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Diffstat (limited to 'src/beta.c')
| -rw-r--r-- | src/beta.c | 345 |
1 files changed, 345 insertions, 0 deletions
diff --git a/src/beta.c b/src/beta.c new file mode 100644 index 000000000..3fb9b9a77 --- /dev/null +++ b/src/beta.c @@ -0,0 +1,345 @@ +/* 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); +} |