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/* mpc_sin -- sine of a complex number.
Copyright (C) 2007 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_sin (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
{
mpfr_t x, y, z;
mp_prec_t prec;
int ok = 0;
/* let op = a + i*b, then sin(op) = sin(a)*cosh(b) + i*cos(a)*sinh(b).
We use the following algorithm (same for the imaginary part),
with rounding to nearest for all operations, and working precision w:
(1) x = o(sin(a))
(2) y = o(cosh(b))
(3) r = o(x*y)
then the error on r is at most 4 ulps, since we can write
r = sin(a)*cosh(b)*(1+t)^3 with |t| <= 2^(-w),
thus for w >= 2, r = sin(a)*cosh(b)*(1+4*t) with |t| <= 2^(-w),
thus the relative error is bounded by 4*2^(-w) <= 4*ulp(r).
*/
/* special case when the input is real: sin(x) = sin(x) */
if (mpfr_cmp_ui (MPC_IM(op), 0) == 0)
{
mpfr_sin (MPC_RE(rop), MPC_RE(op), MPC_RND_RE(rnd));
mpfr_set_ui (MPC_IM(rop), 0, MPC_RND_IM(rnd));
return;
}
/* special case when the input is imaginary: sin(I*y) = sinh(y)*I */
if (mpfr_cmp_ui (MPC_RE(op), 0) == 0)
{
mpfr_set_ui (MPC_RE(rop), 0, MPC_RND_RE(rnd));
mpfr_sinh (MPC_IM(rop), MPC_IM(op), MPC_RND_IM(rnd));
return;
}
prec = MPC_MAX_PREC(rop);
mpfr_init2 (x, 2);
mpfr_init2 (y, 2);
mpfr_init2 (z, 2);
do
{
prec += mpc_ceil_log2 (prec) + 5;
mpfr_set_prec (x, prec);
mpfr_set_prec (y, prec);
mpfr_set_prec (z, prec);
mpfr_sin_cos (x, y, MPC_RE(op), GMP_RNDN);
mpfr_cosh (z, MPC_IM(op), GMP_RNDN);
mpfr_mul (x, x, z, GMP_RNDN);
ok = mpfr_can_round (x, prec - 2, GMP_RNDN, MPC_RND_RE(rnd),
MPFR_PREC(MPC_RE(rop)));
if (ok) /* compute imaginary part */
{
mpfr_sinh (z, MPC_IM(op), GMP_RNDN);
mpfr_mul (y, y, z, GMP_RNDN);
ok = mpfr_can_round (y, prec - 2, GMP_RNDN, MPC_RND_IM(rnd),
MPFR_PREC(MPC_IM(rop)));
}
}
while (ok == 0);
mpfr_set (MPC_RE(rop), x, MPC_RND_RE(rnd));
mpfr_set (MPC_IM(rop), y, MPC_RND_IM(rnd));
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (z);
}
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