1
2 /*
3 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved.
4 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
5 *
6 * This code is free software; you can redistribute it and/or modify it
7 * under the terms of the GNU General Public License version 2 only, as
8 * published by the Free Software Foundation. Oracle designates this
9 * particular file as subject to the "Classpath" exception as provided
10 * by Oracle in the LICENSE file that accompanied this code.
11 *
12 * This code is distributed in the hope that it will be useful, but WITHOUT
13 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
14 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
15 * version 2 for more details (a copy is included in the LICENSE file that
16 * accompanied this code).
17 *
18 * You should have received a copy of the GNU General Public License version
19 * 2 along with this work; if not, write to the Free Software Foundation,
20 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
21 *
22 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
23 * or visit www.oracle.com if you need additional information or have any
24 * questions.
25 */
26
27 /* __ieee754_lgamma_r(x, signgamp)
28 * Reentrant version of the logarithm of the Gamma function
29 * with user provide pointer for the sign of Gamma(x).
30 *
31 * Method:
32 * 1. Argument Reduction for 0 < x <= 8
33 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
34 * reduce x to a number in [1.5,2.5] by
35 * lgamma(1+s) = log(s) + lgamma(s)
36 * for example,
37 * lgamma(7.3) = log(6.3) + lgamma(6.3)
38 * = log(6.3*5.3) + lgamma(5.3)
39 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
40 * 2. Polynomial approximation of lgamma around its
41 * minimun ymin=1.461632144968362245 to maintain monotonicity.
42 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
43 * Let z = x-ymin;
44 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
45 * where
46 * poly(z) is a 14 degree polynomial.
47 * 2. Rational approximation in the primary interval [2,3]
48 * We use the following approximation:
49 * s = x-2.0;
50 * lgamma(x) = 0.5*s + s*P(s)/Q(s)
51 * with accuracy
52 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
53 * Our algorithms are based on the following observation
54 *
55 * zeta(2)-1 2 zeta(3)-1 3
56 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
57 * 2 3
58 *
59 * where Euler = 0.5771... is the Euler constant, which is very
60 * close to 0.5.
61 *
62 * 3. For x>=8, we have
63 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
64 * (better formula:
65 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
66 * Let z = 1/x, then we approximation
67 * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
68 * by
69 * 3 5 11
70 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
71 * where
72 * |w - f(z)| < 2**-58.74
73 *
74 * 4. For negative x, since (G is gamma function)
75 * -x*G(-x)*G(x) = pi/sin(pi*x),
76 * we have
77 * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
78 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
79 * Hence, for x<0, signgam = sign(sin(pi*x)) and
80 * lgamma(x) = log(|Gamma(x)|)
81 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
82 * Note: one should avoid compute pi*(-x) directly in the
83 * computation of sin(pi*(-x)).
84 *
85 * 5. Special Cases
86 * lgamma(2+s) ~ s*(1-Euler) for tiny s
87 * lgamma(1)=lgamma(2)=0
88 * lgamma(x) ~ -log(x) for tiny x
89 * lgamma(0) = lgamma(inf) = inf
90 * lgamma(-integer) = +-inf
91 *
92 */
93
94 #include "fdlibm.h"
95
96 #ifdef __STDC__
97 static const double
98 #else
99 static double
100 #endif
101 two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
102 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
103 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
104 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
105 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
106 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
107 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
108 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
109 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
110 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
111 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
112 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
113 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
114 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
115 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
116 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
117 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
118 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
119 /* tt = -(tail of tf) */
120 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
121 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
122 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
123 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
124 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
125 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
126 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
127 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
128 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
129 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
130 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
131 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
132 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
133 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
134 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
135 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
136 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
137 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
138 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
139 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
140 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
141 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
142 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
143 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
144 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
145 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
146 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
147 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
148 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
149 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
150 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
151 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
152 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
153 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
154 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
155 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
156 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
157 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
158 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
159 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
160 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
161 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
162 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
163 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
164 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
165 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
166 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
167
168 static double zero= 0.00000000000000000000e+00;
169
170 #ifdef __STDC__
171 static double sin_pi(double x)
172 #else
173 static double sin_pi(x)
174 double x;
175 #endif
176 {
177 double y,z;
178 int n,ix;
179
180 ix = 0x7fffffff&__HI(x);
181
182 if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
183 y = -x; /* x is assume negative */
184
185 /*
186 * argument reduction, make sure inexact flag not raised if input
187 * is an integer
188 */
189 z = floor(y);
190 if(z!=y) { /* inexact anyway */
191 y *= 0.5;
192 y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */
193 n = (int) (y*4.0);
194 } else {
195 if(ix>=0x43400000) {
196 y = zero; n = 0; /* y must be even */
197 } else {
198 if(ix<0x43300000) z = y+two52; /* exact */
199 n = __LO(z)&1; /* lower word of z */
200 y = n;
201 n<<= 2;
202 }
203 }
204 switch (n) {
205 case 0: y = __kernel_sin(pi*y,zero,0); break;
206 case 1:
207 case 2: y = __kernel_cos(pi*(0.5-y),zero); break;
208 case 3:
209 case 4: y = __kernel_sin(pi*(one-y),zero,0); break;
210 case 5:
211 case 6: y = -__kernel_cos(pi*(y-1.5),zero); break;
212 default: y = __kernel_sin(pi*(y-2.0),zero,0); break;
213 }
214 return -y;
215 }
216
217
218 #ifdef __STDC__
219 double __ieee754_lgamma_r(double x, int *signgamp)
220 #else
221 double __ieee754_lgamma_r(x,signgamp)
222 double x; int *signgamp;
223 #endif
224 {
225 double t,y,z,nadj=0,p,p1,p2,p3,q,r,w;
226 int i,hx,lx,ix;
227
228 hx = __HI(x);
229 lx = __LO(x);
230
231 /* purge off +-inf, NaN, +-0, and negative arguments */
232 *signgamp = 1;
233 ix = hx&0x7fffffff;
234 if(ix>=0x7ff00000) return x*x;
235 if((ix|lx)==0) return one/zero;
236 if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
237 if(hx<0) {
238 *signgamp = -1;
239 return -__ieee754_log(-x);
240 } else return -__ieee754_log(x);
241 }
242 if(hx<0) {
243 if(ix>=0x43300000) /* |x|>=2**52, must be -integer */
244 return one/zero;
245 t = sin_pi(x);
246 if(t==zero) return one/zero; /* -integer */
247 nadj = __ieee754_log(pi/fabs(t*x));
248 if(t<zero) *signgamp = -1;
249 x = -x;
250 }
251
252 /* purge off 1 and 2 */
253 if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
254 /* for x < 2.0 */
255 else if(ix<0x40000000) {
256 if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
257 r = -__ieee754_log(x);
258 if(ix>=0x3FE76944) {y = one-x; i= 0;}
259 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
260 else {y = x; i=2;}
261 } else {
262 r = zero;
263 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
264 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
265 else {y=x-one;i=2;}
266 }
267 switch(i) {
268 case 0:
269 z = y*y;
270 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
271 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
272 p = y*p1+p2;
273 r += (p-0.5*y); break;
274 case 1:
275 z = y*y;
276 w = z*y;
277 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
278 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
279 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
280 p = z*p1-(tt-w*(p2+y*p3));
281 r += (tf + p); break;
282 case 2:
283 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
284 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
285 r += (-0.5*y + p1/p2);
286 }
287 }
288 else if(ix<0x40200000) { /* x < 8.0 */
289 i = (int)x;
290 t = zero;
291 y = x-(double)i;
292 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
293 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
294 r = half*y+p/q;
295 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
296 switch(i) {
297 case 7: z *= (y+6.0); /* FALLTHRU */
298 case 6: z *= (y+5.0); /* FALLTHRU */
299 case 5: z *= (y+4.0); /* FALLTHRU */
300 case 4: z *= (y+3.0); /* FALLTHRU */
301 case 3: z *= (y+2.0); /* FALLTHRU */
302 r += __ieee754_log(z); break;
303 }
304 /* 8.0 <= x < 2**58 */
305 } else if (ix < 0x43900000) {
306 t = __ieee754_log(x);
307 z = one/x;
308 y = z*z;
309 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
310 r = (x-half)*(t-one)+w;
311 } else
312 /* 2**58 <= x <= inf */
313 r = x*(__ieee754_log(x)-one);
314 if(hx<0) r = nadj - r;
315 return r;
316 }