*/
/* $Header$ */
+
#define __NO_DEFS
#include <math.h>
_atn(x)
double x;
{
- /* The interval [0, infinity) is treated as follows:
- Define partition points Xi
- X0 = 0
- X1 = tan(pi/16)
- X2 = tan(3pi/16)
- X3 = tan(5pi/16)
- X4 = tan(7pi/16)
- X5 = infinity
- and evaluation nodes xi
- x2 = tan(2pi/16)
- x3 = tan(4pi/16)
- x4 = tan(6pi/16)
- x5 = infinity
- An argument x in [Xn-1, Xn] is now reduced to an argument
- t in [-X1, X1] by the following formulas:
-
- t = 1/xn - (1/(xn*xn) + 1)/((1/xn) + x)
-
- arctan(x) = arctan(xi) + arctan(t)
-
- For the interval [0, p/16] an approximation is used:
- arctan(x) = x * P(x*x)/Q(x*x)
+ /* Algorithm and coefficients from:
+ "Software manual for the elementary functions"
+ by W.J. Cody and W. Waite, Prentice-Hall, 1980
*/
- static struct precomputed {
- double X; /* partition point */
- double arctan; /* arctan of evaluation node */
- double one_o_x; /* 1 / xn */
- double one_o_xsq_p_1; /* 1 / (xn*xn) + 1 */
- } prec[5] = {
- { 0.19891236737965800691159762264467622,
- 0.0,
- 0.0, /* these don't matter */
- 0.0 } ,
- { 0.66817863791929891999775768652308076, /* tan(3pi/16) */
- M_PI_8,
- 2.41421356237309504880168872420969808,
- 6.82842712474619009760337744841939616 },
- { 1.49660576266548901760113513494247691, /* tan(5pi/16) */
- M_PI_4,
- 1.0,
- 2.0 },
- { 5.02733949212584810451497507106407238, /* tan(7pi/16) */
- M_3PI_8,
- 0.41421356237309504880168872420969808,
- 1.17157287525380998659662255158060384 },
- { MAXDOUBLE,
- M_PI_2,
- 0.0,
- 1.0 }};
-
- /* Hart & Cheney # 5037 */
- static double p[5] = {
- 0.7698297257888171026986294745e+03,
- 0.1557282793158363491416585283e+04,
- 0.1033384651675161628243434662e+04,
- 0.2485841954911840502660889866e+03,
- 0.1566564964979791769948970100e+02
+ static double p[] = {
+ -0.13688768894191926929e+2,
+ -0.20505855195861651981e+2,
+ -0.84946240351320683534e+1,
+ -0.83758299368150059274e+0
};
-
- static double q[6] = {
- 0.7698297257888171026986294911e+03,
- 0.1813892701754635858982709369e+04,
- 0.1484049607102276827437401170e+04,
- 0.4904645326203706217748848797e+03,
- 0.5593479839280348664778328000e+02,
- 0.1000000000000000000000000000e+01
+ static double q[] = {
+ 0.41066306682575781263e+2,
+ 0.86157349597130242515e+2,
+ 0.59578436142597344465e+2,
+ 0.15024001160028576121e+2,
+ 1.0
+ };
+ static double a[] = {
+ 0.0,
+ 0.52359877559829887307710723554658381, /* pi/6 */
+ M_PI_2,
+ 1.04719755119659774615421446109316763 /* pi/3 */
};
- int negative = x < 0.0;
- register struct precomputed *pr = prec;
+ int neg = x < 0;
+ int n;
+ double g;
- if (negative) {
+ if (neg) {
x = -x;
}
- while (x > pr->X) pr++;
- if (pr != prec) {
- x = pr->arctan +
- _atn(pr->one_o_x - pr->one_o_xsq_p_1/(pr->one_o_x + x));
+ if (x > 1.0) {
+ x = 1.0/x;
+ n = 2;
}
- else {
- double xsq = x*x;
+ else n = 0;
- x = x * POLYNOM4(xsq, p)/POLYNOM5(xsq, q);
+ if (x > 0.26794919243112270647) { /* 2-sqtr(3) */
+ n = n + 1;
+ x = (((0.73205080756887729353*x-0.5)-0.5)+x)/
+ (1.73205080756887729353+x);
}
- return negative ? -x : x;
+
+ /* ??? avoid underflow ??? */
+
+ g = x * x;
+ x += x * g * POLYNOM3(g, p) / POLYNOM4(g, q);
+ if (n > 1) x = -x;
+ x += a[n];
+ return neg ? -x : x;
}
#include <pc_err.h>
extern _trp();
-static double
-floor(x)
- double x;
-{
- extern double _fif();
- double val;
-
- return _fif(x, 1.0, &val) < 0 ? val - 1.0 : val ;
- /* this also works if _fif always returns a positive
- fractional part
- */
-}
-
-static double
-ldexp(fl,exp)
- double fl;
- int exp;
-{
- extern double _fef();
- int sign = 1;
- int currexp;
-
- 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
_exp(x)
double x;
{
- /* 2**x = (Q(x*x)+x*P(x*x))/(Q(x*x)-x*P(x*x)) for x in [0,0.5] */
- /* Hart & Cheney #1069 */
+ /* Algorithm and coefficients from:
+ "Software manual for the elementary functions"
+ by W.J. Cody and W. Waite, Prentice-Hall, 1980
+ */
- static double p[3] = {
- 0.2080384346694663001443843411e+07,
- 0.3028697169744036299076048876e+05,
- 0.6061485330061080841615584556e+02
+ static double p[] = {
+ 0.25000000000000000000e+0,
+ 0.75753180159422776666e-2,
+ 0.31555192765684646356e-4
};
- static double q[4] = {
- 0.6002720360238832528230907598e+07,
- 0.3277251518082914423057964422e+06,
- 0.1749287689093076403844945335e+04,
- 0.1000000000000000000000000000e+01
+ static double q[] = {
+ 0.50000000000000000000e+0,
+ 0.56817302698551221787e-1,
+ 0.63121894374398503557e-3,
+ 0.75104028399870046114e-6
};
+ double xn, g;
+ int n;
+ int negative = x < 0;
- int negative = x < 0;
- int ipart, large = 0;
- double xsqr, xPxx, Qxx;
-
- if (x < M_LN_MIN_D) {
+ if (x <= M_LN_MIN_D) {
return M_MIN_D;
}
if (x >= M_LN_MAX_D) {
}
return M_MAX_D;
}
+ if (negative) x = -x;
- if (negative) {
- x = -x;
+ /* ??? avoid underflow ??? */
+
+ n = x * M_LOG2E + 0.5; /* 1/ln(2) = log2(e), 0.5 added for rounding */
+ xn = n;
+ {
+ double x1 = (long) x;
+ double x2 = x - x1;
+
+ g = ((x1-xn*0.693359375)+x2) - xn*(-2.1219444005469058277e-4);
}
- x /= M_LN2;
- ipart = floor(x);
- x -= ipart;
- if (x > 0.5) {
- large = 1;
- x -= 0.5;
+ if (negative) {
+ g = -g;
+ n = -n;
}
- xsqr = x * x;
- xPxx = x * POLYNOM2(xsqr, p);
- Qxx = POLYNOM3(xsqr, q);
- x = (Qxx + xPxx) / (Qxx - xPxx);
- if (large) x *= M_SQRT2;
- x = ldexp(x, ipart);
- if (negative) return 1.0/x;
- return x;
+ xn = g * g;
+ x = g * POLYNOM2(xn, p);
+ n += 1;
+ return (ldexp(0.5 + x/(POLYNOM3(xn, q) - x), n));
}
*/
/* $Header$ */
+
#define __NO_DEFS
#include <math.h>
#include <pc_err.h>
-extern _trp();
double
_log(x)
- double x;
+ double x;
{
- /* log(x) = z*P(z*z)/Q(z*z), z = (x-1)/(x+1), x in [1/sqrt(2), sqrt(2)]
+ /* Algorithm and coefficients from:
+ "Software manual for the elementary functions"
+ by W.J. Cody and W. Waite, Prentice-Hall, 1980
*/
- /* Hart & Cheney #2707 */
-
- static double p[5] = {
- 0.7504094990777122217455611007e+02,
- -0.1345669115050430235318253537e+03,
- 0.7413719213248602512779336470e+02,
- -0.1277249755012330819984385000e+02,
- 0.3327108381087686938144000000e+00
+ static double a[] = {
+ -0.64124943423745581147e2,
+ 0.16383943563021534222e2,
+ -0.78956112887491257267e0
};
-
- static double q[5] = {
- 0.3752047495388561108727775374e+02,
- -0.7979028073715004879439951583e+02,
- 0.5616126132118257292058560360e+02,
- -0.1450868091858082685362325000e+02,
- 0.1000000000000000000000000000e+01
+ static double b[] = {
+ -0.76949932108494879777e3,
+ 0.31203222091924532844e3,
+ -0.35667977739034646171e2,
+ 1.0
};
- extern double _fef();
- double z, zsqr;
- int exponent;
+ extern double _fef();
+ double znum, zden, z, w;
+ int exponent;
if (x <= 0) {
_trp(ELOG);
}
x = _fef(x, &exponent);
- while (x < M_1_SQRT2) {
- x += x;
+ if (x > M_1_SQRT2) {
+ znum = (x - 0.5) - 0.5;
+ zden = x * 0.5 + 0.5;
+ }
+ else {
+ znum = x - 0.5;
+ zden = znum * 0.5 + 0.5;
exponent--;
}
- z = (x-1)/(x+1);
- zsqr = z*z;
- return z * POLYNOM4(zsqr, p) / POLYNOM4(zsqr, q) + exponent * M_LN2;
+ z = znum/zden; w = z * z;
+ x = z + z * w * (POLYNOM2(w,a)/POLYNOM3(w,b));
+ z = exponent;
+ x += z * (-2.121944400546905827679e-4);
+ return x + z * 0.693359375;
}
#include <math.h>
static double
-sinus(x, quadrant)
+sinus(x, cos_flag)
double x;
{
- /* sin(0.5*pi*x) = x * P(x*x)/Q(x*x) for x in [0,1] */
- /* Hart & Cheney # 3374 */
+ /* Algorithm and coefficients from:
+ "Software manual for the elementary functions"
+ by W.J. Cody and W. Waite, Prentice-Hall, 1980
+ */
- static double p[6] = {
- 0.4857791909822798473837058825e+10,
- -0.1808816670894030772075877725e+10,
- 0.1724314784722489597789244188e+09,
- -0.6351331748520454245913645971e+07,
- 0.1002087631419532326179108883e+06,
- -0.5830988897678192576148973679e+03
+ static double r[] = {
+ -0.16666666666666665052e+0,
+ 0.83333333333331650314e-2,
+ -0.19841269841201840457e-3,
+ 0.27557319210152756119e-5,
+ -0.25052106798274584544e-7,
+ 0.16058936490371589114e-9,
+ -0.76429178068910467734e-12,
+ 0.27204790957888846175e-14
};
- 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;
+ double xsqr;
+ double y;
+ int neg = 0;
if (x < 0) {
- quadrant += 2;
x = -x;
+ neg = 1;
}
- 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 (cos_flag) {
+ neg = 0;
+ y = M_PI_2 + x;
}
- 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;
+ else y = x;
+
+ /* ??? avoid loss of significance, if y is too large, error ??? */
+
+ y = y * M_1_PI + 0.5;
+
+ /* Use extended precision to calculate reduced argument.
+ Here we used 12 bits of the mantissa for a1.
+ Also split x in integer part x1 and fraction part x2.
+ */
+#define A1 3.1416015625
+#define A2 -8.908910206761537356617e-6
+ {
+ double x1, x2;
+ extern double _fif();
+
+ _fif(y, 1.0, &y);
+ if (_fif(y, 0.5, &x1)) neg = !neg;
+ if (cos_flag) y -= 0.5;
+ 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
public / ack.git / commitdiff