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Diffstat (limited to 'src/gammaonethird.c')
| -rw-r--r-- | src/gammaonethird.c | 191 |
1 files changed, 191 insertions, 0 deletions
diff --git a/src/gammaonethird.c b/src/gammaonethird.c new file mode 100644 index 0000000..93fda9c --- /dev/null +++ b/src/gammaonethird.c @@ -0,0 +1,191 @@ +/* Functions for evaluating Gamma(1/3) and Gamma(2/3). Used by mpfr_ai. + +Copyright 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel 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 +#include "mpfr-impl.h" + +#define MPFR_ACC_OR_MUL(v) \ + do \ + { \ + if (v <= ULONG_MAX / acc) \ + acc *= v; \ + else \ + { \ + mpfr_mul_ui (y, y, acc, mode); acc = v; \ + } \ + } \ + while (0) + +#define MPFR_ACC_OR_DIV(v) \ + do \ + { \ + if (v <= ULONG_MAX / acc) \ + acc *= v; \ + else \ + { \ + mpfr_div_ui (y, y, acc, mode); acc = v; \ + } \ + } \ + while (0) + +static void +mpfr_mul_ui5 (mpfr_ptr y, mpfr_srcptr x, + unsigned long int v1, unsigned long int v2, + unsigned long int v3, unsigned long int v4, + unsigned long int v5, mpfr_rnd_t mode) +{ + unsigned long int acc = v1; + mpfr_set (y, x, mode); + MPFR_ACC_OR_MUL (v2); + MPFR_ACC_OR_MUL (v3); + MPFR_ACC_OR_MUL (v4); + MPFR_ACC_OR_MUL (v5); + mpfr_mul_ui (y, y, acc, mode); +} + +void +mpfr_div_ui2 (mpfr_ptr y, mpfr_srcptr x, + unsigned long int v1, unsigned long int v2, mpfr_rnd_t mode) +{ + unsigned long int acc = v1; + mpfr_set (y, x, mode); + MPFR_ACC_OR_DIV (v2); + mpfr_div_ui (y, y, acc, mode); +} + +static void +mpfr_div_ui8 (mpfr_ptr y, mpfr_srcptr x, + unsigned long int v1, unsigned long int v2, + unsigned long int v3, unsigned long int v4, + unsigned long int v5, unsigned long int v6, + unsigned long int v7, unsigned long int v8, mpfr_rnd_t mode) +{ + unsigned long int acc = v1; + mpfr_set (y, x, mode); + MPFR_ACC_OR_DIV (v2); + MPFR_ACC_OR_DIV (v3); + MPFR_ACC_OR_DIV (v4); + MPFR_ACC_OR_DIV (v5); + MPFR_ACC_OR_DIV (v6); + MPFR_ACC_OR_DIV (v7); + MPFR_ACC_OR_DIV (v8); + mpfr_div_ui (y, y, acc, mode); +} + + +/* Gives an approximation of omega = Gamma(1/3)^6 * sqrt(10) / (12pi^4) */ +/* using C. H. Brown's formula. */ +/* The computed value s satisfies |s-omega| <= 2^{1-prec}*omega */ +/* As usual, the variable s is supposed to be initialized. */ +static void +mpfr_Browns_const (mpfr_ptr s, mpfr_prec_t prec) +{ + mpfr_t uk; + unsigned long int k; + + mpfr_prec_t working_prec = prec + 10 + MPFR_INT_CEIL_LOG2 (2 + prec / 10); + + mpfr_init2 (uk, working_prec); + mpfr_set_prec (s, working_prec); + + mpfr_set_ui (uk, 1, MPFR_RNDN); + mpfr_set (s, uk, MPFR_RNDN); + k = 1; + + /* Invariants: uk ~ u(k-1) and s ~ sum(i=0..k-1, u(i)) */ + for (;;) + { + mpfr_mul_ui5 (uk, uk, 6 * k - 5, 6 * k - 4, 6 * k - 3, 6 * k - 2, + 6 * k - 1, MPFR_RNDN); + mpfr_div_ui8 (uk, uk, k, k, 3 * k - 2, 3 * k - 1, 3 * k, 80, 160, 160, + MPFR_RNDN); + MPFR_CHANGE_SIGN (uk); + + mpfr_add (s, s, uk, MPFR_RNDN); + k++; + if (MPFR_GET_EXP (uk) + prec <= MPFR_GET_EXP (s) + 7) + break; + } + + mpfr_clear (uk); + return; +} + +/* Returns y such that |Gamma(1/3)-y| <= 2^{1-prec}*Gamma(1/3) */ +static void +mpfr_gamma_one_third (mpfr_ptr y, mpfr_prec_t prec) +{ + mpfr_t tmp, tmp2, tmp3; + + mpfr_init2 (tmp, prec + 9); + mpfr_init2 (tmp2, prec + 9); + mpfr_init2 (tmp3, prec + 4); + mpfr_set_prec (y, prec + 2); + + mpfr_const_pi (tmp, MPFR_RNDN); + mpfr_sqr (tmp, tmp, MPFR_RNDN); + mpfr_sqr (tmp, tmp, MPFR_RNDN); + mpfr_mul_ui (tmp, tmp, 12, MPFR_RNDN); + + mpfr_Browns_const (tmp2, prec + 9); + mpfr_mul (tmp, tmp, tmp2, MPFR_RNDN); + + mpfr_set_ui (tmp2, 10, MPFR_RNDN); + mpfr_sqrt (tmp2, tmp2, MPFR_RNDN); + mpfr_div (tmp, tmp, tmp2, MPFR_RNDN); + + mpfr_sqrt (tmp3, tmp, MPFR_RNDN); + mpfr_cbrt (y, tmp3, MPFR_RNDN); + + mpfr_clear (tmp); + mpfr_clear (tmp2); + mpfr_clear (tmp3); + return; +} + +/* Computes y1 and y2 such that: */ +/* |y1-Gamma(1/3)| <= 2^{1-prec}Gamma(1/3) */ +/* and |y2-Gamma(2/3)| <= 2^{1-prec}Gamma(2/3) */ +/* */ +/* Uses the formula Gamma(z)Gamma(1-z) = pi / sin(pi*z) */ +/* to compute Gamma(2/3) from Gamma(1/3). */ +void +mpfr_gamma_one_and_two_third (mpfr_ptr y1, mpfr_ptr y2, mpfr_prec_t prec) +{ + mpfr_t temp; + + mpfr_init2 (temp, prec + 4); + mpfr_set_prec (y2, prec + 4); + + mpfr_gamma_one_third (y1, prec + 4); + + mpfr_set_ui (temp, 3, MPFR_RNDN); + mpfr_sqrt (temp, temp, MPFR_RNDN); + mpfr_mul (temp, y1, temp, MPFR_RNDN); + + mpfr_const_pi (y2, MPFR_RNDN); + mpfr_mul_2ui (y2, y2, 1, MPFR_RNDN); + + mpfr_div (y2, y2, temp, MPFR_RNDN); + + mpfr_clear (temp); +} |