1
2 /* @(#)er_lgamma.c 5.1 93/09/24 */
3 /*
4 * ====================================================
5 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 *
13 */
14
15 /* lgamma_r(x)
16 * Reentrant version of the logarithm of the Gamma function
17 * with signgam for the sign of Gamma(x).
18 *
19 * Method:
20 * 1. Argument Reduction for 0 < x <= 8
21 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
22 * reduce x to a number in [1.5,2.5] by
23 * lgamma(1+s) = log(s) + lgamma(s)
24 * for example,
25 * lgamma(7.3) = log(6.3) + lgamma(6.3)
26 * = log(6.3*5.3) + lgamma(5.3)
27 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
28 * 2. Polynomial approximation of lgamma around its
29 * minimun ymin=1.461632144968362245 to maintain monotonicity.
30 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
31 * Let z = x-ymin;
32 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
33 * where
34 * poly(z) is a 14 degree polynomial.
35 * 2. Rational approximation in the primary interval [2,3]
36 * We use the following approximation:
37 * s = x-2.0;
38 * lgamma(x) = 0.5*s + s*P(s)/Q(s)
39 * with accuracy
40 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
41 * Our algorithms are based on the following observation
42 *
43 * zeta(2)-1 2 zeta(3)-1 3
44 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
45 * 2 3
46 *
47 * where Euler = 0.5771... is the Euler constant, which is very
48 * close to 0.5.
49 *
50 * 3. For x>=8, we have
51 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
52 * (better formula:
53 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
54 * Let z = 1/x, then we approximation
55 * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
56 * by
57 * 3 5 11
58 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
59 * where
60 * |w - f(z)| < 2**-58.74
61 *
62 * 4. For negative x, since (G is gamma function)
63 * -x*G(-x)*G(x) = pi/sin(pi*x),
64 * we have
65 * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
66 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
67 * Hence, for x<0, signgam = sign(sin(pi*x)) and
68 * lgamma(x) = log(|Gamma(x)|)
69 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
70 * Note: one should avoid compute pi*(-x) directly in the
71 * computation of sin(pi*(-x)).
72 *
73 * 5. Special Cases
74 * lgamma(2+s) ~ s*(1-Euler) for tiny s
75 * lgamma(1)=lgamma(2)=0
76 * lgamma(x) ~ -log(x) for tiny x
77 * lgamma(0) = lgamma(inf) = inf
78 * lgamma(-integer) = +-inf
79 *
80 */
81
82 #define _ADD_UNDER_R_TO_FUNCS
83
84 #include "fdlibm.h"
85
86 #ifdef _NEED_FLOAT64
87
88 static const __float64 two52 =
89 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
90 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
91 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
92 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
93 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
94 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
95 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
96 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
97 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
98 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
99 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
100 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
101 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
102 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
103 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
104 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
105 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
106 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
107 /* tt = -(tail of tf) */
108 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
109 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
110 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
111 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
112 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
113 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
114 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
115 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
116 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
117 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
118 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
119 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
120 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
121 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
122 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
123 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
124 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
125 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
126 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
127 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
128 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
129 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
130 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
131 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
132 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
133 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
134 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
135 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
136 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
137 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
138 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
139 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
140 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
141 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
142 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
143 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
144 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
145 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
146 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
147 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
148 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
149 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
150 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
151 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
152 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
153 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
154 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
155
156 static const __float64 zero = 0.00000000000000000000e+00;
157
158 static __float64
sin_pi(__float64 x)159 sin_pi(__float64 x)
160 {
161 __float64 y, z;
162 __int32_t n, ix;
163
164 GET_HIGH_WORD(ix, x);
165 ix &= 0x7fffffff;
166
167 if (ix < 0x3fd00000)
168 return __kernel_sin(pi * x, zero, 0);
169 y = -x; /* x is assume negative */
170
171 /*
172 * argument reduction, make sure inexact flag not raised if input
173 * is an integer
174 */
175 z = floor(y);
176 if (z != y) { /* inexact anyway */
177 y *= 0.5;
178 y = 2.0 * (y - floor(y)); /* y = |x| mod 2.0 */
179 n = (__int32_t)(y * 4.0);
180 } else {
181 if (ix >= 0x43400000) {
182 y = zero;
183 n = 0; /* y must be even */
184 } else {
185 if (ix < 0x43300000)
186 z = y + two52; /* exact */
187 GET_LOW_WORD(n, z);
188 n &= 1;
189 y = n;
190 n <<= 2;
191 }
192 }
193 switch (n) {
194 case 0:
195 y = __kernel_sin(pi * y, zero, 0);
196 break;
197 case 1:
198 case 2:
199 y = __kernel_cos(pi * (0.5 - y), zero);
200 break;
201 case 3:
202 case 4:
203 y = __kernel_sin(pi * (one - y), zero, 0);
204 break;
205 case 5:
206 case 6:
207 y = -__kernel_cos(pi * (y - 1.5), zero);
208 break;
209 default:
210 y = __kernel_sin(pi * (y - 2.0), zero, 0);
211 break;
212 }
213 return -y;
214 }
215
216 __float64
__math_lgamma_r(__float64 x,int * signgamp,int * divzero)217 __math_lgamma_r(__float64 x, int *signgamp, int *divzero)
218 {
219 __float64 t, y, z, nadj = 0.0, p, p1, p2, p3, q, r, w;
220 __int32_t i, hx, lx, ix;
221
222 EXTRACT_WORDS(hx, lx, x);
223
224 /* purge off +-inf, NaN, +-0, and negative arguments */
225 *signgamp = 1;
226 ix = hx & 0x7fffffff;
227 if (ix >= 0x7ff00000)
228 return fabs64(x+x);
229 if ((ix | lx) == 0) {
230 if (hx < 0)
231 *signgamp = -1;
232 *divzero = 1;
233 return __math_divzero(0);
234 }
235 if (ix < 0x3b900000) { /* |x|<2**-70, return -log(|x|) */
236 if (hx < 0) {
237 *signgamp = -1;
238 return -log(-x);
239 } else
240 return -log(x);
241 }
242 if (hx < 0) {
243 if (ix >= 0x43300000) { /* |x|>=2**52, must be -integer */
244 *divzero = 1;
245 return __math_divzero(0);
246 }
247 t = sin_pi(x);
248 if (t == zero) {
249 *divzero = 1;
250 return __math_divzero(0);
251 }
252 nadj = log(pi / fabs64(t * x));
253 if (t < zero)
254 *signgamp = -1;
255 x = -x;
256 }
257
258 /* purge off 1 and 2 */
259 if ((((ix - 0x3ff00000) | lx) == 0) || (((ix - 0x40000000) | lx) == 0))
260 r = 0;
261 /* for x < 2.0 */
262 else if (ix < 0x40000000) {
263 if (ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
264 r = -log(x);
265 if (ix >= 0x3FE76944) {
266 y = one - x;
267 i = 0;
268 } else if (ix >= 0x3FCDA661) {
269 y = x - (tc - one);
270 i = 1;
271 } else {
272 y = x;
273 i = 2;
274 }
275 } else {
276 r = zero;
277 if (ix >= 0x3FFBB4C3) {
278 y = 2.0 - x;
279 i = 0;
280 } /* [1.7316,2] */
281 else if (ix >= 0x3FF3B4C4) {
282 y = x - tc;
283 i = 1;
284 } /* [1.23,1.73] */
285 else {
286 y = x - one;
287 i = 2;
288 }
289 }
290 switch (i) {
291 case 0:
292 z = y * y;
293 p1 = a0 + z * (a2 + z * (a4 + z * (a6 + z * (a8 + z * a10))));
294 p2 = z * (a1 + z * (a3 + z * (a5 + z * (a7 + z * (a9 + z * a11)))));
295 p = y * p1 + p2;
296 r += (p - 0.5 * y);
297 break;
298 case 1:
299 z = y * y;
300 w = z * y;
301 p1 = t0 +
302 w * (t3 + w * (t6 + w * (t9 + w * t12))); /* parallel comp */
303 p2 = t1 + w * (t4 + w * (t7 + w * (t10 + w * t13)));
304 p3 = t2 + w * (t5 + w * (t8 + w * (t11 + w * t14)));
305 p = z * p1 - (tt - w * (p2 + y * p3));
306 r += (tf + p);
307 break;
308 case 2:
309 p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * u5)))));
310 p2 = one + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * v5))));
311 r += (-0.5 * y + p1 / p2);
312 }
313 } else if (ix < 0x40200000) { /* x < 8.0 */
314 i = (__int32_t)x;
315 t = zero;
316 y = x - (__float64)i;
317 p = y * (s0 +
318 y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
319 q = one + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * r6)))));
320 r = half * y + p / q;
321 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
322 switch (i) {
323 case 7:
324 z *= (y + 6.0); /* FALLTHRU */
325 case 6:
326 z *= (y + 5.0); /* FALLTHRU */
327 case 5:
328 z *= (y + 4.0); /* FALLTHRU */
329 case 4:
330 z *= (y + 3.0); /* FALLTHRU */
331 case 3:
332 z *= (y + 2.0); /* FALLTHRU */
333 r += log(z);
334 break;
335 }
336 /* 8.0 <= x < 2**58 */
337 } else if (ix < 0x43900000) {
338 t = log(x);
339 z = one / x;
340 y = z * z;
341 w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * w6)))));
342 r = (x - half) * (t - one) + w;
343 } else
344 /* 2**58 <= x <= inf */
345 r = x * (log(x) - one);
346 if (hx < 0)
347 r = nadj - r;
348 return check_oflow(r);
349 }
350
351 __float64
lgamma64_r(__float64 x,int * signgamp)352 lgamma64_r(__float64 x, int *signgamp)
353 {
354 int divzero = 0;
355 return __math_lgamma_r(x, signgamp, &divzero);
356 }
357
358 _MATH_ALIAS_d_dI(lgamma)
359
360 #endif /* _NEED_FLOAT64 */
361
