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206 lines
6.7 KiB
C
206 lines
6.7 KiB
C
/* mpfr_j0, mpfr_j1, mpfr_jn -- Bessel functions of 1st kind, integer order.
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http://www.opengroup.org/onlinepubs/009695399/functions/j0.html
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Copyright 2007 Free Software Foundation, Inc.
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Contributed by the Arenaire and Cacao projects, INRIA.
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This file is part of the MPFR Library.
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The MPFR Library is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License as published by
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the Free Software Foundation; either version 2.1 of the License, or (at your
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option) any later version.
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The MPFR Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with the MPFR Library; see the file COPYING.LIB. If not, write to
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the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston,
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MA 02110-1301, USA. */
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#define MPFR_NEED_LONGLONG_H
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#include "mpfr-impl.h"
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/* Relations: j(-n,z) = (-1)^n j(n,z) */
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static int mpfr_jn_asympt (mpfr_ptr, long, mpfr_srcptr, mp_rnd_t);
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int
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mpfr_j0 (mpfr_ptr res, mpfr_srcptr z, mp_rnd_t r)
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{
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return mpfr_jn (res, 0, z, r);
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}
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int
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mpfr_j1 (mpfr_ptr res, mpfr_srcptr z, mp_rnd_t r)
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{
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return mpfr_jn (res, 1, z, r);
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}
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int
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mpfr_jn (mpfr_ptr res, long n, mpfr_srcptr z, mp_rnd_t r)
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{
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int inex;
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unsigned long absn;
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mp_prec_t prec, err;
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mp_exp_t exps, expT;
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mpfr_t y, s, t;
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unsigned long k, zz;
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MPFR_ZIV_DECL (loop);
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MPFR_LOG_FUNC (("x[%#R]=%R n=%d rnd=%d", z, z, n, r),
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("y[%#R]=%R", res, res));
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absn = SAFE_ABS (unsigned long, n);
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if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (z)))
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{
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if (MPFR_IS_NAN (z))
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{
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MPFR_SET_NAN (res);
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MPFR_RET_NAN;
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}
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/* j(n,z) tends to zero when z goes to +Inf or -Inf, oscillating around
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0. We choose to return +0 in that case. */
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else if (MPFR_IS_INF (z)) /* FIXME: according to j(-n,z) = (-1)^n j(n,z)
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we might want to give a sign depending on
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z and n */
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return mpfr_set_ui (res, 0, r);
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else /* z=0: j(0,0)=1, j(n odd,+/-0) = +/-0 if n > 0, -/+0 if n < 0,
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j(n even,+/-0) = +0 */
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{
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if (n == 0)
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return mpfr_set_ui (res, 1, r);
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else if (absn & 1) /* n odd */
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return (n > 0) ? mpfr_set (res, z, r) : mpfr_neg (res, z, r);
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else /* n even */
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return mpfr_set_ui (res, 0, r);
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}
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}
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/* check for tiny input for j0: j0(z) = 1 - z^2/4 + ..., more precisely
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|j0(z) - 1| <= z^2/4 for -1 <= z <= 1. */
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if (n == 0)
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MPFR_FAST_COMPUTE_IF_SMALL_INPUT (res, __gmpfr_one, -2 * MPFR_GET_EXP (z),
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2, 0, r, return _inexact);
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/* idem for j1: j1(z) = z/2 - z^3/16 + ..., more precisely
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|j1(z) - z/2| <= |z^3|/16 for -1 <= z <= 1, with the sign of j1(z) - z/2
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being the opposite of that of z. */
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if (n == 1)
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/* we first compute 2j1(z) = z - z^3/8 + ..., then divide by 2 using
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the "extra" argument of MPFR_FAST_COMPUTE_IF_SMALL_INPUT. */
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MPFR_FAST_COMPUTE_IF_SMALL_INPUT (res, z, -2 * MPFR_GET_EXP (z), 3,
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0, r, mpfr_div_2ui (res, res, 1, r));
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/* we can use the asymptotic expansion as soon as |z| > p log(2)/2,
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but to get some margin we use it for |z| > p/2 */
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if (mpfr_cmp_ui (z, MPFR_PREC(res) / 2 + 3) > 0 ||
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mpfr_cmp_si (z, - ((long) MPFR_PREC(res) / 2 + 3)) < 0)
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{
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inex = mpfr_jn_asympt (res, n, z, r);
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if (inex != 0)
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return inex;
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}
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mpfr_init2 (y, 32);
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/* check underflow case: |j(n,z)| <= 1/sqrt(2 Pi n) (ze/2n)^n
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(see algorithms.tex) */
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if (absn > 0)
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{
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/* the following is an upper 32-bit approximation of exp(1)/2 */
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mpfr_set_str_binary (y, "1.0101101111110000101010001011001");
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if (MPFR_SIGN(z) > 0)
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mpfr_mul (y, y, z, GMP_RNDU);
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else
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{
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mpfr_mul (y, y, z, GMP_RNDD);
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mpfr_neg (y, y, GMP_RNDU);
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}
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mpfr_div_ui (y, y, absn, GMP_RNDU);
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/* now y is an upper approximation of |ze/2n|: y < 2^EXP(y),
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thus |j(n,z)| < 1/2*y^n < 2^(n*EXP(y)-1).
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If n*EXP(y) < __gmpfr_emin then we have an underflow.
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Warning: absn is an unsigned long. */
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if ((MPFR_EXP(y) < 0 && absn > (unsigned long) (-__gmpfr_emin))
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|| (absn <= (unsigned long) (-MPFR_EMIN_MIN) &&
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MPFR_EXP(y) < __gmpfr_emin / (mp_exp_t) absn))
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{
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mpfr_clear (y);
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return mpfr_underflow (res, (r == GMP_RNDN) ? GMP_RNDZ : r,
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(n % 2) ? ((n > 0) ? MPFR_SIGN(z) : -MPFR_SIGN(z))
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: MPFR_SIGN_POS);
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}
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}
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mpfr_init (s);
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mpfr_init (t);
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prec = MPFR_PREC (res) + MPFR_INT_CEIL_LOG2 (MPFR_PREC (res)) + 3;
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MPFR_ZIV_INIT (loop, prec);
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for (;;)
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{
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mpfr_set_prec (y, prec);
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mpfr_set_prec (s, prec);
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mpfr_set_prec (t, prec);
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mpfr_pow_ui (t, z, absn, GMP_RNDN); /* z^|n| */
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mpfr_mul (y, z, z, GMP_RNDN); /* z^2 */
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zz = mpfr_get_ui (y, GMP_RNDU);
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MPFR_ASSERTN (zz < ULONG_MAX);
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mpfr_div_2ui (y, y, 2, GMP_RNDN); /* z^2/4 */
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mpfr_fac_ui (s, absn, GMP_RNDN); /* |n|! */
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mpfr_div (t, t, s, GMP_RNDN);
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if (absn > 0)
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mpfr_div_2ui (t, t, absn, GMP_RNDN);
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mpfr_set (s, t, GMP_RNDN);
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exps = MPFR_EXP (s);
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expT = exps;
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for (k = 1; ; k++)
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{
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mpfr_mul (t, t, y, GMP_RNDN);
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mpfr_neg (t, t, GMP_RNDN);
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if (k + absn <= ULONG_MAX / k)
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mpfr_div_ui (t, t, k * (k + absn), GMP_RNDN);
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else
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{
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mpfr_div_ui (t, t, k, GMP_RNDN);
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mpfr_div_ui (t, t, k + absn, GMP_RNDN);
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}
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exps = MPFR_EXP (t);
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if (exps > expT)
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expT = exps;
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mpfr_add (s, s, t, GMP_RNDN);
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exps = MPFR_EXP (s);
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if (exps > expT)
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expT = exps;
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if (MPFR_EXP (t) + (mp_exp_t) prec <= MPFR_EXP (s) &&
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zz / (2 * k) < k + n)
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break;
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}
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/* the error is bounded by (4k^2+21/2k+7) ulp(s)*2^(expT-exps)
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<= (k+2)^2 ulp(s)*2^(2+expT-exps) */
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err = 2 * MPFR_INT_CEIL_LOG2(k + 2) + 2 + expT - MPFR_EXP (s);
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if (MPFR_LIKELY (MPFR_CAN_ROUND (s, prec - err, MPFR_PREC(res), r)))
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break;
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MPFR_ZIV_NEXT (loop, prec);
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}
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MPFR_ZIV_FREE (loop);
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inex = ((n >= 0) || ((n & 1) == 0)) ? mpfr_set (res, s, r)
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: mpfr_neg (res, s, r);
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mpfr_clear (y);
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mpfr_clear (s);
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mpfr_clear (t);
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return inex;
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}
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#define MPFR_JN
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#include "jyn_asympt.c"
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