% Arbitrary precision numbers %%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 2014, 2015 Petr Olsak % See the documentation apnum.pdf or apnum.d for more information \def\apnumversion{1.2 } \message{The Arbitrary Precision Numbers, \apnumversion} %%%%%%%%%%%% Internal registers: \newcount\apnumA \newcount\apnumB \newcount\apnumC \newcount\apnumD \newcount\apnumE \newcount\apnumF \newcount\apnumG \newcount\apnumH \newcount\apnumO \newcount\apnumL \newcount\apnumX \newcount\apnumY \newcount\apnumZ \newcount\apSIGNa \newcount\apSIGNb \newcount\apEa \newcount\apEb \newif\ifapX \apnumZ=\catcode`\@ \catcode`\@=12 %%%%%%%%%%%% Public macros, sec. 2.1 in apnum.pdf \def\evaldef{\relax \apEVALa} \def\PLUS{\relax \apPPab\apPLUSa} \def\MINUS#1#2{\relax \apPPab\apPLUSa{#1}{-#2}} \def\MUL{\relax \apPPab\apMULa} \def\DIV{\relax \apPPab\apDIVa} \def\POW{\relax \apPPab\apPOWa} \def\ABS{\relax \apEVALone\apABSa} \def\iDIV{\relax \apEVALtwo\apiDIVa} \def\iMOD{\relax \apEVALtwo\apiMODa} \def\iROUND#1{\relax \evaldef\OUT{#1}\apiROUNDa} \def\iFRAC{\relax \apEVALone\apiFRACa} \def\FAC{\relax \apEVALone\apFACa} \def\ROUND{\apPPs\apROUNDa} \def\ROLL{\apPPs\apROLLa} \def\NORM{\apPPs\apNORMa} \def\addE#1{\edef#1{#1\ifnum\apE=0 \else E\ifnum\apE>0+\fi\the\apE\fi}} \newcount\apSIGN \let\SIGN=\apSIGN \newcount\apE \newcount\apTOT \apTOT=-30 \newcount\apFRAC \apFRAC=20 %%%%%%%%%%%% Evaluation of the expression, sec. 2.2 in apnum.pdf \def\apEVALa#1#2{{\apnumA=0 \apnumE=1 \apEVALb#2\end\expandafter}\tmpb \let#1=\OUT} \def\apEVALb{\def\tmpa{}\apEVALc} \def\apEVALc#1{% \ifx+#1\apEVALd \apEVALc \fi \ifx-#1\edef\tmpa{\tmpa-}\apEVALd\apEVALc \fi \ifx(#1\apEVALd \apEVALe \fi \ifx\the#1\apEVALd \apEVALf\the\fi \ifx\number#1\apEVALd \apEVALf\number\fi \apTESTdigit#1\iftrue \ifx E#1\let\tmpb=\tmpa \expandafter\apEVALd\expandafter\apEVALk \else \edef\tmpb{\tmpa#1}\expandafter\apEVALd\expandafter\apEVALn\fi\fi \edef\tmpb{\tmpa\noexpand#1}\futurelet\apNext\apEVALg } \def\apEVALd#1\fi#2\apNext\apEVALg{\fi#1} \def\apEVALe{% \ifx\tmpa\empty \else \ifnum\tmpa1<0 \def\tmpb{-1}\apEVALp \MUL 4\fi\fi \advance\apnumA by4 \apEVALb } \def\apEVALf#1#2{\expandafter\def\expandafter\tmpb\expandafter{\tmpa#1#2}\apEVALo} \def\apEVALg{\ifx\apNext \bgroup \expandafter\apEVALh \else \expandafter\apEVALo \fi} \def\apEVALh#1{\expandafter\def\expandafter\tmpb\expandafter{\tmpb{#1}}\futurelet\apNext\apEVALg} \def\apEVALk{\afterassignment\apEVALm\apE=} \def\apEVALm{\edef\tmpb{\tmpb E\the\apE}\apEVALo} \def\apEVALn#1{\apTESTdigit#1% \iftrue \ifx E#1\afterassignment\apEVALm\expandafter\expandafter\expandafter\apE \else\edef\tmpb{\tmpb#1}\expandafter\expandafter\expandafter\apEVALn\fi \else \expandafter\apEVALo\expandafter#1\fi } \def\apEVALo#1{\let\apNext=\apEVALb \ifx+#1\apEVALp \apEPLUS 1\fi \ifx-#1\apEVALp \apEMINUS 1\fi \ifx*#1\apEVALp \apEMUL 2\fi \ifx/#1\apEVALp \apEDIV 2\fi \ifx^#1\apEVALp \apEPOW 3\fi \ifx)#1\advance\apnumA by-4 \let\apNext=\apEVALo \let\tmpa=\relax \ifnum\apnumA<0 \apEVALerror{many brackets ")"}\fi \fi \ifx\end#1% \ifnum\apnumA>0 \apEVALerror{missing bracket ")"}% \else \apEVALp\END 0\fi \let\apNext=\relax \fi \ifx\tmpa\relax \else \apEVALerror{unknown operator "\string#1"}\fi \apnumE=0 \apNext } \def\apEVALp#1#2{% \apnumB=#2 \advance\apnumB by\apnumA \toks0=\expandafter{\expandafter{\tmpb}{#1}}% \expandafter\apEVALpush\the\toks0\expandafter{\the\apnumB}% {value}{op}{priority} \let\tmpa=\relax } \let\apEPLUS=\PLUS \let\apEMINUS=\MINUS \let\apEMUL=\MUL \let\apEDIV=\DIV \let\apEPOW=\POW \def\apEVALstack{{}{}{0}.} \def\apEVALpush#1#2#3{% value, operator, priority \toks0={{#1}{#2}{#3}}% \expandafter\def\expandafter\apEVALstack\expandafter{\the\toks0\apEVALstack}% \expandafter\apEVALdo\apEVALstack@% } \def\apEVALdo#1#2#3#4#5#6#7@{% \apnumB=#3 \ifx#2\POW \advance\apnumB by1 \fi \ifnum\apnumB>#6\else \ifnum#6=0 \def\tmpb{#1}%\toks0={#1}\message{RESULT: \the\toks0} \ifnum\apnumE=1 \def\tmpb{\apPPn{#1}}\fi \else \def\apEVALstack{#7}\apEVALpush{#5{#4}{#1}}{#2}{#3}% \fi\fi } \def\apEVALerror#1{\message{\noexpand\evaldef ERROR: #1.}% \def\tmpb{\def\OUT{??}}\def\apNext##1\end{}% } \def\apTESTdigit#1#2{% \ifx E#1\apXtrue \else \ifcat.\noexpand#1% \ifx.#1\apXtrue \else \ifnum`#1<`0 \apXfalse\else \ifnum`#1>`9 \apXfalse\else \apXtrue\fi \fi\fi \else \apXfalse \fi\fi \ifapX } %%%%%%%%%%%% Preparation of the parameter, sec. 2.3 in apnum.pdf \def\apPPa#1#2{\expandafter\apPPb#2@#1} \def\apPPb{\def\tmpc{}\apSIGN=1 \apE=0 \apXfalse \expandafter\expandafter\expandafter\apPPc} \def\apPPc#1{% \ifx+#1\apPPd \fi \ifx-#1\apSIGN=-\apSIGN \apPPd \fi \ifx\relax#1\apPPe \fi \apPPg#1% } \def\apPPd#1\apPPg#2{\fi\expandafter\expandafter\expandafter\apPPc} \def\apPPe#1\apPPg#2#3@{\fi\apXtrue{#3% execution of the parameter in the group \edef\tmpc{\apE=\the\apE\relax\noexpand\apPPf\OUT@}\expandafter}\tmpc } \def\apPPf#1{\ifx-#1\apSIGN=-\apSIGN \expandafter\apPPg\else\expandafter\apPPg\expandafter#1\fi} \def\apPPg#1{% \ifx.#1\def\tmpc{.}\apPPh\fi \ifx\tmpc\empty\else\edef\tmpc{\tmpc#1}\fi \ifx0#1\apPPh\fi \ifx\tmpc\empty\edef\tmpc{#1}\fi \ifx@#1\def\tmpc{@}\fi \expandafter\apPPi\tmpc } \def\apPPh#1\apPPi\tmpc{\fi\apPPg} \def\apPPi{\ifapX \expandafter\apPPj \else \expandafter\apPPk \fi} \def\apPPj#1@#2{\def#2{#1}} \def\apPPk#1@#2{\ifx@#1@\apSIGN=0 \def#2{0}\else \apPPl#1E@#2\fi} \def\apPPl#1E#2@#3{% \ifx@#1@\def#3{1}\else\def#3{#1}\fi \ifx@#2@\else \afterassignment\apPPm \apE=#2\fi } \def\apPPm E{} \def\apPPn#1{\expandafter\apPPb#1@\OUT \edef\OUT{\ifnum\apSIGN<0-\fi\OUT}} \def\apPPab#1#2#3{% \expandafter\apPPb#2@\tmpa \apSIGNa=\apSIGN \apEa=\apE \expandafter\apPPb#3@\tmpb \apSIGNb=\apSIGN \apEb=\apE #1% } \def\apPPs#1#2#3{\def\tmpc{#3}\expandafter\apPPt\expandafter#1#2.@#2} \def\apPPt#1#2{% \ifx-#2\apnumG=-1 \def\apNext{#1}% \else \ifx0#2\apnumG=0 \def\apNext{\apPPu#1}\else \apnumG=1 \def\apNext{#1#2}\fi\fi \apNext } \def\apPPu#1#2.@#3{\ifx@#2@\apnumG=0 \ifx#1\apROUNDa\def\XOUT{}\fi \else\def\apNext{\apPPt#1#2.@#3}\expandafter\apNext\fi } \def\apEVALone#1#2{\evaldef\OUT{#2}\ifnum\apSIGN<0 \expandafter\apNOminus\OUT@\OUT\fi #1} \def\apEVALtwo#1#2#3{% {\evaldef\OUT{#2}\apOUTtmpb}\tmpb \let\tmpa=\OUT \apSIGNa=\apSIGN \apEa=\apE \ifnum\apSIGNa<0 \expandafter\apNOminus\tmpa@\tmpa\fi {\evaldef\OUT{#3}\apOUTtmpb}\tmpb \let\tmpb=\OUT \apSIGNb=\apSIGN \apEb=\apE \ifnum\apSIGNb<0 \expandafter\apNOminus\tmpb@\tmpb\fi #1% } \def\apNOminus-#1@#2{\def#2{#1}} %%%%%%%%%%%% Addition and Subtraction, sec. 2.4 in apnum.pdf \def\apPLUSa{% \ifnum\apEa=\apEb \apE=\apEa \else \apPLUSxE \fi \apDIG\tmpa\relax \apnumA=\apnumD % digits before decimal point \apDIG\tmpb\relax \apnumB=\apnumD \apIVmod \apnumA \apnumE \advance\apnumA by-\apnumE % digits in the first Digit \apIVmod \apnumB \apnumF \advance\apnumB by-\apnumF \apnumC=\apnumB \advance\apnumC by-\apnumA % difference between Digits \ifnum\apSIGNa<0 \def\apPLUSxA{-}\else \def\apPLUSxA{}\fi \ifnum\apSIGNb<0 \def\apPLUSxB{-}\else \def\apPLUSxB{}\fi \apSIGN=0 % \apSIGN=0 means that we are doing subtraction \ifx\apPLUSxA\empty \ifx\apPLUSxB\empty \apSIGN=1 \fi\fi \if\apPLUSxA-\relax \if\apPLUSxB-\relax \apSIGN=-1 \def\apPLUSxA{}\def\apPLUSxB{}\fi\fi \ifnum\apnumC>0 \apPLUSg \apPLUSb \tmpb\apnumF \tmpa\apnumE \apnumB % first pass \else \apnumC=-\apnumC \apPLUSb \tmpa\apnumE \tmpb\apnumF \apnumA \fi \ifnum\apnumG=0 \def\OUT{0}\apSIGN=0 \apE=0 \else \ifnum\apSIGN=0 \apSIGN=\apnumG \let\apNext=\apPLUSm \else \let\apNext=\apPLUSp \fi \apnumX=0 \edef\OUT{\expandafter}\expandafter \apNext \OUT@% second pass \ifnum\apnumD<1 % result in the form .000123 \apnumZ=-\apnumD \def\tmpa{.}% \ifnum\apnumZ>0 \apADDzeros\tmpa \fi % adding dot and left zeros \edef\OUT{\ifnum\apSIGN<0-\fi\tmpa\OUT}% \else \edef\OUT{\expandafter}\expandafter\apPLUSy \OUT@% removing left zeros \fi\fi } \def\apPLUSb#1#2#3#4#5{% \edef\tmpd{\ifcase#4\or{}{}{}\or{}{}\or{}\fi#3}% \edef\tmpc{\ifcase#2\or{}{}{}\or{}{}\or{}\fi}% \let\apNext=\apPLUSc \apnumD=#5\advance\apnumD by4 \apnumG=0 \apnumZ=0 \def\OUT{}% \expandafter\expandafter\expandafter\apPLUSc\expandafter\tmpc#1\apNL\apNL\apNL\apNL@% } \def\apPLUSc#1#2#3#4{\apnumY=\apPLUSxA#1#2#3#4\relax \ifx\apNL#4\let\apNext=\apPLUSd\fi \ifx\apNL#1\relax \ifx\tmpd\empty \expandafter\expandafter\expandafter\apPLUSf \fi\fi \apPLUSe } \def\apPLUSd{\apnumY=0 \ifx\tmpd\empty \expandafter\apPLUSf \else\expandafter \apPLUSe\fi} \def\apPLUSe{% \ifnum\apnumC>0 \advance\apnumC by-4 \else \apIVread\tmpd \advance\apnumY by\apPLUSxB\apnumX \fi \ifnum\apnumZ=0 \apPLUSh \fi \edef\OUT{{\the\apnumY}\OUT}% \advance\apnumD by-4 \apNext } \def\apPLUSf#1@{} \def\apPLUSg{\let\tmpc=\apPLUSxA \let\apPLUSxA=\apPLUSxB \let\apPLUSxB=\tmpc} \def\apPLUSh{\apnumZ=\apnumY \ifnum\apnumY=0 \else \ifnum\apnumY<0 \apnumG=-1 \apnumY=-\apnumY \apPLUSg \else\apnumG=1 \fi\fi } \def\apPLUSm#1{% \ifx@#1\else \apnumA=#1 \advance\apnumA by-\apnumX \ifnum\apnumA<0 \advance\apnumA by\apIVbase \apnumX=1 \else \apnumX=0 \fi \apPLUSw \expandafter\apPLUSm \fi } \def\apPLUSp#1{% \ifx@#1\ifnum\apnumX>0 \apnumA=1 \apPLUSw \fi % .5+.5=.1 bug fixed \else \apnumA=\apnumX \advance\apnumA by#1 \ifnum\apnumA<\apIVbase \apnumX=0 \else \apnumX=1 \advance\apnumA by-\apIVbase \fi \apPLUSw \expandafter\apPLUSp \fi } \def\apPLUSw{% \ifnum\apnumD=0 \ifx\OUT\empty \def\OUT{\empty}\else \edef\OUT{.\OUT}\fi \fi \advance\apnumD by4 \ifx\OUT\empty \edef\tmpa{\apIVwrite\apnumA}\edef\OUT{\apREMzerosR\tmpa}% \else \edef\OUT{\apIVwrite\apnumA\OUT}\fi } \def\apPLUSy#1{\ifx0#1\expandafter\apPLUSy\else \expandafter\apPLUSz\expandafter#1\fi} \def\apPLUSz#1@{\edef\OUT{\ifnum\apSIGN<0-\fi#1}} \def\apPLUSxE{% \apnumE=\apEa \advance\apnumE by-\apEb \ifnum\apEa>\apEb \apPPs\apROLLa\tmpb{-\apnumE}\apE=\apEa \else \apPPs\apROLLa\tmpa{\apnumE}\apE=\apEb \fi } %%%%%%%%%%%% Multiplication, sec. 2.5 in apnum.pdf \def\apMULa{% \apE=\apEa \advance\apE by\apEb \apSIGN=\apSIGNa \multiply\apSIGN by\apSIGNb \ifnum\apSIGN=0 \def\OUT{0}\apE=0 \else \apDIG\tmpa\apnumA \apnumX=\apnumA \advance\apnumA by\apnumD \apDIG\tmpb\apnumB \advance\apnumX by\apnumB \advance\apnumB by\apnumD \apnumD=\apnumX % \apnumD = the number of digits after decimal point in the result \apIVmod \apnumA \apnumF % \apnumF = digits in the first Digit of \tmpa \edef\tmpc{\ifcase\apnumF\or{}{}{}\or{}{}\or{}\fi}\def\OUT{}% \expandafter\expandafter\expandafter \apMULb \expandafter \tmpc \tmpa @@@@% \edef\OUT{*.\OUT}% \apIVmod \apnumB \apnumF % \apnumF = digits in the first Digit of \tmpb \edef\tmpc{\ifcase\apnumF\or{}{}{}\or{}{}\or{}\fi}\def\tmpa{}% \expandafter\expandafter\expandafter \apMULc \expandafter \tmpc \tmpb @@@@% \expandafter\apMULd \tmpa@% \expandafter\apMULg \OUT \edef\tmpa{\ifnum\apSIGN<0-\fi}% \ifnum\apnumD>0 \apnumZ=\apnumD \edef\tmpa{\tmpa.}\apADDzeros\tmpa \fi \ifx\tmpa\empty \else \edef\OUT{\tmpa\OUT}\fi \fi } \def\apMULb#1#2#3#4{\ifx@#4\else \ifx\OUT\empty \edef\OUT{{#1#2#3#4}*}\else\edef\OUT{{#1#2#3#4}0\OUT}\fi \expandafter\apMULb\fi } \def\apMULc#1#2#3#4{\ifx@#4\else \edef\tmpa{{#1#2#3#4}\tmpa}\expandafter\apMULc\fi} \def\apMULd#1{\ifx@#1\else \apnumA=#1 \expandafter\apMULe \OUT \expandafter\apMULd \fi } \def\apMULe#1*#2{\apnumX=0 \def\OUT{#1{#2}*}\def\apOUTl{}\apnumO=1 \apnumL=0 \apMULf} \def\apMULf#1#2{% \advance\apnumO by-1 \ifnum\apnumO=0 \apOUTx \fi \apnumB=#1 \multiply\apnumB by\apnumA \advance\apnumB by\apnumX \ifx*#2% \ifnum\apnumB<\apIVbase \edef\OUT{\OUT\expandafter\apOUTs\apOUTl.,\ifnum\the\apnumB#1=0 \else{\the\apnumB}{#1}\fi*}% \else \apIVtrans \expandafter \edef\csname apOUT:\apOUTn\endcsname {\csname apOUT:\apOUTn\endcsname{\the\apnumB}{#1}}% \apMULf0*\fi \else \advance\apnumB by#2 \ifnum\apnumB<\apIVbase \apnumX=0 \else \apIVtrans \fi \expandafter \edef\csname apOUT:\apOUTn\endcsname{\csname apOUT:\apOUTn\endcsname{\the\apnumB}{#1}}% \expandafter\apMULf \fi } \def\apMULg#1{\def\OUT{}\apMULh} \def\apMULh#1{\ifx*#1\expandafter\apMULi \else \apnumA=#1 \apMULo4{\apIVwrite\apnumA}% \expandafter\apMULh \fi } \def\apMULi#1#2#3{\apnumA=#1 \ifx*#3\apMULo{\apNUMdigits\tmpa}{\the\apnumA}\expandafter\apMULj \else \apMULo4{\apIVwrite\apnumA}\expandafter\apMULi \fi{#3}% } \def\apMULj#1{} \def\apMULo#1#2{\edef\tmpa{#2}% \advance\apnumD by-#1 \ifnum\apnumD<1 \ifnum\apnumD>-4 \apMULt\fi\fi \edef\OUT{\tmpa\OUT}% } \def\apMULt{\edef\tmpa{\apIVdot{-\apnumD}\tmpa}\edef\tmpa{\tmpa}} %%%%%%%%%%%% Division, sec. 2.6 in apnum.pdf \def\apDIVa{% \ifnum\apSIGNb=0 \errmessage{Dividing by zero}\else \apSIGN=\apSIGNa \multiply\apSIGN by\apSIGNb \ifnum\apSIGNa=0 \def\OUT{0}\def\XOUT{0}\apE=0 \apSIGN=0 \else \apE=\apEa \advance\apE by-\apEb \apDIG\tmpb\relax \apnumB=\apnumD \apDIG\tmpa\relax \apnumH=\apnumD \advance\apnumD by-\apnumB % \apnumD = num. of digits before decimal point in the result \apDIVcomp\tmpa\tmpb % apXtrue <=> A>=B, i.e 1 digit from A/B \ifapX \advance\apnumD by1 \advance\apnumH by1 \fi \apnumC=\apTOT \ifnum\apTOT<0 \apnumC=-\apnumC \ifnum\apnumD>\apnumC \apnumC=\apnumD \fi \fi \ifnum\apTOT=0 \apnumC=\apFRAC \advance\apnumC by\apnumD \else \apnumX=\apFRAC \advance\apnumX by\apnumD \ifnum\apnumC>\apnumX \apnumC=\apnumX \fi \fi \ifnum\apnumC>0 % \apnumC = the number of digits in the result \advance\apnumH by-\apnumC % \apnumH = the position of decimal point in the remainder \apIVmod \apnumC \apnumF % \apnumF = the number of digits in the first Digit \apIVread\tmpb \apnumB=\apnumX % \apnumB = partial divisor \apnumX=\apnumF \ifapX \advance\apnumX by-1 \fi \apIVreadX\apnumX\tmpa \apnumA=\apnumX % \apnumA = first Digit of the partial dividend \apIVread\tmpa % \apnumX = second Digit of the partial dividend \edef\apDIVxA{\the\apnumA\apIVwrite\apnumX}% first partial dividend \edef\apDIVxB{\the\apnumB}% partial divisor \edef\XOUT{{\apDIVxB}{\the\apnumX}@{\the\apnumA}}% the \XOUT is initialized \edef\OUT{\ifnum\apSIGN<0-\fi}% \ifnum\apnumD<0 \edef\OUT{\OUT.}\apnumZ=-\apnumD \apADDzeros\OUT \fi \apnumE=1 \apnumZ=0 \let\apNext=\apDIVg \apNext % <--- the main calculation loop is here \ifnum\apnumD>0 \apnumZ=\apnumD \apADDzeros\OUT \fi \ifnum\apnumE=0 \def\XOUT{0}\else % extracting remainder from \XOUT \edef\XOUT{\expandafter}\expandafter\apDIVv\XOUT \def\tmpc{\apnumH}\apnumG=\apSIGNa \expandafter\apROLLa\XOUT.@\XOUT \fi \else \def\OUT{0}\def\XOUT{0}\apE=0 \apSIGN=0 \fi\fi\fi } \def\apDIVcomp#1#2{% \expandafter\def\expandafter\tmpc\expandafter{#1\apNL\apNL\apNL\apNL\apNL\apNL\apNL\apNL@}% \expandafter\def\expandafter\tmpd\expandafter{#2\apNL\apNL\apNL\apNL\apNL\apNL\apNL\apNL@}% \def\apNext{\expandafter\expandafter\expandafter\apDIVcompA\expandafter\tmpc\tmpd}% \apXtrue \apNext } \def\apDIVcompA#1#2#3#4#5#6#7#8#9@{% \ifx#8\apNL \def\tmpc{0000000\apNL@}\else\def\tmpc{#9@}\fi \apnumX=#1#2#3#4#5#6#7#8\relax \apDIVcompB } \def\apDIVcompB#1#2#3#4#5#6#7#8#9@{% \ifnum\apnumX<#1#2#3#4#5#6#7#8 \let\apNext=\relax \apXfalse \else \ifnum\apnumX>#1#2#3#4#5#6#7#8 \let\apNext=\relax \apXtrue \fi\fi \ifx\apNext\relax\else \ifx#8\apNL \def\tmpd{0000000\apNL@}\ifx\tmpc\tmpd\let\apNext=\relax\fi \else\def\tmpd{#9@}\fi \fi \apNext } \def\apDIVg{% \ifx\tmpb\empty \ifx\tmpa\empty \def\apNext{\apDIVi!}\let\apNexti=\apDIVi \else \def\apNext{\expandafter\apDIVh\tmpa\apNL\apNL\apNL\apNL!}\let\apNexti=\apDIVh \fi\fi \ifx\apNext\apDIVg \apIVread\tmpa \apnumA=\apnumX \apIVread\tmpb \edef\XOUT{{\the\apnumX}{\the\apnumA}\XOUT}% \fi \apNext } \def\apDIVh#1#2#3#4{\apnumZ=#1#2#3#4 \ifx\apNL#4\let\apNexti=\apDIVi\fi \apDIVi } \def\apDIVi{% \ifnum\apnumE=0 \apnumC=0 \fi \ifnum\apnumC>0 \expandafter\apDIVp\XOUT \advance\apnumC by-4 \apnumZ=0 \expandafter\apNexti \else \expandafter\apDIVj \fi } \def\apDIVj#1!{} \def\apDIVp{% \apnumA=\apDIVxA \divide\apnumA by\apDIVxB \def\apOUTl{}\apnumO=1 \apnumL=0 \apnumX=0 \apnumB=0 \apnumE=0 \let\apNext=\apDIVq \apNext 0\apnumZ } \def\apDIVq#1#2#3{% B A B \advance\apnumO by-1 \ifnum\apnumO=0 \apOUTx \fi \apnumY=\apnumB \apnumB=#1\multiply\apnumB by-\apnumA \advance\apnumB by#2\advance\apnumB by-\apnumX \ifnum\apnumB<0 \apnumX=\apnumB \advance\apnumX by1 \divide\apnumX by-\apIVbase \advance\apnumX by1 \advance\apnumB by\the\apnumX 0000 \else \apnumX=0 \fi \expandafter \edef\csname apOUT:\apOUTn\endcsname{\csname apOUT:\apOUTn\endcsname{#3}{\the\apnumB}}% \ifnum\apnumE<\apnumB \apnumE=\apnumB \fi \ifx@#3\let\apNext=\apDIVr \fi \apNext{#3}% } \def\apDIVr#1#2{% \ifnum\apnumX=#2 % the calculated Digit is OK, we save it \edef\XOUT{\expandafter\apOUTs\apOUTl.,}% \edef\tmpa{\ifnum\apnumF=4 \expandafter\apIVwrite\else \expandafter\the\fi\apnumA}% \ifnum\apnumD<\apnumF \ifnum\apnumD>-1 \apDIVt \fi\fi %adding dot \ifx\apNexti\apDIVh \apnumE=1 \fi \ifnum\apnumE=0 \apDIVu % removing zeros \advance\apnumD by-\apNUMdigits\tmpa \relax \else \advance\apnumD by-\apnumF \apnumF=4 \fi \edef\OUT{\OUT\tmpa}% save the Digit \edef\apDIVxA{\the\apnumB\apIVwrite\apnumY}% next partial dvividend \else % we need do correction and run the remainder calculation again \advance\apnumA by-1 \apnumX=0 \apnumB=0 \apnumE=0 \def\apOUTl{}\apnumO=1 \apnumL=0 \def\apNext{\let\apNext=\apDIVq \expandafter\apNext\expandafter0\expandafter\apnumZ\XOUT}% \expandafter\apNext \fi } \def\apDIVt{\edef\tmpa{\apIVdot\apnumD\tmpa}\edef\tmpa{\tmpa}} \def\apDIVu{\edef\tmpa{\apREMzerosR\tmpa}\edef\tmpa{\apREMdotR\tmpa}} \def\apDIVv#1#2{\apnumX=#2 \ifx@#1\apDIVw{.\apIVwrite\apnumX}\else\apDIVw{\apIVwrite\apnumX}\expandafter\apDIVv\fi } \def\apDIVw#1{% \ifx\XOUT\empty \ifnum\apnumX=0 \else \edef\tmpa{#1}\edef\XOUT{\apREMzerosR\tmpa\XOUT}% \fi \else \edef\XOUT{#1\XOUT}\fi } %%%%%%%%%%%% Power to the integer, sec. 2.7 in apnum.pdf \def\apPOWa{% \ifnum\apSIGNa=0 \def\OUT{0}\apSIGN=0 \apE=0 \else \ifnum\apSIGNb=0 \def\OUT{1}\apSIGN=1 \apE=0 \else \apDIG\tmpb\apnumB \ifnum\apnumB>0 \errmessage{POW: non-integer exponent is not implemented yet}\apPOWe\fi \ifnum\apEb=0 \else \errmessage{POW: the E notation of exponent isn't allowed}\apPOWe\fi \ifnum\apnumD>8 \errmessage{POW: too big exponent. Do you really need about 10^\the\apnumD\space digits in output?}\apPOWe\fi \apE=\apEa \multiply\apE by\tmpb\relax \apSIGN=\apSIGNa \ifodd\tmpb \else \apSIGN=1 \fi \apDIG\tmpa\apnumA \apnumC=\apnumA \advance\apnumC by\apnumD \apnumD=\apnumA \multiply\apnumD by\tmpb \apIVmod \apnumC \apnumA \edef\tmpc{\ifcase\apnumA\or{}{}{}\or{}{}\or{}\fi}\def\OUT{}% \expandafter\expandafter\expandafter \apMULb \expandafter \tmpc \tmpa @@@@% \edef\OUT{*.\OUT}% \OUT := \tmpa in interleaved format \def\tmpc{*.1*}% \apnumE=\tmpb\relax \apPOWb \expandafter\apPOWg \tmpc % \OUT := \tmpc in human raedable form \ifnum\apnumD=0 \ifnum \apSIGN<0 \edef\OUT{-\OUT}\fi \else \def\tmpc{-\apnumD}\apnumG=\apSIGN \expandafter\apROLLa\OUT.@\OUT\fi \ifnum\apSIGNb<0 \apPPab\apDIVa 1\OUT \fi \relax \fi\fi } \def\apPOWb{% \ifodd\apnumE \def\tmpb{}\expandafter\apPOWd\OUT \let\tmpd=\OUT \let\OUT=\tmpc \expandafter\apMULd \tmpb@\expandafter\apPOWn\OUT@% \let\tmpc=\OUT \let\OUT=\tmpd \fi \divide\apnumE by2 \ifnum\apnumE>0 \expandafter\apPOWt\OUT \expandafter\apPOWn\OUT@% \expandafter\apPOWb \fi } \def\apPOWd#1#2{% \apPOWd => \tmpb (in simple reverse format) \ifx*#1\expandafter\apPOWd \else \edef\tmpb{\tmpb{#1}}% \ifx*#2\else \expandafter\expandafter\expandafter\apPOWd\fi \fi } \def\apPOWe#1\relax{\fi} \def\apPOWg#1#2{\def\OUT{}\apPOWh} % conversion to the human readable form \def\apPOWh#1#2{\apnumA=#1 \ifx*#2\edef\OUT{\the\apnumA\OUT}\else \edef\OUT{\apIVwrite\apnumA\OUT}\expandafter\apPOWh\fi } \def\apPOWn#1{\def\OUT{*}\apPOWna} \def\apPOWna#1{\ifx*#1\expandafter\apPOWnn\else \edef\OUT{\OUT0{#1}}\expandafter\apPOWna\fi} \def\apPOWnn#1#2{\ifx*#1\edef\OUT{\OUT*}\else\edef\OUT{\OUT0{#1}}\expandafter\apPOWnn\fi} \def\apPOWt#1#2{\apPOWu} % power to two \def\apPOWu#1#2{\apnumA=#1 \expandafter\apPOWv\OUT \ifx*#2\else \expandafter\apPOWu\fi } \def\apPOWv#1*#2#3#4{\def\apOUTl{}\apnumO=1 \apnumL=0 \apnumB=\apnumA \multiply\apnumB by\apnumB \multiply\apnumA by2 \ifx*#4\else\advance\apnumB by#4 \fi \ifx\apnumB<\apIVbase \apnumX=0 \else \apIVtrans \fi \edef\OUT{#1{#2}{\the\apnumB}*}% \ifx*#4\apMULf0*\else\expandafter\apMULf\fi } %%%%%%%%%%%% ROLL, ROUND and NORM macros, sec. 2.8 in apnum.pdf \def\apROLLa{\apnumA=\tmpc\relax \ifnum\apnumA<0 \expandafter\apROLLc\else \expandafter\apROLLg\fi} \def\apROLLc{\edef\tmpc{}\edef\tmpd{\ifnum\apnumG<0-\fi}\apnumB=0 \apROLLd} \def\apROLLd#1{% \ifx.#1\expandafter\apROLLe \else \edef\tmpc{\tmpc#1}% \advance\apnumB by1 \expandafter\apROLLd \fi } \def\apROLLe#1{\ifx@#1\edef\tmpc{\tmpc.@}\else\edef\tmpc{\tmpc#1}\fi \advance\apnumB by\apnumA \ifnum\apnumB<0 \apnumZ=-\apnumB \edef\tmpd{\tmpd.}\apADDzeros\tmpd \expandafter\expandafter\expandafter\apROLLf\expandafter\tmpc \else \apnumA=\apnumB \expandafter\expandafter\expandafter\apROLLi\expandafter\tmpc \fi } \def\apROLLf#1.@#2{\edef#2{\tmpd#1}} \def\apROLLg#1{\edef\tmpd{\ifnum\apnumG<0-\fi}\ifx.#1\apnumB=0 \else\apnumB=1 \fi \apROLLh#1} \def\apROLLh#1{\ifx.#1\expandafter\apROLLi\else \edef\tmpd{\tmpd#1}\expandafter\apROLLh\fi} \def\apROLLi#1{\ifx.#1\expandafter\apROLLi\else \ifnum\apnumA>0 \else \apROLLj \apROLLk#1\fi \ifx@#1\apROLLj \apROLLi0@\fi \advance\apnumA by-1 \ifx0#1\else \apnumB=1 \fi \ifnum\apnumB>0 \edef\tmpd{\tmpd#1}\fi \expandafter\apROLLi\fi } \def\apROLLj#1\fi#2\apROLLi\fi{\fi\fi#1} \def\apROLLk#1{\ifx@#1\expandafter\apROLLo\expandafter@\else \def\tmpc{}\apnumB=0 \expandafter\apROLLn\expandafter#1\fi } \def\apROLLn#1{% \ifx.#1\ifnum\apnumB>0 \edef\tmpd{\tmpd.\tmpc}\fi \expandafter\apROLLo \else \edef\tmpc{\tmpc#1}\advance\apnumB by#1 \expandafter\apROLLn \fi } \def\apROLLo@#1{\let#1=\tmpd} \def\apROUNDa{\apnumD=\tmpc\relax \ifnum\apnumD<0 \expandafter\apROUNDe \else \expandafter\apROUNDb \fi } \def\apROUNDb#1.{\edef\tmpc{#1}\apnumX=0 \def\tmpd{}\let\apNext=\apROUNDc \apNext} \def\apROUNDc#1{\ifx@#1\def\apNext{\apROUNDd.@}% \else \advance\apnumD by-1 \ifnum\apnumD<0 \def\apNext{\apROUNDd#1}% \else \ifx.#1\else \advance\apnumX by#1 \edef\tmpd{\tmpd#1}\fi \fi \fi \apNext } \def\apROUNDd#1.@#2{\def\XOUT{#1}% \ifnum\apnumX=0 \def\tmpd{}\fi \ifx\tmpd\empty \ifx\tmpc\empty \def#2{0}% \else \edef#2{\ifnum\apnumG<0-\fi\tmpc}\fi \else\edef#2{\ifnum\apnumG<0-\fi\tmpc.\tmpd}\fi } \def\apROUNDe#1.@#2{\apnumC=\apnumD \apPPs\apROLLa#2{\apnumC}\apPPs\apROUNDa#2{0}\apPPs\apROLLa#2{-\apnumC}% } \def\apNORMa#1.@#2{\ifnum\apnumG<0 \def#2{#1}\fi \expandafter\apNORMb\expandafter#2\tmpc@} \def\apNORMb#1#2#3@{% \ifx.#2\apnumC=#3\relax \apDIG#1\apnumA \apNORMc#1% \else \apnumC=#2#3\relax \apDIG#1\relax \apNORMd#1% \fi } \def\apNORMc#1{\advance\apE by-\apnumA \advance\apE by\apnumC \def\tmpc{-\apnumC}\expandafter\apROLLa#1.@#1% } \def\apNORMd#1{\advance\apE by\apnumD \advance\apE by-\apnumC \def\tmpc{\apnumC}\expandafter\apROLLa\expandafter.#1.@#1% } %%%%%%%%%%%% Function-like macros, sec. 2.9 in apnum.pdf \def\apABSa{\ifnum\apSIGN<0 \apSIGN=1 \fi} \def\apiDIVa{{\apFRAC=0 \apTOT=0 \apDIVa \apOUTtmpb}\tmpb} \def\apiMODa{{\apFRAC=0 \apTOT=0 \apDIVa \let\OUT=\XOUT \apOUTtmpb}\tmpb} \def\apiROUNDa{\apROUNDa\OUT0} \def\apiFRACa{\apROUNDa\OUT0\ifx\XOUT\empty\def\OUT{0}\else\edef\OUT{.\XOUT}\fi} \def\apFACa{{\apnumC=\OUT\relax \loop \ifnum \apnumC>2 \advance\apnumC by-1 \MUL{\OUT}{\the\apnumC}\repeat \global\let\OUT=\OUT}% } %%%%%%%%%%%% Auxiliary macros, sec. 2.10 in apnum.pdf \def\apREV#1{\expandafter\apREVa#1@!} \def\apREVa#1#2!{\ifx@#1\else\apREVa#2!#1\fi} \def\apDIG#1#2{\ifx\relax#2\def\tmpc{}\else #2=0 \def\tmpc{\advance#2 by1 }\fi \apnumD=0 \expandafter\apDIGa#1..@#1% } \def\apDIGa#1{\ifx.#1\csname apDIG\ifnum\apnumD>0 c\else b\fi\expandafter\endcsname \else \advance\apnumD by1 \expandafter\apDIGa\fi} \def\apDIGb#1{% \ifx0#1\advance\apnumD by-1 \tmpc \expandafter\apDIGb \else \expandafter\apDIGc \expandafter#1\fi } \def\apDIGc#1.{\def\tmpd{#1}% \ifx\tmpc\empty \let\apNext=\apDIGe \else \def\apNext{\expandafter\apDIGd\tmpd@}% \fi \apNext } \def\apDIGd#1{\ifx@#1\expandafter\apDIGe \else \tmpc \expandafter\apDIGd \fi} \def\apDIGe#1@#2{% \ifx@#1@\else % #1=empty <=> the param has no dot, we need to do nothing \ifnum\apnumD>0 \edef#2{\expandafter\apDIGf#2@}% the dot plus digits before dot \else \let#2=\tmpd % there are only digits after dot, use \tmpd \fi\fi } \def\apDIGf#1.#2@{#1#2} \def\apNL{0} \def\apIVread#1{\expandafter\apIVreadA#1\apNL\apNL\apNL\apNL\apNL@#1} \def\apIVreadA#1#2#3#4#5\apNL#6@#7{\apnumX=#1#2#3#4\relax \def#7{#5}} \def\apIVreadX#1#2{\edef\tmpc{\ifcase#1{}{}{}0\or{}{}{}\or{}{}\or{}\fi}% \expandafter\expandafter\expandafter\apIVreadA\expandafter\tmpc#2\apNL\apNL\apNL\apNL\apNL@#2% } \def\apIVwrite#1{\ifnum#1<1000 0\ifnum#1<100 0\ifnum#1<10 0\fi\fi\fi\the#1} \mathchardef\apIVbase=10000 \def\apIVtrans{\apnumX=\apnumB \divide\apnumB by\apIVbase \multiply\apnumB by-\apIVbase \advance\apnumB by\apnumX \divide\apnumX by\apIVbase } \def\apIVmod#1#2{#2=#1\divide#2by4 \multiply#2by-4 \advance#2by#1\relax \ifnum#2>0 \else \advance#2by4 \fi } \def\apIVdot#1#2{\noexpand\apIVdotA\ifcase#1....\or...\or..\or.\fi #2....@} \def\apIVdotA#1#2#3#4#5.#6@{\ifx.#1\else#1\fi \ifx.#2\else#2\fi \ifx.#3\else#3\fi \ifx.#4\else#4\fi\ifx.#5.\else.#5\fi } \def\apNUMdigits#1{\expandafter\apNUMdigitsA#1@@@@!} \def\apNUMdigitsA#1#2#3#4#5!{\ifx@#4\ifx@#3\ifx@#2\ifx@#10\else1\fi \else2\fi \else3\fi \else4\fi} \def\apADDzeros#1{\edef#1{#10}\advance\apnumZ by-1 \ifnum\apnumZ>0 \expandafter\apADDzeros\expandafter#1\fi } \def\apREMzerosR#1{\expandafter\apREMzerosRa#1@0@!} \def\apREMzerosRa#10@#2!{\ifx!#2!\apREMzerosRb#1\else\apREMzerosRa#1@0@!\fi} \def\apREMzerosRb#1@{#1} \def\apREMdotR#1{\expandafter\apREMdotRa#1@.@!} \def\apREMdotRa#1.@#2!{\ifx!#2!\apREMzerosRb#1\else#1\fi} \def\apOUTx{\apnumO=7 \edef\apOUTn{\the\apnumL}\edef\apOUTl{\apOUTl\apOUTn,}% \expandafter\def\csname apOUT:\apOUTn\endcsname{}% \advance\apnumL by1 } \def\apOUTs#1,{\ifx.#1\else\csname apOUT:#1\expandafter\endcsname\expandafter\apOUTs\fi} \def\apOUTtmpb{\edef\tmpb{\apSIGN=\the\apSIGN \apE=\the\apE \edef\noexpand\OUT{\OUT}}\expandafter} %%%%%%%%%%%% Conclusion, sec. 2.11 in apnum.pdf \ifx\documentclass\undefined \else % please, don't remove this message \message{SORRY, you are using LaTeX. I don't recommend this. Petr Olsak}\fi \catcode`\@=\apnumZ \endinput 1.0 - First version released 1.1 - POW implemented more simple (by base 2 of exponent) - \next renamed in order to avoid name conflict 1.2 - .5+.5=.1 bug fixed