/*-
* 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, contained the following
* attribution:
*
* This code by P. McIlroy, Oct 1992;
*
* The financial support of UUNET Communications Services is greatfully
* acknowledged.
*
* The algorithm remains, but the code has been re-arranged to facilitate
* porting to other precisions.
*/
/* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
#include <float.h>
#include "math.h"
#include "math_private.h"
/* Used in b_log.c and below. */
struct Double {
double a;
double b;
};
#include "b_log.c"
#include "b_exp.c"
/*
* The range is broken into several subranges. Each is handled by its
* helper functions.
*
* x >= 6.0: large_gam(x)
* 6.0 > x >= xleft: small_gam(x) where xleft = 1 + left + x0.
* xleft > x > iota: smaller_gam(x) where iota = 1e-17.
* iota > x > -itoa: Handle x near 0.
* -iota > x : neg_gam
*
* Special values:
* -Inf: return NaN and raise invalid;
* negative integer: return NaN and raise invalid;
* other x ~< 177.79: return +-0 and raise underflow;
* +-0: return +-Inf and raise divide-by-zero;
* finite x ~> 171.63: return +Inf and raise overflow;
* +Inf: return +Inf;
* NaN: return NaN.
*
* Accuracy: tgamma(x) is accurate to within
* x > 0: error provably < 0.9ulp.
* Maximum observed in 1,000,000 trials was .87ulp.
* x < 0:
* Maximum observed error < 4ulp in 1,000,000 trials.
*/
/*
* Constants for large x approximation (x in [6, Inf])
* (Accurate to 2.8*10^-19 absolute)
*/
static const double zero = 0.;
static const volatile double tiny = 1e-300;
/*
* x >= 6
*
* Use the asymptotic approximation (Stirling's formula) adjusted fof
* 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.
*/
static const double
ln2pi_hi = 0.41894531250000000,
ln2pi_lo = -6.7792953272582197e-6,
Pa0 = 8.3333333333333329e-02, /* 0x3fb55555, 0x55555555 */
Pa1 = -2.7777777777735404e-03, /* 0xbf66c16c, 0x16c145ec */
Pa2 = 7.9365079044114095e-04, /* 0x3f4a01a0, 0x183de82d */
Pa3 = -5.9523715464225254e-04, /* 0xbf438136, 0x0e681f62 */
Pa4 = 8.4161391899445698e-04, /* 0x3f4b93f8, 0x21042a13 */
Pa5 = -1.9065246069191080e-03, /* 0xbf5f3c8b, 0x357cb64e */
Pa6 = 5.9047708485785158e-03, /* 0x3f782f99, 0xdaf5d65f */
Pa7 = -1.6484018705183290e-02; /* 0xbf90e12f, 0xc4fb4df0 */
static struct Double
large_gam(double x)
{
double p, z, thi, tlo, xhi, xlo;
struct Double u;
z = 1 / (x * x);
p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
z * (Pa6 + z * Pa7))))));
p = p / x;
u = __log__D(x);
u.a -= 1;
/* Split (x - 0.5) in high and low parts. */
x -= 0.5;
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 double
#if 0
a0_hi = 8.8560319441088875e-1,
a0_lo = -4.9964270364690197e-17,
#else
a0_hi = 8.8560319441088875e-01, /* 0x3fec56dc, 0x82a74aef */
a0_lo = -4.9642368725563397e-17, /* 0xbc8c9deb, 0xaa64afc3 */
#endif
P0 = 6.2138957182182086e-1,
P1 = 2.6575719865153347e-1,
P2 = 5.5385944642991746e-3,
P3 = 1.3845669830409657e-3,
P4 = 2.4065995003271137e-3,
Q0 = 1.4501953125000000e+0,
Q1 = 1.0625852194801617e+0,
Q2 = -2.0747456194385994e-1,
Q3 = -1.4673413178200542e-1,
Q4 = 3.0787817615617552e-2,
Q5 = 5.1244934798066622e-3,
Q6 = -1.7601274143166700e-3,
Q7 = 9.3502102357378894e-5,
Q8 = 6.1327550747244396e-6;
static struct Double
ratfun_gam(double z, double c)
{
double p, q, thi, tlo;
struct Double r;
q = Q0 + 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)));
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 double
left = -0.3955078125, /* left boundary for rat. approx */
x0 = 4.6163214496836236e-1; /* xmin - 1 */
static double
small_gam(double x)
{
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 double
smaller_gam(double x)
{
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 double
neg_gam(double x)
{
int sgn = 1;
struct Double lg, lsine;
double y, z;
y = ceil(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 == ceil(y))
sgn = -1;
if (z < 0.25)
z = sinpi(z);
else
z = cospi(0.5 - z);
/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
if (x < -170) {
if (x < -190)
return (sgn * tiny * tiny);
y = 1 - x; /* exact: 128 < |x| < 255 */
lg = large_gam(y);
lsine = __log__D(M_PI / z); /* = TRUNC(log(u)) + small */
lg.a -= lsine.a; /* exact (opposite signs) */
lg.b -= lsine.b;
y = -(lg.a + lg.b);
z = (y + lg.a) + lg.b;
y = __exp__D(y, z);
if (sgn < 0) y = -y;
return (y);
}
y = 1 - x;
if (1 - y == x)
y = tgamma(y);
else /* 1-x is inexact */
y = - x * tgamma(-x);
if (sgn < 0) y = -y;
return (M_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 = 171.624376956302725;
static const double iota = 0x1p-56;
double
tgamma(double x)
{
struct Double u;
if (x >= 6) {
if (x > xmax)
return (x / zero);
u = large_gam(x);
return (__exp__D(u.a, u.b));
}
if (x >= 1 + left + x0)
return (small_gam(x));
if (x > iota)
return (smaller_gam(x));
if (x > -iota) {
if (x != 0.)
u.a = 1 - tiny; /* raise inexact */
return (1 / x);
}
if (!isfinite(x))
return (x - x); /* x is NaN or -Inf */
return (neg_gam(x));
}
#if (LDBL_MANT_DIG == 53)
__weak_reference(tgamma, tgammal);
#endif
