vendor/compiler_builtins/libm/src/math/lgamma_r.rs

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  *
  12  */
  13 /* lgamma_r(x, signgamp)
  14  * Reentrant version of the logarithm of the Gamma function
  15  * with user provide pointer for the sign of Gamma(x).
  16  *
  17  * Method:
  18  *   1. Argument Reduction for 0 < x <= 8
  19  *      Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
  20  *      reduce x to a number in [1.5,2.5] by
  21  *              lgamma(1+s) = log(s) + lgamma(s)
  22  *      for example,
  23  *              lgamma(7.3) = log(6.3) + lgamma(6.3)
  24  *                          = log(6.3*5.3) + lgamma(5.3)
  25  *                          = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
  26  *   2. Polynomial approximation of lgamma around its
  27  *      minimun ymin=1.461632144968362245 to maintain monotonicity.
  28  *      On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
  29  *              Let z = x-ymin;
  30  *              lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
  31  *      where
  32  *              poly(z) is a 14 degree polynomial.
  33  *   2. Rational approximation in the primary interval [2,3]
  34  *      We use the following approximation:
  35  *              s = x-2.0;
  36  *              lgamma(x) = 0.5*s + s*P(s)/Q(s)
  37  *      with accuracy
  38  *              |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
  39  *      Our algorithms are based on the following observation
  40  *
  41  *                             zeta(2)-1    2    zeta(3)-1    3
  42  * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
  43  *                                 2                 3
  44  *
  45  *      where Euler = 0.5771... is the Euler constant, which is very
  46  *      close to 0.5.
  47  *
  48  *   3. For x>=8, we have
  49  *      lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
  50  *      (better formula:
  51  *         lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
  52  *      Let z = 1/x, then we approximation
  53  *              f(z) = lgamma(x) - (x-0.5)(log(x)-1)
  54  *      by
  55  *                                  3       5             11
  56  *              w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
  57  *      where
  58  *              |w - f(z)| < 2**-58.74
  59  *
  60  *   4. For negative x, since (G is gamma function)
  61  *              -x*G(-x)*G(x) = PI/sin(PI*x),
  62  *      we have
  63  *              G(x) = PI/(sin(PI*x)*(-x)*G(-x))
  64  *      since G(-x) is positive, sign(G(x)) = sign(sin(PI*x)) for x<0
  65  *      Hence, for x<0, signgam = sign(sin(PI*x)) and
  66  *              lgamma(x) = log(|Gamma(x)|)
  67  *                        = log(PI/(|x*sin(PI*x)|)) - lgamma(-x);
  68  *      Note: one should avoid compute PI*(-x) directly in the
  69  *            computation of sin(PI*(-x)).
  70  *
  71  *   5. Special Cases
  72  *              lgamma(2+s) ~ s*(1-Euler) for tiny s
  73  *              lgamma(1) = lgamma(2) = 0
  74  *              lgamma(x) ~ -log(|x|) for tiny x
  75  *              lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
  76  *              lgamma(inf) = inf
  77  *              lgamma(-inf) = inf (bug for bug compatible with C99!?)
  78  *
  79  */
  80
  81 use super::{floor, k_cos, k_sin, log};
  82
  83 const PI: f64 = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */
  84 const A0: f64 = 7.72156649015328655494e-02; /* 0x3FB3C467, 0xE37DB0C8 */
  85 const A1: f64 = 3.22467033424113591611e-01; /* 0x3FD4A34C, 0xC4A60FAD */
  86 const A2: f64 = 6.73523010531292681824e-02; /* 0x3FB13E00, 0x1A5562A7 */
  87 const A3: f64 = 2.05808084325167332806e-02; /* 0x3F951322, 0xAC92547B */
  88 const A4: f64 = 7.38555086081402883957e-03; /* 0x3F7E404F, 0xB68FEFE8 */
  89 const A5: f64 = 2.89051383673415629091e-03; /* 0x3F67ADD8, 0xCCB7926B */
  90 const A6: f64 = 1.19270763183362067845e-03; /* 0x3F538A94, 0x116F3F5D */
  91 const A7: f64 = 5.10069792153511336608e-04; /* 0x3F40B6C6, 0x89B99C00 */
  92 const A8: f64 = 2.20862790713908385557e-04; /* 0x3F2CF2EC, 0xED10E54D */
  93 const A9: f64 = 1.08011567247583939954e-04; /* 0x3F1C5088, 0x987DFB07 */
  94 const A10: f64 = 2.52144565451257326939e-05; /* 0x3EFA7074, 0x428CFA52 */
  95 const A11: f64 = 4.48640949618915160150e-05; /* 0x3F07858E, 0x90A45837 */
  96 const TC: f64 = 1.46163214496836224576e+00; /* 0x3FF762D8, 0x6356BE3F */
  97 const TF: f64 = -1.21486290535849611461e-01; /* 0xBFBF19B9, 0xBCC38A42 */
  98 /* tt = -(tail of TF) */
  99 const TT: f64 = -3.63867699703950536541e-18; /* 0xBC50C7CA, 0xA48A971F */
 100 const T0: f64 = 4.83836122723810047042e-01; /* 0x3FDEF72B, 0xC8EE38A2 */
 101 const T1: f64 = -1.47587722994593911752e-01; /* 0xBFC2E427, 0x8DC6C509 */
 102 const T2: f64 = 6.46249402391333854778e-02; /* 0x3FB08B42, 0x94D5419B */
 103 const T3: f64 = -3.27885410759859649565e-02; /* 0xBFA0C9A8, 0xDF35B713 */
 104 const T4: f64 = 1.79706750811820387126e-02; /* 0x3F9266E7, 0x970AF9EC */
 105 const T5: f64 = -1.03142241298341437450e-02; /* 0xBF851F9F, 0xBA91EC6A */
 106 const T6: f64 = 6.10053870246291332635e-03; /* 0x3F78FCE0, 0xE370E344 */
 107 const T7: f64 = -3.68452016781138256760e-03; /* 0xBF6E2EFF, 0xB3E914D7 */
 108 const T8: f64 = 2.25964780900612472250e-03; /* 0x3F6282D3, 0x2E15C915 */
 109 const T9: f64 = -1.40346469989232843813e-03; /* 0xBF56FE8E, 0xBF2D1AF1 */
 110 const T10: f64 = 8.81081882437654011382e-04; /* 0x3F4CDF0C, 0xEF61A8E9 */
 111 const T11: f64 = -5.38595305356740546715e-04; /* 0xBF41A610, 0x9C73E0EC */
 112 const T12: f64 = 3.15632070903625950361e-04; /* 0x3F34AF6D, 0x6C0EBBF7 */
 113 const T13: f64 = -3.12754168375120860518e-04; /* 0xBF347F24, 0xECC38C38 */
 114 const T14: f64 = 3.35529192635519073543e-04; /* 0x3F35FD3E, 0xE8C2D3F4 */
 115 const U0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
 116 const U1: f64 = 6.32827064025093366517e-01; /* 0x3FE4401E, 0x8B005DFF */
 117 const U2: f64 = 1.45492250137234768737e+00; /* 0x3FF7475C, 0xD119BD6F */
 118 const U3: f64 = 9.77717527963372745603e-01; /* 0x3FEF4976, 0x44EA8450 */
 119 const U4: f64 = 2.28963728064692451092e-01; /* 0x3FCD4EAE, 0xF6010924 */
 120 const U5: f64 = 1.33810918536787660377e-02; /* 0x3F8B678B, 0xBF2BAB09 */
 121 const V1: f64 = 2.45597793713041134822e+00; /* 0x4003A5D7, 0xC2BD619C */
 122 const V2: f64 = 2.12848976379893395361e+00; /* 0x40010725, 0xA42B18F5 */
 123 const V3: f64 = 7.69285150456672783825e-01; /* 0x3FE89DFB, 0xE45050AF */
 124 const V4: f64 = 1.04222645593369134254e-01; /* 0x3FBAAE55, 0xD6537C88 */
 125 const V5: f64 = 3.21709242282423911810e-03; /* 0x3F6A5ABB, 0x57D0CF61 */
 126 const S0: f64 = -7.72156649015328655494e-02; /* 0xBFB3C467, 0xE37DB0C8 */
 127 const S1: f64 = 2.14982415960608852501e-01; /* 0x3FCB848B, 0x36E20878 */
 128 const S2: f64 = 3.25778796408930981787e-01; /* 0x3FD4D98F, 0x4F139F59 */
 129 const S3: f64 = 1.46350472652464452805e-01; /* 0x3FC2BB9C, 0xBEE5F2F7 */
 130 const S4: f64 = 2.66422703033638609560e-02; /* 0x3F9B481C, 0x7E939961 */
 131 const S5: f64 = 1.84028451407337715652e-03; /* 0x3F5E26B6, 0x7368F239 */
 132 const S6: f64 = 3.19475326584100867617e-05; /* 0x3F00BFEC, 0xDD17E945 */
 133 const R1: f64 = 1.39200533467621045958e+00; /* 0x3FF645A7, 0x62C4AB74 */
 134 const R2: f64 = 7.21935547567138069525e-01; /* 0x3FE71A18, 0x93D3DCDC */
 135 const R3: f64 = 1.71933865632803078993e-01; /* 0x3FC601ED, 0xCCFBDF27 */
 136 const R4: f64 = 1.86459191715652901344e-02; /* 0x3F9317EA, 0x742ED475 */
 137 const R5: f64 = 7.77942496381893596434e-04; /* 0x3F497DDA, 0xCA41A95B */
 138 const R6: f64 = 7.32668430744625636189e-06; /* 0x3EDEBAF7, 0xA5B38140 */
 139 const W0: f64 = 4.18938533204672725052e-01; /* 0x3FDACFE3, 0x90C97D69 */
 140 const W1: f64 = 8.33333333333329678849e-02; /* 0x3FB55555, 0x5555553B */
 141 const W2: f64 = -2.77777777728775536470e-03; /* 0xBF66C16C, 0x16B02E5C */
 142 const W3: f64 = 7.93650558643019558500e-04; /* 0x3F4A019F, 0x98CF38B6 */
 143 const W4: f64 = -5.95187557450339963135e-04; /* 0xBF4380CB, 0x8C0FE741 */
 144 const W5: f64 = 8.36339918996282139126e-04; /* 0x3F4B67BA, 0x4CDAD5D1 */
 145 const W6: f64 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
 146
 147 /* sin(PI*x) assuming x > 2^-100, if sin(PI*x)==0 the sign is arbitrary */
 148 fn sin_pi(mut x: f64) -> f64 {
 149     let mut n: i32;
 150
 151     /* spurious inexact if odd int */
 152     x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */
 153
 154     n = (x * 4.0) as i32;
 155     n = (n + 1) / 2;
 156     x -= (n as f64) * 0.5;
 157     x *= PI;
 158
 159     match n {
 160         1 => k_cos(x, 0.0),
 161         2 => k_sin(-x, 0.0, 0),
 162         3 => -k_cos(x, 0.0),
 163         0 | _ => k_sin(x, 0.0, 0),
 164     }
 165 }
 166
 167 pub fn lgamma_r(mut x: f64) -> (f64, i32) {
 168     let u: u64 = x.to_bits();
 169     let mut t: f64;
 170     let y: f64;
 171     let mut z: f64;
 172     let nadj: f64;
 173     let p: f64;
 174     let p1: f64;
 175     let p2: f64;
 176     let p3: f64;
 177     let q: f64;
 178     let mut r: f64;
 179     let w: f64;
 180     let ix: u32;
 181     let sign: bool;
 182     let i: i32;
 183     let mut signgam: i32;
 184
 185     /* purge off +-inf, NaN, +-0, tiny and negative arguments */
 186     signgam = 1;
 187     sign = (u >> 63) != 0;
 188     ix = ((u >> 32) as u32) & 0x7fffffff;
 189     if ix >= 0x7ff00000 {
 190         return (x * x, signgam);
 191     }
 192     if ix < (0x3ff - 70) << 20 {
 193         /* |x|<2**-70, return -log(|x|) */
 194         if sign {
 195             x = -x;
 196             signgam = -1;
 197         }
 198         return (-log(x), signgam);
 199     }
 200     if sign {
 201         x = -x;
 202         t = sin_pi(x);
 203         if t == 0.0 {
 204             /* -integer */
 205             return (1.0 / (x - x), signgam);
 206         }
 207         if t > 0.0 {
 208             signgam = -1;
 209         } else {
 210             t = -t;
 211         }
 212         nadj = log(PI / (t * x));
 213     } else {
 214         nadj = 0.0;
 215     }
 216
 217     /* purge off 1 and 2 */
 218     if (ix == 0x3ff00000 || ix == 0x40000000) && (u & 0xffffffff) == 0 {
 219         r = 0.0;
 220     }
 221     /* for x < 2.0 */
 222     else if ix < 0x40000000 {
 223         if ix <= 0x3feccccc {
 224             /* lgamma(x) = lgamma(x+1)-log(x) */
 225             r = -log(x);
 226             if ix >= 0x3FE76944 {
 227                 y = 1.0 - x;
 228                 i = 0;
 229             } else if ix >= 0x3FCDA661 {
 230                 y = x - (TC - 1.0);
 231                 i = 1;
 232             } else {
 233                 y = x;
 234                 i = 2;
 235             }
 236         } else {
 237             r = 0.0;
 238             if ix >= 0x3FFBB4C3 {
 239                 /* [1.7316,2] */
 240                 y = 2.0 - x;
 241                 i = 0;
 242             } else if ix >= 0x3FF3B4C4 {
 243                 /* [1.23,1.73] */
 244                 y = x - TC;
 245                 i = 1;
 246             } else {
 247                 y = x - 1.0;
 248                 i = 2;
 249             }
 250         }
 251         match i {
 252             0 => {
 253                 z = y * y;
 254                 p1 = A0 + z * (A2 + z * (A4 + z * (A6 + z * (A8 + z * A10))));
 255                 p2 = z * (A1 + z * (A3 + z * (A5 + z * (A7 + z * (A9 + z * A11)))));
 256                 p = y * p1 + p2;
 257                 r += p - 0.5 * y;
 258             }
 259             1 => {
 260                 z = y * y;
 261                 w = z * y;
 262                 p1 = T0 + w * (T3 + w * (T6 + w * (T9 + w * T12))); /* parallel comp */
 263                 p2 = T1 + w * (T4 + w * (T7 + w * (T10 + w * T13)));
 264                 p3 = T2 + w * (T5 + w * (T8 + w * (T11 + w * T14)));
 265                 p = z * p1 - (TT - w * (p2 + y * p3));
 266                 r += TF + p;
 267             }
 268             2 => {
 269                 p1 = y * (U0 + y * (U1 + y * (U2 + y * (U3 + y * (U4 + y * U5)))));
 270                 p2 = 1.0 + y * (V1 + y * (V2 + y * (V3 + y * (V4 + y * V5))));
 271                 r += -0.5 * y + p1 / p2;
 272             }
 273             #[cfg(debug_assertions)]
 274             _ => unreachable!(),
 275             #[cfg(not(debug_assertions))]
 276             _ => {}
 277         }
 278     } else if ix < 0x40200000 {
 279         /* x < 8.0 */
 280         i = x as i32;
 281         y = x - (i as f64);
 282         p = y * (S0 + y * (S1 + y * (S2 + y * (S3 + y * (S4 + y * (S5 + y * S6))))));
 283         q = 1.0 + y * (R1 + y * (R2 + y * (R3 + y * (R4 + y * (R5 + y * R6)))));
 284         r = 0.5 * y + p / q;
 285         z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */
 286         // TODO: In C, this was implemented using switch jumps with fallthrough.
 287         // Does this implementation have performance problems?
 288         if i >= 7 {
 289             z *= y + 6.0;
 290         }
 291         if i >= 6 {
 292             z *= y + 5.0;
 293         }
 294         if i >= 5 {
 295             z *= y + 4.0;
 296         }
 297         if i >= 4 {
 298             z *= y + 3.0;
 299         }
 300         if i >= 3 {
 301             z *= y + 2.0;
 302             r += log(z);
 303         }
 304     } else if ix < 0x43900000 {
 305         /* 8.0 <= x < 2**58 */
 306         t = log(x);
 307         z = 1.0 / x;
 308         y = z * z;
 309         w = W0 + z * (W1 + y * (W2 + y * (W3 + y * (W4 + y * (W5 + y * W6)))));
 310         r = (x - 0.5) * (t - 1.0) + w;
 311     } else {
 312         /* 2**58 <= x <= inf */
 313         r = x * (log(x) - 1.0);
 314     }
 315     if sign {
 316         r = nadj - r;
 317     }
 318     return (r, signgam);
 319 }