replaced mathematical routines by our own
This commit is contained in:
ceriel
parent
d443f370d2
commit
324c95ae62
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@ -1,74 +1,102 @@
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/*
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* (c) copyright 1988 by the Vrije Universiteit, Amsterdam, The Netherlands.
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* See the copyright notice in the ACK home directory, in the file "Copyright".
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*
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* Author: Ceriel J.H. Jacobs
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*/
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/* $Header$ */
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/*
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floating-point arctangent
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atan returns the value of the arctangent of its
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argument in the range [-pi/2,pi/2].
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there are no error returns.
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coefficients are #5077 from Hart & Cheney. (19.56D)
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*/
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static double sq2p1 = 2.414213562373095048802e0;
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static double sq2m1 = .414213562373095048802e0;
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static double pio2 = 1.570796326794896619231e0;
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static double pio4 = .785398163397448309615e0;
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static double p4 = .161536412982230228262e2;
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static double p3 = .26842548195503973794141e3;
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static double p2 = .11530293515404850115428136e4;
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static double p1 = .178040631643319697105464587e4;
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static double p0 = .89678597403663861959987488e3;
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static double q4 = .5895697050844462222791e2;
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static double q3 = .536265374031215315104235e3;
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static double q2 = .16667838148816337184521798e4;
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static double q1 = .207933497444540981287275926e4;
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static double q0 = .89678597403663861962481162e3;
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/*
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xatan evaluates a series valid in the
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range [-0.414...,+0.414...].
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*/
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static double
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xatan(arg)
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double arg;
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{
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double argsq;
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double value;
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argsq = arg*arg;
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value = ((((p4*argsq + p3)*argsq + p2)*argsq + p1)*argsq + p0);
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value = value/(((((argsq + q4)*argsq + q3)*argsq + q2)*argsq + q1)*argsq + q0);
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return(value*arg);
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}
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static double
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satan(arg)
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double arg;
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{
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if(arg < sq2m1)
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return(xatan(arg));
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else if(arg > sq2p1)
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return(pio2 - xatan(1/arg));
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else
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return(pio4 + xatan((arg-1)/(arg+1)));
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}
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/*
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atan makes its argument positive and
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calls the inner routine satan.
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*/
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#include <math.h>
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double
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_atn(arg)
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double arg;
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_atn(x)
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double x;
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{
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if(arg>0)
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return(satan(arg));
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else
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return(-satan(-arg));
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/* The interval [0, infinity) is treated as follows:
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Define partition points Xi
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X0 = 0
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X1 = tan(pi/16)
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X2 = tan(3pi/16)
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X3 = tan(5pi/16)
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X4 = tan(7pi/16)
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X5 = infinity
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and evaluation nodes xi
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x2 = tan(2pi/16)
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x3 = tan(4pi/16)
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x4 = tan(6pi/16)
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x5 = infinity
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An argument x in [Xn-1, Xn] is now reduced to an argument
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t in [-X1, X1] by the following formulas:
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t = 1/xn - (1/(xn*xn) + 1)/((1/xn) + x)
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arctan(x) = arctan(xi) + arctan(t)
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For the interval [0, p/16] an approximation is used:
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arctan(x) = x * P(x*x)/Q(x*x)
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*/
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static struct precomputed {
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double X; /* partition point */
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double arctan; /* arctan of evaluation node */
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double one_o_x; /* 1 / xn */
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double one_o_xsq_p_1; /* 1 / (xn*xn) + 1 */
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} prec[5] = {
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{ 0.19891236737965800691159762264467622,
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0.0,
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0.0, /* these don't matter */
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0.0 } ,
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{ 0.66817863791929891999775768652308076, /* tan(3pi/16) */
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M_PI_8,
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2.41421356237309504880168872420969808,
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6.82842712474619009760337744841939616 },
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{ 1.49660576266548901760113513494247691, /* tan(5pi/16) */
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M_PI_4,
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1.0,
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2.0 },
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{ 5.02733949212584810451497507106407238, /* tan(7pi/16) */
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M_3PI_8,
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0.41421356237309504880168872420969808,
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1.17157287525380998659662255158060384 },
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{ MAXDOUBLE,
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M_PI_2,
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0.0,
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1.0 }};
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/* Hart & Cheney # 5037 */
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static double p[5] = {
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0.7698297257888171026986294745e+03,
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0.1557282793158363491416585283e+04,
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0.1033384651675161628243434662e+04,
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0.2485841954911840502660889866e+03,
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0.1566564964979791769948970100e+02
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};
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static double q[6] = {
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0.7698297257888171026986294911e+03,
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0.1813892701754635858982709369e+04,
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0.1484049607102276827437401170e+04,
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0.4904645326203706217748848797e+03,
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0.5593479839280348664778328000e+02,
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0.1000000000000000000000000000e+01
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};
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int negative = x < 0.0;
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register struct precomputed *pr = prec;
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if (negative) {
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x = -x;
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}
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while (x > pr->X) pr++;
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if (pr != prec) {
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x = pr->arctan +
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atan(pr->one_o_x - pr->one_o_xsq_p_1/(pr->one_o_x + x));
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}
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else {
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double xsq = x*x;
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x = x * POLYNOM4(xsq, p)/POLYNOM5(xsq, q);
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}
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return negative ? -x : x;
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}
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@ -1,106 +1,112 @@
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/*
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* (c) copyright 1988 by the Vrije Universiteit, Amsterdam, The Netherlands.
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* See the copyright notice in the ACK home directory, in the file "Copyright".
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*
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* Author: Ceriel J.H. Jacobs
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*/
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/* $Header$ */
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#include <pc_err.h>
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extern double _fif();
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extern double _fef();
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extern _trp();
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/*
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exp returns the exponential function of its
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floating-point argument.
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The coefficients are #1069 from Hart and Cheney. (22.35D)
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*/
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#define HUGE 1.701411733192644270e38
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static double p0 = .2080384346694663001443843411e7;
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static double p1 = .3028697169744036299076048876e5;
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static double p2 = .6061485330061080841615584556e2;
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static double q0 = .6002720360238832528230907598e7;
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static double q1 = .3277251518082914423057964422e6;
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static double q2 = .1749287689093076403844945335e4;
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static double log2e = 1.4426950408889634073599247;
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static double sqrt2 = 1.4142135623730950488016887;
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static double maxf = 10000.0;
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#include <math.h>
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#include <pc_err.h>
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extern _trp();
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static double
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floor(d)
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double d;
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floor(x)
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double x;
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{
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if (d<0) {
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d = -d;
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if (_fif(d, 1.0, &d) != 0)
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d += 1;
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d = -d;
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} else
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_fif(d, 1.0, &d);
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return(d);
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extern double _fif();
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double val;
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return _fif(x, 1,0, &val) < 0 ? val - 1.0 : val ;
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/* this also works if _fif always returns a positive
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fractional part
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*/
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}
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static double
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ldexp(fr,exp)
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double fr;
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int exp;
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ldexp(fl,exp)
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double fl;
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int exp;
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{
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int neg,i;
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extern double _fef();
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int sign = 1;
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int currexp;
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neg = 1;
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if (fr < 0) {
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fr = -fr;
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neg = -1;
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if (fl<0) {
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fl = -fl;
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sign = -1;
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}
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fr = _fef(fr, &i);
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/*
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while (fr < 0.5) {
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fr *= 2;
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exp--;
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fl = _fef(fl,&currexp);
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exp += currexp;
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if (exp > 0) {
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while (exp>30) {
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fl *= (double) (1L << 30);
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exp -= 30;
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}
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fl *= (double) (1L << exp);
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}
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*/
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exp += i;
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if (exp > 127) {
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_trp(EEXP);
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return(neg * HUGE);
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else {
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while (exp<-30) {
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fl /= (double) (1L << 30);
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exp += 30;
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}
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fl /= (double) (1L << -exp);
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}
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if (exp < -127)
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return(0);
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while (exp > 14) {
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fr *= (1<<14);
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exp -= 14;
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}
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while (exp < -14) {
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fr /= (1<<14);
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exp += 14;
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}
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if (exp > 0)
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fr *= (1<<exp);
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if (exp < 0)
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fr /= (1<<(-exp));
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return(neg * fr);
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return sign * fl;
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}
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double
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_exp(arg)
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double arg;
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_exp(x)
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double x;
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{
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double fract;
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double temp1, temp2, xsq;
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int ent;
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/* 2**x = (Q(x*x)+x*P(x*x))/(Q(x*x)-x*P(x*x)) for x in [0,0.5] */
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/* Hart & Cheney #1069 */
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if(arg == 0)
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return(1);
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if(arg < -maxf)
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return(0);
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if(arg > maxf) {
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_trp(EEXP);
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return(HUGE);
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static double p[3] = {
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0.2080384346694663001443843411e+07,
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0.3028697169744036299076048876e+05,
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0.6061485330061080841615584556e+02
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};
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static double q[4] = {
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0.6002720360238832528230907598e+07,
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0.3277251518082914423057964422e+06,
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0.1749287689093076403844945335e+04,
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0.1000000000000000000000000000e+01
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};
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int negative = x < 0;
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int ipart, large = 0;
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double xsqr, xPxx, Qxx;
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if (x < M_LN_MIN_D) {
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return M_MIN_D;
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}
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arg *= log2e;
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ent = floor(arg);
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fract = (arg-ent) - 0.5;
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xsq = fract*fract;
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temp1 = ((p2*xsq+p1)*xsq+p0)*fract;
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temp2 = ((xsq+q2)*xsq+q1)*xsq + q0;
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return(ldexp(sqrt2*(temp2+temp1)/(temp2-temp1), ent));
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if (x >= M_LN_MAX_D) {
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if (x > M_LN_MAX_D) {
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_trp(EEXP);
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return HUGE;
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}
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return M_MAX_D;
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}
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if (negative) {
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x = -x;
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}
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x /= M_LN2;
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ipart = floor(x);
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x -= ipart;
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if (x > 0.5) {
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large = 1;
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x -= 0.5;
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}
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xsqr = x * x;
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xPxx = x * POLYNOM2(xsqr, p);
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Qxx = POLYNOM3(xsqr, q);
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x = (Qxx + xPxx) / (Qxx - xPxx);
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if (large) x *= M_SQRT2;
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x = ldexp(x, ipart);
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if (negative) return 1.0/x;
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return x;
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}
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@ -1,59 +1,55 @@
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/*
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* (c) copyright 1988 by the Vrije Universiteit, Amsterdam, The Netherlands.
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* See the copyright notice in the ACK home directory, in the file "Copyright".
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*
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* Author: Ceriel J.H. Jacobs
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*/
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/* $Header$ */
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#include <pc_err.h>
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extern double _fef();
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extern _trp();
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/*
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log returns the natural logarithm of its floating
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point argument.
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The coefficients are #2705 from Hart & Cheney. (19.38D)
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It calls _fef.
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*/
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#define HUGE 1.701411733192644270e38
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static double log2 = 0.693147180559945309e0;
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static double sqrto2 = 0.707106781186547524e0;
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static double p0 = -.240139179559210510e2;
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static double p1 = 0.309572928215376501e2;
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static double p2 = -.963769093368686593e1;
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static double p3 = 0.421087371217979714e0;
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static double q0 = -.120069589779605255e2;
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static double q1 = 0.194809660700889731e2;
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static double q2 = -.891110902798312337e1;
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#include <math.h>
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#include <pc_err.h>
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extern _trp();
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double
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_log(arg)
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double arg;
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_log(x)
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double x;
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{
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double x,z, zsq, temp;
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int exp;
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if(arg <= 0) {
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_trp(ELOG);
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return(-HUGE);
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}
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x = _fef(arg,&exp);
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/*
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while(x < 0.5) {
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x =* 2;
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exp--;
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}
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/* log(x) = z*P(z*z)/Q(z*z), z = (x-1)/(x+1), x in [1/sqrt(2), sqrt(2)]
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*/
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if(x<sqrto2) {
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x *= 2;
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exp--;
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/* Hart & Cheney #2707 */
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static double p[5] = {
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0.7504094990777122217455611007e+02,
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-0.1345669115050430235318253537e+03,
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0.7413719213248602512779336470e+02,
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-0.1277249755012330819984385000e+02,
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0.3327108381087686938144000000e+00
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};
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static double q[5] = {
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0.3752047495388561108727775374e+02,
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-0.7979028073715004879439951583e+02,
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0.5616126132118257292058560360e+02,
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-0.1450868091858082685362325000e+02,
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0.1000000000000000000000000000e+01
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};
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extern double _fef();
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double z, zsqr;
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int exponent;
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if (x <= 0) {
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_trp(ELOG);
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return -HUGE;
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}
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x = _fef(x, &exponent);
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while (x < M_1_SQRT2) {
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x += x;
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exponent--;
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}
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z = (x-1)/(x+1);
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zsq = z*z;
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temp = ((p3*zsq + p2)*zsq + p1)*zsq + p0;
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temp = temp/(((zsq + q2)*zsq + q1)*zsq + q0);
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temp = temp*z + exp*log2;
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return(temp);
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zsqr = z*z;
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return z * POLYNOM4(zsqr, p) / POLYNOM4(zsqr, q) + exponent * M_LN2;
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}
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@ -1,75 +1,112 @@
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/*
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* (c) copyright 1988 by the Vrije Universiteit, Amsterdam, The Netherlands.
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* See the copyright notice in the ACK home directory, in the file "Copyright".
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*
|
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* Author: Ceriel J.H. Jacobs
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*/
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/* $Header$ */
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extern double _fif();
|
||||
|
||||
/*
|
||||
C program for floating point sin/cos.
|
||||
Calls _fif.
|
||||
There are no error exits.
|
||||
Coefficients are #3370 from Hart & Cheney (18.80D).
|
||||
*/
|
||||
|
||||
static double twoopi = 0.63661977236758134308;
|
||||
static double p0 = .1357884097877375669092680e8;
|
||||
static double p1 = -.4942908100902844161158627e7;
|
||||
static double p2 = .4401030535375266501944918e6;
|
||||
static double p3 = -.1384727249982452873054457e5;
|
||||
static double p4 = .1459688406665768722226959e3;
|
||||
static double q0 = .8644558652922534429915149e7;
|
||||
static double q1 = .4081792252343299749395779e6;
|
||||
static double q2 = .9463096101538208180571257e4;
|
||||
static double q3 = .1326534908786136358911494e3;
|
||||
#include <math.h>
|
||||
|
||||
static double
|
||||
sinus(arg, quad)
|
||||
double arg;
|
||||
int quad;
|
||||
sinus(x, quadrant)
|
||||
double x;
|
||||
{
|
||||
double e, f;
|
||||
double ysq;
|
||||
double x,y;
|
||||
int k;
|
||||
double temp1, temp2;
|
||||
/* sin(0.5*pi*x) = x * P(x*x)/Q(x*x) for x in [0,1] */
|
||||
/* Hart & Cheney # 3374 */
|
||||
|
||||
x = arg;
|
||||
if(x<0) {
|
||||
static double p[6] = {
|
||||
0.4857791909822798473837058825e+10,
|
||||
-0.1808816670894030772075877725e+10,
|
||||
0.1724314784722489597789244188e+09,
|
||||
-0.6351331748520454245913645971e+07,
|
||||
0.1002087631419532326179108883e+06,
|
||||
-0.5830988897678192576148973679e+03
|
||||
};
|
||||
|
||||
static double q[6] = {
|
||||
0.3092566379840468199410228418e+10,
|
||||
0.1202384907680254190870913060e+09,
|
||||
0.2321427631602460953669856368e+07,
|
||||
0.2848331644063908832127222835e+05,
|
||||
0.2287602116741682420054505174e+03,
|
||||
0.1000000000000000000000000000e+01
|
||||
};
|
||||
|
||||
double xsqr;
|
||||
int t;
|
||||
|
||||
if (x < 0) {
|
||||
quadrant += 2;
|
||||
x = -x;
|
||||
quad = quad + 2;
|
||||
}
|
||||
x = x*twoopi; /*underflow?*/
|
||||
if(x>32764){
|
||||
y = _fif(x, 10.0, &e);
|
||||
e = e + quad;
|
||||
_fif(0.25, e, &f);
|
||||
quad = e - 4*f;
|
||||
}else{
|
||||
k = x;
|
||||
y = x - k;
|
||||
quad = (quad + k) & 03;
|
||||
if (M_PI_2 - x == M_PI_2) {
|
||||
switch(quadrant) {
|
||||
case 0:
|
||||
case 2:
|
||||
return 0.0;
|
||||
case 1:
|
||||
return 1.0;
|
||||
case 3:
|
||||
return -1.0;
|
||||
}
|
||||
}
|
||||
if (quad & 01)
|
||||
y = 1-y;
|
||||
if(quad > 1)
|
||||
y = -y;
|
||||
if (x >= M_2PI) {
|
||||
if (x <= 0x7fffffff) {
|
||||
/* Use extended precision to calculate reduced argument.
|
||||
Split 2pi in 2 parts a1 and a2, of which the first only
|
||||
uses some bits of the mantissa, so that n * a1 is
|
||||
exactly representable, where n is the integer part of
|
||||
x/pi.
|
||||
Here we used 12 bits of the mantissa for a1.
|
||||
Also split x in integer part x1 and fraction part x2.
|
||||
We then compute x-n*2pi as ((x1 - n*a1) + x2) - n*a2.
|
||||
*/
|
||||
#define A1 6.2822265625
|
||||
#define A2 0.00095874467958647692528676655900576
|
||||
double n = (long) (x / M_2PI);
|
||||
double x1 = (long) x;
|
||||
double x2 = x - x1;
|
||||
x = x1 - n * A1;
|
||||
x += x2;
|
||||
x -= n * A2;
|
||||
#undef A1
|
||||
#undef A2
|
||||
}
|
||||
else {
|
||||
extern double _fif();
|
||||
double dummy;
|
||||
|
||||
ysq = y*y;
|
||||
temp1 = ((((p4*ysq+p3)*ysq+p2)*ysq+p1)*ysq+p0)*y;
|
||||
temp2 = ((((ysq+q3)*ysq+q2)*ysq+q1)*ysq+q0);
|
||||
return(temp1/temp2);
|
||||
x = _fif(x/M_2PI, 1.0, &dummy) * M_2PI;
|
||||
}
|
||||
}
|
||||
x /= M_PI_2;
|
||||
t = x;
|
||||
x -= t;
|
||||
quadrant = (quadrant + (int)(t % 4)) % 4;
|
||||
if (quadrant & 01) {
|
||||
x = 1 - x;
|
||||
}
|
||||
if (quadrant > 1) {
|
||||
x = -x;
|
||||
}
|
||||
xsqr = x * x;
|
||||
x = x * POLYNOM5(xsqr, p) / POLYNOM5(xsqr, q);
|
||||
return x;
|
||||
}
|
||||
|
||||
double
|
||||
_cos(arg)
|
||||
double arg;
|
||||
_sin(x)
|
||||
double x;
|
||||
{
|
||||
if(arg<0)
|
||||
arg = -arg;
|
||||
return(sinus(arg, 1));
|
||||
return sinus(x, 0);
|
||||
}
|
||||
|
||||
double
|
||||
_sin(arg)
|
||||
double arg;
|
||||
_cos(x)
|
||||
double x;
|
||||
{
|
||||
return(sinus(arg, 0));
|
||||
if (x < 0) x = -x;
|
||||
return sinus(x, 1);
|
||||
}
|
||||
|
|
|
|||
|
|
@ -1,60 +1,72 @@
|
|||
/*
|
||||
* (c) copyright 1988 by the Vrije Universiteit, Amsterdam, The Netherlands.
|
||||
* See the copyright notice in the ACK home directory, in the file "Copyright".
|
||||
*
|
||||
* Author: Ceriel J.H. Jacobs
|
||||
*/
|
||||
|
||||
/* $Header$ */
|
||||
|
||||
#include <pc_err.h>
|
||||
#include <math.h>
|
||||
#include <pc_err.h>
|
||||
extern _trp();
|
||||
|
||||
extern double _fef();
|
||||
extern _trp();
|
||||
#define NITER 5
|
||||
|
||||
/*
|
||||
sqrt returns the square root of its floating
|
||||
point argument. Newton's method.
|
||||
static double
|
||||
ldexp(fl,exp)
|
||||
double fl;
|
||||
int exp;
|
||||
{
|
||||
extern double _fef();
|
||||
int sign = 1;
|
||||
int currexp;
|
||||
|
||||
calls _fef
|
||||
*/
|
||||
if (fl<0) {
|
||||
fl = -fl;
|
||||
sign = -1;
|
||||
}
|
||||
fl = _fef(fl,&currexp);
|
||||
exp += currexp;
|
||||
if (exp > 0) {
|
||||
while (exp>30) {
|
||||
fl *= (double) (1L << 30);
|
||||
exp -= 30;
|
||||
}
|
||||
fl *= (double) (1L << exp);
|
||||
}
|
||||
else {
|
||||
while (exp<-30) {
|
||||
fl /= (double) (1L << 30);
|
||||
exp += 30;
|
||||
}
|
||||
fl /= (double) (1L << -exp);
|
||||
}
|
||||
return sign * fl;
|
||||
}
|
||||
|
||||
double
|
||||
_sqt(arg)
|
||||
double arg;
|
||||
_sqt(x)
|
||||
double x;
|
||||
{
|
||||
double x, temp;
|
||||
int exp;
|
||||
int i;
|
||||
extern double _fef();
|
||||
int exponent;
|
||||
double val;
|
||||
|
||||
if(arg <= 0) {
|
||||
if(arg < 0)
|
||||
_trp(ESQT);
|
||||
return(0);
|
||||
if (x <= 0) {
|
||||
if (x < 0) _trp(ESQT);
|
||||
return 0;
|
||||
}
|
||||
x = _fef(arg,&exp);
|
||||
/*
|
||||
while(x < 0.5) {
|
||||
x =* 2;
|
||||
exp--;
|
||||
}
|
||||
*/
|
||||
/*
|
||||
* NOTE
|
||||
* this wont work on 1's comp
|
||||
*/
|
||||
if(exp & 1) {
|
||||
x *= 2;
|
||||
exp--;
|
||||
}
|
||||
temp = 0.5*(1 + x);
|
||||
|
||||
while(exp > 28) {
|
||||
temp *= (1<<14);
|
||||
exp -= 28;
|
||||
val = _fef(x, &exponent);
|
||||
if (exponent & 1) {
|
||||
exponent--;
|
||||
val *= 2;
|
||||
}
|
||||
while(exp < -28) {
|
||||
temp /= (1<<14);
|
||||
exp += 28;
|
||||
val = ldexp(val + 1.0, exponent/2 - 1);
|
||||
/* was: val = (val + 1.0)/2.0; val = ldexp(val, exponent/2); */
|
||||
for (exponent = NITER - 1; exponent >= 0; exponent--) {
|
||||
val = (val + x / val) / 2.0;
|
||||
}
|
||||
if(exp >= 0)
|
||||
temp *= 1 << (exp/2);
|
||||
else
|
||||
temp /= 1 << (-exp/2);
|
||||
for(i=0; i<=4; i++)
|
||||
temp = 0.5*(temp + arg/temp);
|
||||
return(temp);
|
||||
return val;
|
||||
}
|
||||
|
|
|
|||
Loading…
Reference in a new issue