1 /*-
2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
4 *
5 * %sccs.include.redist.c%
6 */
7
8 #ifndef lint
9 static char sccsid[] = "@(#)gamma.c 8.1 (Berkeley) 06/04/93";
10 #endif /* not lint */
11
12 /*
13 * This code by P. McIlroy, Oct 1992;
14 *
15 * The financial support of UUNET Communications Services is greatfully
16 * acknowledged.
17 */
18
19 #include <math.h>
20 #include "mathimpl.h"
21 #include <errno.h>
22
23 /* METHOD:
24 * x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
25 * At negative integers, return +Inf, and set errno.
26 *
27 * x < 6.5:
28 * Use argument reduction G(x+1) = xG(x) to reach the
29 * range [1.066124,2.066124]. Use a rational
30 * approximation centered at the minimum (x0+1) to
31 * ensure monotonicity.
32 *
33 * x >= 6.5: Use the asymptotic approximation (Stirling's formula)
34 * adjusted for equal-ripples:
35 *
36 * log(G(x)) ~= (x-.5)*(log(x)-1) + .5(log(2*pi)-1) + 1/x*P(1/(x*x))
37 *
38 * Keep extra precision in multiplying (x-.5)(log(x)-1), to
39 * avoid premature round-off.
40 *
41 * Special values:
42 * non-positive integer: Set overflow trap; return +Inf;
43 * x > 171.63: Set overflow trap; return +Inf;
44 * NaN: Set invalid trap; return NaN
45 *
46 * Accuracy: Gamma(x) is accurate to within
47 * x > 0: error provably < 0.9ulp.
48 * Maximum observed in 1,000,000 trials was .87ulp.
49 * x < 0:
50 * Maximum observed error < 4ulp in 1,000,000 trials.
51 */
52
53 static double neg_gam __P((double));
54 static double small_gam __P((double));
55 static double smaller_gam __P((double));
56 static struct Double large_gam __P((double));
57 static struct Double ratfun_gam __P((double, double));
58
59 /*
60 * Rational approximation, A0 + x*x*P(x)/Q(x), on the interval
61 * [1.066.., 2.066..] accurate to 4.25e-19.
62 */
63 #define LEFT -.3955078125 /* left boundary for rat. approx */
64 #define x0 .461632144968362356785 /* xmin - 1 */
65
66 #define a0_hi 0.88560319441088874992
67 #define a0_lo -.00000000000000004996427036469019695
68 #define P0 6.21389571821820863029017800727e-01
69 #define P1 2.65757198651533466104979197553e-01
70 #define P2 5.53859446429917461063308081748e-03
71 #define P3 1.38456698304096573887145282811e-03
72 #define P4 2.40659950032711365819348969808e-03
73 #define Q0 1.45019531250000000000000000000e+00
74 #define Q1 1.06258521948016171343454061571e+00
75 #define Q2 -2.07474561943859936441469926649e-01
76 #define Q3 -1.46734131782005422506287573015e-01
77 #define Q4 3.07878176156175520361557573779e-02
78 #define Q5 5.12449347980666221336054633184e-03
79 #define Q6 -1.76012741431666995019222898833e-03
80 #define Q7 9.35021023573788935372153030556e-05
81 #define Q8 6.13275507472443958924745652239e-06
82 /*
83 * Constants for large x approximation (x in [6, Inf])
84 * (Accurate to 2.8*10^-19 absolute)
85 */
86 #define lns2pi_hi 0.418945312500000
87 #define lns2pi_lo -.000006779295327258219670263595
88 #define Pa0 8.33333333333333148296162562474e-02
89 #define Pa1 -2.77777777774548123579378966497e-03
90 #define Pa2 7.93650778754435631476282786423e-04
91 #define Pa3 -5.95235082566672847950717262222e-04
92 #define Pa4 8.41428560346653702135821806252e-04
93 #define Pa5 -1.89773526463879200348872089421e-03
94 #define Pa6 5.69394463439411649408050664078e-03
95 #define Pa7 -1.44705562421428915453880392761e-02
96
97 static const double zero = 0., one = 1.0, tiny = 1e-300;
98 static int endian;
99 /*
100 * TRUNC sets trailing bits in a floating-point number to zero.
101 * is a temporary variable.
102 */
103 #if defined(vax) || defined(tahoe)
104 #define _IEEE 0
105 #define TRUNC(x) x = (double) (float) (x)
106 #else
107 #define _IEEE 1
108 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000
109 #define infnan(x) 0.0
110 #endif
111
112 double
gamma(x)113 gamma(x)
114 double x;
115 {
116 struct Double u;
117 endian = (*(int *) &one) ? 1 : 0;
118
119 if (x >= 6) {
120 if(x > 171.63)
121 return(one/zero);
122 u = large_gam(x);
123 return(__exp__D(u.a, u.b));
124 } else if (x >= 1.0 + LEFT + x0)
125 return (small_gam(x));
126 else if (x > 1.e-17)
127 return (smaller_gam(x));
128 else if (x > -1.e-17) {
129 if (x == 0.0)
130 if (!_IEEE) return (infnan(ERANGE));
131 else return (one/x);
132 one+1e-20; /* Raise inexact flag. */
133 return (one/x);
134 } else if (!finite(x)) {
135 if (_IEEE) /* x = NaN, -Inf */
136 return (x*x);
137 else
138 return (infnan(EDOM));
139 } else
140 return (neg_gam(x));
141 }
142 /*
143 * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
144 */
145 static struct Double
large_gam(x)146 large_gam(x)
147 double x;
148 {
149 double z, p;
150 int i;
151 struct Double t, u, v;
152
153 z = one/(x*x);
154 p = Pa0+z*(Pa1+z*(Pa2+z*(Pa3+z*(Pa4+z*(Pa5+z*(Pa6+z*Pa7))))));
155 p = p/x;
156
157 u = __log__D(x);
158 u.a -= one;
159 v.a = (x -= .5);
160 TRUNC(v.a);
161 v.b = x - v.a;
162 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
163 t.b = v.b*u.a + x*u.b;
164 /* return t.a + t.b + lns2pi_hi + lns2pi_lo + p */
165 t.b += lns2pi_lo; t.b += p;
166 u.a = lns2pi_hi + t.b; u.a += t.a;
167 u.b = t.a - u.a;
168 u.b += lns2pi_hi; u.b += t.b;
169 return (u);
170 }
171 /*
172 * Good to < 1 ulp. (provably .90 ulp; .87 ulp on 1,000,000 runs.)
173 * It also has correct monotonicity.
174 */
175 static double
small_gam(x)176 small_gam(x)
177 double x;
178 {
179 double y, ym1, t, x1;
180 struct Double yy, r;
181 y = x - one;
182 ym1 = y - one;
183 if (y <= 1.0 + (LEFT + x0)) {
184 yy = ratfun_gam(y - x0, 0);
185 return (yy.a + yy.b);
186 }
187 r.a = y;
188 TRUNC(r.a);
189 yy.a = r.a - one;
190 y = ym1;
191 yy.b = r.b = y - yy.a;
192 /* Argument reduction: G(x+1) = x*G(x) */
193 for (ym1 = y-one; ym1 > LEFT + x0; y = ym1--, yy.a--) {
194 t = r.a*yy.a;
195 r.b = r.a*yy.b + y*r.b;
196 r.a = t;
197 TRUNC(r.a);
198 r.b += (t - r.a);
199 }
200 /* Return r*gamma(y). */
201 yy = ratfun_gam(y - x0, 0);
202 y = r.b*(yy.a + yy.b) + r.a*yy.b;
203 y += yy.a*r.a;
204 return (y);
205 }
206 /*
207 * Good on (0, 1+x0+LEFT]. Accurate to 1ulp.
208 */
209 static double
smaller_gam(x)210 smaller_gam(x)
211 double x;
212 {
213 double t, d;
214 struct Double r, xx;
215 if (x < x0 + LEFT) {
216 t = x, TRUNC(t);
217 d = (t+x)*(x-t);
218 t *= t;
219 xx.a = (t + x), TRUNC(xx.a);
220 xx.b = x - xx.a; xx.b += t; xx.b += d;
221 t = (one-x0); t += x;
222 d = (one-x0); d -= t; d += x;
223 x = xx.a + xx.b;
224 } else {
225 xx.a = x, TRUNC(xx.a);
226 xx.b = x - xx.a;
227 t = x - x0;
228 d = (-x0 -t); d += x;
229 }
230 r = ratfun_gam(t, d);
231 d = r.a/x, TRUNC(d);
232 r.a -= d*xx.a; r.a -= d*xx.b; r.a += r.b;
233 return (d + r.a/x);
234 }
235 /*
236 * returns (z+c)^2 * P(z)/Q(z) + a0
237 */
238 static struct Double
ratfun_gam(z,c)239 ratfun_gam(z, c)
240 double z, c;
241 {
242 int i;
243 double p, q;
244 struct Double r, t;
245
246 q = Q0 +z*(Q1+z*(Q2+z*(Q3+z*(Q4+z*(Q5+z*(Q6+z*(Q7+z*Q8)))))));
247 p = P0 + z*(P1 + z*(P2 + z*(P3 + z*P4)));
248
249 /* return r.a + r.b = a0 + (z+c)^2*p/q, with r.a truncated to 26 bits. */
250 p = p/q;
251 t.a = z, TRUNC(t.a); /* t ~= z + c */
252 t.b = (z - t.a) + c;
253 t.b *= (t.a + z);
254 q = (t.a *= t.a); /* t = (z+c)^2 */
255 TRUNC(t.a);
256 t.b += (q - t.a);
257 r.a = p, TRUNC(r.a); /* r = P/Q */
258 r.b = p - r.a;
259 t.b = t.b*p + t.a*r.b + a0_lo;
260 t.a *= r.a; /* t = (z+c)^2*(P/Q) */
261 r.a = t.a + a0_hi, TRUNC(r.a);
262 r.b = ((a0_hi-r.a) + t.a) + t.b;
263 return (r); /* r = a0 + t */
264 }
265
266 static double
neg_gam(x)267 neg_gam(x)
268 double x;
269 {
270 int sgn = 1;
271 struct Double lg, lsine;
272 double y, z;
273
274 y = floor(x + .5);
275 if (y == x) /* Negative integer. */
276 if(!_IEEE)
277 return (infnan(ERANGE));
278 else
279 return (one/zero);
280 z = fabs(x - y);
281 y = .5*ceil(x);
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 = gamma(y);
306 else /* 1-x is inexact */
307 y = -x*gamma(-x);
308 if (sgn < 0) y = -y;
309 return (M_PI / (y*z));
310 }
311