mirror of
https://gitea.nishi.boats/pyrite-dev/milsko
synced 2025-12-31 06:30:52 +00:00
7f523a3d76
git-svn-id: http://svn2.nishi.boats/svn/milsko/trunk@557 b9cfdab3-6d41-4d17-bbe4-086880011989
323 lines
11 KiB
C
323 lines
11 KiB
C
|
|
/* @(#)e_lgamma_r.c 1.3 95/01/18 */
|
|
/*
|
|
* ====================================================
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
|
* Permission to use, copy, modify, and distribute this
|
|
* software is freely granted, provided that this notice
|
|
* is preserved.
|
|
* ====================================================
|
|
*
|
|
*/
|
|
|
|
/* __fdlibm_lgamma_r(x, signgamp)
|
|
* Reentrant version of the logarithm of the Gamma function
|
|
* with user provide pointer for the sign of Gamma(x).
|
|
*
|
|
* Method:
|
|
* 1. Argument Reduction for 0 < x <= 8
|
|
* Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
|
|
* reduce x to a number in [1.5,2.5] by
|
|
* lgamma(1+s) = log(s) + lgamma(s)
|
|
* for example,
|
|
* lgamma(7.3) = log(6.3) + lgamma(6.3)
|
|
* = log(6.3*5.3) + lgamma(5.3)
|
|
* = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
|
|
* 2. Polynomial approximation of lgamma around its
|
|
* minimun ymin=1.461632144968362245 to maintain monotonicity.
|
|
* On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
|
|
* Let z = x-ymin;
|
|
* lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
|
|
* where
|
|
* poly(z) is a 14 degree polynomial.
|
|
* 2. Rational approximation in the primary interval [2,3]
|
|
* We use the following approximation:
|
|
* s = x-2.0;
|
|
* lgamma(x) = 0.5*s + s*P(s)/Q(s)
|
|
* with accuracy
|
|
* |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
|
|
* Our algorithms are based on the following observation
|
|
*
|
|
* zeta(2)-1 2 zeta(3)-1 3
|
|
* lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
|
|
* 2 3
|
|
*
|
|
* where Euler = 0.5771... is the Euler constant, which is very
|
|
* close to 0.5.
|
|
*
|
|
* 3. For x>=8, we have
|
|
* lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
|
|
* (better formula:
|
|
* lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
|
|
* Let z = 1/x, then we approximation
|
|
* f(z) = lgamma(x) - (x-0.5)(log(x)-1)
|
|
* by
|
|
* 3 5 11
|
|
* w = w0 + w1*z + w2*z + w3*z + ... + w6*z
|
|
* where
|
|
* |w - f(z)| < 2**-58.74
|
|
*
|
|
* 4. For negative x, since (G is gamma function)
|
|
* -x*G(-x)*G(x) = pi/sin(pi*x),
|
|
* we have
|
|
* G(x) = pi/(sin(pi*x)*(-x)*G(-x))
|
|
* since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
|
|
* Hence, for x<0, signgam = sign(sin(pi*x)) and
|
|
* lgamma(x) = log(|Gamma(x)|)
|
|
* = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
|
|
* Note: one should avoid compute pi*(-x) directly in the
|
|
* computation of sin(pi*(-x)).
|
|
*
|
|
* 5. Special Cases
|
|
* lgamma(2+s) ~ s*(1-Euler) for tinyv s
|
|
* lgamma(1)=lgamma(2)=0
|
|
* lgamma(x) ~ -log(x) for tinyv x
|
|
* lgamma(0) = lgamma(inf) = inf
|
|
* lgamma(-integer) = +-inf
|
|
*
|
|
*/
|
|
|
|
#include "math.h"
|
|
|
|
static const double
|
|
two52 = 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
|
|
half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
|
|
one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
|
|
pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
|
|
a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
|
|
a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
|
|
a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
|
|
a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
|
|
a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
|
|
a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
|
|
a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
|
|
a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
|
|
a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
|
|
a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
|
|
a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
|
|
a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
|
|
tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
|
|
tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
|
|
/* tt = -(tail of tf) */
|
|
tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
|
|
t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
|
|
t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
|
|
t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
|
|
t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
|
|
t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
|
|
t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
|
|
t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
|
|
t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
|
|
t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
|
|
t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
|
|
t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
|
|
t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
|
|
t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
|
|
t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
|
|
t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
|
|
u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
|
|
u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
|
|
u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
|
|
u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
|
|
u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
|
|
u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
|
|
v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
|
|
v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
|
|
v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
|
|
v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
|
|
v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
|
|
s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
|
|
s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
|
|
s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
|
|
s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
|
|
s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
|
|
s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
|
|
s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
|
|
r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
|
|
r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
|
|
r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
|
|
r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
|
|
r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
|
|
r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
|
|
w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
|
|
w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
|
|
w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
|
|
w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
|
|
w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
|
|
w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
|
|
w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
|
|
|
|
static double zero = 0.00000000000000000000e+00;
|
|
|
|
static double sin_pi(double x) {
|
|
double y, z;
|
|
int n, ix;
|
|
|
|
ix = 0x7fffffff & __HI(x);
|
|
|
|
if(ix < 0x3fd00000) return __fdlibm_kernel_sin(pi * x, zero, 0);
|
|
y = -x; /* x is assume negative */
|
|
|
|
/*
|
|
* argument reduction, make sure inexact flag not raised if input
|
|
* is an integer
|
|
*/
|
|
z = floor(y);
|
|
if(z != y) { /* inexact anyway */
|
|
y *= 0.5;
|
|
y = 2.0 * (y - floor(y)); /* y = |x| mod 2.0 */
|
|
n = (int)(y * 4.0);
|
|
} else {
|
|
if(ix >= 0x43400000) {
|
|
y = zero;
|
|
n = 0; /* y must be even */
|
|
} else {
|
|
if(ix < 0x43300000) z = y + two52; /* exact */
|
|
n = __LO(z) & 1; /* lower word of z */
|
|
y = n;
|
|
n <<= 2;
|
|
}
|
|
}
|
|
switch(n) {
|
|
case 0:
|
|
y = __fdlibm_kernel_sin(pi * y, zero, 0);
|
|
break;
|
|
case 1:
|
|
case 2:
|
|
y = __fdlibm_kernel_cos(pi * (0.5 - y), zero);
|
|
break;
|
|
case 3:
|
|
case 4:
|
|
y = __fdlibm_kernel_sin(pi * (one - y), zero, 0);
|
|
break;
|
|
case 5:
|
|
case 6:
|
|
y = -__fdlibm_kernel_cos(pi * (y - 1.5), zero);
|
|
break;
|
|
default:
|
|
y = __fdlibm_kernel_sin(pi * (y - 2.0), zero, 0);
|
|
break;
|
|
}
|
|
return -y;
|
|
}
|
|
|
|
double __fdlibm_lgamma_r(double x, int* signgamp) {
|
|
double t, y, z, nadj, p, p1, p2, p3, q, r, w;
|
|
int i, hx, lx, ix;
|
|
|
|
hx = __HI(x);
|
|
lx = __LO(x);
|
|
|
|
/* purge off +-inf, NaN, +-0, and negative arguments */
|
|
*signgamp = 1;
|
|
ix = hx & 0x7fffffff;
|
|
if(ix >= 0x7ff00000) return x * x;
|
|
if((ix | lx) == 0) return one / zero;
|
|
if(ix < 0x3b900000) { /* |x|<2**-70, return -log(|x|) */
|
|
if(hx < 0) {
|
|
*signgamp = -1;
|
|
return -__fdlibm_log(-x);
|
|
} else
|
|
return -__fdlibm_log(x);
|
|
}
|
|
if(hx < 0) {
|
|
if(ix >= 0x43300000) /* |x|>=2**52, must be -integer */
|
|
return one / zero;
|
|
t = sin_pi(x);
|
|
if(t == zero) return one / zero; /* -integer */
|
|
nadj = __fdlibm_log(pi / fabs(t * x));
|
|
if(t < zero) *signgamp = -1;
|
|
x = -x;
|
|
}
|
|
|
|
/* purge off 1 and 2 */
|
|
if((((ix - 0x3ff00000) | lx) == 0) || (((ix - 0x40000000) | lx) == 0)) r = 0;
|
|
/* for x < 2.0 */
|
|
else if(ix < 0x40000000) {
|
|
if(ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
|
|
r = -__fdlibm_log(x);
|
|
if(ix >= 0x3FE76944) {
|
|
y = one - x;
|
|
i = 0;
|
|
} else if(ix >= 0x3FCDA661) {
|
|
y = x - (tc - one);
|
|
i = 1;
|
|
} else {
|
|
y = x;
|
|
i = 2;
|
|
}
|
|
} else {
|
|
r = zero;
|
|
if(ix >= 0x3FFBB4C3) {
|
|
y = 2.0 - x;
|
|
i = 0;
|
|
} /* [1.7316,2] */
|
|
else if(ix >= 0x3FF3B4C4) {
|
|
y = x - tc;
|
|
i = 1;
|
|
} /* [1.23,1.73] */
|
|
else {
|
|
y = x - one;
|
|
i = 2;
|
|
}
|
|
}
|
|
switch(i) {
|
|
case 0:
|
|
z = y * y;
|
|
p1 = a0 + z * (a2 + z * (a4 + z * (a6 + z * (a8 + z * a10))));
|
|
p2 = z * (a1 + z * (a3 + z * (a5 + z * (a7 + z * (a9 + z * a11)))));
|
|
p = y * p1 + p2;
|
|
r += (p - 0.5 * y);
|
|
break;
|
|
case 1:
|
|
z = y * y;
|
|
w = z * y;
|
|
p1 = t0 + w * (t3 + w * (t6 + w * (t9 + w * t12))); /* parallel comp */
|
|
p2 = t1 + w * (t4 + w * (t7 + w * (t10 + w * t13)));
|
|
p3 = t2 + w * (t5 + w * (t8 + w * (t11 + w * t14)));
|
|
p = z * p1 - (tt - w * (p2 + y * p3));
|
|
r += (tf + p);
|
|
break;
|
|
case 2:
|
|
p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * u5)))));
|
|
p2 = one + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * v5))));
|
|
r += (-0.5 * y + p1 / p2);
|
|
}
|
|
} else if(ix < 0x40200000) { /* x < 8.0 */
|
|
i = (int)x;
|
|
t = zero;
|
|
y = x - (double)i;
|
|
p = y * (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
|
|
q = one + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * r6)))));
|
|
r = half * y + p / q;
|
|
z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
|
|
switch(i) {
|
|
case 7:
|
|
z *= (y + 6.0); /* FALLTHRU */
|
|
case 6:
|
|
z *= (y + 5.0); /* FALLTHRU */
|
|
case 5:
|
|
z *= (y + 4.0); /* FALLTHRU */
|
|
case 4:
|
|
z *= (y + 3.0); /* FALLTHRU */
|
|
case 3:
|
|
z *= (y + 2.0); /* FALLTHRU */
|
|
r += __fdlibm_log(z);
|
|
break;
|
|
}
|
|
/* 8.0 <= x < 2**58 */
|
|
} else if(ix < 0x43900000) {
|
|
t = __fdlibm_log(x);
|
|
z = one / x;
|
|
y = z * z;
|
|
w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * w6)))));
|
|
r = (x - half) * (t - one) + w;
|
|
} else
|
|
/* 2**58 <= x <= inf */
|
|
r = x * (__fdlibm_log(x) - one);
|
|
if(hx < 0) r = nadj - r;
|
|
return r;
|
|
}
|