diff options
Diffstat (limited to 'Master/texmf-dist/tex')
| -rw-r--r-- | Master/texmf-dist/tex/generic/apnum/apnum.tex | 538 |
1 files changed, 457 insertions, 81 deletions
diff --git a/Master/texmf-dist/tex/generic/apnum/apnum.tex b/Master/texmf-dist/tex/generic/apnum/apnum.tex index 72bfceff9cc..d16f159bc9a 100644 --- a/Master/texmf-dist/tex/generic/apnum/apnum.tex +++ b/Master/texmf-dist/tex/generic/apnum/apnum.tex @@ -4,47 +4,29 @@ % See the documentation apnum.pdf or apnum.d for more information -\def\apnumversion{1.2 <May 2015>} -\message{The Arbitrary Precision Numbers, \apnumversion} +\def\apVERSION{1.4 <Dec 2015>} +\message{The Arbitrary Precision Numbers, \apVERSION} -%%%%%%%%%%%% Internal registers: +%%%%%%%%%%%% 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\apnumL +\newcount\apnumO \newcount\apnumP \newcount\apnumL \newcount\apnumX \newcount\apnumY \newcount\apnumZ \newcount\apSIGNa \newcount\apSIGNb \newcount\apEa \newcount\apEb \newif\ifapX -\apnumZ=\catcode`\@ \catcode`\@=12 - -%%%%%%%%%%%% Public macros, sec. 2.1 in apnum.pdf - -\def\evaldef{\relax \apEVALa} -\def\PLUS{\relax \apPPab\apPLUSa} -\def\MINUS#1#2{\relax \apPPab\apPLUSa{#1}{-#2}} -\def\MUL{\relax \apPPab\apMULa} -\def\DIV{\relax \apPPab\apDIVa} -\def\POW{\relax \apPPab\apPOWa} -\def\ABS{\relax \apEVALone\apABSa} -\def\iDIV{\relax \apEVALtwo\apiDIVa} -\def\iMOD{\relax \apEVALtwo\apiMODa} -\def\iROUND#1{\relax \evaldef\OUT{#1}\apiROUNDa} -\def\iFRAC{\relax \apEVALone\apiFRACa} -\def\FAC{\relax \apEVALone\apFACa} -\def\ROUND{\apPPs\apROUNDa} -\def\ROLL{\apPPs\apROLLa} -\def\NORM{\apPPs\apNORMa} -\def\addE#1{\edef#1{#1\ifnum\apE=0 \else E\ifnum\apE>0+\fi\the\apE\fi}} - -\newcount\apSIGN \let\SIGN=\apSIGN +\newcount\apSIGN \newcount\apE -\newcount\apTOT \apTOT=-30 +\newcount\apTOT \apTOT=0 \newcount\apFRAC \apFRAC=20 +\apnumZ=\catcode`\@ \catcode`\@=12 + %%%%%%%%%%%% Evaluation of the expression, sec. 2.2 in apnum.pdf -\def\apEVALa#1#2{{\apnumA=0 \apnumE=1 \apEVALb#2\end\expandafter}\tmpb \let#1=\OUT} +\def\evaldef{\relax \apEVALa} +\def\apEVALa#1#2{\begingroup \apnumA=0 \apnumE=1 \apEVALb#2\end \tmpb \apEND \let#1=\OUT} \def\apEVALb{\def\tmpa{}\apEVALc} \def\apEVALc#1{% \ifx+#1\apEVALd \apEVALc \fi @@ -55,17 +37,19 @@ \apTESTdigit#1\iftrue \ifx E#1\let\tmpb=\tmpa \expandafter\apEVALd\expandafter\apEVALk \else \edef\tmpb{\tmpa#1}\expandafter\apEVALd\expandafter\apEVALn\fi\fi - \edef\tmpb{\tmpa\noexpand#1}\futurelet\apNext\apEVALg + \edef\tmpb{\tmpa\noexpand#1}\expandafter + \futurelet\expandafter\apNext\expandafter\apEVALg\romannumeral-`\.% } -\def\apEVALd#1\fi#2\apNext\apEVALg{\fi#1} +\def\apEVALd#1\fi#2-`\.{\fi#1} \def\apEVALe{% - \ifx\tmpa\empty \else \ifnum\tmpa1<0 \def\tmpb{-1}\apEVALp \MUL 4\fi\fi + \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}}\futurelet\apNext\apEVALg} +\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% @@ -74,18 +58,17 @@ \else \expandafter\apEVALo\expandafter#1\fi } \def\apEVALo#1{\let\apNext=\apEVALb - \ifx+#1\apEVALp \apEPLUS 1\fi - \ifx-#1\apEVALp \apEMINUS 1\fi - \ifx*#1\apEVALp \apEMUL 2\fi - \ifx/#1\apEVALp \apEDIV 2\fi - \ifx^#1\apEVALp \apEPOW 3\fi + \ifx+#1\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\end#1% - \ifnum\apnumA>0 \apEVALerror{missing bracket ")"}% - \else \apEVALp\END 0\fi - \let\apNext=\relax + \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 @@ -96,7 +79,6 @@ \expandafter\apEVALpush\the\toks0\expandafter{\the\apnumB}% {value}{op}{priority} \let\tmpa=\relax } -\let\apEPLUS=\PLUS \let\apEMINUS=\MINUS \let\apEMUL=\MUL \let\apEDIV=\DIV \let\apEPOW=\POW \def\apEVALstack{{}{}{0}.} \def\apEVALpush#1#2#3{% value, operator, priority \toks0={{#1}{#2}{#3}}% @@ -104,7 +86,7 @@ \expandafter\apEVALdo\apEVALstack@% } \def\apEVALdo#1#2#3#4#5#6#7@{% - \apnumB=#3 \ifx#2\POW \advance\apnumB by1 \fi + \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 @@ -112,7 +94,7 @@ \fi\fi } \def\apEVALerror#1{\message{\noexpand\evaldef ERROR: #1.}% - \def\tmpb{\def\OUT{??}}\def\apNext##1\end{}% + \def\OUT{0}\apE=0\apSIGN=0\def\apNext##1\apEND{\apEND}% } \def\apTESTdigit#1#2{% \ifx E#1\apXtrue \else @@ -137,8 +119,9 @@ \apPPg#1% } \def\apPPd#1\apPPg#2{\fi\expandafter\expandafter\expandafter\apPPc} -\def\apPPe#1\apPPg#2#3@{\fi\apXtrue{#3% execution of the parameter in the group - \edef\tmpc{\apE=\the\apE\relax\noexpand\apPPf\OUT@}\expandafter}\tmpc +\def\apPPe#1\apPPg#2#3@{\fi\apXtrue + \begingroup#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{% @@ -146,7 +129,7 @@ \ifx\tmpc\empty\else\edef\tmpc{\tmpc#1}\fi \ifx0#1\apPPh\fi \ifx\tmpc\empty\edef\tmpc{#1}\fi - \ifx@#1\def\tmpc{@}\fi + \ifx@#1\def\tmpc{@}\apSIGN=0 \fi \expandafter\apPPi\tmpc } \def\apPPh#1\apPPi\tmpc{\fi\apPPg} @@ -158,7 +141,10 @@ \ifx@#2@\else \afterassignment\apPPm \apE=#2\fi } \def\apPPm E{} -\def\apPPn#1{\expandafter\apPPb#1@\OUT \edef\OUT{\ifnum\apSIGN<0-\fi\OUT}} +\def\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 @@ -173,18 +159,11 @@ \def\apPPu#1#2.@#3{\ifx@#2@\apnumG=0 \ifx#1\apROUNDa\def\XOUT{}\fi \else\def\apNext{\apPPt#1#2.@#3}\expandafter\apNext\fi } -\def\apEVALone#1#2{\evaldef\OUT{#2}\ifnum\apSIGN<0 \expandafter\apNOminus\OUT@\OUT\fi #1} -\def\apEVALtwo#1#2#3{% - {\evaldef\OUT{#2}\apOUTtmpb}\tmpb \let\tmpa=\OUT \apSIGNa=\apSIGN \apEa=\apE - \ifnum\apSIGNa<0 \expandafter\apNOminus\tmpa@\tmpa\fi - {\evaldef\OUT{#3}\apOUTtmpb}\tmpb \let\tmpb=\OUT \apSIGNb=\apSIGN \apEb=\apE - \ifnum\apSIGNb<0 \expandafter\apNOminus\tmpb@\tmpb\fi - #1% -} -\def\apNOminus-#1@#2{\def#2{#1}} %%%%%%%%%%%% Addition and Subtraction, sec. 2.4 in apnum.pdf +\def\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 @@ -270,6 +249,7 @@ %%%%%%%%%%%% 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 @@ -339,8 +319,9 @@ %%%%%%%%%%%% Division, sec. 2.6 in apnum.pdf +\def\apDIV{\relax \apPPab\apDIVa} \def\apDIVa{% - \ifnum\apSIGNb=0 \errmessage{Dividing by zero}\else + \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 @@ -377,7 +358,8 @@ \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 + \else + \def\OUT{0}\def\XOUT{0}\apE=0 \apSIGN=0 \fi\fi\fi } \def\apDIVcomp#1#2{% @@ -483,13 +465,14 @@ %%%%%%%%%%%% 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 \errmessage{POW: non-integer exponent is not implemented yet}\apPOWe\fi - \ifnum\apEb=0 \else \errmessage{POW: the E notation of exponent isn't allowed}\apPOWe\fi - \ifnum\apnumD>8 \errmessage{POW: too big exponent. + \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 @@ -549,6 +532,7 @@ %%%%%%%%%%%% 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{% @@ -590,6 +574,7 @@ } \def\apROLLo@#1{\let#1=\tmpd} +\def\apROUND{\apPPs\apROUNDa} \def\apROUNDa{\apnumD=\tmpc\relax \ifnum\apnumD<0 \expandafter\apROUNDe \else \expandafter\apROUNDb @@ -603,7 +588,7 @@ \fi \fi \apNext } -\def\apROUNDd#1.@#2{\def\XOUT{#1}% +\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}% @@ -613,6 +598,7 @@ \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% @@ -625,25 +611,15 @@ \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} -%%%%%%%%%%%% Function-like macros, sec. 2.9 in apnum.pdf +%%%%%%%%%%%% Miscelaneous macros, sec. 2.9 in apnum.pdf -\def\apABSa{\ifnum\apSIGN<0 \apSIGN=1 \fi} -\def\apiDIVa{{\apFRAC=0 \apTOT=0 \apDIVa \apOUTtmpb}\tmpb} -\def\apiMODa{{\apFRAC=0 \apTOT=0 \apDIVa \let\OUT=\XOUT \apOUTtmpb}\tmpb} -\def\apiROUNDa{\apROUNDa\OUT0} -\def\apiFRACa{\apROUNDa\OUT0\ifx\XOUT\empty\def\OUT{0}\else\edef\OUT{.\XOUT}\fi} -\def\apFACa{{\apnumC=\OUT\relax - \loop \ifnum \apnumC>2 \advance\apnumC by-1 - \MUL{\OUT}{\the\apnumC}\repeat - \global\let\OUT=\OUT}% +\def\apEND{\global\let\apENDx=\OUT + \edef\tmpb{\apSIGN=\the\apSIGN \apE=\the\apE}% + \expandafter\endgroup \tmpb \let\OUT=\apENDx } - -%%%%%%%%%%%% Auxiliary macros, sec. 2.10 in apnum.pdf - -\def\apREV#1{\expandafter\apREVa#1@!} -\def\apREVa#1#2!{\ifx@#1\else\apREVa#2!#1\fi} - \def\apDIG#1#2{\ifx\relax#2\def\tmpc{}\else #2=0 \def\tmpc{\advance#2 by1 }\fi \apnumD=0 \expandafter\apDIGa#1..@#1% } @@ -699,6 +675,9 @@ \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{}% @@ -706,17 +685,414 @@ } \def\apOUTs#1,{\ifx.#1\else\csname apOUT:#1\expandafter\endcsname\expandafter\apOUTs\fi} -\def\apOUTtmpb{\edef\tmpb{\apSIGN=\the\apSIGN \apE=\the\apE \edef\noexpand\OUT{\OUT}}\expandafter} +\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 + \evaldef\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 + \evaldef\apAparam{#1}\apEadd\apAparam + \evaldef\apBparam{#2}\apEadd\apBparam % 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 + \evaldef\apAparam{#1}\apEadd\apAparam + \evaldef\apBparam{#2}\apEadd\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\iROUND#1{\relax \evaldef\OUT{#1}\apEnum\OUT \apROUND\OUT0} +\def\iFRAC#1{\relax + \evaldef\OUT{#1}\apEnum\OUT \apROUND\OUT0% % preparing the parameter + \ifx\XOUT\empty \def\OUT{0}\apSIGN=0 % empty fraction part means zero + \else \edef\OUT{.\XOUT}\apSIGN=1 % else OUT = dot+fraction part + \fi +} + +\def\FAC#1{\relax \apINIT % "function-like" in the group, FAC = factorial + \evaldef\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} ... + \evaldef\apAparam{#1}\apEnum\apAparam + \evaldef\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) ... + \evaldef\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) ... + \evaldef\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 by3 + \edef\signX{\the\apSIGN}% + \ifnum\apSIGN<0 \apSIGN=1 \apREMfirst\OUT \fi % remove "minus" sign + \K=0 \N=0 % K = 0, N = 0 + \def\testDot ##1##2\relax##3{\ifx##1.}% + \loop \expandafter \testDot\OUT \relax % loop if (OUT >= 1) + \iftrue \else % OUT = OUT/2 + \apDIV\OUT{2}% % K++ + \advance\K by1 % repeat + \repeat % now: #1 = 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}\apROUND\OUT\apFRAC % OUT = OUT^2 + \advance\N by1 \repeat % N++ + \ifnum\signX<0 \apDIV 1\OUT \fi % if (signX < 0) OUT = 1 / OUT + \apROUND\OUT\digits \apSIGN=1 % EXP is always positive + \apEND +} +\def\apTAYLOR#1{\ifnum\apSIGN=0 \let\OUT=\S \else \apPLUS\S\Sn \let\S=\OUT } + +\def\LN#1{\relax \apINIT % OUT = LN(#1) ... + \evaldef\X{#1}% % X = #1 + \localcounts \M \N \E;% + \E=\apE + \def\round{\apROUND\OUT\apFRAC}% + \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}\round \let\XX=\OUT % XX = OUT^2 + \N=1 % N = 1 + \loop \advance\N by2 % loop N = N + 2 + \do\Kn=\apMUL\Kn\XX\round;% % 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 + \evaldef\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 + \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 + \evaldef\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\message{\OUT/\denom}% + \apDIV{\SIN\X}\denom + \apEND +} +\def\ATAN#1{\relax \apINIT + \advance\apFRAC by3 + \evaldef\X{#1}\apEnum + \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 + \evaldef\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}}} + %%%%%%%%%%%% Conclusion, sec. 2.11 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{SORRY, you are using LaTeX. I don't recommend this. Petr Olsak}\fi \catcode`\@=\apnumZ \endinput -1.0 <Nov. 2014> - First version released -1.1 <Jan. 2015> +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 |