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e_lgamma_r.c
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/* @(#)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.
* ====================================================
*/
/* __ieee754_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 tiny s
* lgamma(1)=lgamma(2)=0
* lgamma(x) ~ -log(x) for tiny x
* lgamma(0) = lgamma(inf) = inf
* lgamma(-integer) = +-inf
*
*/
#ifndef __FDLIBM_H__
#include "fdlibm.h"
#endif
static double sin_pi(double x)
{
double y, z;
int32_t n, ix;
static const double zero = 0.00000000000000000000e+00;
static const double one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
static const double pi = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */
static const double two52 = 4.50359962737049600000e+15; /* 0x43300000, 0x00000000 */
GET_HIGH_WORD(ix, x);
ix &= IC(0x7fffffff);
if (ix < IC(0x3fd00000))
return __ieee754_sin(pi * x);
y = -x; /* x is assume negative */
/*
* argument reduction, make sure inexact flag not raised if input
* is an integer
*/
z = __ieee754_floor(y);
if (z != y)
{ /* inexact anyway */
y *= 0.5;
y = 2.0 * (y - __ieee754_floor(y)); /* y = |x| mod 2.0 */
n = (int32_t) (y * 4.0);
} else
{
if (ix >= IC(0x43400000))
{
y = zero;
n = 0; /* y must be even */
} else
{
if (ix < IC(0x43300000))
z = y + two52; /* exact */
GET_LOW_WORD(n, z);
n &= 1;
y = n;
n <<= 2;
}
}
#ifdef __have_fpu_sin
switch ((int)n)
{
case 0:
y = __ieee754_sin(pi * y);
break;
case 1:
case 2:
y = __ieee754_cos(pi * (0.5 - y));
break;
case 3:
case 4:
y = __ieee754_sin(pi * (one - y));
break;
case 5:
case 6:
y = -__ieee754_cos(pi * (y - 1.5));
break;
default:
y = __ieee754_sin(pi * (y - 2.0));
break;
}
#else
switch ((int)n)
{
case 0:
y = __kernel_sin(pi * y, zero, 0);
break;
case 1:
case 2:
y = __kernel_cos(pi * (0.5 - y), zero);
break;
case 3:
case 4:
y = __kernel_sin(pi * (one - y), zero, 0);
break;
case 5:
case 6:
y = -__kernel_cos(pi * (y - 1.5), zero);
break;
default:
y = __kernel_sin(pi * (y - 2.0), zero, 0);
break;
}
#endif
return -y;
}
double __ieee754_lgamma_r(double x, int *signgamp)
{
double t, y, z, nadj = 0, p, p1, p2, p3, q, r, w;
int32_t i, hx, lx, ix;
static const double zero = 0.00000000000000000000e+00;
static const double half = 5.00000000000000000000e-01; /* 0x3FE00000, 0x00000000 */
static const double one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
static const double pi = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */
static const double a0 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */
static const double a1 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */
static const double a2 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */
static const double a3 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */
static const double a4 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */
static const double a5 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */
static const double a6 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */
static const double a7 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */
static const double a8 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */
static const double a9 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */
static const double a10 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */
static const double a11 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */
static const double tc = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */
static const double tf = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */
/* tt = -(tail of tf) */
static const double tt = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */
static const double t0 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */
static const double t1 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */
static const double t2 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */
static const double t3 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */
static const double t4 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */
static const double t5 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */
static const double t6 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */
static const double t7 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */
static const double t8 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */
static const double t9 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */
static const double t10 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */
static const double t11 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */
static const double t12 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */
static const double t13 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */
static const double t14 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */
static const double u0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
static const double u1 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */
static const double u2 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */
static const double u3 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */
static const double u4 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */
static const double u5 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */
static const double v1 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */
static const double v2 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */
static const double v3 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */
static const double v4 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */
static const double v5 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */
static const double s0 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
static const double s1 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */
static const double s2 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */
static const double s3 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */
static const double s4 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */
static const double s5 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */
static const double s6 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */
static const double r1 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */
static const double r2 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */
static const double r3 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */
static const double r4 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */
static const double r5 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */
static const double r6 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */
static const double w0 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */
static const double w1 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */
static const double w2 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */
static const double w3 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */
static const double w4 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */
static const double w5 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */
static const double w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
GET_DOUBLE_WORDS(hx, lx, x);
/* purge off +-inf, NaN, +-0, and negative arguments */
*signgamp = 1;
ix = hx & IC(0x7fffffff);
if (ix >= IC(0x7ff00000))
return x * x;
if ((ix | lx) == 0)
{
if (hx < 0)
*signgamp = -1;
return one / __ieee754_fabs(x);
}
if (ix < IC(0x3b900000))
{
/* |x|<2**-70, return -log(|x|) */
if (hx < 0)
{
*signgamp = -1;
return -__ieee754_log(-x);
} else
return -__ieee754_log(x);
}
if (hx < 0)
{
if (ix >= IC(0x43300000)) /* |x|>=2**52, must be -integer */
return x / zero;
t = sin_pi(x);
if (t == zero)
return one / __ieee754_fabs(t); /* -integer */
nadj = __ieee754_log(pi / __ieee754_fabs(t * x));
if (t < zero)
*signgamp = -1;
x = -x;
}
/* purge off 1 and 2 */
if ((((ix - IC(0x3ff00000)) | lx) == 0) || (((ix - IC(0x40000000)) | lx) == 0))
r = 0;
/* for x < 2.0 */
else if (ix < IC(0x40000000))
{
if (ix <= IC(0x3feccccc))
{ /* lgamma(x) = lgamma(x+1)-log(x) */
r = -__ieee754_log(x);
if (ix >= IC(0x3FE76944))
{
y = one - x;
i = 0;
} else if (ix >= IC(0x3FCDA661))
{
y = x - (tc - one);
i = 1;
} else
{
y = x;
i = 2;
}
} else
{
r = zero;
if (ix >= IC(0x3FFBB4C3))
{
y = 2.0 - x;
i = 0;
} /* [1.7316,2] */
else if (ix >= IC(0x3FF3B4C4))
{
y = x - tc;
i = 1;
} /* [1.23,1.73] */
else
{
y = x - one;
i = 2;
}
}
switch ((int) 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);
break;
}
} else if (ix < IC(0x40200000))
{ /* x < 8.0 */
i = (int32_t) 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 ((int) 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 += __ieee754_log(z);
break;
}
/* 8.0 <= x < 2**58 */
} else if (ix < IC(0x43900000))
{
t = __ieee754_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 * (__ieee754_log(x) - one);
if (hx < 0)
r = nadj - r;
return r;
}