1 #include "FEATURE/uwin"
2
3 #if !_UWIN || _lib_gamma
4
_STUB_gamma()5 void _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
39 static 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
87 static double neg_gam __P((double));
88 static double small_gam __P((double));
89 static double smaller_gam __P((double));
90 static struct Double large_gam __P((double));
91 static 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
131 static const double zero = 0., one = 1.0, tiny = 1e-300;
132 static 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
146 extern 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 */
178 static struct Double
large_gam(x)179 large_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 */
207 static double
small_gam(x)208 small_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 */
241 static double
smaller_gam(x)242 smaller_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 */
270 static struct Double
ratfun_gam(z,c)271 ratfun_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
297 static double
neg_gam(x)298 neg_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