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diff --git a/Master/texmf-dist/tex/latex/fp/fp-trigo.sty b/Master/texmf-dist/tex/latex/fp/fp-trigo.sty new file mode 100644 index 00000000000..06cfa30f9f8 --- /dev/null +++ b/Master/texmf-dist/tex/latex/fp/fp-trigo.sty @@ -0,0 +1,1119 @@ +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{fp-trigo}[1995/04/14] + +%version information +\def\FP@trigoversion{0.9e} +\message{% + `Fixed Point Trigonometry',% + \space\space\space\space\space\space% + Version \FP@trigoversion% + \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\FPpi{3.141592653589793238} + +%unary functions +\def\FPsin#1#2{\FP@calld\FP@sin#1{#2}} +\def\FPcos#1#2{\FP@calld\FP@cos#1{#2}} +\def\FPsincos#1#2#3{\FP@calle\FP@sincos#1#2{#3}} +\def\FPtan#1#2{\FP@calld\FP@tan#1{#2}} +\def\FPcot#1#2{\FP@calld\FP@cot#1{#2}} +\def\FPtancot#1#2#3{\FP@calle\FP@tancot#1#2{#3}} +\def\FParcsin#1#2{\FP@calld\FP@arcsin#1{#2}} +\def\FParccos#1#2{\FP@calld\FP@arccos#1{#2}} +\def\FParcsincos#1#2#3{\FP@calle\FP@arcsincos#1#2{#3}} +\def\FParctan#1#2{\FP@calld\FP@arctan#1{#2}} +\def\FParccot#1#2{\FP@calld\FP@arccot#1{#2}} +\def\FParctancot#1#2#3{\FP@calle\FP@arctancot#1#2{#3}} + +%%%private fp-area (don't use these macros)%%% + +%allocate registers + +\countdef\FP@ls=50 +\countdef\FP@lfa=51 +\countdef\FP@lfb=52 + +\countdef\FP@loops=53 +\countdef\FP@oct=54 + +\countdef\FP@@arccos=55 +\countdef\FP@oldvala=56 +\countdef\FP@oldvalb=57 +\countdef\FP@subfpih=58 + +%compute 0<=x<2pi for value 2pi*n + x + +\def\FP@modtwopi#1#2{%result is the family {x} + % #1 integer part of value + % #2 fractional part of value + \FP@readvalue{x}{#1}{#2}% + % + \ifnum\ifnum\FP@xib=0 1\else0\fi\ifnum\FP@xia<6 1\else0\fi<11\relax% + %compute modulo without caring about sign + \FP@shift=0\relax% + \FP@ys=1\FP@yia=628318530\FP@yib=717958647\FP@yfa=692528676\FP@yfb=655900577\relax%2*pi + \loop% + \ifnum\FP@xia<100000000\relax% + \FP@multen{x}% + \advance\FP@shift-1\relax% + \repeat% + \FP@loops=18\relax\advance\FP@loops\FP@shift% + \loop% + \ifnum\FP@loops=0\else% + \advance\FP@loops-1\relax% + \FP@counttimes%divides x by \FP@times*y% + \FP@divten{y}% + \repeat% + \loop% + \ifnum\FP@shift<0% + \advance\FP@shift1\relax% + \FP@divten{x}% + \repeat% + \fi% + %correct the result for negative values (modulo is positive) + \loop%should be computed only once + \ifnum\FP@xs<0\relax% + \advance\FP@xfb-179586477\relax\FP@xfb=-\FP@xfb% + \ifnum\FP@xfb<0\relax% + \advance\FP@xfb1000000000\relax% + \advance\FP@xfa1\relax% + \fi% + \advance\FP@xfa-283185307\relax\FP@xfa=-\FP@xfa% + \ifnum\FP@xfa<0\relax% + \advance\FP@xfa1000000000\relax% + \advance\FP@xib1\relax% + \fi% + \advance\FP@xib-6\relax\FP@xib=-\FP@xib% + \ifnum\FP@xib<0% + \FP@errmessage{Error in calculation: This shouldn't happen!}% + \else% + \FP@xs=1\relax% + \fi% + \repeat% +} + +%auxiliary functions + +\def\FP@trigocmpsval#1#2#3{% + % #1 macro, which gets the result + % #2 9 digits integer value + % #3 9 digits integer value + \ifnum#2<#3\relax% + #1=-1\relax% + \else\ifnum#2>#3\relax% + #1=1\relax% + \else% + #1=0\relax% + \fi\fi% +} + +\def\FP@trigocmpval#1#2#3#4#5#6#7{% + % #1 macro, which gets the result + % #2 9 digits integer value + % #3 9 digits integer value + % #4 9 digits integer value + % #5 9 digits integer value + % #6 9 digits integer value + % #7 9 digits integer value + % + \FP@trigocmpsval#1{#2}{#5}% + \ifnum#1=0\relax\FP@trigocmpsval#1{#3}{#6}\relax\fi% + \ifnum#1=0\relax\FP@trigocmpsval#1{#4}{#7}\relax\fi% +} + +\def\FP@trigosub#1#2#3#4#5#6{% + % #1 resulting fractional part one + % #2 resulting fractional part two + % #3 1st value fractional part one + % #4 1st value fractional part two + % #5 2nd value fractional part one + % #6 2nd value fractional part two + \FP@rfb=#4\relax\advance\FP@rfb-#6\relax% + \FP@rfa=#3\relax\advance\FP@rfa-#5\relax% + \ifnum\FP@rfb<0\relax% + \advance\FP@rfa-1\relax% + \advance\FP@rfb1000000000\relax% + \fi% + \ifnum\FP@rfa<0\relax% + \advance\FP@rfa1000000000\relax% + \fi% + #1=\FP@rfa% + #2=\FP@rfb% +} + +%compute octand of value and transform value to one between 0 and pi/4 + +\def\FP@octand#1#2{%resulting values \FP@oct, family {x} + % #1 integer part of value + % #2 fractional part of value + % + \FP@modtwopi{#1}{#2}% + %now it is 0<={x}<2pi + % + \FP@trigocmpval\FP@oct\FP@xib\FP@xfa\FP@xfb{3}{141592653}{589793238}% + \ifnum\FP@oct>0\relax% + \FP@trigocmpval\FP@oct\FP@xib\FP@xfa\FP@xfb{4}{712388980}{384689858}% + \ifnum\FP@oct>0\relax% + \FP@trigocmpval\FP@oct\FP@xib\FP@xfa\FP@xfb{5}{497787143}{782138167}% + \ifnum\FP@oct>0\relax% + \FP@oct=7\relax% + \FP@trigosub\FP@xfa\FP@xfb{283185307}{179586477}\FP@xfa\FP@xfb% + \else% + \FP@oct=6\relax% + \FP@trigosub\FP@xfa\FP@xfb\FP@xfa\FP@xfb{712388980}{384689858}% + \fi% + \else% + \FP@trigocmpval\FP@oct\FP@xib\FP@xfa\FP@xfb{3}{926990816}{987241548}% + \ifnum\FP@oct>0\relax% + \FP@oct=5\relax% + \FP@trigosub\FP@xfa\FP@xfb{712388980}{384689858}\FP@xfa\FP@xfb% + \else% + \FP@oct=4\relax% + \FP@trigosub\FP@xfa\FP@xfb\FP@xfa\FP@xfb{141592653}{589793238}% + \fi% + \fi% + \else% + \FP@trigocmpval\FP@oct\FP@xib\FP@xfa\FP@xfb{1}{570796326}{794896619}% + \ifnum\FP@oct>0\relax% + \FP@trigocmpval\FP@oct\FP@xib\FP@xfa\FP@xfb{2}{356194490}{192344929}% + \ifnum\FP@oct>0\relax% + \FP@oct=3\relax% + \FP@trigosub\FP@xfa\FP@xfb{141592653}{589793238}\FP@xfa\FP@xfb% + \else% + \FP@oct=2\relax% + \FP@trigosub\FP@xfa\FP@xfb\FP@xfa\FP@xfb{570796326}{794896619}% + \fi% + \else% + \FP@trigocmpval\FP@oct\FP@xib\FP@xfa\FP@xfb{0}{785398163}{397448310}% + \ifnum\FP@oct>0\relax% + \FP@oct=1\relax% + \FP@trigosub\FP@xfa\FP@xfb{570796326}{794896619}\FP@xfa\FP@xfb% + \else% + \FP@oct=0\relax% + \fi% + \fi% + \fi% + % + \FP@xib=0\relax% +} + +%auxiliary functions + +\def\FP@trigo@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@trigo@divn#1#2#3#4#5{% + % #1 macro which gets fractional part 1 + % #2 macro which gets fractional part 2 + % #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% +} + +%sine and cosine + +\def\FP@sincos@loop#1#2{% + % #1 fractional part 1 of value + % #2 fractional part 2 of value + \loop% + \FP@ls=-\FP@ls% + \FP@trigo@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb{#1}{#2}% + \FP@trigo@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb{#1}{#2}% + \FP@trigo@divn\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@count% + \advance\FP@count1\relax% + \FP@trigo@divn\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@count% + \advance\FP@count1\relax% + \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% +} + +%sine for values 0<=v<=pi/4 +\def\FP@@sin#1#2{% + % #1 fractional part 1 + % #2 fractional part 2 + \ifnum\ifnum#1=0 0\else1\fi\ifnum#2=0 0\else1\fi=0\relax% + \FP@rs=1\FP@ria=0\FP@rib=0\FP@rfa=0\FP@rfb=0\relax% + \else% + \FP@rs=1\FP@ria=0\FP@rib=0\FP@rfa=#1\FP@rfb=#2\relax% + \FP@ls=1\FP@lfa=#1\FP@lfb=#2\relax% + \FP@count=2\relax% + \FP@sincos@loop{#1}{#2}% + \fi% +} + +%cosine for values 0<=v<=pi/4 +\def\FP@@cos#1#2{% + % #1 fractional part 1 + % #2 fractional part 2 + \ifnum\ifnum#1=0 0\else1\fi\ifnum#2=0 0\else1\fi=0\relax% + \FP@rs=1\FP@ria=0\FP@rib=1\FP@rfa=0\FP@rfb=0\relax% + \else% + \FP@ls=-1\FP@lfa=#1\FP@lfb=#2\relax% + \FP@trigo@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@lfa\FP@lfb% + \FP@trigo@divn\FP@lfa\FP@lfb\FP@lfa\FP@lfb{2}% + \FP@count=3\relax% + \FP@rs=1\FP@ria=0\relax% + \ifnum\FP@lfb=0\relax% + \ifnum\FP@lfa=0\relax% + \FP@rib=1\FP@rfa=0\FP@rfb=0\relax% + \else% + \FP@rib=0\FP@rfa=1000000000\FP@rfb=0\relax% + \fi% + \else% + \FP@rib=0\FP@rfa=999999999\FP@rfb=1000000000\relax% + \fi% + \advance\FP@rfb-\FP@lfb\advance\FP@rfa-\FP@lfa% + \FP@sincos@loop{#1}{#2}% + \fi% +} + +\def\FP@sin#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 everything after the 2nd '.' + % + \FP@beginmessage{SIN}% + % + {\FP@octand{#2}{#3}% + % + \ifnum\FP@oct<4\relax% + \ifnum\FP@oct<2\relax% + \ifnum\FP@oct<1\relax% + \FP@@sin\FP@xfa\FP@xfb% + \else% + \FP@@cos\FP@xfa\FP@xfb% + \fi% + \else% + \ifnum\FP@oct<3\relax% + \FP@@cos\FP@xfa\FP@xfb% + \else% + \FP@@sin\FP@xfa\FP@xfb% + \fi% + \fi% + \else% + \ifnum\FP@oct<6\relax% + \ifnum\FP@oct<5\relax% + \FP@@sin\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \else% + \FP@@cos\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \fi% + \else% + \ifnum\FP@oct<7\relax% + \FP@@cos\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \else% + \FP@@sin\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \fi% + \fi% + \fi% + % + \FP@store\FP@tmp{r}% + \global\let\FP@tmp\FP@tmp% + }% + % + \FP@endmessage{}% + % + \let#1\FP@tmp% +} + +\def\FP@cos#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 everything after the 2nd '.' + % + \FP@beginmessage{COS}% + % + {\FP@octand{#2}{#3}% + % + \ifnum\FP@oct<4\relax% + \ifnum\FP@oct<2\relax% + \ifnum\FP@oct<1\relax% + \FP@@cos\FP@xfa\FP@xfb% + \else% + \FP@@sin\FP@xfa\FP@xfb% + \fi% + \else% + \ifnum\FP@oct<3\relax% + \FP@@sin\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \else% + \FP@@cos\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \fi% + \fi% + \else% + \ifnum\FP@oct<6\relax% + \ifnum\FP@oct<5\relax% + \FP@@cos\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \else% + \FP@@sin\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \fi% + \else% + \ifnum\FP@oct<7\relax% + \FP@@sin\FP@xfa\FP@xfb% + \else% + \FP@@cos\FP@xfa\FP@xfb% + \fi% + \fi% + \fi% + % + \FP@store\FP@tmp{r}% + \global\let\FP@tmp\FP@tmp% + }% + % + \FP@endmessage{}% + % + \let#1\FP@tmp% +} + +\def\FP@@sincos#1#2{% + % #1 integer part of value + % #2 fractional part of value + % + \FP@octand{#1}{#2}% + % + \ifnum\FP@oct<4\relax% + \ifnum\FP@oct<2\relax% + \ifnum\FP@oct<1\relax% + \FP@@sin\FP@xfa\FP@xfb% + \FP@store\FP@tmpc{r}% + \FP@@cos\FP@xfa\FP@xfb% + \FP@store\FP@tmpb{r}% + \else% + \FP@@cos\FP@xfa\FP@xfb% + \FP@store\FP@tmpc{r}% + \FP@@sin\FP@xfa\FP@xfb% + \FP@store\FP@tmpb{r}% + \fi% + \else% + \ifnum\FP@oct<3\relax% + \FP@@cos\FP@xfa\FP@xfb% + \FP@store\FP@tmpc{r}% + \FP@@sin\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \FP@store\FP@tmpb{r}% + \else% + \FP@@sin\FP@xfa\FP@xfb% + \FP@store\FP@tmpc{r}% + \FP@@cos\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \FP@store\FP@tmpb{r}% + \fi% + \fi% + \else% + \ifnum\FP@oct<6\relax% + \ifnum\FP@oct<5\relax% + \FP@@sin\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \FP@store\FP@tmpc{r}% + \FP@@cos\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \FP@store\FP@tmpb{r}% + \else% + \FP@@cos\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \FP@store\FP@tmpc{r}% + \FP@@sin\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \FP@store\FP@tmpb{r}% + \fi% + \else% + \ifnum\FP@oct<7\relax% + \FP@@cos\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \FP@store\FP@tmpc{r}% + \FP@@sin\FP@xfa\FP@xfb% + \FP@store\FP@tmpb{r}% + \else% + \FP@@sin\FP@xfa\FP@xfb% + \FP@rs=-\FP@rs% + \FP@store\FP@tmpc{r}% + \FP@@cos\FP@xfa\FP@xfb% + \FP@store\FP@tmpb{r}% + \fi% + \fi% + \fi% +} + +\def\FP@sincos#1#2#3.#4.#5\relax{% + % #1 macro, which gets the sine result + % #2 macro, which gets the cosine result + % #3 integer part of value + % #4 fractional part of value + % #5 dummy to swallow everything after the 2nd '.' + % + {\FP@beginmessage{SINCOS}% + % + \FP@@sincos{#3}{#4}% + % + \global\let\FP@tmpc\FP@tmpc%sine + \global\let\FP@tmpb\FP@tmpb%cosine + % + \FP@endmessage{}% + }% + % + \let#1\FP@tmpc% + \let#2\FP@tmpb% + } + +%tan and cot + +\def\FP@tan#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 everything after the 2nd '.' + % + \FP@beginmessage{TAN}% + % + {\def\FP@beginmessage##1{}% + \def\FP@endmessage##1{}% + % + \FP@@sincos{#2}{#3}% + % + \FPdiv\FP@tmp\FP@tmpc\FP@tmpb% + % + \global\let\FP@tmp\FP@tmp% + }% + % + \FP@endmessage{}% + % + \let#1\FP@tmp% +} + +\def\FP@cot#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 everything after the 2nd '.' + % + \FP@beginmessage{COT}% + % + {\def\FP@beginmessage##1{}% + \def\FP@endmessage##1{}% + % + \FP@@sincos{#2}{#3}% + % + \FPdiv\FP@tmp\FP@tmpb\FP@tmpc% + % + \global\let\FP@tmp\FP@tmp% + }% + % + \FP@endmessage{}% + % + \let#1\FP@tmp% +} + +\def\FP@tancot#1#2#3.#4.#5\relax{% + % #1 macro, which gets the tan result + % #1 macro, which gets the cot result + % #3 integer part of value + % #4 fractional part of value + % #5 dummy to swallow everything after the 2nd '.' + % + \FP@beginmessage{TANCOT}% + % + {\def\FP@beginmessage##1{}% + \def\FP@endmessage##1{}% + % + \FP@@sincos{#3}{#4}% + % + \FPdiv\FP@tmpd\FP@tmpc\FP@tmpb% + \FPdiv\FP@tmp\FP@tmpb\FP@tmpc% + % + \global\let\FP@tmpd\FP@tmpd% + \global\let\FP@tmp\FP@tmp% + }% + % + \FP@endmessage{}% + % + \let#1\FP@tmpd% + \let#2\FP@tmp% +} + +%auxiliary functions + +% compute x/(x+1) for 0<=x<=1000000 +\def\FP@trigo@xy#1#2#3{% + % #1 macro, which gets fractional part 1 + % #2 macro, which gets fractional part 2 + % #3 natural number + \FP@rega=#3\relax\FP@regb=\FP@rega\advance\FP@regb1\relax% + \multiply\FP@rega1000\relax% + \FP@regc=\FP@rega\divide\FP@regc\FP@regb% + #1=\FP@regc% + \multiply\FP@regc\FP@regb\advance\FP@rega-\FP@regc% + \multiply\FP@rega1000\relax% + \FP@regc=\FP@rega\divide\FP@regc\FP@regb% + \multiply#11000\relax\advance#1\FP@regc% + \multiply\FP@regc\FP@regb\advance\FP@rega-\FP@regc% + \multiply\FP@rega1000\relax% + \FP@regc=\FP@rega\divide\FP@regc\FP@regb% + \multiply#11000\relax\advance#1\FP@regc% + \multiply\FP@regc\FP@regb\advance\FP@rega-\FP@regc% + \multiply\FP@rega1000\relax% + \FP@regc=\FP@rega\divide\FP@regc\FP@regb% + #2=\FP@regc% + \multiply\FP@regc\FP@regb\advance\FP@rega-\FP@regc% + \multiply\FP@rega1000\relax% + \FP@regc=\FP@rega\divide\FP@regc\FP@regb% + \multiply#21000\relax\advance#2\FP@regc% + \multiply\FP@regc\FP@regb\advance\FP@rega-\FP@regc% + \multiply\FP@rega1000\relax% + \FP@regc=\FP@rega\divide\FP@regc\FP@regb% + \multiply#21000\relax\advance#2\FP@regc% + \multiply\FP@regc\FP@regb\advance\FP@rega-\FP@regc% +} + +%if x<=0.707106781 then x:=y else x:=y with x^2 + y^2 = 1 +%sets \FP@@arccos to 0 if x:=y otherwise to 1 +\def\FP@sincos@transform@Rii{% + \ifnum\FP@xfa>707106781\relax% + \FP@@arccos=1\relax% + \FP@trigo@mul\FP@rfa\FP@rfb\FP@xfa\FP@xfb\FP@xfa\FP@xfb% + \FP@lfa=1000000000\relax% + \advance\FP@lfa-\FP@rfa% + \ifnum\FP@rfb>0\relax% + \advance\FP@lfa-1\relax% + \FP@lfb=1000000000\relax% + \advance\FP@lfb-\FP@rfb% + \else% + \FP@lfb=0\relax% + \fi% + \FP@xia=710000000\FP@xib0\relax% + \FP@xfa=0\FP@xfb=0\relax% + \FP@oldvala=0\FP@oldvalb=0\relax% + \loop% + \ifnum\ifnum\FP@xia=\FP@xfa0\else1\fi\ifnum\FP@xib=\FP@xfb0\else1\fi>0\relax% + \FP@rfa\FP@xia\advance\FP@rfa\FP@xfa% + \FP@rfb\FP@xib\advance\FP@rfb\FP@xfb% + \divide\FP@rfb2\relax% + \ifodd\FP@rfa\advance\FP@rfb500000000\relax\fi% + \divide\FP@rfa2\relax% + \ifnum\FP@rfb<1000000000\relax\else% + \advance\FP@rfb-1000000000\relax% + \advance\FP@rfa1\relax% + \fi% + \FP@trigo@mul\FP@ria\FP@rib\FP@rfa\FP@rfb\FP@rfa\FP@rfb% + \ifnum\FP@ria<\FP@lfa% + \FP@oct=1\relax% + \else\ifnum\FP@ria>\FP@lfa% + \FP@oct=2\relax% + \else\ifnum\FP@rib<\FP@lfb% + \FP@oct=1\relax% + \else\ifnum\FP@rib>\FP@lfb% + \FP@oct=2\relax% + \else% + \FP@oct=3\relax% + \fi\fi\fi\fi% + \ifnum\ifnum\FP@ria=\FP@oldvala0\else1\fi\ifnum\FP@rib=\FP@oldvalb0\else1\fi>0\relax% + \FP@oldvala=\FP@ria\FP@oldvalb=\FP@rib% + \else% + \FP@oct=3\relax% + \fi% + \ifodd\FP@oct% + \FP@xfa\FP@rfa\FP@xfb\FP@rfb% + \fi% + \ifnum\FP@oct>1\relax% + \FP@xia\FP@rfa\FP@xib\FP@rfb% + \fi% + \repeat% + \else% + \FP@@arccos=0\relax% + \fi% +} + +%compute arcsin(abs(x)) resp. arccos(abs(x)) for -1<=x<=1 +%\FP@@arccos determines whether arcsin or arccos has been computed +\def\FP@arcsincos@loop{% + \FP@@arccos=0\relax% + \ifnum\ifnum\FP@xia=0 1\else0\fi\ifnum\FP@xib<2 1\else 0\fi>0\relax% + \ifnum\FP@xib=1\relax% + \ifnum\ifnum\FP@xfa=0 1\else0\fi\ifnum\FP@xfb=0 1\else0\fi>10\relax% + \FP@rs=1\FP@ria=0\FP@rib=1\FP@rfa=570796326\FP@rfb=794896619\relax% + \else% + \FP@errmessage{Number too big}% + \fi% + \else% + \ifnum\ifnum\FP@xfa=0 0\else1\fi\ifnum\FP@xfb=0 0\else1\fi=0\relax% + \FP@rs=1\FP@ria=0\FP@rib=0\FP@rfa=0\FP@rfb=0\relax% + \else% + \FP@sincos@transform@Rii% + \FP@rs=1\FP@ria=0\FP@rib=0\FP@rfa=\FP@xfa\FP@rfb=\FP@xfb% + \FP@ls=1\FP@lfa=\FP@xfa\FP@lfb=\FP@xfb% + \FP@count=1\relax% + \loop% + \FP@trigo@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@xfa\FP@xfb% + \FP@trigo@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@xfa\FP@xfb% + \FP@trigo@xy\FP@yfa\FP@yfb\FP@count% + \FP@trigo@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@yfa\FP@yfb% + \advance\FP@count2\relax% + \FP@trigo@divn\FP@yfa\FP@yfb\FP@lfa\FP@lfb\FP@count% + \ifnum\ifnum\FP@yfa=0 0\else1\fi\ifnum\FP@yfb=0 0\else 1\fi>0\relax% + \advance\FP@rfa\FP@yfa\advance\FP@rfb\FP@yfb% + \ifnum\FP@rfb<1000000000\relax\else% + \advance\FP@rfb-1000000000\relax% + \advance\FP@rfa1\relax% + \fi% + \ifnum\FP@rfa<1000000000\relax\else% + \advance\FP@rfa-1000000000\relax% + \advance\FP@rib1\relax% + \fi% + \repeat% + \fi% + \fi% + \else% + \FP@errmessage{Number too big}% + \fi% +} + +%compute sign(r)*r:=#1.#2#3-abs(r) (for abs(r)<=#1.#2#3) +\def\FP@subrfrom#1#2#3{% + % #1 integer part of value to subtract from + % #2 fractional part 1 of value to subtract from + % #3 fractional part 2 of value to subtract from + \FP@ria=#1\relax\advance\FP@ria-\FP@rib\FP@rib\FP@ria% + \FP@ria=#2\relax\advance\FP@ria-\FP@rfa\FP@rfa\FP@ria% + \FP@ria=#3\relax\advance\FP@ria-\FP@rfb\FP@rfb\FP@ria% + \FP@ria=0\relax% + \ifnum\FP@rfb<0\relax% + \advance\FP@rfb1000000000\relax% + \advance\FP@rfa-1\relax% + \fi% + \ifnum\FP@rfa<0\relax% + \advance\FP@rfa1000000000\relax% + \advance\FP@rib-1\relax% + \fi% + \ifnum\FP@rib<0\relax% + \FP@errmessage{Subtraction lead to unaccepted value}% + \fi% +} + +%compute r:= pi/2 - abs(r) (for abs(r)<=pi/2) +\def\FP@pihalf@subr{\FP@subrfrom{1}{570796326}{794896619}} + +%compute sign(r)*r:= pi-abs(r) (for abs(r)<=pi) +\def\FP@pi@subr{\FP@subrfrom{3}{141592653}{589793238}} + +%compute arcsin + +\def\FP@arcsin#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 everything after the 2nd '.' + % + {\FP@beginmessage{ARCSIN}% + % + \FP@readvalue{x}{#2}{#3}% + % + \FP@arcsincos@loop% + % + \ifnum\FP@@arccos=1\relax% + \FP@pihalf@subr% + \fi% + \FP@rs=\FP@xs% + % + \FP@store\FP@tmp{r}% + % + \global\let\FP@tmp\FP@tmp% + % + \FP@endmessage{}% + }% + % + \let#1\FP@tmp% +} + +%compute arccos +\def\FP@arccos#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 everything after the 2nd '.' + % + {\FP@beginmessage{ARCCOS}% + % + \FP@readvalue{x}{#2}{#3}% + % + \FP@arcsincos@loop% + % + \ifnum\FP@@arccos=0\relax% + \FP@pihalf@subr% + \fi% + \FP@rs=1% + \ifnum\FP@xs<0\relax% + \FP@pi@subr% + \fi% + % + \FP@store\FP@tmp{r}% + % + \global\let\FP@tmp\FP@tmp% + % + \FP@endmessage{}% + }% + % + \let#1\FP@tmp% +} + +%compute arcsin and arccos +\def\FP@arcsincos#1#2#3.#4.#5\relax{% + % #1 macro, which gets the arcsin-result + % #2 macro, which gets the arccos-result + % #3 integer part of value + % #4 fractional part of value + % #5 dummy to swallow everything after the 2nd '.' + % + {\FP@beginmessage{ARCSINCOS}% + % + \FP@readvalue{x}{#3}{#4}% + % + \FP@arcsincos@loop% + % + \FP@xia\FP@ria\FP@xib\FP@rib\FP@xfa\FP@rfa\FP@xfb\FP@rfb% + % + \ifnum\FP@@arccos=1\relax% + \FP@pihalf@subr% + \fi% + \FP@rs=\FP@xs% + \FP@store\FP@tmpa{r}% + % + \FP@ria\FP@xia\FP@rib\FP@xib\FP@rfa\FP@xfa\FP@rfb\FP@xfb% + % + \ifnum\FP@@arccos=0\relax% + \FP@pihalf@subr% + \fi% + \FP@rs=1% + \ifnum\FP@xs<0\relax% + \FP@pi@subr% + \fi% + \FP@store\FP@tmpb{r}% + % + \global\let\FP@tmpa\FP@tmpa% + \global\let\FP@tmpb\FP@tmpb% + % + \FP@endmessage{}% + }% + % + \let#1\FP@tmpa% + \let#2\FP@tmpb% +} + +%compute arctan and arccot + +%compute 1/y for y>=1 +\def\FP@trigo@inv{% + \FP@shift=0\relax% + \FP@xia=1000000000\FP@xib=0\FP@xfa=0\FP@xfb=0\relax% + \loop% + \ifnum\FP@yia<100000000\relax% + \FP@multen{y}% + \advance\FP@shift1\relax% + \repeat% + \FP@ria=0\FP@rib=0\FP@rfa=0\FP@rfb=0\relax% + \loop% + \ifFP@zero{y}\else% + \FP@counttimes%divides x by \FP@times*y% + \FP@divten{y}% + \FP@multen{r}% + \advance\FP@rfb\FP@times% + \ifnum\FP@rfb<1000000000\relax\else% + \advance\FP@rfa1\advance\FP@rfb-1000000000\relax% + \ifnum\FP@rfa<1000000000\relax\else% + \advance\FP@rib1\advance\FP@rfa-1000000000\relax% + \ifnum\FP@rib<1000000000\relax\else% + \advance\FP@ria1\advance\FP@rib-1000000000\relax% + \fi% + \fi% + \fi% + \repeat% + \loop% + \ifnum\FP@shift<35\relax% + \advance\FP@shift1\relax% + \FP@divten{r}% + \repeat% +} + +%compute 1/sqrt(1+x^2) for 0<=x<1 +\def\FP@trigo@sq\FP@xfa\FP@xfb{% + \FP@trigo@mul\FP@yfa\FP@yfb\FP@xfa\FP@xfb\FP@xfa\FP@xfb% + \ifnum\FP@yfb=0\relax\else% + \FP@yfb=-\FP@yfb% + \advance\FP@yfb1000000000\relax% + \advance\FP@yfa1\relax% + \fi% + \FP@yfa=-\FP@yfa% + \advance\FP@yfa1250000000\relax% + \FP@trigo@mul\FP@yfa\FP@yfb\FP@yfa\FP@yfb{444444444}{444444444}% + \FP@lfa=666666666\relax\FP@lfb=666666666\relax% + \FP@rs=1\FP@ria=0\FP@rib=0\FP@rfa=\FP@lfa\FP@rfb=\FP@lfb% + \FP@count=0\relax% + \loop% + \advance\FP@count2\relax% + \FP@trigo@divn\FP@yia\FP@yib{1000000000}{0}\FP@count% + \ifnum\ifnum\FP@yia=0 0\else1\fi\ifnum\FP@yib=0 0\else1\fi>0\relax% + \ifnum\FP@yib=0\relax\else% + \FP@yib=-\FP@yib% + \advance\FP@yib1000000000\relax% + \advance\FP@yia1\relax% + \fi% + \ifnum\FP@yia=0\relax\else% + \FP@yia=-\FP@yia% + \advance\FP@yia1000000000\relax% + \fi% + \FP@trigo@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@yia\FP@yib% + \fi% + \FP@trigo@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@yfa\FP@yfb% + \ifnum\ifnum\FP@lfa=0 0\else1\fi\ifnum\FP@lfb=0 0\else1\fi=0\relax\else% + \advance\FP@rfb\FP@lfb% + \ifnum\FP@rfb<1000000000\relax\else% + \advance\FP@rfa1\relax% + \advance\FP@rfb-1000000000\relax% + \fi% + \advance\FP@rfa\FP@lfa% + \ifnum\FP@rfa<1000000000\relax\else% + \advance\FP@rib1\relax% + \advance\FP@rfa-1000000000\relax% + \fi% + \repeat% +} + +%compute arctan for 0<=x<=0.5 +\def\FP@@arctan{% + \ifnum\ifnum\FP@xfa=0 0\else1\fi\ifnum\FP@xfb=0 0\else1\fi=0\relax% + \FP@rs=1\FP@ria=0\FP@rib=0\FP@rfa=0\FP@rfb=0\relax% + \else% + \FP@rs=1\FP@ria=0\FP@rib=0\FP@rfa=\FP@xfa\FP@rfb=\FP@xfb% + \FP@lfa=\FP@xfa\FP@lfb=\FP@xfb% + \FP@trigo@mul\FP@xfa\FP@xfb\FP@xfa\FP@xfb\FP@xfa\FP@xfb% + \FP@count=1\relax% + \FP@regs=1\relax% + \loop% + \FP@regs=-\FP@regs% + \advance\FP@count2\relax% + \FP@trigo@mul\FP@lfa\FP@lfb\FP@lfa\FP@lfb\FP@xfa\FP@xfb% + \FP@trigo@divn\FP@yfa\FP@yfb\FP@lfa\FP@lfb\FP@count% + \ifnum\ifnum\FP@yfa=0 0\else1\fi\ifnum\FP@yfb=0 0\else1\fi>0\relax% + \ifnum\FP@regs<0\relax% + \advance\FP@rfb-\FP@yfb% + \ifnum\FP@rfb<0\relax% + \advance\FP@rfb1000000000\relax% + \advance\FP@rfa-1\relax% + \fi% + \advance\FP@rfa-\FP@yfa% + \ifnum\FP@rfa<0\relax% + \advance\FP@rfa1000000000\relax% + \advance\FP@rib-1\relax% + \fi% + \else% + \advance\FP@rfb\FP@yfb% + \ifnum\FP@rfb<1000000000\relax\else% + \advance\FP@rfb-1000000000\relax% + \advance\FP@rfa1\relax% + \fi% + \advance\FP@rfa\FP@yfa% + \ifnum\FP@rfa<1000000000\relax\else% + \advance\FP@rfa-1000000000\relax% + \advance\FP@rib1\relax% + \fi% + \fi% + \repeat% + \fi% +} + +%compute arctan +\def\FP@@arctancot#1#2{% + % #1 integer part of value + % #2 fractional part of value + % + \FP@readvalue{x}{#1}{#2}% + % + \ifnum\ifnum\FP@xia=0 0\else1\fi\ifnum\FP@xib=1 0\else1\fi\ifnum\FP@xfa=0 0\else1\fi\ifnum\FP@xfb=0 0\else1\fi=0\relax% + \FP@ria=0\FP@rib=0\FP@rfa=785398163\FP@rfb=397448310\relax% + \FP@subfpih=0\relax% + \else% + \ifnum\ifnum\FP@xia=0 0\else1\fi\ifnum\FP@xib=0 0\else1\fi=0\relax% + \FP@subfpih=0\relax% + \else% + \FP@yia=\FP@xia\FP@yib=\FP@xib\FP@yfa=\FP@xfa\FP@yfb=\FP@xfb% + \FP@trigo@inv% + \FP@xia=\FP@ria\FP@xib=\FP@rib\FP@xfa=\FP@rfa\FP@xfb=\FP@rfb% + \FP@subfpih=1\relax% + \fi% + \ifnum\FP@xfa>880000000\relax% + \FP@trigo@sq\FP@xfa\FP@xfb% + \FP@xfa=\FP@rfa\FP@xfb=\FP@rfb% + \FP@arcsincos@loop% + \ifnum\FP@@arccos=0\relax% + \FP@pihalf@subr% + \fi% + \else% + \FP@@arctan\FP@xfa\FP@xfb% + \fi% + \fi% + % + \ifnum\FP@subfpih>0\relax% + \FP@pihalf@subr% + \fi% + % + \FP@rs=\FP@xs% +} + +%compute arctan +\def\FP@arctan#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 everything after the 2nd '.' + % + {\FP@beginmessage{ARCTAN}% + % + \FP@@arctancot{#2}{#3}% + % + \FP@store\FP@tmp{r}% + % + \global\let\FP@tmp\FP@tmp% + % + \FP@endmessage{}% + }% + % + \let#1\FP@tmp% +} + +%compute arccot +\def\FP@arccot#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 everything after the 2nd '.' + % + {\FP@beginmessage{ARCCOT}% + % + \FP@@arctancot{#2}{#3}% + \ifnum\FP@rs<0\relax% + \FP@rs=1\relax% + \FP@pihalf@subr% + \FP@pi@subr% + \else% + \FP@pihalf@subr% + \fi% + % + \FP@store\FP@tmp{r}% + % + \global\let\FP@tmp\FP@tmp% + % + \FP@endmessage{}% + }% + % + \let#1\FP@tmp% +} + +%compute arctan and arccot +\def\FP@arctancot#1#2#3.#4.#5\relax{% + % #1 macro, which gets the arcsin-result + % #2 macro, which gets the arccos-result + % #3 integer part of value + % #4 fractional part of value + % #5 dummy to swallow everything after the 2nd '.' + % + {\FP@beginmessage{ARCTANCOT}% + % + \FP@@arctancot{#3}{#4}% + \FP@xs\FP@rs\FP@xia\FP@ria\FP@xib\FP@rib\FP@xfa\FP@rfa\FP@xfb\FP@rfb% + \FP@store\FP@tmpa{x}% + % + \ifnum\FP@rs<0\relax% + \FP@rs=1\relax% + \FP@pihalf@subr% + \FP@pi@subr% + \else% + \FP@pihalf@subr% + \fi% + \FP@store\FP@tmpb{r}% + % + \global\let\FP@tmpa\FP@tmpa% + \global\let\FP@tmpb\FP@tmpb% + % + \FP@endmessage{}% + }% + % + \let#1\FP@tmpa% + \let#2\FP@tmpb% +} |