e_lgamma_r.c

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/* @(#)er_lgamma.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice 
 * is preserved.
 * ====================================================
 */

/* lgamma_r(x, signgamp)
 * Reentrant version of the logarithm of the Gamma function 
 * with user provide pointer for the sign of Gamma(x). 
 *
 * Method:
 *   1. Argument Reduction for 0 < x <= 8
 *  Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 
 *  reduce x to a number in [1.5,2.5] by
 *      lgamma(1+s) = log(s) + lgamma(s)
 *  for example,
 *      lgamma(7.3) = log(6.3) + lgamma(6.3)
 *              = log(6.3*5.3) + lgamma(5.3)
 *              = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
 *   2. Polynomial approximation of lgamma around its
 *  minimun ymin=1.461632144968362245 to maintain monotonicity.
 *  On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
 *      Let z = x-ymin;
 *      lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
 *  where
 *      poly(z) is a 14 degree polynomial.
 *   2. Rational approximation in the primary interval [2,3]
 *  We use the following approximation:
 *      s = x-2.0;
 *      lgamma(x) = 0.5*s + s*P(s)/Q(s)
 *  with accuracy
 *      |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
 *  Our algorithms are based on the following observation
 *
 *                             zeta(2)-1    2    zeta(3)-1    3
 * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
 *                                 2                 3
 *
 *  where Euler = 0.5771... is the Euler constant, which is very
 *  close to 0.5.
 *
 *   3. For x>=8, we have
 *  lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
 *  (better formula:
 *     lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
 *  Let z = 1/x, then we approximation
 *      f(z) = lgamma(x) - (x-0.5)(log(x)-1)
 *  by
 *                  3       5             11
 *      w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
 *  where 
 *      |w - f(z)| < 2**-58.74
 *      
 *   4. For negative x, since (G is gamma function)
 *      -x*G(-x)*G(x) = pi/sin(pi*x),
 *  we have
 *      G(x) = pi/(sin(pi*x)*(-x)*G(-x))
 *  since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
 *  Hence, for x<0, signgam = sign(sin(pi*x)) and 
 *      lgamma(x) = log(|Gamma(x)|)
 *            = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
 *  Note: one should avoid compute pi*(-x) directly in the 
 *        computation of sin(pi*(-x)).
 *      
 *   5. Special Cases
 *      lgamma(2+s) ~ s*(1-Euler) for tiny s
 *      lgamma(1)=lgamma(2)=0
 *      lgamma(x) ~ -log(x) for tiny x
 *      lgamma(0) = lgamma(inf) = inf
 *      lgamma(-integer) = +-inf
 *  
 */

#include "math.h"
#include "math_private.h"

static const double 
two52=  4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */
half=  5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
one =  1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
pi  =  3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
a0  =  7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
a1  =  3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
a2  =  6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
a3  =  2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
a4  =  7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
a5  =  2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
a6  =  1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
a7  =  5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
a8  =  2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
a9  =  1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
a10 =  2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
a11 =  4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
tc  =  1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
tf  = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
/* tt = -(tail of tf) */
tt  = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
t0  =  4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
t1  = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
t2  =  6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
t3  = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
t4  =  1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
t5  = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
t6  =  6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
t7  = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
t8  =  2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
t9  = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
t10 =  8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
t12 =  3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
t14 =  3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
u0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
u1  =  6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
u2  =  1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
u3  =  9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
u4  =  2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
u5  =  1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
v1  =  2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
v2  =  2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
v3  =  7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
v4  =  1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
v5  =  3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
s0  = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
s1  =  2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
s2  =  3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
s3  =  1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
s4  =  2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
s5  =  1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
s6  =  3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
r1  =  1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
r2  =  7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
r3  =  1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
r4  =  1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
r5  =  7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
r6  =  7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
w0  =  4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
w1  =  8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
w2  = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
w3  =  7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
w4  = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
w5  =  8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
w6  = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */

static const double zero=  0.00000000000000000000e+00;

static double
sin_pi(double x)
{
    double y,z;
    int n,ix;

    GET_HIGH_WORD(ix,x);
    ix &= 0x7fffffff;

    if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0);
    y = -x;     /* x is assume negative */

    /*
     * argument reduction, make sure inexact flag not raised if input
     * is an integer
     */
    z = floor(y);
    if(z!=y) {              /* inexact anyway */
        y  *= 0.5;
        y   = 2.0*(y - floor(y));       /* y = |x| mod 2.0 */
        n   = (int) (y*4.0);
    } else {
            if(ix>=0x43400000) {
                y = zero; n = 0;                 /* y must be even */
            } else {
                if(ix<0x43300000) z = y+two52;  /* exact */
        GET_LOW_WORD(n,z);
        n &= 1;
                y  = n;
                n<<= 2;
            }
        }
    switch (n) {
        case 0:   y =  __kernel_sin(pi*y,zero,0); break;
        case 1:   
        case 2:   y =  __kernel_cos(pi*(0.5-y),zero); break;
        case 3:  
        case 4:   y =  __kernel_sin(pi*(one-y),zero,0); break;
        case 5:
        case 6:   y = -__kernel_cos(pi*(y-1.5),zero); break;
        default:  y =  __kernel_sin(pi*(y-2.0),zero,0); break;
        }
    return -y;
}


double
lgamma_r(double x, int *signgamp)
{
    double t,y,z,nadj,p,p1,p2,p3,q,r,w;
    int i,hx,lx,ix;

    EXTRACT_WORDS(hx,lx,x);

    /* purge off +-inf, NaN, +-0, and negative arguments */
    *signgamp = 1;
    ix = hx&0x7fffffff;
    if(ix>=0x7ff00000) return x*x;
    if((ix|lx)==0) return one/zero;
    if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
        if(hx<0) {
            *signgamp = -1;
            return - log(-x);
        } else return - log(x);
    }
    if(hx<0) {
        if(ix>=0x43300000)  /* |x|>=2**52, must be -integer */
        return one/zero;
        t = sin_pi(x);
        if(t==zero) return one/zero; /* -integer */
        nadj = log(pi/fabs(t*x));
        if(t<zero) *signgamp = -1;
        x = -x;
    }

    /* purge off 1 and 2 */
    if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
    /* for x < 2.0 */
    else if(ix<0x40000000) {
        if(ix<=0x3feccccc) {    /* lgamma(x) = lgamma(x+1)-log(x) */
        r = - log(x);
        if(ix>=0x3FE76944) {y = one-x; i= 0;}
        else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
        else {y = x; i=2;}
        } else {
        r = zero;
            if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
            else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
        else {y=x-one;i=2;}
        }
        switch(i) {
          case 0:
        z = y*y;
        p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
        p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
        p  = y*p1+p2;
        r  += (p-0.5*y); break;
          case 1:
        z = y*y;
        w = z*y;
        p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12)));    /* parallel comp */
        p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
        p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
        p  = z*p1-(tt-w*(p2+y*p3));
        r += (tf + p); break;
          case 2:   
        p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
        p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
        r += (-0.5*y + p1/p2);
        }
    }
    else if(ix<0x40200000) {            /* x < 8.0 */
        i = (int)x;
        t = zero;
        y = x-(double)i;
        p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
        q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
        r = half*y+p/q;
        z = one;    /* lgamma(1+s) = log(s) + lgamma(s) */
        switch(i) {
        case 7: z *= (y+6.0);   /* FALLTHRU */
        case 6: z *= (y+5.0);   /* FALLTHRU */
        case 5: z *= (y+4.0);   /* FALLTHRU */
        case 4: z *= (y+3.0);   /* FALLTHRU */
        case 3: z *= (y+2.0);   /* FALLTHRU */
            r += log(z); break;
        }
    /* 8.0 <= x < 2**58 */
    } else if (ix < 0x43900000) {
        t = log(x);
        z = one/x;
        y = z*z;
        w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
        r = (x-half)*(t-one)+w;
    } else 
    /* 2**58 <= x <= inf */
        r =  x*(log(x)-one);
    if(hx<0) r = nadj - r;
    return r;
}