420 lines
11 KiB
C
420 lines
11 KiB
C
|
/*-
|
||
|
* SPDX-License-Identifier: BSD-3-Clause
|
||
|
*
|
||
|
* Copyright (c) 1992, 1993
|
||
|
* The Regents of the University of California. All rights reserved.
|
||
|
*
|
||
|
* Redistribution and use in source and binary forms, with or without
|
||
|
* modification, are permitted provided that the following conditions
|
||
|
* are met:
|
||
|
* 1. Redistributions of source code must retain the above copyright
|
||
|
* notice, this list of conditions and the following disclaimer.
|
||
|
* 2. Redistributions in binary form must reproduce the above copyright
|
||
|
* notice, this list of conditions and the following disclaimer in the
|
||
|
* documentation and/or other materials provided with the distribution.
|
||
|
* 3. Neither the name of the University nor the names of its contributors
|
||
|
* may be used to endorse or promote products derived from this software
|
||
|
* without specific prior written permission.
|
||
|
*
|
||
|
* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
|
||
|
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
|
||
|
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
|
||
|
* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
|
||
|
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
|
||
|
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
|
||
|
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
|
||
|
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
|
||
|
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
|
||
|
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
|
||
|
* SUCH DAMAGE.
|
||
|
*/
|
||
|
|
||
|
/*
|
||
|
* The original code, FreeBSD's old svn r93211, contain the following
|
||
|
* attribution:
|
||
|
*
|
||
|
* This code by P. McIlroy, Oct 1992;
|
||
|
*
|
||
|
* The financial support of UUNET Communications Services is greatfully
|
||
|
* acknowledged.
|
||
|
*
|
||
|
* bsdrc/b_tgamma.c converted to long double by Steven G. Kargl.
|
||
|
*/
|
||
|
|
||
|
/*
|
||
|
* See bsdsrc/t_tgamma.c for implementation details.
|
||
|
*/
|
||
|
|
||
|
#include <float.h>
|
||
|
|
||
|
#if LDBL_MAX_EXP != 0x4000
|
||
|
#error "Unsupported long double format"
|
||
|
#endif
|
||
|
|
||
|
#ifdef __i386__
|
||
|
#include <ieeefp.h>
|
||
|
#endif
|
||
|
|
||
|
#include "fpmath.h"
|
||
|
#include "math.h"
|
||
|
#include "math_private.h"
|
||
|
|
||
|
/* Used in b_log.c and below. */
|
||
|
struct Double {
|
||
|
long double a;
|
||
|
long double b;
|
||
|
};
|
||
|
|
||
|
#include "b_logl.c"
|
||
|
#include "b_expl.c"
|
||
|
|
||
|
static const double zero = 0.;
|
||
|
static const volatile double tiny = 1e-300;
|
||
|
/*
|
||
|
* x >= 6
|
||
|
*
|
||
|
* Use the asymptotic approximation (Stirling's formula) adjusted for
|
||
|
* equal-ripples:
|
||
|
*
|
||
|
* log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
|
||
|
*
|
||
|
* Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
|
||
|
* premature round-off.
|
||
|
*
|
||
|
* Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
|
||
|
*/
|
||
|
|
||
|
/*
|
||
|
* The following is a decomposition of 0.5 * (log(2*pi) - 1) into the
|
||
|
* first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo. The
|
||
|
* variables are clearly misnamed.
|
||
|
*/
|
||
|
static const union IEEEl2bits
|
||
|
ln2pi_hiu = LD80C(0xd680000000000000, -2, 4.18945312500000000000e-01L),
|
||
|
ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L);
|
||
|
#define ln2pi_hi (ln2pi_hiu.e)
|
||
|
#define ln2pi_lo (ln2pi_lou.e)
|
||
|
|
||
|
static const union IEEEl2bits
|
||
|
Pa0u = LD80C(0xaaaaaaaaaaaaaaaa, -4, 8.33333333333333333288e-02L),
|
||
|
Pa1u = LD80C(0xb60b60b60b5fcd59, -9, -2.77777777777776516326e-03L),
|
||
|
Pa2u = LD80C(0xd00d00cffbb47014, -11, 7.93650793635429639018e-04L),
|
||
|
Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L),
|
||
|
Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11, 8.41749082509607342883e-04L),
|
||
|
Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L),
|
||
|
Pa6u = LD80C(0xd15a4ba04078d3f8, -8, 6.38893788027752396194e-03L),
|
||
|
Pa7u = LD80C(0xe877283110bcad95, -6, -2.83771309846297590312e-02L),
|
||
|
Pa8u = LD80C(0x8da97eed13717af8, -3, 1.38341887683837576925e-01L),
|
||
|
Pa9u = LD80C(0xf093b1c1584e30ce, -2, -4.69876818515470146031e-01L);
|
||
|
#define Pa0 (Pa0u.e)
|
||
|
#define Pa1 (Pa1u.e)
|
||
|
#define Pa2 (Pa2u.e)
|
||
|
#define Pa3 (Pa3u.e)
|
||
|
#define Pa4 (Pa4u.e)
|
||
|
#define Pa5 (Pa5u.e)
|
||
|
#define Pa6 (Pa6u.e)
|
||
|
#define Pa7 (Pa7u.e)
|
||
|
#define Pa8 (Pa8u.e)
|
||
|
#define Pa9 (Pa9u.e)
|
||
|
|
||
|
static struct Double
|
||
|
large_gam(long double x)
|
||
|
{
|
||
|
long double p, z, thi, tlo, xhi, xlo;
|
||
|
long double logx;
|
||
|
struct Double u;
|
||
|
|
||
|
z = 1 / (x * x);
|
||
|
p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
|
||
|
z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9))))))));
|
||
|
p = p / x;
|
||
|
|
||
|
u = __log__D(x);
|
||
|
u.a -= 1;
|
||
|
|
||
|
/* Split (x - 0.5) in high and low parts. */
|
||
|
x -= 0.5L;
|
||
|
xhi = (float)x;
|
||
|
xlo = x - xhi;
|
||
|
|
||
|
/* Compute t = (x-.5)*(log(x)-1) in extra precision. */
|
||
|
thi = xhi * u.a;
|
||
|
tlo = xlo * u.a + x * u.b;
|
||
|
|
||
|
/* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
|
||
|
tlo += ln2pi_lo;
|
||
|
tlo += p;
|
||
|
u.a = ln2pi_hi + tlo;
|
||
|
u.a += thi;
|
||
|
u.b = thi - u.a;
|
||
|
u.b += ln2pi_hi;
|
||
|
u.b += tlo;
|
||
|
return (u);
|
||
|
}
|
||
|
/*
|
||
|
* Rational approximation, A0 + x * x * P(x) / Q(x), on the interval
|
||
|
* [1.066.., 2.066..] accurate to 4.25e-19.
|
||
|
*
|
||
|
* Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
|
||
|
*/
|
||
|
static const union IEEEl2bits
|
||
|
a0_hiu = LD80C(0xe2b6e4153a57746c, -1, 8.85603194410888700265e-01L),
|
||
|
a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L);
|
||
|
#define a0_hi (a0_hiu.e)
|
||
|
#define a0_lo (a0_lou.e)
|
||
|
|
||
|
static const union IEEEl2bits
|
||
|
P0u = LD80C(0xdb629fb9bbdc1c1d, -2, 4.28486815855585429733e-01L),
|
||
|
P1u = LD80C(0xe6f4f9f5641aa6be, -3, 2.25543885805587730552e-01L),
|
||
|
P2u = LD80C(0xead1bd99fdaf7cc1, -6, 2.86644652514293482381e-02L),
|
||
|
P3u = LD80C(0x9ccc8b25838ab1e0, -8, 4.78512567772456362048e-03L),
|
||
|
P4u = LD80C(0x8f0c4383ef9ce72a, -9, 2.18273781132301146458e-03L),
|
||
|
P5u = LD80C(0xe732ab2c0a2778da, -13, 2.20487522485636008928e-04L),
|
||
|
P6u = LD80C(0xce70b27ca822b297, -16, 2.46095923774929264284e-05L),
|
||
|
P7u = LD80C(0xa309e2e16fb63663, -19, 2.42946473022376182921e-06L),
|
||
|
P8u = LD80C(0xaf9c110efb2c633d, -23, 1.63549217667765869987e-07L),
|
||
|
Q1u = LD80C(0xd4d7422719f48f15, -1, 8.31409582658993993626e-01L),
|
||
|
Q2u = LD80C(0xe13138ea404f1268, -5, -5.49785826915643198508e-02L),
|
||
|
Q3u = LD80C(0xd1c6cc91989352c0, -4, -1.02429960435139887683e-01L),
|
||
|
Q4u = LD80C(0xa7e9435a84445579, -7, 1.02484853505908820524e-02L),
|
||
|
Q5u = LD80C(0x83c7c34db89b7bda, -8, 4.02161632832052872697e-03L),
|
||
|
Q6u = LD80C(0xbed06bf6e1c14e5b, -11, -7.27898206351223022157e-04L),
|
||
|
Q7u = LD80C(0xef05bf841d4504c0, -18, 7.12342421869453515194e-06L),
|
||
|
Q8u = LD80C(0xf348d08a1ff53cb1, -19, 3.62522053809474067060e-06L);
|
||
|
#define P0 (P0u.e)
|
||
|
#define P1 (P1u.e)
|
||
|
#define P2 (P2u.e)
|
||
|
#define P3 (P3u.e)
|
||
|
#define P4 (P4u.e)
|
||
|
#define P5 (P5u.e)
|
||
|
#define P6 (P6u.e)
|
||
|
#define P7 (P7u.e)
|
||
|
#define P8 (P8u.e)
|
||
|
#define Q1 (Q1u.e)
|
||
|
#define Q2 (Q2u.e)
|
||
|
#define Q3 (Q3u.e)
|
||
|
#define Q4 (Q4u.e)
|
||
|
#define Q5 (Q5u.e)
|
||
|
#define Q6 (Q6u.e)
|
||
|
#define Q7 (Q7u.e)
|
||
|
#define Q8 (Q8u.e)
|
||
|
|
||
|
static struct Double
|
||
|
ratfun_gam(long double z, long double c)
|
||
|
{
|
||
|
long double p, q, thi, tlo;
|
||
|
struct Double r;
|
||
|
|
||
|
q = 1 + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
|
||
|
z * (Q6 + z * (Q7 + z * Q8)))))));
|
||
|
p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 +
|
||
|
z * (P6 + z * (P7 + z * P8)))))));
|
||
|
p = p / q;
|
||
|
|
||
|
/* Split z into high and low parts. */
|
||
|
thi = (float)z;
|
||
|
tlo = (z - thi) + c;
|
||
|
tlo *= (thi + z);
|
||
|
|
||
|
/* Split (z+c)^2 into high and low parts. */
|
||
|
thi *= thi;
|
||
|
q = thi;
|
||
|
thi = (float)thi;
|
||
|
tlo += (q - thi);
|
||
|
|
||
|
/* Split p/q into high and low parts. */
|
||
|
r.a = (float)p;
|
||
|
r.b = p - r.a;
|
||
|
|
||
|
tlo = tlo * p + thi * r.b + a0_lo;
|
||
|
thi *= r.a; /* t = (z+c)^2*(P/Q) */
|
||
|
r.a = (float)(thi + a0_hi);
|
||
|
r.b = ((a0_hi - r.a) + thi) + tlo;
|
||
|
return (r); /* r = a0 + t */
|
||
|
}
|
||
|
/*
|
||
|
* x < 6
|
||
|
*
|
||
|
* Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
|
||
|
* 2.066124]. Use a rational approximation centered at the minimum
|
||
|
* (x0+1) to ensure monotonicity.
|
||
|
*
|
||
|
* Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
|
||
|
* It also has correct monotonicity.
|
||
|
*/
|
||
|
static const union IEEEl2bits
|
||
|
xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
|
||
|
#define x0 (xm1u.e)
|
||
|
|
||
|
static const double
|
||
|
left = -0.3955078125; /* left boundary for rat. approx */
|
||
|
|
||
|
static long double
|
||
|
small_gam(long double x)
|
||
|
{
|
||
|
long double t, y, ym1;
|
||
|
struct Double yy, r;
|
||
|
|
||
|
y = x - 1;
|
||
|
|
||
|
if (y <= 1 + (left + x0)) {
|
||
|
yy = ratfun_gam(y - x0, 0);
|
||
|
return (yy.a + yy.b);
|
||
|
}
|
||
|
|
||
|
r.a = (float)y;
|
||
|
yy.a = r.a - 1;
|
||
|
y = y - 1 ;
|
||
|
r.b = yy.b = y - yy.a;
|
||
|
|
||
|
/* Argument reduction: G(x+1) = x*G(x) */
|
||
|
for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
|
||
|
t = r.a * yy.a;
|
||
|
r.b = r.a * yy.b + y * r.b;
|
||
|
r.a = (float)t;
|
||
|
r.b += (t - r.a);
|
||
|
}
|
||
|
|
||
|
/* Return r*tgamma(y). */
|
||
|
yy = ratfun_gam(y - x0, 0);
|
||
|
y = r.b * (yy.a + yy.b) + r.a * yy.b;
|
||
|
y += yy.a * r.a;
|
||
|
return (y);
|
||
|
}
|
||
|
/*
|
||
|
* Good on (0, 1+x0+left]. Accurate to 1 ulp.
|
||
|
*/
|
||
|
static long double
|
||
|
smaller_gam(long double x)
|
||
|
{
|
||
|
long double d, rhi, rlo, t, xhi, xlo;
|
||
|
struct Double r;
|
||
|
|
||
|
if (x < x0 + left) {
|
||
|
t = (float)x;
|
||
|
d = (t + x) * (x - t);
|
||
|
t *= t;
|
||
|
xhi = (float)(t + x);
|
||
|
xlo = x - xhi;
|
||
|
xlo += t;
|
||
|
xlo += d;
|
||
|
t = 1 - x0;
|
||
|
t += x;
|
||
|
d = 1 - x0;
|
||
|
d -= t;
|
||
|
d += x;
|
||
|
x = xhi + xlo;
|
||
|
} else {
|
||
|
xhi = (float)x;
|
||
|
xlo = x - xhi;
|
||
|
t = x - x0;
|
||
|
d = - x0 - t;
|
||
|
d += x;
|
||
|
}
|
||
|
|
||
|
r = ratfun_gam(t, d);
|
||
|
d = (float)(r.a / x);
|
||
|
r.a -= d * xhi;
|
||
|
r.a -= d * xlo;
|
||
|
r.a += r.b;
|
||
|
|
||
|
return (d + r.a / x);
|
||
|
}
|
||
|
/*
|
||
|
* x < 0
|
||
|
*
|
||
|
* Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
|
||
|
* At negative integers, return NaN and raise invalid.
|
||
|
*/
|
||
|
static const union IEEEl2bits
|
||
|
piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
|
||
|
#define pi (piu.e)
|
||
|
|
||
|
static long double
|
||
|
neg_gam(long double x)
|
||
|
{
|
||
|
int sgn = 1;
|
||
|
struct Double lg, lsine;
|
||
|
long double y, z;
|
||
|
|
||
|
y = ceill(x);
|
||
|
if (y == x) /* Negative integer. */
|
||
|
return ((x - x) / zero);
|
||
|
|
||
|
z = y - x;
|
||
|
if (z > 0.5)
|
||
|
z = 1 - z;
|
||
|
|
||
|
y = y / 2;
|
||
|
if (y == ceill(y))
|
||
|
sgn = -1;
|
||
|
|
||
|
if (z < 0.25)
|
||
|
z = sinpil(z);
|
||
|
else
|
||
|
z = cospil(0.5 - z);
|
||
|
|
||
|
/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
|
||
|
if (x < -1753) {
|
||
|
|
||
|
if (x < -1760)
|
||
|
return (sgn * tiny * tiny);
|
||
|
y = expl(lgammal(x) / 2);
|
||
|
y *= y;
|
||
|
return (sgn < 0 ? -y : y);
|
||
|
}
|
||
|
|
||
|
|
||
|
y = 1 - x;
|
||
|
if (1 - y == x)
|
||
|
y = tgammal(y);
|
||
|
else /* 1-x is inexact */
|
||
|
y = - x * tgammal(-x);
|
||
|
|
||
|
if (sgn < 0) y = -y;
|
||
|
return (pi / (y * z));
|
||
|
}
|
||
|
/*
|
||
|
* xmax comes from lgamma(xmax) - emax * log(2) = 0.
|
||
|
* static const float xmax = 35.040095f
|
||
|
* static const double xmax = 171.624376956302725;
|
||
|
* ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
|
||
|
* ld128: 1.75554834290446291700388921607020320e+03L,
|
||
|
*
|
||
|
* iota is a sloppy threshold to isolate x = 0.
|
||
|
*/
|
||
|
static const double xmax = 1755.54834290446291689;
|
||
|
static const double iota = 0x1p-116;
|
||
|
|
||
|
long double
|
||
|
tgammal(long double x)
|
||
|
{
|
||
|
struct Double u;
|
||
|
|
||
|
ENTERI();
|
||
|
|
||
|
if (x >= 6) {
|
||
|
if (x > xmax)
|
||
|
RETURNI(x / zero);
|
||
|
u = large_gam(x);
|
||
|
RETURNI(__exp__D(u.a, u.b));
|
||
|
}
|
||
|
|
||
|
if (x >= 1 + left + x0)
|
||
|
RETURNI(small_gam(x));
|
||
|
|
||
|
if (x > iota)
|
||
|
RETURNI(smaller_gam(x));
|
||
|
|
||
|
if (x > -iota) {
|
||
|
if (x != 0)
|
||
|
u.a = 1 - tiny; /* raise inexact */
|
||
|
RETURNI(1 / x);
|
||
|
}
|
||
|
|
||
|
if (!isfinite(x))
|
||
|
RETURNI(x - x); /* x is NaN or -Inf */
|
||
|
|
||
|
RETURNI(neg_gam(x));
|
||
|
}
|