mirror of
https://github.com/vale981/arb
synced 2025-03-05 09:21:38 -05:00
6bf072eb59
This will allow us to not loose the julia session on error. See also https://github.com/wbhart/flint2/pull/243
262 lines
6.9 KiB
C
262 lines
6.9 KiB
C
/*
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Copyright (C) 2017 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 "acb_hypgeom.h"
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static void
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bsplit(acb_ptr VA, const acb_t z, const acb_t z2,
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const acb_t a, const acb_t a1a, slong k, slong h, slong prec)
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{
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if (h - k == 1)
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{
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acb_zero(VA + 0);
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acb_mul_ui(VA + 1, a1a, (k+1)*(k+2), prec);
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acb_mul_si(VA + 2, z2, -k*k, prec);
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acb_mul_ui(VA + 3, a, (k+1)*(2*k+1), prec);
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acb_sub_ui(VA + 3, VA + 3, (k+1)*(k+1), prec);
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acb_mul(VA + 3, VA + 3, z, prec);
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acb_neg(VA + 3, VA + 3);
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acb_set(VA + 4, VA + 1);
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acb_zero(VA + 5);
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acb_set(VA + 6, VA + 1);
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}
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else
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{
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slong m;
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acb_ptr VB;
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if (h <= k) flint_abort();
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m = k + (h - k) / 2;
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VB = _acb_vec_init(7);
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bsplit(VA, z, z2, a, a1a, k, m, prec);
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bsplit(VB, z, z2, a, a1a, m, h, prec);
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acb_mul(VA + 6, VA + 6, VB + 6, prec);
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acb_mul(VA + 4, VA + 4, VB + 6, prec);
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acb_addmul(VA + 4, VA + 0, VB + 4, prec);
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acb_addmul(VA + 4, VA + 2, VB + 5, prec);
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acb_mul(VA + 5, VA + 5, VB + 6, prec);
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acb_addmul(VA + 5, VA + 1, VB + 4, prec);
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acb_addmul(VA + 5, VA + 3, VB + 5, prec);
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acb_set(VB + 6, VA + 3);
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acb_mul(VA + 3, VA + 3, VB + 3, prec);
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acb_addmul(VA + 3, VA + 1, VB + 2, prec);
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acb_set(VB + 5, VA + 2);
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acb_mul(VA + 2, VA + 2, VB + 3, prec);
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acb_addmul(VA + 2, VA + 0, VB + 2, prec);
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acb_mul(VA + 1, VA + 1, VB + 0, prec);
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acb_addmul(VA + 1, VB + 6, VB + 1, prec);
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acb_mul(VA + 0, VA + 0, VB + 0, prec);
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acb_addmul(VA + 0, VB + 5, VB + 1, prec);
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_acb_vec_clear(VB, 7);
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}
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}
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/*
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Some possible approaches to bounding the Taylor coefficients c_k at
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the expansion point a:
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1. Inspection of the symbolic derivatives gives the trivial bound
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|c_k| <= (1+|log(1-a)|) / min(|a|,|a-1|)^k
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which is good enough when not close to 0.
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2. Using Cauchy's integral formula, some explicit computation gives
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|c_k| <= 4/|1-a|^k when |a| <= 1/2 or a = +/- i. The constant
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could certainly be improved.
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3. For k >= 1, c_k = 2F1(k,k,k+1,a) / k^2. Can we use monotonicity to get
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good estimates here when a is complex? Note that 2F1(k,k,k,a) = (1-a)^-k.
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*/
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void
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acb_hypgeom_dilog_continuation(acb_t res, const acb_t a, const acb_t z, slong prec)
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{
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acb_t za, a1, a1a, za2, log1a;
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acb_ptr V;
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slong n;
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double tr;
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mag_t tm, err, am;
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int real;
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if (acb_is_zero(a))
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{
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acb_hypgeom_dilog_zero_taylor(res, z, prec);
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return;
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}
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if (acb_eq(a, z))
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{
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acb_zero(res);
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return;
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}
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acb_init(za);
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acb_init(a1);
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acb_init(a1a);
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acb_init(za2);
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acb_init(log1a);
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mag_init(tm);
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mag_init(err);
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mag_init(am);
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acb_sub(za, z, a, prec); /* z-a */
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acb_sub_ui(a1, a, 1, prec); /* a-1 */
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acb_mul(a1a, a1, a, prec); /* (a-1)a */
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acb_mul(za2, za, za, prec); /* (z-a)^2 */
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acb_neg(log1a, a1);
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acb_log(log1a, log1a, prec); /* log(1-a) */
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acb_get_mag(am, a);
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if (mag_cmp_2exp_si(am, -1) <= 0 ||
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(acb_is_exact(a) && arb_is_zero(acb_realref(a)) &&
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arf_cmpabs_ui(arb_midref(acb_imagref(a)), 1) == 0))
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{
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acb_get_mag_lower(am, a1);
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acb_get_mag(tm, za);
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mag_div(tm, tm, am); /* tm = ratio */
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mag_set_ui(am, 4); /* am = prefactor */
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}
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else
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{
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acb_get_mag_lower(am, a);
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acb_get_mag_lower(tm, a1);
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mag_min(am, am, tm);
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acb_get_mag(tm, za);
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mag_div(tm, tm, am); /* tm = ratio */
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acb_get_mag(am, log1a);
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mag_add_ui(am, am, 1); /* am = prefactor */
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}
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tr = mag_get_d_log2_approx(tm);
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if (tr < -0.1)
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{
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arf_srcptr rr, ii;
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rr = arb_midref(acb_realref(z));
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ii = arb_midref(acb_imagref(z));
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if (arf_cmpabs(ii, rr) > 0)
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rr = ii;
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/* target relative accuracy near 0 */
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if (arf_cmpabs_2exp_si(rr, -2) < 0 && arf_cmpabs_2exp_si(rr, -prec) > 0)
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n = (prec - arf_abs_bound_lt_2exp_si(rr)) / (-tr) + 1;
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else
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n = prec / (-tr) + 1;
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n = FLINT_MAX(n, 2);
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}
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else
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{
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n = 2;
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}
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mag_geom_series(err, tm, n);
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mag_mul(err, err, am);
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real = acb_is_real(a) && acb_is_real(z) &&
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arb_is_negative(acb_realref(a1)) &&
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mag_is_finite(err);
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if (n < 10)
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{
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/* forward recurrence - faster for small n and/or low precision, but
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must be avoided for large n since complex intervals blow up */
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acb_t s, t, u, v;
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slong k;
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acb_init(s);
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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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acb_div(u, log1a, a, prec);
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acb_neg(u, u);
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if (n >= 3)
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{
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acb_inv(v, a1, prec);
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acb_add(v, v, u, prec);
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acb_div(v, v, a, prec);
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acb_mul_2exp_si(v, v, -1);
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acb_neg(v, v);
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acb_mul(v, v, za2, prec);
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}
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acb_mul(u, u, za, prec);
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acb_add(s, u, v, prec);
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for (k = 3; k < n; k++)
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{
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acb_mul_ui(u, u, (k - 2) * (k - 2), prec);
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acb_mul(u, u, za2, prec);
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acb_mul_ui(t, a, (k - 1) * (2 * k - 3), prec);
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acb_sub_ui(t, t, (k - 1) * (k - 1), prec);
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acb_mul(t, t, v, prec);
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acb_addmul(u, t, za, prec);
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acb_mul_ui(t, a1a, (k - 1) * k, prec);
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acb_neg(t, t);
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acb_div(u, u, t, prec);
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acb_add(s, s, u, prec);
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acb_swap(v, u);
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}
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acb_set(res, s);
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acb_clear(s);
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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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else
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{
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/* binary splitting */
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V = _acb_vec_init(7);
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bsplit(V, za, za2, a, a1a, 1, 1 + n, prec);
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acb_mul(V + 1, V + 4, log1a, prec);
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acb_neg(V + 1, V + 1);
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acb_mul(V + 2, V + 5, za2, prec);
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acb_mul_2exp_si(V + 2, V + 2, -1);
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acb_mul(V + 1, V + 1, za, prec);
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acb_div(V + 3, V + 2, a1, prec);
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acb_sub(V + 1, V + 1, V + 3, prec);
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acb_div(V + 0, log1a, a, prec);
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acb_addmul(V + 1, V + 2, V + 0, prec);
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acb_mul(V + 6, V + 6, a, prec);
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acb_div(V + 0, V + 1, V + 6, prec);
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acb_set(res, V + 0);
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_acb_vec_clear(V, 7);
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}
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if (real)
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arb_add_error_mag(acb_realref(res), err);
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else
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acb_add_error_mag(res, err);
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acb_clear(za);
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acb_clear(a1);
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acb_clear(a1a);
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acb_clear(za2);
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acb_clear(log1a);
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mag_clear(tm);
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mag_clear(err);
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mag_clear(am);
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}
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