1#include "FEATURE/uwin"
2
3#if !_UWIN || _lib_gamma
4
5void _STUB_gamma(){}
6
7#else
8
9/*-
10 * Copyright (c) 1992, 1993
11 *	The Regents of the University of California.  All rights reserved.
12 *
13 * Redistribution and use in source and binary forms, with or without
14 * modification, are permitted provided that the following conditions
15 * are met:
16 * 1. Redistributions of source code must retain the above copyright
17 *    notice, this list of conditions and the following disclaimer.
18 * 2. Redistributions in binary form must reproduce the above copyright
19 *    notice, this list of conditions and the following disclaimer in the
20 *    documentation and/or other materials provided with the distribution.
21 * 3. Neither the name of the University nor the names of its contributors
22 *    may be used to endorse or promote products derived from this software
23 *    without specific prior written permission.
24 *
25 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
26 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
27 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
28 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
29 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
30 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
31 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
32 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
33 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
34 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
35 * SUCH DAMAGE.
36 */
37
38#ifndef lint
39static char sccsid[] = "@(#)gamma.c	8.1 (Berkeley) 6/4/93";
40#endif /* not lint */
41
42/*
43 * This code by P. McIlroy, Oct 1992;
44 *
45 * The financial support of UUNET Communications Services is greatfully
46 * acknowledged.
47 */
48
49#define gamma	______gamma
50
51#include <math.h>
52#include <errno.h>
53#include "mathimpl.h"
54
55#undef	gamma
56
57/* METHOD:
58 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
59 * 	At negative integers, return +Inf, and set errno.
60 *
61 * x < 6.5:
62 *	Use argument reduction G(x+1) = xG(x) to reach the
63 *	range [1.066124,2.066124].  Use a rational
64 *	approximation centered at the minimum (x0+1) to
65 *	ensure monotonicity.
66 *
67 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
68 *	adjusted for equal-ripples:
69 *
70 *	log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
71 *
72 *	Keep extra precision in multiplying (x-.5)(log(x)-1), to
73 *	avoid premature round-off.
74 *
75 * Special values:
76 *	non-positive integer:	Set overflow trap; return +Inf;
77 *	x > 171.63:		Set overflow trap; return +Inf;
78 *	NaN: 			Set invalid trap;  return NaN
79 *
80 * Accuracy: Gamma(x) is accurate to within
81 *	x > 0:  error provably < 0.9ulp.
82 *	Maximum observed in 1,000,000 trials was .87ulp.
83 *	x < 0:
84 *	Maximum observed error < 4ulp in 1,000,000 trials.
85 */
86
87static double neg_gam __P((double));
88static double small_gam __P((double));
89static double smaller_gam __P((double));
90static struct Double large_gam __P((double));
91static struct Double ratfun_gam __P((double, double));
92
93/*
94 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
95 * [1.066.., 2.066..] accurate to 4.25e-19.
96 */
97#define LEFT -.3955078125	/* left boundary for rat. approx */
98#define x0 .461632144968362356785	/* xmin - 1 */
99
100#define a0_hi 0.88560319441088874992
101#define a0_lo -.00000000000000004996427036469019695
102#define P0	 6.21389571821820863029017800727e-01
103#define P1	 2.65757198651533466104979197553e-01
104#define P2	 5.53859446429917461063308081748e-03
105#define P3	 1.38456698304096573887145282811e-03
106#define P4	 2.40659950032711365819348969808e-03
107#define Q0	 1.45019531250000000000000000000e+00
108#define Q1	 1.06258521948016171343454061571e+00
109#define Q2	-2.07474561943859936441469926649e-01
110#define Q3	-1.46734131782005422506287573015e-01
111#define Q4	 3.07878176156175520361557573779e-02
112#define Q5	 5.12449347980666221336054633184e-03
113#define Q6	-1.76012741431666995019222898833e-03
114#define Q7	 9.35021023573788935372153030556e-05
115#define Q8	 6.13275507472443958924745652239e-06
116/*
117 * Constants for large x approximation (x in [6, Inf])
118 * (Accurate to 2.8*10^-19 absolute)
119 */
120#define lns2pi_hi 0.418945312500000
121#define lns2pi_lo -.000006779295327258219670263595
122#define Pa0	 8.33333333333333148296162562474e-02
123#define Pa1	-2.77777777774548123579378966497e-03
124#define Pa2	 7.93650778754435631476282786423e-04
125#define Pa3	-5.95235082566672847950717262222e-04
126#define Pa4	 8.41428560346653702135821806252e-04
127#define Pa5	-1.89773526463879200348872089421e-03
128#define Pa6	 5.69394463439411649408050664078e-03
129#define Pa7	-1.44705562421428915453880392761e-02
130
131static const double zero = 0., one = 1.0, tiny = 1e-300;
132static int endian;
133/*
134 * TRUNC sets trailing bits in a floating-point number to zero.
135 * is a temporary variable.
136 */
137#if defined(vax) || defined(tahoe)
138#define _IEEE		0
139#define TRUNC(x)	x = (double) (float) (x)
140#else
141#define _IEEE		1
142#define TRUNC(x)	*(((int *) &x) + endian) &= 0xf8000000
143#define infnan(x)	0.0
144#endif
145
146extern double gamma(x)
147	double x;
148{
149	struct Double u;
150	endian = (*(int *) &one) ? 1 : 0;
151
152	if (x >= 6) {
153		if(x > 171.63)
154			return(one/zero);
155		u = large_gam(x);
156		return(__exp__D(u.a, u.b));
157	} else if (x >= 1.0 + LEFT + x0)
158		return (small_gam(x));
159	else if (x > 1.e-17)
160		return (smaller_gam(x));
161	else if (x > -1.e-17) {
162		if (x == 0.0)
163			if (!_IEEE) return (infnan(ERANGE));
164			else return (one/x);
165		one+1e-20;		/* Raise inexact flag. */
166		return (one/x);
167	} else if (!finite(x)) {
168		if (_IEEE)		/* x = NaN, -Inf */
169			return (x*x);
170		else
171			return (infnan(EDOM));
172	 } else
173		return (neg_gam(x));
174}
175/*
176 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
177 */
178static struct Double
179large_gam(x)
180	double x;
181{
182	double z, p;
183	struct Double t, u, v;
184
185	z = one/(x*x);
186	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
187	p = p/x;
188
189	u = __log__D(x);
190	u.a -= one;
191	v.a = (x -= .5);
192	TRUNC(v.a);
193	v.b = x - v.a;
194	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
195	t.b = v.b*u.a + x*u.b;
196	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
197	t.b += lns2pi_lo; t.b += p;
198	u.a = lns2pi_hi + t.b; u.a += t.a;
199	u.b = t.a - u.a;
200	u.b += lns2pi_hi; u.b += t.b;
201	return (u);
202}
203/*
204 * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
205 * It also has correct monotonicity.
206 */
207static double
208small_gam(x)
209	double x;
210{
211	double y, ym1, t;
212	struct Double yy, r;
213	y = x - one;
214	ym1 = y - one;
215	if (y <= 1.0 + (LEFT + x0)) {
216		yy = ratfun_gam(y - x0, 0);
217		return (yy.a + yy.b);
218	}
219	r.a = y;
220	TRUNC(r.a);
221	yy.a = r.a - one;
222	y = ym1;
223	yy.b = r.b = y - yy.a;
224	/* Argument reduction: G(x+1) = x*G(x) */
225	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
226		t = r.a*yy.a;
227		r.b = r.a*yy.b + y*r.b;
228		r.a = t;
229		TRUNC(r.a);
230		r.b += (t - r.a);
231	}
232	/* Return r*gamma(y). */
233	yy = ratfun_gam(y - x0, 0);
234	y = r.b*(yy.a + yy.b) + r.a*yy.b;
235	y += yy.a*r.a;
236	return (y);
237}
238/*
239 * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
240 */
241static double
242smaller_gam(x)
243	double x;
244{
245	double t, d;
246	struct Double r, xx;
247	if (x < x0 + LEFT) {
248		t = x, TRUNC(t);
249		d = (t+x)*(x-t);
250		t *= t;
251		xx.a = (t + x), TRUNC(xx.a);
252		xx.b = x - xx.a; xx.b += t; xx.b += d;
253		t = (one-x0); t += x;
254		d = (one-x0); d -= t; d += x;
255		x = xx.a + xx.b;
256	} else {
257		xx.a =  x, TRUNC(xx.a);
258		xx.b = x - xx.a;
259		t = x - x0;
260		d = (-x0 -t); d += x;
261	}
262	r = ratfun_gam(t, d);
263	d = r.a/x, TRUNC(d);
264	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
265	return (d + r.a/x);
266}
267/*
268 * returns (z+c)^2 * P(z)/Q(z) + a0
269 */
270static struct Double
271ratfun_gam(z, c)
272	double z, c;
273{
274	double p, q;
275	struct Double r, t;
276
277	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
278	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
279
280	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
281	p = p/q;
282	t.a = z, TRUNC(t.a);		/* t ~= z + c */
283	t.b = (z - t.a) + c;
284	t.b *= (t.a + z);
285	q = (t.a *= t.a);		/* t = (z+c)^2 */
286	TRUNC(t.a);
287	t.b += (q - t.a);
288	r.a = p, TRUNC(r.a);		/* r = P/Q */
289	r.b = p - r.a;
290	t.b = t.b*p + t.a*r.b + a0_lo;
291	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
292	r.a = t.a + a0_hi, TRUNC(r.a);
293	r.b = ((a0_hi-r.a) + t.a) + t.b;
294	return (r);			/* r = a0 + t */
295}
296
297static double
298neg_gam(x)
299	double x;
300{
301	int sgn = 1;
302	struct Double lg, lsine;
303	double y, z;
304
305	y = floor(x + .5);
306	if (y == x)		/* Negative integer. */
307		if(!_IEEE)
308			return (infnan(ERANGE));
309		else
310			return (one/zero);
311	z = fabs(x - y);
312	y = .5*ceil(x);
313	if (y == ceil(y))
314		sgn = -1;
315	if (z < .25)
316		z = sin(M_PI*z);
317	else
318		z = cos(M_PI*(0.5-z));
319	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
320	if (x < -170) {
321		if (x < -190)
322			return ((double)sgn*tiny*tiny);
323		y = one - x;		/* exact: 128 < |x| < 255 */
324		lg = large_gam(y);
325		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
326		lg.a -= lsine.a;		/* exact (opposite signs) */
327		lg.b -= lsine.b;
328		y = -(lg.a + lg.b);
329		z = (y + lg.a) + lg.b;
330		y = __exp__D(y, z);
331		if (sgn < 0) y = -y;
332		return (y);
333	}
334	y = one-x;
335	if (one-y == x)
336		y = gamma(y);
337	else		/* 1-x is inexact */
338		y = -x*gamma(-x);
339	if (sgn < 0) y = -y;
340	return (M_PI / (y*z));
341}
342
343#endif
344