mirror of
https://github.com/vale981/arb
synced 2025-03-05 09:21:38 -05:00
324 lines
7.5 KiB
C
324 lines
7.5 KiB
C
/*
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Copyright (C) 2013 Fredrik Johansson
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This file is part of Arb.
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Arb is free software: you can redistribute it and/or modify it under
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the terms of the GNU Lesser General Public License (LGPL) as published
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by the Free Software Foundation; either version 2.1 of the License, or
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(at your option) any later version. See <http://www.gnu.org/licenses/>.
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*/
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#include "bernoulli.h"
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#include "acb.h"
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void
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acb_gamma_stirling_choose_param(int * reflect, slong * r, slong * n,
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const acb_t z, int use_reflect, int digamma, slong prec);
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void acb_gamma_stirling_bound(mag_ptr err, const acb_t z, slong k0, slong knum, slong n);
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void arb_gamma_stirling_coeff(arb_t b, ulong k, int digamma, slong prec);
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void
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acb_gamma_stirling_eval(acb_t s, const acb_t z, slong nterms, int digamma, slong prec)
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{
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acb_t t, logz, zinv, zinv2;
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arb_t b;
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mag_t err;
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slong k, term_prec;
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double z_mag, term_mag;
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acb_init(t);
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acb_init(logz);
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acb_init(zinv);
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acb_init(zinv2);
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arb_init(b);
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acb_log(logz, z, prec);
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acb_inv(zinv, z, prec);
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nterms = FLINT_MAX(nterms, 1);
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acb_zero(s);
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if (nterms > 1)
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{
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acb_mul(zinv2, zinv, zinv, prec);
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z_mag = arf_get_d(arb_midref(acb_realref(logz)), ARF_RND_UP) * 1.44269504088896;
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for (k = nterms - 1; k >= 1; k--)
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{
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term_mag = bernoulli_bound_2exp_si(2 * k);
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term_mag -= (2 * k - 1) * z_mag;
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term_prec = prec + term_mag;
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term_prec = FLINT_MIN(term_prec, prec);
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term_prec = FLINT_MAX(term_prec, 10);
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arb_gamma_stirling_coeff(b, k, digamma, term_prec);
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if (prec > 2000)
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{
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acb_set_round(t, zinv2, term_prec);
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acb_mul(s, s, t, term_prec);
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}
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else
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acb_mul(s, s, zinv2, term_prec);
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arb_add(acb_realref(s), acb_realref(s), b, term_prec);
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}
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if (digamma)
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acb_mul(s, s, zinv2, prec);
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else
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acb_mul(s, s, zinv, prec);
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}
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/* remainder bound */
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mag_init(err);
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acb_gamma_stirling_bound(err, z, digamma ? 1 : 0, 1, nterms);
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mag_add(arb_radref(acb_realref(s)), arb_radref(acb_realref(s)), err);
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mag_add(arb_radref(acb_imagref(s)), arb_radref(acb_imagref(s)), err);
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mag_clear(err);
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if (digamma)
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{
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acb_neg(s, s);
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acb_mul_2exp_si(zinv, zinv, -1);
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acb_sub(s, s, zinv, prec);
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acb_add(s, s, logz, prec);
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}
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else
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{
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/* (z-0.5)*log(z) - z + log(2*pi)/2 */
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arb_one(b);
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arb_mul_2exp_si(b, b, -1);
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arb_set(acb_imagref(t), acb_imagref(z));
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arb_sub(acb_realref(t), acb_realref(z), b, prec);
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acb_mul(t, logz, t, prec);
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acb_add(s, s, t, prec);
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acb_sub(s, s, z, prec);
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arb_const_log_sqrt2pi(b, prec);
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arb_add(acb_realref(s), acb_realref(s), b, prec);
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}
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acb_clear(t);
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acb_clear(logz);
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acb_clear(zinv);
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acb_clear(zinv2);
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arb_clear(b);
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}
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static void
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_acb_gamma(acb_t y, const acb_t x, slong prec, int inverse)
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{
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int reflect;
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slong r, n, wp;
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acb_t t, u, v;
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double acc;
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wp = prec + FLINT_BIT_COUNT(prec);
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/* todo: for large x (if exact or accurate enough), increase precision */
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acc = acb_rel_accuracy_bits(x);
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acc = FLINT_MAX(acc, 0);
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wp = FLINT_MIN(prec, acc + 20);
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wp = FLINT_MAX(wp, 2);
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wp = wp + FLINT_BIT_COUNT(wp);
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acb_gamma_stirling_choose_param(&reflect, &r, &n, x, 1, 0, wp);
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acb_init(t);
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acb_init(u);
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acb_init(v);
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if (reflect)
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{
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/* gamma(x) = (rf(1-x, r) * pi) / (gamma(1-x+r) sin(pi x)) */
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acb_sub_ui(t, x, 1, wp);
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acb_neg(t, t);
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acb_rising_ui_rec(u, t, r, wp);
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arb_const_pi(acb_realref(v), wp);
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acb_mul_arb(u, u, acb_realref(v), wp);
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acb_add_ui(t, t, r, wp);
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acb_gamma_stirling_eval(v, t, n, 0, wp);
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acb_exp(v, v, wp);
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acb_sin_pi(t, x, wp);
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acb_mul(v, v, t, wp);
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}
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else
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{
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/* gamma(x) = gamma(x+r) / rf(x,r) */
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acb_add_ui(t, x, r, wp);
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acb_gamma_stirling_eval(u, t, n, 0, wp);
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acb_exp(u, u, prec);
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acb_rising_ui_rec(v, x, r, wp);
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}
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if (inverse)
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acb_div(y, v, u, prec);
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else
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acb_div(y, u, v, prec);
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acb_clear(t);
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acb_clear(u);
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acb_clear(v);
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}
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void
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acb_gamma(acb_t y, const acb_t x, slong prec)
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{
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if (acb_is_real(x))
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{
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arb_gamma(acb_realref(y), acb_realref(x), prec);
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arb_zero(acb_imagref(y));
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return;
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}
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_acb_gamma(y, x, prec, 0);
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}
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void
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acb_rgamma(acb_t y, const acb_t x, slong prec)
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{
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if (acb_is_real(x))
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{
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arb_rgamma(acb_realref(y), acb_realref(x), prec);
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arb_zero(acb_imagref(y));
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return;
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}
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_acb_gamma(y, x, prec, 1);
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}
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/* corrects branch cut of sum_{k=0}^{r-1} log(z+k), given the
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logarithm of the product */
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void
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_acb_log_rising_correct_branch(acb_t t,
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const acb_t t_wrong, const acb_t z, ulong r, slong prec)
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{
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acb_t f;
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arb_t pi, u, v;
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fmpz_t pi_mult;
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slong i, argprec;
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acb_init(f);
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arb_init(u);
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arb_init(pi);
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arb_init(v);
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fmpz_init(pi_mult);
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argprec = FLINT_MIN(prec, 40);
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arb_zero(u);
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for (i = 0; i < r; i++)
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{
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acb_add_ui(f, z, i, argprec);
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acb_arg(v, f, argprec);
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arb_add(u, u, v, argprec);
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}
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if (argprec == prec)
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{
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arb_set(acb_imagref(t), u);
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}
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else
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{
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arb_sub(v, u, acb_imagref(t), argprec);
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arb_const_pi(pi, argprec);
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arb_div(v, v, pi, argprec);
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if (arb_get_unique_fmpz(pi_mult, v))
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{
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arb_const_pi(v, prec);
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arb_mul_fmpz(v, v, pi_mult, prec);
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arb_add(acb_imagref(t), acb_imagref(t), v, prec);
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}
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else
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{
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arb_zero(u);
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for (i = 0; i < r; i++)
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{
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acb_add_ui(f, z, i, prec);
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acb_arg(v, f, prec);
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arb_add(u, u, v, prec);
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}
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arb_set(acb_imagref(t), u);
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}
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}
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acb_clear(f);
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arb_clear(u);
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arb_clear(v);
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arb_clear(pi);
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fmpz_clear(pi_mult);
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}
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void
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acb_lgamma(acb_t y, const acb_t x, slong prec)
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{
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int reflect;
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slong r, n, wp;
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acb_t t, u, v;
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if (acb_is_real(x) && arb_is_positive(acb_realref(x)))
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{
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arb_lgamma(acb_realref(y), acb_realref(x), prec);
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arb_zero(acb_imagref(y));
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return;
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}
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wp = prec + FLINT_BIT_COUNT(prec);
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acb_gamma_stirling_choose_param(&reflect, &r, &n, x, 1, 0, wp);
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acb_init(t);
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acb_init(u);
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acb_init(v);
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if (reflect)
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{
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/* log gamma(x) = log rf(1-x, r) - log gamma(1-x+r) - log sin(pi x) + log(pi) */
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acb_sub_ui(u, x, 1, wp);
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acb_neg(u, u);
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acb_rising_ui_rec(t, u, r, prec);
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acb_log(t, t, wp);
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_acb_log_rising_correct_branch(t, t, u, r, wp);
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acb_add_ui(u, u, r, wp);
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acb_gamma_stirling_eval(v, u, n, 0, wp);
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acb_sub(t, t, v, wp);
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acb_log_sin_pi(u, x, wp);
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acb_sub(t, t, u, wp);
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acb_const_pi(u, wp);
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acb_log(u, u, wp);
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acb_add(y, t, u, wp);
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}
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else
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{
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/* log gamma(x) = log gamma(x+r) - log rf(x,r) */
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acb_add_ui(t, x, r, wp);
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acb_gamma_stirling_eval(u, t, n, 0, wp);
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acb_rising_ui_rec(t, x, r, prec);
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acb_log(t, t, wp);
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_acb_log_rising_correct_branch(t, t, x, r, wp);
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acb_sub(y, u, t, prec);
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}
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acb_clear(t);
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acb_clear(u);
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acb_clear(v);
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}
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