1/*	$NetBSD$	*/
2
3/*-
4 * Copyright (c) 1992, 1993
5 *	The Regents of the University of California.  All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 *    notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 *    notice, this list of conditions and the following disclaimer in the
14 *    documentation and/or other materials provided with the distribution.
15 * 3. All advertising materials mentioning features or use of this software
16 *    must display the following acknowledgement:
17 *	This product includes software developed by the University of
18 *	California, Berkeley and its contributors.
19 * 4. Neither the name of the University nor the names of its contributors
20 *    may be used to endorse or promote products derived from this software
21 *    without specific prior written permission.
22 *
23 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
24 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
25 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
26 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
27 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
28 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
29 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
30 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
31 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
32 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
33 * SUCH DAMAGE.
34 */
35
36/* @(#)gamma.c	8.1 (Berkeley) 6/4/93 */
37#include <sys/cdefs.h>
38#if 0
39__FBSDID("$FreeBSD: release/9.0.0/lib/msun/bsdsrc/b_tgamma.c 176449 2008-02-22 02:26:51Z das $");
40#else
41__RCSID("$NetBSD$");
42#endif
43
44/*
45 * This code by P. McIlroy, Oct 1992;
46 *
47 * The financial support of UUNET Communications Services is greatfully
48 * acknowledged.
49 */
50
51#include "math.h"
52#include "math_private.h"
53
54/* METHOD:
55 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
56 * 	At negative integers, return NaN and raise invalid.
57 *
58 * x < 6.5:
59 *	Use argument reduction G(x+1) = xG(x) to reach the
60 *	range [1.066124,2.066124].  Use a rational
61 *	approximation centered at the minimum (x0+1) to
62 *	ensure monotonicity.
63 *
64 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
65 *	adjusted for equal-ripples:
66 *
67 *	log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
68 *
69 *	Keep extra precision in multiplying (x-.5)(log(x)-1), to
70 *	avoid premature round-off.
71 *
72 * Special values:
73 *	-Inf:			return NaN and raise invalid;
74 *	negative integer:	return NaN and raise invalid;
75 *	other x ~< 177.79:	return +-0 and raise underflow;
76 *	+-0:			return +-Inf and raise divide-by-zero;
77 *	finite x ~> 171.63:	return +Inf and raise overflow;
78 *	+Inf:			return +Inf;
79 *	NaN: 			return NaN.
80 *
81 * Accuracy: tgamma(x) is accurate to within
82 *	x > 0:  error provably < 0.9ulp.
83 *	Maximum observed in 1,000,000 trials was .87ulp.
84 *	x < 0:
85 *	Maximum observed error < 4ulp in 1,000,000 trials.
86 */
87
88static double neg_gam(double);
89static double small_gam(double);
90static double smaller_gam(double);
91static struct Double large_gam(double);
92static struct Double ratfun_gam(double, double);
93
94/*
95 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
96 * [1.066.., 2.066..] accurate to 4.25e-19.
97 */
98#define LEFT -.3955078125	/* left boundary for rat. approx */
99#define x0 .461632144968362356785	/* xmin - 1 */
100
101#define a0_hi 0.88560319441088874992
102#define a0_lo -.00000000000000004996427036469019695
103#define P0	 6.21389571821820863029017800727e-01
104#define P1	 2.65757198651533466104979197553e-01
105#define P2	 5.53859446429917461063308081748e-03
106#define P3	 1.38456698304096573887145282811e-03
107#define P4	 2.40659950032711365819348969808e-03
108#define Q0	 1.45019531250000000000000000000e+00
109#define Q1	 1.06258521948016171343454061571e+00
110#define Q2	-2.07474561943859936441469926649e-01
111#define Q3	-1.46734131782005422506287573015e-01
112#define Q4	 3.07878176156175520361557573779e-02
113#define Q5	 5.12449347980666221336054633184e-03
114#define Q6	-1.76012741431666995019222898833e-03
115#define Q7	 9.35021023573788935372153030556e-05
116#define Q8	 6.13275507472443958924745652239e-06
117/*
118 * Constants for large x approximation (x in [6, Inf])
119 * (Accurate to 2.8*10^-19 absolute)
120 */
121#define lns2pi_hi 0.418945312500000
122#define lns2pi_lo -.000006779295327258219670263595
123#define Pa0	 8.33333333333333148296162562474e-02
124#define Pa1	-2.77777777774548123579378966497e-03
125#define Pa2	 7.93650778754435631476282786423e-04
126#define Pa3	-5.95235082566672847950717262222e-04
127#define Pa4	 8.41428560346653702135821806252e-04
128#define Pa5	-1.89773526463879200348872089421e-03
129#define Pa6	 5.69394463439411649408050664078e-03
130#define Pa7	-1.44705562421428915453880392761e-02
131
132static const double zero = 0., one = 1.0, tiny = 1e-300;
133
134double
135tgamma(double x)
136{
137	struct Double u;
138
139	if (x >= 6) {
140		if(x > 171.63)
141			return (x / zero);
142		u = large_gam(x);
143		return(__exp__D(u.a, u.b));
144	} else if (x >= 1.0 + LEFT + x0)
145		return (small_gam(x));
146	else if (x > 1.e-17)
147		return (smaller_gam(x));
148	else if (x > -1.e-17) {
149		if (x != 0.0)
150			u.a = one - tiny;	/* raise inexact */
151		return (one/x);
152	} else if (!finite(x))
153		return (x - x);		/* x is NaN or -Inf */
154	else
155		return (neg_gam(x));
156}
157/*
158 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
159 */
160static struct Double
161large_gam(double x)
162{
163	double z, p;
164	struct Double t, u, v;
165
166	z = one/(x*x);
167	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
168	p = p/x;
169
170	u = __log__D(x);
171	u.a -= one;
172	v.a = (x -= .5);
173	TRUNC(v.a);
174	v.b = x - v.a;
175	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
176	t.b = v.b*u.a + x*u.b;
177	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
178	t.b += lns2pi_lo; t.b += p;
179	u.a = lns2pi_hi + t.b; u.a += t.a;
180	u.b = t.a - u.a;
181	u.b += lns2pi_hi; u.b += t.b;
182	return (u);
183}
184/*
185 * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
186 * It also has correct monotonicity.
187 */
188static double
189small_gam(double x)
190{
191	double y, ym1, t;
192	struct Double yy, r;
193	y = x - one;
194	ym1 = y - one;
195	if (y <= 1.0 + (LEFT + x0)) {
196		yy = ratfun_gam(y - x0, 0);
197		return (yy.a + yy.b);
198	}
199	r.a = y;
200	TRUNC(r.a);
201	yy.a = r.a - one;
202	y = ym1;
203	yy.b = r.b = y - yy.a;
204	/* Argument reduction: G(x+1) = x*G(x) */
205	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
206		t = r.a*yy.a;
207		r.b = r.a*yy.b + y*r.b;
208		r.a = t;
209		TRUNC(r.a);
210		r.b += (t - r.a);
211	}
212	/* Return r*tgamma(y). */
213	yy = ratfun_gam(y - x0, 0);
214	y = r.b*(yy.a + yy.b) + r.a*yy.b;
215	y += yy.a*r.a;
216	return (y);
217}
218/*
219 * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
220 */
221static double
222smaller_gam(double x)
223{
224	double t, d;
225	struct Double r, xx;
226	if (x < x0 + LEFT) {
227		t = x, TRUNC(t);
228		d = (t+x)*(x-t);
229		t *= t;
230		xx.a = (t + x), TRUNC(xx.a);
231		xx.b = x - xx.a; xx.b += t; xx.b += d;
232		t = (one-x0); t += x;
233		d = (one-x0); d -= t; d += x;
234		x = xx.a + xx.b;
235	} else {
236		xx.a =  x, TRUNC(xx.a);
237		xx.b = x - xx.a;
238		t = x - x0;
239		d = (-x0 -t); d += x;
240	}
241	r = ratfun_gam(t, d);
242	d = r.a/x, TRUNC(d);
243	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
244	return (d + r.a/x);
245}
246/*
247 * returns (z+c)^2 * P(z)/Q(z) + a0
248 */
249static struct Double
250ratfun_gam(double z, double 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(double x)
277{
278	int sgn = 1;
279	struct Double lg, lsine;
280	double y, z;
281
282	y = ceil(x);
283	if (y == x)		/* Negative integer. */
284		return ((x - x) / zero);
285	z = y - x;
286	if (z > 0.5)
287		z = one - z;
288	y = 0.5 * y;
289	if (y == ceil(y))
290		sgn = -1;
291	if (z < .25)
292		z = sin(M_PI*z);
293	else
294		z = cos(M_PI*(0.5-z));
295	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
296	if (x < -170) {
297		if (x < -190)
298			return ((double)sgn*tiny*tiny);
299		y = one - x;		/* exact: 128 < |x| < 255 */
300		lg = large_gam(y);
301		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
302		lg.a -= lsine.a;		/* exact (opposite signs) */
303		lg.b -= lsine.b;
304		y = -(lg.a + lg.b);
305		z = (y + lg.a) + lg.b;
306		y = __exp__D(y, z);
307		if (sgn < 0) y = -y;
308		return (y);
309	}
310	y = one-x;
311	if (one-y == x)
312		y = tgamma(y);
313	else		/* 1-x is inexact */
314		y = -x*tgamma(-x);
315	if (sgn < 0) y = -y;
316	return (M_PI / (y*z));
317}
318