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