1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunSoft, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 */
11
12 /*
13 * See e_lgamma_r.c for complete comments.
14 *
15 * Converted to long double by Steven G. Kargl.
16 */
17
18 #ifdef __i386__
19 #include <ieeefp.h>
20 #endif
21
22 #include "fpmath.h"
23 #include "math.h"
24 #include "math_private.h"
25
26 static const volatile double vzero = 0;
27
28 static const double
29 zero= 0,
30 half= 0.5,
31 one = 1;
32
33 static const union IEEEl2bits
34 piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
35 #define pi (piu.e)
36 /*
37 * Domain y in [0x1p-70, 0.27], range ~[-4.5264e-22, 4.5264e-22]:
38 * |(lgamma(2 - y) + y / 2) / y - a(y)| < 2**-70.9
39 */
40 static const union IEEEl2bits
41 a0u = LD80C(0x9e233f1bed863d26, -4, 7.72156649015328606028e-02L),
42 a1u = LD80C(0xa51a6625307d3249, -2, 3.22467033424113218889e-01L),
43 a2u = LD80C(0x89f000d2abafda8c, -4, 6.73523010531979398946e-02L),
44 a3u = LD80C(0xa8991563eca75f26, -6, 2.05808084277991211934e-02L),
45 a4u = LD80C(0xf2027e10634ce6b6, -8, 7.38555102796070454026e-03L),
46 a5u = LD80C(0xbd6eb76dd22187f4, -9, 2.89051035162703932972e-03L),
47 a6u = LD80C(0x9c562ab05e0458ed, -10, 1.19275351624639999297e-03L),
48 a7u = LD80C(0x859baed93ee48e46, -11, 5.09674593842117925320e-04L),
49 a8u = LD80C(0xe9f28a4432949af2, -13, 2.23109648015769155122e-04L),
50 a9u = LD80C(0xd12ad0d9b93c6bb0, -14, 9.97387167479808509830e-05L),
51 a10u= LD80C(0xb7522643c78a219b, -15, 4.37071076331030136818e-05L),
52 a11u= LD80C(0xca024dcdece2cb79, -16, 2.40813493372040143061e-05L),
53 a12u= LD80C(0xbb90fb6968ebdbf9, -19, 2.79495621083634031729e-06L),
54 a13u= LD80C(0xba1c9ffeeae07b37, -17, 1.10931287015513924136e-05L);
55 #define a0 (a0u.e)
56 #define a1 (a1u.e)
57 #define a2 (a2u.e)
58 #define a3 (a3u.e)
59 #define a4 (a4u.e)
60 #define a5 (a5u.e)
61 #define a6 (a6u.e)
62 #define a7 (a7u.e)
63 #define a8 (a8u.e)
64 #define a9 (a9u.e)
65 #define a10 (a10u.e)
66 #define a11 (a11u.e)
67 #define a12 (a12u.e)
68 #define a13 (a13u.e)
69 /*
70 * Domain x in [tc-0.24, tc+0.28], range ~[-6.1205e-22, 6.1205e-22]:
71 * |(lgamma(x) - tf) - t(x - tc)| < 2**-70.5
72 */
73 static const union IEEEl2bits
74 tcu = LD80C(0xbb16c31ab5f1fb71, 0, 1.46163214496836234128e+00L),
75 tfu = LD80C(0xf8cdcde61c520e0f, -4, -1.21486290535849608093e-01L),
76 ttu = LD80C(0xd46ee54b27d4de99, -69, -2.81152980996018785880e-21L),
77 t0u = LD80C(0x80b9406556a62a6b, -68, 3.40728634996055147231e-21L),
78 t1u = LD80C(0xc7e9c6f6df3f8c39, -67, -1.05833162742737073665e-20L),
79 t2u = LD80C(0xf7b95e4771c55d51, -2, 4.83836122723810583532e-01L),
80 t3u = LD80C(0x97213c6e35e119ff, -3, -1.47587722994530691476e-01L),
81 t4u = LD80C(0x845a14a6a81dc94b, -4, 6.46249402389135358063e-02L),
82 t5u = LD80C(0x864d46fa89997796, -5, -3.27885410884846056084e-02L),
83 t6u = LD80C(0x93373cbd00297438, -6, 1.79706751150707171293e-02L),
84 t7u = LD80C(0xa8fcfca7eddc8d1d, -7, -1.03142230361450732547e-02L),
85 t8u = LD80C(0xc7e7015ff4bc45af, -8, 6.10053603296546099193e-03L),
86 t9u = LD80C(0xf178d2247adc5093, -9, -3.68456964904901200152e-03L),
87 t10u = LD80C(0x94188d58f12e5e9f, -9, 2.25976420273774583089e-03L),
88 t11u = LD80C(0xb7cbaef14e1406f1, -10, -1.40224943666225639823e-03L),
89 t12u = LD80C(0xe63a671e6704ea4d, -11, 8.78250640744776944887e-04L),
90 t13u = LD80C(0x914b6c9cae61783e, -11, -5.54255012657716808811e-04L),
91 t14u = LD80C(0xb858f5bdb79276fe, -12, 3.51614951536825927370e-04L),
92 t15u = LD80C(0xea73e744c34b9591, -13, -2.23591563824520112236e-04L),
93 t16u = LD80C(0x99aeabb0d67ba835, -13, 1.46562869351659194136e-04L),
94 t17u = LD80C(0xd7c6938325db2024, -14, -1.02889866046435680588e-04L),
95 t18u = LD80C(0xe24cb1e3b0474775, -15, 5.39540265505221957652e-05L);
96 #define tc (tcu.e)
97 #define tf (tfu.e)
98 #define tt (ttu.e)
99 #define t0 (t0u.e)
100 #define t1 (t1u.e)
101 #define t2 (t2u.e)
102 #define t3 (t3u.e)
103 #define t4 (t4u.e)
104 #define t5 (t5u.e)
105 #define t6 (t6u.e)
106 #define t7 (t7u.e)
107 #define t8 (t8u.e)
108 #define t9 (t9u.e)
109 #define t10 (t10u.e)
110 #define t11 (t11u.e)
111 #define t12 (t12u.e)
112 #define t13 (t13u.e)
113 #define t14 (t14u.e)
114 #define t15 (t15u.e)
115 #define t16 (t16u.e)
116 #define t17 (t17u.e)
117 #define t18 (t18u.e)
118 /*
119 * Domain y in [-0.1, 0.232], range ~[-8.1938e-22, 8.3815e-22]:
120 * |(lgamma(1 + y) + 0.5 * y) / y - u(y) / v(y)| < 2**-71.2
121 */
122 static const union IEEEl2bits
123 u0u = LD80C(0x9e233f1bed863d27, -4, -7.72156649015328606095e-02L),
124 u1u = LD80C(0x98280ee45e4ddd3d, -1, 5.94361239198682739769e-01L),
125 u2u = LD80C(0xe330c8ead4130733, 0, 1.77492629495841234275e+00L),
126 u3u = LD80C(0xd4a213f1a002ec52, 0, 1.66119622514818078064e+00L),
127 u4u = LD80C(0xa5a9ca6f5bc62163, -1, 6.47122051417476492989e-01L),
128 u5u = LD80C(0xc980e49cd5b019e6, -4, 9.83903751718671509455e-02L),
129 u6u = LD80C(0xff636a8bdce7025b, -9, 3.89691687802305743450e-03L),
130 v1u = LD80C(0xbd109c533a19fbf5, 1, 2.95413883330948556544e+00L),
131 v2u = LD80C(0xd295cbf96f31f099, 1, 3.29039286955665403176e+00L),
132 v3u = LD80C(0xdab8bcfee40496cb, 0, 1.70876276441416471410e+00L),
133 v4u = LD80C(0xd2f2dc3638567e9f, -2, 4.12009126299534668571e-01L),
134 v5u = LD80C(0xa07d9b0851070f41, -5, 3.91822868305682491442e-02L),
135 v6u = LD80C(0xe3cd8318f7adb2c4, -11, 8.68998648222144351114e-04L);
136 #define u0 (u0u.e)
137 #define u1 (u1u.e)
138 #define u2 (u2u.e)
139 #define u3 (u3u.e)
140 #define u4 (u4u.e)
141 #define u5 (u5u.e)
142 #define u6 (u6u.e)
143 #define v1 (v1u.e)
144 #define v2 (v2u.e)
145 #define v3 (v3u.e)
146 #define v4 (v4u.e)
147 #define v5 (v5u.e)
148 #define v6 (v6u.e)
149 /*
150 * Domain x in (2, 3], range ~[-3.3648e-22, 3.4416e-22]:
151 * |(lgamma(y+2) - 0.5 * y) / y - s(y)/r(y)| < 2**-72.3
152 * with y = x - 2.
153 */
154 static const union IEEEl2bits
155 s0u = LD80C(0x9e233f1bed863d27, -4, -7.72156649015328606095e-02L),
156 s1u = LD80C(0xd3ff0dcc7fa91f94, -3, 2.07027640921219389860e-01L),
157 s2u = LD80C(0xb2bb62782478ef31, -2, 3.49085881391362090549e-01L),
158 s3u = LD80C(0xb49f7438c4611a74, -3, 1.76389518704213357954e-01L),
159 s4u = LD80C(0x9a957008fa27ecf9, -5, 3.77401710862930008071e-02L),
160 s5u = LD80C(0xda9b389a6ca7a7ac, -9, 3.33566791452943399399e-03L),
161 s6u = LD80C(0xbc7a2263faf59c14, -14, 8.98728786745638844395e-05L),
162 r1u = LD80C(0xbf5cff5b11477d4d, 0, 1.49502555796294337722e+00L),
163 r2u = LD80C(0xd9aec89de08e3da6, -1, 8.50323236984473285866e-01L),
164 r3u = LD80C(0xeab7ae5057c443f9, -3, 2.29216312078225806131e-01L),
165 r4u = LD80C(0xf29707d9bd2b1e37, -6, 2.96130326586640089145e-02L),
166 r5u = LD80C(0xd376c2f09736c5a3, -10, 1.61334161411590662495e-03L),
167 r6u = LD80C(0xc985983d0cd34e3d, -16, 2.40232770710953450636e-05L),
168 r7u = LD80C(0xe5c7a4f7fc2ef13d, -25, -5.34997929289167573510e-08L);
169 #define s0 (s0u.e)
170 #define s1 (s1u.e)
171 #define s2 (s2u.e)
172 #define s3 (s3u.e)
173 #define s4 (s4u.e)
174 #define s5 (s5u.e)
175 #define s6 (s6u.e)
176 #define r1 (r1u.e)
177 #define r2 (r2u.e)
178 #define r3 (r3u.e)
179 #define r4 (r4u.e)
180 #define r5 (r5u.e)
181 #define r6 (r6u.e)
182 #define r7 (r7u.e)
183 /*
184 * Domain z in [8, 0x1p70], range ~[-3.0235e-22, 3.0563e-22]:
185 * |lgamma(x) - (x - 0.5) * (log(x) - 1) - w(1/x)| < 2**-71.7
186 */
187 static const union IEEEl2bits
188 w0u = LD80C(0xd67f1c864beb4a69, -2, 4.18938533204672741776e-01L),
189 w1u = LD80C(0xaaaaaaaaaaaaaaa1, -4, 8.33333333333333332678e-02L),
190 w2u = LD80C(0xb60b60b60b5491c9, -9, -2.77777777777760927870e-03L),
191 w3u = LD80C(0xd00d00cf58aede4c, -11, 7.93650793490637233668e-04L),
192 w4u = LD80C(0x9c09bf626783d4a5, -11, -5.95238023926039051268e-04L),
193 w5u = LD80C(0xdca7cadc5baa517b, -11, 8.41733700408000822962e-04L),
194 w6u = LD80C(0xfb060e361e1ffd07, -10, -1.91515849570245136604e-03L),
195 w7u = LD80C(0xcbd5101bb58d1f2b, -8, 6.22046743903262649294e-03L),
196 w8u = LD80C(0xad27a668d32c821b, -6, -2.11370706734662081843e-02L);
197 #define w0 (w0u.e)
198 #define w1 (w1u.e)
199 #define w2 (w2u.e)
200 #define w3 (w3u.e)
201 #define w4 (w4u.e)
202 #define w5 (w5u.e)
203 #define w6 (w6u.e)
204 #define w7 (w7u.e)
205 #define w8 (w8u.e)
206
207 static long double
sin_pil(long double x)208 sin_pil(long double x)
209 {
210 volatile long double vz;
211 long double y,z;
212 uint64_t n;
213 uint16_t hx;
214
215 y = -x;
216
217 vz = y+0x1p63;
218 z = vz-0x1p63;
219 if (z == y)
220 return zero;
221
222 vz = y+0x1p61;
223 EXTRACT_LDBL80_WORDS(hx,n,vz);
224 z = vz-0x1p61;
225 if (z > y) {
226 z -= 0.25; /* adjust to round down */
227 n--;
228 }
229 n &= 7; /* octant of y mod 2 */
230 y = y - z + n * 0.25; /* y mod 2 */
231
232 switch (n) {
233 case 0: y = __kernel_sinl(pi*y,zero,0); break;
234 case 1:
235 case 2: y = __kernel_cosl(pi*(0.5-y),zero); break;
236 case 3:
237 case 4: y = __kernel_sinl(pi*(one-y),zero,0); break;
238 case 5:
239 case 6: y = -__kernel_cosl(pi*(y-1.5),zero); break;
240 default: y = __kernel_sinl(pi*(y-2.0),zero,0); break;
241 }
242 return -y;
243 }
244
245 long double
lgammal_r(long double x,int * signgamp)246 lgammal_r(long double x, int *signgamp)
247 {
248 long double nadj,p,p1,p2,q,r,t,w,y,z;
249 uint64_t lx;
250 int i;
251 uint16_t hx,ix;
252
253 EXTRACT_LDBL80_WORDS(hx,lx,x);
254
255 /* purge +-Inf and NaNs */
256 *signgamp = 1;
257 ix = hx&0x7fff;
258 if(ix==0x7fff) return x*x;
259
260 ENTERI();
261
262 /* purge +-0 and tiny arguments */
263 *signgamp = 1-2*(hx>>15);
264 if(ix<0x3fff-67) { /* |x|<2**-(p+3), return -log(|x|) */
265 if((ix|lx)==0)
266 RETURNI(one/vzero);
267 RETURNI(-logl(fabsl(x)));
268 }
269
270 /* purge negative integers and start evaluation for other x < 0 */
271 if(hx&0x8000) {
272 *signgamp = 1;
273 if(ix>=0x3fff+63) /* |x|>=2**(p-1), must be -integer */
274 RETURNI(one/vzero);
275 t = sin_pil(x);
276 if(t==zero) RETURNI(one/vzero); /* -integer */
277 nadj = logl(pi/fabsl(t*x));
278 if(t<zero) *signgamp = -1;
279 x = -x;
280 }
281
282 /* purge 1 and 2 */
283 if((ix==0x3fff || ix==0x4000) && lx==0x8000000000000000ULL) r = 0;
284 /* for x < 2.0 */
285 else if(ix<0x4000) {
286 /*
287 * XXX Supposedly, one can use the following information to replace the
288 * XXX FP rational expressions. A similar approach is appropriate
289 * XXX for ld128, but one (may need?) needs to consider llx, too.
290 *
291 * 8.9999961853027344e-01 3ffe e666600000000000
292 * 7.3159980773925781e-01 3ffe bb4a200000000000
293 * 2.3163998126983643e-01 3ffc ed33080000000000
294 * 1.7316312789916992e+00 3fff dda6180000000000
295 * 1.2316322326660156e+00 3fff 9da6200000000000
296 */
297 if(x<8.9999961853027344e-01) {
298 r = -logl(x);
299 if(x>=7.3159980773925781e-01) {y = 1-x; i= 0;}
300 else if(x>=2.3163998126983643e-01) {y= x-(tc-1); i=1;}
301 else {y = x; i=2;}
302 } else {
303 r = 0;
304 if(x>=1.7316312789916992e+00) {y=2-x;i=0;}
305 else if(x>=1.2316322326660156e+00) {y=x-tc;i=1;}
306 else {y=x-1;i=2;}
307 }
308 switch(i) {
309 case 0:
310 z = y*y;
311 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*(a10+z*a12)))));
312 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*(a11+z*a13))))));
313 p = y*p1+p2;
314 r += p-y/2; break;
315 case 1:
316 p = t0+y*t1+tt+y*y*(t2+y*(t3+y*(t4+y*(t5+y*(t6+y*(t7+y*(t8+
317 y*(t9+y*(t10+y*(t11+y*(t12+y*(t13+y*(t14+y*(t15+y*(t16+
318 y*(t17+y*t18))))))))))))))));
319 r += tf + p; break;
320 case 2:
321 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*(u5+y*u6))))));
322 p2 = 1+y*(v1+y*(v2+y*(v3+y*(v4+y*(v5+y*v6)))));
323 r += p1/p2-y/2;
324 }
325 }
326 /* x < 8.0 */
327 else if(ix<0x4002) {
328 i = x;
329 y = x-i;
330 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
331 q = 1+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*(r6+y*r7))))));
332 r = y/2+p/q;
333 z = 1; /* lgamma(1+s) = log(s) + lgamma(s) */
334 switch(i) {
335 case 7: z *= (y+6); /* FALLTHRU */
336 case 6: z *= (y+5); /* FALLTHRU */
337 case 5: z *= (y+4); /* FALLTHRU */
338 case 4: z *= (y+3); /* FALLTHRU */
339 case 3: z *= (y+2); /* FALLTHRU */
340 r += logl(z); break;
341 }
342 /* 8.0 <= x < 2**(p+3) */
343 } else if (ix<0x3fff+67) {
344 t = logl(x);
345 z = one/x;
346 y = z*z;
347 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*(w6+y*(w7+y*w8)))))));
348 r = (x-half)*(t-one)+w;
349 /* 2**(p+3) <= x <= inf */
350 } else
351 r = x*(logl(x)-1);
352 if(hx&0x8000) r = nadj - r;
353 RETURNI(r);
354 }
355