1/*      $NetBSD: n_gamma.c,v 1.6 2006/11/24 21:15:54 wiz Exp $ */
2/*-
3 * Copyright (c) 1992, 1993
4 *	The Regents of the University of California.  All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 *    notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 *    notice, this list of conditions and the following disclaimer in the
13 *    documentation and/or other materials provided with the distribution.
14 * 3. Neither the name of the University nor the names of its contributors
15 *    may be used to endorse or promote products derived from this software
16 *    without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
21 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * SUCH DAMAGE.
29 */
30
31#ifndef lint
32#if 0
33static char sccsid[] = "@(#)gamma.c	8.1 (Berkeley) 6/4/93";
34#endif
35#endif /* not lint */
36
37/*
38 * This code by P. McIlroy, Oct 1992;
39 *
40 * The financial support of UUNET Communications Services is gratefully
41 * acknowledged.
42 */
43
44#include <math.h>
45#include "mathimpl.h"
46#include <errno.h>
47
48/* METHOD:
49 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
50 * 	At negative integers, return +Inf, and set errno.
51 *
52 * x < 6.5:
53 *	Use argument reduction G(x+1) = xG(x) to reach the
54 *	range [1.066124,2.066124].  Use a rational
55 *	approximation centered at the minimum (x0+1) to
56 *	ensure monotonicity.
57 *
58 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
59 *	adjusted for equal-ripples:
60 *
61 *	log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
62 *
63 *	Keep extra precision in multiplying (x-.5)(log(x)-1), to
64 *	avoid premature round-off.
65 *
66 * Special values:
67 *	non-positive integer:	Set overflow trap; return +Inf;
68 *	x > 171.63:		Set overflow trap; return +Inf;
69 *	NaN: 			Set invalid trap;  return NaN
70 *
71 * Accuracy: Gamma(x) is accurate to within
72 *	x > 0:  error provably < 0.9ulp.
73 *	Maximum observed in 1,000,000 trials was .87ulp.
74 *	x < 0:
75 *	Maximum observed error < 4ulp in 1,000,000 trials.
76 */
77
78static double neg_gam (double);
79static double small_gam (double);
80static double smaller_gam (double);
81static struct Double large_gam (double);
82static struct Double ratfun_gam (double, double);
83
84/*
85 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
86 * [1.066.., 2.066..] accurate to 4.25e-19.
87 */
88#define LEFT -.3955078125	/* left boundary for rat. approx */
89#define x0 .461632144968362356785	/* xmin - 1 */
90
91#define a0_hi 0.88560319441088874992
92#define a0_lo -.00000000000000004996427036469019695
93#define P0	 6.21389571821820863029017800727e-01
94#define P1	 2.65757198651533466104979197553e-01
95#define P2	 5.53859446429917461063308081748e-03
96#define P3	 1.38456698304096573887145282811e-03
97#define P4	 2.40659950032711365819348969808e-03
98#define Q0	 1.45019531250000000000000000000e+00
99#define Q1	 1.06258521948016171343454061571e+00
100#define Q2	-2.07474561943859936441469926649e-01
101#define Q3	-1.46734131782005422506287573015e-01
102#define Q4	 3.07878176156175520361557573779e-02
103#define Q5	 5.12449347980666221336054633184e-03
104#define Q6	-1.76012741431666995019222898833e-03
105#define Q7	 9.35021023573788935372153030556e-05
106#define Q8	 6.13275507472443958924745652239e-06
107/*
108 * Constants for large x approximation (x in [6, Inf])
109 * (Accurate to 2.8*10^-19 absolute)
110 */
111#define lns2pi_hi 0.418945312500000
112#define lns2pi_lo -.000006779295327258219670263595
113#define Pa0	 8.33333333333333148296162562474e-02
114#define Pa1	-2.77777777774548123579378966497e-03
115#define Pa2	 7.93650778754435631476282786423e-04
116#define Pa3	-5.95235082566672847950717262222e-04
117#define Pa4	 8.41428560346653702135821806252e-04
118#define Pa5	-1.89773526463879200348872089421e-03
119#define Pa6	 5.69394463439411649408050664078e-03
120#define Pa7	-1.44705562421428915453880392761e-02
121
122static const double zero = 0., one = 1.0, tiny = _TINY;
123/*
124 * TRUNC sets trailing bits in a floating-point number to zero.
125 * is a temporary variable.
126 */
127#if defined(__vax__) || defined(tahoe)
128#define _IEEE		0
129#define TRUNC(x)	x = (double) (float) (x)
130#else
131static int endian;
132#define _IEEE		1
133#define TRUNC(x)	*(((int *) &x) + endian) &= 0xf8000000
134#define infnan(x)	0.0
135#endif
136
137double
138gamma(x)
139	double x;
140{
141	double b;
142	struct Double u;
143#if _IEEE
144	int endian = (*(int *) &one) ? 1 : 0;
145#endif
146
147	if (x >= 6) {
148		if(x > 171.63)
149			return(one/zero);
150		u = large_gam(x);
151		return(__exp__D(u.a, u.b));
152	} else if (x >= 1.0 + LEFT + x0) {
153		return (small_gam(x));
154	} else if (x > 1.e-17) {
155		return (smaller_gam(x));
156	} else if (x > -1.e-17) {
157		if (x == 0.0) {
158			if (!_IEEE) return (infnan(ERANGE));
159			else return (one/x);
160		}
161		b =one+1e-20;		/* Raise inexact flag. ??? -ragge */
162		return (one/x);
163	} else if (!finite(x)) {
164		if (_IEEE)		/* x = NaN, -Inf */
165			return (x*x);
166		else
167			return (infnan(EDOM));
168	 } else
169		return (neg_gam(x));
170}
171/*
172 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
173 */
174static struct Double
175large_gam(double x)
176{
177	double z, p;
178	struct Double t, u, v;
179
180	z = one/(x*x);
181	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
182	p = p/x;
183
184	u = __log__D(x);
185	u.a -= one;
186	v.a = (x -= .5);
187	TRUNC(v.a);
188	v.b = x - v.a;
189	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
190	t.b = v.b*u.a + x*u.b;
191	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
192	t.b += lns2pi_lo; t.b += p;
193	u.a = lns2pi_hi + t.b; u.a += t.a;
194	u.b = t.a - u.a;
195	u.b += lns2pi_hi; u.b += t.b;
196	return (u);
197}
198/*
199 * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
200 * It also has correct monotonicity.
201 */
202static double
203small_gam(double x)
204{
205	double y, ym1, t;
206	struct Double yy, r;
207	y = x - one;
208	ym1 = y - one;
209	if (y <= 1.0 + (LEFT + x0)) {
210		yy = ratfun_gam(y - x0, 0);
211		return (yy.a + yy.b);
212	}
213	r.a = y;
214	TRUNC(r.a);
215	yy.a = r.a - one;
216	y = ym1;
217	yy.b = r.b = y - yy.a;
218	/* Argument reduction: G(x+1) = x*G(x) */
219	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
220		t = r.a*yy.a;
221		r.b = r.a*yy.b + y*r.b;
222		r.a = t;
223		TRUNC(r.a);
224		r.b += (t - r.a);
225	}
226	/* Return r*gamma(y). */
227	yy = ratfun_gam(y - x0, 0);
228	y = r.b*(yy.a + yy.b) + r.a*yy.b;
229	y += yy.a*r.a;
230	return (y);
231}
232/*
233 * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
234 */
235static double
236smaller_gam(double x)
237{
238	double t, d;
239	struct Double r, xx;
240	if (x < x0 + LEFT) {
241		t = x, TRUNC(t);
242		d = (t+x)*(x-t);
243		t *= t;
244		xx.a = (t + x), TRUNC(xx.a);
245		xx.b = x - xx.a; xx.b += t; xx.b += d;
246		t = (one-x0); t += x;
247		d = (one-x0); d -= t; d += x;
248		x = xx.a + xx.b;
249	} else {
250		xx.a =  x, TRUNC(xx.a);
251		xx.b = x - xx.a;
252		t = x - x0;
253		d = (-x0 -t); d += x;
254	}
255	r = ratfun_gam(t, d);
256	d = r.a/x, TRUNC(d);
257	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
258	return (d + r.a/x);
259}
260/*
261 * returns (z+c)^2 * P(z)/Q(z) + a0
262 */
263static struct Double
264ratfun_gam(double z, double c)
265{
266	double p, q;
267	struct Double r, t;
268
269	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
270	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
271
272	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
273	p = p/q;
274	t.a = z, TRUNC(t.a);		/* t ~= z + c */
275	t.b = (z - t.a) + c;
276	t.b *= (t.a + z);
277	q = (t.a *= t.a);		/* t = (z+c)^2 */
278	TRUNC(t.a);
279	t.b += (q - t.a);
280	r.a = p, TRUNC(r.a);		/* r = P/Q */
281	r.b = p - r.a;
282	t.b = t.b*p + t.a*r.b + a0_lo;
283	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
284	r.a = t.a + a0_hi, TRUNC(r.a);
285	r.b = ((a0_hi-r.a) + t.a) + t.b;
286	return (r);			/* r = a0 + t */
287}
288
289static double
290neg_gam(double x)
291{
292	int sgn = 1;
293	struct Double lg, lsine;
294	double y, z;
295
296	y = floor(x + .5);
297	if (y == x) {		/* Negative integer. */
298		if(!_IEEE)
299			return (infnan(ERANGE));
300		else
301			return (one/zero);
302	}
303	z = fabs(x - y);
304	y = .5*ceil(x);
305	if (y == ceil(y))
306		sgn = -1;
307	if (z < .25)
308		z = sin(M_PI*z);
309	else
310		z = cos(M_PI*(0.5-z));
311	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
312	if (x < -170) {
313		if (x < -190)
314			return ((double)sgn*tiny*tiny);
315		y = one - x;		/* exact: 128 < |x| < 255 */
316		lg = large_gam(y);
317		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
318		lg.a -= lsine.a;		/* exact (opposite signs) */
319		lg.b -= lsine.b;
320		y = -(lg.a + lg.b);
321		z = (y + lg.a) + lg.b;
322		y = __exp__D(y, z);
323		if (sgn < 0) y = -y;
324		return (y);
325	}
326	y = one-x;
327	if (one-y == x)
328		y = gamma(y);
329	else		/* 1-x is inexact */
330		y = -x*gamma(-x);
331	if (sgn < 0) y = -y;
332	return (M_PI / (y*z));
333}
334