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+/* mpfr_beta -- beta function
+
+Copyright 2017 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
+http://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 /* for MPFR_INT_CEIL_LOG2 */
+#include "mpfr-impl.h"
+
+/* use formula (6.2.2) from Abramowitz & Stegun:
+ beta(z,w) = gamma(z)*gamma(w)/gamma(z+w) */
+int
+mpfr_beta (mpfr_ptr r, mpfr_srcptr z, mpfr_srcptr w, mpfr_rnd_t rnd_mode)
+{
+ mpfr_exp_t emin, emax;
+ mpfr_uexp_t pmin;
+ mpfr_prec_t prec;
+ mpfr_t z_plus_w, tmp, tmp2;
+ int inex, w_integer;
+ MPFR_GROUP_DECL (group);
+ MPFR_ZIV_DECL (loop);
+ MPFR_SAVE_EXPO_DECL (expo);
+
+ if (mpfr_less_p (z, w))
+ return mpfr_beta (r, w, z, rnd_mode);
+
+ /* Now, either z and w are unordered (at least one is a NaN), or z >= w. */
+
+ if (MPFR_ARE_SINGULAR (z, w))
+ {
+ /* if z or w is NaN, return NaN */
+ if (MPFR_IS_NAN (z) || MPFR_IS_NAN (w))
+ {
+ MPFR_SET_NAN (r);
+ MPFR_RET_NAN;
+ }
+ else if (MPFR_IS_INF (z) || MPFR_IS_INF (w))
+ {
+ /* Since we have z >= w:
+ if z = +Inf and w > 0, then r = +0 (including w = +Inf);
+ if z = +Inf and w = 0, then r = NaN
+ [beta(z,1/log(z)) tends to +Inf whereas
+ beta(z,1/log(log(z))) tends to +0]
+ if z = +Inf and w < 0:
+ if w is an integer or -Inf: r = NaN
+ if -2k-1 < w < -2k: r = -Inf
+ if -2k-2 < w < -2k-1: r = +Inf
+ if w = -Inf and z is finite and not an integer:
+ beta(z,t) for t going to -Inf oscillates between positive and
+ negative values, with poles around integer values of t, thus
+ beta(z,w) gives NaN;
+ if w = -Inf and z is an integer:
+ beta(z,w) gives +0 for z even > 0, -0 for z odd > 0,
+ NaN for z <= 0;
+ if z = -Inf (then w = -Inf too): r = NaN */
+ if (MPFR_IS_INF (z) && MPFR_IS_POS(z)) /* z = +Inf */
+ {
+ if (mpfr_cmp_ui (w, 0) > 0)
+ {
+ MPFR_SET_ZERO(r);
+ MPFR_SET_POS(r);
+ MPFR_RET(0);
+ }
+ else if (MPFR_IS_ZERO(w) || MPFR_IS_INF(w) || mpfr_integer_p (w))
+ {
+ MPFR_SET_NAN(r);
+ MPFR_RET_NAN;
+ }
+ else
+ {
+ long q;
+ mpfr_t t;
+
+ MPFR_SAVE_EXPO_MARK (expo);
+ mpfr_init2 (t, MPFR_PREC_MIN);
+ mpfr_set_ui (t, 1, MPFR_RNDN);
+ mpfr_fmodquo (t, &q, w, t, MPFR_RNDD);
+ mpfr_clear (t);
+ MPFR_SAVE_EXPO_FREE (expo);
+ /* q contains the low bits of trunc(w) where trunc() rounds
+ towards zero, thus if q is odd, then -2k-2 < w < -2k-1 */
+ MPFR_SET_INF(r);
+ if ((unsigned long) q & 1)
+ MPFR_SET_NEG(r);
+ else
+ MPFR_SET_POS(r);
+ MPFR_RET(0);
+ }
+ }
+ else if (MPFR_IS_INF(w)) /* w = -Inf */
+ {
+ if (mpfr_cmp_ui (z, 0) <= 0 || !mpfr_integer_p (z))
+ {
+ MPFR_SET_NAN(r);
+ MPFR_RET_NAN;
+ }
+ else
+ {
+ MPFR_SET_ZERO(r);
+ if (mpfr_odd_p (z))
+ MPFR_SET_NEG(r);
+ else
+ MPFR_SET_POS(r);
+ MPFR_RET(0);
+ }
+ }
+ }
+ else /* z or w is 0 */
+ {
+ /* If x is not a nonpositive integer, Gamma(x) is regular, so that
+ when y -> 0 with either y >= 0 or y <= 0,
+ Beta(x,y) ~ Gamma(x) * Gamma(y) / Gamma(x) = Gamma(y)
+ Gamma(y) tends to an infinity of the same sign as y.
+ Thus Beta(x,y) should be an infinity of the same sign as y.
+ */
+ if (mpfr_cmp_ui (z, 0) != 0) /* then w is +0 or -0 and z > 0 */
+ {
+ /* beta(z,+0) = +Inf, beta(z,-0) = -Inf (see above) */
+ MPFR_SET_INF(r);
+ MPFR_SET_SAME_SIGN(r,w);
+ MPFR_SET_DIVBY0 ();
+ MPFR_RET(0);
+ }
+ else if (mpfr_cmp_ui (w, 0) != 0) /* then z is +0 or -0 and w < 0 */
+ {
+ if (mpfr_integer_p (w))
+ {
+ /* For small u > 0, Beta(2u,w+u) and Beta(2u,w-u) have
+ opposite signs, so that they tend to infinities of
+ opposite signs when u -> 0. Thus the result is NaN. */
+ MPFR_SET_NAN(r);
+ MPFR_RET_NAN;
+ }
+ else
+ {
+ /* beta(+0,w) = +Inf, beta(-0,w) = -Inf (see above) */
+ MPFR_SET_INF(r);
+ MPFR_SET_SAME_SIGN(r,z);
+ MPFR_SET_DIVBY0 ();
+ MPFR_RET(0);
+ }
+ }
+ else /* w = z = 0:
+ beta(+0,+0) = +Inf
+ beta(-0,-0) = -Inf
+ beta(+0,-0) = NaN */
+ {
+ if (MPFR_SIGN(z) == MPFR_SIGN(w))
+ {
+ MPFR_SET_INF(r);
+ MPFR_SET_SAME_SIGN(r,z);
+ MPFR_SET_DIVBY0 ();
+ MPFR_RET(0);
+ }
+ else
+ {
+ MPFR_SET_NAN(r);
+ MPFR_RET_NAN;
+ }
+ }
+ }
+ }
+
+ /* special case when w is a negative integer */
+ w_integer = mpfr_integer_p (w);
+ if (w_integer && MPFR_IS_NEG(w))
+ {
+ /* if z < 0 or z+w > 0, or z is not an integer, return NaN */
+ if (MPFR_IS_NEG(z) || mpfr_cmpabs (z, w) > 0 || !mpfr_integer_p (z))
+ {
+ MPFR_SET_NAN(r);
+ MPFR_RET_NAN;
+ }
+ /* If z+w = 0, the result is 1/z. */
+ if (mpfr_cmpabs (z, w) == 0)
+ return mpfr_ui_div (r, 1, z, rnd_mode);
+ /* Now z is an integer and z+w <= 0: return (-1)^z*beta(z,1-w-z).
+ Since z and w are of opposite signs, |z+w| <= max(|z|,|w|). */
+ emax = MAX (MPFR_EXP(z), MPFR_EXP(w));
+ mpfr_init2 (z_plus_w, (mpfr_prec_t) emax);
+ inex = mpfr_add (z_plus_w, z, w, MPFR_RNDN);
+ MPFR_ASSERTN(inex == 0);
+ inex = mpfr_ui_sub (z_plus_w, 1, z_plus_w, MPFR_RNDN);
+ MPFR_ASSERTN(inex == 0);
+ if (mpfr_odd_p (z))
+ {
+ inex = -mpfr_beta (r, z, z_plus_w, MPFR_INVERT_RND (rnd_mode));
+ MPFR_CHANGE_SIGN(r);
+ }
+ else
+ inex = mpfr_beta (r, z, z_plus_w, rnd_mode);
+ mpfr_clear (z_plus_w);
+ return inex;
+ }
+
+ /* special case when z is a negative integer: here w < z and w is not an
+ integer */
+ if (mpfr_integer_p (z) && MPFR_IS_NEG(z))
+ {
+ MPFR_SET_NAN(r);
+ MPFR_RET_NAN;
+ }
+
+ MPFR_SAVE_EXPO_MARK (expo);
+
+ /* compute the smallest precision such that z + w is exact */
+ emax = MAX (MPFR_EXP(z), MPFR_EXP(w));
+ emin = MIN (MPFR_EXP(z) - MPFR_PREC(z), MPFR_EXP(w) - MPFR_PREC(w));
+ MPFR_ASSERTD (emax >= emin);
+ /* Thus the math value of emax - emin is representable in mpfr_uexp_t. */
+ pmin = (mpfr_uexp_t) emax - emin;
+ /* If z and w have same sign, their sum can have exponent emax + 1. */
+ pmin += 1;
+ if (pmin > MPFR_PREC_MAX) /* FIXME: check if result can differ from NaN. */
+ {
+ MPFR_SAVE_EXPO_FREE (expo);
+ MPFR_SET_NAN(r);
+ MPFR_RET_NAN;
+ }
+ MPFR_ASSERTN (pmin <= MPFR_PREC_MAX); /* detect integer overflow */
+ mpfr_init2 (z_plus_w, (mpfr_prec_t) pmin);
+ inex = mpfr_add (z_plus_w, z, w, MPFR_RNDN);
+ /* if z+w overflows with rounding to nearest, then w must be larger than
+ 1/2*ulp(z), thus we have an underflow. */
+ if (MPFR_IS_INF(z_plus_w))
+ {
+ mpfr_clear (z_plus_w);
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_underflow (r, rnd_mode, 1);
+ }
+ MPFR_ASSERTN(inex == 0);
+
+ /* If z+w is 0 or a negative integer, return +0 when w (and thus z) is not
+ an integer. Indeed, gamma(z) and gamma(w) are regular numbers, and
+ gamma(z+w) is Inf, thus 1/gamma(z+w) is zero. Unless there is a rule
+ to choose the sign of 0, we choose +0. */
+ if (mpfr_cmp_ui (z_plus_w, 0) <= 0 && !w_integer
+ && mpfr_integer_p (z_plus_w))
+ {
+ mpfr_clear (z_plus_w);
+ MPFR_SAVE_EXPO_FREE (expo);
+ MPFR_SET_ZERO(r);
+ MPFR_SET_POS(r);
+ MPFR_RET(0);
+ }
+
+ prec = MPFR_PREC(r);
+ prec += MPFR_INT_CEIL_LOG2 (prec);
+ MPFR_GROUP_INIT_2 (group, prec, tmp, tmp2);
+ MPFR_ZIV_INIT (loop, prec);
+ for (;;)
+ {
+ unsigned int inex2; /* unsigned due to bitwise operations */
+
+ MPFR_GROUP_REPREC_2 (group, prec, tmp, tmp2);
+ inex2 = mpfr_gamma (tmp, z, MPFR_RNDN);
+ /* tmp = gamma(z) * (1 + theta) with |theta| <= 2^-prec */
+ inex2 |= mpfr_gamma (tmp2, w, MPFR_RNDN);
+ /* tmp2 = gamma(w) * (1 + theta2) with |theta2| <= 2^-prec */
+ inex2 |= mpfr_mul (tmp, tmp, tmp2, MPFR_RNDN);
+ /* tmp = gamma(z)*gamma(w) * (1 + theta3)^3 with |theta3| <= 2^-prec */
+ inex2 |= mpfr_gamma (tmp2, z_plus_w, MPFR_RNDN);
+ /* tmp2 = gamma(z+w) * (1 + theta4) with |theta4| <= 2^-prec */
+ inex2 |= mpfr_div (tmp, tmp, tmp2, MPFR_RNDN);
+ /* tmp = gamma(z)*gamma(w)/gamma(z+w) * (1 + theta5)^5
+ with |theta5| <= 2^-prec. For prec >= 3, we have
+ |(1 + theta5)^5 - 1| <= 7 * 2^(-prec), thus the error is bounded
+ by 7 ulps */
+
+ if (MPFR_IS_NAN(tmp)) /* FIXME: most probably gamma(z)*gamma(w) = +-Inf,
+ and gamma(z+w) = +-Inf, can we do better? */
+ {
+ mpfr_clear (z_plus_w);
+ MPFR_ZIV_FREE (loop);
+ MPFR_GROUP_CLEAR (group);
+ MPFR_SAVE_EXPO_FREE (expo);
+ MPFR_SET_NAN(r);
+ MPFR_RET_NAN;
+ }
+
+ MPFR_ASSERTN(mpfr_regular_p (tmp));
+
+ /* if inex2 = 0, then tmp is exactly beta(z,w) */
+ if (inex2 == 0 ||
+ MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - 3, MPFR_PREC(r), rnd_mode)))
+ break;
+
+ /* beta(1,+/-2^(-k)) = +/-2^k is exact, and cannot be detected above
+ since gamma(+/-2^(-k)) is not exact */
+ if (mpfr_cmp_ui (z, 1) == 0)
+ {
+ mpfr_exp_t expw = mpfr_get_exp (w);
+ if (mpfr_cmp_ui_2exp (w, 1, expw - 1) == 0)
+ {
+ /* since z >= w, this will only match w <= 1 */
+ mpfr_set_ui_2exp (tmp, 1, 1 - expw, MPFR_RNDN);
+ break;
+ }
+ else if (mpfr_cmp_si_2exp (w, -1, expw - 1) == 0)
+ {
+ mpfr_set_si_2exp (tmp, -1, 1 - expw, MPFR_RNDN);
+ break;
+ }
+ }
+
+ /* beta(2^k,1) = 1/2^k for k > 0 (k <= 0 was already tested above) */
+ if (mpfr_cmp_ui (w, 1) == 0 &&
+ mpfr_cmp_ui_2exp (z, 1, MPFR_EXP(z) - 1) == 0)
+ {
+ mpfr_set_ui_2exp (tmp, 1, 1 - MPFR_EXP(z), MPFR_RNDN);
+ break;
+ }
+
+ /* beta(2,-0.5) = -4 */
+ if (mpfr_cmp_ui (z, 2) == 0 && mpfr_cmp_si_2exp (w, -1, -1) == 0)
+ {
+ mpfr_set_si_2exp (tmp, -1, 2, MPFR_RNDN);
+ break;
+ }
+
+ MPFR_ZIV_NEXT (loop, prec);
+ }
+ MPFR_ZIV_FREE (loop);
+ inex = mpfr_set (r, tmp, rnd_mode);
+ MPFR_GROUP_CLEAR (group);
+ mpfr_clear (z_plus_w);
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (r, inex, rnd_mode);
+}