File size: 12,004 Bytes
7885a28 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 |
/* Translated into C++ by SciPy developers in 2024.
* Original header with Copyright information appears below.
*/
/*
* Gamma function
*
*
*
* SYNOPSIS:
*
* double x, y, Gamma();
*
* y = Gamma( x );
*
*
*
* DESCRIPTION:
*
* Returns Gamma function of the argument. The result is
* correctly signed.
*
* Arguments |x| <= 34 are reduced by recurrence and the function
* approximated by a rational function of degree 6/7 in the
* interval (2,3). Large arguments are handled by Stirling's
* formula. Large negative arguments are made positive using
* a reflection formula.
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE -170,-33 20000 2.3e-15 3.3e-16
* IEEE -33, 33 20000 9.4e-16 2.2e-16
* IEEE 33, 171.6 20000 2.3e-15 3.2e-16
*
* Error for arguments outside the test range will be larger
* owing to error amplification by the exponential function.
*
*/
/* lgam()
*
* Natural logarithm of Gamma function
*
*
*
* SYNOPSIS:
*
* double x, y, lgam();
*
* y = lgam( x );
*
*
*
* DESCRIPTION:
*
* Returns the base e (2.718...) logarithm of the absolute
* value of the Gamma function of the argument.
*
* For arguments greater than 13, the logarithm of the Gamma
* function is approximated by the logarithmic version of
* Stirling's formula using a polynomial approximation of
* degree 4. Arguments between -33 and +33 are reduced by
* recurrence to the interval [2,3] of a rational approximation.
* The cosecant reflection formula is employed for arguments
* less than -33.
*
* Arguments greater than MAXLGM return INFINITY and an error
* message. MAXLGM = 2.556348e305 for IEEE arithmetic.
*
*
*
* ACCURACY:
*
*
* arithmetic domain # trials peak rms
* IEEE 0, 3 28000 5.4e-16 1.1e-16
* IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17
* The error criterion was relative when the function magnitude
* was greater than one but absolute when it was less than one.
*
* The following test used the relative error criterion, though
* at certain points the relative error could be much higher than
* indicated.
* IEEE -200, -4 10000 4.8e-16 1.3e-16
*
*/
/*
* Cephes Math Library Release 2.2: July, 1992
* Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#pragma once
#include "../config.h"
#include "../error.h"
#include "const.h"
#include "polevl.h"
#include "trig.h"
namespace xsf {
namespace cephes {
namespace detail {
constexpr double gamma_P[] = {1.60119522476751861407E-4, 1.19135147006586384913E-3, 1.04213797561761569935E-2,
4.76367800457137231464E-2, 2.07448227648435975150E-1, 4.94214826801497100753E-1,
9.99999999999999996796E-1};
constexpr double gamma_Q[] = {-2.31581873324120129819E-5, 5.39605580493303397842E-4, -4.45641913851797240494E-3,
1.18139785222060435552E-2, 3.58236398605498653373E-2, -2.34591795718243348568E-1,
7.14304917030273074085E-2, 1.00000000000000000320E0};
/* Stirling's formula for the Gamma function */
constexpr double gamma_STIR[5] = {
7.87311395793093628397E-4, -2.29549961613378126380E-4, -2.68132617805781232825E-3,
3.47222221605458667310E-3, 8.33333333333482257126E-2,
};
constexpr double MAXSTIR = 143.01608;
/* Gamma function computed by Stirling's formula.
* The polynomial STIR is valid for 33 <= x <= 172.
*/
XSF_HOST_DEVICE inline double stirf(double x) {
double y, w, v;
if (x >= MAXGAM) {
return (std::numeric_limits<double>::infinity());
}
w = 1.0 / x;
w = 1.0 + w * xsf::cephes::polevl(w, gamma_STIR, 4);
y = std::exp(x);
if (x > MAXSTIR) { /* Avoid overflow in pow() */
v = std::pow(x, 0.5 * x - 0.25);
y = v * (v / y);
} else {
y = std::pow(x, x - 0.5) / y;
}
y = SQRTPI * y * w;
return (y);
}
} // namespace detail
XSF_HOST_DEVICE inline double Gamma(double x) {
double p, q, z;
int i;
int sgngam = 1;
if (!std::isfinite(x)) {
if (x > 0) {
// gamma(+inf) = +inf
return x;
}
// gamma(NaN) and gamma(-inf) both should equal NaN.
return std::numeric_limits<double>::quiet_NaN();
}
if (x == 0) {
/* For pole at zero, value depends on sign of zero.
* +inf when approaching from right, -inf when approaching
* from left. */
return std::copysign(std::numeric_limits<double>::infinity(), x);
}
q = std::abs(x);
if (q > 33.0) {
if (x < 0.0) {
p = std::floor(q);
if (p == q) {
// x is a negative integer. This is a pole.
set_error("Gamma", SF_ERROR_SINGULAR, NULL);
return (std::numeric_limits<double>::quiet_NaN());
}
i = p;
if ((i & 1) == 0) {
sgngam = -1;
}
z = q - p;
if (z > 0.5) {
p += 1.0;
z = q - p;
}
z = q * sinpi(z);
if (z == 0.0) {
return (sgngam * std::numeric_limits<double>::infinity());
}
z = std::abs(z);
z = M_PI / (z * detail::stirf(q));
} else {
z = detail::stirf(x);
}
return (sgngam * z);
}
z = 1.0;
while (x >= 3.0) {
x -= 1.0;
z *= x;
}
while (x < 0.0) {
if (x > -1.E-9) {
goto small;
}
z /= x;
x += 1.0;
}
while (x < 2.0) {
if (x < 1.e-9) {
goto small;
}
z /= x;
x += 1.0;
}
if (x == 2.0) {
return (z);
}
x -= 2.0;
p = polevl(x, detail::gamma_P, 6);
q = polevl(x, detail::gamma_Q, 7);
return (z * p / q);
small:
if (x == 0.0) {
/* For this to have happened, x must have started as a negative integer. */
set_error("Gamma", SF_ERROR_SINGULAR, NULL);
return (std::numeric_limits<double>::quiet_NaN());
} else
return (z / ((1.0 + 0.5772156649015329 * x) * x));
}
namespace detail {
/* A[]: Stirling's formula expansion of log Gamma
* B[], C[]: log Gamma function between 2 and 3
*/
constexpr double gamma_A[] = {8.11614167470508450300E-4, -5.95061904284301438324E-4, 7.93650340457716943945E-4,
-2.77777777730099687205E-3, 8.33333333333331927722E-2};
constexpr double gamma_B[] = {-1.37825152569120859100E3, -3.88016315134637840924E4, -3.31612992738871184744E5,
-1.16237097492762307383E6, -1.72173700820839662146E6, -8.53555664245765465627E5};
constexpr double gamma_C[] = {
/* 1.00000000000000000000E0, */
-3.51815701436523470549E2, -1.70642106651881159223E4, -2.20528590553854454839E5,
-1.13933444367982507207E6, -2.53252307177582951285E6, -2.01889141433532773231E6};
/* log( sqrt( 2*pi ) ) */
constexpr double LS2PI = 0.91893853320467274178;
constexpr double MAXLGM = 2.556348e305;
/* Disable optimizations for this function on 32 bit systems when compiling with GCC.
* We've found that enabling optimizations can result in degraded precision
* for this asymptotic approximation in that case. */
#if defined(__GNUC__) && defined(__i386__)
#pragma GCC push_options
#pragma GCC optimize("00")
#endif
XSF_HOST_DEVICE inline double lgam_large_x(double x) {
double q = (x - 0.5) * std::log(x) - x + LS2PI;
if (x > 1.0e8) {
return (q);
}
double p = 1.0 / (x * x);
p = ((7.9365079365079365079365e-4 * p - 2.7777777777777777777778e-3) * p + 0.0833333333333333333333) / x;
return q + p;
}
#if defined(__GNUC__) && defined(__i386__)
#pragma GCC pop_options
#endif
XSF_HOST_DEVICE inline double lgam_sgn(double x, int *sign) {
double p, q, u, w, z;
int i;
*sign = 1;
if (!std::isfinite(x)) {
return x;
}
if (x < -34.0) {
q = -x;
w = lgam_sgn(q, sign);
p = std::floor(q);
if (p == q) {
lgsing:
set_error("lgam", SF_ERROR_SINGULAR, NULL);
return (std::numeric_limits<double>::infinity());
}
i = p;
if ((i & 1) == 0) {
*sign = -1;
} else {
*sign = 1;
}
z = q - p;
if (z > 0.5) {
p += 1.0;
z = p - q;
}
z = q * sinpi(z);
if (z == 0.0) {
goto lgsing;
}
/* z = log(M_PI) - log( z ) - w; */
z = LOGPI - std::log(z) - w;
return (z);
}
if (x < 13.0) {
z = 1.0;
p = 0.0;
u = x;
while (u >= 3.0) {
p -= 1.0;
u = x + p;
z *= u;
}
while (u < 2.0) {
if (u == 0.0) {
goto lgsing;
}
z /= u;
p += 1.0;
u = x + p;
}
if (z < 0.0) {
*sign = -1;
z = -z;
} else {
*sign = 1;
}
if (u == 2.0) {
return (std::log(z));
}
p -= 2.0;
x = x + p;
p = x * polevl(x, gamma_B, 5) / p1evl(x, gamma_C, 6);
return (std::log(z) + p);
}
if (x > MAXLGM) {
return (*sign * std::numeric_limits<double>::infinity());
}
if (x >= 1000.0) {
return lgam_large_x(x);
}
q = (x - 0.5) * std::log(x) - x + LS2PI;
p = 1.0 / (x * x);
return q + polevl(p, gamma_A, 4) / x;
}
} // namespace detail
/* Logarithm of Gamma function */
XSF_HOST_DEVICE inline double lgam(double x) {
int sign;
return detail::lgam_sgn(x, &sign);
}
/* Sign of the Gamma function */
XSF_HOST_DEVICE inline double gammasgn(double x) {
double fx;
if (std::isnan(x)) {
return x;
}
if (x > 0) {
return 1.0;
}
if (x == 0) {
return std::copysign(1.0, x);
}
if (std::isinf(x)) {
// x > 0 case handled, so x must be negative infinity.
return std::numeric_limits<double>::quiet_NaN();
}
fx = std::floor(x);
if (x - fx == 0.0) {
return std::numeric_limits<double>::quiet_NaN();
}
// sign of gamma for x in (-n, -n+1) for positive integer n is (-1)^n.
if (static_cast<int>(fx) % 2) {
return -1.0;
}
return 1.0;
}
} // namespace cephes
} // namespace xsf
|