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+/* Functions for evaluating Gamma(1/3) and Gamma(2/3). Used by mpfr_ai.
+
+Copyright 2010-2016 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
+#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);
+}