xref: /minix3/lib/libm/noieee_src/n_gamma.c (revision 84d9c625bfea59e274550651111ae9edfdc40fbd)
1 /*      $NetBSD: n_gamma.c,v 1.9 2013/11/09 21:41:03 christos 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
33 static char sccsid[] = "@(#)gamma.c	8.1 (Berkeley) 6/4/93";
34 #endif
35 #endif /* not lint */
36 
37 /*
38  * This code by P. McIlroy, Oct 1992;
39  *
40  * The financial support of UUNET Communications Services is gratefully
41  * acknowledged.
42  */
43 
44 #include <math.h>
45 #include "mathimpl.h"
46 #include <errno.h>
47 
48 /* METHOD:
49  * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
50  * 	At negative integers, return +Inf, and set errno.
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  *	non-positive integer:	Set overflow trap; return +Inf;
68  *	x > 171.63:		Set overflow trap; return +Inf;
69  *	NaN: 			Set invalid trap;  return NaN
70  *
71  * Accuracy: Gamma(x) is accurate to within
72  *	x > 0:  error provably < 0.9ulp.
73  *	Maximum observed in 1,000,000 trials was .87ulp.
74  *	x < 0:
75  *	Maximum observed error < 4ulp in 1,000,000 trials.
76  */
77 
78 static double neg_gam (double);
79 static double small_gam (double);
80 static double smaller_gam (double);
81 static struct Double large_gam (double);
82 static struct Double ratfun_gam (double, double);
83 
84 /*
85  * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
86  * [1.066.., 2.066..] accurate to 4.25e-19.
87  */
88 #define LEFT -.3955078125	/* left boundary for rat. approx */
89 #define x0 .461632144968362356785	/* xmin - 1 */
90 
91 #define a0_hi 0.88560319441088874992
92 #define a0_lo -.00000000000000004996427036469019695
93 #define P0	 6.21389571821820863029017800727e-01
94 #define P1	 2.65757198651533466104979197553e-01
95 #define P2	 5.53859446429917461063308081748e-03
96 #define P3	 1.38456698304096573887145282811e-03
97 #define P4	 2.40659950032711365819348969808e-03
98 #define Q0	 1.45019531250000000000000000000e+00
99 #define Q1	 1.06258521948016171343454061571e+00
100 #define Q2	-2.07474561943859936441469926649e-01
101 #define Q3	-1.46734131782005422506287573015e-01
102 #define Q4	 3.07878176156175520361557573779e-02
103 #define Q5	 5.12449347980666221336054633184e-03
104 #define Q6	-1.76012741431666995019222898833e-03
105 #define Q7	 9.35021023573788935372153030556e-05
106 #define Q8	 6.13275507472443958924745652239e-06
107 /*
108  * Constants for large x approximation (x in [6, Inf])
109  * (Accurate to 2.8*10^-19 absolute)
110  */
111 #define lns2pi_hi 0.418945312500000
112 #define lns2pi_lo -.000006779295327258219670263595
113 #define Pa0	 8.33333333333333148296162562474e-02
114 #define Pa1	-2.77777777774548123579378966497e-03
115 #define Pa2	 7.93650778754435631476282786423e-04
116 #define Pa3	-5.95235082566672847950717262222e-04
117 #define Pa4	 8.41428560346653702135821806252e-04
118 #define Pa5	-1.89773526463879200348872089421e-03
119 #define Pa6	 5.69394463439411649408050664078e-03
120 #define Pa7	-1.44705562421428915453880392761e-02
121 
122 static const double zero = 0., one = 1.0, tiny = _TINY;
123 /*
124  * TRUNC sets trailing bits in a floating-point number to zero.
125  * is a temporary variable.
126  */
127 #if defined(__vax__) || defined(tahoe)
128 #define _IEEE		0
129 #define TRUNC(x)	x = (double) (float) (x)
130 #else
131 static int endian;
132 #define _IEEE		1
133 #define TRUNC(x)	*(((int *) &x) + endian) &= 0xf8000000
134 #define infnan(x)	0.0
135 #endif
136 
137 double
gamma(double x)138 gamma(double x)
139 {
140 	double b;
141 	struct Double u;
142 #if _IEEE
143 	int endian = (*(int *) &one) ? 1 : 0;
144 #endif
145 
146 	if (x >= 6) {
147 		if(x > 171.63)
148 			return(one/zero);
149 		u = large_gam(x);
150 		return(__exp__D(u.a, u.b));
151 	} else if (x >= 1.0 + LEFT + x0) {
152 		return (small_gam(x));
153 	} else if (x > 1.e-17) {
154 		return (smaller_gam(x));
155 	} else if (x > -1.e-17) {
156 		if (x == 0.0) {
157 			if (!_IEEE) return (infnan(ERANGE));
158 			else return (one/x);
159 		}
160 		b =one+1e-20;		/* Raise inexact flag. ??? -ragge */
161 		__USE(b);
162 		return (one/x);
163 	} else if (!finite(x)) {
164 		if (_IEEE)		/* x = NaN, -Inf */
165 			return (x*x);
166 		else
167 			return (infnan(EDOM));
168 	 } else
169 		return (neg_gam(x));
170 }
171 /*
172  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
173  */
174 static struct Double
large_gam(double x)175 large_gam(double x)
176 {
177 	double z, p;
178 	struct Double t, u, v;
179 
180 	z = one/(x*x);
181 	p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
182 	p = p/x;
183 
184 	u = __log__D(x);
185 	u.a -= one;
186 	v.a = (x -= .5);
187 	TRUNC(v.a);
188 	v.b = x - v.a;
189 	t.a = v.a*u.a;			/* t = (x-.5)*(log(x)-1) */
190 	t.b = v.b*u.a + x*u.b;
191 	/* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
192 	t.b += lns2pi_lo; t.b += p;
193 	u.a = lns2pi_hi + t.b; u.a += t.a;
194 	u.b = t.a - u.a;
195 	u.b += lns2pi_hi; u.b += t.b;
196 	return (u);
197 }
198 /*
199  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
200  * It also has correct monotonicity.
201  */
202 static double
small_gam(double x)203 small_gam(double x)
204 {
205 	double y, ym1, t;
206 	struct Double yy, r;
207 	y = x - one;
208 	ym1 = y - one;
209 	if (y <= 1.0 + (LEFT + x0)) {
210 		yy = ratfun_gam(y - x0, 0);
211 		return (yy.a + yy.b);
212 	}
213 	r.a = y;
214 	TRUNC(r.a);
215 	yy.a = r.a - one;
216 	y = ym1;
217 	yy.b = r.b = y - yy.a;
218 	/* Argument reduction: G(x+1) = x*G(x) */
219 	for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
220 		t = r.a*yy.a;
221 		r.b = r.a*yy.b + y*r.b;
222 		r.a = t;
223 		TRUNC(r.a);
224 		r.b += (t - r.a);
225 	}
226 	/* Return r*gamma(y). */
227 	yy = ratfun_gam(y - x0, 0);
228 	y = r.b*(yy.a + yy.b) + r.a*yy.b;
229 	y += yy.a*r.a;
230 	return (y);
231 }
232 /*
233  * Good on (0, 1+x0+LEFT].  Accurate to 1ulp.
234  */
235 static double
smaller_gam(double x)236 smaller_gam(double x)
237 {
238 	double t, d;
239 	struct Double r, xx;
240 	if (x < x0 + LEFT) {
241 		t = x, TRUNC(t);
242 		d = (t+x)*(x-t);
243 		t *= t;
244 		xx.a = (t + x), TRUNC(xx.a);
245 		xx.b = x - xx.a; xx.b += t; xx.b += d;
246 		t = (one-x0); t += x;
247 		d = (one-x0); d -= t; d += x;
248 		x = xx.a + xx.b;
249 	} else {
250 		xx.a =  x, TRUNC(xx.a);
251 		xx.b = x - xx.a;
252 		t = x - x0;
253 		d = (-x0 -t); d += x;
254 	}
255 	r = ratfun_gam(t, d);
256 	d = r.a/x, TRUNC(d);
257 	r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
258 	return (d + r.a/x);
259 }
260 /*
261  * returns (z+c)^2 * P(z)/Q(z) + a0
262  */
263 static struct Double
ratfun_gam(double z,double c)264 ratfun_gam(double z, double c)
265 {
266 	double p, q;
267 	struct Double r, t;
268 
269 	q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
270 	p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
271 
272 	/* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
273 	p = p/q;
274 	t.a = z, TRUNC(t.a);		/* t ~= z + c */
275 	t.b = (z - t.a) + c;
276 	t.b *= (t.a + z);
277 	q = (t.a *= t.a);		/* t = (z+c)^2 */
278 	TRUNC(t.a);
279 	t.b += (q - t.a);
280 	r.a = p, TRUNC(r.a);		/* r = P/Q */
281 	r.b = p - r.a;
282 	t.b = t.b*p + t.a*r.b + a0_lo;
283 	t.a *= r.a;			/* t = (z+c)^2*(P/Q) */
284 	r.a = t.a + a0_hi, TRUNC(r.a);
285 	r.b = ((a0_hi-r.a) + t.a) + t.b;
286 	return (r);			/* r = a0 + t */
287 }
288 
289 static double
neg_gam(double x)290 neg_gam(double x)
291 {
292 	int sgn = 1;
293 	struct Double lg, lsine;
294 	double y, z;
295 
296 	y = floor(x + .5);
297 	if (y == x) {		/* Negative integer. */
298 		if(!_IEEE)
299 			return (infnan(ERANGE));
300 		else
301 			return (one/zero);
302 	}
303 	z = fabs(x - y);
304 	y = .5*ceil(x);
305 	if (y == ceil(y))
306 		sgn = -1;
307 	if (z < .25)
308 		z = sin(M_PI*z);
309 	else
310 		z = cos(M_PI*(0.5-z));
311 	/* Special case: G(1-x) = Inf; G(x) may be nonzero. */
312 	if (x < -170) {
313 		if (x < -190)
314 			return ((double)sgn*tiny*tiny);
315 		y = one - x;		/* exact: 128 < |x| < 255 */
316 		lg = large_gam(y);
317 		lsine = __log__D(M_PI/z);	/* = TRUNC(log(u)) + small */
318 		lg.a -= lsine.a;		/* exact (opposite signs) */
319 		lg.b -= lsine.b;
320 		y = -(lg.a + lg.b);
321 		z = (y + lg.a) + lg.b;
322 		y = __exp__D(y, z);
323 		if (sgn < 0) y = -y;
324 		return (y);
325 	}
326 	y = one-x;
327 	if (one-y == x)
328 		y = gamma(y);
329 	else		/* 1-x is inexact */
330 		y = -x*gamma(-x);
331 	if (sgn < 0) y = -y;
332 	return (M_PI / (y*z));
333 }
334