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diff --git a/Master/texmf-dist/tex/generic/apnum/apnum.tex b/Master/texmf-dist/tex/generic/apnum/apnum.tex new file mode 100644 index 00000000000..cb891d55bcc --- /dev/null +++ b/Master/texmf-dist/tex/generic/apnum/apnum.tex @@ -0,0 +1,730 @@ +% Arbitrary precision numbers +%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% 2014 Petr Olsak + +% See the documentation apnum.pdf or apnum.d for more information + +\def\apnumversion{1.0 <Nov. 2014>} +\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\next\apEVALg +} +\def\apEVALd#1\fi#2\next\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\next \bgroup \expandafter\apEVALh \else \expandafter\apEVALo \fi} +\def\apEVALh#1{\expandafter\def\expandafter\tmpb\expandafter{\tmpb{#1}}\futurelet\next\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\next=\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\next=\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\next=\relax + \fi + \ifx\tmpa\relax \else \apEVALerror{unknown operator "\string#1"}\fi + \apnumE=0 \next +} +\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\next##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\next{#1}% + \else \ifx0#2\apnumG=0 \def\next{\apPPu#1}\else \apnumG=1 \def\next{#1#2}\fi\fi + \next +} +\def\apPPu#1#2.@#3{\ifx@#2@\apnumG=0 \ifx#1\apROUNDa\def\XOUT{}\fi + \else\def\next{\apPPt#1#2.@#3}\expandafter\next\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\next=\apPLUSm \else \let\next=\apPLUSp \fi + \apnumX=0 \edef\OUT{\expandafter}\expandafter \next \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\next=\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\next=\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 + \next +} +\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 \edef\OUT{1\OUT}\fi + \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\next=\apDIVg \next % <--- 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\next{\expandafter\expandafter\expandafter\apDIVcompA\expandafter\tmpc\tmpd}% + \apXtrue \next +} +\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\next=\relax \apXfalse \else + \ifnum\apnumX>#1#2#3#4#5#6#7#8 \let\next=\relax \apXtrue + \fi\fi + \ifx\next\relax\else + \ifx#8\apNL \def\tmpd{0000000\apNL@}\ifx\tmpc\tmpd\let\next=\relax\fi \else\def\tmpd{#9@}\fi + \fi + \next +} +\def\apDIVg{% + \ifx\tmpb\empty + \ifx\tmpa\empty \def\next{\apDIVi!}\let\nexti=\apDIVi + \else \def\next{\expandafter\apDIVh\tmpa\apNL\apNL\apNL\apNL!}\let\nexti=\apDIVh + \fi\fi + \ifx\next\apDIVg + \apIVread\tmpa \apnumA=\apnumX + \apIVread\tmpb + \edef\XOUT{{\the\apnumX}{\the\apnumA}\XOUT}% + \fi + \next +} +\def\apDIVh#1#2#3#4{\apnumZ=#1#2#3#4 + \ifx\apNL#4\let\nexti=\apDIVi\fi + \apDIVi +} +\def\apDIVi{% + \ifnum\apnumE=0 \apnumC=0 \fi + \ifnum\apnumC>0 + \expandafter\apDIVp\XOUT + \advance\apnumC by-4 + \apnumZ=0 + \expandafter\nexti + \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\next=\apDIVq \next 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\next=\apDIVr \fi + \next{#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\nexti\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\next{\let\next=\apDIVq + \expandafter\next\expandafter0\expandafter\apnumZ\XOUT}% + \expandafter\next + \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 + \let\tmpd=\tmpa \def\tmpa{}% + \expandafter\expandafter\expandafter \apMULc \expandafter \tmpc \tmpd @@@@% + \def\tmpc{*.1*}\apnumE=10 + \edef\tmpb{\apREV\tmpb}% + \expandafter\apPOWb\tmpb@% do multiplications + \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#1{% + \ifx@#1\else + \ifnum\apnumE<10 + \expandafter\apPOWc\ifcase\apnumE + ppmp.\or ppmp.\or pmp.\or mmp.\or mp.\or p.\or mmmm.\or mmm.\or mm.\or m.\or .\fi + \def\tmpb{}\expandafter\apPOWd\OUT \let\tmpa=\tmpb + \fi + \apnumE=#1 + \ifnum\apnumE=0 \else + \expandafter\apPOWc\ifcase\apnumE + \or .\or p.\or pm.\or pp.\or ppm.\or pmp.\or pmpm.\or ppp.\or pppm.\fi + \def\tmpb{}\expandafter\apPOWd\OUT + \let\tmpd=\OUT \let\OUT=\tmpc + \expandafter\apMULd \tmpb@\expandafter\apPOWn\OUT@% + \let\tmpc=\OUT \let\OUT=\tmpd + \fi + \expandafter\apPOWb + \fi +} +\def\apPOWc#1{\ifx.#1\else \csname apPOW#1\endcsname \expandafter\apPOWc\fi} +\def\apPOWd#1#2{% \apPOWd <spec format> => \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\apPOWm{\expandafter\apMULd\tmpa@\expandafter\apPOWn\OUT@} +\def\apPOWp{\expandafter\apPOWt\OUT \expandafter\apPOWn\OUT@} +\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\next=\apROUNDc \next} +\def\apROUNDc#1{\ifx@#1\def\next{\apROUNDd.@}% + \else \advance\apnumD by-1 + \ifnum\apnumD<0 \def\next{\apROUNDd#1}% + \else \ifx.#1\else \advance\apnumX by#1 \edef\tmpd{\tmpd#1}\fi + \fi + \fi \next +} +\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\next=\apDIGe + \else \def\next{\expandafter\apDIGd\tmpd@}% + \fi \next +} +\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 + |