\NeedsTeXFormat{LaTeX2e} \ProvidesPackage{fp-exp}[1995/04/03] %version information \def\FP@expversion{0.7e} \message{% `Fixed Point Exponentiation',% \space\space\space\space% Version \FP@expversion% \space (C) Michael Mehlich% \space\space\space\space\space\space\space% \space\space\space\space\space\space% } %resolve dependencies \RequirePackage{fp-basic} %%%public area (macros which may be used)%%% %constants \edef\FPe{2.718281828459045235} %unary functions \def\FPexp#1#2{\FP@calld\FP@exp#1{#2}} \def\FPln#1#2{\FP@calld\FP@ln#1{#2}} %binary functions \def\FPpow#1#2#3{\FP@pow#1{#2}{#3}} \def\FProot#1#2#3{\FP@root#1{#2}{#3}} %%%private fp-area (don't use these macros)%%% %allocation of registers \countdef\FP@ls=50 \countdef\FP@lfa=51 \countdef\FP@lfb=52 \countdef\FP@regd=53 \countdef\FP@rege=54 %compute e pow x \expandafter\edef\csname FP@e0\endcsname{1.000000000000000000} \expandafter\edef\csname FP@e1\endcsname{2.718281828459045235} \expandafter\edef\csname FP@e2\endcsname{7.389056098930650227} \expandafter\edef\csname FP@e3\endcsname{20.085536923187667741} \expandafter\edef\csname FP@e4\endcsname{54.598150033144239078} \expandafter\edef\csname FP@e5\endcsname{148.413159102576603421} \expandafter\edef\csname FP@e6\endcsname{403.428793492735122608} \expandafter\edef\csname FP@e7\endcsname{1096.633158428458599264} \expandafter\edef\csname FP@e8\endcsname{2980.957987041728274744} \expandafter\edef\csname FP@e9\endcsname{8103.083927575384007710} \expandafter\edef\csname FP@e10\endcsname{22026.465794806716516958} \expandafter\edef\csname FP@e11\endcsname{59874.141715197818455326} \expandafter\edef\csname FP@e12\endcsname{162754.791419003920808005} \expandafter\edef\csname FP@e13\endcsname{442413.392008920503326103} \expandafter\edef\csname FP@e14\endcsname{1202604.284164776777749237} \expandafter\edef\csname FP@e15\endcsname{3269017.372472110639301855} \expandafter\edef\csname FP@e16\endcsname{8886110.520507872636763024} \expandafter\edef\csname FP@e17\endcsname{24154952.753575298214775435} \expandafter\edef\csname FP@e18\endcsname{65659969.137330511138786503} \expandafter\edef\csname FP@e19\endcsname{178482300.963187260844910034} \expandafter\edef\csname FP@e20\endcsname{485165195.409790277969106831} \expandafter\edef\csname FP@e21\endcsname{1318815734.483214697209998884} \expandafter\edef\csname FP@e22\endcsname{3584912846.131591561681159946} \expandafter\edef\csname FP@e23\endcsname{9744803446.248902600034632685} \expandafter\edef\csname FP@e24\endcsname{26489122129.843472294139162153} \expandafter\edef\csname FP@e25\endcsname{72004899337.385872524161351466} \expandafter\edef\csname FP@e26\endcsname{195729609428.838764269776397876} \expandafter\edef\csname FP@e27\endcsname{532048240601.798616683747304341} \expandafter\edef\csname FP@e28\endcsname{1446257064291.475173677047422997} \expandafter\edef\csname FP@e29\endcsname{3931334297144.042074388620580844} \expandafter\edef\csname FP@e30\endcsname{10686474581524.462146990468650741} \expandafter\edef\csname FP@e31\endcsname{29048849665247.425231085682111680} \expandafter\edef\csname FP@e32\endcsname{78962960182680.695160978022635108} \expandafter\edef\csname FP@e33\endcsname{214643579785916.064624297761531261} \expandafter\edef\csname FP@e34\endcsname{583461742527454.881402902734610391} \expandafter\edef\csname FP@e35\endcsname{1586013452313430.728129644625774660} \expandafter\edef\csname FP@e36\endcsname{4311231547115195.227113422292856925} \expandafter\edef\csname FP@e37\endcsname{11719142372802611.308772939791190195} \expandafter\edef\csname FP@e38\endcsname{31855931757113756.220328671701298646} \expandafter\edef\csname FP@e39\endcsname{86593400423993746.953606932719264934} \expandafter\edef\csname FP@e40\endcsname{235385266837019985.407899910749034805} \expandafter\edef\csname FP@e41\endcsname{639843493530054949.222663403515570819} \expandafter\edef\csname FP@e-0\endcsname{1.000000000000000000} \expandafter\edef\csname FP@e-1\endcsname{0.367879441171442322} \expandafter\edef\csname FP@e-2\endcsname{0.135335283236612692} \expandafter\edef\csname FP@e-3\endcsname{0.049787068367863943} \expandafter\edef\csname FP@e-4\endcsname{0.018315638888734180} \expandafter\edef\csname FP@e-5\endcsname{0.006737946999085467} \expandafter\edef\csname FP@e-6\endcsname{0.002478752176666358} \expandafter\edef\csname FP@e-7\endcsname{0.000911881965554516} \expandafter\edef\csname FP@e-8\endcsname{0.000335462627902512} \expandafter\edef\csname FP@e-9\endcsname{0.000123409804086680} \expandafter\edef\csname FP@e-10\endcsname{0.000045399929762485} \expandafter\edef\csname FP@e-11\endcsname{0.000016701700790246} \expandafter\edef\csname FP@e-12\endcsname{0.000006144212353328} \expandafter\edef\csname FP@e-13\endcsname{0.000002260329406981} \expandafter\edef\csname FP@e-14\endcsname{0.000000831528719104} \expandafter\edef\csname FP@e-15\endcsname{0.000000305902320502} \expandafter\edef\csname FP@e-16\endcsname{0.000000112535174719} \expandafter\edef\csname FP@e-17\endcsname{0.000000041399377188} \expandafter\edef\csname FP@e-18\endcsname{0.000000015229979745} \expandafter\edef\csname FP@e-19\endcsname{0.000000005602796438} \expandafter\edef\csname FP@e-20\endcsname{0.000000002061153622} \expandafter\edef\csname FP@e-21\endcsname{0.000000000758256043} \expandafter\edef\csname FP@e-22\endcsname{0.000000000278946809} \expandafter\edef\csname FP@e-23\endcsname{0.000000000102618796} \expandafter\edef\csname FP@e-24\endcsname{0.000000000037751345} \expandafter\edef\csname FP@e-25\endcsname{0.000000000013887944} \expandafter\edef\csname FP@e-26\endcsname{0.000000000005109089} \expandafter\edef\csname FP@e-27\endcsname{0.000000000001879529} \expandafter\edef\csname FP@e-28\endcsname{0.000000000000691440} \expandafter\edef\csname FP@e-29\endcsname{0.000000000000254367} \expandafter\edef\csname FP@e-30\endcsname{0.000000000000093576} \expandafter\edef\csname FP@e-31\endcsname{0.000000000000034425} \expandafter\edef\csname FP@e-32\endcsname{0.000000000000012664} \expandafter\edef\csname FP@e-33\endcsname{0.000000000000004659} \expandafter\edef\csname FP@e-34\endcsname{0.000000000000001714} \expandafter\edef\csname FP@e-35\endcsname{0.000000000000000631} \expandafter\edef\csname FP@e-36\endcsname{0.000000000000000232} \expandafter\edef\csname FP@e-37\endcsname{0.000000000000000085} \expandafter\edef\csname FP@e-38\endcsname{0.000000000000000031} \expandafter\edef\csname FP@e-39\endcsname{0.000000000000000012} \expandafter\edef\csname FP@e-40\endcsname{0.000000000000000004} \expandafter\edef\csname FP@e-41\endcsname{0.000000000000000002} \expandafter\edef\csname FP@e-42\endcsname{0.000000000000000001} \def\FP@exp@mul#1#2#3#4#5#6{% % #1 macro which gets fractional part 1 % #2 macro which fractional part 1 % #3 first value part 1 % #4 first value part 2 % #5 second value part 1 % #6 second value part 2 \FP@split\FP@xq\FP@xr\FP@xz{#3}\FP@split\FP@xt\FP@xu\FP@xv{#4}% \FP@split\FP@yq\FP@yr\FP@yz{#5}\FP@split\FP@yt\FP@yu\FP@yv{#6}% \FP@prod=0\relax% \edef\FP@rd{}% \FP@@mul vv \relax\FP@saveshift% \FP@@mul vu uv \relax\FP@saveshift% \FP@@mul uu vt tv \relax\FP@saveshift% \FP@@mul ut tu vz zv \relax\FP@saveshift% \FP@@mul tt uz zu vr rv \relax\FP@saveshift% \FP@@mul zt tz ur ru qv vq \relax\FP@saveshift% \FP@@mul zz rt tr qu uq \relax\FP@saveshift% \FP@@mul rz zr qt tq \relax\FP@saveshift% \FP@@mul rr qz zq \relax\FP@saveshift% \FP@@mul qr rq \relax\FP@saveshift% \FP@@mul qq \relax\FP@saveshift\FP@saveshift% \FP@ninesplit\FP@rd#1=\FP@res% \FP@ninesplit\FP@rd#2=\FP@res% } \def\FP@exp@divn#1#2#3#4#5{% % #1 macro which gets fractional part 1 % #2 macro which fractional part 1 % #3 value part 1 % #4 value part 2 % #5 integer (0,..,1000) \FP@rega=#3\divide\FP@rega#5% \FP@regd=\FP@rega% \multiply\FP@rega#5\relax% \advance\FP@rega-#3\relax% \multiply\FP@rega-1000\relax% \FP@regb=#4\divide\FP@regb1000000\relax% \advance\FP@rega\FP@regb% \FP@regc=\FP@rega\divide\FP@regc#5\relax% \FP@rege=\FP@regc\multiply\FP@rege1000000\relax% \multiply\FP@regc#5\advance\FP@rega-\FP@regc\multiply\FP@rega1000000\relax% \multiply\FP@regb1000000\relax\advance\FP@regb-#4\relax% \advance\FP@rega-\FP@regb% \divide\FP@rega#5\relax% \advance\FP@rege\FP@rega% #1=\FP@regd% #2=\FP@rege% } %exp (for values between -1 and 1) \def\FP@@exp#1#2#3#4{% % #1 macro which gets the result % #2 sign % #3 fractional part 1 % #4 fractional part 2 \ifnum\ifnum#3=0 0\else1\fi\ifnum#4=0 0\else1\fi=0\relax% \edef#1{1}% \else% \FP@rs=1\relax% \FP@ria=0\relax% \ifnum#2>0\relax% \FP@rib=1\relax% \FP@rfa=#3\relax% \FP@rfb=#4\relax% \else% \FP@rib=0\relax% \ifnum#4=0\relax% \FP@rfa=1000000000\relax\FP@rfb=0\relax% \else% \FP@rfa=999999999\relax\FP@rfb=1000000000\relax% \fi% \advance\FP@rfa-#3\relax% \advance\FP@rfb-#4\relax% \fi% \FP@ls=#2\relax% \FP@lfa=#3\relax% \FP@lfb=#4\relax% \FP@count=1\relax% \loop% \advance\FP@count1\relax% \multiply\FP@ls#2\relax% \FP@exp@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb#3#4\relax% \FP@exp@divn\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@count% \ifnum\ifnum\FP@lfa=0 0\else1\fi\ifnum\FP@lfb=0 0\else1\fi>0\relax% \ifnum\FP@ls>0\relax% \advance\FP@rfa\FP@lfa% \advance\FP@rfb\FP@lfb% \ifnum\FP@rfb<1000000000\relax\else% \advance\FP@rfa1\relax% \advance\FP@rfb-1000000000\relax% \fi% \ifnum\FP@rfa<1000000000\relax\else% \advance\FP@rib1\relax% \advance\FP@rfa-1000000000\relax% \fi% \else% \advance\FP@rfa-\FP@lfa% \advance\FP@rfb-\FP@lfb% \ifnum\FP@rfb<0\relax% \advance\FP@rfa-1\relax% \advance\FP@rfb1000000000\relax% \fi% \ifnum\FP@rfa<0\relax% \advance\FP@rib-1\relax% \advance\FP@rfa1000000000\relax% \fi% \fi% \repeat% \FP@store#1{r}% \fi% } \def\FP@exp#1#2.#3.#4\relax{% % #1 macro which gets the result % #2 integer part of value % #3 fractional part of value % #4 dummy to swallow everthing after the 2nd '.' % \FP@beginmessage{EXP}% % {\def\FP@beginmessage##1{}% \def\FP@endmessage##1{}% \FP@readvalue{x}{#2}{#3}% \ifnum\FP@xia=0\relax% \ifnum\FP@xs<0\relax% \ifnum\FP@xib>42\relax% \edef\FP@tmp{0}% \else% \edef\FP@tmp{\csname FP@e-\the\FP@xib\endcsname}% \fi% \else% \ifnum\FP@xib>41\relax% \FP@errmessage{Overflow}% \else% \edef\FP@tmp{\csname FP@e\the\FP@xib\endcsname}% \fi% \fi% \FP@@exp\FP@tmpa\FP@xs\FP@xfa\FP@xfb% \FPmul\FP@tmp\FP@tmpa\FP@tmp% \global\let\FP@tmp=\FP@tmp% \else% \FP@errmessage{Overflow}% \fi% }% % \FP@endmessage{}% % \let#1\FP@tmp% } %compute ln x \expandafter\edef\csname FP@ln2-0\endcsname{0.000000000000000000} \expandafter\edef\csname FP@ln2-1\endcsname{0.693147180559945309} \expandafter\edef\csname FP@ln2-2\endcsname{1.386294361119890618} \expandafter\edef\csname FP@ln2-3\endcsname{2.079441541679835928} \expandafter\edef\csname FP@ln10-0\endcsname{0.000000000000000000} \expandafter\edef\csname FP@ln10-1\endcsname{2.302585092994045684} \expandafter\edef\csname FP@ln10-2\endcsname{4.605170185988091368} \expandafter\edef\csname FP@ln10-3\endcsname{6.907755278982137052} \expandafter\edef\csname FP@ln10-4\endcsname{9.210340371976182736} \expandafter\edef\csname FP@ln10-5\endcsname{11.512925464970228420} \expandafter\edef\csname FP@ln10-6\endcsname{13.815510557964274104} \expandafter\edef\csname FP@ln10-7\endcsname{16.118095650958319788} \expandafter\edef\csname FP@ln10-8\endcsname{18.420680743952365472} \expandafter\edef\csname FP@ln10-9\endcsname{20.723265836946411156} \expandafter\edef\csname FP@ln10-10\endcsname{23.025850929940456840} \expandafter\edef\csname FP@ln10-11\endcsname{25.328436022934502524} \expandafter\edef\csname FP@ln10-12\endcsname{27.631021115928548208} \expandafter\edef\csname FP@ln10-13\endcsname{29.933606208922593892} \expandafter\edef\csname FP@ln10-14\endcsname{32.236191301916639576} \expandafter\edef\csname FP@ln10-15\endcsname{34.538776394910685260} \expandafter\edef\csname FP@ln10-16\endcsname{36.841361487904730944} \expandafter\edef\csname FP@ln10-17\endcsname{39.143946580898776628} \expandafter\edef\csname FP@ln10-18\endcsname{41.446531673892822312} \def\FP@ln@divtwo#1#2#3#4#5#6{% % #1 macro which gets integer part % #2 macro which gets fractional part 1 % #3 macro which fractional part 2 % #4 integer part 1 % #5 fractional part 1 % #6 fractional part 2 #1=#4\relax% #2=#5\relax\ifodd#4\advance#21000000000\relax\fi% #3=#6\relax\ifodd#5\advance#31000000000\relax\fi% \divide#32\relax% \divide#22\relax% \divide#12\relax% } %ln for values between (1 and 2) \def\FP@@ln#1#2#3#4{% % #1 macro which gets the result % #2 integer part (1 or 2) % #3 fractional part 1 % #4 fractional part 2 {\FP@xs=1\FP@xia=0\FP@xib=#2\FP@xfa=#3\FP@xfb=#4% {\advance\FP@xib-1\relax\FP@store\FP@tmpa{x}\global\let\FP@tmpa\FP@tmpa}% {\advance\FP@xib+1\relax\FP@store\FP@tmpb{x}\global\let\FP@tmpb\FP@tmpb}% \FPdiv\FP@tmpa\FP@tmpa\FP@tmpb% % \def\next##1.##2.##3\relax{\FP@readvalue{x}{##1}{##2}}% \expandafter\next\FP@tmpa..\relax% % \FP@lfa=\FP@xfa% \FP@lfb=\FP@xfb% % \FP@rs=1\relax% \FP@ria=0\relax% \FP@rib=0\relax% \FP@rfa=\FP@xfa% \FP@rfb=\FP@xfb% % \FP@count=1\relax% \FP@exp@mul\FP@xfa\FP@xfb\FP@xfa\FP@xfb\FP@xfa\FP@xfb% % \loop% \advance\FP@count2\relax% \FP@exp@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@xfa\FP@xfb% \ifnum\ifnum\FP@lfa=0 0\else1\fi\ifnum\FP@lfb=0 0\else1\fi=0\relax\else% \FP@exp@divn\FP@yfa\FP@yfb\FP@lfa\FP@lfb\FP@count% \advance\FP@rfb\FP@yfb% \advance\FP@rfa\FP@yfa% \ifnum\FP@rfb<1000000000\relax\else% \advance\FP@rfa1\relax% \advance\FP@rfb-1000000000\relax% \fi% \ifnum\FP@rfa<1000000000\relax\else% \advance\FP@rib1\relax% \advance\FP@rfa-1000000000\relax% \fi% \repeat% % \advance\FP@rfa\FP@rfa% \ifnum\FP@rfb<500000000\relax\else% \advance\FP@rfa1\relax% \advance\FP@rfb-500000000\relax% \fi% \advance\FP@rfb\FP@rfb% % \FP@store\FP@tmpa{r}% % \global\let\FP@tmpa\FP@tmpa% }% % \let#1\FP@tmpa% } \def\FP@ln#1#2.#3.#4\relax{% % #1 macro which gets the result % #2 integer part of value % #3 fractional part of value % #4 dummy to swallow everthing after the 2nd '.' % \FP@beginmessage{LN}% % {\def\FP@beginmessage##1{}% \def\FP@endmessage##1{}% % \FP@readvalue{x}{#2}{#3}% % \ifFP@zero{x}% \FP@errmessage{Logarithm of zero}% \else\ifnum\FP@xs<0\relax% \FP@errmessage{Logarithm of negative value}% \else% \FP@count=0\relax% \loop% \ifnum\ifnum\FP@xia>0 1\else0\fi\ifnum\FP@xib>9 1\else0\fi>0\relax% \advance\FP@count1\relax% \FP@divten{x}% \repeat% \loop% \ifnum\ifnum\FP@xia=0 0\else1\fi\ifnum\FP@xib=0 0\else1\fi=0\relax% \advance\FP@count-1\relax% \FP@multen{x}% \repeat% \ifnum\FP@count<0\relax% \FP@count=-\FP@count% \edef\FP@tmpc{-\csname FP@ln10-\the\FP@count\endcsname}% \else% \edef\FP@tmpc{\csname FP@ln10-\the\FP@count\endcsname}% \fi% \FP@count=0\relax% \loop% \ifnum\FP@xib<2\relax\else% \advance\FP@count1\relax% \FP@ln@divtwo\FP@xib\FP@xfa\FP@xfb\FP@xib\FP@xfa\FP@xfb% \repeat% \FPadd\FP@tmpc\FP@tmpc{\csname FP@ln2-\the\FP@count\endcsname}% \FP@@ln\FP@tmpb\FP@xib\FP@xfa\FP@xfb% \FPadd\FP@tmp\FP@tmpc\FP@tmpb% \fi\fi% % \global\let\FP@tmp\FP@tmp% }% % \FP@endmessage{}% % \let#1\FP@tmp% } %compute x pow y \def\FP@pow@zero#1{% \FPifzero{#1}% \FP@errmessage{0-th pow of 0 not defined}% \else% \edef\FP@tmp{0}% \fi% } \def\FP@pow#1#2#3{% % #1 macro, which gets the result % #2 base % #3 exponent % \FP@beginmessage{POW}% % {\def\FP@beginmessage##1{}% \def\FP@endmessage##1{}% % \FPifzero{#2}% \FP@pow@zero{#3}% \else% \FPln\FP@tmpd{#2}% \FPmul\FP@tmpd\FP@tmpd{#3}% \FPexp\FP@tmp\FP@tmpd% \fi% % \global\let\FP@tmp\FP@tmp% }% % \FP@endmessage{}% % \let#1\FP@tmp% } %compute y-th root of x \def\FP@root#1#2#3{% % #1 macro, which gets the result % #2 base % #3 which root % \FP@beginmessage{ROOT}% % {\def\FP@beginmessage##1{}% \def\FP@endmessage##1{}% % \FPifzero{#3}% \FP@errmessage{0-th root not defined}% \fi% % \FPifzero{#2}% \edef\FP@tmp{0}% \else% \FPln\FP@tmpd{#2}% \FPdiv\FP@tmpd\FP@tmpd{#3}% \FPexp\FP@tmp\FP@tmpd% \fi% % \global\let\FP@tmp\FP@tmp% }% % \FP@endmessage{}% % \let#1\FP@tmp% }