use new math algorithms
This commit is contained in:
ceriel
parent
8b702734cf
commit
761312d0dd
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@ -6,6 +6,7 @@
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*/
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/* $Header$ */
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#define __NO_DEFS
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#include <math.h>
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@ -13,90 +14,55 @@ double
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_atn(x)
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double x;
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{
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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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/* Algorithm and coefficients from:
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"Software manual for the elementary functions"
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by W.J. Cody and W. Waite, Prentice-Hall, 1980
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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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static double p[] = {
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-0.13688768894191926929e+2,
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-0.20505855195861651981e+2,
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-0.84946240351320683534e+1,
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-0.83758299368150059274e+0
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};
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static double q[] = {
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0.41066306682575781263e+2,
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0.86157349597130242515e+2,
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0.59578436142597344465e+2,
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0.15024001160028576121e+2,
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1.0
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};
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static double a[] = {
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0.0,
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0.52359877559829887307710723554658381, /* pi/6 */
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M_PI_2,
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1.04719755119659774615421446109316763 /* pi/3 */
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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 neg = x < 0;
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int n;
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double g;
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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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if (neg) {
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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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_atn(pr->one_o_x - pr->one_o_xsq_p_1/(pr->one_o_x + x));
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if (x > 1.0) {
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x = 1.0/x;
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n = 2;
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}
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else {
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double xsq = x*x;
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else n = 0;
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x = x * POLYNOM4(xsq, p)/POLYNOM5(xsq, q);
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if (x > 0.26794919243112270647) { /* 2-sqtr(3) */
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n = n + 1;
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x = (((0.73205080756887729353*x-0.5)-0.5)+x)/
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(1.73205080756887729353+x);
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}
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return negative ? -x : x;
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/* ??? avoid underflow ??? */
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g = x * x;
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x += x * g * POLYNOM3(g, p) / POLYNOM4(g, q);
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if (n > 1) x = -x;
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x += a[n];
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return neg ? -x : x;
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}
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@ -11,76 +11,32 @@
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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(x)
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double x;
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{
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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(fl,exp)
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double fl;
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int exp;
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{
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extern double _fef();
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int sign = 1;
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int currexp;
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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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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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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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return sign * fl;
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}
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double
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_exp(x)
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double x;
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{
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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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/* Algorithm and coefficients from:
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"Software manual for the elementary functions"
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by W.J. Cody and W. Waite, Prentice-Hall, 1980
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*/
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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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static double p[] = {
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0.25000000000000000000e+0,
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0.75753180159422776666e-2,
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0.31555192765684646356e-4
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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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static double q[] = {
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0.50000000000000000000e+0,
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0.56817302698551221787e-1,
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0.63121894374398503557e-3,
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0.75104028399870046114e-6
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};
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double xn, g;
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int n;
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int negative = x < 0;
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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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if (x <= M_LN_MIN_D) {
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return M_MIN_D;
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}
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if (x >= M_LN_MAX_D) {
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@ -90,23 +46,24 @@ _exp(x)
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}
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return M_MAX_D;
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}
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if (negative) x = -x;
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/* ??? avoid underflow ??? */
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n = x * M_LOG2E + 0.5; /* 1/ln(2) = log2(e), 0.5 added for rounding */
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xn = n;
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{
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double x1 = (long) x;
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double x2 = x - x1;
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g = ((x1-xn*0.693359375)+x2) - xn*(-2.1219444005469058277e-4);
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}
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if (negative) {
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x = -x;
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g = -g;
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n = -n;
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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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xn = g * g;
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x = g * POLYNOM2(xn, p);
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n += 1;
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return (ldexp(0.5 + x/(POLYNOM3(xn, q) - x), n));
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}
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@ -6,38 +6,34 @@
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*/
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/* $Header$ */
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#define __NO_DEFS
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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(x)
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double x;
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double x;
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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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/* Algorithm and coefficients from:
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"Software manual for the elementary functions"
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by W.J. Cody and W. Waite, Prentice-Hall, 1980
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*/
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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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static double a[] = {
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-0.64124943423745581147e2,
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0.16383943563021534222e2,
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-0.78956112887491257267e0
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};
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static double b[] = {
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-0.76949932108494879777e3,
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0.31203222091924532844e3,
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-0.35667977739034646171e2,
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1.0
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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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extern double _fef();
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double znum, zden, z, w;
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int exponent;
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if (x <= 0) {
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_trp(ELOG);
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@ -45,11 +41,18 @@ _log(x)
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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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if (x > M_1_SQRT2) {
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znum = (x - 0.5) - 0.5;
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zden = x * 0.5 + 0.5;
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}
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else {
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znum = x - 0.5;
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zden = znum * 0.5 + 0.5;
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exponent--;
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}
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z = (x-1)/(x+1);
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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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z = znum/zden; w = z * z;
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x = z + z * w * (POLYNOM2(w,a)/POLYNOM3(w,b));
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z = exponent;
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x += z * (-2.121944400546905827679e-4);
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return x + z * 0.693359375;
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}
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@ -11,90 +11,74 @@
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#include <math.h>
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static double
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sinus(x, quadrant)
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sinus(x, cos_flag)
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double x;
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{
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/* sin(0.5*pi*x) = x * P(x*x)/Q(x*x) for x in [0,1] */
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/* Hart & Cheney # 3374 */
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/* Algorithm and coefficients from:
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"Software manual for the elementary functions"
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by W.J. Cody and W. Waite, Prentice-Hall, 1980
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*/
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static double p[6] = {
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0.4857791909822798473837058825e+10,
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-0.1808816670894030772075877725e+10,
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0.1724314784722489597789244188e+09,
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-0.6351331748520454245913645971e+07,
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0.1002087631419532326179108883e+06,
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-0.5830988897678192576148973679e+03
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static double r[] = {
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-0.16666666666666665052e+0,
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0.83333333333331650314e-2,
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-0.19841269841201840457e-3,
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0.27557319210152756119e-5,
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-0.25052106798274584544e-7,
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0.16058936490371589114e-9,
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-0.76429178068910467734e-12,
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0.27204790957888846175e-14
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};
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static double q[6] = {
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0.3092566379840468199410228418e+10,
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0.1202384907680254190870913060e+09,
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0.2321427631602460953669856368e+07,
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0.2848331644063908832127222835e+05,
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0.2287602116741682420054505174e+03,
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0.1000000000000000000000000000e+01
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};
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double xsqr;
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int t;
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double xsqr;
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double y;
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int neg = 0;
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if (x < 0) {
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quadrant += 2;
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x = -x;
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neg = 1;
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}
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if (M_PI_2 - x == M_PI_2) {
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switch(quadrant) {
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case 0:
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case 2:
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return 0.0;
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case 1:
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return 1.0;
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case 3:
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return -1.0;
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}
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if (cos_flag) {
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neg = 0;
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y = M_PI_2 + x;
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}
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if (x >= M_2PI) {
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if (x <= 0x7fffffff) {
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/* Use extended precision to calculate reduced argument.
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Split 2pi in 2 parts a1 and a2, of which the first only
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uses some bits of the mantissa, so that n * a1 is
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exactly representable, where n is the integer part of
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x/pi.
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Here we used 12 bits of the mantissa for a1.
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Also split x in integer part x1 and fraction part x2.
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We then compute x-n*2pi as ((x1 - n*a1) + x2) - n*a2.
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*/
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#define A1 6.2822265625
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#define A2 0.00095874467958647692528676655900576
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double n = (long) (x / M_2PI);
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double x1 = (long) x;
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double x2 = x - x1;
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x = x1 - n * A1;
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else y = x;
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/* ??? avoid loss of significance, if y is too large, error ??? */
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y = y * M_1_PI + 0.5;
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/* Use extended precision to calculate reduced argument.
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Here we used 12 bits of the mantissa for a1.
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Also split x in integer part x1 and fraction part x2.
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*/
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#define A1 3.1416015625
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#define A2 -8.908910206761537356617e-6
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{
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double x1, x2;
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extern double _fif();
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_fif(y, 1.0, &y);
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if (_fif(y, 0.5, &x1)) neg = !neg;
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if (cos_flag) y -= 0.5;
|
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x2 = _fif(x, 1.0, &x1);
|
||||
x = x1 - y * A1;
|
||||
x += x2;
|
||||
x -= n * A2;
|
||||
x -= y * A2;
|
||||
#undef A1
|
||||
#undef A2
|
||||
}
|
||||
else {
|
||||
extern double _fif();
|
||||
double dummy;
|
||||
}
|
||||
|
||||
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) {
|
||||
if (x < 0) {
|
||||
neg = !neg;
|
||||
x = -x;
|
||||
}
|
||||
xsqr = x * x;
|
||||
x = x * POLYNOM5(xsqr, p) / POLYNOM5(xsqr, q);
|
||||
return x;
|
||||
|
||||
/* ??? avoid underflow ??? */
|
||||
|
||||
y = x * x;
|
||||
x += x * y * POLYNOM7(y, r);
|
||||
return neg ? -x : x;
|
||||
}
|
||||
|
||||
double
|
||||
|
|
|
|||
Loading…
Reference in a new issue