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/* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
/*
* ====================================================
* 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.
* ====================================================
*
*/
/* 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(neg.integer) = inf and raise divide-by-zero
* lgamma(inf) = inf
* lgamma(-inf) = inf (bug for bug compatible with C99!?)
*
*/
#include "libm.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 const double zero = 0.00000000000000000000e+00;
static double sin_pi(double x)
{
double y,z;
int n,ix;
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
if (ix < 0x3fd00000)
return __sin(pi*x, zero, 0);
y = -x; /* negative x is assumed */
/*
* 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; /* y must be even */
n = 0;
} else {
if (ix < 0x43300000)
z = y + two52; /* exact */
GET_LOW_WORD(n, z);
n &= 1;
y = n;
n <<= 2;
}
}
switch (n) {
case 0: y = __sin(pi*y, zero, 0); break;
case 1:
case 2: y = __cos(pi*(0.5-y), zero); break;
case 3:
case 4: y = __sin(pi*(one-y), zero, 0); break;
case 5:
case 6: y = -__cos(pi*(y-1.5), zero); break;
default: y = __sin(pi*(y-2.0), zero, 0); break;
}
return -y;
}
double lgamma_r(double x, int *signgamp)
{
double t,y,z,nadj,p,p1,p2,p3,q,r,w;
int32_t hx;
int i,lx,ix;
EXTRACT_WORDS(hx, lx, x);
/* purge off +-inf, NaN, +-0, tiny 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 -log(-x);
}
return -log(x);
}
if (hx < 0) {
if (ix >= 0x43300000) /* |x|>=2**52, must be -integer */
return one/zero;
t = sin_pi(x);
if (t == zero) /* -integer */
return one/zero;
nadj = 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 = -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) { /* [1.7316,2] */
y = 2.0 - x;
i = 0;
} else if(ix >= 0x3FF3B4C4) { /* [1.23,1.73] */
y = x - tc;
i = 1;
} 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;
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 += log(z);
break;
}
} else if (ix < 0x43900000) { /* 8.0 <= x < 2**58 */
t = 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*(log(x)-one);
if (hx < 0)
r = nadj - r;
return r;
}
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