newlib/libm/mathfp/er_lgamma.c

   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 /*
  16 FUNCTION
  17         <<gamma>>, <<gammaf>>, <<lgamma>>, <<lgammaf>>, <<gamma_r>>,
  18         <<gammaf_r>>, <<lgamma_r>>, <<lgammaf_r>>---logarithmic gamma
  19         function
  20 INDEX
  21 gamma
  22 INDEX
  23 gammaf
  24 INDEX
  25 lgamma
  26 INDEX
  27 lgammaf
  28 INDEX
  29 gamma_r
  30 INDEX
  31 gammaf_r
  32 INDEX
  33 lgamma_r
  34 INDEX
  35 lgammaf_r
  36
  37 SYNOPSIS
  38 #include <math.h>
  39 double gamma(double <[x]>);
  40 float gammaf(float <[x]>);
  41 double lgamma(double <[x]>);
  42 float lgammaf(float <[x]>);
  43 double gamma_r(double <[x]>, int *<[signgamp]>);
  44 float gammaf_r(float <[x]>, int *<[signgamp]>);
  45 double lgamma_r(double <[x]>, int *<[signgamp]>);
  46 float lgammaf_r(float <[x]>, int *<[signgamp]>);
  47
  48 DESCRIPTION
  49 <<gamma>> calculates
  50 @tex
  51 $\mit ln\bigl(\Gamma(x)\bigr)$,
  52 @end tex
  53 the natural logarithm of the gamma function of <[x]>.  The gamma function
  54 (<<exp(gamma(<[x]>))>>) is a generalization of factorial, and retains
  55 the property that
  56 @ifnottex
  57 <<exp(gamma(N))>> is equivalent to <<N*exp(gamma(N-1))>>.
  58 @end ifnottex
  59 @tex
  60 $\mit \Gamma(N)\equiv N\times\Gamma(N-1)$.
  61 @end tex
  62 Accordingly, the results of the gamma function itself grow very
  63 quickly.  <<gamma>> is defined as
  64 @tex
  65 $\mit ln\bigl(\Gamma(x)\bigr)$ rather than simply $\mit \Gamma(x)$
  66 @end tex
  67 @ifnottex
  68 the natural log of the gamma function, rather than the gamma function
  69 itself,
  70 @end ifnottex
  71 to extend the useful range of results representable.
  72
  73 The sign of the result is returned in the global variable <<signgam>>,
  74 which is declared in math.h.
  75
  76 <<gammaf>> performs the same calculation as <<gamma>>, but uses and
  77 returns <<float>> values.
  78
  79 <<lgamma>> and <<lgammaf>> are alternate names for <<gamma>> and
  80 <<gammaf>>.  The use of <<lgamma>> instead of <<gamma>> is a reminder
  81 that these functions compute the log of the gamma function, rather
  82 than the gamma function itself.
  83
  84 The functions <<gamma_r>>, <<gammaf_r>>, <<lgamma_r>>, and
  85 <<lgammaf_r>> are just like <<gamma>>, <<gammaf>>, <<lgamma>>, and
  86 <<lgammaf>>, respectively, but take an additional argument.  This
  87 additional argument is a pointer to an integer.  This additional
  88 argument is used to return the sign of the result, and the global
  89 variable <<signgam>> is not used.  These functions may be used for
  90 reentrant calls (but they will still set the global variable <<errno>>
  91 if an error occurs).
  92
  93 RETURNS
  94 Normally, the computed result is returned.
  95
  96 When <[x]> is a nonpositive integer, <<gamma>> returns <<HUGE_VAL>>
  97 and <<errno>> is set to <<EDOM>>.  If the result overflows, <<gamma>>
  98 returns <<HUGE_VAL>> and <<errno>> is set to <<ERANGE>>.
  99
 100 PORTABILITY
 101 Neither <<gamma>> nor <<gammaf>> is ANSI C.  */
 102
 103 /* lgamma_r(x, signgamp)
 104  * Reentrant version of the logarithm of the Gamma function
 105  * with user provide pointer for the sign of Gamma(x).
 106  *
 107  * Method:
 108  *   1. Argument Reduction for 0 < x <= 8
 109  *      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
 110  *      reduce x to a number in [1.5,2.5] by
 111  *              lgamma(1+s) = log(s) + lgamma(s)
 112  *      for example,
 113  *              lgamma(7.3) = log(6.3) + lgamma(6.3)
 114  *                          = log(6.3*5.3) + lgamma(5.3)
 115  *                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
 116  *   2. Polynomial approximation of lgamma around its
 117  *      minimun ymin=1.461632144968362245 to maintain monotonicity.
 118  *      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
 119  *              Let z = x-ymin;
 120  *              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
 121  *      where
 122  *              poly(z) is a 14 degree polynomial.
 123  *   2. Rational approximation in the primary interval [2,3]
 124  *      We use the following approximation:
 125  *              s = x-2.0;
 126  *              lgamma(x) = 0.5*s + s*P(s)/Q(s)
 127  *      with accuracy
 128  *              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
 129  *      Our algorithms are based on the following observation
 130  *
 131  *                             zeta(2)-1    2    zeta(3)-1    3
 132  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
 133  *                                 2                 3
 134  *
 135  *      where Euler = 0.5771... is the Euler constant, which is very
 136  *      close to 0.5.
 137  *
 138  *   3. For x>=8, we have
 139  *      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
 140  *      (better formula:
 141  *         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
 142  *      Let z = 1/x, then we approximation
 143  *              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
 144  *      by
 145  *                                  3       5             11
 146  *              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
 147  *      where
 148  *              |w - f(z)| < 2**-58.74
 149  *
 150  *   4. For negative x, since (G is gamma function)
 151  *              -x*G(-x)*G(x) = pi/sin(pi*x),
 152  *      we have
 153  *              G(x) = pi/(sin(pi*x)*(-x)*G(-x))
 154  *      since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
 155  *      Hence, for x<0, signgam = sign(sin(pi*x)) and
 156  *              lgamma(x) = log(|Gamma(x)|)
 157  *                        = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
 158  *      Note: one should avoid compute pi*(-x) directly in the
 159  *            computation of sin(pi*(-x)).
 160  *
 161  *   5. Special Cases
 162  *              lgamma(2+s) ~ s*(1-Euler) for tiny s
 163  *              lgamma(1)=lgamma(2)=0
 164  *              lgamma(x) ~ -log(x) for tiny x
 165  *              lgamma(0) = lgamma(inf) = inf
 166  *              lgamma(-integer) = +-inf
 167  *
 168  */
 169
 170 #include "fdlibm.h"
 171
 172 #ifdef __STDC__
 173 static const double
 174 #else
 175 static double
 176 #endif
 177 two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
 178 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
 179 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
 180 pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
 181 a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
 182 a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
 183 a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
 184 a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
 185 a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
 186 a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
 187 a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
 188 a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
 189 a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
 190 a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
 191 a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
 192 a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
 193 tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
 194 tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
 195 /* tt = -(tail of tf) */
 196 tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
 197 t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
 198 t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
 199 t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
 200 t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
 201 t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
 202 t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
 203 t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
 204 t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
 205 t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
 206 t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
 207 t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
 208 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
 209 t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
 210 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
 211 t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
 212 u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 213 u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
 214 u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
 215 u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
 216 u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
 217 u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
 218 v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
 219 v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
 220 v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
 221 v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
 222 v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
 223 s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
 224 s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
 225 s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
 226 s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
 227 s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
 228 s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
 229 s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
 230 r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
 231 r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
 232 r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
 233 r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
 234 r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
 235 r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
 236 w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
 237 w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
 238 w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
 239 w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
 240 w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
 241 w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
 242 w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
 243
 244 #ifdef __STDC__
 245 static const double zero=  0.00000000000000000000e+00;
 246 #else
 247 static double zero=  0.00000000000000000000e+00;
 248 #endif
 249
 250 #ifdef __STDC__
 251         static double sin_pi(double x)
 252 #else
 253         static double sin_pi(x)
 254         double x;
 255 #endif
 256 {
 257         double y,z;
 258         __int32_t n,ix;
 259
 260         GET_HIGH_WORD(ix,x);
 261         ix &= 0x7fffffff;
 262
 263         if(ix<0x3fd00000) return sin(pi*x);
 264         y = -x;         /* x is assume negative */
 265
 266     /*
 267      * argument reduction, make sure inexact flag not raised if input
 268      * is an integer
 269      */
 270         z = floor(y);
 271         if(z!=y) {                              /* inexact anyway */
 272             y  *= 0.5;
 273             y   = 2.0*(y - floor(y));           /* y = |x| mod 2.0 */
 274             n   = (__int32_t) (y*4.0);
 275         } else {
 276             if(ix>=0x43400000) {
 277                 y = zero; n = 0;                 /* y must be even */
 278             } else {
 279                 if(ix<0x43300000) z = y+two52;  /* exact */
 280                 GET_LOW_WORD(n,z);
 281                 n &= 1;
 282                 y  = n;
 283                 n<<= 2;
 284             }
 285         }
 286         switch (n) {
 287             case 0:   y =  sin(pi*y); break;
 288             case 1:
 289             case 2:   y =  cos(pi*(0.5-y)); break;
 290             case 3:
 291             case 4:   y =  sin(pi*(one-y)); break;
 292             case 5:
 293             case 6:   y = -cos(pi*(y-1.5)); break;
 294             default:  y =  sin(pi*(y-2.0)); break;
 295             }
 296         return -y;
 297 }
 298
 299
 300 #ifdef __STDC__
 301         double lgamma_r(double x, int *signgamp)
 302 #else
 303         double lgamma_r(x,signgamp)
 304         double x; int *signgamp;
 305 #endif
 306 {
 307         double t,y,z,nadj,p,p1,p2,p3,q,r,w;
 308         __int32_t i,hx,lx,ix;
 309
 310         nadj = 0;
 311
 312         EXTRACT_WORDS(hx,lx,x);
 313
 314     /* purge off +-inf, NaN, +-0, and negative arguments */
 315         *signgamp = 1;
 316         ix = hx&0x7fffffff;
 317         if(ix>=0x7ff00000) return x*x;
 318         if((ix|lx)==0) return one/zero;
 319         if(ix<0x3b900000) {     /* |x|<2**-70, return -log(|x|) */
 320             if(hx<0) {
 321                 *signgamp = -1;
 322                 return -log(-x);
 323             } else return -log(x);
 324         }
 325         if(hx<0) {
 326             if(ix>=0x43300000)  /* |x|>=2**52, must be -integer */
 327                 return one/zero;
 328             t = sin_pi(x);
 329             if(t==zero) return one/zero; /* -integer */
 330             nadj = log(pi/fabs(t*x));
 331             if(t<zero) *signgamp = -1;
 332             x = -x;
 333         }
 334
 335     /* purge off 1 and 2 */
 336         if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
 337     /* for x < 2.0 */
 338         else if(ix<0x40000000) {
 339             if(ix<=0x3feccccc) {        /* lgamma(x) = lgamma(x+1)-log(x) */
 340                 r = -log(x);
 341                 if(ix>=0x3FE76944) {y = one-x; i= 0;}
 342                 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
 343                 else {y = x; i=2;}
 344             } else {
 345                 r = zero;
 346                 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
 347                 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
 348                 else {y=x-one;i=2;}
 349             }
 350             switch(i) {
 351               case 0:
 352                 z = y*y;
 353                 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
 354                 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
 355                 p  = y*p1+p2;
 356                 r  += (p-0.5*y); break;
 357               case 1:
 358                 z = y*y;
 359                 w = z*y;
 360                 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
 361                 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
 362                 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
 363                 p  = z*p1-(tt-w*(p2+y*p3));
 364                 r += (tf + p); break;
 365               case 2:
 366                 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
 367                 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
 368                 r += (-0.5*y + p1/p2);
 369             }
 370         }
 371         else if(ix<0x40200000) {                        /* x < 8.0 */
 372             i = (__int32_t)x;
 373             t = zero;
 374             y = x-(double)i;
 375             p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
 376             q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
 377             r = half*y+p/q;
 378             z = one;    /* lgamma(1+s) = log(s) + lgamma(s) */
 379             switch(i) {
 380             case 7: z *= (y+6.0);       /* FALLTHRU */
 381             case 6: z *= (y+5.0);       /* FALLTHRU */
 382             case 5: z *= (y+4.0);       /* FALLTHRU */
 383             case 4: z *= (y+3.0);       /* FALLTHRU */
 384             case 3: z *= (y+2.0);       /* FALLTHRU */
 385                     r += log(z); break;
 386             }
 387     /* 8.0 <= x < 2**58 */
 388         } else if (ix < 0x43900000) {
 389             t = log(x);
 390             z = one/x;
 391             y = z*z;
 392             w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
 393             r = (x-half)*(t-one)+w;
 394         } else
 395     /* 2**58 <= x <= inf */
 396             r =  x*(log(x)-one);
 397         if(hx<0) r = nadj - r;
 398         return r;
 399 }
 400
 401 double
 402 lgamma(double x)
 403 {
 404   return lgamma_r(x, &(_REENT_SIGNGAM(_REENT)));
 405 }