RTAI API: base/math/e_lgamma

00001 /* @(#)er_lgamma.c 5.1 93/09/24 */
00002 /*
00003  * ====================================================
00004  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00005  *
00006  * Developed at SunPro, a Sun Microsystems, Inc. business.
00007  * Permission to use, copy, modify, and distribute this
00008  * software is freely granted, provided that this notice 
00009  * is preserved.
00010  * ====================================================
00011  */
00012 
00013 #if defined(LIBM_SCCS) && !defined(lint)
00014 static char rcsid[] = "$NetBSD: e_lgamma_r.c,v 1.7 1995/05/10 20:45:42 jtc Exp $";
00015 #endif
00016 
00017 /* __ieee754_lgamma_r(x, signgamp)
00018  * Reentrant version of the logarithm of the Gamma function 
00019  * with user provide pointer for the sign of Gamma(x). 
00020  *
00021  * Method:
00022  *   1. Argument Reduction for 0 < x <= 8
00023  *  Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 
00024  *  reduce x to a number in [1.5,2.5] by
00025  *      lgamma(1+s) = log(s) + lgamma(s)
00026  *  for example,
00027  *      lgamma(7.3) = log(6.3) + lgamma(6.3)
00028  *              = log(6.3*5.3) + lgamma(5.3)
00029  *              = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
00030  *   2. Polynomial approximation of lgamma around its
00031  *  minimun ymin=1.461632144968362245 to maintain monotonicity.
00032  *  On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
00033  *      Let z = x-ymin;
00034  *      lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
00035  *  where
00036  *      poly(z) is a 14 degree polynomial.
00037  *   2. Rational approximation in the primary interval [2,3]
00038  *  We use the following approximation:
00039  *      s = x-2.0;
00040  *      lgamma(x) = 0.5*s + s*P(s)/Q(s)
00041  *  with accuracy
00042  *      |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
00043  *  Our algorithms are based on the following observation
00044  *
00045  *                             zeta(2)-1    2    zeta(3)-1    3
00046  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
00047  *                                 2                 3
00048  *
00049  *  where Euler = 0.5771... is the Euler constant, which is very
00050  *  close to 0.5.
00051  *
00052  *   3. For x>=8, we have
00053  *  lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
00054  *  (better formula:
00055  *     lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
00056  *  Let z = 1/x, then we approximation
00057  *      f(z) = lgamma(x) - (x-0.5)(log(x)-1)
00058  *  by
00059  *                  3       5             11
00060  *      w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
00061  *  where 
00062  *      |w - f(z)| < 2**-58.74
00063  *      
00064  *   4. For negative x, since (G is gamma function)
00065  *      -x*G(-x)*G(x) = pi/sin(pi*x),
00066  *  we have
00067  *      G(x) = pi/(sin(pi*x)*(-x)*G(-x))
00068  *  since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
00069  *  Hence, for x<0, signgam = sign(sin(pi*x)) and 
00070  *      lgamma(x) = log(|Gamma(x)|)
00071  *            = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
00072  *  Note: one should avoid compute pi*(-x) directly in the 
00073  *        computation of sin(pi*(-x)).
00074  *      
00075  *   5. Special Cases
00076  *      lgamma(2+s) ~ s*(1-Euler) for tiny s
00077  *      lgamma(1)=lgamma(2)=0
00078  *      lgamma(x) ~ -log(x) for tiny x
00079  *      lgamma(0) = lgamma(inf) = inf
00080  *      lgamma(-integer) = +-inf
00081  *  
00082  */
00083 
00084 #include "math.h"
00085 #include "mathP.h"
00086 
00087 #ifdef __STDC__
00088 static const double 
00089 #else
00090 static double 
00091 #endif
00092 two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
00093 half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
00094 one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
00095 pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
00096 a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
00097 a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
00098 a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
00099 a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
00100 a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
00101 a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
00102 a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
00103 a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
00104 a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
00105 a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
00106 a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
00107 a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
00108 tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
00109 tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
00110 /* tt = -(tail of tf) */
00111 tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
00112 t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
00113 t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
00114 t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
00115 t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
00116 t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
00117 t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
00118 t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
00119 t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
00120 t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
00121 t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
00122 t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
00123 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
00124 t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
00125 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
00126 t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
00127 u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
00128 u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
00129 u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
00130 u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
00131 u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
00132 u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
00133 v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
00134 v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
00135 v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
00136 v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
00137 v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
00138 s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
00139 s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
00140 s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
00141 s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
00142 s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
00143 s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
00144 s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
00145 r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
00146 r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
00147 r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
00148 r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
00149 r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
00150 r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
00151 w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
00152 w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
00153 w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
00154 w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
00155 w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
00156 w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
00157 w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
00158 
00159 #ifdef __STDC__
00160 static const double zero=  0.00000000000000000000e+00;
00161 #else
00162 static double zero=  0.00000000000000000000e+00;
00163 #endif
00164 
00165 static
00166 #ifdef __GNUC__
00167 __inline__
00168 #endif
00169 #ifdef __STDC__
00170     double sin_pi(double x)
00171 #else
00172     double sin_pi(x)
00173     double x;
00174 #endif
00175 {
00176     double y,z;
00177     int n,ix;
00178 
00179     GET_HIGH_WORD(ix,x);
00180     ix &= 0x7fffffff;
00181 
00182     if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
00183     y = -x;     /* x is assume negative */
00184 
00185     /*
00186      * argument reduction, make sure inexact flag not raised if input
00187      * is an integer
00188      */
00189     z = floor(y);
00190     if(z!=y) {              /* inexact anyway */
00191         y  *= 0.5;
00192         y   = 2.0*(y - floor(y));       /* y = |x| mod 2.0 */
00193         n   = (int) (y*4.0);
00194     } else {
00195             if(ix>=0x43400000) {
00196                 y = zero; n = 0;                 /* y must be even */
00197             } else {
00198                 if(ix<0x43300000) z = y+two52;  /* exact */
00199         GET_LOW_WORD(n,z);
00200         n &= 1;
00201                 y  = n;
00202                 n<<= 2;
00203             }
00204         }
00205     switch (n) {
00206         case 0:   y =  __kernel_sin(pi*y,zero,0); break;
00207         case 1:   
00208         case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
00209         case 3:  
00210         case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
00211         case 5:
00212         case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
00213         default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
00214         }
00215     return -y;
00216 }
00217 
00218 
00219 #ifdef __STDC__
00220     double __ieee754_lgamma_r(double x, int *signgamp)
00221 #else
00222     double __ieee754_lgamma_r(x,signgamp)
00223     double x; int *signgamp;
00224 #endif
00225 {
00226     double t,y,z,nadj=0.0,p,p1,p2,p3,q,r,w;
00227     int i,hx,lx,ix;
00228 
00229     EXTRACT_WORDS(hx,lx,x);
00230 
00231     /* purge off +-inf, NaN, +-0, and negative arguments */
00232     *signgamp = 1;
00233     ix = hx&0x7fffffff;
00234     if(ix>=0x7ff00000) return x*x;
00235     if((ix|lx)==0) return one/zero;
00236     if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
00237         if(hx<0) {
00238             *signgamp = -1;
00239             return -__ieee754_log(-x);
00240         } else return -__ieee754_log(x);
00241     }
00242     if(hx<0) {
00243         if(ix>=0x43300000)  /* |x|>=2**52, must be -integer */
00244         return one/zero;
00245         t = sin_pi(x);
00246         if(t==zero) return one/zero; /* -integer */
00247         nadj = __ieee754_log(pi/fabs(t*x));
00248         if(t<zero) *signgamp = -1;
00249         x = -x;
00250     }
00251 
00252     /* purge off 1 and 2 */
00253     if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
00254     /* for x < 2.0 */
00255     else if(ix<0x40000000) {
00256         if(ix<=0x3feccccc) {    /* lgamma(x) = lgamma(x+1)-log(x) */
00257         r = -__ieee754_log(x);
00258         if(ix>=0x3FE76944) {y = one-x; i= 0;}
00259         else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
00260         else {y = x; i=2;}
00261         } else {
00262         r = zero;
00263             if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
00264             else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
00265         else {y=x-one;i=2;}
00266         }
00267         switch(i) {
00268           case 0:
00269         z = y*y;
00270         p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
00271         p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
00272         p  = y*p1+p2;
00273         r  += (p-0.5*y); break;
00274           case 1:
00275         z = y*y;
00276         w = z*y;
00277         p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
00278         p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
00279         p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
00280         p  = z*p1-(tt-w*(p2+y*p3));
00281         r += (tf + p); break;
00282           case 2:   
00283         p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
00284         p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
00285         r += (-0.5*y + p1/p2);
00286         }
00287     }
00288     else if(ix<0x40200000) {            /* x < 8.0 */
00289         i = (int)x;
00290         t = zero;
00291         y = x-(double)i;
00292         p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
00293         q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
00294         r = half*y+p/q;
00295         z = one;    /* lgamma(1+s) = log(s) + lgamma(s) */
00296         switch(i) {
00297         case 7: z *= (y+6.0);   /* FALLTHRU */
00298         case 6: z *= (y+5.0);   /* FALLTHRU */
00299         case 5: z *= (y+4.0);   /* FALLTHRU */
00300         case 4: z *= (y+3.0);   /* FALLTHRU */
00301         case 3: z *= (y+2.0);   /* FALLTHRU */
00302             r += __ieee754_log(z); break;
00303         }
00304     /* 8.0 <= x < 2**58 */
00305     } else if (ix < 0x43900000) {
00306         t = __ieee754_log(x);
00307         z = one/x;
00308         y = z*z;
00309         w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
00310         r = (x-half)*(t-one)+w;
00311     } else 
00312     /* 2**58 <= x <= inf */
00313         r =  x*(__ieee754_log(x)-one);
00314     if(hx<0) r = nadj - r;
00315     return r;
00316 }
base/math/e_lgamma_r.c
