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/* mpc_exp -- exponential of a complex number.
Copyright (C) 2002 Andreas Enge, Paul Zimmermann
This file is part of the MPC Library.
The MPC 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 2.1 of the License, or (at your
option) any later version.
The MPC 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 MPC Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include "gmp.h"
#include "mpfr.h"
#include "mpc.h"
#include "mpc-impl.h"
void
mpc_exp (mpc_ptr rop, mpc_srcptr op, mp_rnd_t rnd)
{
mpfr_t x, y, z;
mp_prec_t prec;
int ok = 0;
/* let op = a + i*b, then exp(op) = exp(a)*[cos(b) + i*sin(b)]
= exp(a)*cos(b) + i*exp(a)*sin(b).
We use the following algorithm (same for the imaginary part):
(1) x = o(exp(a)) rounded towards +infinity:
(2) y = o(cos(b)) rounded to nearest
(3) r = o(x*y)
then the error on r for the real part is at most 4 ulps:
|r - exp(a)*cos(b)| <= ulp(r) + |x*y - exp(a)*cos(b)|
<= ulp(r) + |x*y - exp(a)*y| + exp(a) * |y - cos(b)|
<= ulp(r) + |y| ulp(x) + 1/2 * x * ulp(y)
<= ulp(r) + 2 * ulp(x*y) + ulp(x*y) [Rule 4]
<= 4 * ulp(r) [Rule 8]
*/
/* special case when the input is real */
if (mpfr_cmp_ui (MPC_IM(op), 0) == 0)
{
mpfr_exp (MPC_RE(rop), MPC_RE(op), MPC_RND_RE(rnd));
mpfr_set_ui (MPC_IM(rop), 0, MPC_RND_IM(rnd));
return;
}
prec = MPC_MAX_PREC(rop);
mpfr_init2 (x, 2);
mpfr_init2 (y, 2);
mpfr_init2 (z, 2);
do
{
prec += _mpfr_ceil_log2 ((double) prec) + 5;
mpfr_set_prec (x, prec);
mpfr_set_prec (y, prec);
mpfr_set_prec (z, prec);
mpfr_exp (x, MPC_RE(op), GMP_RNDU);
mpfr_sin_cos (z, y, MPC_IM(op), GMP_RNDN);
mpfr_mul (y, y, x, GMP_RNDN);
ok = mpfr_can_round (y, prec - 2, GMP_RNDN, MPC_RND_RE(rnd),
MPFR_PREC(MPC_RE(rop)));
if (ok) /* compute imaginary part */
{
mpfr_mul (z, z, x, GMP_RNDN);
ok = mpfr_can_round (z, prec - 2, GMP_RNDN, MPC_RND_IM(rnd),
MPFR_PREC(MPC_IM(rop)));
}
}
while (ok == 0);
mpfr_set (MPC_RE(rop), y, MPC_RND_RE(rnd));
mpfr_set (MPC_IM(rop), z, MPC_RND_IM(rnd));
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (z);
}
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