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/* @(#)gamma.c	8.1 (Berkeley) 6/4/93 */
35#include <sys/cdefs.h>
36__FBSDID("$FreeBSD$");
37
38/*
39 * This code by P. McIlroy, Oct 1992;
40 *
41 * The financial support of UUNET Communications Services is greatfully
42 * acknowledged.
43 */
44
45#include <math.h>
46#include "mathimpl.h"
47
48/* METHOD:
49 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
50 * 	At negative integers, return NaN and raise invalid.
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 *	-Inf:			return NaN and raise invalid;
68 *	negative integer:	return NaN and raise invalid;
69 *	other x ~< 177.79:	return +-0 and raise underflow;
70 *	+-0:			return +-Inf and raise divide-by-zero;
71 *	finite x ~> 171.63:	return +Inf and raise overflow;
72 *	+Inf:			return +Inf;
73 *	NaN: 			return NaN.
74 *
75 * Accuracy: tgamma(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 (x / 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			u.a = one - tiny;	/* raise inexact */
146		return (one/x);
147	} else if (!finite(x))
148		return (x - x);		/* x is NaN or -Inf */
149	else
150		return (neg_gam(x));
151}
152/*
153 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
154 */
155static struct Double
156large_gam(x)
157	double x;
158{
159	double z, p;
160	struct Double t, u, v;
161
162	z = one/(x*x);
163	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
164	p = p/x;
165
166	u = __log__D(x);
167	u.a -= one;
168	v.a = (x -= .5);
169	TRUNC(v.a);
170	v.b = x - v.a;
171	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
172	t.b = v.b*u.a + x*u.b;
173	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
174	t.b += lns2pi_lo; t.b += p;
175	u.a = lns2pi_hi + t.b; u.a += t.a;
176	u.b = t.a - u.a;
177	u.b += lns2pi_hi; u.b += t.b;
178	return (u);
179}
180/*
181 * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
182 * It also has correct monotonicity.
183 */
184static double
185small_gam(x)
186	double x;
187{
188	double y, ym1, t;
189	struct Double yy, r;
190	y = x - one;
191	ym1 = y - one;
192	if (y <= 1.0 + (LEFT + x0)) {
193		yy = ratfun_gam(y - x0, 0);
194		return (yy.a + yy.b);
195	}
196	r.a = y;
197	TRUNC(r.a);
198	yy.a = r.a - one;
199	y = ym1;
200	yy.b = r.b = y - yy.a;
201	/* Argument reduction: G(x+1) = x*G(x) */
202	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
203		t = r.a*yy.a;
204		r.b = r.a*yy.b + y*r.b;
205		r.a = t;
206		TRUNC(r.a);
207		r.b += (t - r.a);
208	}
209	/* Return r*tgamma(y). */
210	yy = ratfun_gam(y - x0, 0);
211	y = r.b*(yy.a + yy.b) + r.a*yy.b;
212	y += yy.a*r.a;
213	return (y);
214}
215/*
216 * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
217 */
218static double
219smaller_gam(x)
220	double x;
221{
222	double t, d;
223	struct Double r, xx;
224	if (x < x0 + LEFT) {
225		t = x, TRUNC(t);
226		d = (t+x)*(x-t);
227		t *= t;
228		xx.a = (t + x), TRUNC(xx.a);
229		xx.b = x - xx.a; xx.b += t; xx.b += d;
230		t = (one-x0); t += x;
231		d = (one-x0); d -= t; d += x;
232		x = xx.a + xx.b;
233	} else {
234		xx.a =  x, TRUNC(xx.a);
235		xx.b = x - xx.a;
236		t = x - x0;
237		d = (-x0 -t); d += x;
238	}
239	r = ratfun_gam(t, d);
240	d = r.a/x, TRUNC(d);
241	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
242	return (d + r.a/x);
243}
244/*
245 * returns (z+c)^2 * P(z)/Q(z) + a0
246 */
247static struct Double
248ratfun_gam(z, c)
249	double z, c;
250{
251	double p, q;
252	struct Double r, t;
253
254	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
255	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
256
257	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
258	p = p/q;
259	t.a = z, TRUNC(t.a);		/* t ~= z + c */
260	t.b = (z - t.a) + c;
261	t.b *= (t.a + z);
262	q = (t.a *= t.a);		/* t = (z+c)^2 */
263	TRUNC(t.a);
264	t.b += (q - t.a);
265	r.a = p, TRUNC(r.a);		/* r = P/Q */
266	r.b = p - r.a;
267	t.b = t.b*p + t.a*r.b + a0_lo;
268	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
269	r.a = t.a + a0_hi, TRUNC(r.a);
270	r.b = ((a0_hi-r.a) + t.a) + t.b;
271	return (r);			/* r = a0 + t */
272}
273
274static double
275neg_gam(x)
276	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