n_lgamma.c revision 1.6 
1/*      $NetBSD: n_lgamma.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[] = "@(#)lgamma.c	8.2 (Berkeley) 11/30/93";
34#endif
35#endif /* not lint */
36
37/*
38 * Coded by Peter McIlroy, Nov 1992;
39 *
40 * The financial support of UUNET Communications Services is gratefully
41 * acknowledged.
42 */
43
44#include <math.h>
45#include <errno.h>
46
47#include "mathimpl.h"
48
49/* Log gamma function.
50 * Error:  x > 0 error < 1.3ulp.
51 *	   x > 4, error < 1ulp.
52 *	   x > 9, error < .6ulp.
53 * 	   x < 0, all bets are off. (When G(x) ~ 1, log(G(x)) ~ 0)
54 * Method:
55 *	x > 6:
56 *		Use the asymptotic expansion (Stirling's Formula)
57 *	0 < x < 6:
58 *		Use gamma(x+1) = x*gamma(x) for argument reduction.
59 *		Use rational approximation in
60 *		the range 1.2, 2.5
61 *		Two approximations are used, one centered at the
62 *		minimum to ensure monotonicity; one centered at 2
63 *		to maintain small relative error.
64 *	x < 0:
65 *		Use the reflection formula,
66 *		G(1-x)G(x) = PI/sin(PI*x)
67 * Special values:
68 *	non-positive integer	returns +Inf.
69 *	NaN			returns NaN
70*/
71#if defined(__vax__) || defined(tahoe)
72#define _IEEE		0
73/* double and float have same size exponent field */
74#define TRUNC(x)	x = (double) (float) (x)
75#else
76static int endian;
77#define _IEEE		1
78#define TRUNC(x)	*(((int *) &x) + endian) &= 0xf8000000
79#define infnan(x)	0.0
80#endif
81
82static double small_lgam(double);
83static double large_lgam(double);
84static double neg_lgam(double);
85static const double one = 1.0;
86int signgam;
87
88#define UNDERFL (1e-1020 * 1e-1020)
89
90#define LEFT	(1.0 - (x0 + .25))
91#define RIGHT	(x0 - .218)
92/*
93 * Constants for approximation in [1.244,1.712]
94*/
95#define x0	0.461632144968362356785
96#define x0_lo	-.000000000000000015522348162858676890521
97#define a0_hi	-0.12148629128932952880859
98#define a0_lo	.0000000007534799204229502
99#define r0	-2.771227512955130520e-002
100#define r1	-2.980729795228150847e-001
101#define r2	-3.257411333183093394e-001
102#define r3	-1.126814387531706041e-001
103#define r4	-1.129130057170225562e-002
104#define r5	-2.259650588213369095e-005
105#define s0	 1.714457160001714442e+000
106#define s1	 2.786469504618194648e+000
107#define s2	 1.564546365519179805e+000
108#define s3	 3.485846389981109850e-001
109#define s4	 2.467759345363656348e-002
110/*
111 * Constants for approximation in [1.71, 2.5]
112*/
113#define a1_hi	4.227843350984671344505727574870e-01
114#define a1_lo	4.670126436531227189e-18
115#define p0	3.224670334241133695662995251041e-01
116#define p1	3.569659696950364669021382724168e-01
117#define p2	1.342918716072560025853732668111e-01
118#define p3	1.950702176409779831089963408886e-02
119#define p4	8.546740251667538090796227834289e-04
120#define q0	1.000000000000000444089209850062e+00
121#define q1	1.315850076960161985084596381057e+00
122#define q2	6.274644311862156431658377186977e-01
123#define q3	1.304706631926259297049597307705e-01
124#define q4	1.102815279606722369265536798366e-02
125#define q5	2.512690594856678929537585620579e-04
126#define q6	-1.003597548112371003358107325598e-06
127/*
128 * Stirling's Formula, adjusted for equal-ripple. x in [6,Inf].
129*/
130#define lns2pi	.418938533204672741780329736405
131#define pb0	 8.33333333333333148296162562474e-02
132#define pb1	-2.77777777774548123579378966497e-03
133#define pb2	 7.93650778754435631476282786423e-04
134#define pb3	-5.95235082566672847950717262222e-04
135#define pb4	 8.41428560346653702135821806252e-04
136#define pb5	-1.89773526463879200348872089421e-03
137#define pb6	 5.69394463439411649408050664078e-03
138#define pb7	-1.44705562421428915453880392761e-02
139
140__pure double
141lgamma(double x)
142{
143	double r;
144
145	signgam = 1;
146#if _IEEE
147	endian = ((*(int *) &one)) ? 1 : 0;
148#endif
149
150	if (!finite(x)) {
151		if (_IEEE)
152			return (x+x);
153		else return (infnan(EDOM));
154	}
155
156	if (x > 6 + RIGHT) {
157		r = large_lgam(x);
158		return (r);
159	} else if (x > 1e-16)
160		return (small_lgam(x));
161	else if (x > -1e-16) {
162		if (x < 0)
163			signgam = -1, x = -x;
164		return (-log(x));
165	} else
166		return (neg_lgam(x));
167}
168
169static double
170large_lgam(double x)
171{
172	double z, p, x1;
173	struct Double t, u, v;
174	u = __log__D(x);
175	u.a -= 1.0;
176	if (x > 1e15) {
177		v.a = x - 0.5;
178		TRUNC(v.a);
179		v.b = (x - v.a) - 0.5;
180		t.a = u.a*v.a;
181		t.b = x*u.b + v.b*u.a;
182		if (_IEEE == 0 && !finite(t.a))
183			return(infnan(ERANGE));
184		return(t.a + t.b);
185	}
186	x1 = 1./x;
187	z = x1*x1;
188	p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7))))));
189					/* error in approximation = 2.8e-19 */
190
191	p = p*x1;			/* error < 2.3e-18 absolute */
192					/* 0 < p < 1/64 (at x = 5.5) */
193	v.a = x = x - 0.5;
194	TRUNC(v.a);			/* truncate v.a to 26 bits. */
195	v.b = x - v.a;
196	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
197	t.b = v.b*u.a + x*u.b;
198	t.b += p; t.b += lns2pi;	/* return t + lns2pi + p */
199	return (t.a + t.b);
200}
201
202static double
203small_lgam(double x)
204{
205	int x_int;
206	double y, z, t, r = 0, p, q, hi, lo;
207	struct Double rr;
208	x_int = (x + .5);
209	y = x - x_int;
210	if (x_int <= 2 && y > RIGHT) {
211		t = y - x0;
212		y--; x_int++;
213		goto CONTINUE;
214	} else if (y < -LEFT) {
215		t = y +(1.0-x0);
216CONTINUE:
217		z = t - x0_lo;
218		p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5))));
219		q = s0+z*(s1+z*(s2+z*(s3+z*s4)));
220		r = t*(z*(p/q) - x0_lo);
221		t = .5*t*t;
222		z = 1.0;
223		switch (x_int) {
224		case 6:	z  = (y + 5);
225		case 5:	z *= (y + 4);
226		case 4:	z *= (y + 3);
227		case 3:	z *= (y + 2);
228			rr = __log__D(z);
229			rr.b += a0_lo; rr.a += a0_hi;
230			return(((r+rr.b)+t+rr.a));
231		case 2: return(((r+a0_lo)+t)+a0_hi);
232		case 0: r -= log1p(x);
233		default: rr = __log__D(x);
234			rr.a -= a0_hi; rr.b -= a0_lo;
235			return(((r - rr.b) + t) - rr.a);
236		}
237	} else {
238		p = p0+y*(p1+y*(p2+y*(p3+y*p4)));
239		q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6)))));
240		p = p*(y/q);
241		t = (double)(float) y;
242		z = y-t;
243		hi = (double)(float) (p+a1_hi);
244		lo = a1_hi - hi; lo += p; lo += a1_lo;
245		r = lo*y + z*hi;	/* q + r = y*(a0+p/q) */
246		q = hi*t;
247		z = 1.0;
248		switch (x_int) {
249		case 6:	z  = (y + 5);
250		case 5:	z *= (y + 4);
251		case 4:	z *= (y + 3);
252		case 3:	z *= (y + 2);
253			rr = __log__D(z);
254			r += rr.b; r += q;
255			return(rr.a + r);
256		case 2:	return (q+ r);
257		case 0: rr = __log__D(x);
258			r -= rr.b; r -= log1p(x);
259			r += q; r-= rr.a;
260			return(r);
261		default: rr = __log__D(x);
262			r -= rr.b;
263			q -= rr.a;
264			return (r+q);
265		}
266	}
267}
268
269static double
270neg_lgam(double x)
271{
272	int xi;
273	double y, z, zero = 0.0;
274
275	/* avoid destructive cancellation as much as possible */
276	if (x > -170) {
277		xi = x;
278		if (xi == x) {
279			if (_IEEE)
280				return(one/zero);
281			else
282				return(infnan(ERANGE));
283		}
284		y = gamma(x);
285		if (y < 0)
286			y = -y, signgam = -1;
287		return (log(y));
288	}
289	z = floor(x + .5);
290	if (z == x) {		/* convention: G(-(integer)) -> +Inf */
291		if (_IEEE)
292			return (one/zero);
293		else
294			return (infnan(ERANGE));
295	}
296	y = .5*ceil(x);
297	if (y == ceil(y))
298		signgam = -1;
299	x = -x;
300	z = fabs(x + z);	/* 0 < z <= .5 */
301	if (z < .25)
302		z = sin(M_PI*z);
303	else
304		z = cos(M_PI*(0.5-z));
305	z = log(M_PI/(z*x));
306	y = large_lgam(x);
307	return (z - y);
308}
309