src/math/lgamma_r.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
   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 /* lgamma_r(x, signgamp)
  14  * Reentrant version of the logarithm of the Gamma function
  15  * with user provide pointer for the sign of Gamma(x).
  16  *
  17  * Method:
  18  *   1. Argument Reduction for 0 < x <= 8
  19  *      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
  20  *      reduce x to a number in [1.5,2.5] by
  21  *              lgamma(1+s) = log(s) + lgamma(s)
  22  *      for example,
  23  *              lgamma(7.3) = log(6.3) + lgamma(6.3)
  24  *                          = log(6.3*5.3) + lgamma(5.3)
  25  *                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
  26  *   2. Polynomial approximation of lgamma around its
  27  *      minimun ymin=1.461632144968362245 to maintain monotonicity.
  28  *      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
  29  *              Let z = x-ymin;
  30  *              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
  31  *      where
  32  *              poly(z) is a 14 degree polynomial.
  33  *   2. Rational approximation in the primary interval [2,3]
  34  *      We use the following approximation:
  35  *              s = x-2.0;
  36  *              lgamma(x) = 0.5*s + s*P(s)/Q(s)
  37  *      with accuracy
  38  *              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
  39  *      Our algorithms are based on the following observation
  40  *
  41  *                             zeta(2)-1    2    zeta(3)-1    3
  42  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
  43  *                                 2                 3
  44  *
  45  *      where Euler = 0.5771... is the Euler constant, which is very
  46  *      close to 0.5.
  47  *
  48  *   3. For x>=8, we have
  49  *      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
  50  *      (better formula:
  51  *         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
  52  *      Let z = 1/x, then we approximation
  53  *              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
  54  *      by
  55  *                                  3       5             11
  56  *              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
  57  *      where
  58  *              |w - f(z)| < 2**-58.74
  59  *
  60  *   4. For negative x, since (G is gamma function)
  61  *              -x*G(-x)*G(x) = pi/sin(pi*x),
  62  *      we have
  63  *              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
  64  *      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
  65  *      Hence, for x<0, signgam = sign(sin(pi*x)) and
  66  *              lgamma(x) = log(|Gamma(x)|)
  67  *                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
  68  *      Note: one should avoid compute pi*(-x) directly in the
  69  *            computation of sin(pi*(-x)).
  70  *
  71  *   5. Special Cases
  72  *              lgamma(2+s) ~ s*(1-Euler) for tiny s
  73  *              lgamma(1) = lgamma(2) = 0
  74  *              lgamma(x) ~ -log(|x|) for tiny x
  75  *              lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
  76  *              lgamma(inf) = inf
  77  *              lgamma(-inf) = inf (bug for bug compatible with C99!?)
  78  *
  79  */
  80
  81 #include "libm.h"
  82
  83 static const double
  84 two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
  85 pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
  86 a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
  87 a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
  88 a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
  89 a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
  90 a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
  91 a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
  92 a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
  93 a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
  94 a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
  95 a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
  96 a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
  97 a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
  98 tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
  99 tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
 100 /* tt = -(tail of tf) */
 101 tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
 102 t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
 103 t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
 104 t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
 105 t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
 106 t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
 107 t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
 108 t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
 109 t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
 110 t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
 111 t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
 112 t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
 113 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
 114 t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
 115 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
 116 t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
 117 u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 118 u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
 119 u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
 120 u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
 121 u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
 122 u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
 123 v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
 124 v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
 125 v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
 126 v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
 127 v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
 128 s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 129 s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
 130 s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
 131 s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
 132 s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
 133 s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
 134 s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
 135 r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
 136 r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
 137 r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
 138 r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
 139 r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
 140 r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
 141 w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
 142 w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
 143 w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
 144 w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
 145 w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
 146 w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
 147 w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
 148
 149 static double sin_pi(double x)
 150 {
 151         double y,z;
 152         int n,ix;
 153
 154         GET_HIGH_WORD(ix, x);
 155         ix &= 0x7fffffff;
 156
 157         if (ix < 0x3fd00000)
 158                 return __sin(pi*x, 0.0, 0);
 159
 160         y = -x;  /* negative x is assumed */
 161
 162         /*
 163          * argument reduction, make sure inexact flag not raised if input
 164          * is an integer
 165          */
 166         z = floor(y);
 167         if (z != y) {    /* inexact anyway */
 168                 y *= 0.5;
 169                 y  = 2.0*(y - floor(y));   /* y = |x| mod 2.0 */
 170                 n  = (int)(y*4.0);
 171         } else {
 172                 if (ix >= 0x43400000) {
 173                         y = 0.0;    /* y must be even */
 174                         n = 0;
 175                 } else {
 176                         if (ix < 0x43300000)
 177                                 z = y + two52;  /* exact */
 178                         GET_LOW_WORD(n, z);
 179                         n &= 1;
 180                         y = n;
 181                         n <<= 2;
 182                 }
 183         }
 184         switch (n) {
 185         case 0:  y =  __sin(pi*y, 0.0, 0); break;
 186         case 1:
 187         case 2:  y =  __cos(pi*(0.5-y), 0.0); break;
 188         case 3:
 189         case 4:  y =  __sin(pi*(1.0-y), 0.0, 0); break;
 190         case 5:
 191         case 6:  y = -__cos(pi*(y-1.5), 0.0); break;
 192         default: y =  __sin(pi*(y-2.0), 0.0, 0); break;
 193         }
 194         return -y;
 195 }
 196
 197
 198 double __lgamma_r(double x, int *signgamp)
 199 {
 200         double t,y,z,nadj,p,p1,p2,p3,q,r,w;
 201         int32_t hx;
 202         int i,lx,ix;
 203
 204         EXTRACT_WORDS(hx, lx, x);
 205
 206         /* purge off +-inf, NaN, +-0, tiny and negative arguments */
 207         *signgamp = 1;
 208         ix = hx & 0x7fffffff;
 209         if (ix >= 0x7ff00000)
 210                 return x*x;
 211         if ((ix|lx) == 0)
 212                 return 1.0/0.0;
 213         if (ix < 0x3b900000) {  /* |x|<2**-70, return -log(|x|) */
 214                 if(hx < 0) {
 215                         *signgamp = -1;
 216                         return -log(-x);
 217                 }
 218                 return -log(x);
 219         }
 220         if (hx < 0) {
 221                 if (ix >= 0x43300000)  /* |x|>=2**52, must be -integer */
 222                         return 1.0/0.0;
 223                 t = sin_pi(x);
 224                 if (t == 0.0) /* -integer */
 225                         return 1.0/0.0;
 226                 nadj = log(pi/fabs(t*x));
 227                 if (t < 0.0)
 228                         *signgamp = -1;
 229                 x = -x;
 230         }
 231
 232         /* purge off 1 and 2 */
 233         if (((ix - 0x3ff00000)|lx) == 0 || ((ix - 0x40000000)|lx) == 0)
 234                 r = 0;
 235         /* for x < 2.0 */
 236         else if (ix < 0x40000000) {
 237                 if (ix <= 0x3feccccc) {   /* lgamma(x) = lgamma(x+1)-log(x) */
 238                         r = -log(x);
 239                         if (ix >= 0x3FE76944) {
 240                                 y = 1.0 - x;
 241                                 i = 0;
 242                         } else if (ix >= 0x3FCDA661) {
 243                                 y = x - (tc-1.0);
 244                                 i = 1;
 245                         } else {
 246                                 y = x;
 247                                 i = 2;
 248                         }
 249                 } else {
 250                         r = 0.0;
 251                         if (ix >= 0x3FFBB4C3) {  /* [1.7316,2] */
 252                                 y = 2.0 - x;
 253                                 i = 0;
 254                         } else if(ix >= 0x3FF3B4C4) {  /* [1.23,1.73] */
 255                                 y = x - tc;
 256                                 i = 1;
 257                         } else {
 258                                 y = x - 1.0;
 259                                 i = 2;
 260                         }
 261                 }
 262                 switch (i) {
 263                 case 0:
 264                         z = y*y;
 265                         p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
 266                         p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
 267                         p = y*p1+p2;
 268                         r += (p-0.5*y);
 269                         break;
 270                 case 1:
 271                         z = y*y;
 272                         w = z*y;
 273                         p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
 274                         p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
 275                         p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
 276                         p = z*p1-(tt-w*(p2+y*p3));
 277                         r += tf + p;
 278                         break;
 279                 case 2:
 280                         p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
 281                         p2 = 1.0+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
 282                         r += -0.5*y + p1/p2;
 283                 }
 284         } else if (ix < 0x40200000) {  /* x < 8.0 */
 285                 i = (int)x;
 286                 y = x - (double)i;
 287                 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
 288                 q = 1.0+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
 289                 r = 0.5*y+p/q;
 290                 z = 1.0;    /* lgamma(1+s) = log(s) + lgamma(s) */
 291                 switch (i) {
 292                 case 7: z *= y + 6.0;  /* FALLTHRU */
 293                 case 6: z *= y + 5.0;  /* FALLTHRU */
 294                 case 5: z *= y + 4.0;  /* FALLTHRU */
 295                 case 4: z *= y + 3.0;  /* FALLTHRU */
 296                 case 3: z *= y + 2.0;  /* FALLTHRU */
 297                         r += log(z);
 298                         break;
 299                 }
 300         } else if (ix < 0x43900000) {  /* 8.0 <= x < 2**58 */
 301                 t = log(x);
 302                 z = 1.0/x;
 303                 y = z*z;
 304                 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
 305                 r = (x-0.5)*(t-1.0)+w;
 306         } else                         /* 2**58 <= x <= inf */
 307                 r =  x*(log(x)-1.0);
 308         if (hx < 0)
 309                 r = nadj - r;
 310         return r;
 311 }
 312
 313 weak_alias(__lgamma_r, lgamma_r);