xref: /csrg-svn/lib/libm/common_source/gamma.c (revision 57452)

/*-
 * Copyright (c) 1992 The Regents of the University of California.
 * All rights reserved.
 *
 * %sccs.include.redist.c%
 */

#ifndef lint
static char sccsid[] = "@(#)gamma.c	5.4 (Berkeley) 01/10/93";
#endif /* not lint */

/*
 * This code by P. McIlroy, Oct 1992;
 *
 * The financial support of UUNET Communications Services is greatfully
 * acknowledged.
 */

#include <math.h>
#include "mathimpl.h"
#include <errno.h>

/* METHOD:
 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
 * 	At negative integers, return +Inf, and set errno.
 *
 * 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:
 *	non-positive integer:	Set overflow trap; return +Inf;
 *	x > 171.63:		Set overflow trap; return +Inf;
 *	NaN: 			Set invalid trap;  return NaN
 *
 * Accuracy: Gamma(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 __P((double));
static double small_gam __P((double));
static double smaller_gam __P((double));
static struct Double large_gam __P((double));
static struct Double ratfun_gam __P((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;
static int endian;
/*
 * TRUNC sets trailing bits in a floating-point number to zero.
 * is a temporary variable.
 */
#if defined(vax) || defined(tahoe)
#define _IEEE		0
#define TRUNC(x)	x = (double) (float) (x)
#else
#define _IEEE		1
#define TRUNC(x)	*(((int *) &x) + endian) &= 0xf8000000
#define infnan(x)	0.0
#endif

double
gamma(x)
	double x;
{
	struct Double u;
	endian = (*(int *) &one) ? 1 : 0;

	if (x >= 6) {
		if(x > 171.63)
			return(one/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)
			if (!_IEEE) return (infnan(ERANGE));
			else return (one/x);
		one+1e-20;		/* Raise inexact flag. */
		return (one/x);
	} else if (!finite(x)) {
		if (_IEEE)		/* x = NaN, -Inf */
			return (x*x);
		else
			return (infnan(EDOM));
	 } else
		return (neg_gam(x));
}
/*
 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
 */
static struct Double
large_gam(x)
	double x;
{
	double z, p;
	int i;
	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(x)
	double x;
{
	double y, ym1, t, x1;
	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*gamma(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(x)
	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(z, c)
	double z, c;
{
	int i;
	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(x)
	double x;
{
	int sgn = 1;
	struct Double lg, lsine;
	double y, z;

	y = floor(x + .5);
	if (y == x)		/* Negative integer. */
		if(!_IEEE)
			return (infnan(ERANGE));
		else
			return (one/zero);
	z = fabs(x - y);
	y = .5*ceil(x);
	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 = gamma(y);
	else		/* 1-x is inexact */
		y = -x*gamma(-x);
	if (sgn < 0) y = -y;
	return (M_PI / (y*z));
}