/*-
* 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));
}