libm/e_lgamma_r.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, 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 /* __ieee754_lgamma_r(x, signgamp)
  13  * Reentrant version of the logarithm of the Gamma function
  14  * with user provide pointer for the sign of Gamma(x).
  15  *
  16  * Method:
  17  *   1. Argument Reduction for 0 < x <= 8
  18  *      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
  19  *      reduce x to a number in [1.5,2.5] by
  20  *              lgamma(1+s) = log(s) + lgamma(s)
  21  *      for example,
  22  *              lgamma(7.3) = log(6.3) + lgamma(6.3)
  23  *                          = log(6.3*5.3) + lgamma(5.3)
  24  *                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
  25  *   2. Polynomial approximation of lgamma around its
  26  *      minimun ymin=1.461632144968362245 to maintain monotonicity.
  27  *      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
  28  *              Let z = x-ymin;
  29  *              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
  30  *      where
  31  *              poly(z) is a 14 degree polynomial.
  32  *   2. Rational approximation in the primary interval [2,3]
  33  *      We use the following approximation:
  34  *              s = x-2.0;
  35  *              lgamma(x) = 0.5*s + s*P(s)/Q(s)
  36  *      with accuracy
  37  *              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
  38  *      Our algorithms are based on the following observation
  39  *
  40  *                             zeta(2)-1    2    zeta(3)-1    3
  41  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
  42  *                                 2                 3
  43  *
  44  *      where Euler = 0.5771... is the Euler constant, which is very
  45  *      close to 0.5.
  46  *
  47  *   3. For x>=8, we have
  48  *      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
  49  *      (better formula:
  50  *         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
  51  *      Let z = 1/x, then we approximation
  52  *              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
  53  *      by
  54  *                                  3       5             11
  55  *              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
  56  *      where
  57  *              |w - f(z)| < 2**-58.74
  58  *
  59  *   4. For negative x, since (G is gamma function)
  60  *              -x*G(-x)*G(x) = pi/sin(pi*x),
  61  *      we have
  62  *              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
  63  *      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
  64  *      Hence, for x<0, signgam = sign(sin(pi*x)) and
  65  *              lgamma(x) = log(|Gamma(x)|)
  66  *                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
  67  *      Note: one should avoid compute pi*(-x) directly in the
  68  *            computation of sin(pi*(-x)).
  69  *
  70  *   5. Special Cases
  71  *              lgamma(2+s) ~ s*(1-Euler) for tiny s
  72  *              lgamma(1)=lgamma(2)=0
  73  *              lgamma(x) ~ -log(x) for tiny x
  74  *              lgamma(0) = lgamma(inf) = inf
  75  *              lgamma(-integer) = +-inf
  76  *
  77  */
  78
  79 #include "math.h"
  80 #include "math_private.h"
  81
  82 static const double
  83 two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
  84 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
  85 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
  86 pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
  87 a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
  88 a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
  89 a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
  90 a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
  91 a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
  92 a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
  93 a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
  94 a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
  95 a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
  96 a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
  97 a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
  98 a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
  99 tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
 100 tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
 101 /* tt = -(tail of tf) */
 102 tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
 103 t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
 104 t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
 105 t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
 106 t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
 107 t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
 108 t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
 109 t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
 110 t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
 111 t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
 112 t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
 113 t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
 114 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
 115 t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
 116 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
 117 t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
 118 u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 119 u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
 120 u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
 121 u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
 122 u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
 123 u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
 124 v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
 125 v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
 126 v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
 127 v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
 128 v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
 129 s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 130 s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
 131 s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
 132 s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
 133 s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
 134 s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
 135 s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
 136 r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
 137 r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
 138 r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
 139 r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
 140 r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
 141 r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
 142 w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
 143 w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
 144 w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
 145 w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
 146 w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
 147 w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
 148 w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
 149
 150 static const double zero=  0.00000000000000000000e+00;
 151
 152 static
 153 #ifdef __GNUC__
 154 __inline__
 155 #endif
 156 double sin_pi(double x)
 157 {
 158         double y,z;
 159         int n,ix;
 160
 161         GET_HIGH_WORD(ix,x);
 162         ix &= 0x7fffffff;
 163
 164         if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
 165         y = -x;         /* x is assume negative */
 166
 167     /*
 168      * argument reduction, make sure inexact flag not raised if input
 169      * is an integer
 170      */
 171         z = floor(y);
 172         if(z!=y) {                              /* inexact anyway */
 173             y  *= 0.5;
 174             y   = 2.0*(y - floor(y));           /* y = |x| mod 2.0 */
 175             n   = (int) (y*4.0);
 176         } else {
 177             if(ix>=0x43400000) {
 178                 y = zero; n = 0;                 /* y must be even */
 179             } else {
 180                 if(ix<0x43300000) z = y+two52;  /* exact */
 181                 GET_LOW_WORD(n,z);
 182                 n &= 1;
 183                 y  = n;
 184                 n<<= 2;
 185             }
 186         }
 187         switch (n) {
 188             case 0:   y =  __kernel_sin(pi*y,zero,0); break;
 189             case 1:
 190             case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
 191             case 3:
 192             case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
 193             case 5:
 194             case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
 195             default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
 196             }
 197         return -y;
 198 }
 199
 200 double __ieee754_lgamma_r(double x, int *signgamp)
 201 {
 202         double t,y,z,nadj=0,p,p1,p2,p3,q,r,w;
 203         int i,hx,lx,ix;
 204
 205         EXTRACT_WORDS(hx,lx,x);
 206
 207     /* purge off +-inf, NaN, +-0, and negative arguments */
 208         *signgamp = 1;
 209         ix = hx&0x7fffffff;
 210         if(ix>=0x7ff00000) return x*x;
 211         if((ix|lx)==0) {
 212             if (signbit(x))
 213                 *signgamp = -1;
 214             return one/zero;
 215         }
 216         if(ix<0x3b900000) {     /* |x|<2**-70, return -log(|x|) */
 217             if(hx<0) {
 218                 *signgamp = -1;
 219                 return -__ieee754_log(-x);
 220             } else return -__ieee754_log(x);
 221         }
 222         if(hx<0) {
 223             if(ix>=0x43300000)  /* |x|>=2**52, must be -integer */
 224                 return one/zero;
 225             t = sin_pi(x);
 226             if(t==zero) return one/zero; /* -integer */
 227             nadj = __ieee754_log(pi/fabs(t*x));
 228             if(t<zero) *signgamp = -1;
 229             x = -x;
 230         }
 231
 232     /* purge off 1 and 2 */
 233         if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
 234     /* for x < 2.0 */
 235         else if(ix<0x40000000) {
 236             if(ix<=0x3feccccc) {        /* lgamma(x) = lgamma(x+1)-log(x) */
 237                 r = -__ieee754_log(x);
 238                 if(ix>=0x3FE76944) {y = one-x; i= 0;}
 239                 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
 240                 else {y = x; i=2;}
 241             } else {
 242                 r = zero;
 243                 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
 244                 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
 245                 else {y=x-one;i=2;}
 246             }
 247             switch(i) {
 248               case 0:
 249                 z = y*y;
 250                 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
 251                 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
 252                 p  = y*p1+p2;
 253                 r  += (p-0.5*y); break;
 254               case 1:
 255                 z = y*y;
 256                 w = z*y;
 257                 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
 258                 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
 259                 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
 260                 p  = z*p1-(tt-w*(p2+y*p3));
 261                 r += (tf + p); break;
 262               case 2:
 263                 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
 264                 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
 265                 r += (-0.5*y + p1/p2);
 266             }
 267         }
 268         else if(ix<0x40200000) {                        /* x < 8.0 */
 269             i = (int)x;
 270             t = zero;
 271             y = x-(double)i;
 272             p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
 273             q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
 274             r = half*y+p/q;
 275             z = one;    /* lgamma(1+s) = log(s) + lgamma(s) */
 276             switch(i) {
 277             case 7: z *= (y+6.0);       /* FALLTHRU */
 278             case 6: z *= (y+5.0);       /* FALLTHRU */
 279             case 5: z *= (y+4.0);       /* FALLTHRU */
 280             case 4: z *= (y+3.0);       /* FALLTHRU */
 281             case 3: z *= (y+2.0);       /* FALLTHRU */
 282                     r += __ieee754_log(z); break;
 283             }
 284     /* 8.0 <= x < 2**58 */
 285         } else if (ix < 0x43900000) {
 286             t = __ieee754_log(x);
 287             z = one/x;
 288             y = z*z;
 289             w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
 290             r = (x-half)*(t-one)+w;
 291         } else
 292     /* 2**58 <= x <= inf */
 293             r =  x*(__ieee754_log(x)-one);
 294         if(hx<0) r = nadj - r;
 295         return r;
 296 }
 297
 298 strong_alias(__ieee754_lgamma_r, lgamma_r)
 299 libm_hidden_def(lgamma_r)
 300
 301 /* __ieee754_lgamma(x)
 302  * Return the logarithm of the Gamma function of x.
 303  */
 304 double __ieee754_lgamma(double x)
 305 {
 306         return __ieee754_lgamma_r(x, &signgam);
 307 }
 308
 309 strong_alias(__ieee754_lgamma, lgamma);
 310 libm_hidden_def(lgamma)
 311
 312
 313 /* NB: gamma function is an old name for lgamma.
 314  * It is deprecated.
 315  * Some C math libraries redefine it as a "true gamma", i.e.,
 316  * not a ln(|Gamma(x)|) but just Gamma(x), but standards
 317  * introduced tgamma name for that.
 318  */
 319 strong_alias(__ieee754_lgamma_r, gamma_r)
 320 strong_alias(__ieee754_lgamma, gamma)
 321 libm_hidden_def(gamma)
 322
 323
 324 /* double tgamma(double x)
 325  * Return the Gamma function of x.
 326  */
 327 double tgamma(double x)
 328 {
 329         int sign_of_gamma;
 330         int32_t hx;
 331         u_int32_t lx;
 332
 333         /* We don't have a real gamma implementation now.  We'll use lgamma
 334            and the exp function.  But due to the required boundary
 335            conditions we must check some values separately.  */
 336
 337         EXTRACT_WORDS(hx, lx, x);
 338
 339         if (((hx & 0x7fffffff) | lx) == 0) {
 340                 /* Return value for x == 0 is Inf with divide by zero exception.  */
 341                 return 1.0 / x;
 342         }
 343         if (hx < 0 && (u_int32_t)hx < 0xfff00000 && rint(x) == x) {
 344                 /* Return value for integer x < 0 is NaN with invalid exception.  */
 345                 return (x - x) / (x - x);
 346         }
 347         if ((u_int32_t)hx == 0xfff00000 && lx == 0) {
 348                 /* x == -Inf.  According to ISO this is NaN.  */
 349                 return x - x;
 350         }
 351
 352         x = exp(lgamma_r(x, &sign_of_gamma));
 353         return sign_of_gamma >= 0 ? x : -x;
 354 }
 355 libm_hidden_def(tgamma)