264 lines
5.8 KiB
C
264 lines
5.8 KiB
C
/*=============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2012 Sebastian Pancratz
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Copyright (C) 2012 Fredrik Johansson
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******************************************************************************/
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#include "padic.h"
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/*
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Computes the sum
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\begin{equation*}
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(a-1)! x^{1-a} \sum_{i=a}^{b-1} \frac{x^i}{i!}.
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\end{equation*}
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in the rational $(T, Q)$.
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Assumes that $1 \leq a < b$.
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If $a + 1 = b$, sets $P = x$, $Q = a$, and $T = x$.
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If $a + 2 = b$, sets $P = x^2$, $Q = a (a + 1)$, $T = x (a + 1) + x^2$.
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In general, sets
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\begin{align*}
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P & = x^{b-a}, \\
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Q & = \frac{(b-1)!}{(a-1)!}, \\
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T & = (b-1)! x^{1-a} \sum_{i=a}^{b-1} \frac{x^i}{i!}.
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\end{align*}
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*/
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static void
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_padic_exp_bsplit_series(fmpz_t P, fmpz_t Q, fmpz_t T,
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const fmpz_t x, slong a, slong b)
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{
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if (b - a == 1)
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{
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fmpz_set(P, x);
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fmpz_set_ui(Q, a);
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fmpz_set(T, x);
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}
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else if (b - a == 2)
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{
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fmpz_mul(P, x, x);
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fmpz_set_ui(Q, a);
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fmpz_mul_ui(Q, Q, a + 1);
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fmpz_mul_ui(T, x, a + 1);
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fmpz_add(T, T, P);
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}
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else
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{
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const slong m = (a + b) / 2;
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fmpz_t PR, QR, TR;
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fmpz_init(PR);
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fmpz_init(QR);
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fmpz_init(TR);
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_padic_exp_bsplit_series(P, Q, T, x, a, m);
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_padic_exp_bsplit_series(PR, QR, TR, x, m, b);
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fmpz_mul(T, T, QR);
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fmpz_addmul(T, P, TR);
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fmpz_mul(P, P, PR);
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fmpz_mul(Q, Q, QR);
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fmpz_clear(PR);
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fmpz_clear(QR);
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fmpz_clear(TR);
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}
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}
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/*
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Assumes that $x$ is such that $\exp(x)$ converges.
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Assumes that $v = \ord_p(x)$ with $v < N$,
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which also forces $N$ to positive.
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The result $y$ might not be reduced modulo $p^N$.
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Supports aliasing between $x$ and $y$.
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*/
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static void
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_padic_exp_bsplit(fmpz_t y, const fmpz_t x, slong v, const fmpz_t p, slong N)
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{
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const slong n = _padic_exp_bound(v, N, p);
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if (n == 1)
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{
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fmpz_one(y);
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}
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else
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{
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fmpz_t P, Q, T;
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fmpz_init(P);
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fmpz_init(Q);
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fmpz_init(T);
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_padic_exp_bsplit_series(P, Q, T, x, 1, n);
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fmpz_add(T, T, Q); /* (T,Q) := (T,Q) + 1 */
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/* Note exp(x) is a unit so val(T) == val(Q). */
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if (fmpz_remove(T, T, p))
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fmpz_remove(Q, Q, p);
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_padic_inv(Q, Q, p, N);
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fmpz_mul(y, T, Q);
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fmpz_clear(P);
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fmpz_clear(Q);
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fmpz_clear(T);
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}
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}
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void _padic_exp_balanced_2(fmpz_t rop, const fmpz_t xu, slong xv, slong N)
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{
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const fmpz_t p = {WORD(2)};
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fmpz_t r, t;
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slong w;
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fmpz_init(r);
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fmpz_init(t);
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w = 1;
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fmpz_mul_2exp(t, xu, xv);
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fmpz_fdiv_r_2exp(t, t, N);
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fmpz_one(rop);
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while (!fmpz_is_zero(t))
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{
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fmpz_fdiv_r_2exp(r, t, 2*w);
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fmpz_sub(t, t, r);
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if (!fmpz_is_zero(r))
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{
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_padic_exp_bsplit(r, r, w, p, N);
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fmpz_mul(rop, rop, r);
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fmpz_fdiv_r_2exp(rop, rop, N);
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}
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w *= 2;
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}
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fmpz_clear(r);
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fmpz_clear(t);
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}
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void _padic_exp_balanced_p(fmpz_t rop, const fmpz_t xu, slong xv,
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const fmpz_t p, slong N)
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{
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fmpz_t r, t, pw, pN;
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slong w;
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fmpz_init(r);
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fmpz_init(t);
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fmpz_init(pw);
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fmpz_init(pN);
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fmpz_set(pw, p);
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fmpz_pow_ui(pN, p, N);
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w = 1;
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fmpz_pow_ui(t, p, xv);
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fmpz_mul(t, t, xu);
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fmpz_mod(t, t, pN);
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fmpz_one(rop);
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while (!fmpz_is_zero(t))
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{
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fmpz_mul(pw, pw, pw);
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fmpz_fdiv_r(r, t, pw);
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fmpz_sub(t, t, r);
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if (!fmpz_is_zero(r))
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{
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_padic_exp_bsplit(r, r, w, p, N);
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fmpz_mul(rop, rop, r);
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fmpz_mod(rop, rop, pN);
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}
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w *= 2;
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}
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fmpz_clear(r);
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fmpz_clear(t);
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fmpz_clear(pw);
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fmpz_clear(pN);
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}
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/*
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Assumes that the exponential series converges at $x \neq 0$,
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and that $\ord_p(x) < N$.
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Supports aliasing between $x$ and $y$.
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TODO: Take advantage of additional factors of $p$ in $x$.
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*/
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void _padic_exp_balanced(fmpz_t rop, const fmpz_t u, slong v,
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const fmpz_t p, slong N)
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{
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if (fmpz_equal_ui(p, 2))
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_padic_exp_balanced_2(rop, u, v, N);
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else
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_padic_exp_balanced_p(rop, u, v, p, N);
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}
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int padic_exp_balanced(padic_t rop, const padic_t op, const padic_ctx_t ctx)
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{
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const slong N = padic_prec(rop);
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const slong v = padic_val(op);
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const fmpz *p = ctx->p;
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if (padic_is_zero(op))
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{
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padic_one(rop);
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return 1;
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}
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if ((fmpz_equal_ui(p, 2) && v <= 1) || (v <= 0))
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{
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return 0;
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}
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else
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{
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if (v < N)
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{
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_padic_exp_balanced(padic_unit(rop),
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padic_unit(op), padic_val(op), p, N);
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padic_val(rop) = 0;
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}
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else
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{
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padic_one(rop);
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}
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return 1;
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}
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}
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