5980be9b3c
This work is based in part on code from NetBSD libm, libc and kernel. The library is partly public domain and partly BSD-style licensed.
314 lines
8.7 KiB
C
314 lines
8.7 KiB
C
/* $NetBSD: b_tgamma.c,v 1.1.2.2 2012/05/09 18:22:36 riz Exp $ */
|
|
|
|
/*-
|
|
* 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.
|
|
* 4. 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.
|
|
*/
|
|
|
|
/* @(#)gamma.c 8.1 (Berkeley) 6/4/93 */
|
|
#include <sys/cdefs.h>
|
|
#if 0
|
|
__FBSDID("$FreeBSD: release/9.0.0/lib/msun/bsdsrc/b_tgamma.c 176449 2008-02-22 02:26:51Z das $");
|
|
#else
|
|
__RCSID("$NetBSD: b_tgamma.c,v 1.1.2.2 2012/05/09 18:22:36 riz Exp $");
|
|
#endif
|
|
|
|
/*
|
|
* This code by P. McIlroy, Oct 1992;
|
|
*
|
|
* The financial support of UUNET Communications Services is greatfully
|
|
* acknowledged.
|
|
*/
|
|
|
|
#include "math.h"
|
|
#include "math_private.h"
|
|
|
|
/* METHOD:
|
|
* x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
|
|
* At negative integers, return NaN and raise invalid.
|
|
*
|
|
* x < 6.5:
|
|
* 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.
|
|
*
|
|
* x >= 6.5: Use the asymptotic approximation (Stirling's formula)
|
|
* adjusted for equal-ripples:
|
|
*
|
|
* log(G(x)) ~= (x-.5)*(log(x)-1) + .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.
|
|
*
|
|
* 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.
|
|
*/
|
|
|
|
static double neg_gam(double);
|
|
static double small_gam(double);
|
|
static double smaller_gam(double);
|
|
static struct Double large_gam(double);
|
|
static struct Double ratfun_gam(double, double);
|
|
|
|
/*
|
|
* Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
|
|
* [1.066.., 2.066..] accurate to 4.25e-19.
|
|
*/
|
|
#define LEFT -.3955078125 /* left boundary for rat. approx */
|
|
#define x0 .461632144968362356785 /* xmin - 1 */
|
|
|
|
#define a0_hi 0.88560319441088874992
|
|
#define a0_lo -.00000000000000004996427036469019695
|
|
#define P0 6.21389571821820863029017800727e-01
|
|
#define P1 2.65757198651533466104979197553e-01
|
|
#define P2 5.53859446429917461063308081748e-03
|
|
#define P3 1.38456698304096573887145282811e-03
|
|
#define P4 2.40659950032711365819348969808e-03
|
|
#define Q0 1.45019531250000000000000000000e+00
|
|
#define Q1 1.06258521948016171343454061571e+00
|
|
#define Q2 -2.07474561943859936441469926649e-01
|
|
#define Q3 -1.46734131782005422506287573015e-01
|
|
#define Q4 3.07878176156175520361557573779e-02
|
|
#define Q5 5.12449347980666221336054633184e-03
|
|
#define Q6 -1.76012741431666995019222898833e-03
|
|
#define Q7 9.35021023573788935372153030556e-05
|
|
#define Q8 6.13275507472443958924745652239e-06
|
|
/*
|
|
* Constants for large x approximation (x in [6, Inf])
|
|
* (Accurate to 2.8*10^-19 absolute)
|
|
*/
|
|
#define lns2pi_hi 0.418945312500000
|
|
#define lns2pi_lo -.000006779295327258219670263595
|
|
#define Pa0 8.33333333333333148296162562474e-02
|
|
#define Pa1 -2.77777777774548123579378966497e-03
|
|
#define Pa2 7.93650778754435631476282786423e-04
|
|
#define Pa3 -5.95235082566672847950717262222e-04
|
|
#define Pa4 8.41428560346653702135821806252e-04
|
|
#define Pa5 -1.89773526463879200348872089421e-03
|
|
#define Pa6 5.69394463439411649408050664078e-03
|
|
#define Pa7 -1.44705562421428915453880392761e-02
|
|
|
|
static const double zero = 0., one = 1.0, tiny = 1e-300;
|
|
|
|
double
|
|
tgamma(double x)
|
|
{
|
|
struct Double u;
|
|
|
|
if (x >= 6) {
|
|
if(x > 171.63)
|
|
return (x / zero);
|
|
u = large_gam(x);
|
|
return(__exp__D(u.a, u.b));
|
|
} else if (x >= 1.0 + LEFT + x0)
|
|
return (small_gam(x));
|
|
else if (x > 1.e-17)
|
|
return (smaller_gam(x));
|
|
else if (x > -1.e-17) {
|
|
if (x != 0.0)
|
|
u.a = one - tiny; /* raise inexact */
|
|
return (one/x);
|
|
} else if (!finite(x))
|
|
return (x - x); /* x is NaN or -Inf */
|
|
else
|
|
return (neg_gam(x));
|
|
}
|
|
/*
|
|
* Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
|
|
*/
|
|
static struct Double
|
|
large_gam(double x)
|
|
{
|
|
double z, p;
|
|
struct Double t, u, v;
|
|
|
|
z = one/(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 -= one;
|
|
v.a = (x -= .5);
|
|
TRUNC(v.a);
|
|
v.b = x - v.a;
|
|
t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
|
|
t.b = v.b*u.a + x*u.b;
|
|
/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
|
|
t.b += lns2pi_lo; t.b += p;
|
|
u.a = lns2pi_hi + t.b; u.a += t.a;
|
|
u.b = t.a - u.a;
|
|
u.b += lns2pi_hi; u.b += t.b;
|
|
return (u);
|
|
}
|
|
/*
|
|
* Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
|
|
* It also has correct monotonicity.
|
|
*/
|
|
static double
|
|
small_gam(double x)
|
|
{
|
|
double y, ym1, t;
|
|
struct Double yy, r;
|
|
y = x - one;
|
|
ym1 = y - one;
|
|
if (y <= 1.0 + (LEFT + x0)) {
|
|
yy = ratfun_gam(y - x0, 0);
|
|
return (yy.a + yy.b);
|
|
}
|
|
r.a = y;
|
|
TRUNC(r.a);
|
|
yy.a = r.a - one;
|
|
y = ym1;
|
|
yy.b = r.b = y - yy.a;
|
|
/* Argument reduction: G(x+1) = x*G(x) */
|
|
for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
|
|
t = r.a*yy.a;
|
|
r.b = r.a*yy.b + y*r.b;
|
|
r.a = t;
|
|
TRUNC(r.a);
|
|
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 1ulp.
|
|
*/
|
|
static double
|
|
smaller_gam(double x)
|
|
{
|
|
double t, d;
|
|
struct Double r, xx;
|
|
if (x < x0 + LEFT) {
|
|
t = x, TRUNC(t);
|
|
d = (t+x)*(x-t);
|
|
t *= t;
|
|
xx.a = (t + x), TRUNC(xx.a);
|
|
xx.b = x - xx.a; xx.b += t; xx.b += d;
|
|
t = (one-x0); t += x;
|
|
d = (one-x0); d -= t; d += x;
|
|
x = xx.a + xx.b;
|
|
} else {
|
|
xx.a = x, TRUNC(xx.a);
|
|
xx.b = x - xx.a;
|
|
t = x - x0;
|
|
d = (-x0 -t); d += x;
|
|
}
|
|
r = ratfun_gam(t, d);
|
|
d = r.a/x, TRUNC(d);
|
|
r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
|
|
return (d + r.a/x);
|
|
}
|
|
/*
|
|
* returns (z+c)^2 * P(z)/Q(z) + a0
|
|
*/
|
|
static struct Double
|
|
ratfun_gam(double z, double c)
|
|
{
|
|
double p, q;
|
|
struct Double r, t;
|
|
|
|
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)));
|
|
|
|
/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
|
|
p = p/q;
|
|
t.a = z, TRUNC(t.a); /* t ~= z + c */
|
|
t.b = (z - t.a) + c;
|
|
t.b *= (t.a + z);
|
|
q = (t.a *= t.a); /* t = (z+c)^2 */
|
|
TRUNC(t.a);
|
|
t.b += (q - t.a);
|
|
r.a = p, TRUNC(r.a); /* r = P/Q */
|
|
r.b = p - r.a;
|
|
t.b = t.b*p + t.a*r.b + a0_lo;
|
|
t.a *= r.a; /* t = (z+c)^2*(P/Q) */
|
|
r.a = t.a + a0_hi, TRUNC(r.a);
|
|
r.b = ((a0_hi-r.a) + t.a) + t.b;
|
|
return (r); /* r = a0 + t */
|
|
}
|
|
|
|
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 = one - z;
|
|
y = 0.5 * y;
|
|
if (y == ceil(y))
|
|
sgn = -1;
|
|
if (z < .25)
|
|
z = sin(M_PI*z);
|
|
else
|
|
z = cos(M_PI*(0.5-z));
|
|
/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
|
|
if (x < -170) {
|
|
if (x < -190)
|
|
return ((double)sgn*tiny*tiny);
|
|
y = one - 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 = one-x;
|
|
if (one-y == x)
|
|
y = tgamma(y);
|
|
else /* 1-x is inexact */
|
|
y = -x*tgamma(-x);
|
|
if (sgn < 0) y = -y;
|
|
return (M_PI / (y*z));
|
|
}
|