% 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)