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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
commit | e0c6872cf40896c7be36b11dcc744620f10adf1d (patch) | |
tree | 60335e10d2f4354b0674ec22d7b53f0f8abee672 /macros/generic/apnum/apnum.tex |
Initial commit
Diffstat (limited to 'macros/generic/apnum/apnum.tex')
-rw-r--r-- | macros/generic/apnum/apnum.tex | 1203 |
1 files changed, 1203 insertions, 0 deletions
diff --git a/macros/generic/apnum/apnum.tex b/macros/generic/apnum/apnum.tex new file mode 100644 index 0000000000..91fc3bfde2 --- /dev/null +++ b/macros/generic/apnum/apnum.tex @@ -0,0 +1,1203 @@ +% Arbitrary precision numbers +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% 2014, 2015, 2016, 2018 Petr Olsak + +% See the documentation apnum.pdf or apnum.d for more information + +\def\apVERSION{1.7 <Apr 2018>} +\message{The Arbitrary Precision Numbers, \apVERSION} + +%%%%%%%%%%%% Internal registers, sec. 2.1 in apnum.pdf + +\newcount\apnumA \newcount\apnumB \newcount\apnumC \newcount\apnumD +\newcount\apnumE \newcount\apnumF \newcount\apnumG \newcount\apnumH +\newcount\apnumO \newcount\apnumP \newcount\apnumL +\newcount\apnumX \newcount\apnumY \newcount\apnumZ +\newcount\apSIGNa \newcount\apSIGNb \newcount\apEa \newcount\apEb +\newif\ifapX + +\newcount\apSIGN +\newcount\apE +\newcount\apTOT \apTOT=0 +\newcount\apFRAC \apFRAC=20 +\newcount\apEX \apEX=10 + +\apnumZ=\catcode`\@ \catcode`\@=12 + +%%%%%%%%%%%% Evaluation of the expression, sec. 2.2 in apnum.pdf + +\def\evaldef{\relax \apEVALa{\apEadd\OUT}} +\def\evalmdef{\relax \apEVALa{}} +\def\apEVALa#1#2#3{\begingroup \apnumA=0 \apnumE=1 \apEVALb#3\limits \tmpb \apEND #1\let#2=\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}\expandafter + \futurelet\expandafter\apNext\expandafter\apEVALg\romannumeral-`\.% +} +\def\apEVALd#1\fi#2-`\.{\fi#1} +\def\apEVALe{% + \ifx\tmpa\empty \else \ifnum\tmpa1<0 \def\tmpb{-1}\apEVALp \apMUL 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}}\expandafter + \futurelet\expandafter\apNext\expandafter\apEVALg\romannumeral-`\.} +\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 \apPLUS 1\fi + \ifx-#1\apEVALp \apMINUS 1\fi + \ifx*#1\apEVALp \apMUL 2\fi + \ifx/#1\apEVALp \apDIV 2\fi + \ifx^#1\apEVALp \apPOWx 3\fi + \ifx)#1\advance\apnumA by-4 \let\apNext=\apEVALo \let\tmpa=\relax + \ifnum\apnumA<0 \apEVALerror{many brackets ")"}\fi + \fi + \ifx\limits#1% + \ifnum\apnumA>0 \apEVALerror{missing bracket ")"}\let\tmpa=\relax + \else \apEVALp\END 0\let\apNext=\relax \fi + \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 +} +\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\apPOWx \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\OUT{0}\apE=0\apSIGN=0\def\apNext##1\apEND{\apEND}% +} +\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 \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 + \begingroup\apE=0 #3% execution of the parameter in the group + \edef\tmpb{\apE=\the\apE\relax\noexpand\apPPf\OUT@}\expandafter\endgroup\tmpb +} +\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{@}\apSIGN=0 \fi + \expandafter\apPPi\tmpc +} +\def\apPPh#1\apPPi\tmpc{\fi\apPPg} +\def\apPPi{\ifnum\apE=0 \expandafter\apPPk \else \expandafter\apPPj \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 + \ifnum\apSIGN=0 \def\OUT{0}\fi + \ifnum\apSIGN<0 \edef\OUT{-\OUT}\fi +} +\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 +} + +%%%%%%%%%%%% Addition and Subtraction, sec. 2.4 in apnum.pdf + +\def\apPLUS{\relax \apPPab\apPLUSa} +\def\apMINUS#1#2{\relax \apPPab\apPLUSa{#1}{-#2}} +\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\apMUL{\relax \apPPab\apMULa} +\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\apDIV{\relax \apPPab\apDIVa} +\def\apDIVa{% + \ifnum\apSIGNb=0 \apERR{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\apPOW{\relax \apPPab\apPOWa} \let\apPOWx=\apPOW % for usage as ^ operator +\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 \apERR{POW: non-integer exponent is not implemented yet}\apPOWe\fi + \ifnum\apEb=0 \else \apERR{POW: the E notation of exponent isn't allowed}\apPOWe\fi + \ifnum\apnumD>8 \apERR{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 <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\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\apROLL{\apPPs\apROLLa} +\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\apROUND{\apPPs\apROUNDa} +\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}\edef\XOUT{\apREMzerosR\XOUT}% + \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\apNORM{\apPPs\apNORMa} +\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% +} +\def\apEadd#1{\ifnum\apE=0 \else\edef#1{#1E\ifnum\apE>0+\fi\the\apE}\apE=0 \fi} +\def\apEnum#1{\ifnum\apE=0 \else\apROLL#1\apE \apE=0 \fi} + +%%%%%%%%%%%% Miscelaneous macros, sec. 2.9 in apnum.pdf + +\def\apEND{\global\let\apENDx=\OUT + \edef\tmpb{\apSIGN=\the\apSIGN \apE=\the\apE}% + \expandafter\endgroup \tmpb \let\OUT=\apENDx +} +\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\apREMfirst#1{\expandafter\apREMfirsta#1@#1} +\def\apREMfirsta#1#2@#3{\def#3{#2}} + +\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\apINIT{\begingroup \let\do=\apEVALxdo \let\localcounts=\apCOUNTS} +\def\apCOUNTS#1{\ifx;#1\else + \advance\count10 by1 \countdef#1=\count10 + \expandafter\apCOUNTS\fi +} +\def\apEVALxdo#1=#2;{#2\let#1=\OUT} + +\def\apRETURN#1\apEND{\fi\apEND} +\def\apERR#1{\errmessage{#1}} + +{\lccode`\?=`\p \lccode`\!=`\t \lowercase{\gdef\apNOPT#1?!{#1}}} + +\def\loop#1\repeat{\def\body{#1\relax\expandafter\body\fi}\body} + +%%%%%%%%%%%% Function-like macros, sec. 2.10 in apnum.pdf + +\def\ABS#1{\relax % mandatory \relax for "function-like" macros + \evalmdef\OUT{#1}% % evaluation of the input parameter + \ifnum\apSIGN<0 % if (input < 0) + \apSIGN=1 % sign = 1 + \apREMfirst\OUT % remove first "minus" from OUT + \fi % fi +} +\def\SGN#1{\relax \evaldef\OUT{#1}\edef\OUT{\the\apSIGN}\apE=0 } +\def\iDIV#1#2{\relax \apINIT % calculation in group + \evalmdef\apAparam{#1}\apEnum\apAparam + \evalmdef\apBparam{#2}\apEnum\apAparam % evaluation of the parameters + \apTOT=0 \apFRAC=0 \apDIV\apAparam\apBparam % integer division + \apEND % end of group +} +\def\iMOD#1#2{\relax \apINIT % calculation in group + \evalmdef\apAparam{#1}\apEnum\apAparam + \evalmdef\apBparam{#2}\apEnum\apBparam % evaluation of the parameters + \apTOT=0 \apFRAC=0 \apDIV\apAparam\apBparam % integer division + \let\OUT=\XOUT % remainder is the output + \apEND % end of group +} +\def\iFLOOR#1{\relax \evalmdef\OUT{#1}\apEnum\OUT \apROUND\OUT0% + \ifnum\apSIGN<0 \ifx\XOUT\empty \else \apPLUS\OUT{-1}\fi\fi + \def\tmp{0}\ifx\tmp\OUT \apSIGN=0 \fi +} +\def\iFRAC#1{\relax + \evalmdef\OUT{#1}\apEnum\OUT \apROUND\OUT0% % preparing the parameter + \ifx\XOUT\empty \def\OUT{0}\apSIGN=0 % empty fraction part means zero + \else \ifnum\apSIGN<0 + \edef\XOUT{-.\XOUT}\apPLUS1\XOUT % OUT = 1 - .\XOUT + \else \edef\OUT{.\XOUT}\apSIGN=1 % else OUT = .\XOUT + \fi \fi +} + +\def\FAC#1{\relax \apINIT % "function-like" in the group, FAC = factorial + \evalmdef\OUT{#1}\apEnum\OUT % preparing the parameter + \localcounts \N;% % local \newcount + \ifnum\apSIGN<0 \apERR{\string\FAC: argument {\OUT} cannot be negative}\apRETURN\fi + \let\tmp=\OUT \apROUND\tmp0% % test, if parameter is integer + \ifx\XOUT\empty \else \apERR{\string\FAC: argument {\OUT} must be integer}\apRETURN\fi + \N=\OUT\relax % N = param (error here if it is an big integer) + \ifnum\N=0\def\OUT{1}\apSIGN=1 \fi % special definition for factorial(0) + \loop \ifnum \N>2 \advance\N by-1 % loop if (N>2) N-- + \apMUL{\OUT}{\the\N}\repeat % OUT = OUT * N , repeat + \apEND % end of group +} +\def\BINOM#1#2{\relax \apINIT % BINOM = {#1 \choose #2} ... + \evalmdef\apAparam{#1}\apEnum\apAparam + \evalmdef\apBparam{#2}\apEnum\apBparam % preparation of the parameters + \localcounts \A \B \C ;% % local \newcounts + \let\OUT=\apBparam \apROUND\OUT0% % test if B is integer + \ifx\XOUT\empty\else\apERR{\string\BINOM: second arg. {\apBparam} must be integer}\apRETURN\fi + \let\OUT=\apAparam \apROUND\OUT0% % test if A is integer + \ifx\XOUT\empty % A is integer: + \A=\apAparam \B=\apBparam % A = #1, B = #2 + \C=\A \advance\C by-\B % C = A - B + \ifnum\C>\B \C=\B \fi % if (C > B) C = B fi + \ifnum\A<0 \C=\B % if (A < 0) C = B fi + \else \ifnum\A<\B \def\OUT{0}\apSIGN=0 % if (0 <= A < B) OUT = 0 return + \expandafter\expandafter\expandafter \apRETURN \fi\fi + \def\step{\advance\A by-1 \apMUL\OUT{\the\A}}% + \else \C=\apBparam % A is not integer + \def\step{\let\apBparam\OUT \do\apAparam=\apPLUS\apAparam{-1};% + \let\OUT=\apBparam \apMUL\OUT\apAparam}% + \fi + \ifnum\C=0 \def\OUT{1}\apSIGN=1 \apRETURN\fi + \do\D=\FAC{\the\C};% % D = C! + \let\OUT=\apAparam % OUT = #1 + \loop \advance\C by-1 % loop C-- + \ifnum\C>0 \step \repeat % if (C > 0) A--, OUT = OUT * A, repeat + \apDIV{\OUT}{\D}% % OUT = OUT / D + \apEND +} +\def\SQRT#1{\relax \apINIT % OUT = SQRT(#1) ... + \evalmdef\A{#1}% % parameter preparation + \localcounts \M \E ;% % local counters + \E=\apE \apE=0 + \ifnum\apSIGN=0 \apRETURN\fi % SQRT(0) = 0 (OUT is set to 0 by previous \evaldef) + \ifnum\apSIGN<0 \apERR{\string\SQRT: argument {\A} is out of range}\apRETURN\fi + \ifodd\E \apROLL\A{-1}\advance\E by1 \fi % we need the E representation with even exponent + \let\B=\A \let\C=\A + \apDIG\C\relax \M=\apnumD % M is the number of digits before decimal point + \advance\M by-2 \ifodd\M \advance\M by1 \fi % M = M - 2 , M must be even + \ifx\apSQRTxo\undefined % we need to calculate Xo + \ifnum\M=0 \else \apROLL\B{-\M}\divide\M by2 \fi % shift decimal point by -M, M = M / 2 + \apSQRTr\B \let\Xn=\OUT % Xn = estimate of SQRT + \ifnum\M<0 \let\A=\B \fi % if (A < 1) calculate with B where decimal point is shifted + \ifnum\M>0 \apROLL\Xn \M \fi % if (A >= 100) shift the decial point of initial guess + \else \let\Xn=\apSQRTxo \fi + \loop % loop ... Newton's method + \apDIV{\apPLUS{\Xn}{\apDIV{\A}{\Xn}}}{2}% % OUT = (Xn + A/Xn) / 2 + \ifx\OUT\Xn \else % if (OUT != Xn) + \let\Xn=\OUT \repeat % Xn = OUT, repeat + \ifnum\M<0 \apROLL\OUT\M \fi % shift the decimal point by M back + \apE=\E \divide\apE by2 % correct the E exponent + \apEND +} +\def\apSQRTr#1{\dimen0=#1pt \apnumB=1 \apnumC=1 \apSQRTra} +\def\apSQRTra{\advance\apnumB by2 \advance\apnumC by\apnumB % B = difference, C = x_i + \ifnum\apnumC>100 \def\OUT{10}\else + \ifdim\dimen0<\apnumC pt \apSQRTrb \else + \expandafter\expandafter\expandafter\apSQRTra\fi\fi +} +\def\apSQRTrb{% x = dimen0, B = x_i - x_{i-1}, C = x_i = i + \ifdim\dimen0<4pt + \ifdim\dimen0>2pt \dimen1=4pt \advance\dimen1 by-\dimen0 \divide\dimen1 by2 + \else \dimen1=\dimen0 \advance\dimen1 by-1pt \fi + \dimen1=.080884\dimen1 % dimen1 = additional linear correction + \else \dimen1=0pt \fi + \advance\apnumC by-\apnumB % C = x_{i-1} + \advance\dimen0 by-\apnumC pt % dimen0 = (x - x_{i-1}) + \divide\dimen0 by\apnumB % dimen0 = (x - x_{i-1}) / difference + \divide\apnumB by2 % B = i-1 = g(x_{i-1}) + \advance\dimen0 by\apnumB pt % dimen0 = g(x_{i-1}) + (x - x_{i-1} / (x_i-x_{i-1}) + \advance\dimen0 by\dimen1 % dimen0 += additional linear correction + \edef\OUT{\expandafter\apNOPT\the\dimen0}% OUT = dimen0 +} +\def\EXP#1{\relax\apINIT % OUT = EXP(#1) ... + \evalmdef\OUT{#1}\apEnum\OUT % OUT = #1 + \localcounts \N \K ;% + \ifnum\apSIGN=0 \def\OUT{1}\apSIGN=1 \apRETURN \fi + \edef\digits{\the\apFRAC}\advance\apFRAC by4 + \edef\signX{\the\apSIGN}% + \ifnum\apSIGN<0 \apSIGN=1 \apREMfirst\OUT \fi % remove "minus" sign + \def\testBig ##1##2##3\relax##4{\ifx##1.\apXfalse \else + \ifx##2.\ifnum##1<4 \apXfalse \else \apXtrue \fi \else \apXtrue + \fi \fi \ifapX}% + \expandafter\testBig \OUT.\relax + \iftrue \apEXPb \else \apEXPa \fi % OUT = e^OUT + \ifnum\signX<0 \K=-\apE \apDIV 1\OUT \apE=\K \fi % if (signX < 0) OUT = 1 / OUT + \apSIGN=1 % EXP is always positive + \apEND +} +\def\apEXPa{% + \def\testDot ##1##2\relax##3{\ifx##1.}% + \K=0 \N=0 % K = 0, N = 0 + \loop \expandafter \testDot\OUT \relax % loop if (OUT >= 1) + \iftrue \else % OUT = OUT/2 + \apDIV\OUT{2}% % K++ + \advance\K by1 % repeat + \repeat % oriOUT = 2^K * OUT, OUT < 1 + \advance\apFRAC by\K + \def\S{1}\def\Sn{1}\N=0 \let\X=\OUT % S = 1, Sn = 1, N = 0, X = OUT + \loop \advance\N by1 % loop N++ + \do\Sn=\apDIV{\apMUL\Sn\X}{\the\N};% % Sn = Sn * X / N + \apTAYLOR\iftrue \repeat % S = S + Sn (... Taylor) + \N=0 + \loop \ifnum\N < \K % loop if (N < K) + \apPOW\OUT{2}% % OUT = OUT^2 + \advance\N by1 \repeat % N++ + \apFRAC=\digits\relax \apROUND\OUT\apFRAC +} +\def\apTAYLOR#1{\ifnum\apSIGN=0 \let\OUT=\S \else \apPLUS\S\Sn \let\S=\OUT } +\def\apEXPb{% + \let\X=\OUT \apLNtenexec \apDIV\X\apLNten \let\D=\OUT + \apROUND\D{0}% % D = floor( X/ln(X) ) + \ifnum\D<\apEX \advance\apFRAC by\D \relax \apLNtenexec \fi + \EXP{\X-\D*\apLNten}% % mantissa = EXP(X-D*LN(10)) + \ifnum\D<\apEX \apROLL\OUT\D \apE=0 \else \apE=\D \relax \fi + \apFRAC=\digits \apROUND\OUT\apFRAC % OUT = mantissa * 10^D +} +\def\LN#1{\relax \apINIT % OUT = LN(#1) ... + \evalmdef\X{#1}% % X = #1 + \localcounts \M \N \E;% + \E=\apE + \edef\digits{\the\apFRAC}\advance\apFRAC by4 + \ifnum\apSIGN>0 \else \apERR{\string\LN: argument {\X} is out of range}\apRETURN\fi + \apDIG\OUT\relax \M=\apnumD % find M: X = mantissa * 10^M + \ifnum\M>-\E \def\sgnout{1}\else % if X in (0,1): + \def\sgnout{-1}% % sgnout = -1 + \do\X=\apDIV 1\X;\E=-\E % X = 1/X + \apDIG\OUT\relax \M=\apnumD % find M: X = mantissa * 10^M + \fi % else sgnout = 1 + \advance\M by-1 % M = M - 1 + \ifnum\M=0 \else\apROLL\X{-\M}\fi % X = X * 10^(-M), now X in (1,10) + \advance\M by\E % M = M + E (sientific format of numbers) + \do\lnX=\apLNr\X;% % lnX = LN(X) ... roughly estimate + \do\A=\apDIV\X{\EXP\lnX};% % A = X / EXP(lnX) ... A =approx= 1 + \apLNtaylor % OUT = LN(A) + \do\LNOUT=\apPLUS\OUT\lnX;% % LNOUT = OUT + LNrOUT + \ifnum\M>0 % if M > 0 + \apLNtenexec % LNtenOUT = ln(10) + \apPLUS\LNOUT{\apMUL{\the\M}{\apLNten}}% OUT = LNOUT + M * LNten + \fi + \ifnum\apSIGN=0 \else \apSIGN=\sgnout \fi % if (OUT != 0) apSIGN = saved sign + \apROUND\OUT\digits % round result to desired precision + \ifnum\apSIGN<0 \xdef\OUT{-\OUT}\else \global\let\OUT=\OUT \fi + \apEND +} +\def\apLNtaylor{% + \apDIV{\apPLUS{\A}{-1}}{\apPLUS{\A}{1}}% % OUT = (A-1) / (A+1) + \ifnum\apSIGN=0 \def\OUT{0}\else % ln 1 = 0 else: + \let\Sn=\OUT \let\Kn=\OUT \let\S=\OUT % Sn = OUT, Kn = OUT, S = OUT + \apPOW\OUT{2}\apROUND\OUT\apFRAC \let\XX=\OUT % XX = OUT^2 + \N=1 % N = 1 + \loop \advance\N by2 % loop N = N + 2 + \do\Kn=\apMUL\Kn\XX\apROUND\OUT\apFRAC;% Kn = Kn * XX + \do\Sn=\apDIV\Kn{\the\N};% % Sn = Kn / N + \apTAYLOR\iftrue \repeat % S = S + Sn (Taylor) + \apMUL\S{2}% % OUT = 2 * OUT + \fi +} +\def\apLNr#1{\dimen0=#1pt \apnumC=1 + \apLNra {0}{.69}{1.098}{1.386}{1.609}{1.791}{1.9459}{2.079}{2.197}{\apLNrten}{}\relax +} +\def\apLNra #1#2{\advance\apnumC by1 + \ifx\relax#2\relax \let\OUT=\apLNrten \let\apNext=\relax + \else + \ifdim\dimen0<\apnumC pt % linear interpolation: + \advance\dimen0 by-\apnumC pt \advance\dimen0 by1pt % dimen0 = x - x_{i-1} + \dimen1=#2pt \advance\dimen1 by-#1pt % dimen1 = f(x_i) - f(x_{i-1}) + \dimen1=\expandafter\apNOPT\the\dimen0 \dimen1 % dimen1 = (x - x_{i-1}) * dimen1 + \advance\dimen1 by#1pt % dimen1 = f(x_{i-1}) + dimen1 + \edef\OUT{\expandafter\apNOPT\the\dimen1}% % OUT = dimen1 + \def\apNext##1\relax{}% + \else \def\apNext{\apLNra{#2}}% + \fi\fi \apNext +} +\def\apLNrten{2.302585} % apLNrten = ln 10 (roughly) +\def\apLNtenexec{% % OUT = ln 10 ... + \expandafter\ifx\csname LNten:\the\apFRAC\endcsname \relax + \begingroup \apTOT=0 + \do\A=\apDIV{10}{\EXP\apLNrten};% % A = 10 / exp(LNrten) + \apLNtaylor % OUT = ln A + \apPLUS\OUT\apLNrten % OUT = OUT + LNrten + \global\expandafter\let\csname LNten:\the\apFRAC\endcsname=\OUT + \endgroup + \fi + \expandafter\let\expandafter \apLNten \csname LNten:\the\apFRAC\endcsname +} +\def\apPIvalue{3.141592653589793238462643383279} +\def\apPIdigits{30} +\def\apPIexec{% + \expandafter\ifx\csname apPI:\the\apFRAC\endcsname \relax \apPIexecA \else + \expandafter\let\expandafter\apPI\csname apPI:\the\apFRAC\endcsname + \expandafter\let\expandafter\apPIhalf\csname apPIh:\the\apFRAC\endcsname + \fi +} +\def\apPIexecA{% + \ifnum\apPIdigits<\apFRAC \apPIexecB \fi + \let\apPI=\apPIvalue + \ifnum\apPIdigits>\apFRAC \apROUND\apPI\apFRAC \fi + \apnumP=\apTOT \apTOT=0 \apDIV\apPI2\let\apPIhalf=\OUT \apTOT=\apnumP + \global\expandafter\let\csname apPI:\the\apFRAC\endcsname=\apPI + \global\expandafter\let\csname apPIh:\the\apFRAC\endcsname=\apPIhalf +} +\def\apPIexecB{\apINIT + \localcounts \N \a \c;% + \apTOT=0 \advance\apFRAC by2 + \def\apSQRTxo{800.199975006248}% initial value for Newton method for SQRT + \SQRT{640320}% + \let\sqrtval=\OUT + \N=0 \def\An{1}\def\Bn{1}\def\Cn{1}\def\S{13591409}% + \loop + \advance\N by 1 + \a=\N \multiply\a by6 \advance\a by-1 \c=\a + \advance\a by-2 \multiply\c by\a % An = An * 8 * (6N-5) * + \advance\a by-2 \multiply\a by8 % * (6N-3) * (6N-1) + \apMUL\An{\apMUL{\the\a}{\the\c}}\let\An=\OUT + \c=\N \multiply\c by\N % Bn = Bn * n^3 + \apMUL\Bn{\apMUL{\the\c}{\the\N}}\let\Bn=\OUT + \apMUL\Cn{-262537412640768000}\let\Cn=\OUT % Cn = Cn * K3 + \apDIV{\apMUL\An{\apPLUS{13591409}{\apMUL{545140134}{\the\N}}}}{\apMUL\Bn\Cn}% + \let\Sn=\OUT % Sn = An * (K1 + K2 * N) / (Bn * Cn) + \apTAYLOR \iftrue \repeat + \advance\apFRAC by-2 + \apDIV{\apMUL{\sqrtval}{53360}}\S + \global\let\apPIvalue=\OUT + \xdef\apPIdigits{\the\apFRAC}% + \apEND +} +\def\PI{\relax \apPIexec \let\OUT=\apPI} +\def\PIhalf{\relax \apPIexec \let\OUT=\apPIhalf} + +\def\SIN{\relax \let\apSINCOSx=\apSINx \apSINCOSa} +\def\COS{\relax \let\apSINCOSx=\apCOSx \apSINCOSa} +\def\apSINCOSa#1{\apINIT + \advance\apFRAC by3 + \evalmdef\X{#1}\apEnum\X + \def\signK{1}\apSINCOSo\apCOSx + \ifnum\apSIGN<0 \apREMfirst\X \def\sign{-}\else\def\sign{+}\fi + \ifx\apSINCOSx\apCOSx \def\sign{+}\fi + \edef\apFRACsave{\the\apFRAC}% + \apPIexec + \apFRAC=0 \apDIV\X\apPI % OUT = X div PI + \ifnum\apSIGN=0 \apSIGN=1 \else + \let\K=\OUT + \do\X=\apPLUS\X{-\apMUL\K\apPI};% X := X - K * PI + \apROLL\K{-1}\apROUND\K{0}% + \ifodd 0\XOUT\space \def\signK{-1}\else\def\signK{1}\fi + \fi + \apSINCOSo\apCOSx + \apFRAC=\apFRACsave \relax + \do\XmPIh=\apPLUS\X{-\apPIhalf};% XmPIh = | X - PI/2 | + \apSINCOSo\apSINx + \ifnum\apSIGN<0 \apREMfirst\XmPIh + \else % X in (PI/2, PI) + \do\X=\apPLUS\apPI{-\X};% + \ifx\apSINCOSx\apCOSx \apSIGN=-\signK \edef\signK{\the\apSIGN}\fi + \fi % X in (0, PI/2): + \apMINUS\X{.78}% % OUT = X - cca PI/4 + \ifnum\apSIGN<0 \else % if X in (PI/4, PI/2) : + \let\X=\XmPIh % X = | X - PI/2 |; SIN <-> COS + \ifx\apSINCOSx\apSINx \let\apSINCOSx=\apCOSx \else \let\apSINCOSx=\apSINx \fi + \fi + \localcounts \N \NN;% + \do\XX=\apPOW\X{2}\ROUND\OUT\apFRAC;% + \apSINCOSx % X in (0, PI/4), initialize Taylor SIN X or COS X + \loop + \advance\N by1 \NN=\N + \advance\N by1 \multiply\NN by\N + \do\Sn=\apDIV{\apMUL\Sn\XX}{-\the\NN};% Sn = - Sn * X^2 / N*(N+1) + \apTAYLOR \iftrue\repeat + \apSIGN=\sign\signK + \ifnum\apTOT=0 \advance\apFRAC by-3 \else \apFRAC=\apTOT \fi + \ifnum\apFRAC<0 \apFRAC=-\apFRAC \fi + \apROUND\OUT\apFRAC + \def\X{0}\ifx\OUT\X \apSIGN=0 \fi + \ifnum\apSIGN<0 \edef\OUT{-\OUT}\fi + \apEND +} +\def\apSINx{\let\S=\X \N=1 \let\Sn=\X} +\def\apCOSx{\def\S{1}\N=0 \let\Sn=\S} +\def\apSINCOSo#1{\ifnum\apSIGN=0 \ifx#1\SCgo \apSIGN=\signK \let\OUT=\signK \fi \apRETURN\fi} +\def\TAN#1{\relax \apINIT + \advance\apFRAC by3 + \evalmdef\X{#1}\apEnum\X + \advance\apFRAC by-3 + \do\denom=\COS\X;% + \ifnum\apSIGN=0 \apERR{\string\TAN: argument {\X} is out of range}\apRETURN\fi + \SIN\X + \apDIV{\SIN\X}\denom + \apEND +} +\def\ATAN#1{\relax \apINIT + \advance\apFRAC by3 + \evalmdef\X{#1}\apEnum\X + \ifnum\apSIGN=0 \def\OUT{0}\apRETURN\fi + \ifnum\apSIGN<0 \def\sign{-}\apREMfirst\X \else\def\sign{}\fi + \let\tmp=\X \apDIG\tmp\relax + \ifnum\apnumD>0 % if X > 1: + \apPIexec % OUT = apPIhalf - apATANox + \def\tmp{1}\ifx\tmp\X \apDIV\apPIhalf2\else \apATANox \apPLUS\apPIhalf{-\OUT}\fi + \else % else + \do\X=\apDIV{1}\X;% X := 1/X + \apATANox % OUT = apATANox + \fi + \ifnum\apTOT=0 \advance\apFRAC by-3 \else \apFRAC=\apTOT \fi + \ifnum\apFRAC<0 \apFRAC=-\apFRAC \fi + \apROUND\OUT\apFRAC + \ifx\sign\empty\apSIGN=1 \else \edef\OUT{-\OUT}\apSIGN=-1 \fi + \apEND +} +\def\apATANox{% + \localcounts \N;% + \do\XX=\apPLUS{1}{\apPOW\X{2}}\apROUND\OUT\apFRAC;% XX = 1 + X^2 + \do\Sn=\apDIV\X\XX \apROUND\OUT\apFRAC;% % Sn = X / (1+X^2) + \N=1 \let\S=\Sn + \loop + \advance\N by1 + \do\Sn=\apMUL{\the\N}\Sn;% + \advance\N by1 + \do\Sn=\apDIV\Sn{\apMUL{\the\N}\XX};% Sn = Sn * N / ((N+1) * (1+X^2)) + \apTAYLOR \iftrue \repeat +} +\def\ASIN#1{\relax \apINIT + \evalmdef\X{#1}\apEnum\X \edef\sign{\the\apSIGN}% + \apPLUS 1{-\apPOW\X2}% OUT = 1 - X^2 + \ifnum\apSIGN<0 \apERR{\string\ASIN: argument {\X} is out of range}\apRETURN\fi + \do\sqrt=\SQRT\OUT;% sqrt = SRQT {1 - X^1} + \ifnum\apSIGN=0 \apPIexec + \ifnum\sign<0 \edef\OUT{-\apPIhalf}\apSIGN=-1 % ASIN(-1) = -PI/2 + \else \let\OUT=\apPIhalf \apSIGN=1 \fi % ASIN(1) = PI/2 + \apRETURN \fi + \ATAN{\X/\sqrt}% OUT = arctan ( X / SQRT {1 - X^2} ) + \apEND +} +\def\ACOS#1{\relax \apPIexec \apPLUS\apPIhalf{-\ASIN{#1}}} + + +%%%%%%%%%%%% Printing expressions, sec 2.11 in apnum.pdf + +\def\eprint#1#2{\bgroup \apnumA=0 \apnumE=1 \apEVALb#1\limits + \let\apEPe=\relax \apEPi #2\tmpb \apEPe \egroup +} +\def\apEPi{\let\apPLUS=\apEPplus \let\apMINUS=\apEPminus + \let\apMUL=\apEPmul \let\apDIV=\apEPdiv \let\apPOWx=\apEPpow \def\apPPn##1{##1}% + \let\EXP=\apEPexp \def\LN{\apEPf{ln}}\let\SQRT=\apEPsqrt + \def\SIN{\apEPf{sin}}\def\COS{\apEPf{cos}}\def\TAN{\apEPf{tan}}% + \def\ASIN{\apEPf{arcsin}}\def\ACOS{\apEPf{arccos}}\def\ATAN{\apEPf{arctan}}% + \let\PI=\pi \def\PIhalf{{\pi\over2}}% + \let\ABS=\apEPabs \let\FAC=\apEPfac \let\BINOM=\apEPbinom + \let\SGN=\apEPsgn \let\iDIV=\apEPidiv \let\iMOD=\apEPimod + \let\iFLOOR=\apEPifloor \let\iFRAC=\apEPifrac + \let\apEPk=\empty \let\apEPy=\empty \def\apEPx{.}% + \let\apEPi=\relax \apEPj +} +\def\apEPj{} +\def\apEPplus#1#2{\apEPp{#1}{?...}+\apEPp{#2}{!...}} +\def\apEPminus#1#2{\apEPp{#1}{?...}-\apEPp{#2}{!!..}} +\def\apEPmul#1#2{\def\tmpa{#1}\def\tmpb{-1}% + \ifx\tmpa\tmpb \if\apEPx!\left(-\apEPp{#2}{!!..}\right)\else + -\apEPp{#2}{!!..}\fi + \else \apEPp{#1}{?!..}\apMULop \apEPp{#2}{!!..}\fi +} +\def\apEPdiv#1#2{{\def\apEPx{.}#1}\over{\def\apEPx{.}#2}} +\def\apEPpow#1#2{% + \let\apEPy=\empty \apEPpowa{#1}\end{#2}% + \ifx\apEPy\empty \apEPp{#1}{!!!!}^{\def\apEPx{.}#2}\else#1\fi +} +\def\apEPpowa#1{\expandafter\apEPpowb#1;} +\def\apEPpowb#1#2;\end#3{\ifx#1\apEPf \def\apEPy{\let\apEPy=\empty\def\apEPx{.}#3}\fi} +\def\apEPf#1#2{\begingroup + \mathop{\rm#1}\nolimits + \ifx\apEPy\empty \else ^{\apEPy}\let\apEPy=\empty \fi + \def\apEPk{\mskip-\thinmuskip}% + \def\apEPx{.}% + \eprint{#2}{\expandafter\apEPb}\endgroup +} +\def\apEPb#1#2{\def\next{\apEPk\left(\def\apEPe{\right)}}% + \ifx\apPPn#1\expandafter\apEPd#2.\end{}{\let\next=\relax}.\fi + \ifx\apDIV#1\let\next=\relax \fi + \next\let\apEPk=\empty #1{#2}% +} +\def\apEPp#1#2{\apEPq#1\end\bgroup{\left(\def\apEPx{.}}#2#1\apEPq#1\end\egroup{\right)}#2} +\def\apEPq#1#2\end#3#4#5#6#7#8{ + \ifx#5!\def\apEPx{!}\fi + \ifx#1\apEPplus \ifx#6!#4\else#3\fi\else + \ifx#1\apEPminus \ifx#6!#4\else#3\fi\else + \ifx#1\apEPmul \ifx#7!#4\else#3\fi\else + \ifx#1\apEPdiv \ifx#8!#4\else#3\fi\else + \ifx#1\apEPpow \ifx#8!#4\else#3\fi\else + \expandafter\apEPd#1.\end#3{}\apEPx\fi\fi\fi\fi\fi +} +\def\apEPd#1#2\end#3#4#5{\ifx-#1\if#5!\ifx#3\bgroup\left(\else\right)\fi\fi\else#4\fi} + +\let\apMULop=\cdot +\def\apEPabs#1{\left|\eprint{#1}{}\right|} +\def\apEPfac#1{\eprint{#1}{\expandafter\apEPb}\,!} +\def\apEPbinom#1#2{{\eprint{#1}{}\choose\eprint{#2}{}}} +\def\apEPsqrt#1{\sqrt{\eprint{#1}{}}} +\def\apEPexp#1{{\rm e}^{\eprint{#1}{}}} +\def\apEPsgn#1{\mathop{\rm sign}\eprint{#1}{\expandafter\apEPb}} +\def\apEPdivmod#1#2#3{\left[\eprint{#2}{\expandafter\apEPb}% + \mathbin{\rm #1}\eprint{#3}{\expandafter\apEPb}\right]} +\def\apEPidiv{\apEPdivmod{div}} +\def\apEPimod{\apEPdivmod{mod}} +\def\apEPifloor#1{\left\lfloor\eprint{#1}{}\right\rfloor} +\def\apEPifrac#1{\left\{\eprint{#1}{}\right\}} + +\def\corrnum#1{\edef#1{\expandafter\apEPc#1\end}} +\def\apEPc#1#2\end{\ifx#1-{-}\apEPc#2\end\else \ifx#1.0.#2\else #1#2\fi\fi} + +%%%%%%%%%%%% Conclusion, sec. 2.12 in apnum.pdf + +\let\PLUS=\apPLUS \let\MINUS=\apMINUS \let\MUL=\apMUL \let\DIV=\apDIV \let\POW=\apPOW +\let\SIGN=\apSIGN \let\ROUND=\apROUND \let\NORM=\apNORM \let\ROLL=\apROLL + +\ifx\documentclass\undefined \else % please, don't remove this message +\message{WARNING: the author of apnum package recommends: Never use LaTeX.}\fi +\catcode`\@=\apnumZ +\endinput + +1.0 <Nov 2014> - First version released +1.1 <Jan 2015> + - POW implemented more simple (by base 2 of exponent) + - \next renamed in order to avoid name conflict +1.2 <May 2015> - .5+.5=.1 bug fixed +1.3 <Dec 2015> - + - \apPPn corrected (empty \OUT bug fixed) + - \apEVAL: spaces ignored between parameters of function-like macros + - \apEVAL: in one group, \apEND introduced, \apOUTtmpb removed + - \apPLUS, etc. instead \PLUS introduced + - \apSTRIPfirst introduced + - \apEVALone, \apEVALtwo removed + - \addE renamed to \apEadd + - \ROLL, \NORM, \ROUND renamed to \apROLL, \apNORM, \apROUND + - \apREV removed + - \ABS, \iDIV, \iMOD, \iROUND, \iFRAC, \FAC rewriten + - \XOUT is empty, no "0000" after \apROLL\a0 (2.0000 bug fixed) + - \def#1{} corrected + - \localcount after \evaldef in order to avoid name conflict + - \PI added + - \apEnorm to \apEnum renamed +1.4 <Dec 2015> + - \ATAN, \ASIN, \ACOS added + - \SIN, \COS, \TAN added + - \apTOT=0 by default +1.4a \end -> \limits, internal change in \evaldef because LaTeX redefines \end +1.4b \apSINCOS: \ifx\OUT eq 0 added after \apROUND (bug fixed). +1.5 <Jan 2016> + - \iROUND replaced by \iFLOOR, \iFRAC corrected (for negative numbers) + - \eprint introduced +1.6 <Feb 2016> + - \evalmdef introduced + - \EXP for arg>=4 rewritten, \apEX register introduced +1.7 <Apr 2018> + - \eprint: bug removed (round brackets around negative constants) |