/* mpc_tan -- tangent of a complex number.
Copyright (C) INRIA, 2008, 2009, 2010, 2011
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 <stdio.h> /* for MPC_ASSERT */
#include <limits.h>
#include "mpc-impl.h"
int
mpc_tan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
{
mpc_t x, y;
mpfr_prec_t prec;
mpfr_exp_t err;
int ok = 0;
int inex;
/* special values */
if (!mpc_fin_p (op))
{
if (mpfr_nan_p (MPC_RE (op)))
{
if (mpfr_inf_p (MPC_IM (op)))
/* tan(NaN -i*Inf) = +/-0 -i */
/* tan(NaN +i*Inf) = +/-0 +i */
{
/* exact unless 1 is not in exponent range */
inex = mpc_set_si_si (rop, 0,
(MPFR_SIGN (MPC_IM (op)) < 0) ? -1 : +1,
rnd);
}
else
/* tan(NaN +i*y) = NaN +i*NaN, when y is finite */
/* tan(NaN +i*NaN) = NaN +i*NaN */
{
mpfr_set_nan (MPC_RE (rop));
mpfr_set_nan (MPC_IM (rop));
inex = MPC_INEX (0, 0); /* always exact */
}
}
else if (mpfr_nan_p (MPC_IM (op)))
{
if (mpfr_cmp_ui (MPC_RE (op), 0) == 0)
/* tan(-0 +i*NaN) = -0 +i*NaN */
/* tan(+0 +i*NaN) = +0 +i*NaN */
{
mpc_set (rop, op, rnd);
inex = MPC_INEX (0, 0); /* always exact */
}
else
/* tan(x +i*NaN) = NaN +i*NaN, when x != 0 */
{
mpfr_set_nan (MPC_RE (rop));
mpfr_set_nan (MPC_IM (rop));
inex = MPC_INEX (0, 0); /* always exact */
}
}
else if (mpfr_inf_p (MPC_RE (op)))
{
if (mpfr_inf_p (MPC_IM (op)))
/* tan(-Inf -i*Inf) = -/+0 -i */
/* tan(-Inf +i*Inf) = -/+0 +i */
/* tan(+Inf -i*Inf) = +/-0 -i */
/* tan(+Inf +i*Inf) = +/-0 +i */
{
const int sign_re = mpfr_signbit (MPC_RE (op));
int inex_im;
mpfr_set_ui (MPC_RE (rop), 0, MPC_RND_RE (rnd));
mpfr_setsign (MPC_RE (rop), MPC_RE (rop), sign_re, GMP_RNDN);
/* exact, unless 1 is not in exponent range */
inex_im = mpfr_set_si (MPC_IM (rop),
mpfr_signbit (MPC_IM (op)) ? -1 : +1,
MPC_RND_IM (rnd));
inex = MPC_INEX (0, inex_im);
}
else
/* tan(-Inf +i*y) = tan(+Inf +i*y) = NaN +i*NaN, when y is
finite */
{
mpfr_set_nan (MPC_RE (rop));
mpfr_set_nan (MPC_IM (rop));
inex = MPC_INEX (0, 0); /* always exact */
}
}
else
/* tan(x -i*Inf) = +0*sin(x)*cos(x) -i, when x is finite */
/* tan(x +i*Inf) = +0*sin(x)*cos(x) +i, when x is finite */
{
mpfr_t c;
mpfr_t s;
int inex_im;
mpfr_init (c);
mpfr_init (s);
mpfr_sin_cos (s, c, MPC_RE (op), GMP_RNDN);
mpfr_set_ui (MPC_RE (rop), 0, MPC_RND_RE (rnd));
mpfr_setsign (MPC_RE (rop), MPC_RE (rop),
mpfr_signbit (c) != mpfr_signbit (s), GMP_RNDN);
/* exact, unless 1 is not in exponent range */
inex_im = mpfr_set_si (MPC_IM (rop),
(mpfr_signbit (MPC_IM (op)) ? -1 : +1),
MPC_RND_IM (rnd));
inex = MPC_INEX (0, inex_im);
mpfr_clear (s);
mpfr_clear (c);
}
return inex;
}
if (mpfr_zero_p (MPC_RE (op)))
/* tan(-0 -i*y) = -0 +i*tanh(y), when y is finite. */
/* tan(+0 +i*y) = +0 +i*tanh(y), when y is finite. */
{
int inex_im;
mpfr_set (MPC_RE (rop), MPC_RE (op), MPC_RND_RE (rnd));
inex_im = mpfr_tanh (MPC_IM (rop), MPC_IM (op), MPC_RND_IM (rnd));
return MPC_INEX (0, inex_im);
}
if (mpfr_zero_p (MPC_IM (op)))
/* tan(x -i*0) = tan(x) -i*0, when x is finite. */
/* tan(x +i*0) = tan(x) +i*0, when x is finite. */
{
int inex_re;
inex_re = mpfr_tan (MPC_RE (rop), MPC_RE (op), MPC_RND_RE (rnd));
mpfr_set (MPC_IM (rop), MPC_IM (op), MPC_RND_IM (rnd));
return MPC_INEX (inex_re, 0);
}
/* ordinary (non-zero) numbers */
/* tan(op) = sin(op) / cos(op).
We use the following algorithm with rounding away from 0 for all
operations, and working precision w:
(1) x = A(sin(op))
(2) y = A(cos(op))
(3) z = A(x/y)
the error on Im(z) is at most 81 ulp,
the error on Re(z) is at most
7 ulp if k < 2,
8 ulp if k = 2,
else 5+k ulp, where
k = Exp(Re(x))+Exp(Re(y))-2min{Exp(Re(y)), Exp(Im(y))}-Exp(Re(x/y))
see proof in algorithms.tex.
*/
prec = MPC_MAX_PREC(rop);
mpc_init2 (x, 2);
mpc_init2 (y, 2);
err = 7;
do
{
mpfr_exp_t k, exr, eyr, eyi, ezr;
ok = 0;
/* FIXME: prevent addition overflow */
prec += mpc_ceil_log2 (prec) + err;
mpc_set_prec (x, prec);
mpc_set_prec (y, prec);
/* rounding away from zero: except in the cases x=0 or y=0 (processed
above), sin x and cos y are never exact, so rounding away from 0 is
rounding towards 0 and adding one ulp to the absolute value */
mpc_sin_cos (x, y, op, MPC_RNDZZ, MPC_RNDZZ);
MPFR_ADD_ONE_ULP (MPC_RE (x));
MPFR_ADD_ONE_ULP (MPC_IM (x));
MPFR_ADD_ONE_ULP (MPC_RE (y));
MPFR_ADD_ONE_ULP (MPC_IM (y));
MPC_ASSERT (mpfr_zero_p (MPC_RE (x)) == 0);
if ( mpfr_inf_p (MPC_RE (x)) || mpfr_inf_p (MPC_IM (x))
|| mpfr_inf_p (MPC_RE (y)) || mpfr_inf_p (MPC_IM (y))) {
/* If the real or imaginary part of x is infinite, it means that
Im(op) was large, in which case the result is
sign(tan(Re(op)))*0 + sign(Im(op))*I,
where sign(tan(Re(op))) = sign(Re(x))*sign(Re(y)). */
int inex_re, inex_im;
mpfr_set_ui (MPC_RE (rop), 0, GMP_RNDN);
if (mpfr_sgn (MPC_RE (x)) * mpfr_sgn (MPC_RE (y)) < 0)
{
mpfr_neg (MPC_RE (rop), MPC_RE (rop), GMP_RNDN);
inex_re = 1;
}
else
inex_re = -1; /* +0 is rounded down */
if (mpfr_sgn (MPC_IM (op)) > 0)
{
mpfr_set_ui (MPC_IM (rop), 1, GMP_RNDN);
inex_im = 1;
}
else
{
mpfr_set_si (MPC_IM (rop), -1, GMP_RNDN);
inex_im = -1;
}
inex = MPC_INEX(inex_re, inex_im);
goto end;
}
exr = mpfr_get_exp (MPC_RE (x));
eyr = mpfr_get_exp (MPC_RE (y));
eyi = mpfr_get_exp (MPC_IM (y));
/* some parts of the quotient may be exact */
inex = mpc_div (x, x, y, MPC_RNDZZ);
/* OP is no pure real nor pure imaginary, so in theory the real and
imaginary parts of its tangent cannot be null. However due to
rouding errors this might happen. Consider for example
tan(1+14*I) = 1.26e-10 + 1.00*I. For small precision sin(op) and
cos(op) differ only by a factor I, thus after mpc_div x = I and
its real part is zero. */
if (mpfr_zero_p (MPC_RE (x)) || mpfr_zero_p (MPC_IM (x)))
{
err = prec; /* double precision */
continue;
}
if (MPC_INEX_RE (inex))
MPFR_ADD_ONE_ULP (MPC_RE (x));
if (MPC_INEX_IM (inex))
MPFR_ADD_ONE_ULP (MPC_IM (x));
MPC_ASSERT (mpfr_zero_p (MPC_RE (x)) == 0);
ezr = mpfr_get_exp (MPC_RE (x));
/* FIXME: compute
k = Exp(Re(x))+Exp(Re(y))-2min{Exp(Re(y)), Exp(Im(y))}-Exp(Re(x/y))
avoiding overflow */
k = exr - ezr + MPC_MAX(-eyr, eyr - 2 * eyi);
err = k < 2 ? 7 : (k == 2 ? 8 : (5 + k));
/* Can the real part be rounded? */
ok = (!mpfr_number_p (MPC_RE (x)))
|| mpfr_can_round (MPC_RE(x), prec - err, GMP_RNDN, GMP_RNDZ,
MPC_PREC_RE(rop) + (MPC_RND_RE(rnd) == GMP_RNDN));
if (ok)
{
/* Can the imaginary part be rounded? */
ok = (!mpfr_number_p (MPC_IM (x)))
|| mpfr_can_round (MPC_IM(x), prec - 6, GMP_RNDN, GMP_RNDZ,
MPC_PREC_IM(rop) + (MPC_RND_IM(rnd) == GMP_RNDN));
}
}
while (ok == 0);
inex = mpc_set (rop, x, rnd);
end:
mpc_clear (x);
mpc_clear (y);
return inex;
}