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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504
|
/* mpfr_gamma_inc -- incomplete gamma function
Copyright 2016-2019 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU MPFR Library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation; either version 3 of the License, or (at your
option) any later version.
The GNU MPFR Library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* The incomplete gamma function is defined for x >= 0 and a not a negative
integer by:
gamma_inc(a,x) := Gamma(a,x) = int(t^(a-1) * exp(-t), t=x..infinity)
= Gamma(a) - gamma(a,x) with:
gamma(a,x) = int(t^(a-1) * exp(-t), t=0..x).
The function gamma(a,x) satisfies the Taylor expansions (we use the second
one in the code below):
gamma(a,x) = x^a * sum((-x)^k/k!/(a+k), k=0..infinity)
gamma(a,x) = x^a * exp(-x) * sum(x^k/(a*(a+1)*...*(a+k)), k=0..infinity)
*/
static int
mpfr_gamma_inc_negint (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t r);
int
mpfr_gamma_inc (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x, mpfr_rnd_t rnd)
{
mpfr_prec_t w;
mpfr_t s, t, u;
int inex;
unsigned long k;
mpfr_exp_t e0, e1, e2, err;
MPFR_GROUP_DECL(group);
MPFR_ZIV_DECL(loop);
MPFR_SAVE_EXPO_DECL (expo);
if (MPFR_ARE_SINGULAR (a, x))
{
/* if a or x is NaN, return NaN */
if (MPFR_IS_NAN (a) || MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* Note: for x < 0, gamma_inc (a, x) is a complex number */
if (MPFR_IS_INF (a) || MPFR_IS_INF (x))
{
if (MPFR_IS_INF (a) && MPFR_IS_INF (x))
{
if ((MPFR_IS_POS (a) && MPFR_IS_POS (x)) || MPFR_IS_NEG (x))
{
/* (a) gamma_inc(+Inf,+Inf) = NaN because
gamma_inc(x,x) tends to +Inf but
gamma_inc(x,x^2) tends to +0.
(b) gamma_inc(+/-Inf,-Inf) = NaN, for example
gamma_inc (a, -a) is a complex number
for a not an integer */
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else
{
/* gamma_inc(-Inf,+Inf) = +0 */
MPFR_SET_ZERO (y);
MPFR_SET_POS (y);
MPFR_RET (0); /* exact */
}
}
else /* only one of a, x is infinite */
{
if (MPFR_IS_INF (a))
{
MPFR_ASSERTD (MPFR_IS_INF (a) && MPFR_IS_FP (x));
if (MPFR_IS_POS (a))
{
/* gamma_inc(+Inf, x) = +Inf */
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0); /* exact */
}
else /* a = -Inf */
{
/* gamma_inc(-Inf, x) = NaN for x < 0
+Inf for 0 <= x < 1
+0 for 1 <= x */
if (mpfr_cmp_ui (x, 0) < 0)
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (mpfr_cmp_ui (x, 1) < 0)
{
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0); /* exact */
}
else
{
MPFR_SET_ZERO (y);
MPFR_SET_POS (y);
MPFR_RET (0); /* exact */
}
}
}
else
{
MPFR_ASSERTD (MPFR_IS_FP (a) && MPFR_IS_INF (x));
if (MPFR_IS_POS (x))
{
/* x is +Inf: integral tends to zero */
MPFR_SET_ZERO (y);
MPFR_SET_POS (y);
MPFR_RET (0); /* exact */
}
else /* NaN for x < 0 */
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
}
}
}
if (MPFR_IS_ZERO (a) || MPFR_IS_ZERO (x))
{
if (MPFR_IS_ZERO (a))
{
if (mpfr_cmp_ui (x, 0) < 0)
{
/* gamma_inc(a,x) = NaN for x < 0 */
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (MPFR_IS_ZERO (x))
/* gamma_inc(a,0) = gamma(a) */
return mpfr_gamma (y, a, rnd); /* a=+0->+Inf, a=-0->-Inf */
else
{
/* gamma_inc (0, x) = int (exp(-t), t=x..infinity) = E1(x) */
mpfr_t minus_x;
MPFR_TMP_INIT_NEG(minus_x, x);
/* mpfr_eint(x) for x < 0 returns -E1(-x) */
inex = mpfr_eint (y, minus_x, MPFR_INVERT_RND(rnd));
MPFR_CHANGE_SIGN(y);
return -inex;
}
}
else /* x = 0: gamma_inc(a,0) = gamma(a) */
return mpfr_gamma (y, a, rnd);
}
}
/* for x < 0 return NaN */
if (MPFR_SIGN(x) < 0)
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
if (mpfr_integer_p (a) && MPFR_SIGN(a) < 0)
return mpfr_gamma_inc_negint (y, a, x, rnd);
MPFR_SAVE_EXPO_MARK (expo);
w = MPFR_PREC(y) + 13; /* working precision */
MPFR_GROUP_INIT_2(group, w, s, t);
mpfr_init2 (u, 2); /* u is special (see below) */
MPFR_ZIV_INIT (loop, w);
for (;;)
{
mpfr_exp_t expu, precu, exps;
mpfr_t s_abs;
mpfr_exp_t decay = 0;
MPFR_BLOCK_DECL (flags);
/* Note: in the error analysis below, theta represents any value of
absolute value less than 2^(-w) where w is the working precision (two
instances of theta may represent different values), cf Higham's book.
*/
/* to ensure that u = a + k is exact, we have three cases:
(1) EXP(a) <= 0, then we need PREC(u) >= 1 - EXP(a) + PREC(a)
(2) EXP(a) - PREC(a) <= 0 < E(a), then PREC(u) >= PREC(a)
(3) 0 < EXP(a) - PREC(a), then PREC(u) >= EXP(a) */
precu = MPFR_GET_EXP(a) <= 0 ?
MPFR_ADD_PREC (MPFR_PREC(a), 1 - MPFR_EXP(a))
: (MPFR_EXP(a) <= MPFR_PREC(a)) ? MPFR_PREC(a) : MPFR_EXP(a);
MPFR_ASSERTD (precu + 1 <= MPFR_PREC_MAX);
mpfr_set_prec (u, precu + 1);
expu = (MPFR_EXP(a) > 0) ? MPFR_EXP(a) : 1;
/* estimate Taylor series */
mpfr_ui_div (t, 1, a, MPFR_RNDA); /* t = 1/a * (1 + theta) */
mpfr_set (s, t, MPFR_RNDA); /* s = 1/a * (1 + theta) */
if (MPFR_IS_NEG(a))
{
mpfr_init2 (s_abs, 32);
mpfr_abs (s_abs, s, MPFR_RNDU);
}
for (k = 1;; k++)
{
mpfr_mul (t, t, x, MPFR_RNDU); /* t = x^k/(a * ... * (a+k-1))
* (1 + theta)^(2k) */
inex = mpfr_add_ui (u, a, k, MPFR_RNDZ); /* u = a+k exactly */
MPFR_ASSERTD(inex == 0);
mpfr_div (t, t, u, MPFR_RNDA); /* t = x^k/(a * ... * (a+k))
* (1 + theta)^(2k+1) */
mpfr_add (s, s, t, MPFR_RNDZ);
/* when s is zero, we consider ulp(s) = ulp(t) */
exps = (MPFR_IS_ZERO(s)) ? MPFR_GET_EXP(t) : MPFR_GET_EXP(s);
if (MPFR_IS_NEG(a))
{
if (MPFR_IS_POS(t))
mpfr_add (s_abs, s_abs, t, MPFR_RNDU);
else
mpfr_sub (s_abs, s_abs, t, MPFR_RNDU);
}
/* we stop when |t| < ulp(s), u > 0 and |x/u| < 1/2, which ensures
that the tail is at most 2*ulp(s) */
MPFR_ASSERTD (MPFR_NOTZERO(t));
if (MPFR_GET_EXP(t) + w <= exps && MPFR_IS_POS(u) &&
MPFR_GET_EXP(x) + 1 < MPFR_GET_EXP(u))
break;
/* if there was an exponent shift in u, increase the precision of
u so that mpfr_add_ui (u, a, k) remains exact */
if (MPFR_EXP(u) > expu) /* exponent shift in u */
{
MPFR_ASSERTD(MPFR_EXP(u) == expu + 1);
expu = MPFR_EXP(u);
mpfr_set_prec (u, mpfr_get_prec (u) + 1);
}
}
if (MPFR_IS_NEG(a))
{
decay = MPFR_GET_EXP(s_abs) - MPFR_GET_EXP(s);
mpfr_clear (s_abs);
}
/* For a > 0, since all terms are positive, we have
s = S * (1 + theta)^(2k+3) with S being the infinite Taylor series.
For a < 0, the error is bounded by that on the sum s_abs of absolute
values of the terms, i.e., S_abs * [(1 + theta)^(2k+3) - 1]. Thus we
can simply use the same error analysis as for a > 0, adding an error
corresponding to the decay of exponent between s_abs and s. */
/* multiply by exp(-x) */
mpfr_exp (t, x, MPFR_RNDZ); /* t = exp(x) * (1+theta) */
mpfr_div (s, s, t, MPFR_RNDZ); /* s = <exact value> * (1+theta)^(2k+5) */
/* multiply by x^a */
mpfr_pow (t, x, a, MPFR_RNDZ); /* t = x^a * (1+theta) */
mpfr_mul (s, s, t, MPFR_RNDZ); /* s = Gamma(a,x) * (1+theta)^(2k+7) */
/* Since |theta| < 2^(-w) using the Taylor expansion of log(1+x)
we have log(1+theta) = theta1 with |theta1| < 1.16*2^(-w) for w >= 2,
thus (1+theta)^(2k+7) = exp((2k+7)*theta1).
Assuming 2k+7 = t*2^w for |t| < 0.5, we have
|(2k+7)*theta1| = |t*2^w*theta1| < 0.58.
For |u| < 0.58 we have |exp(u)-1| < 1.36*|u|
thus |(1+theta)^(2k+7) - 1| < 1.36*0.58*(2k+7)/2^w < 0.79*(2k+7)/2^w.
Since one ulp is at worst a relative error of 2^(1-w),
the error on s is at most 2^(decay+1)*(2k+7) ulps. */
/* subtract from gamma(a) */
MPFR_BLOCK (flags, mpfr_gamma (t, a, MPFR_RNDZ));
MPFR_ASSERTN (!MPFR_OVERFLOW (flags)); /* FIXME: support overflow */
/* t = gamma(a) * (1+theta) */
e0 = MPFR_GET_EXP (t);
e1 = (MPFR_IS_ZERO(s)) ? __gmpfr_emin : MPFR_GET_EXP (s);
mpfr_sub (s, t, s, MPFR_RNDZ);
/* if s is zero, we can assume ulp(s) = ulp(t), but anyway we won't
be able to round */
e2 = (MPFR_IS_ZERO(s)) ? e0 : MPFR_GET_EXP (s);
/* the final error is at most 1 ulp (for the final subtraction)
+ 2^(e0-e2) ulps # for the error in t
+ 2^(decay+1)*(2k+7) ulps * 2^(e1-e2) # for the error in gamma(a,x) */
e1 += decay + 1 + MPFR_INT_CEIL_LOG2 (2*k+7);
/* Now the error is <= 1 + 2^(e0-e2) + 2^(e1-e2).
Since the formula is symmetric in e0 and e1, we can assume without
loss of generality e0 >= e1, then:
if e0 = e1: err <= 1 + 2*2^(e0-e2) <= 2^(e0-e2+2)
if e0 > e1: err <= 1 + 1.5*2^(e0-e2)
<= 2^(e0-e2+1) if e0 > e2
<= 2^2 otherwise */
if (e0 == e1)
{
/* Check that e0 - e2 + 2 <= MPFR_EXP_MAX */
MPFR_ASSERTD (e2 >= 2 || e0 <= (MPFR_EXP_MAX - 2) + e2);
/* Check that e0 - e2 + 2 >= MPFR_EXP_MIN */
MPFR_ASSERTD (e2 <= 2 || e0 >= MPFR_EXP_MIN + (e2 - 2));
err = e0 - e2 + 2;
}
else
{
e0 = (e0 > e1) ? e0 : e1; /* max(e0,e1) */
MPFR_ASSERTD (e0 <= e2 || e2 >= 1 || e0 <= (MPFR_EXP_MAX - 1) + e2);
err = (e0 > e2) ? e0 - e2 + 1 : 2;
}
if (MPFR_LIKELY (MPFR_CAN_ROUND (s, w - err, MPFR_PREC(y), rnd)))
break;
MPFR_ZIV_NEXT (loop, w);
MPFR_GROUP_REPREC_2(group, w, s, t);
}
MPFR_ZIV_FREE (loop);
mpfr_clear (u);
inex = mpfr_set (y, s, rnd);
MPFR_GROUP_CLEAR(group);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd);
}
/* For a negative integer, we have (formula 6.5.19):
gamma(-n,x) = (-1)^n/n! [E_1(x) - exp(-x) sum((-1)^j*j!/x^(j+1), j=0..n-1)]
See also http://arxiv.org/pdf/1407.0349v1.pdf.
Assumes 'a' is a negative integer.
*/
static int
mpfr_gamma_inc_negint (mpfr_ptr y, mpfr_srcptr a, mpfr_srcptr x,
mpfr_rnd_t rnd)
{
mpfr_t s, t, abs_a, neg_x;
unsigned long j;
mpfr_prec_t w;
int inex;
mpfr_exp_t exp_s, new_exp_s, exp_t, err_s, logj;
MPFR_GROUP_DECL(group);
MPFR_ZIV_DECL(loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ASSERTD(mpfr_integer_p (a));
MPFR_ASSERTD(mpfr_cmp_ui (a, 0) < 0);
MPFR_TMP_INIT_ABS(abs_a, a);
/* below, theta represents any value such that |theta| <= 2^(-w) */
w = MPFR_PREC(y) + 10; /* initial working precision */
MPFR_SAVE_EXPO_MARK (expo);
MPFR_GROUP_INIT_2(group, w, s, t);
MPFR_ZIV_INIT (loop, w);
for (;;)
{
/* we require |a| <= 2^(w-3) for the error analysis below */
if (MPFR_GET_EXP(a) + 3 > w)
w = MPFR_GET_EXP(a) + 3;
mpfr_ui_div (t, 1, x, MPFR_RNDN); /* t = 1/x * (1 + theta) */
mpfr_set (s, t, MPFR_RNDN);
MPFR_ASSERTD (MPFR_NOTZERO(s));
exp_t = exp_s = MPFR_GET_EXP(s); /* max. exponent of s/t during loop */
new_exp_s = exp_s;
for (j = 1; mpfr_cmp_ui (abs_a, j) > 0; j++)
{
/* invariant: t = (-1)^(j-1)*(j-1)!/x^j * (1 + theta)^(2j-1) */
mpfr_mul_ui (t, t, j, MPFR_RNDN);
mpfr_neg (t, t, MPFR_RNDN); /* exact */
mpfr_div (t, t, x, MPFR_RNDN);
/* now t = (-1)^j*j!/x^(j+1) * (1 + theta)^(2j+1).
We have (1 + theta)^(2j+1) = exp((2j+1)*log(1+theta)).
For |u| <= 1/2, we have |log(1+u)| <= 1.4 |u| thus:
|(1+theta)^(2j+1)-1| <= max |exp(1.4*(2j+1)*u)-1| for |u|<=2^(-w).
Now for |v| <= 1/2 we have |exp(v)-1| <= 0.7*|v| thus:
|(1+theta)^(2j+1) - 1| <= 2*(2j+1)*2^(-w)
as long as 1.4*(2j+1)*2^(-w) <= 1/2, which is true when j<2^(w-3).
Since j < |a| it suffices that |a| <= 2^(w-3).
In that case the rel. error on t is bounded by 2*(2j+1)*2^(-w),
thus the error in ulps is bounded by 2*(2j+1) ulp(t). */
if (MPFR_IS_ZERO(t)) /* underflow on t */
break;
if (MPFR_GET_EXP(t) > exp_t)
exp_t = MPFR_GET_EXP(t);
mpfr_add (s, s, t, MPFR_RNDN);
/* if s is zero, we can assume its ulp is that of t */
new_exp_s = (MPFR_IS_ZERO(s)) ? MPFR_GET_EXP(t) : MPFR_GET_EXP(s);
if (new_exp_s > exp_s)
exp_s = new_exp_s;
}
/* the error on s is bounded by (j-1) * 2^(exp_s - EXP(s)) * 1/2
for the mpfr_add roundings, plus
sum(2*(2i+1), i=1..j-1) * 2^(exp_t - EXP(s)) for the error on t.
The latter sum is (2*j^2-2) * 2^(exp_t - EXP(s)). */
logj = MPFR_INT_CEIL_LOG2(j);
exp_s += logj - 1;
exp_t += 1 + 2 * logj;
/* now the error on s is bounded by 2^(exp_s-EXP(s))+2^(exp_t-EXP(s)) */
exp_s = (exp_s >= exp_t) ? exp_s + 1 : exp_t + 1;
err_s = exp_s - new_exp_s;
/* now the error on the sum S := sum((-1)^j*j!/x^(j+1), j=0..n-1)
is bounded by 2^err_s ulp(s) */
MPFR_TMP_INIT_NEG(neg_x, x);
mpfr_exp (t, neg_x, MPFR_RNDN); /* t = exp(-x) * (1 + theta) */
mpfr_mul (s, s, t, MPFR_RNDN);
if (MPFR_IS_ZERO(s))
{
MPFR_ASSERTD (MPFR_NOTZERO(t));
new_exp_s += MPFR_GET_EXP(t);
}
/* s = exp(-x) * (S +/- 2^err_s ulp(S)) * (1 + theta)^2.
= exp(-x) * (S +/- 2^err_s ulp(S)) * (1 +/- 3 ulp(1))
The error on s is bounded by:
exp(-x) * [2^err_s*ulp(S) + S*3*ulp(1) + 2^err_s*ulp(S)*3*ulp(1)]
<= ulp(s) * [2^(err_s+1) + 6 + 1]
<= ulp(s) * 2^(err_s+2) as long as err_s >= 2. */
err_s = (err_s >= 2) ? err_s + 2 : 4;
/* now the error on s is bounded by 2^err_s ulp(s) */
mpfr_eint (t, neg_x, MPFR_RNDN); /* t = -E1(-x) * (1 + theta) */
mpfr_neg (t, t, MPFR_RNDN); /* exact */
exp_s = (MPFR_IS_ZERO(s)) ? new_exp_s : MPFR_GET_EXP(s);
MPFR_ASSERTD (MPFR_NOTZERO(t));
exp_t = MPFR_GET_EXP(t);
mpfr_sub (s, t, s, MPFR_RNDN); /* E_1(x) - exp(-x) * S */
if (MPFR_IS_ZERO(s)) /* cancellation: increase working precision */
goto next_w;
/* err(s) <= 1/2 * ulp(s) [mpfr_sub]
+ 2^err_s * 2^(exp_s-EXP(s)) * ulp(s) [previous error on s]
+ 1/2 * 2^(exp_t-EXP(s)) * ulp(s) [error on t] */
exp_s += err_s;
exp_t -= 1;
exp_s = (exp_s >= exp_t) ? exp_s + 1 : exp_t + 1;
MPFR_ASSERTD (MPFR_NOTZERO(s));
err_s = exp_s - MPFR_GET_EXP(s);
/* err(s) <= 1/2 * ulp(s) + 2^err_s * ulp(s) */
/* divide by n! */
mpfr_gamma (t, abs_a, MPFR_RNDN); /* t = (n-1)! * (1 + theta) */
mpfr_mul (t, t, abs_a, MPFR_RNDN); /* t = n! * (1 + theta)^2 */
mpfr_div (s, s, t, MPFR_RNDN);
/* since (1 + theta)^2 converts to an error of at most 3 ulps
for w >= 2, the final error is at most:
2 * (1/2 + 2^err_s) * ulp(s) [error on previous s]
+ 2 * 3 * ulp(s) [error on t]
+ 1 * ulp(s) [product of errors]
= ulp(s) * (2^(err_s+1) + 8) */
err_s = (err_s >= 2) ? err_s + 1 : 4;
/* the final error is bounded by 2^err_s * ulp(s) */
/* Is there a better way to compute (-1)^n? */
mpfr_set_si (t, -1, MPFR_RNDN);
mpfr_pow (t, t, abs_a, MPFR_RNDN);
if (MPFR_IS_NEG(t))
mpfr_neg (s, s, MPFR_RNDN);
if (MPFR_LIKELY (MPFR_CAN_ROUND (s, w - err_s, MPFR_PREC(y), rnd)))
break;
next_w:
MPFR_ZIV_NEXT (loop, w);
MPFR_GROUP_REPREC_2(group, w, s, t);
}
MPFR_ZIV_FREE (loop);
inex = mpfr_set (y, s, rnd);
MPFR_GROUP_CLEAR(group);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inex, rnd);
}
|