libm & rtld: Ansify some remaining functions.
[dragonfly.git] / lib / libm / src / b_tgamma.c
CommitLineData
1a3b704c
JM
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 * @(#)gamma.c 8.1 (Berkeley) 6/4/93
34 * FreeBSD SVN: 176449 (2008-02-22)
35 */
36
37/*
38 * This code by P. McIlroy, Oct 1992;
39 *
40 * The financial support of UUNET Communications Services is greatfully
41 * acknowledged.
42 */
43
44#include <math.h>
45#include "mathimpl.h"
46
47/* METHOD:
48 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
49 * At negative integers, return NaN and raise invalid.
50 *
51 * x < 6.5:
52 * Use argument reduction G(x+1) = xG(x) to reach the
53 * range [1.066124,2.066124]. Use a rational
54 * approximation centered at the minimum (x0+1) to
55 * ensure monotonicity.
56 *
57 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
58 * adjusted for equal-ripples:
59 *
60 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
61 *
62 * Keep extra precision in multiplying (x-.5)(log(x)-1), to
63 * avoid premature round-off.
64 *
65 * Special values:
66 * -Inf: return NaN and raise invalid;
67 * negative integer: return NaN and raise invalid;
68 * other x ~< 177.79: return +-0 and raise underflow;
69 * +-0: return +-Inf and raise divide-by-zero;
70 * finite x ~> 171.63: return +Inf and raise overflow;
71 * +Inf: return +Inf;
72 * NaN: return NaN.
73 *
74 * Accuracy: tgamma(x) is accurate to within
75 * x > 0: error provably < 0.9ulp.
76 * Maximum observed in 1,000,000 trials was .87ulp.
77 * x < 0:
78 * Maximum observed error < 4ulp in 1,000,000 trials.
79 */
80
81static double neg_gam(double);
82static double small_gam(double);
83static double smaller_gam(double);
84static struct Double large_gam(double);
85static struct Double ratfun_gam(double, double);
86
87/*
88 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
89 * [1.066.., 2.066..] accurate to 4.25e-19.
90 */
91#define LEFT -.3955078125 /* left boundary for rat. approx */
92#define x0 .461632144968362356785 /* xmin - 1 */
93
94#define a0_hi 0.88560319441088874992
95#define a0_lo -.00000000000000004996427036469019695
96#define P0 6.21389571821820863029017800727e-01
97#define P1 2.65757198651533466104979197553e-01
98#define P2 5.53859446429917461063308081748e-03
99#define P3 1.38456698304096573887145282811e-03
100#define P4 2.40659950032711365819348969808e-03
101#define Q0 1.45019531250000000000000000000e+00
102#define Q1 1.06258521948016171343454061571e+00
103#define Q2 -2.07474561943859936441469926649e-01
104#define Q3 -1.46734131782005422506287573015e-01
105#define Q4 3.07878176156175520361557573779e-02
106#define Q5 5.12449347980666221336054633184e-03
107#define Q6 -1.76012741431666995019222898833e-03
108#define Q7 9.35021023573788935372153030556e-05
109#define Q8 6.13275507472443958924745652239e-06
110/*
111 * Constants for large x approximation (x in [6, Inf])
112 * (Accurate to 2.8*10^-19 absolute)
113 */
114#define lns2pi_hi 0.418945312500000
115#define lns2pi_lo -.000006779295327258219670263595
116#define Pa0 8.33333333333333148296162562474e-02
117#define Pa1 -2.77777777774548123579378966497e-03
118#define Pa2 7.93650778754435631476282786423e-04
119#define Pa3 -5.95235082566672847950717262222e-04
120#define Pa4 8.41428560346653702135821806252e-04
121#define Pa5 -1.89773526463879200348872089421e-03
122#define Pa6 5.69394463439411649408050664078e-03
123#define Pa7 -1.44705562421428915453880392761e-02
124
125static const double zero = 0., one = 1.0, tiny = 1e-300;
126
127double
472de6d1 128tgamma(double x)
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129{
130 struct Double u;
131
132 if (x >= 6) {
133 if(x > 171.63)
134 return (x / zero);
135 u = large_gam(x);
136 return(__exp__D(u.a, u.b));
137 } else if (x >= 1.0 + LEFT + x0)
138 return (small_gam(x));
139 else if (x > 1.e-17)
140 return (smaller_gam(x));
141 else if (x > -1.e-17) {
142 if (x != 0.0)
143 u.a = one - tiny; /* raise inexact */
144 return (one/x);
145 } else if (!finite(x))
146 return (x - x); /* x is NaN or -Inf */
147 else
148 return (neg_gam(x));
149}
150/*
151 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
152 */
153static struct Double
472de6d1 154large_gam(double x)
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155{
156 double z, p;
157 struct Double t, u, v;
158
159 z = one/(x*x);
160 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
161 p = p/x;
162
163 u = __log__D(x);
164 u.a -= one;
165 v.a = (x -= .5);
166 TRUNC(v.a);
167 v.b = x - v.a;
168 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
169 t.b = v.b*u.a + x*u.b;
170 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
171 t.b += lns2pi_lo; t.b += p;
172 u.a = lns2pi_hi + t.b; u.a += t.a;
173 u.b = t.a - u.a;
174 u.b += lns2pi_hi; u.b += t.b;
175 return (u);
176}
177/*
178 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
179 * It also has correct monotonicity.
180 */
181static double
472de6d1 182small_gam(double x)
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183{
184 double y, ym1, t;
185 struct Double yy, r;
186 y = x - one;
187 ym1 = y - one;
188 if (y <= 1.0 + (LEFT + x0)) {
189 yy = ratfun_gam(y - x0, 0);
190 return (yy.a + yy.b);
191 }
192 r.a = y;
193 TRUNC(r.a);
194 yy.a = r.a - one;
195 y = ym1;
196 yy.b = r.b = y - yy.a;
197 /* Argument reduction: G(x+1) = x*G(x) */
198 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
199 t = r.a*yy.a;
200 r.b = r.a*yy.b + y*r.b;
201 r.a = t;
202 TRUNC(r.a);
203 r.b += (t - r.a);
204 }
205 /* Return r*tgamma(y). */
206 yy = ratfun_gam(y - x0, 0);
207 y = r.b*(yy.a + yy.b) + r.a*yy.b;
208 y += yy.a*r.a;
209 return (y);
210}
211/*
212 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
213 */
214static double
472de6d1 215smaller_gam(double x)
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216{
217 double t, d;
218 struct Double r, xx;
219 if (x < x0 + LEFT) {
220 t = x, TRUNC(t);
221 d = (t+x)*(x-t);
222 t *= t;
223 xx.a = (t + x), TRUNC(xx.a);
224 xx.b = x - xx.a; xx.b += t; xx.b += d;
225 t = (one-x0); t += x;
226 d = (one-x0); d -= t; d += x;
227 x = xx.a + xx.b;
228 } else {
229 xx.a = x, TRUNC(xx.a);
230 xx.b = x - xx.a;
231 t = x - x0;
232 d = (-x0 -t); d += x;
233 }
234 r = ratfun_gam(t, d);
235 d = r.a/x, TRUNC(d);
236 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
237 return (d + r.a/x);
238}
239/*
240 * returns (z+c)^2 * P(z)/Q(z) + a0
241 */
242static struct Double
472de6d1 243ratfun_gam(double z, double c)
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244{
245 double p, q;
246 struct Double r, t;
247
248 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
249 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
250
251 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
252 p = p/q;
253 t.a = z, TRUNC(t.a); /* t ~= z + c */
254 t.b = (z - t.a) + c;
255 t.b *= (t.a + z);
256 q = (t.a *= t.a); /* t = (z+c)^2 */
257 TRUNC(t.a);
258 t.b += (q - t.a);
259 r.a = p, TRUNC(r.a); /* r = P/Q */
260 r.b = p - r.a;
261 t.b = t.b*p + t.a*r.b + a0_lo;
262 t.a *= r.a; /* t = (z+c)^2*(P/Q) */
263 r.a = t.a + a0_hi, TRUNC(r.a);
264 r.b = ((a0_hi-r.a) + t.a) + t.b;
265 return (r); /* r = a0 + t */
266}
267
268static double
472de6d1 269neg_gam(double x)
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270{
271 int sgn = 1;
272 struct Double lg, lsine;
273 double y, z;
274
275 y = ceil(x);
276 if (y == x) /* Negative integer. */
277 return ((x - x) / zero);
278 z = y - x;
279 if (z > 0.5)
280 z = one - z;
281 y = 0.5 * y;
282 if (y == ceil(y))
283 sgn = -1;
284 if (z < .25)
285 z = sin(M_PI*z);
286 else
287 z = cos(M_PI*(0.5-z));
288 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
289 if (x < -170) {
290 if (x < -190)
291 return ((double)sgn*tiny*tiny);
292 y = one - x; /* exact: 128 < |x| < 255 */
293 lg = large_gam(y);
294 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */
295 lg.a -= lsine.a; /* exact (opposite signs) */
296 lg.b -= lsine.b;
297 y = -(lg.a + lg.b);
298 z = (y + lg.a) + lg.b;
299 y = __exp__D(y, z);
300 if (sgn < 0) y = -y;
301 return (y);
302 }
303 y = one-x;
304 if (one-y == x)
305 y = tgamma(y);
306 else /* 1-x is inexact */
307 y = -x*tgamma(-x);
308 if (sgn < 0) y = -y;
309 return (M_PI / (y*z));
310}