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/* mpc_atan -- arctangent of a complex number.
Copyright (C) 2009 Philippe Th\'eveny
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 "mpc-impl.h"
/* set rop to
-pi/2 if s < 0
+pi/2 else
rounded in the direction rnd
*/
static int
set_pi_over_2 (mpfr_ptr rop, int s, mpfr_rnd_t rnd)
{
int inex;
inex = mpfr_const_pi (rop, s ? INV_RND (rnd) : rnd);
mpfr_div_2ui (rop, rop, 1, GMP_RNDN);
if (s)
{
inex = -inex;
mpfr_neg (rop, rop, GMP_RNDN);
}
return inex;
}
int
mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
{
int s_re;
int s_im;
s_re = mpfr_signbit (MPC_RE (op));
s_im = mpfr_signbit (MPC_IM (op));
/* special values */
if (mpfr_nan_p (MPC_RE (op)) || mpfr_nan_p (MPC_IM (op)))
{
int inex_re;
inex_re = 0;
if (mpfr_nan_p (MPC_RE (op)))
{
mpfr_set_nan (MPC_RE (rop));
if (mpfr_zero_p (MPC_IM (op)) || mpfr_inf_p (MPC_IM (op)))
{
mpfr_set_ui (MPC_IM (rop), 0, GMP_RNDN);
if (s_im)
mpc_conj (rop, rop, MPC_RNDNN);
}
else
mpfr_set_nan (MPC_IM (rop));
}
else
{
if (mpfr_inf_p (MPC_RE (op)))
{
inex_re = set_pi_over_2 (MPC_RE (rop), s_re, MPC_RND_RE (rnd));
mpfr_set_ui (MPC_IM (rop), 0, GMP_RNDN);
}
else
{
mpfr_set_nan (MPC_RE (rop));
mpfr_set_nan (MPC_IM (rop));
}
}
return MPC_INEX (inex_re, 0);
}
if (mpfr_inf_p (MPC_RE (op)) || mpfr_inf_p (MPC_IM (op)))
{
int inex_re;
inex_re = set_pi_over_2 (MPC_RE (rop), s_re, MPC_RND_RE (rnd));
mpfr_set_ui (MPC_IM (rop), 0, GMP_RNDN);
if (s_im)
mpc_conj (rop, rop, GMP_RNDN);
return MPC_INEX (inex_re, 0);
}
/* pure real argument */
if (mpfr_zero_p (MPC_IM (op)))
{
int inex_re;
inex_re = mpfr_atan (MPC_RE (rop), MPC_RE (op), MPC_RND_RE (rnd));
mpfr_set_ui (MPC_IM (rop), 0, GMP_RNDN);
if (s_im)
mpc_conj (rop, rop, GMP_RNDN);
return MPC_INEX (inex_re, 0);
}
/* pure imaginary argument */
if (mpfr_zero_p (MPC_RE (op)))
{
int inex_re;
int inex_im;
int cmp_1;
inex_re = 0;
inex_im = 0;
if (s_im)
cmp_1 = -mpfr_cmp_si (MPC_IM (op), -1);
else
cmp_1 = mpfr_cmp_ui (MPC_IM (op), +1);
if (cmp_1 < 0)
{
/* atan(+0+iy) = +0 +i*atanh(y), if |y| < 1
atan(-0+iy) = -0 +i*atanh(y), if |y| < 1 */
mpfr_set_ui (MPC_RE (rop), 0, GMP_RNDN);
if (s_re)
mpfr_neg (MPC_RE (rop), MPC_RE (rop), GMP_RNDN);
inex_im = mpfr_atanh (MPC_IM (rop), MPC_IM (op), MPC_RND_IM (rnd));
}
else if (cmp_1 == 0)
{
/* atan(+/-0+i) = NaN +i*inf
atan(+/-0-i) = NaN -i*inf */
mpfr_set_nan (MPC_RE (rop));
mpfr_set_inf (MPC_IM (rop), s_im ? -1 : +1);
}
else
{
/* atan(+0+iy) = +pi/2 +i*atanh(1/y), if |y| > 1
atan(-0+iy) = -pi/2 +i*atanh(1/y), if |y| > 1 */
mp_rnd_t rnd_im;
mpfr_t y;
mp_prec_t p, p_im;
mp_prec_t err;
int ok;
int cmp_2;
rnd_im = MPC_RND_IM (rnd);
mpfr_init (y);
p_im = mpfr_get_prec (MPC_IM (op));
p = p_im;
/* a = o(1/y) with error(a) < 1 ulp(a)
b = o(atanh(a)) with error(b) < (1+2^{1+Exp(a)-Exp(b)}) ulp(b)
* if 1 < |y| <= 2 then Exp(a) = 0 and Exp(b) >= 0, so, at most, 2
bits of precision are lost.
* if |y| > 2, no simplification. */
if (s_im)
cmp_2 = -mpfr_cmp_si (MPC_IM (op), -2);
else
cmp_2 = mpfr_cmp_ui (MPC_IM (op), +2);
err = 2;
do
{
p += mpc_ceil_log2 (p) + err;
mpfr_set_prec (y, p);
inex_im = mpfr_ui_div (y, 1, MPC_IM (op), GMP_RNDZ);
if (mpfr_zero_p (y))
break; /* underflow. */
/* FIXME: should we consider the case with unreasonably huge
precision prec(y)>3*exp_min, where atanh(1/Im(op)) could be
representable while 1/Im(op) underflows ? */
if (cmp_2 > 0)
err = 2 + (mp_prec_t)MPC_MAX (MPFR_EXP (y), 0);
/* atanh cannot underflow: |atanh(x)| > |x| for |x| < 1 */
inex_im |= mpfr_atanh (y, y, GMP_RNDZ);
if (cmp_2 > 0)
err += MPC_MAX (SAFE_ABS (mp_prec_t, MPFR_EXP (y)), 0);
ok = inex_im == 0
|| mpfr_can_round (y, p-err, GMP_RNDZ, rnd_im,
p_im + (rnd_im == GMP_RNDN));
} while (ok == 0);
inex_re = set_pi_over_2 (MPC_RE (rop), s_re, MPC_RND_RE (rnd));
inex_im = mpfr_set (MPC_IM (rop), y, rnd_im);
mpfr_clear (y);
}
return MPC_INEX (inex_re, inex_im);
}
/* regular number argument */
/* TODO */
mpfr_set_nan (MPC_RE (rop));
mpfr_set_nan (MPC_IM (rop));
return 0;
}
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