newlib/libm/ld80/e_lgammal_r.c

   1 /* @(#)e_lgamma_r.c 1.3 95/01/18 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12
  13 #include <sys/cdefs.h>
  14 __FBSDID("$FreeBSD$");
  15
  16 /*
  17  * See e_lgamma_r.c for complete comments.
  18  *
  19  * Converted to long double by Steven G. Kargl.
  20  */
  21
  22 #ifdef __i386__
  23 #include <ieeefp.h>
  24 #endif
  25
  26 #include "../ld/fpmath.h"
  27 #include "math.h"
  28 #include "../ld/math_private.h"
  29
  30 static const volatile double vzero = 0;
  31
  32 static const double
  33 zero=  0,
  34 half=  0.5,
  35 one =  1;
  36
  37 static const union IEEEl2bits
  38 piu = LD80C(0xc90fdaa22168c235, 1,  3.14159265358979323851e+00L);
  39 #define pi      (piu.e)
  40 /*
  41  * Domain y in [0x1p-70, 0.27], range ~[-4.5264e-22, 4.5264e-22]:
  42  * |(lgamma(2 - y) + y / 2) / y - a(y)| < 2**-70.9
  43  */
  44 static const union IEEEl2bits
  45 a0u = LD80C(0x9e233f1bed863d26, -4,  7.72156649015328606028e-02L),
  46 a1u = LD80C(0xa51a6625307d3249, -2,  3.22467033424113218889e-01L),
  47 a2u = LD80C(0x89f000d2abafda8c, -4,  6.73523010531979398946e-02L),
  48 a3u = LD80C(0xa8991563eca75f26, -6,  2.05808084277991211934e-02L),
  49 a4u = LD80C(0xf2027e10634ce6b6, -8,  7.38555102796070454026e-03L),
  50 a5u = LD80C(0xbd6eb76dd22187f4, -9,  2.89051035162703932972e-03L),
  51 a6u = LD80C(0x9c562ab05e0458ed, -10,  1.19275351624639999297e-03L),
  52 a7u = LD80C(0x859baed93ee48e46, -11,  5.09674593842117925320e-04L),
  53 a8u = LD80C(0xe9f28a4432949af2, -13,  2.23109648015769155122e-04L),
  54 a9u = LD80C(0xd12ad0d9b93c6bb0, -14,  9.97387167479808509830e-05L),
  55 a10u= LD80C(0xb7522643c78a219b, -15,  4.37071076331030136818e-05L),
  56 a11u= LD80C(0xca024dcdece2cb79, -16,  2.40813493372040143061e-05L),
  57 a12u= LD80C(0xbb90fb6968ebdbf9, -19,  2.79495621083634031729e-06L),
  58 a13u= LD80C(0xba1c9ffeeae07b37, -17,  1.10931287015513924136e-05L);
  59 #define a0      (a0u.e)
  60 #define a1      (a1u.e)
  61 #define a2      (a2u.e)
  62 #define a3      (a3u.e)
  63 #define a4      (a4u.e)
  64 #define a5      (a5u.e)
  65 #define a6      (a6u.e)
  66 #define a7      (a7u.e)
  67 #define a8      (a8u.e)
  68 #define a9      (a9u.e)
  69 #define a10     (a10u.e)
  70 #define a11     (a11u.e)
  71 #define a12     (a12u.e)
  72 #define a13     (a13u.e)
  73 /*
  74  * Domain x in [tc-0.24, tc+0.28], range ~[-6.1205e-22, 6.1205e-22]:
  75  * |(lgamma(x) - tf) -  t(x - tc)| < 2**-70.5
  76  */
  77 static const union IEEEl2bits
  78 tcu  = LD80C(0xbb16c31ab5f1fb71, 0,  1.46163214496836234128e+00L),
  79 tfu  = LD80C(0xf8cdcde61c520e0f, -4, -1.21486290535849608093e-01L),
  80 ttu  = LD80C(0xd46ee54b27d4de99, -69, -2.81152980996018785880e-21L),
  81 t0u  = LD80C(0x80b9406556a62a6b, -68,  3.40728634996055147231e-21L),
  82 t1u  = LD80C(0xc7e9c6f6df3f8c39, -67, -1.05833162742737073665e-20L),
  83 t2u  = LD80C(0xf7b95e4771c55d51, -2,  4.83836122723810583532e-01L),
  84 t3u  = LD80C(0x97213c6e35e119ff, -3, -1.47587722994530691476e-01L),
  85 t4u  = LD80C(0x845a14a6a81dc94b, -4,  6.46249402389135358063e-02L),
  86 t5u  = LD80C(0x864d46fa89997796, -5, -3.27885410884846056084e-02L),
  87 t6u  = LD80C(0x93373cbd00297438, -6,  1.79706751150707171293e-02L),
  88 t7u  = LD80C(0xa8fcfca7eddc8d1d, -7, -1.03142230361450732547e-02L),
  89 t8u  = LD80C(0xc7e7015ff4bc45af, -8,  6.10053603296546099193e-03L),
  90 t9u  = LD80C(0xf178d2247adc5093, -9, -3.68456964904901200152e-03L),
  91 t10u = LD80C(0x94188d58f12e5e9f, -9,  2.25976420273774583089e-03L),
  92 t11u = LD80C(0xb7cbaef14e1406f1, -10, -1.40224943666225639823e-03L),
  93 t12u = LD80C(0xe63a671e6704ea4d, -11,  8.78250640744776944887e-04L),
  94 t13u = LD80C(0x914b6c9cae61783e, -11, -5.54255012657716808811e-04L),
  95 t14u = LD80C(0xb858f5bdb79276fe, -12,  3.51614951536825927370e-04L),
  96 t15u = LD80C(0xea73e744c34b9591, -13, -2.23591563824520112236e-04L),
  97 t16u = LD80C(0x99aeabb0d67ba835, -13,  1.46562869351659194136e-04L),
  98 t17u = LD80C(0xd7c6938325db2024, -14, -1.02889866046435680588e-04L),
  99 t18u = LD80C(0xe24cb1e3b0474775, -15,  5.39540265505221957652e-05L);
 100 #define tc      (tcu.e)
 101 #define tf      (tfu.e)
 102 #define tt      (ttu.e)
 103 #define t0      (t0u.e)
 104 #define t1      (t1u.e)
 105 #define t2      (t2u.e)
 106 #define t3      (t3u.e)
 107 #define t4      (t4u.e)
 108 #define t5      (t5u.e)
 109 #define t6      (t6u.e)
 110 #define t7      (t7u.e)
 111 #define t8      (t8u.e)
 112 #define t9      (t9u.e)
 113 #define t10     (t10u.e)
 114 #define t11     (t11u.e)
 115 #define t12     (t12u.e)
 116 #define t13     (t13u.e)
 117 #define t14     (t14u.e)
 118 #define t15     (t15u.e)
 119 #define t16     (t16u.e)
 120 #define t17     (t17u.e)
 121 #define t18     (t18u.e)
 122 /*
 123  * Domain y in [-0.1, 0.232], range ~[-8.1938e-22, 8.3815e-22]:
 124  * |(lgamma(1 + y) + 0.5 * y) / y - u(y) / v(y)| < 2**-71.2
 125  */
 126 static const union IEEEl2bits
 127 u0u = LD80C(0x9e233f1bed863d27, -4, -7.72156649015328606095e-02L),
 128 u1u = LD80C(0x98280ee45e4ddd3d, -1,  5.94361239198682739769e-01L),
 129 u2u = LD80C(0xe330c8ead4130733, 0,  1.77492629495841234275e+00L),
 130 u3u = LD80C(0xd4a213f1a002ec52, 0,  1.66119622514818078064e+00L),
 131 u4u = LD80C(0xa5a9ca6f5bc62163, -1,  6.47122051417476492989e-01L),
 132 u5u = LD80C(0xc980e49cd5b019e6, -4,  9.83903751718671509455e-02L),
 133 u6u = LD80C(0xff636a8bdce7025b, -9,  3.89691687802305743450e-03L),
 134 v1u = LD80C(0xbd109c533a19fbf5, 1,  2.95413883330948556544e+00L),
 135 v2u = LD80C(0xd295cbf96f31f099, 1,  3.29039286955665403176e+00L),
 136 v3u = LD80C(0xdab8bcfee40496cb, 0,  1.70876276441416471410e+00L),
 137 v4u = LD80C(0xd2f2dc3638567e9f, -2,  4.12009126299534668571e-01L),
 138 v5u = LD80C(0xa07d9b0851070f41, -5,  3.91822868305682491442e-02L),
 139 v6u = LD80C(0xe3cd8318f7adb2c4, -11,  8.68998648222144351114e-04L);
 140 #define u0      (u0u.e)
 141 #define u1      (u1u.e)
 142 #define u2      (u2u.e)
 143 #define u3      (u3u.e)
 144 #define u4      (u4u.e)
 145 #define u5      (u5u.e)
 146 #define u6      (u6u.e)
 147 #define v1      (v1u.e)
 148 #define v2      (v2u.e)
 149 #define v3      (v3u.e)
 150 #define v4      (v4u.e)
 151 #define v5      (v5u.e)
 152 #define v6      (v6u.e)
 153 /*
 154  * Domain x in (2, 3], range ~[-3.3648e-22, 3.4416e-22]:
 155  * |(lgamma(y+2) - 0.5 * y) / y - s(y)/r(y)| < 2**-72.3
 156  * with y = x - 2.
 157  */
 158 static const union IEEEl2bits
 159 s0u = LD80C(0x9e233f1bed863d27, -4, -7.72156649015328606095e-02L),
 160 s1u = LD80C(0xd3ff0dcc7fa91f94, -3,  2.07027640921219389860e-01L),
 161 s2u = LD80C(0xb2bb62782478ef31, -2,  3.49085881391362090549e-01L),
 162 s3u = LD80C(0xb49f7438c4611a74, -3,  1.76389518704213357954e-01L),
 163 s4u = LD80C(0x9a957008fa27ecf9, -5,  3.77401710862930008071e-02L),
 164 s5u = LD80C(0xda9b389a6ca7a7ac, -9,  3.33566791452943399399e-03L),
 165 s6u = LD80C(0xbc7a2263faf59c14, -14,  8.98728786745638844395e-05L),
 166 r1u = LD80C(0xbf5cff5b11477d4d, 0,  1.49502555796294337722e+00L),
 167 r2u = LD80C(0xd9aec89de08e3da6, -1,  8.50323236984473285866e-01L),
 168 r3u = LD80C(0xeab7ae5057c443f9, -3,  2.29216312078225806131e-01L),
 169 r4u = LD80C(0xf29707d9bd2b1e37, -6,  2.96130326586640089145e-02L),
 170 r5u = LD80C(0xd376c2f09736c5a3, -10,  1.61334161411590662495e-03L),
 171 r6u = LD80C(0xc985983d0cd34e3d, -16,  2.40232770710953450636e-05L),
 172 r7u = LD80C(0xe5c7a4f7fc2ef13d, -25, -5.34997929289167573510e-08L);
 173 #define s0      (s0u.e)
 174 #define s1      (s1u.e)
 175 #define s2      (s2u.e)
 176 #define s3      (s3u.e)
 177 #define s4      (s4u.e)
 178 #define s5      (s5u.e)
 179 #define s6      (s6u.e)
 180 #define r1      (r1u.e)
 181 #define r2      (r2u.e)
 182 #define r3      (r3u.e)
 183 #define r4      (r4u.e)
 184 #define r5      (r5u.e)
 185 #define r6      (r6u.e)
 186 #define r7      (r7u.e)
 187 /*
 188  * Domain z in [8, 0x1p70], range ~[-3.0235e-22, 3.0563e-22]:
 189  * |lgamma(x) - (x - 0.5) * (log(x) - 1) - w(1/x)| < 2**-71.7
 190  */
 191 static const union IEEEl2bits
 192 w0u = LD80C(0xd67f1c864beb4a69, -2,  4.18938533204672741776e-01L),
 193 w1u = LD80C(0xaaaaaaaaaaaaaaa1, -4,  8.33333333333333332678e-02L),
 194 w2u = LD80C(0xb60b60b60b5491c9, -9, -2.77777777777760927870e-03L),
 195 w3u = LD80C(0xd00d00cf58aede4c, -11,  7.93650793490637233668e-04L),
 196 w4u = LD80C(0x9c09bf626783d4a5, -11, -5.95238023926039051268e-04L),
 197 w5u = LD80C(0xdca7cadc5baa517b, -11,  8.41733700408000822962e-04L),
 198 w6u = LD80C(0xfb060e361e1ffd07, -10, -1.91515849570245136604e-03L),
 199 w7u = LD80C(0xcbd5101bb58d1f2b, -8,  6.22046743903262649294e-03L),
 200 w8u = LD80C(0xad27a668d32c821b, -6, -2.11370706734662081843e-02L);
 201 #define w0      (w0u.e)
 202 #define w1      (w1u.e)
 203 #define w2      (w2u.e)
 204 #define w3      (w3u.e)
 205 #define w4      (w4u.e)
 206 #define w5      (w5u.e)
 207 #define w6      (w6u.e)
 208 #define w7      (w7u.e)
 209 #define w8      (w8u.e)
 210
 211 static long double
 212 sin_pil(long double x)
 213 {
 214         volatile long double vz;
 215         long double y,z;
 216         uint64_t n;
 217         uint16_t hx;
 218
 219         y = -x;
 220
 221         vz = y+0x1p63;
 222         z = vz-0x1p63;
 223         if (z == y)
 224             return zero;
 225
 226         vz = y+0x1p61;
 227         EXTRACT_LDBL80_WORDS(hx,n,vz);
 228         z = vz-0x1p61;
 229         if (z > y) {
 230             z -= 0.25;                  /* adjust to round down */
 231             n--;
 232         }
 233         n &= 7;                         /* octant of y mod 2 */
 234         y = y - z + n * 0.25;           /* y mod 2 */
 235
 236         switch (n) {
 237             case 0:   y =  __kernel_sinl(pi*y,zero,0); break;
 238             case 1:
 239             case 2:   y =  __kernel_cosl(pi*(0.5-y),zero); break;
 240             case 3:
 241             case 4:   y =  __kernel_sinl(pi*(one-y),zero,0); break;
 242             case 5:
 243             case 6:   y = -__kernel_cosl(pi*(y-1.5),zero); break;
 244             default:  y =  __kernel_sinl(pi*(y-2.0),zero,0); break;
 245             }
 246         return -y;
 247 }
 248
 249 long double
 250 lgammal_r(long double x, int *signgamp)
 251 {
 252         long double nadj,p,p1,p2,q,r,t,w,y,z;
 253         uint64_t lx;
 254         int i;
 255         uint16_t hx,ix;
 256
 257         EXTRACT_LDBL80_WORDS(hx,lx,x);
 258
 259     /* purge +-Inf and NaNs */
 260         *signgamp = 1;
 261         ix = hx&0x7fff;
 262         if(ix==0x7fff) return x*x;
 263
 264         ENTERI();
 265
 266     /* purge +-0 and tiny arguments */
 267         *signgamp = 1-2*(hx>>15);
 268         if(ix<0x3fff-67) {              /* |x|<2**-(p+3), return -log(|x|) */
 269             if((ix|lx)==0)
 270                 RETURNI(one/vzero);
 271             RETURNI(-logl(fabsl(x)));
 272         }
 273
 274     /* purge negative integers and start evaluation for other x < 0 */
 275         if(hx&0x8000) {
 276             *signgamp = 1;
 277             if(ix>=0x3fff+63)           /* |x|>=2**(p-1), must be -integer */
 278                 RETURNI(one/vzero);
 279             t = sin_pil(x);
 280             if(t==zero) RETURNI(one/vzero); /* -integer */
 281             nadj = logl(pi/fabsl(t*x));
 282             if(t<zero) *signgamp = -1;
 283             x = -x;
 284         }
 285
 286     /* purge 1 and 2 */
 287         if((ix==0x3fff || ix==0x4000) && lx==0x8000000000000000ULL) r = 0;
 288     /* for x < 2.0 */
 289         else if(ix<0x4000) {
 290     /*
 291      * XXX Supposedly, one can use the following information to replace the
 292      * XXX FP rational expressions.  A similar approach is appropriate
 293      * XXX for ld128, but one (may need?) needs to consider llx, too.
 294      *
 295      * 8.9999961853027344e-01 3ffe e666600000000000
 296      * 7.3159980773925781e-01 3ffe bb4a200000000000
 297      * 2.3163998126983643e-01 3ffc ed33080000000000
 298      * 1.7316312789916992e+00 3fff dda6180000000000
 299      * 1.2316322326660156e+00 3fff 9da6200000000000
 300      */
 301             if(x<8.9999961853027344e-01) {
 302                 r = -logl(x);
 303                 if(x>=7.3159980773925781e-01) {y = 1-x; i= 0;}
 304                 else if(x>=2.3163998126983643e-01) {y= x-(tc-1); i=1;}
 305                 else {y = x; i=2;}
 306             } else {
 307                 r = 0;
 308                 if(x>=1.7316312789916992e+00) {y=2-x;i=0;}
 309                 else if(x>=1.2316322326660156e+00) {y=x-tc;i=1;}
 310                 else {y=x-1;i=2;}
 311             }
 312             switch(i) {
 313               case 0:
 314                 z = y*y;
 315                 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*(a10+z*a12)))));
 316                 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*(a11+z*a13))))));
 317                 p  = y*p1+p2;
 318                 r  += p-y/2; break;
 319               case 1:
 320                 p = t0+y*t1+tt+y*y*(t2+y*(t3+y*(t4+y*(t5+y*(t6+y*(t7+y*(t8+
 321                     y*(t9+y*(t10+y*(t11+y*(t12+y*(t13+y*(t14+y*(t15+y*(t16+
 322                     y*(t17+y*t18))))))))))))))));
 323                 r += tf + p; break;
 324               case 2:
 325                 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*(u5+y*u6))))));
 326                 p2 = 1+y*(v1+y*(v2+y*(v3+y*(v4+y*(v5+y*v6)))));
 327                 r += p1/p2-y/2;
 328             }
 329         }
 330     /* x < 8.0 */
 331         else if(ix<0x4002) {
 332             i = x;
 333             y = x-i;
 334             p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
 335             q = 1+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*(r6+y*r7))))));
 336             r = y/2+p/q;
 337             z = 1;      /* lgamma(1+s) = log(s) + lgamma(s) */
 338             switch(i) {
 339             case 7: z *= (y+6);         /* FALLTHRU */
 340             case 6: z *= (y+5);         /* FALLTHRU */
 341             case 5: z *= (y+4);         /* FALLTHRU */
 342             case 4: z *= (y+3);         /* FALLTHRU */
 343             case 3: z *= (y+2);         /* FALLTHRU */
 344                     r += logl(z); break;
 345             }
 346     /* 8.0 <= x < 2**(p+3) */
 347         } else if (ix<0x3fff+67) {
 348             t = logl(x);
 349             z = one/x;
 350             y = z*z;
 351             w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*(w6+y*(w7+y*w8)))))));
 352             r = (x-half)*(t-one)+w;
 353     /* 2**(p+3) <= x <= inf */
 354         } else
 355             r =  x*(logl(x)-1);
 356         if(hx&0x8000) r = nadj - r;
 357         RETURNI(r);
 358 }