fusl/src/math/lgamma_r.c

/* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, 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(neg.integer) = inf and raise divide-by-zero
 *              lgamma(inf) = inf
 *              lgamma(-inf) = inf (bug for bug compatible with C99!?)
 *
 */

#include "libm.h"
#include "libc.h"

static const double 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 */

/* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
static double sin_pi(double x) {
  int n;

  /* spurious inexact if odd int */
  x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */

  n = (int)(x * 4.0);
  n = (n + 1) / 2;
  x -= n * 0.5f;
  x *= pi;

  switch (n) {
    default: /* case 4: */
    case 0:
      return __sin(x, 0.0, 0);
    case 1:
      return __cos(x, 0.0);
    case 2:
      return __sin(-x, 0.0, 0);
    case 3:
      return -__cos(x, 0.0);
  }
}

double __lgamma_r(double x, int* signgamp) {
  union {
    double f;
    uint64_t i;
  } u = {x};
  double_t t, y, z, nadj, p, p1, p2, p3, q, r, w;
  uint32_t ix;
  int sign, i;

  /* purge off +-inf, NaN, +-0, tiny and negative arguments */
  *signgamp = 1;
  sign = u.i >> 63;
  ix = u.i >> 32 & 0x7fffffff;
  if (ix >= 0x7ff00000)
    return x * x;
  if (ix < (0x3ff - 70) << 20) { /* |x|<2**-70, return -log(|x|) */
    if (sign) {
      x = -x;
      *signgamp = -1;
    }
    return -log(x);
  }
  if (sign) {
    x = -x;
    t = sin_pi(x);
    if (t == 0.0) /* -integer */
      return 1.0 / (x - x);
    if (t > 0.0)
      *signgamp = -1;
    else
      t = -t;
    nadj = log(pi / (t * x));
  }

  /* purge off 1 and 2 */
  if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t)u.i == 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 = 1.0 - x;
        i = 0;
      } else if (ix >= 0x3FCDA661) {
        y = x - (tc - 1.0);
        i = 1;
      } else {
        y = x;
        i = 2;
      }
    } else {
      r = 0.0;
      if (ix >= 0x3FFBB4C3) { /* [1.7316,2] */
        y = 2.0 - x;
        i = 0;
      } else if (ix >= 0x3FF3B4C4) { /* [1.23,1.73] */
        y = x - tc;
        i = 1;
      } else {
        y = x - 1.0;
        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 = 1.0 + 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;
    y = x - (double)i;
    p = y *
        (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
    q = 1.0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * r6)))));
    r = 0.5 * y + p / q;
    z = 1.0; /* 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;
    }
  } else if (ix < 0x43900000) { /* 8.0 <= x < 2**58 */
    t = log(x);
    z = 1.0 / x;
    y = z * z;
    w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * w6)))));
    r = (x - 0.5) * (t - 1.0) + w;
  } else /* 2**58 <= x <= inf */
    r = x * (log(x) - 1.0);
  if (sign)
    r = nadj - r;
  return r;
}

weak_alias(__lgamma_r, lgamma_r);