\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% }