1 /* $OpenBSD: b_tgamma.c,v 1.10 2016/09/12 19:47:02 guenther 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 /*
32 * This code by P. McIlroy, Oct 1992;
33 *
34 * The financial support of UUNET Communications Services is greatfully
35 * acknowledged.
36 */
37
38 #include <float.h>
39 #include <math.h>
40
41 #include "math_private.h"
42
43 /* METHOD:
44 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
45 * At negative integers, return NaN and raise invalid.
46 *
47 * x < 6.5:
48 * Use argument reduction G(x+1) = xG(x) to reach the
49 * range [1.066124,2.066124]. Use a rational
50 * approximation centered at the minimum (x0+1) to
51 * ensure monotonicity.
52 *
53 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
54 * adjusted for equal-ripples:
55 *
56 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
57 *
58 * Keep extra precision in multiplying (x-.5)(log(x)-1), to
59 * avoid premature round-off.
60 *
61 * Special values:
62 * -Inf: return NaN and raise invalid;
63 * negative integer: return NaN and raise invalid;
64 * other x ~< -177.79: return +-0 and raise underflow;
65 * +-0: return +-Inf and raise divide-by-zero;
66 * finite x ~> 171.63: return +Inf and raise overflow;
67 * +Inf: return +Inf;
68 * NaN: return NaN.
69 *
70 * Accuracy: tgamma(x) is accurate to within
71 * x > 0: error provably < 0.9ulp.
72 * Maximum observed in 1,000,000 trials was .87ulp.
73 * x < 0:
74 * Maximum observed error < 4ulp in 1,000,000 trials.
75 */
76
77 static double neg_gam(double);
78 static double small_gam(double);
79 static double smaller_gam(double);
80 static struct Double large_gam(double);
81 static struct Double ratfun_gam(double, double);
82
83 /*
84 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
85 * [1.066.., 2.066..] accurate to 4.25e-19.
86 */
87 #define LEFT -.3955078125 /* left boundary for rat. approx */
88 #define x0 .461632144968362356785 /* xmin - 1 */
89
90 #define a0_hi 0.88560319441088874992
91 #define a0_lo -.00000000000000004996427036469019695
92 #define P0 6.21389571821820863029017800727e-01
93 #define P1 2.65757198651533466104979197553e-01
94 #define P2 5.53859446429917461063308081748e-03
95 #define P3 1.38456698304096573887145282811e-03
96 #define P4 2.40659950032711365819348969808e-03
97 #define Q0 1.45019531250000000000000000000e+00
98 #define Q1 1.06258521948016171343454061571e+00
99 #define Q2 -2.07474561943859936441469926649e-01
100 #define Q3 -1.46734131782005422506287573015e-01
101 #define Q4 3.07878176156175520361557573779e-02
102 #define Q5 5.12449347980666221336054633184e-03
103 #define Q6 -1.76012741431666995019222898833e-03
104 #define Q7 9.35021023573788935372153030556e-05
105 #define Q8 6.13275507472443958924745652239e-06
106 /*
107 * Constants for large x approximation (x in [6, Inf])
108 * (Accurate to 2.8*10^-19 absolute)
109 */
110 #define lns2pi_hi 0.418945312500000
111 #define lns2pi_lo -.000006779295327258219670263595
112 #define Pa0 8.33333333333333148296162562474e-02
113 #define Pa1 -2.77777777774548123579378966497e-03
114 #define Pa2 7.93650778754435631476282786423e-04
115 #define Pa3 -5.95235082566672847950717262222e-04
116 #define Pa4 8.41428560346653702135821806252e-04
117 #define Pa5 -1.89773526463879200348872089421e-03
118 #define Pa6 5.69394463439411649408050664078e-03
119 #define Pa7 -1.44705562421428915453880392761e-02
120
121 static const double zero = 0., one = 1.0, tiny = 1e-300;
122
123 double
tgamma(double x)124 tgamma(double x)
125 {
126 struct Double u;
127
128 if (x >= 6) {
129 if(x > 171.63)
130 return(x/zero);
131 u = large_gam(x);
132 return(__exp__D(u.a, u.b));
133 } else if (x >= 1.0 + LEFT + x0)
134 return (small_gam(x));
135 else if (x > 1.e-17)
136 return (smaller_gam(x));
137 else if (x > -1.e-17) {
138 if (x != 0.0)
139 u.a = one - tiny; /* raise inexact */
140 return (one/x);
141 } else if (!isfinite(x)) {
142 return (x - x); /* x = NaN, -Inf */
143 } else
144 return (neg_gam(x));
145 }
146 DEF_STD(tgamma);
147 LDBL_MAYBE_UNUSED_CLONE(tgamma);
148
149 /*
150 * We simply call tgamma() rather than bloating the math library
151 * with a float-optimized version of it. The reason is that tgammaf()
152 * is essentially useless, since the function is superexponential
153 * and floats have very limited range. -- das@freebsd.org
154 */
155
156 float
tgammaf(float x)157 tgammaf(float x)
158 {
159 return tgamma(x);
160 }
161
162 /*
163 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
164 */
165
166 static struct Double
large_gam(double x)167 large_gam(double x)
168 {
169 double z, p;
170 struct Double t, u, v;
171
172 z = one/(x*x);
173 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
174 p = p/x;
175
176 u = __log__D(x);
177 u.a -= one;
178 v.a = (x -= .5);
179 TRUNC(v.a);
180 v.b = x - v.a;
181 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
182 t.b = v.b*u.a + x*u.b;
183 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
184 t.b += lns2pi_lo; t.b += p;
185 u.a = lns2pi_hi + t.b; u.a += t.a;
186 u.b = t.a - u.a;
187 u.b += lns2pi_hi; u.b += t.b;
188 return (u);
189 }
190
191 /*
192 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
193 * It also has correct monotonicity.
194 */
195
196 static double
small_gam(double x)197 small_gam(double x)
198 {
199 double y, ym1, t;
200 struct Double yy, r;
201 y = x - one;
202 ym1 = y - one;
203 if (y <= 1.0 + (LEFT + x0)) {
204 yy = ratfun_gam(y - x0, 0);
205 return (yy.a + yy.b);
206 }
207 r.a = y;
208 TRUNC(r.a);
209 yy.a = r.a - one;
210 y = ym1;
211 yy.b = r.b = y - yy.a;
212 /* Argument reduction: G(x+1) = x*G(x) */
213 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
214 t = r.a*yy.a;
215 r.b = r.a*yy.b + y*r.b;
216 r.a = t;
217 TRUNC(r.a);
218 r.b += (t - r.a);
219 }
220 /* Return r*tgamma(y). */
221 yy = ratfun_gam(y - x0, 0);
222 y = r.b*(yy.a + yy.b) + r.a*yy.b;
223 y += yy.a*r.a;
224 return (y);
225 }
226
227 /*
228 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
229 */
230
231 static double
smaller_gam(double x)232 smaller_gam(double x)
233 {
234 double t, d;
235 struct Double r, xx;
236 if (x < x0 + LEFT) {
237 t = x;
238 TRUNC(t);
239 d = (t+x)*(x-t);
240 t *= t;
241 xx.a = (t + x);
242 TRUNC(xx.a);
243 xx.b = x - xx.a; xx.b += t; xx.b += d;
244 t = (one-x0); t += x;
245 d = (one-x0); d -= t; d += x;
246 x = xx.a + xx.b;
247 } else {
248 xx.a = x;
249 TRUNC(xx.a);
250 xx.b = x - xx.a;
251 t = x - x0;
252 d = (-x0 -t); d += x;
253 }
254 r = ratfun_gam(t, d);
255 d = r.a/x;
256 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 /*
262 * returns (z+c)^2 * P(z)/Q(z) + a0
263 */
264
265 static struct Double
ratfun_gam(double z,double c)266 ratfun_gam(double z, double c)
267 {
268 double p, q;
269 struct Double r, t;
270
271 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
272 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
273
274 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
275 p = p/q;
276 t.a = z;
277 TRUNC(t.a); /* t ~= z + c */
278 t.b = (z - t.a) + c;
279 t.b *= (t.a + z);
280 q = (t.a *= t.a); /* t = (z+c)^2 */
281 TRUNC(t.a);
282 t.b += (q - t.a);
283 r.a = p;
284 TRUNC(r.a); /* r = P/Q */
285 r.b = p - r.a;
286 t.b = t.b*p + t.a*r.b + a0_lo;
287 t.a *= r.a; /* t = (z+c)^2*(P/Q) */
288 r.a = t.a + a0_hi;
289 TRUNC(r.a);
290 r.b = ((a0_hi-r.a) + t.a) + t.b;
291 return (r); /* r = a0 + t */
292 }
293
294 static double
neg_gam(double x)295 neg_gam(double x)
296 {
297 int sgn = 1;
298 struct Double lg, lsine;
299 double y, z;
300
301 y = ceil(x);
302 if (y == x) /* Negative integer. */
303 return ((x - x) / zero);
304 z = y - x;
305 if (z > 0.5)
306 z = one - z;
307 y = 0.5 * y;
308 if (y == ceil(y))
309 sgn = -1;
310 if (z < .25)
311 z = sin(M_PI*z);
312 else
313 z = cos(M_PI*(0.5-z));
314 /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
315 if (x < -170) {
316 if (x < -190)
317 return ((double)sgn*tiny*tiny);
318 y = one - x; /* exact: 128 < |x| < 255 */
319 lg = large_gam(y);
320 lsine = __log__D(M_PI/z); /* = TRUNC(log(u)) + small */
321 lg.a -= lsine.a; /* exact (opposite signs) */
322 lg.b -= lsine.b;
323 y = -(lg.a + lg.b);
324 z = (y + lg.a) + lg.b;
325 y = __exp__D(y, z);
326 if (sgn < 0) y = -y;
327 return (y);
328 }
329 y = one-x;
330 if (one-y == x)
331 y = tgamma(y);
332 else /* 1-x is inexact */
333 y = -x*tgamma(-x);
334 if (sgn < 0) y = -y;
335 return (M_PI / (y*z));
336 }
337