newlib/libm/ld80/b_tgammal.c

   1 /*-
   2  * SPDX-License-Identifier: BSD-3-Clause
   3  *
   4  * Copyright (c) 1992, 1993
   5  *      The Regents of the University of California.  All rights reserved.
   6  *
   7  * Redistribution and use in source and binary forms, with or without
   8  * modification, are permitted provided that the following conditions
   9  * are met:
  10  * 1. Redistributions of source code must retain the above copyright
  11  *    notice, this list of conditions and the following disclaimer.
  12  * 2. Redistributions in binary form must reproduce the above copyright
  13  *    notice, this list of conditions and the following disclaimer in the
  14  *    documentation and/or other materials provided with the distribution.
  15  * 3. Neither the name of the University nor the names of its contributors
  16  *    may be used to endorse or promote products derived from this software
  17  *    without specific prior written permission.
  18  *
  19  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  20  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  21  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  22  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  23  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  24  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  25  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  26  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  27  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  28  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  29  * SUCH DAMAGE.
  30  */
  31
  32 /*
  33  * The original code, FreeBSD's old svn r93211, contain the following
  34  * attribution:
  35  *
  36  *    This code by P. McIlroy, Oct 1992;
  37  *
  38  *    The financial support of UUNET Communications Services is greatfully
  39  *    acknowledged.
  40  *
  41  * bsdrc/b_tgamma.c converted to long double by Steven G. Kargl.
  42  */
  43
  44 /*
  45  * See bsdsrc/t_tgamma.c for implementation details.
  46  */
  47
  48 #include <float.h>
  49
  50 #if LDBL_MAX_EXP != 0x4000
  51 #error "Unsupported long double format"
  52 #endif
  53
  54 #ifdef __i386__
  55 #include <ieeefp.h>
  56 #endif
  57
  58 #include "../ld/fpmath.h"
  59 #include "math.h"
  60 #include "../ld/math_private.h"
  61
  62 long double sinpil(long double x);
  63 long double cospil(long double x);
  64
  65 /* Used in b_log.c and below. */
  66 struct Double {
  67         long double a;
  68         long double b;
  69 };
  70
  71 #include "b_logl.c"
  72 #include "b_expl.c"
  73
  74 static const double zero = 0.;
  75 static const volatile double tiny = 1e-300;
  76 /*
  77  * x >= 6
  78  *
  79  * Use the asymptotic approximation (Stirling's formula) adjusted for
  80  * equal-ripples:
  81  *
  82  * log(G(x)) ~= (x-0.5)*(log(x)-1) + 0.5(log(2*pi)-1) + 1/x*P(1/(x*x))
  83  *
  84  * Keep extra precision in multiplying (x-.5)(log(x)-1), to avoid
  85  * premature round-off.
  86  *
  87  * Accurate to max(ulp(1/128) absolute, 2^-66 relative) error.
  88  */
  89
  90 /*
  91  * The following is a decomposition of 0.5 * (log(2*pi) - 1) into the
  92  * first 12 bits in ln2pi_hi and the trailing 64 bits in ln2pi_lo.  The
  93  * variables are clearly misnamed.
  94  */
  95 static const union IEEEl2bits
  96 ln2pi_hiu = LD80C(0xd680000000000000,  -2,  4.18945312500000000000e-01L),
  97 ln2pi_lou = LD80C(0xe379b414b596d687, -18, -6.77929532725821967032e-06L);
  98 #define ln2pi_hi        (ln2pi_hiu.e)
  99 #define ln2pi_lo        (ln2pi_lou.e)
 100
 101 static const union IEEEl2bits
 102     Pa0u = LD80C(0xaaaaaaaaaaaaaaaa,  -4,  8.33333333333333333288e-02L),
 103     Pa1u = LD80C(0xb60b60b60b5fcd59,  -9, -2.77777777777776516326e-03L),
 104     Pa2u = LD80C(0xd00d00cffbb47014, -11,  7.93650793635429639018e-04L),
 105     Pa3u = LD80C(0x9c09c07c0805343e, -11, -5.95238087960599252215e-04L),
 106     Pa4u = LD80C(0xdca8d31f8e6e5e8f, -11,  8.41749082509607342883e-04L),
 107     Pa5u = LD80C(0xfb4d4289632f1638, -10, -1.91728055205541624556e-03L),
 108     Pa6u = LD80C(0xd15a4ba04078d3f8,  -8,  6.38893788027752396194e-03L),
 109     Pa7u = LD80C(0xe877283110bcad95,  -6, -2.83771309846297590312e-02L),
 110     Pa8u = LD80C(0x8da97eed13717af8,  -3,  1.38341887683837576925e-01L),
 111     Pa9u = LD80C(0xf093b1c1584e30ce,  -2, -4.69876818515470146031e-01L);
 112 #define Pa0     (Pa0u.e)
 113 #define Pa1     (Pa1u.e)
 114 #define Pa2     (Pa2u.e)
 115 #define Pa3     (Pa3u.e)
 116 #define Pa4     (Pa4u.e)
 117 #define Pa5     (Pa5u.e)
 118 #define Pa6     (Pa6u.e)
 119 #define Pa7     (Pa7u.e)
 120 #define Pa8     (Pa8u.e)
 121 #define Pa9     (Pa9u.e)
 122
 123 static struct Double
 124 large_gam(long double x)
 125 {
 126         long double p, z, thi, tlo, xhi, xlo;
 127         long double logx;
 128         struct Double u;
 129
 130         z = 1 / (x * x);
 131         p = Pa0 + z * (Pa1 + z * (Pa2 + z * (Pa3 + z * (Pa4 + z * (Pa5 +
 132             z * (Pa6 + z * (Pa7 + z * (Pa8 + z * Pa9))))))));
 133         p = p / x;
 134
 135         u = __log__D(x);
 136         u.a -= 1;
 137
 138         /* Split (x - 0.5) in high and low parts. */
 139         x -= 0.5L;
 140         xhi = (float)x;
 141         xlo = x - xhi;
 142
 143         /* Compute  t = (x-.5)*(log(x)-1) in extra precision. */
 144         thi = xhi * u.a;
 145         tlo = xlo * u.a + x * u.b;
 146
 147         /* Compute thi + tlo + ln2pi_hi + ln2pi_lo + p. */
 148         tlo += ln2pi_lo;
 149         tlo += p;
 150         u.a = ln2pi_hi + tlo;
 151         u.a += thi;
 152         u.b = thi - u.a;
 153         u.b += ln2pi_hi;
 154         u.b += tlo;
 155         return (u);
 156 }
 157 /*
 158  * Rational approximation, A0 + x * x * P(x) / Q(x), on the interval
 159  * [1.066.., 2.066..] accurate to 4.25e-19.
 160  *
 161  * Returns r.a + r.b = a0 + (z + c)^2 * p / q, with r.a truncated.
 162  */
 163 static const union IEEEl2bits
 164     a0_hiu = LD80C(0xe2b6e4153a57746c,  -1, 8.85603194410888700265e-01L),
 165     a0_lou = LD80C(0x851566d40f32c76d, -66, 1.40907742727049706207e-20L);
 166 #define a0_hi   (a0_hiu.e)
 167 #define a0_lo   (a0_lou.e)
 168
 169 static const union IEEEl2bits
 170 P0u = LD80C(0xdb629fb9bbdc1c1d,    -2,  4.28486815855585429733e-01L),
 171 P1u = LD80C(0xe6f4f9f5641aa6be,    -3,  2.25543885805587730552e-01L),
 172 P2u = LD80C(0xead1bd99fdaf7cc1,    -6,  2.86644652514293482381e-02L),
 173 P3u = LD80C(0x9ccc8b25838ab1e0,    -8,  4.78512567772456362048e-03L),
 174 P4u = LD80C(0x8f0c4383ef9ce72a,    -9,  2.18273781132301146458e-03L),
 175 P5u = LD80C(0xe732ab2c0a2778da,   -13,  2.20487522485636008928e-04L),
 176 P6u = LD80C(0xce70b27ca822b297,   -16,  2.46095923774929264284e-05L),
 177 P7u = LD80C(0xa309e2e16fb63663,   -19,  2.42946473022376182921e-06L),
 178 P8u = LD80C(0xaf9c110efb2c633d,   -23,  1.63549217667765869987e-07L),
 179 Q1u = LD80C(0xd4d7422719f48f15,    -1,  8.31409582658993993626e-01L),
 180 Q2u = LD80C(0xe13138ea404f1268,    -5, -5.49785826915643198508e-02L),
 181 Q3u = LD80C(0xd1c6cc91989352c0,    -4, -1.02429960435139887683e-01L),
 182 Q4u = LD80C(0xa7e9435a84445579,    -7,  1.02484853505908820524e-02L),
 183 Q5u = LD80C(0x83c7c34db89b7bda,    -8,  4.02161632832052872697e-03L),
 184 Q6u = LD80C(0xbed06bf6e1c14e5b,   -11, -7.27898206351223022157e-04L),
 185 Q7u = LD80C(0xef05bf841d4504c0,   -18,  7.12342421869453515194e-06L),
 186 Q8u = LD80C(0xf348d08a1ff53cb1,   -19,  3.62522053809474067060e-06L);
 187 #define P0      (P0u.e)
 188 #define P1      (P1u.e)
 189 #define P2      (P2u.e)
 190 #define P3      (P3u.e)
 191 #define P4      (P4u.e)
 192 #define P5      (P5u.e)
 193 #define P6      (P6u.e)
 194 #define P7      (P7u.e)
 195 #define P8      (P8u.e)
 196 #define Q1      (Q1u.e)
 197 #define Q2      (Q2u.e)
 198 #define Q3      (Q3u.e)
 199 #define Q4      (Q4u.e)
 200 #define Q5      (Q5u.e)
 201 #define Q6      (Q6u.e)
 202 #define Q7      (Q7u.e)
 203 #define Q8      (Q8u.e)
 204
 205 static struct Double
 206 ratfun_gam(long double z, long double c)
 207 {
 208         long double p, q, thi, tlo;
 209         struct Double r;
 210
 211         q = 1  + z * (Q1 + z * (Q2 + z * (Q3 + z * (Q4 + z * (Q5 +
 212             z * (Q6 + z * (Q7 + z * Q8)))))));
 213         p = P0 + z * (P1 + z * (P2 + z * (P3 + z * (P4 + z * (P5 +
 214             z * (P6 + z * (P7 + z * P8)))))));
 215         p = p / q;
 216
 217         /* Split z into high and low parts. */
 218         thi = (float)z;
 219         tlo = (z - thi) + c;
 220         tlo *= (thi + z);
 221
 222         /* Split (z+c)^2 into high and low parts. */
 223         thi *= thi;
 224         q = thi;
 225         thi = (float)thi;
 226         tlo += (q - thi);
 227
 228         /* Split p/q into high and low parts. */
 229         r.a = (float)p;
 230         r.b = p - r.a;
 231
 232         tlo = tlo * p + thi * r.b + a0_lo;
 233         thi *= r.a;                             /* t = (z+c)^2*(P/Q) */
 234         r.a = (float)(thi + a0_hi);
 235         r.b = ((a0_hi - r.a) + thi) + tlo;
 236         return (r);                             /* r = a0 + t */
 237 }
 238 /*
 239  * x < 6
 240  *
 241  * Use argument reduction G(x+1) = xG(x) to reach the range [1.066124,
 242  * 2.066124].  Use a rational approximation centered at the minimum
 243  * (x0+1) to ensure monotonicity.
 244  *
 245  * Good to < 1 ulp.  (provably .90 ulp; .87 ulp on 1,000,000 runs.)
 246  * It also has correct monotonicity.
 247  */
 248 static const union IEEEl2bits
 249   xm1u = LD80C(0xec5b0c6ad7c7edc3, -2, 4.61632144968362341254e-01L);
 250 #define x0      (xm1u.e)
 251
 252 static const double
 253     left = -0.3955078125;       /* left boundary for rat. approx */
 254
 255 static long double
 256 small_gam(long double x)
 257 {
 258         long double t, y, ym1;
 259         struct Double yy, r;
 260
 261         y = x - 1;
 262
 263         if (y <= 1 + (left + x0)) {
 264                 yy = ratfun_gam(y - x0, 0);
 265                 return (yy.a + yy.b);
 266         }
 267
 268         r.a = (float)y;
 269         yy.a = r.a - 1;
 270         y = y - 1 ;
 271         r.b = yy.b = y - yy.a;
 272
 273         /* Argument reduction: G(x+1) = x*G(x) */
 274         for (ym1 = y - 1; ym1 > left + x0; y = ym1--, yy.a--) {
 275                 t = r.a * yy.a;
 276                 r.b = r.a * yy.b + y * r.b;
 277                 r.a = (float)t;
 278                 r.b += (t - r.a);
 279         }
 280
 281         /* Return r*tgamma(y). */
 282         yy = ratfun_gam(y - x0, 0);
 283         y = r.b * (yy.a + yy.b) + r.a * yy.b;
 284         y += yy.a * r.a;
 285         return (y);
 286 }
 287 /*
 288  * Good on (0, 1+x0+left].  Accurate to 1 ulp.
 289  */
 290 static long double
 291 smaller_gam(long double x)
 292 {
 293         long double d, rhi, rlo, t, xhi, xlo;
 294         struct Double r;
 295
 296         if (x < x0 + left) {
 297                 t = (float)x;
 298                 d = (t + x) * (x - t);
 299                 t *= t;
 300                 xhi = (float)(t + x);
 301                 xlo = x - xhi;
 302                 xlo += t;
 303                 xlo += d;
 304                 t = 1 - x0;
 305                 t += x;
 306                 d = 1 - x0;
 307                 d -= t;
 308                 d += x;
 309                 x = xhi + xlo;
 310         } else {
 311                 xhi = (float)x;
 312                 xlo = x - xhi;
 313                 t = x - x0;
 314                 d = - x0 - t;
 315                 d += x;
 316         }
 317
 318         r = ratfun_gam(t, d);
 319         d = (float)(r.a / x);
 320         r.a -= d * xhi;
 321         r.a -= d * xlo;
 322         r.a += r.b;
 323
 324         return (d + r.a / x);
 325 }
 326 /*
 327  * x < 0
 328  *
 329  * Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x)).
 330  * At negative integers, return NaN and raise invalid.
 331  */
 332 static const union IEEEl2bits
 333 piu = LD80C(0xc90fdaa22168c235, 1, 3.14159265358979323851e+00L);
 334 #define pi      (piu.e)
 335
 336 static long double
 337 neg_gam(long double x)
 338 {
 339         int sgn = 1;
 340         struct Double lg, lsine;
 341         long double y, z;
 342
 343         y = ceill(x);
 344         if (y == x)             /* Negative integer. */
 345                 return ((x - x) / zero);
 346
 347         z = y - x;
 348         if (z > 0.5)
 349                 z = 1 - z;
 350
 351         y = y / 2;
 352         if (y == ceill(y))
 353                 sgn = -1;
 354
 355         if (z < 0.25)
 356                 z = sinpil(z);
 357         else
 358                 z = cospil(0.5 - z);
 359
 360         /* Special case: G(1-x) = Inf; G(x) may be nonzero. */
 361         if (x < -1753) {
 362
 363                 if (x < -1760)
 364                         return (sgn * tiny * tiny);
 365                 y = expl(lgammal(x) / 2);
 366                 y *= y;
 367                 return (sgn < 0 ? -y : y);
 368         }
 369
 370
 371         y = 1 - x;
 372         if (1 - y == x)
 373                 y = tgammal(y);
 374         else            /* 1-x is inexact */
 375                 y = - x * tgammal(-x);
 376
 377         if (sgn < 0) y = -y;
 378         return (pi / (y * z));
 379 }
 380 /*
 381  * xmax comes from lgamma(xmax) - emax * log(2) = 0.
 382  * static const float  xmax = 35.040095f
 383  * static const double xmax = 171.624376956302725;
 384  * ld80: LD80C(0xdb718c066b352e20, 10, 1.75554834290446291689e+03L),
 385  * ld128: 1.75554834290446291700388921607020320e+03L,
 386  *
 387  * iota is a sloppy threshold to isolate x = 0.
 388  */
 389 static const double xmax = 1755.54834290446291689;
 390 static const double iota = 0x1p-116;
 391
 392 long double
 393 tgammal(long double x)
 394 {
 395         struct Double u;
 396
 397         ENTERI();
 398
 399         if (x >= 6) {
 400                 if (x > xmax)
 401                         RETURNI(x / zero);
 402                 u = large_gam(x);
 403                 RETURNI(__exp__D(u.a, u.b));
 404         }
 405
 406         if (x >= 1 + left + x0)
 407                 RETURNI(small_gam(x));
 408
 409         if (x > iota)
 410                 RETURNI(smaller_gam(x));
 411
 412         if (x > -iota) {
 413                 if (x != 0)
 414                         u.a = 1 - tiny; /* raise inexact */
 415                 RETURNI(1 / x);
 416         }
 417
 418         if (!isfinite(x))
 419                 RETURNI(x - x);         /* x is NaN or -Inf */
 420
 421         RETURNI(neg_gam(x));
 422 }