xint@pow@add@cz - \W\XINT@pow@add@CD -{\expandafter - \XINT@pow@add@CDw\the\numexpr 1+10#1#2#3\relax -}% -\def\XINT@pow@add@CDw #1#2#3#4#5#6% -{% - \xint@quatrezeros #2#3#4#5\XINT@pow@add@endDw@zeros - 0000\XINT@pow@add@endDw #2#3#4#5% -}% -\def\XINT@pow@add@endDw@zeros 0000\XINT@pow@add@endDw 0000#1\X\Y\Z{ #1}% -\def\XINT@pow@add@endDw #1#2#3#4#5\X\Y\Z{ #5#4#3#2#1}% -\def\xint@pow@add@cx - #1\xint@pow@add@cy - #2\xint@pow@add@cz - \W\XINT@pow@add@CD -{\expandafter - \XINT@pow@add@CDx\the\numexpr 1+100#1#2\relax -}% -\def\XINT@pow@add@CDx #1#2#3#4#5#6% -{% - \xint@quatrezeros #2#3#4#5\XINT@pow@add@endDx@zeros - 0000\XINT@pow@add@endDx #2#3#4#5% -}% -\def\XINT@pow@add@endDx@zeros 0000\XINT@pow@add@endDx 0000#1\Y\Z{ #1}% -\def\XINT@pow@add@endDx #1#2#3#4#5\Y\Z{ #5#4#3#2#1}% -\def\xint@pow@add@cy - #1\xint@pow@add@cz - \W\XINT@pow@add@CD -{\expandafter - \XINT@pow@add@CDy\the\numexpr 1+1000#1\relax -}% -\def\XINT@pow@add@CDy #1#2#3#4#5#6% -{% - \xint@quatrezeros #2#3#4#5\XINT@pow@add@endDy@zeros - 0000\XINT@pow@add@endDy #2#3#4#5% -}% -\def\XINT@pow@add@endDy@zeros 0000\XINT@pow@add@endDy 0000#1\Z{ #1}% -\def\XINT@pow@add@endDy #1#2#3#4#5\Z{ #5#4#3#2#1}% -\def\xint@pow@add@cz\W\XINT@pow@add@CD #1#2{ #21000}% -\def\XINT@pow@add@F #1#2#3#4#5% -{% - \xint@w - #5\xint@pow@add@Gw - #4\xint@pow@add@Gx - #3\xint@pow@add@Gy - #2\xint@pow@add@Gz - \W\XINT@pow@add@G {#2#3#4#5}{#1}% -}% -\def\XINT@pow@add@G #1#2% -{% - \XINT@pow@add@F {#2#1}% -}% -\def\xint@pow@add@Gw - #1\xint@pow@add@Gx - #2\xint@pow@add@Gy - #3\xint@pow@add@Gz - \W\XINT@pow@add@G #4% -{% - \xint@quatrezeros #3#2#10\XINT@pow@add@endGw@zeros - 0000\XINT@pow@add@endGw #3#2#10% -}% -\def\XINT@pow@add@endGw@zeros 0000\XINT@pow@add@endGw 0000#1\X\Y\Z{ #1}% -\def\XINT@pow@add@endGw #1#2#3#4#5\X\Y\Z{ #5#1#2#3#4}% -\def\xint@pow@add@Gx - #1\xint@pow@add@Gy - #2\xint@pow@add@Gz - \W\XINT@pow@add@G #3% -{% - \xint@quatrezeros #2#100\XINT@pow@add@endGx@zeros - 0000\XINT@pow@add@endGx #2#100% -}% -\def\XINT@pow@add@endGx@zeros 0000\XINT@pow@add@endGx 0000#1\Y\Z{ #1}% -\def\XINT@pow@add@endGx #1#2#3#4#5\Y\Z{ #5#1#2#3#4}% -\def\xint@pow@add@Gy - #1\xint@pow@add@Gz - \W\XINT@pow@add@G #2% -{% - \xint@quatrezeros #1000\XINT@pow@add@endGy@zeros - 0000\XINT@pow@add@endGy #1000% -}% -\def\XINT@pow@add@endGy@zeros 0000\XINT@pow@add@endGy 0000#1\Z{ #1}% -\def\XINT@pow@add@endGy #1#2#3#4#5\Z{ #5#1#2#3#4}% -\def\xint@pow@add@Gz\W\XINT@pow@add@G #1#2{ #2}% -% \end{macrocode} -% \subsection{\ch{xintDivision}, \ch{xintQuo}, \ch{xintRem}} +% \subsection{\csh{xintDivision}, \csh{xintQuo}, \csh{xintRem}} % \begin{macrocode} \def\xintQuo {\romannumeral0\xintquo }% \def\xintRem {\romannumeral0\xintrem }% @@ -4195,6 +5163,7 @@ and modify it to what is needed. % \vspace*{-.5\baselineskip} % \begin{verbatim} % #1 = A, #2 = B. On calcule le quotient de A par B +% 1.03 adds the detection of 1 for B. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -4282,7 +5251,7 @@ and modify it to what is needed. }% \def\XINT@div@AisNegative@post #1#2% {% - \ifcase\xintSgn {#2} + \ifcase\xintiSgn {#2} \expandafter \XINT@div@AisNegative@zerorem \or \expandafter \XINT@div@AisNegative@posrem @@ -4310,10 +5279,8 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@AisNegative@posrem #1% {% - \expandafter - \XINT@div@AisNegative@posrem@b - \expandafter - {\romannumeral0\xintopp {\XINT@Add{#1}{1}}}% + \expandafter \XINT@div@AisNegative@posrem@b \expandafter + {\romannumeral0\xintiopp {\XINT@Add{#1}{1}}}% }% % \end{macrocode} % \vspace*{-.5\baselineskip} @@ -4325,9 +5292,7 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@AisNegative@posrem@b #1#2#3% {% - \expandafter - \xint@exchangetwo@keepbraces@andstop - \expandafter + \expandafter \xint@exchangetwo@keepbraces@andstop \expandafter {\romannumeral0\XINT@sub {#3}{#2}}{#1}% }% % \end{macrocode} @@ -4340,22 +5305,39 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@prepare #1% {% - \expandafter - \XINT@div@prepareB@a - \expandafter - {\romannumeral0\XINT@length {#1}}{#1}% B > 0 ici + \expandafter \XINT@div@prepareB@aa \expandafter + {\romannumeral0\XINT@length {#1}}{#1}% B > 0 ici }% % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} % Calcul du plus petit K = 4n >= longueur de B +% 1.03 adds the interception of B=1 % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} +\def\XINT@div@prepareB@aa #1% +{% + \ifnum #1=1 + \expandafter\XINT@div@prepareB@ab + \else + \expandafter\XINT@div@prepareB@a + \fi + {#1}% +}% +\def\XINT@div@prepareB@ab #1#2% +{% + \ifnum #2=1 + \expandafter\XINT@div@prepareB@BisOne + \else + \xint@afterfi{\XINT@div@prepareB@e {000}{3}{4}{#2}}% + \fi +}% +\def\XINT@div@prepareB@BisOne #1{ {#1}{0}}% \def\XINT@div@prepareB@a #1% {% - \expandafter\XINT@div@prepareB@b\expandafter - {\the\numexpr 4*((#1+1)/4)\relax}{#1}% + \expandafter \XINT@div@prepareB@b \expandafter + {\the\numexpr 4*((#1+1)/4)\relax}{#1}% }% % \end{macrocode} % \vspace*{-.5\baselineskip} @@ -4366,7 +5348,7 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@prepareB@b #1#2% {% - \expandafter\XINT@div@prepareB@c \expandafter + \expandafter \XINT@div@prepareB@c \expandafter {\the\numexpr #1-#2\relax}{#1}% }% % \end{macrocode} @@ -4410,10 +5392,8 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@prepareB@f #1#2#3#4#5\Z {% - \expandafter - \XINT@div@prepareB@g - \expandafter - {\romannumeral0\XINT@rev {#1#2#3#4#5}}{#1#2#3#4}% + \expandafter \XINT@div@prepareB@g \expandafter + {\romannumeral0\XINT@rev {#1#2#3#4#5}}{#1#2#3#4}% }% % \end{macrocode} % \vspace*{-.5\baselineskip} @@ -4438,10 +5418,8 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@prepareA@a #1% {% - \expandafter - \XINT@div@prepareA@b - \expandafter - {\romannumeral0\XINT@length {#1}}{#1}% A >0 ici + \expandafter \XINT@div@prepareA@b \expandafter + {\romannumeral0\XINT@length {#1}}{#1}% A >0 ici }% % \end{macrocode} % \vspace*{-.5\baselineskip} @@ -4521,9 +5499,7 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@finished@a #1#2#3% {% - \expandafter - \XINT@div@finished@b - \expandafter + \expandafter \XINT@div@finished@b \expandafter {\romannumeral0\XINT@cuz {#1}}% }% % \end{macrocode} @@ -4555,9 +5531,7 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@finished@c #1#2% {% - \expandafter - \space - \expandafter + \expandafter \space \expandafter {\romannumeral0\expandafter\xint@cleanupzeros@andstop \romannumeral0\XINT@rev {#2}}{#1}% }% @@ -4578,8 +5552,7 @@ and modify it to what is needed. \xint@quatrezeros #1#2#3#4\xint@div@final@c0000% \XINT@div@final@c {#1#2#3#4}{#1#2#3#4#5}% }% -\def\xint@div@final@c0000\XINT@div@final@c #1% - {\XINT@div@finished@a }% +\def\xint@div@final@c0000\XINT@div@final@c #1{\XINT@div@finished@a }% % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} @@ -4589,9 +5562,7 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@final@c #1#2#3#4% {% - \expandafter - \XINT@div@final@d - \expandafter + \expandafter \XINT@div@final@d \expandafter {\the\numexpr #1/#4\relax}{#2}% }% % \end{macrocode} @@ -4604,9 +5575,7 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@final@d #1#2#3#4#5% q,A,Q,L,B puis c {% - \expandafter - \XINT@div@final@da - \expandafter + \expandafter \XINT@div@final@da \expandafter {\romannumeral0\XINT@mul@M {#1}#5\Z\Z\Z\Z }% {\romannumeral0\xint@cleanupzeros@andstop #2}% {#1}{#3}{#5}% @@ -4626,8 +5595,8 @@ and modify it to what is needed. \def\XINT@div@final@e #1#2#3#4#5% A,qB,q,Q,B,puis c {% \expandafter\XINT@div@final@f - \expandafter{\romannumeral0\xintsub {#1}{#2}}% - {\romannumeral0\xintadd {\XINT@Rev@andcleanupzeros{#4}}{#3}}% + \expandafter{\romannumeral0\xintisub {#1}{#2}}% + {\romannumeral0\xintiadd {\XINT@Rev@andcleanupzeros{#4}}{#3}}% }% \def\XINT@div@final@dc #1#2#3% A sans leading zeros,trash,q,Q,B,c {% @@ -4637,9 +5606,9 @@ and modify it to what is needed. \def\XINT@div@final@dd #1#2#3#4% q,A,Q,B puis c {% \expandafter\XINT@div@final@f - \expandafter{\romannumeral0\xintsub + \expandafter{\romannumeral0\xintisub {#2}{\romannumeral0\XINT@mul@M {#1}#4\Z\Z\Z\Z }}% - {\romannumeral0\xintadd {\XINT@Rev@andcleanupzeros{#3}}{#1}}% + {\romannumeral0\xintiadd {\XINT@Rev@andcleanupzeros{#3}}{#1}}% }% \def\XINT@div@final@f #1#2#3% R,Q à développer,c {% @@ -4659,31 +5628,29 @@ and modify it to what is needed. % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% Boucle Principale +% Boucle Principale (on reviendra en div@body@b pas div@body@a) % A, K, x, Q, L, B, c % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} \def\XINT@div@body@a #1% {% - \XINT@div@body@b #1\Z + \XINT@div@body@b #1\Z {#1}% }% \def\XINT@div@body@b #1#2#3#4#5#6#7#8#9\Z {% - \XINT@div@body@c - {#1#2#3#4#5#6#7#8#9}% - {#1#2#3#4#5#6#7#8}% + \XINT@div@body@c {#1#2#3#4#5#6#7#8}% }% % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% A, a, K, x, Q, L, B, c +% a, A, K, x, Q, L, B, c % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} \def\XINT@div@body@c #1#2#3% {% - \XINT@div@body@d {#3}{}#1\Z {#2}{#3}% + \XINT@div@body@d {#3}{}#2\Z {#1}{#3}% }% \def\XINT@div@body@d #1#2#3#4#5#6% {% @@ -4705,37 +5672,58 @@ and modify it to what is needed. % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% a, alpha, alpha', K, x, Q, L, B, c +% a, alpha (à l'envers), alpha' (à l'endroit), K, x, Q, L, B (à l'envers), c % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} \def\XINT@div@body@f #1#2#3#4#5#6#7#8% {% - \expandafter\XINT@div@body@g - \expandafter - {\the\numexpr (#1+(#5+1)/2)/(#5+1)-1\relax }% - {#2}{#8}{#4}{#5}{#3}{#6}{#7}{#8}% + \expandafter\XINT@div@body@gg + \the\numexpr (#1+(#5+1)/2)/(#5+1)+99999\relax + {#8}{#2}{#8}{#4}{#5}{#3}{#6}{#7}{#8}% }% % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% q1, alpha, B, K, x, alpha', Q, L, B, c +% q1 sur six chiffres (il en a 5 au max), B, alpha, B, K, x, alpha', Q, L, B, c % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} -\def\XINT@div@body@g #1#2#3% +\def\XINT@div@body@gg #1#2#3#4#5#6% +{% + \xint@UDzerofork + #2\dummy \XINT@div@body@gk + 0\dummy {\XINT@div@body@ggk #2}% + \xint@UDkrof + {#3#4#5#6}% +}% +\def\XINT@div@body@gk #1#2#3% {% \expandafter \XINT@div@body@h + \romannumeral0\XINT@div@sub@xpxp + {\romannumeral0\XINT@mul@Mr {#1}#2\Z\Z\Z\Z }{#3}\Z {#1}% +}% +\def\XINT@div@body@ggk #1#2#3% +{% + \expandafter \XINT@div@body@gggk \expandafter + {\romannumeral0\XINT@mul@Mr {#1}0000#3\Z\Z\Z\Z }% + {\romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z }% + {#1#2}% +}% +\def\XINT@div@body@gggk #1#2#3#4% +{% + \expandafter + \XINT@div@body@h \romannumeral0\XINT@div@sub@xpxp - {\romannumeral0\XINT@pow@mul@enter #3\W\X\Y\Z #1\W\X\Y\Z }% - {#2}\Z - {#3}{#1}% + {\romannumeral0\expandafter\XINT@mul@Ar + \expandafter0\expandafter{\expandafter}#2\W\X\Y\Z #1\W\X\Y\Z }% + {#4}\Z {#3}% }% % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% alpha1 = alpha-q1 B, \Z, B, q1, K, x, alpha', Q, L, B, c +% alpha1 = alpha-q1 B, \Z, q1, B, K, x, alpha', Q, L, B, c % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -4748,10 +5736,14 @@ and modify it to what is needed. \fi {#1#2#3#4#5#6#7#8#9}% }% +\def\XINT@div@body@k #1#2#3% +{% + \XINT@div@body@l {#1}{#2}% +}% % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% a1, alpha1, B, q1, K, x, alpha', Q, L, B, c +% a1, alpha1 (à l'endroit), q1, B, K, x, alpha', Q, L, B, c % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -4761,39 +5753,17 @@ and modify it to what is needed. \expandafter{\the\numexpr (#1+(#6+1)/2)/(#6+1)-1\relax }% {#2}{#3}{#4}{#5}{#6}% }% -% \end{macrocode} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% q2, alpha1, B, q1, K, x, alpha', Q, L, B, c -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} \def\XINT@div@body@j #1#2#3#4% {% - \expandafter - \XINT@div@body@l - \expandafter{\romannumeral0\XINT@div@sub@xpxp - {\romannumeral0\XINT@pow@mul@enter #3\W\X\Y\Z #1\W\X\Y\Z }% - {\XINT@Rev{#2}}}% - {#4+#1}% -}% -% \end{macrocode} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% alpha2, q1+q2, K, x, alpha', Q, L, B, c -% attention body@j -> body@l -% alpha1, B, q=q1, K, x, alpha', Q, L, B, c -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} -\def\XINT@div@body@k #1#2% -{% - \XINT@div@body@l {#1}% + \expandafter \XINT@div@body@l \expandafter + {\romannumeral0\XINT@div@sub@xpxp + {\romannumeral0\XINT@mul@Mr {#1}#4\Z\Z\Z\Z }{\XINT@Rev{#2}}}% + {#3+#1}% }% % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% alpha2, q= q1+q2, K, x, alpha', Q, L, B, c +% alpha2 (à l'endroit, ou alpha1), q1+q2 (ou q1), K, x, alpha', Q, L, B, c % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -4807,7 +5777,7 @@ and modify it to what is needed. % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% chiffres de q, Q, K, L, A', x, B, c +% chiffres de q, Q, K, L, A'=nouveau A, x, B, c % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -4829,7 +5799,7 @@ and modify it to what is needed. \def\XINT@div@body@n #1#2% {% \expandafter\XINT@div@body@o\expandafter - {\romannumeral0\XINT@sum@A 0{}#1\W\X\Y\Z #2\W\X\Y\Z }% + {\romannumeral0\XINT@addr@A 0{}#1\W\X\Y\Z #2\W\X\Y\Z }% }% % \end{macrocode} % \vspace*{-.5\baselineskip} @@ -4872,9 +5842,7 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@div@body@repeatp #1#2#3#4#5#6#7% {% - \expandafter - \XINT@div@body@p - \expandafter + \expandafter \XINT@div@body@p \expandafter {\the\numexpr #1-4\relax}{#2}{0000#3}% }% % \end{macrocode} @@ -4882,13 +5850,13 @@ and modify it to what is needed. % \begin{verbatim} % L -> L-4, zeros->zeros+0000, répéter jusqu'à ce que soit L=K % soit on ne trouve plus 0000 -% nouveau L, K, zeros, nouveau A=#4, Q+q, x, B, c +% nouveau L, K, zeros, nouveau A=#4, \Z, Q+q (à l'envers), x, B, c % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} \def\XINT@div@body@q #1#2#3#4\Z #5#6% {% - \XINT@div@body@a {#4}{#2}{#6}{#3#5}{#1}% + \XINT@div@body@b #4\Z {#4}{#2}{#6}{#3#5}{#1}% }% % \end{macrocode} % \vspace*{-.5\baselineskip} @@ -4906,13 +5874,6 @@ and modify it to what is needed. {% \XINT@div@gotofinal@b #3\Z {#4}{#1}% }% -% \end{macrocode} -% \vspace*{-.5\baselineskip} -% \begin{verbatim} -% zeros\Z, A, L=K, Q, x, B, c -% \end{verbatim} -% \vspace*{-1.5\baselineskip} -% \begin{macrocode} \def\XINT@div@gotofinal@b 0000#1\Z #2#3#4#5% {% \XINT@div@final@a {#2}{#3}{#5}{#1#4}{#3}% @@ -4920,17 +5881,18 @@ and modify it to what is needed. % \end{macrocode} % \vspace*{-.5\baselineskip} % \begin{verbatim} -% A, L=K, x, Q avec zéros, L, B, c -% La soustraction spéciale. Étendre deux fois les arguments -% pour \XINT@div@sub@enter longueur multiple de 4 on sait que #2>#1, +% La soustraction spéciale. +% Elle fait l'expansion (une fois pour le premier, deux fois pour le second) de +% ses arguments. Ceux-ci doivent être à l'envers sur 4n. De plus on sait a +% priori que le second est > le premier. Et le résultat de la différence est +% renvoyé **avec la même longueur que le second** (donc avec des leading zéros +% éventuels), et *à l'endroit*. % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} \def\XINT@div@sub@xpxp #1% {% - \expandafter - \XINT@div@sub@xpxp@ - \expandafter + \expandafter \XINT@div@sub@xpxp@ \expandafter {#1}% }% \def\XINT@div@sub@xpxp@ #1#2% @@ -5023,7 +5985,7 @@ and modify it to what is needed. % MULTIPLICATION BY POWER OF TEN, SPLIT OPERATION. % \end{verbatim} % \vspace*{-2\baselineskip} -% \subsection{\ch{xintFDg}} +% \subsection{\csh{xintFDg}} % \begin{verbatim} % FIRST DIGIT % \end{verbatim} @@ -5047,7 +6009,7 @@ and modify it to what is needed. \xint@z }% % \end{macrocode} -% \subsection{\ch{xintLDg}} +% \subsection{\csh{xintLDg}} % \begin{verbatim} % LAST DIGIT % \end{verbatim} @@ -5073,7 +6035,32 @@ and modify it to what is needed. \expandafter\space\expandafter #1\xint@z }% % \end{macrocode} -% \subsection{\ch{xintOdd}} +% \subsection{\csh{xintMON}} +% \begin{verbatim} +% MINUS ONE TO THE POWER N +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\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}} % \begin{verbatim} % ODDNESS % \end{verbatim} @@ -5097,7 +6084,7 @@ and modify it to what is needed. \fi }% % \end{macrocode} -% \subsection{\ch{xintDSL}} +% \subsection{\csh{xintDSL}} % \begin{verbatim} % DECIMAL SHIFT LEFT (=MULTIPLICATION PAR 10) % \end{verbatim} @@ -5117,7 +6104,7 @@ and modify it to what is needed. \def\xint@dsl@zero 0\XINT@dsl@ 0#1\Z { 0}% \def\XINT@dsl@ #1\Z { #10}% % \end{macrocode} -% \subsection{\ch{xintDSR}} +% \subsection{\csh{xintDSR}} % \begin{verbatim} % DECIMAL SHIFT RIGHT (=DIVISION PAR 10) % \end{verbatim} @@ -5149,15 +6136,15 @@ and modify it to what is needed. \def\xint@dsr@onedigit #1\XINT@rev #2{ 0}% \def\XINT@dsr@removew #1\W { }% % \end{macrocode} -% \subsection{\ch{xintDSH}, \ch{xintDSHr}} +% \subsection{\csh{xintDSH}, \csh{xintDSHr}} % \begin{verbatim} % DECIMAL SHIFTS % \xintDSH {x}{A} -% si x <= 0, fait A -> A.10^(|x|) +% 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'. +% \xintDSHr donne le `reste' (si x<=0 donne zéro). % \end{verbatim} % \vspace*{-1.5\baselineskip} % \begin{macrocode} @@ -5192,14 +6179,13 @@ and modify it to what is needed. \def\xint@dsh #1#2% {% \expandafter\expandafter\expandafter - \XINT@dsh@checksignx - #2\Z {#1}% + \XINT@dsh@checksignx #2\Z {#1}% }% \def\XINT@dsh@checksignx #1% {% \xint@UDzerominusfork #1-\dummy \XINT@dsh@xiszero - 0#1\dummy \XINT@dsx@xisNeg + 0#1\dummy \XINT@dsx@xisNeg@checkA % on passe direct dans DSx 0-\dummy {\XINT@dsh@xisPos #1}% \xint@UDkrof }% @@ -5208,18 +6194,19 @@ and modify it to what is needed. {% \expandafter \xint@firstoftwo@andstop - \romannumeral0\XINT@dsx@checksignA #2\Z {#1}% + \romannumeral0\XINT@dsx@checksignA #2\Z {#1}% via DSx }% % \end{macrocode} -% \subsection{\ch{xintDSx}} +% \subsection{\csh{xintDSx}} % \begin{verbatim} % 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. -% 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))} +% --> 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 @@ -5239,8 +6226,7 @@ and modify it to what is needed. \def\xint@dsx #1#2% {% \expandafter\expandafter\expandafter - \XINT@dsx@checksignx - #2\Z {#1}% + \XINT@dsx@checksignx #2\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}}% @@ -5248,40 +6234,50 @@ and modify it to what is needed. {% \xint@UDzerominusfork #1-\dummy \XINT@dsx@xisZero - 0#1\dummy \XINT@dsx@xisNeg + 0#1\dummy \XINT@dsx@xisNeg@checkA 0-\dummy {\XINT@dsx@xisPos #1}% \xint@UDkrof }% -\def\XINT@dsx@xisZero #1\Z #2{ {#2}{0}}% -\def\XINT@dsx@xisNeg #1\Z +\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% {% - \ifnum\XINT@Len {#1} > 9 + \xint@zero #1\XINT@dsx@xisNeg@Azero 0\expandafter + \XINT@dsx@xisNeg@checkx\expandafter + {\romannumeral0\XINT@length {#3}}{#3}\Z {#1#2}% +}% +\def\XINT@dsx@xisNeg@Azero #1#2#3#4#5#6#7#8{ 0}% +\def\XINT@dsx@xisNeg@checkx #1% +{% + \ifnum #1> 9 \xint@afterfi {\xintError:TooBigDecimalShift \XINT@dsx@toobigx }% \else \expandafter \XINT@dsx@zeroloop \fi - {#1}\Z }% -\def\XINT@dsx@toobigx #1\Z #2{ #2}% +\def\XINT@dsx@toobigx #1#2#3{ 0}% \def\XINT@dsx@zeroloop #1% {% \ifcase #1 - \expandafter \XINT@dsx@exit + \XINT@dsx@exit \or - \expandafter \XINT@dsx@exiti + \XINT@dsx@exiti \or - \expandafter \XINT@dsx@exitii + \XINT@dsx@exitii \or - \expandafter \XINT@dsx@exitiii + \XINT@dsx@exitiii \or - \expandafter \XINT@dsx@exitiv + \XINT@dsx@exitiv \or - \expandafter \XINT@dsx@exitv + \XINT@dsx@exitv \or - \expandafter \XINT@dsx@exitvi + \XINT@dsx@exitvi \or - \expandafter \XINT@dsx@exitvii + \XINT@dsx@exitvii \else \xint@afterfi {\expandafter @@ -5290,22 +6286,14 @@ and modify it to what is needed. }% \fi }% -\def\XINT@dsx@exit #1\Z - {\XINT@dsx@addzeros {#1}}% -\def\XINT@dsx@exiti #1\Z - {\XINT@dsx@addzeros {0#1}}% -\def\XINT@dsx@exitii #1\Z - {\XINT@dsx@addzeros {00#1}}% -\def\XINT@dsx@exitiii #1\Z - {\XINT@dsx@addzeros {000#1}}% -\def\XINT@dsx@exitiv #1\Z - {\XINT@dsx@addzeros {0000#1}}% -\def\XINT@dsx@exitv #1\Z - {\XINT@dsx@addzeros {00000#1}}% -\def\XINT@dsx@exitvi #1\Z - {\XINT@dsx@addzeros {000000#1}}% -\def\XINT@dsx@exitvii #1\Z - {\XINT@dsx@addzeros {0000000#1}}% +\def\XINT@dsx@exit #1\fi #2\Z {\fi \XINT@dsx@addzeros {#2}}% +\def\XINT@dsx@exiti #1\fi #2\Z {\fi \XINT@dsx@addzeros {0#2}}% +\def\XINT@dsx@exitii #1\fi #2\Z {\fi \XINT@dsx@addzeros {00#2}}% +\def\XINT@dsx@exitiii #1\fi #2\Z {\fi \XINT@dsx@addzeros {000#2}}% +\def\XINT@dsx@exitiv #1\fi #2\Z {\fi \XINT@dsx@addzeros {0000#2}}% +\def\XINT@dsx@exitv #1\fi #2\Z {\fi \XINT@dsx@addzeros {00000#2}}% +\def\XINT@dsx@exitvi #1\fi #2\Z {\fi \XINT@dsx@addzeros {000000#2}}% +\def\XINT@dsx@exitvii #1\fi #2\Z {\fi \XINT@dsx@addzeros {0000000#2}}% \def\XINT@dsx@addzeros #1#2{ #2#1}% \def\XINT@dsx@xisPos #1\Z #2% {% @@ -5366,7 +6354,7 @@ and modify it to what is needed. \expandafter\space\expandafter{#2}{#1}% }% % \end{macrocode} -% \subsection{\ch{xintDecSplit}, \ch{xintDecSplitL}, \ch{xintDecSplitR}} +% \subsection{\csh{xintDecSplit}, \csh{xintDecSplitL}, \csh{xintDecSplitR}} % \begin{verbatim} % DECIMAL SPLIT % v1.01: **New** behavior, for use in future extensions of the xint bundle: @@ -5400,10 +6388,8 @@ and modify it to what is needed. \def\xintDecSplit {\romannumeral0\xintdecsplit }% \def\xintdecsplit #1#2% {% - \expandafter - \xint@split - \expandafter - {\romannumeral0\xintabs {#2}}{#1}% fait expansion de A + \expandafter \xint@split \expandafter + {\romannumeral0\xintiabs {#2}}{#1}% fait expansion de A }% \def\xint@split #1#2% {% @@ -5446,32 +6432,31 @@ and modify it to what is needed. \def\XINT@split@fromleft@loop #1% {% \ifcase #1 - \expandafter\XINT@split@fromleft@endsplit + \XINT@split@fromleft@endsplit \or - \expandafter\XINT@split@fromleft@one@andend + \XINT@split@fromleft@one@andend \or - \expandafter\XINT@split@fromleft@two@andend + \XINT@split@fromleft@two@andend \or - \expandafter\XINT@split@fromleft@three@andend + \XINT@split@fromleft@three@andend \or - \expandafter\XINT@split@fromleft@four@andend + \XINT@split@fromleft@four@andend \or - \expandafter\XINT@split@fromleft@five@andend + \XINT@split@fromleft@five@andend \or - \expandafter\XINT@split@fromleft@six@andend + \XINT@split@fromleft@six@andend \or - \expandafter\XINT@split@fromleft@seven@andend + \XINT@split@fromleft@seven@andend \else \expandafter \XINT@split@fromleft@loop@perhaps \expandafter - {\the\numexpr - #1-8\expandafter\expandafter\expandafter }% + {\the\numexpr #1-8\expandafter\expandafter\expandafter }% \expandafter \XINT@split@fromleft@eight \fi }% -\def\XINT@split@fromleft@endsplit #1#2\W #3\Z - { {#1}{#2}}% +\def\XINT@split@fromleft@endsplit #1\fi #2#3\W #4\Z + {\expandafter\space\fi {#2}{#3}}% \def\XINT@split@fromleft@eight #1#2#3#4#5#6#7#8#9% {% #9{#1#2#3#4#5#6#7#8#9}% @@ -5489,26 +6474,26 @@ and modify it to what is needed. {% \space {#1}{}% }% -\def\XINT@split@fromleft@one@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@one }% +\def\XINT@split@fromleft@one@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@one }% \def\XINT@split@fromleft@one #1#2{#2{#1#2}}% -\def\XINT@split@fromleft@two@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@two }% +\def\XINT@split@fromleft@two@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@two }% \def\XINT@split@fromleft@two #1#2#3{#3{#1#2#3}}% -\def\XINT@split@fromleft@three@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@three }% +\def\XINT@split@fromleft@three@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@three }% \def\XINT@split@fromleft@three #1#2#3#4{#4{#1#2#3#4}}% -\def\XINT@split@fromleft@four@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@four }% +\def\XINT@split@fromleft@four@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@four }% \def\XINT@split@fromleft@four #1#2#3#4#5{#5{#1#2#3#4#5}}% -\def\XINT@split@fromleft@five@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@five }% +\def\XINT@split@fromleft@five@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@five }% \def\XINT@split@fromleft@five #1#2#3#4#5#6{#6{#1#2#3#4#5#6}}% -\def\XINT@split@fromleft@six@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@six }% +\def\XINT@split@fromleft@six@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@six }% \def\XINT@split@fromleft@six #1#2#3#4#5#6#7{#7{#1#2#3#4#5#6#7}}% -\def\XINT@split@fromleft@seven@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@seven }% +\def\XINT@split@fromleft@seven@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@seven }% \def\XINT@split@fromleft@seven #1#2#3#4#5#6#7#8{#8{#1#2#3#4#5#6#7#8}}% \def\XINT@split@fromleft@checkiftoofar #1#2#3\W #4\Z {% @@ -5525,9 +6510,7 @@ and modify it to what is needed. }% \def\XINT@split@fromright #1\Z #2% {% - \expandafter - \XINT@split@fromright@a - \expandafter + \expandafter \XINT@split@fromright@a \expandafter {\romannumeral0\XINT@rev {#2}}{#1}{#2}% }% \def\XINT@split@fromright@a #1#2% @@ -5539,19 +6522,19 @@ and modify it to what is needed. \ifcase #1 \expandafter\XINT@split@fromright@endsplit \or - \expandafter\XINT@split@fromright@one@andend + \XINT@split@fromright@one@andend \or - \expandafter\XINT@split@fromright@two@andend + \XINT@split@fromright@two@andend \or - \expandafter\XINT@split@fromright@three@andend + \XINT@split@fromright@three@andend \or - \expandafter\XINT@split@fromright@four@andend + \XINT@split@fromright@four@andend \or - \expandafter\XINT@split@fromright@five@andend + \XINT@split@fromright@five@andend \or - \expandafter\XINT@split@fromright@six@andend + \XINT@split@fromright@six@andend \or - \expandafter\XINT@split@fromright@seven@andend + \XINT@split@fromright@seven@andend \else \expandafter \XINT@split@fromright@loop@perhaps \expandafter @@ -5563,9 +6546,7 @@ and modify it to what is needed. }% \def\XINT@split@fromright@endsplit #1#2\W #3\Z #4% {% - \expandafter - \space - \expandafter {\romannumeral0\XINT@rev{#2}}{#1}% + \expandafter\space\expandafter {\romannumeral0\XINT@rev{#2}}{#1}% }% \def\XINT@split@fromright@eight #1#2#3#4#5#6#7#8#9% {% @@ -5576,28 +6557,27 @@ and modify it to what is needed. \xint@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@one@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@one }% +\def\XINT@split@fromright@toofar\W\XINT@split@fromright@loop #1#2#3\Z { {}}% +\def\XINT@split@fromright@one@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@one }% \def\XINT@split@fromright@one #1#2{#2{#2#1}}% -\def\XINT@split@fromright@two@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@two }% +\def\XINT@split@fromright@two@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@two }% \def\XINT@split@fromright@two #1#2#3{#3{#3#2#1}}% -\def\XINT@split@fromright@three@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@three }% +\def\XINT@split@fromright@three@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@three }% \def\XINT@split@fromright@three #1#2#3#4{#4{#4#3#2#1}}% -\def\XINT@split@fromright@four@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@four }% +\def\XINT@split@fromright@four@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@four }% \def\XINT@split@fromright@four #1#2#3#4#5{#5{#5#4#3#2#1}}% -\def\XINT@split@fromright@five@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@five }% +\def\XINT@split@fromright@five@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@five }% \def\XINT@split@fromright@five #1#2#3#4#5#6{#6{#6#5#4#3#2#1}}% -\def\XINT@split@fromright@six@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@six }% +\def\XINT@split@fromright@six@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@six }% \def\XINT@split@fromright@six #1#2#3#4#5#6#7{#7{#7#6#5#4#3#2#1}}% -\def\XINT@split@fromright@seven@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@seven }% +\def\XINT@split@fromright@seven@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@seven }% \def\XINT@split@fromright@seven #1#2#3#4#5#6#7#8{#8{#8#7#6#5#4#3#2#1}}% \def\XINT@split@fromright@checkiftoofar #1% {% @@ -5617,6 +6597,7 @@ and modify it to what is needed. % % The commenting is currently (\docdate) very sparse. % +% \localtableofcontents % \subsection{Catcodes, \eTeX{} detection, reload detection} % % The code for reload detection is copied from \textsc{Heiko @@ -5676,7 +6657,7 @@ and modify it to what is needed. \fi \z% % \end{macrocode} -% \subsection{Validation of \xintname loading} +% \subsection{Confirmation of \xintname loading} % \begin{macrocode} \begingroup\catcode61\catcode48\catcode32=10\relax% \catcode13=5 % ^^M @@ -5777,23 +6758,21 @@ and modify it to what is needed. \fi \expandafter\x\csname ver@xintgcd.sty\endcsname \ProvidesPackage{xintgcd}% - [2013/04/05 v1.02 Euclide algorithm with xint package (jfB)]% + [2013/04/14 v1.03 Euclide algorithm with xint package (jfB)]% % \end{macrocode} -% \subsection{\ch{xintGCD}} +% \subsection{\csh{xintGCD}} % \begin{macrocode} \def\xintGCD {\romannumeral0\xintgcd }% \def\xintgcd #1% {% - \expandafter - \XINT@gcd - \expandafter - {\romannumeral0\xintabs {#1}}% + \expandafter \XINT@gcd \expandafter + {\romannumeral0\xintiabs {#1}}% }% \def\XINT@gcd #1#2% {% \expandafter \XINT@gcd@fork - \romannumeral0\xintabs {#2}\Z #1\Z + \romannumeral0\xintiabs {#2}\Z #1\Z }% % \end{macrocode} % \vspace*{-.5\baselineskip} @@ -5834,7 +6813,7 @@ and modify it to what is needed. {#1}% }% % \end{macrocode} -% \subsection{\ch{xintBezout}} +% \subsection{\csh{xintBezout}} % \begin{macrocode} \def\xintBezout {\romannumeral0\xintbezout }% \def\xintbezout #1% @@ -5919,10 +6898,8 @@ and modify it to what is needed. }% \def\XINT@bezout@mm@post #1#2% {% - \expandafter - \XINT@bezout@mm@postb - \expandafter - {\romannumeral0\xintopp{#2}}{\romannumeral0\xintopp{#1}}% + \expandafter \XINT@bezout@mm@postb \expandafter + {\romannumeral0\xintiopp{#2}}{\romannumeral0\xintiopp{#1}}% }% \def\XINT@bezout@mm@postb #1#2% {% @@ -5948,10 +6925,8 @@ and modify it to what is needed. }% \def\XINT@bezout@mp@post #1#2% {% - \expandafter - \XINT@bezout@mp@postb - \expandafter - {\romannumeral0\xintopp {#2}}{#1}% + \expandafter \XINT@bezout@mp@postb \expandafter + {\romannumeral0\xintiopp {#2}}{#1}% }% \def\XINT@bezout@mp@postb #1#2#3#4#5% {% @@ -5971,10 +6946,8 @@ and modify it to what is needed. }% \def\XINT@bezout@pm@post #1% {% - \expandafter - \XINT@bezout@pm@postb - \expandafter - {\romannumeral0\xintopp{#1}}% + \expandafter \XINT@bezout@pm@postb \expandafter + {\romannumeral0\xintiopp{#1}}% }% \def\XINT@bezout@pm@postb #1#2#3#4#5% {% @@ -6032,11 +7005,9 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@bezout@loop@b #1#2#3#4#5#6#7#8% {% - \expandafter - \XINT@bezout@loop@c - \expandafter - {\romannumeral0\xintadd{\XINT@Mul{#5}{#2}}{#7}}% - {\romannumeral0\xintadd{\XINT@Mul{#6}{#2}}{#8}}% + \expandafter \XINT@bezout@loop@c \expandafter + {\romannumeral0\xintiadd{\XINT@Mul{#5}{#2}}{#7}}% + {\romannumeral0\xintiadd{\XINT@Mul{#6}{#2}}{#8}}% {#1}{#3}{#4}{#5}{#6}% }% % \end{macrocode} @@ -6048,9 +7019,7 @@ and modify it to what is needed. % \begin{macrocode} \def\XINT@bezout@loop@c #1#2% {% - \expandafter - \XINT@bezout@loop@d - \expandafter + \expandafter \XINT@bezout@loop@d \expandafter {#2}{#1}% }% % \end{macrocode} @@ -6111,7 +7080,7 @@ and modify it to what is needed. \space {-#5}{-#4}{#1}% }% % \end{macrocode} -% \subsection{\ch{xintEuclideAlgorithm}} +% \subsection{\csh{xintEuclideAlgorithm}} % \begin{verbatim} % Pour Euclide: % {N}{A}{D=r(n)}{B}{q1}{r1}{q2}{r2}{q3}{r3}....{qN}{rN=0} @@ -6122,16 +7091,14 @@ and modify it to what is needed. \def\xintEuclideAlgorithm {\romannumeral0\xinteuclidealgorithm }% \def\xinteuclidealgorithm #1% {% - \expandafter - \XINT@euc - \expandafter - {\romannumeral0\xintabs {#1}}% + \expandafter \XINT@euc \expandafter + {\romannumeral0\xintiabs {#1}}% }% \def\XINT@euc #1#2% {% \expandafter \XINT@euc@fork - \romannumeral0\xintabs {#2}\Z #1\Z + \romannumeral0\xintiabs {#2}\Z #1\Z }% % \end{macrocode} % \vspace*{-.5\baselineskip} @@ -6226,7 +7193,7 @@ and modify it to what is needed. \space {#1}{#3}{#2}% }% % \end{macrocode} -% \subsection{\ch{xintBezoutAlgorithm}} +% \subsection{\csh{xintBezoutAlgorithm}} % \begin{verbatim} % Pour Bezout: objectif, renvoyer % alpha0=1, beta0=0 @@ -6239,16 +7206,14 @@ and modify it to what is needed. \def\xintBezoutAlgorithm {\romannumeral0\xintbezoutalgorithm }% \def\xintbezoutalgorithm #1% {% - \expandafter - \XINT@bezalg - \expandafter - {\romannumeral0\xintabs {#1}}% + \expandafter \XINT@bezalg \expandafter + {\romannumeral0\xintiabs {#1}}% }% \def\XINT@bezalg #1#2% {% \expandafter \XINT@bezalg@fork - \romannumeral0\xintabs {#2}\Z #1\Z + \romannumeral0\xintiabs {#2}\Z #1\Z }% % \end{macrocode} % \vspace*{-.5\baselineskip} @@ -6295,8 +7260,8 @@ and modify it to what is needed. \def\XINT@bezalg@b #1#2#3#4#5#6#7#8% {% \expandafter\XINT@bezalg@c\expandafter - {\romannumeral0\xintadd {\xintMul {#6}{#2}}{#8}}% - {\romannumeral0\xintadd {\xintMul {#5}{#2}}{#7}}% + {\romannumeral0\xintiadd {\xintiMul {#6}{#2}}{#8}}% + {\romannumeral0\xintiadd {\xintiMul {#5}{#2}}{#7}}% {#1}{#2}{#3}{#4}{#5}{#6}% }% % \end{macrocode} @@ -6374,7 +7339,7 @@ and modify it to what is needed. \space {#1}{#3}{0}{1}{#2}{#4}{1}{0}% }% % \end{macrocode} -% \subsection{\ch{xintTypesetEuclideAlgorithm}} +% \subsection{\csh{xintTypesetEuclideAlgorithm}} % \begin{verbatim} % TYPESETTING % Organisation: @@ -6410,7 +7375,7 @@ and modify it to what is needed. \endgroup }% % \end{macrocode} -% \subsection{\ch{xintTypesetBezoutAlgorithm}} +% \subsection{\csh{xintTypesetBezoutAlgorithm}} % \begin{verbatim} % Pour Bezout on a: % {N}{A}{0}{1}{D=r(n)}{B}{1}{0}{q1}{r1}{alpha1=q1}{beta1=1} @@ -6465,9 +7430,1571 @@ and modify it to what is needed. }% \XINT@gcd@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> +% \section{Package \xintfracname implementation} +% +% The commenting is currently (\docdate) very sparse. +% +% \localtableofcontents +% \subsection{Catcodes, \eTeX{} detection, 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}{Package xint is required}% + \y{xintfrac}{Will try \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}{Package xint is required}% + \y{xintfrac}{Will try \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 \xintname 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 % : + \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 + \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} +% +% Perhaps catcodes have changed after the loading of \xintname +% and prior to the current loading of \xintfracname, so we can not employ +% the |\XINT@restorecatcodes@endinput| in this style file. But +% there is no problem using |\XINT@setcatcodes|. +% +% \begin{macrocode} +\begingroup\catcode61\catcode48\catcode32=10\relax% + \catcode13=5 % ^^M + \endlinechar=13 % + \catcode123=1 % { + \catcode125=2 % } + \catcode64=11 % @ + \def\x + {% + \endgroup + \edef\XINT@frac@restorecatcodes@endinput + {% + \catcode94=\the\catcode94 % ^ + \catcode93=\the\catcode93 % ] + \catcode91=\the\catcode91 % [ + \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 % # + \catcode64=\the\catcode64 % @ + \catcode125=\the\catcode125 % } + \catcode123=\the\catcode123 % { + \endlinechar=\the\endlinechar + \catcode13=\the\catcode13 % ^^M + \catcode32=\the\catcode32 % + \catcode61=\the\catcode61 % = + \noexpand\endinput + }% + \XINT@setcatcodes + \catcode91=12 % [ + \catcode93=12 % ] + \catcode94=7 % ^ + }% +\x +% \end{macrocode} +% \subsection{Package identification} +% \begin{macrocode} +\begingroup + \catcode58=12 % : + \expandafter\ifx\csname ProvidesPackage\endcsname\relax + \def\x#1#2#3[#4]{\endgroup + \immediate\write-1{Package: #3 #4}% + \xdef#1{#4}% + }% + \else + \def\x#1#2[#3]{\endgroup + #2[{#3}]% + \ifx#1\@undefined + \xdef#1{#3}% + \fi + \ifx#1\relax + \xdef#1{#3}% + \fi + }% + \fi +\expandafter\x\csname ver@xintfrac.sty\endcsname +\ProvidesPackage{xintfrac}% + [2013/04/14 v1.03 Expandable operations on fractions (jfB)]% +% \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@outfrac}} +% \begin{macrocode} +\def\XINT@outfrac #1#2#3% +{% + \ifcase\XINT@Sgn{#3} + \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}% +\def\XINT@outfrac@P #1#2% +{% + \ifcase\XINT@Sgn{#1} + \expandafter\XINT@outfrac@Zero + \fi + \space #1/#2% +}% +\def\XINT@outfrac@Zero #1[#2]{ 0[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{xintNumerator}} +% \begin{macrocode} +\def\xintNumerator {\romannumeral0\xintnumerator }% +\def\xintnumerator +{% + \expandafter\XINT@numer\romannumeral0\XINT@infrac +}% +\def\XINT@numer #1% +{% + \ifcase\XINT@Sgn {#1} + \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@Sgn {#1} + \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@@frac@A\romannumeral0\XINT@infrac {#1}% +}% +\def\XINT@@frac@A #1{\XINT@@frac@B #1\Z }% +\def\XINT@@frac@B #1#2\Z +{% + \xint@zero #1\XINT@@frac@C 0\XINT@@frac@D {10^{#1#2}}% +}% +\def\XINT@@frac@C #1#2#3#4#5% +{% + \ifcase\XINT@isOne {#5} + \or \xint@afterfi {\expandafter\xint@firstoftwo@andstop\xint@gobble@two }% + \fi + \space + \frac {#4}{#5}% +}% +\def\XINT@@frac@D #1#2#3% +{% + \ifcase\XINT@isOne {#3} + \or \XINT@@frac@E + \fi + \space + \frac {#2}{#3}#1% +}% +\def\XINT@@frac@E \fi #1#2#3#4{\fi \space #3\cdot }% +% \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@zero #1\XINT@fwover@C 0\XINT@fwover@D {10^{#1#2}}% +}% +\def\XINT@fwover@C #1#2#3#4#5% +{% + \ifcase\XINT@isOne {#5} + \xint@afterfi { {#4\over #5}}% + \or + \xint@afterfi { #4\cdot }% + \fi +}% +\def\XINT@fwover@D #1#2#3% +{% + \ifcase\XINT@isOne {#3} + \xint@afterfi { {#2\over #3}}% + \or + \xint@afterfi { #2\cdot }% + \fi + #1% +}% +% \end{macrocode} +% \subsection{\csh{xintSum}, \csh{xintSumExpr}} +% \begin{macrocode} +\def\xintSum {\romannumeral0\xintsum }% +\def\xintsum #1{\xintsumexpr #1\relax }% +\def\xintSumExpr {\romannumeral0\xintsumexpr }% +\def\xintsumexpr {\expandafter\expandafter\expandafter\XINT@fsumexpr }% +\def\XINT@fsumexpr {\XINT@fsum@loop {0}}% +\def\XINT@fsum@loop #1#2% +{% + \xint@relax #2\XINT@fsum@finished\relax + \expandafter\XINT@fsum@loop\expandafter + {\romannumeral0\xintadd {#1}{#2}}% +}% +\def\XINT@fsum@finished #1#2#3#4#5% +{% + \XINT@fsum@finished@ #5% +}% +\def\XINT@fsum@finished@ #1#2#3#4#5{ #4}% +% \end{macrocode} +% \subsection{\csh{xintPrd}, \csh{xintProductExpr}} +% \begin{macrocode} +\def\xintPrd {\romannumeral0\xintprd }% +\def\xintprd #1{\xintproductexpr #1\relax }% +\def\xintProductExpr {\romannumeral0\xintproductexpr }% +\def\xintproductexpr{\expandafter\expandafter\expandafter\XINT@fproductexpr }% +\def\XINT@fproductexpr {\XINT@fprod@loop {1}}% +\def\XINT@fprod@loop #1#2% +{% + \xint@relax #2\XINT@fprod@finished\relax + \expandafter\XINT@fprod@loop\expandafter + {\romannumeral0\xintmul {#2}{#1}}% +}% +\def\XINT@fprod@finished #1#2#3#4#5% +{% + \XINT@fprod@finished@ #5% +}% +\def\XINT@fprod@finished@ #1#2#3#4#5{ #5}% +% \end{macrocode} +% \subsection{\csh{XINT@inFrac}} +% \begin{macrocode} +\def\XINT@inFrac {\romannumeral0\XINT@infrac }% +\def\XINT@infrac #1% +{% + \expandafter\expandafter\expandafter + \XINT@infrac@ #1[\W]\Z\T +}% +\def\XINT@infrac@ #1[#2#3]#4\Z +{% + \xint@UDwfork + #2\dummy \XINT@infrac@A + \W\dummy \XINT@infrac@B + \xint@UDkrof + #1[#2#3]#4% +}% +\def\XINT@infrac@A #1[\W]\T +{% + \XINT@frac #1/\W\Z +}% +\def\XINT@infrac@B #1% +{% + \xint@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\dummy \XINT@infrac@BCa + \W\dummy {\expandafter\expandafter\expandafter\XINT@infrac@BCb #2}% + \xint@UDkrof + #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}} +% \begin{macrocode} +\def\XINT@frac #1/#2#3\Z +{% + \xint@UDwfork + #2\dummy \XINT@frac@A + \W\dummy {\expandafter\expandafter\expandafter\XINT@frac@B #2}% + \xint@UDkrof + #3.\W\Z #1.\W\Z +}% +\def\XINT@frac@B #1.#2#3\Z +{% + \xint@UDwfork + #2\dummy \XINT@frac@Ba + \W\dummy {\XINT@frac@Bb #2}% + \xint@UDkrof + #3\Z #1\Z +}% +\def\XINT@frac@Bb #1/\W.\W\Z #2\Z +{% + \expandafter \XINT@frac@C \expandafter + {\romannumeral0\XINT@length {#1}}{#2#1}% +}% +\def\XINT@frac@Ba \Z #1/\W\Z {\XINT@frac@C {0}{#1}}% +\def\XINT@frac@A .\W\Z {\XINT@frac@C {0}{1}}% +\def\XINT@frac@C #1#2#3.#4#5\Z +{% + \xint@UDwfork + #4\dummy \XINT@frac@Ca + \W\dummy {\XINT@frac@Cb #4}% + \xint@UDkrof + #5\Z #3\Z {#1}{#2}% +}% +\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\XINT@length {#1}}{#2#1}% +}% +\def\XINT@frac@D #1#2#3#4% +{% + \expandafter \XINT@frac@E \expandafter + {\the\numexpr -#1+#3\expandafter}\expandafter + {\romannumeral0\XINT@num@loop #2\R\R\R\R\R\R\R\R\Z }% + {\romannumeral0\XINT@num@loop #4\R\R\R\R\R\R\R\R\Z }% +}% +\def\XINT@frac@E #1#2#3% +{% + \expandafter \XINT@frac@F #3\Z {#2}{#1}% +}% +\def\XINT@frac@F #1% +{% + \xint@UDzerominusfork + #1-\dummy \XINT@frac@Gdivisionbyzero + 0#1\dummy \XINT@frac@Gneg + 0-\dummy {\XINT@frac@Gpos #1}% + \xint@UDkrof +}% +\def\XINT@frac@Gdivisionbyzero #1\Z #2#3% +{% + \xintError:DivisionByZero + \expandafter\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}} +% \begin{macrocode} +\def\XINT@factortens #1% +{% + \expandafter\XINT@cuz@cnt@loop\expandafter + {\expandafter}\romannumeral0\XINT@rord@main {}#1% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + \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@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 #1#2#3#4#5#6% +{% + \XINT@cuz@cnt@toofarb {#5}#6% +}% +\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@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@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@zero #1\expandafter\XINT@cuz@cnt@loop\xint@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@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\relax}#3% +}% +\def\XINT@cuz@cnt@stopd #1#2\R #3\Z +{% + \expandafter\space\expandafter + {\romannumeral0\XINT@rord@main {}#2% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF }{#1}% +}% +% \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-\dummy \XINT@rez@zero + 0#1\dummy \XINT@rez@neg + 0-\dummy {\XINT@rez@B #1}% + \xint@UDkrof +}% +\def\XINT@rez@zero #1\Z #2#3{ 0/1[0]}% +\def\XINT@rez@neg +{\expandafter\space\expandafter-\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{xintIrr}} +% \begin{macrocode} +\def\XINT@@bts #1#2#3% +{% + \ifcase\XINT@isOne {#2} + \xint@afterfi {#3#1/#2}% + \or + \xint@afterfi {#3#1}% + \fi +}% +\def\xintIrr {\romannumeral0\xintirr }% +\def\xintirr +{% + \expandafter\XINT@@bts + \romannumeral0\expandafter\XINT@irr + \romannumeral0\XINT@infrac +}% +\def\XINT@irr #1% +{% + \ifcase\XINT@Sgn {#1} + \expandafter\XINT@irr@B + \or + \expandafter\XINT@irr@A + \else + \expandafter\XINT@irr@B + \fi + {#1}% +}% +\def\XINT@irr@A #1#2% +{% + \expandafter \XINT@irr@AC \expandafter + {\romannumeral0\xint@dsh {#2}{-#1}}% +}% +\def\XINT@irr@AC #1#2{\XINT@irr@C #2\Z #1\Z }% +\def\XINT@irr@B #1#2#3% +{% + \expandafter \XINT@irr@C \romannumeral0\xint@dsh {#3}{#1}\Z #2\Z +}% +\def\XINT@irr@C #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@irr@minusminus + #1-\dummy \XINT@irr@minusplus + #3-\dummy \XINT@irr@plusminus + --\dummy \XINT@irr@plusplus + \xint@UDkrof + {#4}{#2}#3#1% +}% +\def\XINT@irr@minusminus #1#2#3#4{\XINT@irr@D #1\Z #2\Z \space}% +\def\XINT@irr@minusplus #1#2#3#4{\XINT@irr@D #3#1\Z #2\Z \XINT@opp}% +\def\XINT@irr@plusminus #1#2#3#4{\XINT@irr@D #1\Z #4#2\Z \XINT@opp}% +\def\XINT@irr@plusplus #1#2#3#4{\XINT@irr@D #3#1\Z #4#2\Z \space}% +\def\XINT@irr@D #1#2\Z #3#4\Z +{% + \xint@UDzerosfork + #3#1\dummy \XINT@irr@indeterminate + #30\dummy \XINT@irr@divisionbyzero + #10\dummy \XINT@irr@zero + 00\dummy \XINT@irr@nonzero@checkifone + \xint@UDkrof + {#3#4}{#1#2}{#3#4}{#1#2}% +}% +\def\XINT@irr@indeterminate #1#2#3#4{\expandafter\xintError:ZeroOverZero + \space 00}% +\def\XINT@irr@divisionbyzero #1#2#3#4{\expandafter\xintError:DivisionByZero + \space {#2}0}% +\def\XINT@irr@zero #1#2#3#4{ 0/1}% +\def\XINT@irr@nonzero@checkifone #1% +{% + \ifcase\XINT@isOne {#1} + \xint@afterfi {\XINT@irr@loop@a {#1}}% + \or + \expandafter \XINT@irr@denomisone + \fi +}% +\def\XINT@irr@denomisone #1#2#3{ {#1}1}% +\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#3% +{% + \XINT@irr@loop@e #2\Z {#3}% +}% +\def\XINT@irr@loop@e #1#2\Z +{% + \xint@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\xintquo {#3}{#2}}% + {\romannumeral0\xintquo {#4}{#2}}% +}% +\def\XINT@irr@loop@exitb #1#2% +{% + \expandafter\space\expandafter {#2}{#1}% +}% +% \end{macrocode} +% \subsection{\csh{xintJrr}} +% \begin{macrocode} +\def\xintJrr {\romannumeral0\xintjrr }% +\def\xintjrr +{% + \expandafter\XINT@@bts + \romannumeral0\expandafter\XINT@jrr@start + \romannumeral0\xintrez +}% +\def\XINT@jrr@start #1/#2[#3]% +{% + \ifcase\XINT@Sgn {#3} + \expandafter\XINT@jrr@B + \or + \expandafter\XINT@jrr@A + \else + \expandafter\XINT@jrr@B + \fi + {#3}{#1}{#2}% +}% +\def\XINT@jrr@A #1#2% +{% + \expandafter \XINT@jrr@AC \expandafter + {\romannumeral0\xint@dsh {#2}{-#1}}% +}% +\def\XINT@jrr@AC #1#2{\XINT@jrr@C #2\Z #1\Z }% +\def\XINT@jrr@B #1#2#3% +{% + \expandafter \XINT@jrr@C \romannumeral0\xint@dsh {#3}{#1}\Z #2\Z +}% +\def\XINT@jrr@C #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@jrr@minusminus + #1-\dummy \XINT@jrr@minusplus + #3-\dummy \XINT@jrr@plusminus + --\dummy \XINT@jrr@plusplus + \xint@UDkrof + {#4}{#2}#3#1% +}% +\def\XINT@jrr@minusminus #1#2#3#4{\XINT@jrr@D #1\Z #2\Z \space }% +\def\XINT@jrr@minusplus #1#2#3#4{\XINT@jrr@D #3#1\Z #2\Z \XINT@opp }% +\def\XINT@jrr@plusminus #1#2#3#4{\XINT@jrr@D #1\Z #4#2\Z \XINT@opp }% +\def\XINT@jrr@plusplus #1#2#3#4{\XINT@jrr@D #3#1\Z #4#2\Z \space }% +\def\XINT@jrr@D #1#2\Z #3#4\Z +{% + \xint@UDzerosfork + #3#1\dummy \XINT@jrr@indeterminate + #30\dummy \XINT@jrr@divisionbyzero + #10\dummy \XINT@jrr@zero + 00\dummy \XINT@jrr@nonzero@checkifone + \xint@UDkrof + {#3#4}{#1#2}1001% +}% +\def\XINT@jrr@indeterminate #1#2#3#4#5#6{\expandafter\xintError:ZeroOverZero + \space 00}% +\def\XINT@jrr@divisionbyzero #1#2#3#4#5#6{\expandafter\xintError:DivisionByZero + \space {#2}0}% +\def\XINT@jrr@zero #1#2#3#4#5#6{ 0/1}% +\def\XINT@jrr@nonzero@checkifone #1% +{% + \ifcase\XINT@isOne {#1} + \xint@afterfi {\XINT@jrr@loop@a {#1}}% + \or + \expandafter \XINT@jrr@denomisone + \fi +}% +\def\XINT@jrr@denomisone #1#2#3#4#5{ {#1}1}% +\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\xintiadd{\XINT@Mul{#4}{#1}}{#6}}% + {\romannumeral0\xintiadd{\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@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% +{% + \space {#3}{#4}% +}% +% \end{macrocode} +% \subsection{\csh{xintTrunc}, \csh{xintiTrunc}} +% \begin{macrocode} +\def\xintTrunc {\romannumeral0\xinttrunc }% +\def\xintiTrunc {\romannumeral0\xintitrunc }% +\def\xinttrunc #1% +{% + \expandafter\expandafter\expandafter + \xint@trunc + \expandafter\expandafter\expandafter + {#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\expandafter\expandafter + \xint@itrunc + \expandafter\expandafter\expandafter + {#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\relax}#2\Z {#3}% +}% +\def\XINT@trunc@checkifzero #1#2#3\Z +{% + \xint@zero #2\XINT@trunc@iszero0\XINT@trunc@B {#1}{#2#3}% +}% +\def\XINT@trunc@iszero #1#2#3#4#5{ 0\Z 0}% +\def\XINT@trunc@B #1% +{% + \ifcase\XINT@Sgn {#1} + \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@E + \romannumeral0\xint@dsh {#3}{#1}\Z #2\Z +}% +\def\XINT@trunc@D #1#2% +{% + \expandafter \XINT@trunc@DE \expandafter + {\romannumeral0\xint@dsh {#2}{-#1}}% +}% +\def\XINT@trunc@DE #1#2{\XINT@trunc@E #2\Z #1\Z }% +\def\XINT@trunc@E #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@trunc@minusminus + #1-\dummy {\XINT@trunc@minusplus #3}% + #3-\dummy {\XINT@trunc@plusminus #1}% + --\dummy {\XINT@trunc@plusplus #3#1}% + \xint@UDkrof + {#4}{#2}% +}% +\def\XINT@trunc@minusminus #1#2{\xintquo {#1}{#2}\Z \space}% +\def\XINT@trunc@minusplus #1#2#3{\xintquo {#1#2}{#3}\Z \xint@minus@andstop}% +\def\XINT@trunc@plusminus #1#2#3{\xintquo {#2}{#1#3}\Z \xint@minus@andstop}% +\def\XINT@trunc@plusplus #1#2#3#4{\xintquo {#1#3}{#2#4}\Z \space}% +\def\XINT@itrunc@G #1\Z #2#3% +{% + \xint@zero #2\XINT@trunc@zero 0\xint@firstoftwo {#2#1}0% +}% +\def\XINT@trunc@G #1\Z #2#3% +{% + \xint@zero #2\XINT@trunc@zero 0% + \expandafter\XINT@trunc@H\expandafter + {\the\numexpr\romannumeral0\XINT@length {#1}-#3\relax}{#3}{#1}#2% +}% +\def\XINT@trunc@zero 0#10{ 0}% +\def\XINT@trunc@H #1#2% +{% + \ifnum #1 > 0 + \xint@afterfi {\XINT@trunc@Ha {#2}}% + \else + \xint@afterfi {\XINT@trunc@Hb {-#1}}% + \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% +}% +% \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\xintimul {#6}{#3}}% + {\romannumeral0\xintimul {#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}} +% \begin{macrocode} +\def\xintPow {\romannumeral0\xintpow }% +\def\xintpow #1% +{% + \expandafter\xint@fpow\expandafter {\romannumeral0\XINT@infrac {#1}}% +}% +\def\xint@fpow #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@fpow@fork + #2\Z #1% +}% +\def\XINT@fpow@fork #1#2\Z +{% + \xint@UDzerominusfork + #1-\dummy \XINT@fpow@zero + 0#1\dummy \XINT@fpow@neg + 0-\dummy {\XINT@fpow@pos #1}% + \xint@UDkrof + {#2}% +}% +\def\XINT@fpow@zero #1#2#3#4% +{% + \space 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\xintipow {#5}{#1#2}}% + {\romannumeral0\xintipow {#4}{#1#2}}% +}% +\def\XINT@fpow@neg #1#2#3#4% +{% + \expandafter\XINT@fpow@pos@A\expandafter + {\the\numexpr -#1*#2\expandafter}\expandafter + {\romannumeral0\xintipow {#3}{#1}}% + {\romannumeral0\xintipow {#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{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\xintimul {#2}{#6}}% + {\romannumeral0\xintimul {#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{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\xintimul {#7}{#5}}% + {\romannumeral0\xintiadd + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#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\xintimul {#7}{#5}}% + {\romannumeral0\xintisub + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#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{xintCmp}} +% \begin{macrocode} +\def\xintCmp {\romannumeral0\xintcmp }% +\def\xintcmp #1% +{% + \expandafter\xint@fcmp\expandafter {\romannumeral0\XINT@infrac {#1}}% +}% +\def\xint@fcmp #1#2{\expandafter\XINT@fcmp@A\romannumeral0\XINT@infrac {#2}#1}% +\def\XINT@fcmp@A #1#2#3#4% +{% + \ifnum #4 > #1 + \xint@afterfi {\XINT@fcmp@B {#1}}% + \else + \xint@afterfi {\XINT@fcmp@B {#4}}% + \fi + {#1}{#4}{#2}{#3}% +}% +\def\XINT@fcmp@B #1#2#3#4#5#6#7% +{% + \xinticmp + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}% +}% +% \end{macrocode} +% \subsection{\csh{xintMax}} +% \begin{macrocode} +\def\xintMax {\romannumeral0\xintmax }% +\def\xintmax #1% +{% + \expandafter\xint@fmax\expandafter {\romannumeral0\XINT@infrac {#1}}% +}% +\def\xint@fmax #1#2{\expandafter\XINT@outfrac + \romannumeral0\expandafter\XINT@fmax@A + \romannumeral0\XINT@infrac {#2}#1}% +\def\XINT@fmax@A #1#2#3#4#5#6% +{% + \ifnum #4 > #1 + \xint@afterfi {\XINT@fmax@B {#1}}% + \else + \xint@afterfi {\XINT@fmax@B {#4}}% + \fi + {#1}{#4}{#2}{#3}{#5}{#6}{{#4}{#5}{#6}}{{#1}{#2}{#3}}% +}% +\def\XINT@fmax@B #1#2#3#4#5#6#7% +{% + \expandafter\XINT@fmax@C\expandafter + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}% +}% +\def\XINT@fmax@C #1#2% +{% + \expandafter\XINT@max@fork #2\Z #1\Z +}% +% \end{macrocode} +% \subsection{\csh{xintMin}} +% \begin{macrocode} +\def\xintMin {\romannumeral0\xintmin }% +\def\xintmin #1% +{% + \expandafter\xint@fmin\expandafter {\romannumeral0\XINT@infrac {#1}}% +}% +\def\xint@fmin #1#2{\expandafter\XINT@outfrac + \romannumeral0\expandafter\XINT@fmin@A + \romannumeral0\XINT@infrac {#2}#1}% +\def\XINT@fmin@A #1#2#3#4#5#6% +{% + \ifnum #4 > #1 + \xint@afterfi {\XINT@fmin@B {#1}}% + \else + \xint@afterfi {\XINT@fmin@B {#4}}% + \fi + {#1}{#4}{#2}{#3}{#5}{#6}{{#4}{#5}{#6}}{{#1}{#2}{#3}}% +}% +\def\XINT@fmin@B #1#2#3#4#5#6#7% +{% + \expandafter\XINT@fmin@C\expandafter + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}% +}% +\def\XINT@fmin@C #1#2% +{% + \expandafter\XINT@min@fork #2\Z #1\Z +}% +% \end{macrocode} +% \subsection{\csh{xintSgn}} +% \begin{macrocode} +\def\xintSgn {\romannumeral0\xintsgn }% +\def\xintsgn #1% +{% + \expandafter\xint@fsgn\romannumeral0\XINT@infrac {#1}% +}% +\def\xint@fsgn #1#2#3{\xintisgn {#2}}% +% \end{macrocode} +% \subsection{\csh{xintOpp}} +% \begin{macrocode} +\def\xintOpp {\romannumeral0\xintopp }% +\def\xintopp #1% +{% + \expandafter\xint@fopp\romannumeral0\XINT@infrac {#1}% +}% +\def\xint@fopp #1#2{\expandafter\XINT@outfrac\expandafter + {\the\numexpr #1\expandafter}\expandafter + {\romannumeral0\XINT@opp #2}}% +% \end{macrocode} +% \subsection{\csh{xintAbs}} +% \begin{macrocode} +\def\xintAbs {\romannumeral0\xintabs }% +\def\xintabs #1% +{% + \expandafter\xint@fabs\romannumeral0\XINT@infrac {#1}% +}% +\def\xint@fabs #1#2{\expandafter\XINT@outfrac\expandafter + {\the\numexpr #1\expandafter}\expandafter + {\romannumeral0\XINT@abs #2}}% +\XINT@frac@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> +% \section{Package \xintseriesname implementation} +% +% The commenting is currently (\docdate) very sparse. +% +% \localtableofcontents +% \subsection{Catcodes, \eTeX{} detection, 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}{Package xintfrac is required}% + \y{xintseries}{Will try \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}{Package xintfrac is required}% + \y{xintseries}{Will try \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 \xintfracname 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 % : + \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 + \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} +% +% Perhaps catcodes have changed after the loading of \xintname and +% \xintfracname and prior to the current loading of \xintseriesname, +% so we can not employ the |\XINT@restorecatcodes@endinput| in this style +% file. But there is no problem using |\XINT@setcatcodes|. +% +% \begin{macrocode} +\begingroup\catcode61\catcode48\catcode32=10\relax% + \catcode13=5 % ^^M + \endlinechar=13 % + \catcode123=1 % { + \catcode125=2 % } + \catcode64=11 % @ + \def\x + {% + \endgroup + \edef\XINT@series@restorecatcodes@endinput + {% + \catcode93=\the\catcode93 % ] + \catcode91=\the\catcode91 % [ + \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 % # + \catcode64=\the\catcode64 % @ + \catcode125=\the\catcode125 % } + \catcode123=\the\catcode123 % { + \endlinechar=\the\endlinechar + \catcode13=\the\catcode13 % ^^M + \catcode32=\the\catcode32 % + \catcode61=\the\catcode61 % = + \noexpand\endinput + }% + \XINT@setcatcodes + \catcode91=12 % [ + \catcode93=12 % ] + }% +\x +% \end{macrocode} +% \subsection{Package identification} +% \begin{macrocode} +\begingroup + \catcode58=12 % : + \expandafter\ifx\csname ProvidesPackage\endcsname\relax + \def\x#1#2#3[#4]{\endgroup + \immediate\write-1{Package: #3 #4}% + \xdef#1{#4}% + }% + \else + \def\x#1#2[#3]{\endgroup + #2[{#3}]% + \ifx#1\@undefined + \xdef#1{#3}% + \fi + \ifx#1\relax + \xdef#1{#3}% + \fi + }% + \fi +\expandafter\x\csname ver@xintseries.sty\endcsname +\ProvidesPackage{xintseries}% + [2013/04/14 v1.03 Expandable partial sums with xint package (jfB)]% +% \end{macrocode} +% \subsection{\csh{xintSeries}} +% \begin{macrocode} +\def\xintSeries {\romannumeral0\xintseries }% +\def\xintseries #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@series@i + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@series@i #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@series@ii + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@series@ii #1#2#3% +{% + \ifnum #2<#1 + \xint@afterfi { 0[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@two #6% +}% +% \end{macrocode} +% \subsection{\csh{xintiSeries}} +% \begin{macrocode} +\def\xintiSeries {\romannumeral0\xintiseries }% +\def\xintiseries #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@iseries@i + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@iseries@i #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@iseries@ii + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@iseries@ii #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\xintiadd {#2}{#4{#1}}}% + {#3}{#4}% +}% +\def\XINT@iseries@exit \fi #1#2#3#4#5#6#7#8% +{% + \fi\xint@gobble@two #6% +}% +% \end{macrocode} +% \subsection{\csh{xintPowerSeries}} +% \begin{macrocode} +\def\xintPowerSeries {\romannumeral0\xintpowerseries }% +\def\xintpowerseries #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@powseries@i + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@powseries@i #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@powseries@ii + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@powseries@ii #1#2#3#4% +{% + \ifnum #2<#1 + \xint@afterfi { 0[0]}% + \else + \xint@afterfi + {\expandafter\XINT@powseries@loop@pre\expandafter + {\romannumeral0\xintpow {#4}{#1}}{#1}{#4}{#2}{#3}% + }% + \fi +}% +\def\XINT@powseries@loop@pre #1#2#3#4#5% +{% + \ifnum #4>#2 \else\XINT@powseries@dont@i \fi + \expandafter\XINT@powseries@loop@i\expandafter + {\the\numexpr #2+1\expandafter}\expandafter + {\romannumeral0\xintmul {#5{#2}}{#1}}{#1}{#3}{#4}{#5}% +}% +\def\XINT@powseries@dont@i \fi\expandafter\XINT@powseries@loop@i + {\fi \expandafter\XINT@powseries@dont@ii }% +\def\XINT@powseries@dont@ii #1#2#3#4#5#6{ #2}% +\def\XINT@powseries@loop@i #1#2#3#4#5% +{% + \ifnum #5>#1 \else \XINT@powseries@exit@i \fi + \expandafter\XINT@powseries@loop@ii\expandafter + {\romannumeral0\xintmul {#3}{#4}}{#1}{#4}{#2}{#5}% +}% +\def\XINT@powseries@loop@ii #1#2#3#4#5#6% +{% + \expandafter\XINT@powseries@loop@i\expandafter + {\the\numexpr #2+1\expandafter}\expandafter + {\romannumeral0\xintadd {#4}{\xintMul {#6{#2}}{#1}}}% + {#1}{#3}{#5}{#6}% +}% +\def\XINT@powseries@exit@i\fi \expandafter\XINT@powseries@loop@ii + {\fi \expandafter\XINT@powseries@exit@ii }% +\def\XINT@powseries@exit@ii #1#2#3#4#5#6% + {\xintadd {#4}{\xintMul {#6{#2}}{#1}}}% +% \end{macrocode} +% \subsection{\csh{xintFxPtPowerSeries}} +% \begin{verbatim} +% I am not two happy with this piece of code. Will make it more economical +% another day. +% \end{verbatim} +% \vspace*{-1.5\baselineskip} +% \begin{macrocode} +\def\xintFxPtPowerSeries {\romannumeral0\xintfxptpowerseries }% +\def\xintfxptpowerseries #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@fppowseries@i + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@fppowseries@i #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@fppowseries@ii + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@fppowseries@ii #1#2#3#4#5% +{% + \ifnum #2<#1 + \xint@afterfi {\xinttrunc {#5}{0[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+1\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+1\expandafter}\expandafter + {\romannumeral0\xintiadd {#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}% + {\xintiAdd {#4}{\xintiTrunc {#7}{\xintMul {#6{#2}}{#1}}}[-#7]}}% +\XINT@series@restorecatcodes@endinput% +% \end{macrocode} % \DeleteShortVerb{\|} % \MakePercentComment -%</xintgcd> +%</xintseries> %<*none> \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 @@ -6485,7 +9012,7 @@ and modify it to what is needed. Grave accent \` Left brace \{ Vertical bar \| Right brace \} Tilde \~} -\CheckSum{6549} +\CheckSum{9336} \Finale %% diff --git a/Master/texmf-dist/source/generic/xint/xint.ins b/Master/texmf-dist/source/generic/xint/xint.ins index bb4722ead3f..52151674d3c 100644 --- a/Master/texmf-dist/source/generic/xint/xint.ins +++ b/Master/texmf-dist/source/generic/xint/xint.ins @@ -1,12 +1,13 @@ %% %%---------------------------------------------------------------- -%% The xint bundle (version 1.02 of April 5, 2013) +%% The xint bundle (version 1.03 of April 14, 2013) %% Copyright (C) 2013 by Jean-Francois Burnol %%---------------------------------------------------------------- %% %% %% This is a generated file. Run tex or latex on this file to -%% extract xint.sty and xintgcd.sty from xint.dtx +%% extract xint.sty, xintgcd.sty, xintfrac.sty and xintseries.sty +%% from xint.dtx %% %% See xint.dtx for the statements of copyright and conditions of %% distribution and/or modification of this work. @@ -15,7 +16,9 @@ \askforoverwritefalse \generate{\usepreamble\defaultpreamble \file{xint.sty}{\from{xint.dtx}{xint}} -\file{xintgcd.sty}{\from{xint.dtx}{xintgcd}}} +\file{xintgcd.sty}{\from{xint.dtx}{xintgcd}} +\file{xintfrac.sty}{\from{xint.dtx}{xintfrac}} +\file{xintseries.sty}{\from{xint.dtx}{xintseries}}} \endbatchfile \endinput %% diff --git a/Master/texmf-dist/tex/generic/xint/xint.sty b/Master/texmf-dist/tex/generic/xint/xint.sty index b8489389530..e121c9bcffb 100644 --- a/Master/texmf-dist/tex/generic/xint/xint.sty +++ b/Master/texmf-dist/tex/generic/xint/xint.sty @@ -22,7 +22,7 @@ %% in the same archive or directory.) %% %%---------------------------------------------------------------- -%% The xint bundle (version 1.02 of April 5, 2013) +%% The xint bundle (version 1.03 of April 14, 2013) %% xint: Expandable operations on long numbers %% Copyright (C) 2013 by Jean-Francois Burnol %%---------------------------------------------------------------- @@ -134,7 +134,7 @@ \fi \expandafter\x\csname ver@xint.sty\endcsname \ProvidesPackage{xint}% - [2013/04/05 v1.02 Expandable operations on long numbers (jfB)]% + [2013/04/14 v1.03 Expandable operations on long numbers (jfB)]% \def\xint@gobble #1{}% \def\xint@gobble@one #1{}% \def\xint@gobble@two #1#2{}% @@ -144,12 +144,14 @@ \def\xint@gobble@six #1#2#3#4#5#6{}% \def\xint@gobble@seven #1#2#3#4#5#6#7{}% \def\xint@gobble@eight #1#2#3#4#5#6#7#8{}% +\def\xint@firstoftwo #1#2{#1}% \def\xint@secondoftwo #1#2{#2}% \def\xint@firstoftwo@andstop #1#2{ #1}% \def\xint@secondoftwo@andstop #1#2{ #2}% \def\xint@exchangetwo@keepbraces #1#2{{#2}{#1}}% \def\xint@exchangetwo@keepbraces@andstop #1#2{ {#2}{#1}}% \def\xint@xpxp@andstop {\expandafter\expandafter\expandafter\space }% +\def\xint@minus@andstop { -}% \def\xint@r #1\R {}% \def\xint@w #1\W {}% \def\xint@z #1\Z {}% @@ -161,6 +163,7 @@ \def\xint@bracedundef {\xint@undef }% \def\xint@UDzerofork #10\dummy #2#3\xint@UDkrof {#2}% \def\xint@UDsignfork #1-\dummy #2#3\xint@UDkrof {#2}% +\def\xint@UDwfork #1\W\dummy #2#3\xint@UDkrof {#2}% \def\xint@UDzerosfork #100\dummy #2#3\xint@UDkrof {#2}% \def\xint@UDonezerofork #110\dummy #2#3\xint@UDkrof {#2}% \def\xint@UDzerominusfork #10-\dummy #2#3\xint@UDkrof {#2}% @@ -188,11 +191,7 @@ }% \def\XINT@rev@negative #1#2\Z {% - \expandafter - \space - \expandafter - -% - \romannumeral0\XINT@rev {#2}% + \expandafter \space \expandafter -\romannumeral0\XINT@rev {#2}% }% \def\XINT@rev@nonnegative #1\Z {% @@ -296,6 +295,16 @@ }% \def\XINT@cuz@Stop #1\W #2\Z{ #1}% \def\xint@cuz@backtoloop 0\XINT@cuz@Stop 0{\XINT@cuz@loop }% +\def\XINT@isOne #1{\romannumeral0\XINT@isone #1\W\Z }% +\def\XINT@isone #1#2% +{% + \xint@one #1\XINT@isone@b 1\expandafter\space\expandafter 0\xint@z #2% +}% +\def\XINT@isone@b #1\xint@z #2% +{% + \xint@w #2\XINT@isone@yes\W\expandafter\space\expandafter 0\xint@z +}% +\def\XINT@isone@yes #1\Z{ 1}% \def\xintNum {\romannumeral0\xintnum }% \def\xintnum #1% {% @@ -331,12 +340,13 @@ \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\R #2\Z { #1}% -\def\xintLen {\romannumeral0\xintlen }% -\def\xintlen #1% +\def\xintiLen {\romannumeral0\xintilen }% +\def\xintilen #1% {% \expandafter\expandafter\expandafter \XINT@length@fork #1\R\R\R\R\R\R\R\R\Z }% +\let\xintLen\xintiLen \let\xintlen\xintilen \def\XINT@Len #1{\romannumeral0\XINT@length@fork #1\R\R\R\R\R\R\R\R\Z }% \def\XINT@length@fork #1% {% @@ -495,12 +505,13 @@ }% }% \let\xintDigitsOf\xintAssignArray -\def\xintSgn {\romannumeral0\xintsgn }% -\def\xintsgn #1% +\def\xintiSgn {\romannumeral0\xintisgn }% +\def\xintisgn #1% {% \expandafter\expandafter\expandafter \XINT@sgn #1\Z% }% +\let\xintSgn\xintiSgn \let\xintsgn\xintisgn \def\XINT@Sgn #1{\romannumeral0\XINT@sgn #1\Z }% \def\XINT@sgn #1% {% @@ -512,12 +523,13 @@ \xint@UDkrof \xint@z }% -\def\xintOpp {\romannumeral0\xintopp }% -\def\xintopp #1% +\def\xintiOpp {\romannumeral0\xintiopp }% +\def\xintiopp #1% {% \expandafter\expandafter\expandafter \XINT@opp #1% }% +\let\xintOpp\xintiOpp \let\xintopp\xintiopp \def\XINT@Opp #1{\romannumeral0\XINT@opp #1}% \def\XINT@opp #1% {% @@ -528,13 +540,14 @@ 0-\dummy {-#1}% positive \xint@UDkrof }% -\def\xintAbs {\romannumeral0\xintabs }% -\def\xintabs #1% +\def\xintiAbs {\romannumeral0\xintiabs }% +\def\xintiabs #1% {% \expandafter\expandafter\expandafter \XINT@abs #1% }% -\def\XINT@Abs {\romannumeral0\XINT@abs }% +\let\xintAbs\xintiAbs \let\xintabs\xintiabs +\def\XINT@Abs #1{\romannumeral0\XINT@abs #1}% \def\XINT@abs #1% {% \xint@UDsignfork @@ -542,14 +555,348 @@ -\dummy { #1}% \xint@UDkrof }% -\def\xintAdd {\romannumeral0\xintadd }% -\def\xintadd #1% +\def\XINT@add@A #1#2#3#4#5#6% +{% + \xint@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% +}% +\def\XINT@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8% +{% + \xint@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\relax.% +}% +\def\XINT@add@ABEA #1#2#3.#4% +{% + \XINT@add@A #2{#3#4}% +}% +\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\relax.% +}% +\def\XINT@add@CC #1#2#3.#4% +{% + \XINT@add@AC@checkcarry #2{#3#4}% on va examiner et \'eliminer #2 +}% +\def\XINT@add@AC@checkcarry #1% +{% + \xint@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@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + #1% +}% +\def\XINT@add@C #1#2#3#4#5% +{% + \xint@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\relax.% +}% +\def\xint@add@cz\W\XINT@add@CD #1#2{ 1#2}% +\def\XINT@addr@A #1#2#3#4#5#6% +{% + \xint@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@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@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@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}% +\def\XINT@addm@A #1#2#3#4#5#6% +{% + \xint@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\relax.% +}% +\def\XINT@addm@ABEA #1#2#3.#4% +{% + \XINT@addm@A #2{#3#4}% +}% +\def\XINT@addm@AC@checkcarry #1% +{% + \xint@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@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + #1% +}% +\def\XINT@addm@C #1#2#3#4#5% +{% + \xint@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\relax.% +}% +\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\relax.% +}% +\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\relax.% +}% +\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\relax.% +}% +\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}% +\def\XINT@addm@end #1#2#3#4#5% +{\expandafter\space\the\numexpr #1#2#3#4#5\relax +}% +\def\XINT@addp@A #1#2#3#4#5#6% +{% + \xint@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@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@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 1+10#1#2#3\relax +}% +\def\XINT@addp@CDw #1#2#3#4#5#6% +{% + \xint@quatrezeros #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 1+100#1#2\relax +}% +\def\XINT@addp@CDx #1#2#3#4#5#6% +{% + \xint@quatrezeros #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 1+1000#1\relax +}% +\def\XINT@addp@CDy #1#2#3#4#5#6% +{% + \xint@quatrezeros #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@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@quatrezeros #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@quatrezeros #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@quatrezeros #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}% +\def\xintiAdd {\romannumeral0\xintiadd }% +\def\xintiadd #1% {% \expandafter\expandafter\expandafter \xint@add \expandafter\expandafter\expandafter {#1}% }% +\let\xintAdd\xintiAdd \let\xintadd\xintiadd \def\xint@add #1#2% {% \expandafter\expandafter\expandafter @@ -604,73 +951,15 @@ \romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z \W\X\Y\Z #1\W\X\Y\Z }% -\def\XINT@add@A #1#2#3#4#5#6% -{% - \xint@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% -}% -\def\XINT@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint@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\relax.% -}% -\def\XINT@add@ABEA #1#2#3.#4% -{% - \XINT@add@A #2{#3#4}% -}% -\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\relax.% -}% -\def\XINT@add@CC #1#2#3.#4% -{% - \XINT@add@AC@checkcarry #2{#3#4}% on va examiner et \'eliminer #2 -}% -\def\XINT@add@AC@checkcarry #1% -{% - \xint@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@UNDEF - \xint@undef\xint@undef\xint@undef\xint@undef - \xint@undef\xint@undef\xint@undef\xint@undef - \xint@UNDEF - #1% -}% -\def\XINT@add@C #1#2#3#4#5% -{% - \xint@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\relax.% -}% -\def\xint@add@cz\W\XINT@add@CD #1#2{ 1#2}% -\def\xintSub {\romannumeral0\xintsub }% -\def\xintsub #1% +\def\xintiSub {\romannumeral0\xintisub }% +\def\xintisub #1% {% \expandafter\expandafter\expandafter \xint@sub \expandafter\expandafter\expandafter {#1}% }% +\let\xintSub\xintiSub \let\xintsub\xintisub \def\xint@sub #1#2% {% \expandafter\expandafter\expandafter @@ -912,7 +1201,7 @@ }% \def\XINT@sub@KK@onestep #1#2% {\expandafter - \XINT@sub@backtoKK\the\numexpr 110000-#2+#1-1\relax.% + \XINT@sub@backtoKK\the\numexpr 109999-#2+#1\relax.% }% \def\XINT@sub@backtoKK #1#2#3.#4% {% @@ -923,14 +1212,15 @@ \expandafter\space\expandafter-\romannumeral 0\XINT@cuz@loop #3\W\W\W\W\W\W\W\Z }% -\def\xintCmp {\romannumeral0\xintcmp }% -\def\xintcmp #1% +\def\xintiCmp {\romannumeral0\xinticmp }% +\def\xinticmp #1% {% \expandafter\expandafter\expandafter \xint@cmp \expandafter\expandafter\expandafter {#1}% }% +\let\xintCmp\xintiCmp \let\xintcmp\xinticmp \def\xint@cmp #1#2% {% \expandafter\expandafter\expandafter @@ -1125,20 +1415,23 @@ 0\dummy { 1} % pas de retenue \xint@UDkrof }% -\def\xintMax {\romannumeral0\xintmax }% -\def\xintmax #1% +\def\xintiMax {\romannumeral0\xintimax }% +\def\xintimax #1% {% \expandafter\expandafter\expandafter \xint@max \expandafter\expandafter\expandafter {#1}% }% +\let\xintMax\xintiMax \let\xintmax\xintimax \def\xint@max #1#2% {% \expandafter\expandafter\expandafter - \XINT@max@fork #2\Z #1\Z + \XINT@max@pre + \expandafter\expandafter\expandafter {#2}{#1}% }% -\def\XINT@Max #1#2{\romannumeral0\XINT@max@fork #2\Z #1\Z }% +\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}}% \def\XINT@max@fork #1#2\Z #3#4\Z {% \xint@UDsignsfork @@ -1154,41 +1447,44 @@ \xint@UDkrof {#2}{#4}#1#3% }% -\def\XINT@max@zerozero #1#2#3#4{ 0}% -\def\XINT@max@zeroplus #1#2#3#4{ #4#2}% -\def\XINT@max@pluszero #1#2#3#4{ #3#1}% -\def\XINT@max@minusplus #1#2#3#4{ #4#2}% -\def\XINT@max@plusminus #1#2#3#4{ #3#1}% -\def\XINT@max@plusplus #1#2#3#4% +\def\XINT@max@zerozero #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@max@zeroplus #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@max@pluszero #1#2#3#4{\xint@secondoftwo@andstop }% +\def\XINT@max@minusplus #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@max@plusminus #1#2#3#4{\xint@secondoftwo@andstop }% +\def\XINT@max@plusplus #1#2#3#4% {% \ifodd\XINT@Geq {#4#2}{#3#1} - \xint@afterfi { #4#2}% + \expandafter\xint@firstoftwo@andstop \else - \xint@afterfi { #3#1}% + \expandafter\xint@secondoftwo@andstop \fi }% \def\XINT@max@minusminus #1#2#3#4% {% \ifodd\XINT@Geq {#1}{#2} - \xint@afterfi { -#2}% + \expandafter\xint@firstoftwo@andstop \else - \xint@afterfi { -#1}% + \expandafter\xint@secondoftwo@andstop \fi }% -\def\xintMin {\romannumeral0\xintmin }% -\def\xintmin #1% +\def\xintiMin {\romannumeral0\xintimin }% +\def\xintimin #1% {% \expandafter\expandafter\expandafter \xint@min \expandafter\expandafter\expandafter {#1}% }% +\let\xintMin\xintiMin \let\xintmin\xintimin \def\xint@min #1#2% {% \expandafter\expandafter\expandafter - \XINT@min@fork #2\Z #1\Z + \XINT@min@pre + \expandafter\expandafter\expandafter {#2}{#1}% }% -\def\XINT@Min #1#2{\romannumeral0\XINT@min@fork #2\Z #1\Z }% +\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}}% \def\XINT@min@fork #1#2\Z #3#4\Z {% \xint@UDsignsfork @@ -1204,166 +1500,82 @@ \xint@UDkrof {#2}{#4}#1#3% }% -\def\XINT@min@zerozero #1#2#3#4{ 0}% -\def\XINT@min@zeroplus #1#2#3#4{ 0}% -\def\XINT@min@pluszero #1#2#3#4{ 0}% -\def\XINT@min@minusplus #1#2#3#4{ #3#1}% -\def\XINT@min@plusminus #1#2#3#4{ #4#2}% +\def\XINT@min@zerozero #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@min@zeroplus #1#2#3#4{\xint@secondoftwo@andstop }% +\def\XINT@min@pluszero #1#2#3#4{\xint@firstoftwo@andstop }% +\def\XINT@min@minusplus #1#2#3#4{\xint@secondoftwo@andstop }% +\def\XINT@min@plusminus #1#2#3#4{\xint@firstoftwo@andstop }% \def\XINT@min@plusplus #1#2#3#4% {% \ifodd\XINT@Geq {#4#2}{#3#1} - \xint@afterfi { #3#1}% + \expandafter\xint@secondoftwo@andstop \else - \xint@afterfi { #4#2}% + \expandafter\xint@firstoftwo@andstop \fi }% \def\XINT@min@minusminus #1#2#3#4% {% \ifodd\XINT@Geq {#1}{#2} - \xint@afterfi { -#1}% + \expandafter\xint@secondoftwo@andstop \else - \xint@afterfi { -#2}% + \expandafter\xint@firstoftwo@andstop \fi }% -\def\XINT@psum #1% -{% - \romannumeral0\XINT@psum@checkifemptysum #1\Z -}% -\def\XINT@psum@checkifemptysum #1% -{% - \xint@relax #1\XINT@psum@returnzero\relax \XINT@psum@RQfirst #1% -}% -\def\XINT@psum@returnzero #1\Z { 0}% -\def\XINT@psum@RQfirst #1\Z -{% - \expandafter\XINT@psum@loop\expandafter - {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z}% avant: #1\Z -}% -\def\XINT@psum@loop #1#2% -{% - \xint@relax #2\XINT@psum@end\relax - \expandafter - \XINT@psum@loop\expandafter - {\romannumeral0\expandafter\XINT@sum@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 }% -}% -\def\XINT@psum@end\relax\expandafter - \XINT@psum@loop\expandafter #1% - {\XINT@psum@end@ #1}% -\def\XINT@psum@end@ #1\W\X\Y\Z #2\W\X\Y\Z -{% - \expandafter - \xint@cleanupzeros@andstop\romannumeral0\XINT@rev {#2}% -}% -\def\xintSumExpr {\romannumeral0\xintsumexpr }% -\def\xintSum {\romannumeral0\xintsum }% -\def\xintsum #1% +\def\xintiSum {\romannumeral0\xintisum }% +\def\xintisum #1{\xintisumexpr #1\relax }% +\def\xintiSumExpr {\romannumeral0\xintisumexpr }% +\def\xintisumexpr {% - \expandafter\expandafter\expandafter - \xintsumexpr #1\relax + \expandafter\expandafter\expandafter\XINT@sumexpr }% -\def\xintsumexpr #1% +\let\xintSum\xintiSum \let\xintsum\xintisum +\let\xintSumExpr\xintiSumExpr \let\xintsumexpr\xintisumexpr +\def\XINT@sumexpr {\XINT@sum@loop {0000}{0000}}% +\def\XINT@sum@loop #1#2#3% {% + \xint@relax #3\XINT@sum@finished\relax \expandafter\expandafter\expandafter - \XINT@sum@checkifempty #1\Z {\XINT@psum }{\XINT@psum }% -}% -\def\XINT@sum@checkifempty #1% -{% - \xint@relax #1\XINT@sum@returnzero\relax - \XINT@sum@checksign #1% + \XINT@sum@checksign #3\Z {#1}{#2}% }% -\def\XINT@sum@returnzero #1\Z #2#3{ 0}% \def\XINT@sum@checksign #1% {% \xint@zero #1\XINT@sum@skipzeroinput0% \xint@UDsignfork - #1\dummy \XINT@sum@pushneg - -\dummy \XINT@sum@pushpos + #1\dummy \XINT@sum@N + -\dummy {\XINT@sum@P #1}% \xint@UDkrof - #1% }% -\def\XINT@sum@skipzeroinput #1\xint@UDkrof #2\Z #3#4% +\def\XINT@sum@skipzeroinput #1\xint@UDkrof #2\Z {\XINT@sum@loop }% +\def\XINT@sum@P #1\Z #2% {% - \XINT@sum@xpxpnext {#3}{#4}% + \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@pushpos #1#2\Z #3#4% +\def\XINT@sum@N #1\Z #2#3% {% - \XINT@sum@xpxpnext {#3{#1#2}}{#4}% + \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@pushneg #1#2\Z #3#4% +\def\XINT@sum@NN #1#2{\XINT@sum@loop {#2}{#1}}% +\def\XINT@sum@finished #1\Z #2#3% {% - \XINT@sum@xpxpnext {#3}{#4{#2}}% + \XINT@sub@A 1{}#3\W\X\Y\Z #2\W\X\Y\Z }% -\def\XINT@sum@xpxpnext #1#2#3% -{% - \expandafter\expandafter\expandafter - \XINT@sum@checkiffinished #3\Z {#1}{#2}% -}% -\def\XINT@sum@checkiffinished #1% -{% - \xint@relax #1\XINT@sum@end\relax - \XINT@sum@checksign #1% -}% -\def\XINT@sum@end\relax\XINT@sum@checksign\relax #1\Z #2#3% - {\xintsub{#2\relax}{#3\relax}}% -\def\XINT@sum@A #1#2#3#4#5#6% -{% - \xint@w - #3\xint@sum@az - \W\XINT@sum@B #1{#3#4#5#6}{#2}% -}% -\def\xint@sum@az\W\XINT@sum@B #1#2% -{% - \XINT@sum@AC@checkcarry #1% -}% -\def\XINT@sum@B #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \xint@w - #5\xint@sum@bz - \W\XINT@sum@E #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT@sum@E #1#2#3#4#5#6% -{\expandafter - \XINT@sum@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax -}% -\def\XINT@sum@ABEA #1#2#3#4#5#6#7% -{% - \XINT@sum@A #2{#7#6#5#4#3}% -}% -\def\xint@sum@bz\W\XINT@sum@E #1#2#3#4#5#6% -{\expandafter - \XINT@sum@CC\the\numexpr #1+10#5#4#3#2\relax -}% -\def\XINT@sum@CC #1#2#3#4#5#6#7% -{% - \XINT@sum@AC@checkcarry #2{#7#6#5#4#3}% -}% -\def\XINT@sum@AC@checkcarry #1% -{% - \xint@zero #1\xint@sum@AC@nocarry 0\XINT@sum@C -}% -\def\xint@sum@AC@nocarry 0\XINT@sum@C #1#2\W\X\Y\Z { #1#2}% -\def\XINT@sum@C #1#2#3#4#5% -{% - \xint@w - #2\xint@sum@cz - \W\XINT@sum@D {#5#4#3#2}{#1}% -}% -\def\XINT@sum@D #1% -{\expandafter - \XINT@sum@CC\the\numexpr 1+10#1\relax -}% -\def\xint@sum@cz\W\XINT@sum@D #1#2{ #21000}% -\def\xintMul {\romannumeral0\xintmul }% -\def\xintmul #1% +\def\xintiMul {\romannumeral0\xintimul }% +\def\xintimul #1% {% \expandafter\expandafter\expandafter \xint@mul \expandafter\expandafter\expandafter {#1}% }% +\let\xintMul\xintiMul \let\xintmul\xintimul \def\xint@mul #1#2% {\expandafter\expandafter\expandafter \XINT@mul@fork #2\Z #1\Z @@ -1377,136 +1589,129 @@ 0\dummy {\xint@UDsignsfork #1#3\dummy \XINT@mul@minusminus % #1 = #3 = - - #1-\dummy \XINT@mul@minusplus % #1 = - - #3-\dummy \XINT@mul@plusminus % #3 = - - --\dummy \XINT@mul@plusplus + #1-\dummy {\XINT@mul@minusplus #3}% % #1 = - + #3-\dummy {\XINT@mul@plusminus #1}% % #3 = - + --\dummy {\XINT@mul@plusplus #1#3}% \xint@UDkrof }% \xint@UDkrof - {#2}{#4}#1#3% + {#2}{#4}% }% -\def\XINT@mul@zero #1#2#3#4{ 0}% -\def\XINT@mul@minusminus #1#2#3#4% +\def\XINT@mul@zero #1#2{ 0}% +\def\XINT@mul@minusminus #1#2% {% - \expandafter - \XINT@mul@enter\romannumeral0% - \XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z + \expandafter\XINT@mul@choice@a + \expandafter{\romannumeral0\XINT@length {#2}}% + {\romannumeral0\XINT@length {#1}}{#1}{#2}% }% -\def\XINT@mul@minusplus #1#2#3#4% +\def\XINT@mul@minusplus #1#2#3% {% - \expandafter\space\expandafter-% - \romannumeral0\expandafter - \XINT@mul@enter\romannumeral0% - \XINT@RQ {}#4#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z + \expandafter\space\expandafter-\romannumeral0\expandafter + \XINT@mul@choice@a + \expandafter{\romannumeral0\XINT@length {#1#3}}% + {\romannumeral0\XINT@length {#2}}{#2}{#1#3}% }% -\def\XINT@mul@plusminus #1#2#3#4% +\def\XINT@mul@plusminus #1#2#3% {% - \expandafter\space\expandafter-% - \romannumeral0\expandafter - \XINT@mul@enter\romannumeral0% - \XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #3#1\W\X\Y\Z + \expandafter\space\expandafter-\romannumeral0\expandafter + \XINT@mul@choice@a + \expandafter{\romannumeral0\XINT@length {#3}}% + {\romannumeral0\XINT@length {#1#2}}{#1#2}{#3}% }% \def\XINT@mul@plusplus #1#2#3#4% {% - \expandafter - \XINT@mul@enter\romannumeral0% - \XINT@RQ {}#4#2\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #3#1\W\X\Y\Z -}% -\def\XINT@mul@add@A #1#2#3#4#5#6% -{% - \xint@w - #3\xint@mul@add@az - \W\XINT@mul@add@AB #1{#3#4#5#6}{#2}% + \expandafter\XINT@mul@choice@a + \expandafter{\romannumeral0\XINT@length {#2#4}}% + {\romannumeral0\XINT@length {#1#3}}{#1#3}{#2#4}% }% -\def\xint@mul@add@az\W\XINT@mul@add@AB #1#2% +\def\XINT@mul@choice@a #1#2% {% - \XINT@mul@add@AC@checkcarry #1% + \expandafter\XINT@mul@choice@b \expandafter{#2}{#1}% }% -\def\XINT@mul@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8% +\def\XINT@mul@choice@b #1#2% {% - \XINT@mul@add@ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT@mul@add@ABE #1#2#3#4#5#6% -{\expandafter - \XINT@mul@add@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax.% -}% -\def\XINT@mul@add@ABEA #1#2#3.#4% -{% - \XINT@mul@add@A #2{#3#4}% -}% -\def\XINT@mul@add@AC@checkcarry #1% -{% - \xint@zero #1\xint@mul@add@AC@nocarry 0\XINT@mul@add@C + \ifnum #1<5 + \expandafter\XINT@mul@choice@littlebyfirst + \else + \ifnum #2<5 + \expandafter\expandafter\expandafter + \XINT@mul@choice@littlebysecond + \else + \expandafter\expandafter\expandafter + \XINT@mul@choice@compare + \fi + \fi + {#1}{#2}% }% -\def\xint@mul@add@AC@nocarry 0\XINT@mul@add@C #1#2\W\X\Y\Z +\def\XINT@mul@choice@littlebyfirst #1#2#3#4% {% - \expandafter - \xint@cleanupzeros@andstop - \romannumeral0% - \XINT@rord@main {}#2% - \xint@UNDEF - \xint@undef\xint@undef\xint@undef\xint@undef - \xint@undef\xint@undef\xint@undef\xint@undef - \xint@UNDEF - #1% + \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@add@C #1#2#3#4#5% +\def\XINT@mul@choice@littlebysecond #1#2#3#4% {% - \xint@w - #5\xint@mul@add@cw - #4\xint@mul@add@cx - #3\xint@mul@add@cy - #2\xint@mul@add@cz - \W\XINT@mul@add@CD {#5#4#3#2}{#1}% -}% -\def\XINT@mul@add@CD #1% -{\expandafter - \XINT@mul@add@CC\the\numexpr 1+10#1\relax.% + \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@add@CC #1#2#3.#4% +\def\XINT@mul@choice@compare #1#2% {% - \XINT@mul@add@AC@checkcarry #2{#3#4}% -}% -\def\xint@mul@add@cw - #1\xint@mul@add@cx - #2\xint@mul@add@cy - #3\xint@mul@add@cz - \W\XINT@mul@add@CD -{\expandafter - \XINT@mul@add@CDw\the\numexpr 1+#1#2#3\relax.% + \ifnum #1>#2 + \expandafter \XINT@mul@choice@i + \else + \expandafter \XINT@mul@choice@ii + \fi + {#1}{#2}% }% -\def\XINT@mul@add@CDw #1.#2#3\X\Y\Z -{% - \XINT@mul@add@end #1#3% +\def\XINT@mul@choice@i #1#2% +{% + \ifcase \numexpr (#2-3)/4\relax + \or \xint@afterfi {\ifnum #1<330 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<168 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<109 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<80 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<66 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \or \xint@afterfi {\ifnum #1<52 \expandafter \XINT@mul@choice@same + \else \expandafter \XINT@mul@choice@permute \fi}% + \else \expandafter \XINT@mul@choice@permute + \fi }% -\def\xint@mul@add@cx - #1\xint@mul@add@cy - #2\xint@mul@add@cz - \W\XINT@mul@add@CD -{\expandafter - \XINT@mul@add@CDx\the\numexpr 1+#1#2\relax.% +\def\XINT@mul@choice@ii #1#2% +{% + \ifcase \numexpr (#1-3)/4\relax + \or \xint@afterfi {\ifnum #2<330 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<168 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<109 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<80 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<66 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \or \xint@afterfi {\ifnum #2<52 \expandafter \XINT@mul@choice@permute + \else \expandafter \XINT@mul@choice@same \fi}% + \else \expandafter \XINT@mul@choice@same + \fi }% -\def\XINT@mul@add@CDx #1.#2#3\Y\Z +\def\XINT@mul@choice@same #1#2% {% - \XINT@mul@add@end #1#3% -}% -\def\xint@mul@add@cy - #1\xint@mul@add@cz - \W\XINT@mul@add@CD -{\expandafter - \XINT@mul@add@CDy\the\numexpr 1+#1\relax.% + \expandafter + \XINT@mul@enter\romannumeral0% + \XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #2\W\X\Y\Z }% -\def\XINT@mul@add@CDy #1.#2#3\Z +\def\XINT@mul@choice@permute #1#2% {% - \XINT@mul@add@end #1#3% -}% -\def\xint@mul@add@cz\W\XINT@mul@add@CD #1#2#3{\XINT@mul@add@end #1#3}% -\def\XINT@mul@add@end #1#2#3#4#5% -{\expandafter\space - \the\numexpr #1#2#3#4#5\relax + \expandafter + \XINT@mul@enter\romannumeral0% + \XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z + \W\X\Y\Z #1\W\X\Y\Z }% \def\XINT@mul@Ar #1#2#3#4#5#6% {% @@ -1514,7 +1719,7 @@ }% \def\xint@mul@br\Z\XINT@mul@Br #1#2% {% - \XINT@sum@AC@checkcarry #1% + \XINT@addr@AC@checkcarry #1% }% \def\XINT@mul@Br #1#2#3#4\W\X\Y\Z #5#6#7#8% {\expandafter @@ -1579,7 +1784,8 @@ {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 {% +\def\XINT@mul@M@one #1#2#3#4\Z\Z\Z\Z +{% \expandafter \xint@cleanupzeros@andstop \romannumeral0\XINT@rev{#4}% @@ -1653,8 +1859,8 @@ \XINT@mul@main \expandafter {\romannumeral0\expandafter \XINT@mul@Ar \expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z \W\X\Y\Z 0000#1\W\X\Y\Z - }#3\W\X\Y\Z + \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z + \W\X\Y\Z 0000#1\W\X\Y\Z }#3\W\X\Y\Z }% \def\xint@mul@mainw #1\xint@mul@mainx @@ -1663,7 +1869,7 @@ \W\XINT@mul@compute #4#5#6\W\X\Y\Z \X\Y\Z {% \expandafter - \XINT@mul@add@A \expandafter0\expandafter{\expandafter}% + \XINT@addm@A \expandafter0\expandafter{\expandafter}% \romannumeral0% \XINT@mul@Mr {#3#2#1}#6\Z\Z\Z\Z \W\X\Y\Z 000#4\W\X\Y\Z @@ -1674,7 +1880,7 @@ \W\XINT@mul@compute #3#4#5\W\X\Y\Z \Y\Z {% \expandafter - \XINT@mul@add@A \expandafter0\expandafter{\expandafter}% + \XINT@addm@A \expandafter0\expandafter{\expandafter}% \romannumeral0% \XINT@mul@Mr {#2#1}#5\Z\Z\Z\Z \W\X\Y\Z 00#3\W\X\Y\Z @@ -1684,7 +1890,7 @@ \W\XINT@mul@compute #2#3#4\W\X\Y\Z \Z {% \expandafter - \XINT@mul@add@A \expandafter0\expandafter{\expandafter}% + \XINT@addm@A \expandafter0\expandafter{\expandafter}% \romannumeral0% \XINT@mul@Mr {#1}#4\Z\Z\Z\Z \W\X\Y\Z 0#2\W\X\Y\Z @@ -1694,153 +1900,128 @@ \expandafter \xint@cleanupzeros@andstop\romannumeral0\XINT@rev{#1}% }% -\def\xintSqr {\romannumeral0\xintsqr }% -\def\xintsqr #1% -{% - \expandafter\expandafter\expandafter - \XINT@sqr - \expandafter\expandafter\expandafter - {\xintAbs{#1}}% fait l'expansion de #1 et se d\'ebarrasse du signe -}% -\def\XINT@sqr #1% -{\expandafter - \XINT@mul@enter - \romannumeral0% - \XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z - \W\X\Y\Z #1\W\X\Y\Z -}% -\def\XINT@posprod #1% +\def\XINT@mulr@enter #1\W\X\Y\Z #2#3#4#5% {% - \XINT@pprod@checkifempty #1\Z -}% -\def\XINT@pprod@checkifempty #1% + \xint@w + #5\xint@mulr@enterw + #4\xint@mulr@enterx + #3\xint@mulr@entery + #2\xint@mulr@enterz + \W\XINT@mulr@start {#2#3#4#5}#1\W\X\Y\Z +}% +\def\xint@mulr@enterw + #1\xint@mulr@enterx + #2\xint@mulr@entery + #3\xint@mulr@enterz + \W\XINT@mulr@start #4#5\W\X\Y\Z \X\Y\Z {% - \xint@relax #1\XINT@pprod@emptyproduct\relax - \XINT@pprod@first #1% + \XINT@mul@Mr {#3#2#1}#5\Z\Z\Z\Z }% -\def\XINT@pprod@emptyproduct #1\Z { 1}% -\def\XINT@pprod@first #1\Z +\def\xint@mulr@enterx + #1\xint@mulr@entery + #2\xint@mulr@enterz + \W\XINT@mulr@start #3#4\W\X\Y\Z \Y\Z {% - \XINT@pprod@getnext {#1}% + \XINT@mul@Mr {#2#1}#4\Z\Z\Z\Z }% -\def\XINT@pprod@getnext #1#2% +\def\xint@mulr@entery + #1\xint@mulr@enterz + \W\XINT@mulr@start #2#3\W\X\Y\Z \Z {% - \XINT@pprod@checkiffinished #2\Z {#1}% + \XINT@mul@Mr {#1}#3\Z\Z\Z\Z }% -\def\XINT@pprod@checkiffinished #1% -{% - \xint@relax #1\XINT@pprod@end\relax - \XINT@pprod@RQnew #1% +\def\XINT@mulr@start #1#2\W\X\Y\Z +{\expandafter + \XINT@mulr@main \expandafter + {\romannumeral0\XINT@mul@Mr {#1}#2\Z\Z\Z\Z }#2\W\X\Y\Z }% -\def\XINT@pprod@RQnew #1\Z +\def\XINT@mulr@main #1#2\W\X\Y\Z #3#4#5#6% {% - \expandafter\XINT@pprod@compute - \expandafter - {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z }% + \xint@w + #6\xint@mulr@mainw + #5\xint@mulr@mainx + #4\xint@mulr@mainy + #3\xint@mulr@mainz + \W\XINT@mulr@compute {#1}{#3#4#5#6}#2\W\X\Y\Z }% -\def\XINT@pprod@compute #1#2% -{% - \expandafter - \XINT@pprod@getnext - \expandafter - {\romannumeral0\XINT@mul@enter #1\W\X\Y\Z #2\W\X\Y\Z }% +\def\XINT@mulr@compute #1#2#3\W\X\Y\Z +{\expandafter + \XINT@mulr@main \expandafter + {\romannumeral0\expandafter + \XINT@mul@Ar \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z \W\X\Y\Z 0000#1\W\X\Y\Z + }#3\W\X\Y\Z +}% +\def\xint@mulr@mainw + #1\xint@mulr@mainx + #2\xint@mulr@mainy + #3\xint@mulr@mainz + \W\XINT@mulr@compute #4#5#6\W\X\Y\Z \X\Y\Z +{\expandafter + \XINT@addp@A + \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@mul@Mr {#3#2#1}#6\Z\Z\Z\Z + \W\X\Y\Z 000#4\W\X\Y\Z }% -\def\XINT@pprod@end\relax\XINT@pprod@RQnew #1\Z #2{ #2}% -\def\xintProductExpr {\romannumeral0\xintproductexpr }% -\def\xintPrd {\romannumeral0\xintprd }% -\def\xintprd #1% -{% - \expandafter\expandafter\expandafter - \xintproductexpr #1\relax +\def\xint@mulr@mainx + #1\xint@mulr@mainy + #2\xint@mulr@mainz + \W\XINT@mulr@compute #3#4#5\W\X\Y\Z \Y\Z +{\expandafter + \XINT@addp@A + \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@mul@Mr {#2#1}#5\Z\Z\Z\Z + \W\X\Y\Z 00#3\W\X\Y\Z +}% +\def\xint@mulr@mainy + #1\xint@mulr@mainz + \W\XINT@mulr@compute #2#3#4\W\X\Y\Z \Z +{\expandafter + \XINT@addp@A + \expandafter0\expandafter{\expandafter}% + \romannumeral0\XINT@mul@Mr {#1}#4\Z\Z\Z\Z + \W\X\Y\Z 0#2\W\X\Y\Z }% -\def\xintproductexpr #1% +\def\xint@mulr@mainz\W\XINT@mulr@compute #1#2#3\W\X\Y\Z { #1}% +\def\xintiSqr {\romannumeral0\xintisqr }% +\def\xintisqr #1% {% \expandafter\expandafter\expandafter - \XINT@prod@checkifempty #1\Z -}% -\def\XINT@prod@checkifempty #1% -{% - \xint@relax #1\XINT@prod@emptyproduct\relax - \XINT@prod@checkfirstsign #1% -}% -\def\XINT@prod@emptyproduct #1\Z { 1}% -\def\XINT@prod@checkfirstsign #1% -{% - \xint@zero #1\XINT@prod@returnzero0% - \xint@UDsignfork - #1\dummy \XINT@prod@firstisneg - -\dummy \XINT@prod@firstispos - \xint@UDkrof - #1% -}% -\def\XINT@prod@returnzero #1\relax { 0}% -\def\XINT@prod@firstisneg #1#2\Z -{% - \XINT@prod@xpxpnext 0{#2}% -}% -\def\XINT@prod@firstispos #1\Z -{% - \XINT@prod@xpxpnext 1{#1}% -}% -\def\XINT@prod@xpxpnext #1#2#3% -{% + \XINT@sqr \expandafter\expandafter\expandafter - \XINT@prod@checkiffinished #3\Z {#2}#1% -}% -\def\XINT@prod@checkiffinished #1% -{% - \xint@relax #1\XINT@prod@end\relax - \XINT@prod@checksign #1% -}% -\def\XINT@prod@checksign #1% -{% - \xint@zero #1\XINT@prod@returnzero0% - \xint@UDsignfork - #1\dummy \XINT@prod@neg@RQnew - -\dummy \XINT@prod@pos@RQnew - \xint@UDkrof - #1% -}% -\def\XINT@prod@pos@RQnew #1\Z -{% - \expandafter - \XINT@prod@pos - \expandafter - {\romannumeral0\XINT@RQ {}#1\R\R\R\R\R\R\R\R\Z }% + {\xintiAbs{#1}}% fait l'expansion de #1 et se d\'ebarrasse du signe }% -\def\XINT@prod@pos #1#2#3% -{% - \expandafter - \XINT@prod@xpxpnext - \expandafter - #3% - \expandafter - {\romannumeral0\XINT@mul@enter #1\W\X\Y\Z #2\W\X\Y\Z }% -}% -\def\XINT@prod@neg@RQnew #1#2\Z -{% - \expandafter - \XINT@prod@neg - \expandafter - {\romannumeral0\XINT@RQ {}#2\R\R\R\R\R\R\R\R\Z }% +\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 + \W\X\Y\Z #1\W\X\Y\Z }% -\def\XINT@prod@neg #1#2#3% +\def\xintiPrd {\romannumeral0\xintiprd }% +\def\xintiprd #1{\xintiproductexpr #1\relax }% +\let\xintPrd\xintiPrd +\let\xintprd\xintiprd +\def\xintiProductExpr {\romannumeral0\xintiproductexpr }% +\def\xintiproductexpr {% - \expandafter - \XINT@prod@xpxpnext - \expandafter - {\the\numexpr 1-#3\expandafter}% - \expandafter - {\romannumeral0\XINT@mul@enter #1\W\X\Y\Z #2\W\X\Y\Z }% + \expandafter\expandafter\expandafter\XINT@productexpr }% -\def\XINT@prod@end\relax\XINT@prod@checksign #1\Z #2#3% +\let\xintProductExpr\xintiProductExpr +\let\xintproductexpr\xintiproductexpr +\def\XINT@productexpr {\XINT@prod@loop {1}}% +\def\XINT@prod@loop #1#2% {% - \xint@prod@cleanupzeros #3#2% + \xint@relax #2\XINT@prod@finished\relax + \expandafter\XINT@prod@loop\expandafter + {\romannumeral0\xintimul {#2}{#1}}% }% -\def\xint@prod@cleanupzeros #1#2#3#4#5% +\def\XINT@prod@finished #1#2#3#4#5% {% - \expandafter\space\the\numexpr (2*#1-1)*#2#3#4#5\relax + \XINT@prod@finished@ #5% }% +\def\XINT@prod@finished@ #1#2#3#4#5{ #5}% \def\xintFac {\romannumeral0\xintfac }% \def\xintfac #1% {% @@ -1852,7 +2033,7 @@ \def\XINT@Fac {\romannumeral0\XINT@fac@fork }% \def\XINT@fac@fork #1% {% - \ifcase\xintSgn {#1} + \ifcase\xintiSgn {#1} \xint@afterfi{\expandafter\space\expandafter 1\xint@gobble }% \or \expandafter\XINT@fac@checklength @@ -1891,57 +2072,50 @@ }% \def\XINT@fac@big@docomputation #1#2% {% - \expandafter - \XINT@pprod@getnext - \expandafter + \expandafter \XINT@fac@bigcompute@loop \expandafter {\romannumeral0\XINT@fac@loop {9999}}#2\relax }% -\def\XINT@fac@loop #1{\XINT@fac@loop@main 1{#1}{}}% -\def\XINT@fac@loop@main #1#2#3% +\def\XINT@fac@bigcompute@loop #1#2% {% - \ifnum #1<#2 - \expandafter - \XINT@fac@loop@main - \expandafter - {\the\numexpr #1+1\expandafter }% - \else - \expandafter\XINT@fac@docomputation - \fi - {#2}{#3{#1}}% + \xint@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 + \W\X\Y\Z #1\W\X\Y\Z }% }% -\def\XINT@fac@docomputation #1#2% +\def\XINT@fac@bigcompute@end #1#2#3#4#5% {% - \XINT@fprod@getnext {1000}#2\relax + \XINT@fac@bigcompute@end@ #5% }% -\def\XINT@fprod@getnext #1#2% -{% - \XINT@fprod@checkiffinished #2\Z {#1}% -}% -\def\XINT@fprod@checkiffinished #1% +\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% {% - \xint@relax #1\XINT@fprod@end\relax - \XINT@fprod@compute #1% + \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@fprod@compute #1\Z #2% +\def\XINT@fac@loop@exit #1#2#3#4#5#6#7% {% - \expandafter - \XINT@fprod@getnext - \expandafter - {\romannumeral0\XINT@mul@Mr {#1}#2\Z\Z\Z\Z }% + \XINT@fac@loop@exit@ #6% }% -\def\XINT@fprod@end\relax\XINT@fprod@compute #1\Z #2% +\def\XINT@fac@loop@exit@ #1#2#3% {% - \expandafter - \xint@cleanupzeros@andstop - \romannumeral0\XINT@rev {#2}% + \XINT@mul@M }% -\def\xintPow {\romannumeral0\xintpow }% -\def\xintpow #1% +\def\xintiPow {\romannumeral0\xintipow }% +\def\xintipow #1% {% \expandafter\expandafter\expandafter \xint@pow #1\Z% }% +\let\xintPow\xintiPow \let\xintpow\xintipow \def\xint@pow #1#2\Z {% \xint@UDsignfork @@ -2012,7 +2186,7 @@ \def\XINT@pow@BisZero #1#2{ 1}% \def\XINT@pow@checkBlength #1#2% {% - \ifnum\xintLen{#1} >9 + \ifnum\xintiLen{#1} >9 \expandafter\XINT@pow@BtooBig \else \expandafter\XINT@pow@loop @@ -2033,7 +2207,7 @@ \xint@afterfi{\expandafter\XINT@pow@loop@a \expandafter{\the\numexpr 2*(#1/2)-#1\expandafter }% b mod 2 \expandafter{\the\numexpr #1-#1/2\expandafter }% [b/2] - \expandafter{\romannumeral0\xintsqr{#2}}}% + \expandafter{\romannumeral0\xintisqr{#2}}}% \fi {{#2}}% }% @@ -2076,10 +2250,8 @@ }% \def\XINT@pow@pprod@compute #1\Z #2% {% - \expandafter - \XINT@pow@pprod@getnext - \expandafter - {\romannumeral0\XINT@pow@mul@enter #2\W\X\Y\Z #1\W\X\Y\Z}% + \expandafter \XINT@pow@pprod@getnext \expandafter + {\romannumeral0\XINT@mulr@enter #2\W\X\Y\Z #1\W\X\Y\Z}% }% \def\XINT@pow@pprod@end\relax\XINT@pow@pprod@compute #1\Z #2% {% @@ -2087,229 +2259,6 @@ \xint@cleanupzeros@andstop \romannumeral0\XINT@rev {#2}% }% -\def\XINT@pow@mul@enter #1\W\X\Y\Z #2#3#4#5% -{% - \xint@w - #5\xint@pow@mul@enterw - #4\xint@pow@mul@enterx - #3\xint@pow@mul@entery - #2\xint@pow@mul@enterz - \W\XINT@pow@mul@start {#2#3#4#5}#1\W\X\Y\Z -}% -\def\xint@pow@mul@enterw - #1\xint@pow@mul@enterx - #2\xint@pow@mul@entery - #3\xint@pow@mul@enterz - \W\XINT@pow@mul@start #4#5\W\X\Y\Z \X\Y\Z -{% - \XINT@mul@Mr {#3#2#1}#5\Z\Z\Z\Z -}% -\def\xint@pow@mul@enterx - #1\xint@pow@mul@entery - #2\xint@pow@mul@enterz - \W\XINT@pow@mul@start #3#4\W\X\Y\Z \Y\Z -{% - \XINT@mul@Mr {#2#1}#4\Z\Z\Z\Z -}% -\def\xint@pow@mul@entery - #1\xint@pow@mul@enterz - \W\XINT@pow@mul@start #2#3\W\X\Y\Z \Z -{% - \XINT@mul@Mr {#1}#3\Z\Z\Z\Z -}% -\def\XINT@pow@mul@start #1#2\W\X\Y\Z -{\expandafter - \XINT@pow@mul@main \expandafter - {\romannumeral0% - \XINT@mul@Mr {#1}#2\Z\Z\Z\Z - }#2\W\X\Y\Z -}% -\def\XINT@pow@mul@main #1#2\W\X\Y\Z #3#4#5#6% -{% - \xint@w - #6\xint@pow@mul@mainw - #5\xint@pow@mul@mainx - #4\xint@pow@mul@mainy - #3\xint@pow@mul@mainz - \W\XINT@pow@mul@compute {#1}{#3#4#5#6}#2\W\X\Y\Z -}% -\def\XINT@pow@mul@compute #1#2#3\W\X\Y\Z -{\expandafter - \XINT@pow@mul@main \expandafter - {\romannumeral0\expandafter - \XINT@mul@Ar \expandafter0\expandafter{\expandafter}% - \romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z \W\X\Y\Z 0000#1\W\X\Y\Z - }#3\W\X\Y\Z -}% -\def\xint@pow@mul@mainw - #1\xint@pow@mul@mainx - #2\xint@pow@mul@mainy - #3\xint@pow@mul@mainz - \W\XINT@pow@mul@compute #4#5#6\W\X\Y\Z \X\Y\Z -{% - \expandafter - \XINT@pow@add@A \expandafter0\expandafter{\expandafter}% - \romannumeral0% - \XINT@mul@Mr {#3#2#1}#6\Z\Z\Z\Z - \W\X\Y\Z 000#4\W\X\Y\Z -}% -\def\xint@pow@mul@mainx - #1\xint@pow@mul@mainy - #2\xint@pow@mul@mainz - \W\XINT@pow@mul@compute #3#4#5\W\X\Y\Z \Y\Z -{% - \expandafter - \XINT@pow@add@A \expandafter0\expandafter{\expandafter}% - \romannumeral0% - \XINT@mul@Mr {#2#1}#5\Z\Z\Z\Z - \W\X\Y\Z 00#3\W\X\Y\Z -}% -\def\xint@pow@mul@mainy - #1\xint@pow@mul@mainz - \W\XINT@pow@mul@compute #2#3#4\W\X\Y\Z \Z -{% - \expandafter - \XINT@pow@add@A \expandafter0\expandafter{\expandafter}% - \romannumeral0% - \XINT@mul@Mr {#1}#4\Z\Z\Z\Z - \W\X\Y\Z 0#2\W\X\Y\Z -}% -\def\xint@pow@mul@mainz\W\XINT@pow@mul@compute #1#2#3\W\X\Y\Z -{ #1}% -\def\XINT@pow@add@A #1#2#3#4#5#6% -{% - \xint@w - #3\xint@pow@add@az - \W\XINT@pow@add@AB #1{#3#4#5#6}{#2}% -}% -\def\xint@pow@add@az\W\XINT@pow@add@AB #1#2% -{% - \XINT@pow@add@AC@checkcarry #1% -}% -\def\XINT@pow@add@AC@checkcarry #1% -{% - \xint@zero #1\xint@pow@add@AC@nocarry 0\XINT@pow@add@C -}% -\def\xint@pow@add@AC@nocarry 0\XINT@pow@add@C -{% - \XINT@pow@add@F -}% -\def\XINT@pow@add@AB #1#2#3#4\W\X\Y\Z #5#6#7#8% -{% - \XINT@pow@add@ABE #1#2{#8#7#6#5}{#3}#4\W\X\Y\Z -}% -\def\XINT@pow@add@ABE #1#2#3#4#5#6% -{\expandafter - \XINT@pow@add@ABEA\the\numexpr #1+10#5#4#3#2+#6\relax -}% -\def\XINT@pow@add@ABEA #1#2#3#4#5#6#7% -{% - \XINT@pow@add@A #2{#7#6#5#4#3}%<-- attention on met donc \`a droite -}% -\def\XINT@pow@add@C #1#2#3#4#5% -{% - \xint@w - #5\xint@pow@add@cw - #4\xint@pow@add@cx - #3\xint@pow@add@cy - #2\xint@pow@add@cz - \W\XINT@pow@add@CD {#5#4#3#2}{#1}% -}% -\def\XINT@pow@add@CD #1% -{\expandafter - \XINT@pow@add@CC\the\numexpr 1+10#1\relax -}% -\def\XINT@pow@add@CC #1#2#3#4#5#6#7% -{% - \XINT@pow@add@AC@checkcarry #2{#7#6#5#4#3}% -}% -\def\xint@pow@add@cw - #1\xint@pow@add@cx - #2\xint@pow@add@cy - #3\xint@pow@add@cz - \W\XINT@pow@add@CD -{\expandafter - \XINT@pow@add@CDw\the\numexpr 1+10#1#2#3\relax -}% -\def\XINT@pow@add@CDw #1#2#3#4#5#6% -{% - \xint@quatrezeros #2#3#4#5\XINT@pow@add@endDw@zeros - 0000\XINT@pow@add@endDw #2#3#4#5% -}% -\def\XINT@pow@add@endDw@zeros 0000\XINT@pow@add@endDw 0000#1\X\Y\Z{ #1}% -\def\XINT@pow@add@endDw #1#2#3#4#5\X\Y\Z{ #5#4#3#2#1}% -\def\xint@pow@add@cx - #1\xint@pow@add@cy - #2\xint@pow@add@cz - \W\XINT@pow@add@CD -{\expandafter - \XINT@pow@add@CDx\the\numexpr 1+100#1#2\relax -}% -\def\XINT@pow@add@CDx #1#2#3#4#5#6% -{% - \xint@quatrezeros #2#3#4#5\XINT@pow@add@endDx@zeros - 0000\XINT@pow@add@endDx #2#3#4#5% -}% -\def\XINT@pow@add@endDx@zeros 0000\XINT@pow@add@endDx 0000#1\Y\Z{ #1}% -\def\XINT@pow@add@endDx #1#2#3#4#5\Y\Z{ #5#4#3#2#1}% -\def\xint@pow@add@cy - #1\xint@pow@add@cz - \W\XINT@pow@add@CD -{\expandafter - \XINT@pow@add@CDy\the\numexpr 1+1000#1\relax -}% -\def\XINT@pow@add@CDy #1#2#3#4#5#6% -{% - \xint@quatrezeros #2#3#4#5\XINT@pow@add@endDy@zeros - 0000\XINT@pow@add@endDy #2#3#4#5% -}% -\def\XINT@pow@add@endDy@zeros 0000\XINT@pow@add@endDy 0000#1\Z{ #1}% -\def\XINT@pow@add@endDy #1#2#3#4#5\Z{ #5#4#3#2#1}% -\def\xint@pow@add@cz\W\XINT@pow@add@CD #1#2{ #21000}% -\def\XINT@pow@add@F #1#2#3#4#5% -{% - \xint@w - #5\xint@pow@add@Gw - #4\xint@pow@add@Gx - #3\xint@pow@add@Gy - #2\xint@pow@add@Gz - \W\XINT@pow@add@G {#2#3#4#5}{#1}% -}% -\def\XINT@pow@add@G #1#2% -{% - \XINT@pow@add@F {#2#1}% -}% -\def\xint@pow@add@Gw - #1\xint@pow@add@Gx - #2\xint@pow@add@Gy - #3\xint@pow@add@Gz - \W\XINT@pow@add@G #4% -{% - \xint@quatrezeros #3#2#10\XINT@pow@add@endGw@zeros - 0000\XINT@pow@add@endGw #3#2#10% -}% -\def\XINT@pow@add@endGw@zeros 0000\XINT@pow@add@endGw 0000#1\X\Y\Z{ #1}% -\def\XINT@pow@add@endGw #1#2#3#4#5\X\Y\Z{ #5#1#2#3#4}% -\def\xint@pow@add@Gx - #1\xint@pow@add@Gy - #2\xint@pow@add@Gz - \W\XINT@pow@add@G #3% -{% - \xint@quatrezeros #2#100\XINT@pow@add@endGx@zeros - 0000\XINT@pow@add@endGx #2#100% -}% -\def\XINT@pow@add@endGx@zeros 0000\XINT@pow@add@endGx 0000#1\Y\Z{ #1}% -\def\XINT@pow@add@endGx #1#2#3#4#5\Y\Z{ #5#1#2#3#4}% -\def\xint@pow@add@Gy - #1\xint@pow@add@Gz - \W\XINT@pow@add@G #2% -{% - \xint@quatrezeros #1000\XINT@pow@add@endGy@zeros - 0000\XINT@pow@add@endGy #1000% -}% -\def\XINT@pow@add@endGy@zeros 0000\XINT@pow@add@endGy 0000#1\Z{ #1}% -\def\XINT@pow@add@endGy #1#2#3#4#5\Z{ #5#1#2#3#4}% -\def\xint@pow@add@Gz\W\XINT@pow@add@G #1#2{ #2}% \def\xintQuo {\romannumeral0\xintquo }% \def\xintRem {\romannumeral0\xintrem }% \def\xintquo {\expandafter @@ -2370,7 +2319,7 @@ }% \def\XINT@div@AisNegative@post #1#2% {% - \ifcase\xintSgn {#2} + \ifcase\xintiSgn {#2} \expandafter \XINT@div@AisNegative@zerorem \or \expandafter \XINT@div@AisNegative@posrem @@ -2384,33 +2333,45 @@ }% \def\XINT@div@AisNegative@posrem #1% {% - \expandafter - \XINT@div@AisNegative@posrem@b - \expandafter - {\romannumeral0\xintopp {\XINT@Add{#1}{1}}}% + \expandafter \XINT@div@AisNegative@posrem@b \expandafter + {\romannumeral0\xintiopp {\XINT@Add{#1}{1}}}% }% \def\XINT@div@AisNegative@posrem@b #1#2#3% {% - \expandafter - \xint@exchangetwo@keepbraces@andstop - \expandafter + \expandafter \xint@exchangetwo@keepbraces@andstop \expandafter {\romannumeral0\XINT@sub {#3}{#2}}{#1}% }% \def\XINT@div@prepare #1% {% - \expandafter - \XINT@div@prepareB@a - \expandafter - {\romannumeral0\XINT@length {#1}}{#1}% B > 0 ici + \expandafter \XINT@div@prepareB@aa \expandafter + {\romannumeral0\XINT@length {#1}}{#1}% B > 0 ici +}% +\def\XINT@div@prepareB@aa #1% +{% + \ifnum #1=1 + \expandafter\XINT@div@prepareB@ab + \else + \expandafter\XINT@div@prepareB@a + \fi + {#1}% +}% +\def\XINT@div@prepareB@ab #1#2% +{% + \ifnum #2=1 + \expandafter\XINT@div@prepareB@BisOne + \else + \xint@afterfi{\XINT@div@prepareB@e {000}{3}{4}{#2}}% + \fi }% +\def\XINT@div@prepareB@BisOne #1{ {#1}{0}}% \def\XINT@div@prepareB@a #1% {% - \expandafter\XINT@div@prepareB@b\expandafter - {\the\numexpr 4*((#1+1)/4)\relax}{#1}% + \expandafter \XINT@div@prepareB@b \expandafter + {\the\numexpr 4*((#1+1)/4)\relax}{#1}% }% \def\XINT@div@prepareB@b #1#2% {% - \expandafter\XINT@div@prepareB@c \expandafter + \expandafter \XINT@div@prepareB@c \expandafter {\the\numexpr #1-#2\relax}{#1}% }% \def\XINT@div@prepareB@c #1% @@ -2432,10 +2393,8 @@ }% \def\XINT@div@prepareB@f #1#2#3#4#5\Z {% - \expandafter - \XINT@div@prepareB@g - \expandafter - {\romannumeral0\XINT@rev {#1#2#3#4#5}}{#1#2#3#4}% + \expandafter \XINT@div@prepareB@g \expandafter + {\romannumeral0\XINT@rev {#1#2#3#4#5}}{#1#2#3#4}% }% \def\XINT@div@prepareB@g #1#2#3#4#5#6% {% @@ -2443,10 +2402,8 @@ }% \def\XINT@div@prepareA@a #1% {% - \expandafter - \XINT@div@prepareA@b - \expandafter - {\romannumeral0\XINT@length {#1}}{#1}% A >0 ici + \expandafter \XINT@div@prepareA@b \expandafter + {\romannumeral0\XINT@length {#1}}{#1}% A >0 ici }% \def\XINT@div@prepareA@b #1% {% @@ -2490,9 +2447,7 @@ }% \def\XINT@div@finished@a #1#2#3% {% - \expandafter - \XINT@div@finished@b - \expandafter + \expandafter \XINT@div@finished@b \expandafter {\romannumeral0\XINT@cuz {#1}}% }% \def\XINT@div@finished@b #1#2#3#4#5% @@ -2508,9 +2463,7 @@ }% \def\XINT@div@finished@c #1#2% {% - \expandafter - \space - \expandafter + \expandafter \space \expandafter {\romannumeral0\expandafter\xint@cleanupzeros@andstop \romannumeral0\XINT@rev {#2}}{#1}% }% @@ -2523,20 +2476,15 @@ \xint@quatrezeros #1#2#3#4\xint@div@final@c0000% \XINT@div@final@c {#1#2#3#4}{#1#2#3#4#5}% }% -\def\xint@div@final@c0000\XINT@div@final@c #1% - {\XINT@div@finished@a }% +\def\xint@div@final@c0000\XINT@div@final@c #1{\XINT@div@finished@a }% \def\XINT@div@final@c #1#2#3#4% {% - \expandafter - \XINT@div@final@d - \expandafter + \expandafter \XINT@div@final@d \expandafter {\the\numexpr #1/#4\relax}{#2}% }% \def\XINT@div@final@d #1#2#3#4#5% q,A,Q,L,B puis c {% - \expandafter - \XINT@div@final@da - \expandafter + \expandafter \XINT@div@final@da \expandafter {\romannumeral0\XINT@mul@M {#1}#5\Z\Z\Z\Z }% {\romannumeral0\xint@cleanupzeros@andstop #2}% {#1}{#3}{#5}% @@ -2556,8 +2504,8 @@ \def\XINT@div@final@e #1#2#3#4#5% A,qB,q,Q,B,puis c {% \expandafter\XINT@div@final@f - \expandafter{\romannumeral0\xintsub {#1}{#2}}% - {\romannumeral0\xintadd {\XINT@Rev@andcleanupzeros{#4}}{#3}}% + \expandafter{\romannumeral0\xintisub {#1}{#2}}% + {\romannumeral0\xintiadd {\XINT@Rev@andcleanupzeros{#4}}{#3}}% }% \def\XINT@div@final@dc #1#2#3% A sans leading zeros,trash,q,Q,B,c {% @@ -2567,9 +2515,9 @@ \def\XINT@div@final@dd #1#2#3#4% q,A,Q,B puis c {% \expandafter\XINT@div@final@f - \expandafter{\romannumeral0\xintsub + \expandafter{\romannumeral0\xintisub {#2}{\romannumeral0\XINT@mul@M {#1}#4\Z\Z\Z\Z }}% - {\romannumeral0\xintadd {\XINT@Rev@andcleanupzeros{#3}}{#1}}% + {\romannumeral0\xintiadd {\XINT@Rev@andcleanupzeros{#3}}{#1}}% }% \def\XINT@div@final@f #1#2#3% R,Q à développer,c {% @@ -2588,17 +2536,15 @@ }% \def\XINT@div@body@a #1% {% - \XINT@div@body@b #1\Z + \XINT@div@body@b #1\Z {#1}% }% \def\XINT@div@body@b #1#2#3#4#5#6#7#8#9\Z {% - \XINT@div@body@c - {#1#2#3#4#5#6#7#8#9}% - {#1#2#3#4#5#6#7#8}% + \XINT@div@body@c {#1#2#3#4#5#6#7#8}% }% \def\XINT@div@body@c #1#2#3% {% - \XINT@div@body@d {#3}{}#1\Z {#2}{#3}% + \XINT@div@body@d {#3}{}#2\Z {#1}{#3}% }% \def\XINT@div@body@d #1#2#3#4#5#6% {% @@ -2619,19 +2565,40 @@ }% \def\XINT@div@body@f #1#2#3#4#5#6#7#8% {% - \expandafter\XINT@div@body@g - \expandafter - {\the\numexpr (#1+(#5+1)/2)/(#5+1)-1\relax }% - {#2}{#8}{#4}{#5}{#3}{#6}{#7}{#8}% + \expandafter\XINT@div@body@gg + \the\numexpr (#1+(#5+1)/2)/(#5+1)+99999\relax + {#8}{#2}{#8}{#4}{#5}{#3}{#6}{#7}{#8}% }% -\def\XINT@div@body@g #1#2#3% +\def\XINT@div@body@gg #1#2#3#4#5#6% +{% + \xint@UDzerofork + #2\dummy \XINT@div@body@gk + 0\dummy {\XINT@div@body@ggk #2}% + \xint@UDkrof + {#3#4#5#6}% +}% +\def\XINT@div@body@gk #1#2#3% {% \expandafter \XINT@div@body@h \romannumeral0\XINT@div@sub@xpxp - {\romannumeral0\XINT@pow@mul@enter #3\W\X\Y\Z #1\W\X\Y\Z }% - {#2}\Z - {#3}{#1}% + {\romannumeral0\XINT@mul@Mr {#1}#2\Z\Z\Z\Z }{#3}\Z {#1}% +}% +\def\XINT@div@body@ggk #1#2#3% +{% + \expandafter \XINT@div@body@gggk \expandafter + {\romannumeral0\XINT@mul@Mr {#1}0000#3\Z\Z\Z\Z }% + {\romannumeral0\XINT@mul@Mr {#2}#3\Z\Z\Z\Z }% + {#1#2}% +}% +\def\XINT@div@body@gggk #1#2#3#4% +{% + \expandafter + \XINT@div@body@h + \romannumeral0\XINT@div@sub@xpxp + {\romannumeral0\expandafter\XINT@mul@Ar + \expandafter0\expandafter{\expandafter}#2\W\X\Y\Z #1\W\X\Y\Z }% + {#4}\Z {#3}% }% \def\XINT@div@body@h #1#2#3#4#5#6#7#8#9\Z {% @@ -2642,6 +2609,10 @@ \fi {#1#2#3#4#5#6#7#8#9}% }% +\def\XINT@div@body@k #1#2#3% +{% + \XINT@div@body@l {#1}{#2}% +}% \def\XINT@div@body@i #1#2#3#4#5#6% {% \expandafter\XINT@div@body@j @@ -2650,16 +2621,10 @@ }% \def\XINT@div@body@j #1#2#3#4% {% - \expandafter - \XINT@div@body@l - \expandafter{\romannumeral0\XINT@div@sub@xpxp - {\romannumeral0\XINT@pow@mul@enter #3\W\X\Y\Z #1\W\X\Y\Z }% - {\XINT@Rev{#2}}}% - {#4+#1}% -}% -\def\XINT@div@body@k #1#2% -{% - \XINT@div@body@l {#1}% + \expandafter \XINT@div@body@l \expandafter + {\romannumeral0\XINT@div@sub@xpxp + {\romannumeral0\XINT@mul@Mr {#1}#4\Z\Z\Z\Z }{\XINT@Rev{#2}}}% + {#3+#1}% }% \def\XINT@div@body@l #1#2#3#4#5#6#7% {% @@ -2679,7 +2644,7 @@ \def\XINT@div@body@n #1#2% {% \expandafter\XINT@div@body@o\expandafter - {\romannumeral0\XINT@sum@A 0{}#1\W\X\Y\Z #2\W\X\Y\Z }% + {\romannumeral0\XINT@addr@A 0{}#1\W\X\Y\Z #2\W\X\Y\Z }% }% \def\XINT@div@body@o #1#2#3#4% {% @@ -2701,14 +2666,12 @@ }% \def\XINT@div@body@repeatp #1#2#3#4#5#6#7% {% - \expandafter - \XINT@div@body@p - \expandafter + \expandafter \XINT@div@body@p \expandafter {\the\numexpr #1-4\relax}{#2}{0000#3}% }% \def\XINT@div@body@q #1#2#3#4\Z #5#6% {% - \XINT@div@body@a {#4}{#2}{#6}{#3#5}{#1}% + \XINT@div@body@b #4\Z {#4}{#2}{#6}{#3#5}{#1}% }% \def\XINT@div@gotofinal@a #1#2#3#4\Z % {% @@ -2720,9 +2683,7 @@ }% \def\XINT@div@sub@xpxp #1% {% - \expandafter - \XINT@div@sub@xpxp@ - \expandafter + \expandafter \XINT@div@sub@xpxp@ \expandafter {#1}% }% \def\XINT@div@sub@xpxp@ #1#2% @@ -2842,6 +2803,24 @@ {% \expandafter\space\expandafter #1\xint@z }% +\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 +}% \def\xintOdd {\romannumeral0\xintodd }% \def\xintodd #1% {% @@ -2928,14 +2907,13 @@ \def\xint@dsh #1#2% {% \expandafter\expandafter\expandafter - \XINT@dsh@checksignx - #2\Z {#1}% + \XINT@dsh@checksignx #2\Z {#1}% }% \def\XINT@dsh@checksignx #1% {% \xint@UDzerominusfork #1-\dummy \XINT@dsh@xiszero - 0#1\dummy \XINT@dsx@xisNeg + 0#1\dummy \XINT@dsx@xisNeg@checkA % on passe direct dans DSx 0-\dummy {\XINT@dsh@xisPos #1}% \xint@UDkrof }% @@ -2944,7 +2922,7 @@ {% \expandafter \xint@firstoftwo@andstop - \romannumeral0\XINT@dsx@checksignA #2\Z {#1}% + \romannumeral0\XINT@dsx@checksignA #2\Z {#1}% via DSx }% \def\xintDSx {\romannumeral0\xintdsx }% \def\xintdsx #1#2% @@ -2957,8 +2935,7 @@ \def\xint@dsx #1#2% {% \expandafter\expandafter\expandafter - \XINT@dsx@checksignx - #2\Z {#1}% + \XINT@dsx@checksignx #2\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}}% @@ -2966,40 +2943,50 @@ {% \xint@UDzerominusfork #1-\dummy \XINT@dsx@xisZero - 0#1\dummy \XINT@dsx@xisNeg + 0#1\dummy \XINT@dsx@xisNeg@checkA 0-\dummy {\XINT@dsx@xisPos #1}% \xint@UDkrof }% -\def\XINT@dsx@xisZero #1\Z #2{ {#2}{0}}% -\def\XINT@dsx@xisNeg #1\Z +\def\XINT@dsx@xisZero #1\Z #2{ {#2}{0}}% attention comme x > 0 +\def\XINT@dsx@xisNeg@checkA #1\Z #2% {% - \ifnum\XINT@Len {#1} > 9 + \XINT@dsx@xisNeg@checkA@ #2\Z {#1}% +}% +\def\XINT@dsx@xisNeg@checkA@ #1#2\Z #3% +{% + \xint@zero #1\XINT@dsx@xisNeg@Azero 0\expandafter + \XINT@dsx@xisNeg@checkx\expandafter + {\romannumeral0\XINT@length {#3}}{#3}\Z {#1#2}% +}% +\def\XINT@dsx@xisNeg@Azero #1#2#3#4#5#6#7#8{ 0}% +\def\XINT@dsx@xisNeg@checkx #1% +{% + \ifnum #1> 9 \xint@afterfi {\xintError:TooBigDecimalShift \XINT@dsx@toobigx }% \else \expandafter \XINT@dsx@zeroloop \fi - {#1}\Z }% -\def\XINT@dsx@toobigx #1\Z #2{ #2}% +\def\XINT@dsx@toobigx #1#2#3{ 0}% \def\XINT@dsx@zeroloop #1% {% \ifcase #1 - \expandafter \XINT@dsx@exit + \XINT@dsx@exit \or - \expandafter \XINT@dsx@exiti + \XINT@dsx@exiti \or - \expandafter \XINT@dsx@exitii + \XINT@dsx@exitii \or - \expandafter \XINT@dsx@exitiii + \XINT@dsx@exitiii \or - \expandafter \XINT@dsx@exitiv + \XINT@dsx@exitiv \or - \expandafter \XINT@dsx@exitv + \XINT@dsx@exitv \or - \expandafter \XINT@dsx@exitvi + \XINT@dsx@exitvi \or - \expandafter \XINT@dsx@exitvii + \XINT@dsx@exitvii \else \xint@afterfi {\expandafter @@ -3008,22 +2995,14 @@ }% \fi }% -\def\XINT@dsx@exit #1\Z - {\XINT@dsx@addzeros {#1}}% -\def\XINT@dsx@exiti #1\Z - {\XINT@dsx@addzeros {0#1}}% -\def\XINT@dsx@exitii #1\Z - {\XINT@dsx@addzeros {00#1}}% -\def\XINT@dsx@exitiii #1\Z - {\XINT@dsx@addzeros {000#1}}% -\def\XINT@dsx@exitiv #1\Z - {\XINT@dsx@addzeros {0000#1}}% -\def\XINT@dsx@exitv #1\Z - {\XINT@dsx@addzeros {00000#1}}% -\def\XINT@dsx@exitvi #1\Z - {\XINT@dsx@addzeros {000000#1}}% -\def\XINT@dsx@exitvii #1\Z - {\XINT@dsx@addzeros {0000000#1}}% +\def\XINT@dsx@exit #1\fi #2\Z {\fi \XINT@dsx@addzeros {#2}}% +\def\XINT@dsx@exiti #1\fi #2\Z {\fi \XINT@dsx@addzeros {0#2}}% +\def\XINT@dsx@exitii #1\fi #2\Z {\fi \XINT@dsx@addzeros {00#2}}% +\def\XINT@dsx@exitiii #1\fi #2\Z {\fi \XINT@dsx@addzeros {000#2}}% +\def\XINT@dsx@exitiv #1\fi #2\Z {\fi \XINT@dsx@addzeros {0000#2}}% +\def\XINT@dsx@exitv #1\fi #2\Z {\fi \XINT@dsx@addzeros {00000#2}}% +\def\XINT@dsx@exitvi #1\fi #2\Z {\fi \XINT@dsx@addzeros {000000#2}}% +\def\XINT@dsx@exitvii #1\fi #2\Z {\fi \XINT@dsx@addzeros {0000000#2}}% \def\XINT@dsx@addzeros #1#2{ #2#1}% \def\XINT@dsx@xisPos #1\Z #2% {% @@ -3100,10 +3079,8 @@ \def\xintDecSplit {\romannumeral0\xintdecsplit }% \def\xintdecsplit #1#2% {% - \expandafter - \xint@split - \expandafter - {\romannumeral0\xintabs {#2}}{#1}% fait expansion de A + \expandafter \xint@split \expandafter + {\romannumeral0\xintiabs {#2}}{#1}% fait expansion de A }% \def\xint@split #1#2% {% @@ -3146,32 +3123,31 @@ \def\XINT@split@fromleft@loop #1% {% \ifcase #1 - \expandafter\XINT@split@fromleft@endsplit + \XINT@split@fromleft@endsplit \or - \expandafter\XINT@split@fromleft@one@andend + \XINT@split@fromleft@one@andend \or - \expandafter\XINT@split@fromleft@two@andend + \XINT@split@fromleft@two@andend \or - \expandafter\XINT@split@fromleft@three@andend + \XINT@split@fromleft@three@andend \or - \expandafter\XINT@split@fromleft@four@andend + \XINT@split@fromleft@four@andend \or - \expandafter\XINT@split@fromleft@five@andend + \XINT@split@fromleft@five@andend \or - \expandafter\XINT@split@fromleft@six@andend + \XINT@split@fromleft@six@andend \or - \expandafter\XINT@split@fromleft@seven@andend + \XINT@split@fromleft@seven@andend \else \expandafter \XINT@split@fromleft@loop@perhaps \expandafter - {\the\numexpr - #1-8\expandafter\expandafter\expandafter }% + {\the\numexpr #1-8\expandafter\expandafter\expandafter }% \expandafter \XINT@split@fromleft@eight \fi }% -\def\XINT@split@fromleft@endsplit #1#2\W #3\Z - { {#1}{#2}}% +\def\XINT@split@fromleft@endsplit #1\fi #2#3\W #4\Z + {\expandafter\space\fi {#2}{#3}}% \def\XINT@split@fromleft@eight #1#2#3#4#5#6#7#8#9% {% #9{#1#2#3#4#5#6#7#8#9}% @@ -3189,26 +3165,26 @@ {% \space {#1}{}% }% -\def\XINT@split@fromleft@one@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@one }% +\def\XINT@split@fromleft@one@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@one }% \def\XINT@split@fromleft@one #1#2{#2{#1#2}}% -\def\XINT@split@fromleft@two@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@two }% +\def\XINT@split@fromleft@two@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@two }% \def\XINT@split@fromleft@two #1#2#3{#3{#1#2#3}}% -\def\XINT@split@fromleft@three@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@three }% +\def\XINT@split@fromleft@three@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@three }% \def\XINT@split@fromleft@three #1#2#3#4{#4{#1#2#3#4}}% -\def\XINT@split@fromleft@four@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@four }% +\def\XINT@split@fromleft@four@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@four }% \def\XINT@split@fromleft@four #1#2#3#4#5{#5{#1#2#3#4#5}}% -\def\XINT@split@fromleft@five@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@five }% +\def\XINT@split@fromleft@five@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@five }% \def\XINT@split@fromleft@five #1#2#3#4#5#6{#6{#1#2#3#4#5#6}}% -\def\XINT@split@fromleft@six@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@six }% +\def\XINT@split@fromleft@six@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@six }% \def\XINT@split@fromleft@six #1#2#3#4#5#6#7{#7{#1#2#3#4#5#6#7}}% -\def\XINT@split@fromleft@seven@andend - {\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@seven }% +\def\XINT@split@fromleft@seven@andend #1\fi +{\fi\expandafter\XINT@split@fromleft@checkiftoofar\XINT@split@fromleft@seven }% \def\XINT@split@fromleft@seven #1#2#3#4#5#6#7#8{#8{#1#2#3#4#5#6#7#8}}% \def\XINT@split@fromleft@checkiftoofar #1#2#3\W #4\Z {% @@ -3225,9 +3201,7 @@ }% \def\XINT@split@fromright #1\Z #2% {% - \expandafter - \XINT@split@fromright@a - \expandafter + \expandafter \XINT@split@fromright@a \expandafter {\romannumeral0\XINT@rev {#2}}{#1}{#2}% }% \def\XINT@split@fromright@a #1#2% @@ -3239,19 +3213,19 @@ \ifcase #1 \expandafter\XINT@split@fromright@endsplit \or - \expandafter\XINT@split@fromright@one@andend + \XINT@split@fromright@one@andend \or - \expandafter\XINT@split@fromright@two@andend + \XINT@split@fromright@two@andend \or - \expandafter\XINT@split@fromright@three@andend + \XINT@split@fromright@three@andend \or - \expandafter\XINT@split@fromright@four@andend + \XINT@split@fromright@four@andend \or - \expandafter\XINT@split@fromright@five@andend + \XINT@split@fromright@five@andend \or - \expandafter\XINT@split@fromright@six@andend + \XINT@split@fromright@six@andend \or - \expandafter\XINT@split@fromright@seven@andend + \XINT@split@fromright@seven@andend \else \expandafter \XINT@split@fromright@loop@perhaps \expandafter @@ -3263,9 +3237,7 @@ }% \def\XINT@split@fromright@endsplit #1#2\W #3\Z #4% {% - \expandafter - \space - \expandafter {\romannumeral0\XINT@rev{#2}}{#1}% + \expandafter\space\expandafter {\romannumeral0\XINT@rev{#2}}{#1}% }% \def\XINT@split@fromright@eight #1#2#3#4#5#6#7#8#9% {% @@ -3276,28 +3248,27 @@ \xint@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@one@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@one }% +\def\XINT@split@fromright@toofar\W\XINT@split@fromright@loop #1#2#3\Z { {}}% +\def\XINT@split@fromright@one@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@one }% \def\XINT@split@fromright@one #1#2{#2{#2#1}}% -\def\XINT@split@fromright@two@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@two }% +\def\XINT@split@fromright@two@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@two }% \def\XINT@split@fromright@two #1#2#3{#3{#3#2#1}}% -\def\XINT@split@fromright@three@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@three }% +\def\XINT@split@fromright@three@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@three }% \def\XINT@split@fromright@three #1#2#3#4{#4{#4#3#2#1}}% -\def\XINT@split@fromright@four@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@four }% +\def\XINT@split@fromright@four@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@four }% \def\XINT@split@fromright@four #1#2#3#4#5{#5{#5#4#3#2#1}}% -\def\XINT@split@fromright@five@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@five }% +\def\XINT@split@fromright@five@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@five }% \def\XINT@split@fromright@five #1#2#3#4#5#6{#6{#6#5#4#3#2#1}}% -\def\XINT@split@fromright@six@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@six }% +\def\XINT@split@fromright@six@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@six }% \def\XINT@split@fromright@six #1#2#3#4#5#6#7{#7{#7#6#5#4#3#2#1}}% -\def\XINT@split@fromright@seven@andend - {\expandafter\XINT@split@fromright@checkiftoofar\XINT@split@fromright@seven }% +\def\XINT@split@fromright@seven@andend #1\fi {\fi\expandafter + \XINT@split@fromright@checkiftoofar\XINT@split@fromright@seven }% \def\XINT@split@fromright@seven #1#2#3#4#5#6#7#8{#8{#8#7#6#5#4#3#2#1}}% \def\XINT@split@fromright@checkiftoofar #1% {% diff --git a/Master/texmf-dist/tex/generic/xint/xintfrac.sty b/Master/texmf-dist/tex/generic/xint/xintfrac.sty new file mode 100644 index 00000000000..895d408430b --- /dev/null +++ b/Master/texmf-dist/tex/generic/xint/xintfrac.sty @@ -0,0 +1,1107 @@ +%% +%% This is file `xintfrac.sty', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% xint.dtx (with options: `xintfrac') +%% +%% IMPORTANT NOTICE: +%% +%% For the copyright see the source file. +%% +%% Any modified versions of this file must be renamed +%% with new filenames distinct from xintfrac.sty. +%% +%% For distribution of the original source see the terms +%% for copying and modification in the file xint.dtx. +%% +%% This generated file may be distributed as long as the +%% original source files, as listed above, are part of the +%% same distribution. (The sources need not necessarily be +%% in the same archive or directory.) +%% +%%---------------------------------------------------------------- +%% The xint bundle (version 1.03 of April 14, 2013) +%% xintfrac: Expandable operations on fractions +%% Copyright (C) 2013 by Jean-Francois Burnol +%%---------------------------------------------------------------- +%% +\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}{Package xint is required}% + \y{xintfrac}{Will try \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}{Package xint is required}% + \y{xintfrac}{Will try \string\RequirePackage{xint}}% + \def\z{\endgroup\RequirePackage{xint}}% + \fi + \else + \y{xintfrac}{I was already loaded, aborting input}% + \aftergroup\endinput + \fi + \fi + \fi +\z% +\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 % : + \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 + \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% +\begingroup\catcode61\catcode48\catcode32=10\relax% + \catcode13=5 % ^^M + \endlinechar=13 % + \catcode123=1 % { + \catcode125=2 % } + \catcode64=11 % @ + \def\x + {% + \endgroup + \edef\XINT@frac@restorecatcodes@endinput + {% + \catcode94=\the\catcode94 % ^ + \catcode93=\the\catcode93 % ] + \catcode91=\the\catcode91 % [ + \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 % # + \catcode64=\the\catcode64 % @ + \catcode125=\the\catcode125 % } + \catcode123=\the\catcode123 % { + \endlinechar=\the\endlinechar + \catcode13=\the\catcode13 % ^^M + \catcode32=\the\catcode32 % + \catcode61=\the\catcode61 % = + \noexpand\endinput + }% + \XINT@setcatcodes + \catcode91=12 % [ + \catcode93=12 % ] + \catcode94=7 % ^ + }% +\x +\begingroup + \catcode58=12 % : + \expandafter\ifx\csname ProvidesPackage\endcsname\relax + \def\x#1#2#3[#4]{\endgroup + \immediate\write-1{Package: #3 #4}% + \xdef#1{#4}% + }% + \else + \def\x#1#2[#3]{\endgroup + #2[{#3}]% + \ifx#1\@undefined + \xdef#1{#3}% + \fi + \ifx#1\relax + \xdef#1{#3}% + \fi + }% + \fi +\expandafter\x\csname ver@xintfrac.sty\endcsname +\ProvidesPackage{xintfrac}% + [2013/04/14 v1.03 Expandable operations on fractions (jfB)]% +\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 +}% +\def\XINT@outfrac #1#2#3% +{% + \ifcase\XINT@Sgn{#3} + \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}% +\def\XINT@outfrac@P #1#2% +{% + \ifcase\XINT@Sgn{#1} + \expandafter\XINT@outfrac@Zero + \fi + \space #1/#2% +}% +\def\XINT@outfrac@Zero #1[#2]{ 0[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}% +}% +\def\xintNumerator {\romannumeral0\xintnumerator }% +\def\xintnumerator +{% + \expandafter\XINT@numer\romannumeral0\XINT@infrac +}% +\def\XINT@numer #1% +{% + \ifcase\XINT@Sgn {#1} + \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}% +\def\xintDenominator {\romannumeral0\xintdenominator }% +\def\xintdenominator +{% + \expandafter\XINT@denom\romannumeral0\XINT@infrac +}% +\def\XINT@denom #1% +{% + \ifcase\XINT@Sgn {#1} + \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}}% +\def\xintFrac {\romannumeral0\xintfrac }% +\def\xintfrac #1% +{% + \expandafter\XINT@@frac@A\romannumeral0\XINT@infrac {#1}% +}% +\def\XINT@@frac@A #1{\XINT@@frac@B #1\Z }% +\def\XINT@@frac@B #1#2\Z +{% + \xint@zero #1\XINT@@frac@C 0\XINT@@frac@D {10^{#1#2}}% +}% +\def\XINT@@frac@C #1#2#3#4#5% +{% + \ifcase\XINT@isOne {#5} + \or \xint@afterfi {\expandafter\xint@firstoftwo@andstop\xint@gobble@two }% + \fi + \space + \frac {#4}{#5}% +}% +\def\XINT@@frac@D #1#2#3% +{% + \ifcase\XINT@isOne {#3} + \or \XINT@@frac@E + \fi + \space + \frac {#2}{#3}#1% +}% +\def\XINT@@frac@E \fi #1#2#3#4{\fi \space #3\cdot }% +\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@zero #1\XINT@fwover@C 0\XINT@fwover@D {10^{#1#2}}% +}% +\def\XINT@fwover@C #1#2#3#4#5% +{% + \ifcase\XINT@isOne {#5} + \xint@afterfi { {#4\over #5}}% + \or + \xint@afterfi { #4\cdot }% + \fi +}% +\def\XINT@fwover@D #1#2#3% +{% + \ifcase\XINT@isOne {#3} + \xint@afterfi { {#2\over #3}}% + \or + \xint@afterfi { #2\cdot }% + \fi + #1% +}% +\def\xintSum {\romannumeral0\xintsum }% +\def\xintsum #1{\xintsumexpr #1\relax }% +\def\xintSumExpr {\romannumeral0\xintsumexpr }% +\def\xintsumexpr {\expandafter\expandafter\expandafter\XINT@fsumexpr }% +\def\XINT@fsumexpr {\XINT@fsum@loop {0}}% +\def\XINT@fsum@loop #1#2% +{% + \xint@relax #2\XINT@fsum@finished\relax + \expandafter\XINT@fsum@loop\expandafter + {\romannumeral0\xintadd {#1}{#2}}% +}% +\def\XINT@fsum@finished #1#2#3#4#5% +{% + \XINT@fsum@finished@ #5% +}% +\def\XINT@fsum@finished@ #1#2#3#4#5{ #4}% +\def\xintPrd {\romannumeral0\xintprd }% +\def\xintprd #1{\xintproductexpr #1\relax }% +\def\xintProductExpr {\romannumeral0\xintproductexpr }% +\def\xintproductexpr{\expandafter\expandafter\expandafter\XINT@fproductexpr }% +\def\XINT@fproductexpr {\XINT@fprod@loop {1}}% +\def\XINT@fprod@loop #1#2% +{% + \xint@relax #2\XINT@fprod@finished\relax + \expandafter\XINT@fprod@loop\expandafter + {\romannumeral0\xintmul {#2}{#1}}% +}% +\def\XINT@fprod@finished #1#2#3#4#5% +{% + \XINT@fprod@finished@ #5% +}% +\def\XINT@fprod@finished@ #1#2#3#4#5{ #5}% +\def\XINT@inFrac {\romannumeral0\XINT@infrac }% +\def\XINT@infrac #1% +{% + \expandafter\expandafter\expandafter + \XINT@infrac@ #1[\W]\Z\T +}% +\def\XINT@infrac@ #1[#2#3]#4\Z +{% + \xint@UDwfork + #2\dummy \XINT@infrac@A + \W\dummy \XINT@infrac@B + \xint@UDkrof + #1[#2#3]#4% +}% +\def\XINT@infrac@A #1[\W]\T +{% + \XINT@frac #1/\W\Z +}% +\def\XINT@infrac@B #1% +{% + \xint@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\dummy \XINT@infrac@BCa + \W\dummy {\expandafter\expandafter\expandafter\XINT@infrac@BCb #2}% + \xint@UDkrof + #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}}% +\def\XINT@frac #1/#2#3\Z +{% + \xint@UDwfork + #2\dummy \XINT@frac@A + \W\dummy {\expandafter\expandafter\expandafter\XINT@frac@B #2}% + \xint@UDkrof + #3.\W\Z #1.\W\Z +}% +\def\XINT@frac@B #1.#2#3\Z +{% + \xint@UDwfork + #2\dummy \XINT@frac@Ba + \W\dummy {\XINT@frac@Bb #2}% + \xint@UDkrof + #3\Z #1\Z +}% +\def\XINT@frac@Bb #1/\W.\W\Z #2\Z +{% + \expandafter \XINT@frac@C \expandafter + {\romannumeral0\XINT@length {#1}}{#2#1}% +}% +\def\XINT@frac@Ba \Z #1/\W\Z {\XINT@frac@C {0}{#1}}% +\def\XINT@frac@A .\W\Z {\XINT@frac@C {0}{1}}% +\def\XINT@frac@C #1#2#3.#4#5\Z +{% + \xint@UDwfork + #4\dummy \XINT@frac@Ca + \W\dummy {\XINT@frac@Cb #4}% + \xint@UDkrof + #5\Z #3\Z {#1}{#2}% +}% +\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\XINT@length {#1}}{#2#1}% +}% +\def\XINT@frac@D #1#2#3#4% +{% + \expandafter \XINT@frac@E \expandafter + {\the\numexpr -#1+#3\expandafter}\expandafter + {\romannumeral0\XINT@num@loop #2\R\R\R\R\R\R\R\R\Z }% + {\romannumeral0\XINT@num@loop #4\R\R\R\R\R\R\R\R\Z }% +}% +\def\XINT@frac@E #1#2#3% +{% + \expandafter \XINT@frac@F #3\Z {#2}{#1}% +}% +\def\XINT@frac@F #1% +{% + \xint@UDzerominusfork + #1-\dummy \XINT@frac@Gdivisionbyzero + 0#1\dummy \XINT@frac@Gneg + 0-\dummy {\XINT@frac@Gpos #1}% + \xint@UDkrof +}% +\def\XINT@frac@Gdivisionbyzero #1\Z #2#3% +{% + \xintError:DivisionByZero + \expandafter\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}}% +\def\XINT@factortens #1% +{% + \expandafter\XINT@cuz@cnt@loop\expandafter + {\expandafter}\romannumeral0\XINT@rord@main {}#1% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF + \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@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 #1#2#3#4#5#6% +{% + \XINT@cuz@cnt@toofarb {#5}#6% +}% +\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@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@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@zero #1\expandafter\XINT@cuz@cnt@loop\xint@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@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\relax}#3% +}% +\def\XINT@cuz@cnt@stopd #1#2\R #3\Z +{% + \expandafter\space\expandafter + {\romannumeral0\XINT@rord@main {}#2% + \xint@UNDEF + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@undef\xint@undef\xint@undef\xint@undef + \xint@UNDEF }{#1}% +}% +\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-\dummy \XINT@rez@zero + 0#1\dummy \XINT@rez@neg + 0-\dummy {\XINT@rez@B #1}% + \xint@UDkrof +}% +\def\XINT@rez@zero #1\Z #2#3{ 0/1[0]}% +\def\XINT@rez@neg +{\expandafter\space\expandafter-\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]}% +\def\XINT@@bts #1#2#3% +{% + \ifcase\XINT@isOne {#2} + \xint@afterfi {#3#1/#2}% + \or + \xint@afterfi {#3#1}% + \fi +}% +\def\xintIrr {\romannumeral0\xintirr }% +\def\xintirr +{% + \expandafter\XINT@@bts + \romannumeral0\expandafter\XINT@irr + \romannumeral0\XINT@infrac +}% +\def\XINT@irr #1% +{% + \ifcase\XINT@Sgn {#1} + \expandafter\XINT@irr@B + \or + \expandafter\XINT@irr@A + \else + \expandafter\XINT@irr@B + \fi + {#1}% +}% +\def\XINT@irr@A #1#2% +{% + \expandafter \XINT@irr@AC \expandafter + {\romannumeral0\xint@dsh {#2}{-#1}}% +}% +\def\XINT@irr@AC #1#2{\XINT@irr@C #2\Z #1\Z }% +\def\XINT@irr@B #1#2#3% +{% + \expandafter \XINT@irr@C \romannumeral0\xint@dsh {#3}{#1}\Z #2\Z +}% +\def\XINT@irr@C #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@irr@minusminus + #1-\dummy \XINT@irr@minusplus + #3-\dummy \XINT@irr@plusminus + --\dummy \XINT@irr@plusplus + \xint@UDkrof + {#4}{#2}#3#1% +}% +\def\XINT@irr@minusminus #1#2#3#4{\XINT@irr@D #1\Z #2\Z \space}% +\def\XINT@irr@minusplus #1#2#3#4{\XINT@irr@D #3#1\Z #2\Z \XINT@opp}% +\def\XINT@irr@plusminus #1#2#3#4{\XINT@irr@D #1\Z #4#2\Z \XINT@opp}% +\def\XINT@irr@plusplus #1#2#3#4{\XINT@irr@D #3#1\Z #4#2\Z \space}% +\def\XINT@irr@D #1#2\Z #3#4\Z +{% + \xint@UDzerosfork + #3#1\dummy \XINT@irr@indeterminate + #30\dummy \XINT@irr@divisionbyzero + #10\dummy \XINT@irr@zero + 00\dummy \XINT@irr@nonzero@checkifone + \xint@UDkrof + {#3#4}{#1#2}{#3#4}{#1#2}% +}% +\def\XINT@irr@indeterminate #1#2#3#4{\expandafter\xintError:ZeroOverZero + \space 00}% +\def\XINT@irr@divisionbyzero #1#2#3#4{\expandafter\xintError:DivisionByZero + \space {#2}0}% +\def\XINT@irr@zero #1#2#3#4{ 0/1}% +\def\XINT@irr@nonzero@checkifone #1% +{% + \ifcase\XINT@isOne {#1} + \xint@afterfi {\XINT@irr@loop@a {#1}}% + \or + \expandafter \XINT@irr@denomisone + \fi +}% +\def\XINT@irr@denomisone #1#2#3{ {#1}1}% +\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#3% +{% + \XINT@irr@loop@e #2\Z {#3}% +}% +\def\XINT@irr@loop@e #1#2\Z +{% + \xint@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\xintquo {#3}{#2}}% + {\romannumeral0\xintquo {#4}{#2}}% +}% +\def\XINT@irr@loop@exitb #1#2% +{% + \expandafter\space\expandafter {#2}{#1}% +}% +\def\xintJrr {\romannumeral0\xintjrr }% +\def\xintjrr +{% + \expandafter\XINT@@bts + \romannumeral0\expandafter\XINT@jrr@start + \romannumeral0\xintrez +}% +\def\XINT@jrr@start #1/#2[#3]% +{% + \ifcase\XINT@Sgn {#3} + \expandafter\XINT@jrr@B + \or + \expandafter\XINT@jrr@A + \else + \expandafter\XINT@jrr@B + \fi + {#3}{#1}{#2}% +}% +\def\XINT@jrr@A #1#2% +{% + \expandafter \XINT@jrr@AC \expandafter + {\romannumeral0\xint@dsh {#2}{-#1}}% +}% +\def\XINT@jrr@AC #1#2{\XINT@jrr@C #2\Z #1\Z }% +\def\XINT@jrr@B #1#2#3% +{% + \expandafter \XINT@jrr@C \romannumeral0\xint@dsh {#3}{#1}\Z #2\Z +}% +\def\XINT@jrr@C #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@jrr@minusminus + #1-\dummy \XINT@jrr@minusplus + #3-\dummy \XINT@jrr@plusminus + --\dummy \XINT@jrr@plusplus + \xint@UDkrof + {#4}{#2}#3#1% +}% +\def\XINT@jrr@minusminus #1#2#3#4{\XINT@jrr@D #1\Z #2\Z \space }% +\def\XINT@jrr@minusplus #1#2#3#4{\XINT@jrr@D #3#1\Z #2\Z \XINT@opp }% +\def\XINT@jrr@plusminus #1#2#3#4{\XINT@jrr@D #1\Z #4#2\Z \XINT@opp }% +\def\XINT@jrr@plusplus #1#2#3#4{\XINT@jrr@D #3#1\Z #4#2\Z \space }% +\def\XINT@jrr@D #1#2\Z #3#4\Z +{% + \xint@UDzerosfork + #3#1\dummy \XINT@jrr@indeterminate + #30\dummy \XINT@jrr@divisionbyzero + #10\dummy \XINT@jrr@zero + 00\dummy \XINT@jrr@nonzero@checkifone + \xint@UDkrof + {#3#4}{#1#2}1001% +}% +\def\XINT@jrr@indeterminate #1#2#3#4#5#6{\expandafter\xintError:ZeroOverZero + \space 00}% +\def\XINT@jrr@divisionbyzero #1#2#3#4#5#6{\expandafter\xintError:DivisionByZero + \space {#2}0}% +\def\XINT@jrr@zero #1#2#3#4#5#6{ 0/1}% +\def\XINT@jrr@nonzero@checkifone #1% +{% + \ifcase\XINT@isOne {#1} + \xint@afterfi {\XINT@jrr@loop@a {#1}}% + \or + \expandafter \XINT@jrr@denomisone + \fi +}% +\def\XINT@jrr@denomisone #1#2#3#4#5{ {#1}1}% +\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\xintiadd{\XINT@Mul{#4}{#1}}{#6}}% + {\romannumeral0\xintiadd{\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@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% +{% + \space {#3}{#4}% +}% +\def\xintTrunc {\romannumeral0\xinttrunc }% +\def\xintiTrunc {\romannumeral0\xintitrunc }% +\def\xinttrunc #1% +{% + \expandafter\expandafter\expandafter + \xint@trunc + \expandafter\expandafter\expandafter + {#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\expandafter\expandafter + \xint@itrunc + \expandafter\expandafter\expandafter + {#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\relax}#2\Z {#3}% +}% +\def\XINT@trunc@checkifzero #1#2#3\Z +{% + \xint@zero #2\XINT@trunc@iszero0\XINT@trunc@B {#1}{#2#3}% +}% +\def\XINT@trunc@iszero #1#2#3#4#5{ 0\Z 0}% +\def\XINT@trunc@B #1% +{% + \ifcase\XINT@Sgn {#1} + \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@E + \romannumeral0\xint@dsh {#3}{#1}\Z #2\Z +}% +\def\XINT@trunc@D #1#2% +{% + \expandafter \XINT@trunc@DE \expandafter + {\romannumeral0\xint@dsh {#2}{-#1}}% +}% +\def\XINT@trunc@DE #1#2{\XINT@trunc@E #2\Z #1\Z }% +\def\XINT@trunc@E #1#2\Z #3#4\Z +{% + \xint@UDsignsfork + #1#3\dummy \XINT@trunc@minusminus + #1-\dummy {\XINT@trunc@minusplus #3}% + #3-\dummy {\XINT@trunc@plusminus #1}% + --\dummy {\XINT@trunc@plusplus #3#1}% + \xint@UDkrof + {#4}{#2}% +}% +\def\XINT@trunc@minusminus #1#2{\xintquo {#1}{#2}\Z \space}% +\def\XINT@trunc@minusplus #1#2#3{\xintquo {#1#2}{#3}\Z \xint@minus@andstop}% +\def\XINT@trunc@plusminus #1#2#3{\xintquo {#2}{#1#3}\Z \xint@minus@andstop}% +\def\XINT@trunc@plusplus #1#2#3#4{\xintquo {#1#3}{#2#4}\Z \space}% +\def\XINT@itrunc@G #1\Z #2#3% +{% + \xint@zero #2\XINT@trunc@zero 0\xint@firstoftwo {#2#1}0% +}% +\def\XINT@trunc@G #1\Z #2#3% +{% + \xint@zero #2\XINT@trunc@zero 0% + \expandafter\XINT@trunc@H\expandafter + {\the\numexpr\romannumeral0\XINT@length {#1}-#3\relax}{#3}{#1}#2% +}% +\def\XINT@trunc@zero 0#10{ 0}% +\def\XINT@trunc@H #1#2% +{% + \ifnum #1 > 0 + \xint@afterfi {\XINT@trunc@Ha {#2}}% + \else + \xint@afterfi {\XINT@trunc@Hb {-#1}}% + \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% +}% +\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\xintimul {#6}{#3}}% + {\romannumeral0\xintimul {#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}}% +\def\xintSqr {\romannumeral0\xintsqr }% +\def\xintsqr #1% +{% + \expandafter\xint@fsqr\expandafter{\romannumeral0\XINT@infrac {#1}}% +}% +\def\xint@fsqr #1{\XINT@fmul@A #1#1}% +\def\xintPow {\romannumeral0\xintpow }% +\def\xintpow #1% +{% + \expandafter\xint@fpow\expandafter {\romannumeral0\XINT@infrac {#1}}% +}% +\def\xint@fpow #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@fpow@fork + #2\Z #1% +}% +\def\XINT@fpow@fork #1#2\Z +{% + \xint@UDzerominusfork + #1-\dummy \XINT@fpow@zero + 0#1\dummy \XINT@fpow@neg + 0-\dummy {\XINT@fpow@pos #1}% + \xint@UDkrof + {#2}% +}% +\def\XINT@fpow@zero #1#2#3#4% +{% + \space 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\xintipow {#5}{#1#2}}% + {\romannumeral0\xintipow {#4}{#1#2}}% +}% +\def\XINT@fpow@neg #1#2#3#4% +{% + \expandafter\XINT@fpow@pos@A\expandafter + {\the\numexpr -#1*#2\expandafter}\expandafter + {\romannumeral0\xintipow {#3}{#1}}% + {\romannumeral0\xintipow {#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}}% +\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\xintimul {#2}{#6}}% + {\romannumeral0\xintimul {#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}}% +\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\xintimul {#7}{#5}}% + {\romannumeral0\xintiadd + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#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}}% +\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\xintimul {#7}{#5}}% + {\romannumeral0\xintisub + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#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}}% +\def\xintCmp {\romannumeral0\xintcmp }% +\def\xintcmp #1% +{% + \expandafter\xint@fcmp\expandafter {\romannumeral0\XINT@infrac {#1}}% +}% +\def\xint@fcmp #1#2{\expandafter\XINT@fcmp@A\romannumeral0\XINT@infrac {#2}#1}% +\def\XINT@fcmp@A #1#2#3#4% +{% + \ifnum #4 > #1 + \xint@afterfi {\XINT@fcmp@B {#1}}% + \else + \xint@afterfi {\XINT@fcmp@B {#4}}% + \fi + {#1}{#4}{#2}{#3}% +}% +\def\XINT@fcmp@B #1#2#3#4#5#6#7% +{% + \xinticmp + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}% +}% +\def\xintMax {\romannumeral0\xintmax }% +\def\xintmax #1% +{% + \expandafter\xint@fmax\expandafter {\romannumeral0\XINT@infrac {#1}}% +}% +\def\xint@fmax #1#2{\expandafter\XINT@outfrac + \romannumeral0\expandafter\XINT@fmax@A + \romannumeral0\XINT@infrac {#2}#1}% +\def\XINT@fmax@A #1#2#3#4#5#6% +{% + \ifnum #4 > #1 + \xint@afterfi {\XINT@fmax@B {#1}}% + \else + \xint@afterfi {\XINT@fmax@B {#4}}% + \fi + {#1}{#4}{#2}{#3}{#5}{#6}{{#4}{#5}{#6}}{{#1}{#2}{#3}}% +}% +\def\XINT@fmax@B #1#2#3#4#5#6#7% +{% + \expandafter\XINT@fmax@C\expandafter + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}% +}% +\def\XINT@fmax@C #1#2% +{% + \expandafter\XINT@max@fork #2\Z #1\Z +}% +\def\xintMin {\romannumeral0\xintmin }% +\def\xintmin #1% +{% + \expandafter\xint@fmin\expandafter {\romannumeral0\XINT@infrac {#1}}% +}% +\def\xint@fmin #1#2{\expandafter\XINT@outfrac + \romannumeral0\expandafter\XINT@fmin@A + \romannumeral0\XINT@infrac {#2}#1}% +\def\XINT@fmin@A #1#2#3#4#5#6% +{% + \ifnum #4 > #1 + \xint@afterfi {\XINT@fmin@B {#1}}% + \else + \xint@afterfi {\XINT@fmin@B {#4}}% + \fi + {#1}{#4}{#2}{#3}{#5}{#6}{{#4}{#5}{#6}}{{#1}{#2}{#3}}% +}% +\def\XINT@fmin@B #1#2#3#4#5#6#7% +{% + \expandafter\XINT@fmin@C\expandafter + {\romannumeral0\xintimul {\xintDSH {\the\numexpr -#3+#1\relax}{#6}}{#5}}% + {\romannumeral0\xintimul {#7}{\xintDSH {\the\numexpr -#2+#1\relax}{#4}}}% +}% +\def\XINT@fmin@C #1#2% +{% + \expandafter\XINT@min@fork #2\Z #1\Z +}% +\def\xintSgn {\romannumeral0\xintsgn }% +\def\xintsgn #1% +{% + \expandafter\xint@fsgn\romannumeral0\XINT@infrac {#1}% +}% +\def\xint@fsgn #1#2#3{\xintisgn {#2}}% +\def\xintOpp {\romannumeral0\xintopp }% +\def\xintopp #1% +{% + \expandafter\xint@fopp\romannumeral0\XINT@infrac {#1}% +}% +\def\xint@fopp #1#2{\expandafter\XINT@outfrac\expandafter + {\the\numexpr #1\expandafter}\expandafter + {\romannumeral0\XINT@opp #2}}% +\def\xintAbs {\romannumeral0\xintabs }% +\def\xintabs #1% +{% + \expandafter\xint@fabs\romannumeral0\XINT@infrac {#1}% +}% +\def\xint@fabs #1#2{\expandafter\XINT@outfrac\expandafter + {\the\numexpr #1\expandafter}\expandafter + {\romannumeral0\XINT@abs #2}}% +\XINT@frac@restorecatcodes@endinput% +\endinput +%% +%% End of file `xintfrac.sty'. diff --git a/Master/texmf-dist/tex/generic/xint/xintgcd.sty b/Master/texmf-dist/tex/generic/xint/xintgcd.sty index a270186f4ed..24bd6ef2109 100644 --- a/Master/texmf-dist/tex/generic/xint/xintgcd.sty +++ b/Master/texmf-dist/tex/generic/xint/xintgcd.sty @@ -22,7 +22,7 @@ %% in the same archive or directory.) %% %%---------------------------------------------------------------- -%% The xint bundle (version 1.02 of April 5, 2013) +%% The xint bundle (version 1.03 of April 14, 2013) %% xintgcd: Euclidean algorithm with xint package %% Copyright (C) 2013 by Jean-Francois Burnol %%---------------------------------------------------------------- @@ -162,20 +162,18 @@ \fi \expandafter\x\csname ver@xintgcd.sty\endcsname \ProvidesPackage{xintgcd}% - [2013/04/05 v1.02 Euclide algorithm with xint package (jfB)]% + [2013/04/14 v1.03 Euclide algorithm with xint package (jfB)]% \def\xintGCD {\romannumeral0\xintgcd }% \def\xintgcd #1% {% - \expandafter - \XINT@gcd - \expandafter - {\romannumeral0\xintabs {#1}}% + \expandafter \XINT@gcd \expandafter + {\romannumeral0\xintiabs {#1}}% }% \def\XINT@gcd #1#2% {% \expandafter \XINT@gcd@fork - \romannumeral0\xintabs {#2}\Z #1\Z + \romannumeral0\xintiabs {#2}\Z #1\Z }% \def\XINT@gcd@fork #1#2\Z #3#4\Z {% @@ -255,10 +253,8 @@ }% \def\XINT@bezout@mm@post #1#2% {% - \expandafter - \XINT@bezout@mm@postb - \expandafter - {\romannumeral0\xintopp{#2}}{\romannumeral0\xintopp{#1}}% + \expandafter \XINT@bezout@mm@postb \expandafter + {\romannumeral0\xintiopp{#2}}{\romannumeral0\xintiopp{#1}}% }% \def\XINT@bezout@mm@postb #1#2% {% @@ -277,10 +273,8 @@ }% \def\XINT@bezout@mp@post #1#2% {% - \expandafter - \XINT@bezout@mp@postb - \expandafter - {\romannumeral0\xintopp {#2}}{#1}% + \expandafter \XINT@bezout@mp@postb \expandafter + {\romannumeral0\xintiopp {#2}}{#1}% }% \def\XINT@bezout@mp@postb #1#2#3#4#5% {% @@ -293,10 +287,8 @@ }% \def\XINT@bezout@pm@post #1% {% - \expandafter - \XINT@bezout@pm@postb - \expandafter - {\romannumeral0\xintopp{#1}}% + \expandafter \XINT@bezout@pm@postb \expandafter + {\romannumeral0\xintiopp{#1}}% }% \def\XINT@bezout@pm@postb #1#2#3#4#5% {% @@ -319,18 +311,14 @@ }% \def\XINT@bezout@loop@b #1#2#3#4#5#6#7#8% {% - \expandafter - \XINT@bezout@loop@c - \expandafter - {\romannumeral0\xintadd{\XINT@Mul{#5}{#2}}{#7}}% - {\romannumeral0\xintadd{\XINT@Mul{#6}{#2}}{#8}}% + \expandafter \XINT@bezout@loop@c \expandafter + {\romannumeral0\xintiadd{\XINT@Mul{#5}{#2}}{#7}}% + {\romannumeral0\xintiadd{\XINT@Mul{#6}{#2}}{#8}}% {#1}{#3}{#4}{#5}{#6}% }% \def\XINT@bezout@loop@c #1#2% {% - \expandafter - \XINT@bezout@loop@d - \expandafter + \expandafter \XINT@bezout@loop@d \expandafter {#2}{#1}% }% \def\XINT@bezout@loop@d #1#2#3#4#5% @@ -364,16 +352,14 @@ \def\xintEuclideAlgorithm {\romannumeral0\xinteuclidealgorithm }% \def\xinteuclidealgorithm #1% {% - \expandafter - \XINT@euc - \expandafter - {\romannumeral0\xintabs {#1}}% + \expandafter \XINT@euc \expandafter + {\romannumeral0\xintiabs {#1}}% }% \def\XINT@euc #1#2% {% \expandafter \XINT@euc@fork - \romannumeral0\xintabs {#2}\Z #1\Z + \romannumeral0\xintiabs {#2}\Z #1\Z }% \def\XINT@euc@fork #1#2\Z #3#4\Z {% @@ -418,16 +404,14 @@ \def\xintBezoutAlgorithm {\romannumeral0\xintbezoutalgorithm }% \def\xintbezoutalgorithm #1% {% - \expandafter - \XINT@bezalg - \expandafter - {\romannumeral0\xintabs {#1}}% + \expandafter \XINT@bezalg \expandafter + {\romannumeral0\xintiabs {#1}}% }% \def\XINT@bezalg #1#2% {% \expandafter \XINT@bezalg@fork - \romannumeral0\xintabs {#2}\Z #1\Z + \romannumeral0\xintiabs {#2}\Z #1\Z }% \def\XINT@bezalg@fork #1#2\Z #3#4\Z {% @@ -450,8 +434,8 @@ \def\XINT@bezalg@b #1#2#3#4#5#6#7#8% {% \expandafter\XINT@bezalg@c\expandafter - {\romannumeral0\xintadd {\xintMul {#6}{#2}}{#8}}% - {\romannumeral0\xintadd {\xintMul {#5}{#2}}{#7}}% + {\romannumeral0\xintiadd {\xintiMul {#6}{#2}}{#8}}% + {\romannumeral0\xintiadd {\xintiMul {#5}{#2}}{#7}}% {#1}{#2}{#3}{#4}{#5}{#6}% }% \def\XINT@bezalg@c #1#2#3#4#5#6% diff --git a/Master/texmf-dist/tex/generic/xint/xintseries.sty b/Master/texmf-dist/tex/generic/xint/xintseries.sty new file mode 100644 index 00000000000..6cde51e7928 --- /dev/null +++ b/Master/texmf-dist/tex/generic/xint/xintseries.sty @@ -0,0 +1,349 @@ +%% +%% This is file `xintseries.sty', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% xint.dtx (with options: `xintseries') +%% +%% IMPORTANT NOTICE: +%% +%% For the copyright see the source file. +%% +%% Any modified versions of this file must be renamed +%% with new filenames distinct from xintseries.sty. +%% +%% For distribution of the original source see the terms +%% for copying and modification in the file xint.dtx. +%% +%% This generated file may be distributed as long as the +%% original source files, as listed above, are part of the +%% same distribution. (The sources need not necessarily be +%% in the same archive or directory.) +%% +%%---------------------------------------------------------------- +%% The xint bundle (version 1.03 of April 14, 2013) +%% xintseries: Expandable partial sums with xint package +%% Copyright (C) 2013 by Jean-Francois Burnol +%%---------------------------------------------------------------- +%% +\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}{Package xintfrac is required}% + \y{xintseries}{Will try \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}{Package xintfrac is required}% + \y{xintseries}{Will try \string\RequirePackage{xintfrac}}% + \def\z{\endgroup\RequirePackage{xintfrac}}% + \fi + \else + \y{xintseries}{I was already loaded, aborting input}% + \aftergroup\endinput + \fi + \fi + \fi +\z% +\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 % : + \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 + \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% +\begingroup\catcode61\catcode48\catcode32=10\relax% + \catcode13=5 % ^^M + \endlinechar=13 % + \catcode123=1 % { + \catcode125=2 % } + \catcode64=11 % @ + \def\x + {% + \endgroup + \edef\XINT@series@restorecatcodes@endinput + {% + \catcode93=\the\catcode93 % ] + \catcode91=\the\catcode91 % [ + \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 % # + \catcode64=\the\catcode64 % @ + \catcode125=\the\catcode125 % } + \catcode123=\the\catcode123 % { + \endlinechar=\the\endlinechar + \catcode13=\the\catcode13 % ^^M + \catcode32=\the\catcode32 % + \catcode61=\the\catcode61 % = + \noexpand\endinput + }% + \XINT@setcatcodes + \catcode91=12 % [ + \catcode93=12 % ] + }% +\x +\begingroup + \catcode58=12 % : + \expandafter\ifx\csname ProvidesPackage\endcsname\relax + \def\x#1#2#3[#4]{\endgroup + \immediate\write-1{Package: #3 #4}% + \xdef#1{#4}% + }% + \else + \def\x#1#2[#3]{\endgroup + #2[{#3}]% + \ifx#1\@undefined + \xdef#1{#3}% + \fi + \ifx#1\relax + \xdef#1{#3}% + \fi + }% + \fi +\expandafter\x\csname ver@xintseries.sty\endcsname +\ProvidesPackage{xintseries}% + [2013/04/14 v1.03 Expandable partial sums with xint package (jfB)]% +\def\xintSeries {\romannumeral0\xintseries }% +\def\xintseries #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@series@i + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@series@i #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@series@ii + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@series@ii #1#2#3% +{% + \ifnum #2<#1 + \xint@afterfi { 0[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@two #6% +}% +\def\xintiSeries {\romannumeral0\xintiseries }% +\def\xintiseries #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@iseries@i + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@iseries@i #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@iseries@ii + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@iseries@ii #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\xintiadd {#2}{#4{#1}}}% + {#3}{#4}% +}% +\def\XINT@iseries@exit \fi #1#2#3#4#5#6#7#8% +{% + \fi\xint@gobble@two #6% +}% +\def\xintPowerSeries {\romannumeral0\xintpowerseries }% +\def\xintpowerseries #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@powseries@i + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@powseries@i #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@powseries@ii + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@powseries@ii #1#2#3#4% +{% + \ifnum #2<#1 + \xint@afterfi { 0[0]}% + \else + \xint@afterfi + {\expandafter\XINT@powseries@loop@pre\expandafter + {\romannumeral0\xintpow {#4}{#1}}{#1}{#4}{#2}{#3}% + }% + \fi +}% +\def\XINT@powseries@loop@pre #1#2#3#4#5% +{% + \ifnum #4>#2 \else\XINT@powseries@dont@i \fi + \expandafter\XINT@powseries@loop@i\expandafter + {\the\numexpr #2+1\expandafter}\expandafter + {\romannumeral0\xintmul {#5{#2}}{#1}}{#1}{#3}{#4}{#5}% +}% +\def\XINT@powseries@dont@i \fi\expandafter\XINT@powseries@loop@i + {\fi \expandafter\XINT@powseries@dont@ii }% +\def\XINT@powseries@dont@ii #1#2#3#4#5#6{ #2}% +\def\XINT@powseries@loop@i #1#2#3#4#5% +{% + \ifnum #5>#1 \else \XINT@powseries@exit@i \fi + \expandafter\XINT@powseries@loop@ii\expandafter + {\romannumeral0\xintmul {#3}{#4}}{#1}{#4}{#2}{#5}% +}% +\def\XINT@powseries@loop@ii #1#2#3#4#5#6% +{% + \expandafter\XINT@powseries@loop@i\expandafter + {\the\numexpr #2+1\expandafter}\expandafter + {\romannumeral0\xintadd {#4}{\xintMul {#6{#2}}{#1}}}% + {#1}{#3}{#5}{#6}% +}% +\def\XINT@powseries@exit@i\fi \expandafter\XINT@powseries@loop@ii + {\fi \expandafter\XINT@powseries@exit@ii }% +\def\XINT@powseries@exit@ii #1#2#3#4#5#6% + {\xintadd {#4}{\xintMul {#6{#2}}{#1}}}% +\def\xintFxPtPowerSeries {\romannumeral0\xintfxptpowerseries }% +\def\xintfxptpowerseries #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@fppowseries@i + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@fppowseries@i #1#2% +{% + \expandafter\expandafter\expandafter + \XINT@fppowseries@ii + \expandafter\expandafter\expandafter + {#2}{#1}% +}% +\def\XINT@fppowseries@ii #1#2#3#4#5% +{% + \ifnum #2<#1 + \xint@afterfi {\xinttrunc {#5}{0[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+1\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+1\expandafter}\expandafter + {\romannumeral0\xintiadd {#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}% + {\xintiAdd {#4}{\xintiTrunc {#7}{\xintMul {#6{#2}}{#1}}}[-#7]}}% +\XINT@series@restorecatcodes@endinput% +\endinput +%% +%% End of file `xintseries.sty'. |