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authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
commite0c6872cf40896c7be36b11dcc744620f10adf1d (patch)
tree60335e10d2f4354b0674ec22d7b53f0f8abee672 /macros/generic/apnum/apnum.tex
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+% 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)