a4e4b355f4
lgamma(x) = -log(x) - log(1+x) + x*(1-g) + x**2*P(x) with g = 0.57... being the Euler constant and P(x) a polynomial. Substitution of small into the RHS shows that the last 3 terms are negligible in comparison to the leading term. The choice of 3 may be conservative. The value large=2**(p+3) is detemined from Stirling's approximation lgamma(x) = x*(log(x)-1) - log(x)/2 + log(2*pi)/2 + P(1/x)/x Again, substitution of large into the RHS reveals the last 3 terms are negligible in comparison to the leading term. Move the x=+-0 special case into the |x|<small block. In the ld80 and ld128 implementaion, use fdlibm compatible comparisons involving ix, lx, and llx. This replaces several floating point comparisons (some involving fabsl()) and also fixes the special cases x=1 and x=2. While here . Remove unnecessary parentheses. . Fix/improve comments due to the above changes. . Fix nearby whitespace. * src/e_lgamma_r.c: . Sort declaration. . Remove unneeded explicit cast for type conversion. . Replace a double literal constant by an integer literal constant. * src/e_lgammaf_r.c: . Sort declaration. * ld128/e_lgammal_r.c: . Replace a long double literal constant by a double literal constant. * ld80/e_lgammal_r.c: . Remove unused '#include float.h' . Replace a long double literal constant by a double literal constant. Requested by: bde
359 lines
12 KiB
C
359 lines
12 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.
|
|
* ====================================================
|
|
*/
|
|
|
|
#include <sys/cdefs.h>
|
|
__FBSDID("$FreeBSD$");
|
|
|
|
/*
|
|
* See e_lgamma_r.c for complete comments.
|
|
*
|
|
* Converted to long double by Steven G. Kargl.
|
|
*/
|
|
|
|
#ifdef __i386__
|
|
#include <ieeefp.h>
|
|
#endif
|
|
|
|
#include "fpmath.h"
|
|
#include "math.h"
|
|
#include "math_private.h"
|
|
|
|
static const volatile double vzero = 0;
|
|
|
|
static const double
|
|
zero= 0,
|
|
half= 0.5,
|
|
one = 1;
|
|
|
|
static const union IEEEl2bits
|
|
piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
|
|
#define pi (piu.e)
|
|
/*
|
|
* Domain y in [0x1p-70, 0.27], range ~[-4.5264e-22, 4.5264e-22]:
|
|
* |(lgamma(2 - y) + y / 2) / y - a(y)| < 2**-70.9
|
|
*/
|
|
static const union IEEEl2bits
|
|
a0u = LD80C(0x9e233f1bed863d26, -4, 7.72156649015328606028e-02L),
|
|
a1u = LD80C(0xa51a6625307d3249, -2, 3.22467033424113218889e-01L),
|
|
a2u = LD80C(0x89f000d2abafda8c, -4, 6.73523010531979398946e-02L),
|
|
a3u = LD80C(0xa8991563eca75f26, -6, 2.05808084277991211934e-02L),
|
|
a4u = LD80C(0xf2027e10634ce6b6, -8, 7.38555102796070454026e-03L),
|
|
a5u = LD80C(0xbd6eb76dd22187f4, -9, 2.89051035162703932972e-03L),
|
|
a6u = LD80C(0x9c562ab05e0458ed, -10, 1.19275351624639999297e-03L),
|
|
a7u = LD80C(0x859baed93ee48e46, -11, 5.09674593842117925320e-04L),
|
|
a8u = LD80C(0xe9f28a4432949af2, -13, 2.23109648015769155122e-04L),
|
|
a9u = LD80C(0xd12ad0d9b93c6bb0, -14, 9.97387167479808509830e-05L),
|
|
a10u= LD80C(0xb7522643c78a219b, -15, 4.37071076331030136818e-05L),
|
|
a11u= LD80C(0xca024dcdece2cb79, -16, 2.40813493372040143061e-05L),
|
|
a12u= LD80C(0xbb90fb6968ebdbf9, -19, 2.79495621083634031729e-06L),
|
|
a13u= LD80C(0xba1c9ffeeae07b37, -17, 1.10931287015513924136e-05L);
|
|
#define a0 (a0u.e)
|
|
#define a1 (a1u.e)
|
|
#define a2 (a2u.e)
|
|
#define a3 (a3u.e)
|
|
#define a4 (a4u.e)
|
|
#define a5 (a5u.e)
|
|
#define a6 (a6u.e)
|
|
#define a7 (a7u.e)
|
|
#define a8 (a8u.e)
|
|
#define a9 (a9u.e)
|
|
#define a10 (a10u.e)
|
|
#define a11 (a11u.e)
|
|
#define a12 (a12u.e)
|
|
#define a13 (a13u.e)
|
|
/*
|
|
* Domain x in [tc-0.24, tc+0.28], range ~[-6.1205e-22, 6.1205e-22]:
|
|
* |(lgamma(x) - tf) - t(x - tc)| < 2**-70.5
|
|
*/
|
|
static const union IEEEl2bits
|
|
tcu = LD80C(0xbb16c31ab5f1fb71, 0, 1.46163214496836234128e+00L),
|
|
tfu = LD80C(0xf8cdcde61c520e0f, -4, -1.21486290535849608093e-01L),
|
|
ttu = LD80C(0xd46ee54b27d4de99, -69, -2.81152980996018785880e-21L),
|
|
t0u = LD80C(0x80b9406556a62a6b, -68, 3.40728634996055147231e-21L),
|
|
t1u = LD80C(0xc7e9c6f6df3f8c39, -67, -1.05833162742737073665e-20L),
|
|
t2u = LD80C(0xf7b95e4771c55d51, -2, 4.83836122723810583532e-01L),
|
|
t3u = LD80C(0x97213c6e35e119ff, -3, -1.47587722994530691476e-01L),
|
|
t4u = LD80C(0x845a14a6a81dc94b, -4, 6.46249402389135358063e-02L),
|
|
t5u = LD80C(0x864d46fa89997796, -5, -3.27885410884846056084e-02L),
|
|
t6u = LD80C(0x93373cbd00297438, -6, 1.79706751150707171293e-02L),
|
|
t7u = LD80C(0xa8fcfca7eddc8d1d, -7, -1.03142230361450732547e-02L),
|
|
t8u = LD80C(0xc7e7015ff4bc45af, -8, 6.10053603296546099193e-03L),
|
|
t9u = LD80C(0xf178d2247adc5093, -9, -3.68456964904901200152e-03L),
|
|
t10u = LD80C(0x94188d58f12e5e9f, -9, 2.25976420273774583089e-03L),
|
|
t11u = LD80C(0xb7cbaef14e1406f1, -10, -1.40224943666225639823e-03L),
|
|
t12u = LD80C(0xe63a671e6704ea4d, -11, 8.78250640744776944887e-04L),
|
|
t13u = LD80C(0x914b6c9cae61783e, -11, -5.54255012657716808811e-04L),
|
|
t14u = LD80C(0xb858f5bdb79276fe, -12, 3.51614951536825927370e-04L),
|
|
t15u = LD80C(0xea73e744c34b9591, -13, -2.23591563824520112236e-04L),
|
|
t16u = LD80C(0x99aeabb0d67ba835, -13, 1.46562869351659194136e-04L),
|
|
t17u = LD80C(0xd7c6938325db2024, -14, -1.02889866046435680588e-04L),
|
|
t18u = LD80C(0xe24cb1e3b0474775, -15, 5.39540265505221957652e-05L);
|
|
#define tc (tcu.e)
|
|
#define tf (tfu.e)
|
|
#define tt (ttu.e)
|
|
#define t0 (t0u.e)
|
|
#define t1 (t1u.e)
|
|
#define t2 (t2u.e)
|
|
#define t3 (t3u.e)
|
|
#define t4 (t4u.e)
|
|
#define t5 (t5u.e)
|
|
#define t6 (t6u.e)
|
|
#define t7 (t7u.e)
|
|
#define t8 (t8u.e)
|
|
#define t9 (t9u.e)
|
|
#define t10 (t10u.e)
|
|
#define t11 (t11u.e)
|
|
#define t12 (t12u.e)
|
|
#define t13 (t13u.e)
|
|
#define t14 (t14u.e)
|
|
#define t15 (t15u.e)
|
|
#define t16 (t16u.e)
|
|
#define t17 (t17u.e)
|
|
#define t18 (t18u.e)
|
|
/*
|
|
* Domain y in [-0.1, 0.232], range ~[-8.1938e-22, 8.3815e-22]:
|
|
* |(lgamma(1 + y) + 0.5 * y) / y - u(y) / v(y)| < 2**-71.2
|
|
*/
|
|
static const union IEEEl2bits
|
|
u0u = LD80C(0x9e233f1bed863d27, -4, -7.72156649015328606095e-02L),
|
|
u1u = LD80C(0x98280ee45e4ddd3d, -1, 5.94361239198682739769e-01L),
|
|
u2u = LD80C(0xe330c8ead4130733, 0, 1.77492629495841234275e+00L),
|
|
u3u = LD80C(0xd4a213f1a002ec52, 0, 1.66119622514818078064e+00L),
|
|
u4u = LD80C(0xa5a9ca6f5bc62163, -1, 6.47122051417476492989e-01L),
|
|
u5u = LD80C(0xc980e49cd5b019e6, -4, 9.83903751718671509455e-02L),
|
|
u6u = LD80C(0xff636a8bdce7025b, -9, 3.89691687802305743450e-03L),
|
|
v1u = LD80C(0xbd109c533a19fbf5, 1, 2.95413883330948556544e+00L),
|
|
v2u = LD80C(0xd295cbf96f31f099, 1, 3.29039286955665403176e+00L),
|
|
v3u = LD80C(0xdab8bcfee40496cb, 0, 1.70876276441416471410e+00L),
|
|
v4u = LD80C(0xd2f2dc3638567e9f, -2, 4.12009126299534668571e-01L),
|
|
v5u = LD80C(0xa07d9b0851070f41, -5, 3.91822868305682491442e-02L),
|
|
v6u = LD80C(0xe3cd8318f7adb2c4, -11, 8.68998648222144351114e-04L);
|
|
#define u0 (u0u.e)
|
|
#define u1 (u1u.e)
|
|
#define u2 (u2u.e)
|
|
#define u3 (u3u.e)
|
|
#define u4 (u4u.e)
|
|
#define u5 (u5u.e)
|
|
#define u6 (u6u.e)
|
|
#define v1 (v1u.e)
|
|
#define v2 (v2u.e)
|
|
#define v3 (v3u.e)
|
|
#define v4 (v4u.e)
|
|
#define v5 (v5u.e)
|
|
#define v6 (v6u.e)
|
|
/*
|
|
* Domain x in (2, 3], range ~[-3.3648e-22, 3.4416e-22]:
|
|
* |(lgamma(y+2) - 0.5 * y) / y - s(y)/r(y)| < 2**-72.3
|
|
* with y = x - 2.
|
|
*/
|
|
static const union IEEEl2bits
|
|
s0u = LD80C(0x9e233f1bed863d27, -4, -7.72156649015328606095e-02L),
|
|
s1u = LD80C(0xd3ff0dcc7fa91f94, -3, 2.07027640921219389860e-01L),
|
|
s2u = LD80C(0xb2bb62782478ef31, -2, 3.49085881391362090549e-01L),
|
|
s3u = LD80C(0xb49f7438c4611a74, -3, 1.76389518704213357954e-01L),
|
|
s4u = LD80C(0x9a957008fa27ecf9, -5, 3.77401710862930008071e-02L),
|
|
s5u = LD80C(0xda9b389a6ca7a7ac, -9, 3.33566791452943399399e-03L),
|
|
s6u = LD80C(0xbc7a2263faf59c14, -14, 8.98728786745638844395e-05L),
|
|
r1u = LD80C(0xbf5cff5b11477d4d, 0, 1.49502555796294337722e+00L),
|
|
r2u = LD80C(0xd9aec89de08e3da6, -1, 8.50323236984473285866e-01L),
|
|
r3u = LD80C(0xeab7ae5057c443f9, -3, 2.29216312078225806131e-01L),
|
|
r4u = LD80C(0xf29707d9bd2b1e37, -6, 2.96130326586640089145e-02L),
|
|
r5u = LD80C(0xd376c2f09736c5a3, -10, 1.61334161411590662495e-03L),
|
|
r6u = LD80C(0xc985983d0cd34e3d, -16, 2.40232770710953450636e-05L),
|
|
r7u = LD80C(0xe5c7a4f7fc2ef13d, -25, -5.34997929289167573510e-08L);
|
|
#define s0 (s0u.e)
|
|
#define s1 (s1u.e)
|
|
#define s2 (s2u.e)
|
|
#define s3 (s3u.e)
|
|
#define s4 (s4u.e)
|
|
#define s5 (s5u.e)
|
|
#define s6 (s6u.e)
|
|
#define r1 (r1u.e)
|
|
#define r2 (r2u.e)
|
|
#define r3 (r3u.e)
|
|
#define r4 (r4u.e)
|
|
#define r5 (r5u.e)
|
|
#define r6 (r6u.e)
|
|
#define r7 (r7u.e)
|
|
/*
|
|
* Domain z in [8, 0x1p70], range ~[-3.0235e-22, 3.0563e-22]:
|
|
* |lgamma(x) - (x - 0.5) * (log(x) - 1) - w(1/x)| < 2**-71.7
|
|
*/
|
|
static const union IEEEl2bits
|
|
w0u = LD80C(0xd67f1c864beb4a69, -2, 4.18938533204672741776e-01L),
|
|
w1u = LD80C(0xaaaaaaaaaaaaaaa1, -4, 8.33333333333333332678e-02L),
|
|
w2u = LD80C(0xb60b60b60b5491c9, -9, -2.77777777777760927870e-03L),
|
|
w3u = LD80C(0xd00d00cf58aede4c, -11, 7.93650793490637233668e-04L),
|
|
w4u = LD80C(0x9c09bf626783d4a5, -11, -5.95238023926039051268e-04L),
|
|
w5u = LD80C(0xdca7cadc5baa517b, -11, 8.41733700408000822962e-04L),
|
|
w6u = LD80C(0xfb060e361e1ffd07, -10, -1.91515849570245136604e-03L),
|
|
w7u = LD80C(0xcbd5101bb58d1f2b, -8, 6.22046743903262649294e-03L),
|
|
w8u = LD80C(0xad27a668d32c821b, -6, -2.11370706734662081843e-02L);
|
|
#define w0 (w0u.e)
|
|
#define w1 (w1u.e)
|
|
#define w2 (w2u.e)
|
|
#define w3 (w3u.e)
|
|
#define w4 (w4u.e)
|
|
#define w5 (w5u.e)
|
|
#define w6 (w6u.e)
|
|
#define w7 (w7u.e)
|
|
#define w8 (w8u.e)
|
|
|
|
static long double
|
|
sin_pil(long double x)
|
|
{
|
|
volatile long double vz;
|
|
long double y,z;
|
|
uint64_t n;
|
|
uint16_t hx;
|
|
|
|
y = -x;
|
|
|
|
vz = y+0x1p63;
|
|
z = vz-0x1p63;
|
|
if (z == y)
|
|
return zero;
|
|
|
|
vz = y+0x1p61;
|
|
EXTRACT_LDBL80_WORDS(hx,n,vz);
|
|
z = vz-0x1p61;
|
|
if (z > y) {
|
|
z -= 0.25; /* adjust to round down */
|
|
n--;
|
|
}
|
|
n &= 7; /* octant of y mod 2 */
|
|
y = y - z + n * 0.25; /* y mod 2 */
|
|
|
|
switch (n) {
|
|
case 0: y = __kernel_sinl(pi*y,zero,0); break;
|
|
case 1:
|
|
case 2: y = __kernel_cosl(pi*(0.5-y),zero); break;
|
|
case 3:
|
|
case 4: y = __kernel_sinl(pi*(one-y),zero,0); break;
|
|
case 5:
|
|
case 6: y = -__kernel_cosl(pi*(y-1.5),zero); break;
|
|
default: y = __kernel_sinl(pi*(y-2.0),zero,0); break;
|
|
}
|
|
return -y;
|
|
}
|
|
|
|
long double
|
|
lgammal_r(long double x, int *signgamp)
|
|
{
|
|
long double nadj,p,p1,p2,p3,q,r,t,w,y,z;
|
|
uint64_t lx;
|
|
int i;
|
|
uint16_t hx,ix;
|
|
|
|
EXTRACT_LDBL80_WORDS(hx,lx,x);
|
|
|
|
/* purge +-Inf and NaNs */
|
|
*signgamp = 1;
|
|
ix = hx&0x7fff;
|
|
if(ix==0x7fff) return x*x;
|
|
|
|
ENTERI();
|
|
|
|
/* purge +-0 and tiny arguments */
|
|
*signgamp = 1-2*(hx>>15);
|
|
if(ix<0x3fff-67) { /* |x|<2**-(p+3), return -log(|x|) */
|
|
if((ix|lx)==0)
|
|
RETURNI(one/vzero);
|
|
RETURNI(-logl(fabsl(x)));
|
|
}
|
|
|
|
/* purge negative integers and start evaluation for other x < 0 */
|
|
if(hx&0x8000) {
|
|
*signgamp = 1;
|
|
if(ix>=0x3fff+63) /* |x|>=2**(p-1), must be -integer */
|
|
RETURNI(one/vzero);
|
|
t = sin_pil(x);
|
|
if(t==zero) RETURNI(one/vzero); /* -integer */
|
|
nadj = logl(pi/fabsl(t*x));
|
|
if(t<zero) *signgamp = -1;
|
|
x = -x;
|
|
}
|
|
|
|
/* purge 1 and 2 */
|
|
if((ix==0x3fff || ix==0x4000) && lx==0x8000000000000000ULL) r = 0;
|
|
/* for x < 2.0 */
|
|
else if(ix<0x4000) {
|
|
/*
|
|
* XXX Supposedly, one can use the following information to replace the
|
|
* XXX FP rational expressions. A similar approach is appropriate
|
|
* XXX for ld128, but one (may need?) needs to consider llx, too.
|
|
*
|
|
* 8.9999961853027344e-01 3ffe e666600000000000
|
|
* 7.3159980773925781e-01 3ffe bb4a200000000000
|
|
* 2.3163998126983643e-01 3ffc ed33080000000000
|
|
* 1.7316312789916992e+00 3fff dda6180000000000
|
|
* 1.2316322326660156e+00 3fff 9da6200000000000
|
|
*/
|
|
if(x<8.9999961853027344e-01) {
|
|
r = -logl(x);
|
|
if(x>=7.3159980773925781e-01) {y = 1-x; i= 0;}
|
|
else if(x>=2.3163998126983643e-01) {y= x-(tc-1); i=1;}
|
|
else {y = x; i=2;}
|
|
} else {
|
|
r = 0;
|
|
if(x>=1.7316312789916992e+00) {y=2-x;i=0;}
|
|
else if(x>=1.2316322326660156e+00) {y=x-tc;i=1;}
|
|
else {y=x-1;i=2;}
|
|
}
|
|
switch(i) {
|
|
case 0:
|
|
z = y*y;
|
|
p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*(a10+z*a12)))));
|
|
p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*(a11+z*a13))))));
|
|
p = y*p1+p2;
|
|
r += p-y/2; break;
|
|
case 1:
|
|
p = t0+y*t1+tt+y*y*(t2+y*(t3+y*(t4+y*(t5+y*(t6+y*(t7+y*(t8+
|
|
y*(t9+y*(t10+y*(t11+y*(t12+y*(t13+y*(t14+y*(t15+y*(t16+
|
|
y*(t17+y*t18))))))))))))))));
|
|
r += tf + p; break;
|
|
case 2:
|
|
p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*(u5+y*u6))))));
|
|
p2 = 1+y*(v1+y*(v2+y*(v3+y*(v4+y*(v5+y*v6)))));
|
|
r += p1/p2-y/2;
|
|
}
|
|
}
|
|
/* x < 8.0 */
|
|
else if(ix<0x4002) {
|
|
i = x;
|
|
y = x-i;
|
|
p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
|
|
q = 1+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*(r6+y*r7))))));
|
|
r = y/2+p/q;
|
|
z = 1; /* lgamma(1+s) = log(s) + lgamma(s) */
|
|
switch(i) {
|
|
case 7: z *= (y+6); /* FALLTHRU */
|
|
case 6: z *= (y+5); /* FALLTHRU */
|
|
case 5: z *= (y+4); /* FALLTHRU */
|
|
case 4: z *= (y+3); /* FALLTHRU */
|
|
case 3: z *= (y+2); /* FALLTHRU */
|
|
r += logl(z); break;
|
|
}
|
|
/* 8.0 <= x < 2**(p+3) */
|
|
} else if (ix<0x3fff+67) {
|
|
t = logl(x);
|
|
z = one/x;
|
|
y = z*z;
|
|
w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*(w6+y*(w7+y*w8)))))));
|
|
r = (x-half)*(t-one)+w;
|
|
/* 2**(p+3) <= x <= inf */
|
|
} else
|
|
r = x*(logl(x)-1);
|
|
if(hx&0x8000) r = nadj - r;
|
|
RETURNI(r);
|
|
}
|