1/*-
2 * Copyright (c) 1992, 1993
3 *	The Regents of the University of California.  All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 *    notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 *    notice, this list of conditions and the following disclaimer in the
12 *    documentation and/or other materials provided with the distribution.
13 * 3. All advertising materials mentioning features or use of this software
14 *    must display the following acknowledgement:
15 *	This product includes software developed by the University of
16 *	California, Berkeley and its contributors.
17 * 4. Neither the name of the University nor the names of its contributors
18 *    may be used to endorse or promote products derived from this software
19 *    without specific prior written permission.
20 *
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
31 * SUCH DAMAGE.
32 */
33
34#ifndef lint
35static char sccsid[] = "@(#)gamma.c	8.1 (Berkeley) 6/4/93";
36#endif /* not lint */
37#include <sys/cdefs.h>
38/* __FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_tgamma.c,v 1.7 2005/09/19 11:28:19 bde Exp $"); */
39
40/*
41 * This code by P. McIlroy, Oct 1992;
42 *
43 * The financial support of UUNET Communications Services is greatfully
44 * acknowledged.
45 */
46
47//#include <math.h>
48#include "../include/math.h"
49#include "mathimpl.h"
50#include <errno.h>
51
52/* METHOD:
53 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
54 * 	At negative integers, return +Inf, and set errno.
55 *
56 * x < 6.5:
57 *	Use argument reduction G(x+1) = xG(x) to reach the
58 *	range [1.066124,2.066124].  Use a rational
59 *	approximation centered at the minimum (x0+1) to
60 *	ensure monotonicity.
61 *
62 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
63 *	adjusted for equal-ripples:
64 *
65 *	log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
66 *
67 *	Keep extra precision in multiplying (x-.5)(log(x)-1), to
68 *	avoid premature round-off.
69 *
70 * Special values:
71 *	non-positive integer:	Set overflow trap; return +Inf;
72 *	x > 171.63:		Set overflow trap; return +Inf;
73 *	NaN: 			Set invalid trap;  return NaN
74 *
75 * Accuracy: Gamma(x) is accurate to within
76 *	x > 0:  error provably < 0.9ulp.
77 *	Maximum observed in 1,000,000 trials was .87ulp.
78 *	x < 0:
79 *	Maximum observed error < 4ulp in 1,000,000 trials.
80 */
81
82static double neg_gam(double);
83static double small_gam(double);
84static double smaller_gam(double);
85static struct Double large_gam(double);
86static struct Double ratfun_gam(double, double);
87
88/*
89 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
90 * [1.066.., 2.066..] accurate to 4.25e-19.
91 */
92#define LEFT -.3955078125	/* left boundary for rat. approx */
93#define x0 .461632144968362356785	/* xmin - 1 */
94
95#define a0_hi 0.88560319441088874992
96#define a0_lo -.00000000000000004996427036469019695
97#define P0	 6.21389571821820863029017800727e-01
98#define P1	 2.65757198651533466104979197553e-01
99#define P2	 5.53859446429917461063308081748e-03
100#define P3	 1.38456698304096573887145282811e-03
101#define P4	 2.40659950032711365819348969808e-03
102#define Q0	 1.45019531250000000000000000000e+00
103#define Q1	 1.06258521948016171343454061571e+00
104#define Q2	-2.07474561943859936441469926649e-01
105#define Q3	-1.46734131782005422506287573015e-01
106#define Q4	 3.07878176156175520361557573779e-02
107#define Q5	 5.12449347980666221336054633184e-03
108#define Q6	-1.76012741431666995019222898833e-03
109#define Q7	 9.35021023573788935372153030556e-05
110#define Q8	 6.13275507472443958924745652239e-06
111/*
112 * Constants for large x approximation (x in [6, Inf])
113 * (Accurate to 2.8*10^-19 absolute)
114 */
115#define lns2pi_hi 0.418945312500000
116#define lns2pi_lo -.000006779295327258219670263595
117#define Pa0	 8.33333333333333148296162562474e-02
118#define Pa1	-2.77777777774548123579378966497e-03
119#define Pa2	 7.93650778754435631476282786423e-04
120#define Pa3	-5.95235082566672847950717262222e-04
121#define Pa4	 8.41428560346653702135821806252e-04
122#define Pa5	-1.89773526463879200348872089421e-03
123#define Pa6	 5.69394463439411649408050664078e-03
124#define Pa7	-1.44705562421428915453880392761e-02
125
126static const double zero = 0., one = 1.0, tiny = 1e-300;
127
128double
129tgamma(x)
130	double x;
131{
132	struct Double u;
133
134	if (x >= 6) {
135		if(x > 171.63)
136			return(one/zero);
137		u = large_gam(x);
138		return(__exp__D(u.a, u.b));
139	} else if (x >= 1.0 + LEFT + x0)
140		return (small_gam(x));
141	else if (x > 1.e-17)
142		return (smaller_gam(x));
143	else if (x > -1.e-17) {
144		if (x == 0.0)
145			return (one/x);
146		one+1e-20;		/* Raise inexact flag. */
147		return (one/x);
148	} else if (!finite(x))
149		return (x*x);		/* x = NaN, -Inf */
150	else
151		return (neg_gam(x));
152}
153/*
154 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
155 */
156static struct Double
157large_gam(x)
158	double x;
159{
160	double z, p;
161	struct Double t, u, v;
162
163	z = one/(x*x);
164	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
165	p = p/x;
166
167	u = __log__D(x);
168	u.a -= one;
169	v.a = (x -= .5);
170	TRUNC(v.a);
171	v.b = x - v.a;
172	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
173	t.b = v.b*u.a + x*u.b;
174	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
175	t.b += lns2pi_lo; t.b += p;
176	u.a = lns2pi_hi + t.b; u.a += t.a;
177	u.b = t.a - u.a;
178	u.b += lns2pi_hi; u.b += t.b;
179	return (u);
180}
181/*
182 * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
183 * It also has correct monotonicity.
184 */
185static double
186small_gam(x)
187	double x;
188{
189	double y, ym1, t;
190	struct Double yy, r;
191	y = x - one;
192	ym1 = y - one;
193	if (y <= 1.0 + (LEFT + x0)) {
194		yy = ratfun_gam(y - x0, 0);
195		return (yy.a + yy.b);
196	}
197	r.a = y;
198	TRUNC(r.a);
199	yy.a = r.a - one;
200	y = ym1;
201	yy.b = r.b = y - yy.a;
202	/* Argument reduction: G(x+1) = x*G(x) */
203	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
204		t = r.a*yy.a;
205		r.b = r.a*yy.b + y*r.b;
206		r.a = t;
207		TRUNC(r.a);
208		r.b += (t - r.a);
209	}
210	/* Return r*tgamma(y). */
211	yy = ratfun_gam(y - x0, 0);
212	y = r.b*(yy.a + yy.b) + r.a*yy.b;
213	y += yy.a*r.a;
214	return (y);
215}
216/*
217 * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
218 */
219static double
220smaller_gam(x)
221	double x;
222{
223	double t, d;
224	struct Double r, xx;
225	if (x < x0 + LEFT) {
226		t = x, TRUNC(t);
227		d = (t+x)*(x-t);
228		t *= t;
229		xx.a = (t + x), TRUNC(xx.a);
230		xx.b = x - xx.a; xx.b += t; xx.b += d;
231		t = (one-x0); t += x;
232		d = (one-x0); d -= t; d += x;
233		x = xx.a + xx.b;
234	} else {
235		xx.a =  x, TRUNC(xx.a);
236		xx.b = x - xx.a;
237		t = x - x0;
238		d = (-x0 -t); d += x;
239	}
240	r = ratfun_gam(t, d);
241	d = r.a/x, TRUNC(d);
242	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
243	return (d + r.a/x);
244}
245/*
246 * returns (z+c)^2 * P(z)/Q(z) + a0
247 */
248static struct Double
249ratfun_gam(z, c)
250	double z, c;
251{
252	double p, q;
253	struct Double r, t;
254
255	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
256	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
257
258	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
259	p = p/q;
260	t.a = z, TRUNC(t.a);		/* t ~= z + c */
261	t.b = (z - t.a) + c;
262	t.b *= (t.a + z);
263	q = (t.a *= t.a);		/* t = (z+c)^2 */
264	TRUNC(t.a);
265	t.b += (q - t.a);
266	r.a = p, TRUNC(r.a);		/* r = P/Q */
267	r.b = p - r.a;
268	t.b = t.b*p + t.a*r.b + a0_lo;
269	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
270	r.a = t.a + a0_hi, TRUNC(r.a);
271	r.b = ((a0_hi-r.a) + t.a) + t.b;
272	return (r);			/* r = a0 + t */
273}
274
275static double
276neg_gam(x)
277	double x;
278{
279	int sgn = 1;
280	struct Double lg, lsine;
281	double y, z;
282
283	y = floor(x + .5);
284	if (y == x)		/* Negative integer. */
285		return (one/zero);
286	z = fabs(x - y);
287	y = .5*ceil(x);
288	if (y == ceil(y))
289		sgn = -1;
290	if (z < .25)
291		z = sin(M_PI*z);
292	else
293		z = cos(M_PI*(0.5-z));
294	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
295	if (x < -170) {
296		if (x < -190)
297			return ((double)sgn*tiny*tiny);
298		y = one - x;		/* exact: 128 < |x| < 255 */
299		lg = large_gam(y);
300		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
301		lg.a -= lsine.a;		/* exact (opposite signs) */
302		lg.b -= lsine.b;
303		y = -(lg.a + lg.b);
304		z = (y + lg.a) + lg.b;
305		y = __exp__D(y, z);
306		if (sgn < 0) y = -y;
307		return (y);
308	}
309	y = one-x;
310	if (one-y == x)
311		y = tgamma(y);
312	else		/* 1-x is inexact */
313		y = -x*tgamma(-x);
314	if (sgn < 0) y = -y;
315	return (M_PI / (y*z));
316}
317