/* mpc_sin_cos -- combined sine and cosine of a complex number.
Copyright (C) 2010, 2011, 2012 INRIA
This file is part of GNU MPC.
GNU MPC 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 3 of the License, or (at your
option) any later version.
GNU MPC 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 this program. If not, see http://www.gnu.org/licenses/ .
*/
#include <stdio.h>
#include "mpc-impl.h"
static int
mpc_sin_cos_nonfinite (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op,
mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos)
/* assumes that op (that is, its real or imaginary part) is not finite */
{
int overlap;
mpc_t op_loc;
overlap = (rop_sin == op || rop_cos == op);
if (overlap) {
mpc_init3 (op_loc, MPC_PREC_RE (op), MPC_PREC_IM (op));
mpc_set (op_loc, op, MPC_RNDNN);
}
else
op_loc [0] = op [0];
if (rop_sin != NULL) {
if (mpfr_nan_p (mpc_realref (op_loc)) || mpfr_nan_p (mpc_imagref (op_loc))) {
mpc_set (rop_sin, op_loc, rnd_sin);
if (mpfr_nan_p (mpc_imagref (op_loc))) {
/* sin(x +i*NaN) = NaN +i*NaN, except for x=0 */
/* sin(-0 +i*NaN) = -0 +i*NaN */
/* sin(+0 +i*NaN) = +0 +i*NaN */
if (!mpfr_zero_p (mpc_realref (op_loc)))
mpfr_set_nan (mpc_realref (rop_sin));
}
else /* op = NaN + i*y */
if (!mpfr_inf_p (mpc_imagref (op_loc)) && !mpfr_zero_p (mpc_imagref (op_loc)))
/* sin(NaN -i*Inf) = NaN -i*Inf */
/* sin(NaN -i*0) = NaN -i*0 */
/* sin(NaN +i*0) = NaN +i*0 */
/* sin(NaN +i*Inf) = NaN +i*Inf */
/* sin(NaN +i*y) = NaN +i*NaN, when 0<|y|<Inf */
mpfr_set_nan (mpc_imagref (rop_sin));
}
else if (mpfr_inf_p (mpc_realref (op_loc))) {
mpfr_set_nan (mpc_realref (rop_sin));
if (!mpfr_inf_p (mpc_imagref (op_loc)) && !mpfr_zero_p (mpc_imagref (op_loc)))
/* sin(+/-Inf +i*y) = NaN +i*NaN, when 0<|y|<Inf */
mpfr_set_nan (mpc_imagref (rop_sin));
else
/* sin(+/-Inf -i*Inf) = NaN -i*Inf */
/* sin(+/-Inf +i*Inf) = NaN +i*Inf */
/* sin(+/-Inf -i*0) = NaN -i*0 */
/* sin(+/-Inf +i*0) = NaN +i*0 */
mpfr_set (mpc_imagref (rop_sin), mpc_imagref (op_loc), MPC_RND_IM (rnd_sin));
}
else if (mpfr_zero_p (mpc_realref (op_loc))) {
/* sin(-0 -i*Inf) = -0 -i*Inf */
/* sin(+0 -i*Inf) = +0 -i*Inf */
/* sin(-0 +i*Inf) = -0 +i*Inf */
/* sin(+0 +i*Inf) = +0 +i*Inf */
mpc_set (rop_sin, op_loc, rnd_sin);
}
else {
/* sin(x -i*Inf) = +Inf*(sin(x) -i*cos(x)) */
/* sin(x +i*Inf) = +Inf*(sin(x) +i*cos(x)) */
mpfr_t s, c;
mpfr_init2 (s, 2);
mpfr_init2 (c, 2);
mpfr_sin_cos (s, c, mpc_realref (op_loc), MPFR_RNDZ);
mpfr_set_inf (mpc_realref (rop_sin), MPFR_SIGN (s));
mpfr_set_inf (mpc_imagref (rop_sin), MPFR_SIGN (c)*MPFR_SIGN (mpc_imagref (op_loc)));
mpfr_clear (s);
mpfr_clear (c);
}
}
if (rop_cos != NULL) {
if (mpfr_nan_p (mpc_realref (op_loc))) {
/* cos(NaN + i * NaN) = NaN + i * NaN */
/* cos(NaN - i * Inf) = +Inf + i * NaN */
/* cos(NaN + i * Inf) = +Inf + i * NaN */
/* cos(NaN - i * 0) = NaN - i * 0 */
/* cos(NaN + i * 0) = NaN + i * 0 */
/* cos(NaN + i * y) = NaN + i * NaN, when y != 0 */
if (mpfr_inf_p (mpc_imagref (op_loc)))
mpfr_set_inf (mpc_realref (rop_cos), +1);
else
mpfr_set_nan (mpc_realref (rop_cos));
if (mpfr_zero_p (mpc_imagref (op_loc)))
mpfr_set (mpc_imagref (rop_cos), mpc_imagref (op_loc), MPC_RND_IM (rnd_cos));
else
mpfr_set_nan (mpc_imagref (rop_cos));
}
else if (mpfr_nan_p (mpc_imagref (op_loc))) {
/* cos(-Inf + i * NaN) = NaN + i * NaN */
/* cos(+Inf + i * NaN) = NaN + i * NaN */
/* cos(-0 + i * NaN) = NaN - i * 0 */
/* cos(+0 + i * NaN) = NaN + i * 0 */
/* cos(x + i * NaN) = NaN + i * NaN, when x != 0 */
if (mpfr_zero_p (mpc_realref (op_loc)))
mpfr_set (mpc_imagref (rop_cos), mpc_realref (op_loc), MPC_RND_IM (rnd_cos));
else
mpfr_set_nan (mpc_imagref (rop_cos));
mpfr_set_nan (mpc_realref (rop_cos));
}
else if (mpfr_inf_p (mpc_realref (op_loc))) {
/* cos(-Inf -i*Inf) = cos(+Inf +i*Inf) = -Inf +i*NaN */
/* cos(-Inf +i*Inf) = cos(+Inf -i*Inf) = +Inf +i*NaN */
/* cos(-Inf -i*0) = cos(+Inf +i*0) = NaN -i*0 */
/* cos(-Inf +i*0) = cos(+Inf -i*0) = NaN +i*0 */
/* cos(-Inf +i*y) = cos(+Inf +i*y) = NaN +i*NaN, when y != 0 */
const int same_sign =
mpfr_signbit (mpc_realref (op_loc)) == mpfr_signbit (mpc_imagref (op_loc));
if (mpfr_inf_p (mpc_imagref (op_loc)))
mpfr_set_inf (mpc_realref (rop_cos), (same_sign ? -1 : +1));
else
mpfr_set_nan (mpc_realref (rop_cos));
if (mpfr_zero_p (mpc_imagref (op_loc)))
mpfr_setsign (mpc_imagref (rop_cos), mpc_imagref (op_loc), same_sign,
MPC_RND_IM(rnd_cos));
else
mpfr_set_nan (mpc_imagref (rop_cos));
}
else if (mpfr_zero_p (mpc_realref (op_loc))) {
/* cos(-0 -i*Inf) = cos(+0 +i*Inf) = +Inf -i*0 */
/* cos(-0 +i*Inf) = cos(+0 -i*Inf) = +Inf +i*0 */
const int same_sign =
mpfr_signbit (mpc_realref (op_loc)) == mpfr_signbit (mpc_imagref (op_loc));
mpfr_setsign (mpc_imagref (rop_cos), mpc_realref (op_loc), same_sign,
MPC_RND_IM (rnd_cos));
mpfr_set_inf (mpc_realref (rop_cos), +1);
}
else {
/* cos(x -i*Inf) = +Inf*cos(x) +i*Inf*sin(x), when x != 0 */
/* cos(x +i*Inf) = +Inf*cos(x) -i*Inf*sin(x), when x != 0 */
mpfr_t s, c;
mpfr_init2 (c, 2);
mpfr_init2 (s, 2);
mpfr_sin_cos (s, c, mpc_realref (op_loc), MPFR_RNDN);
mpfr_set_inf (mpc_realref (rop_cos), mpfr_sgn (c));
mpfr_set_inf (mpc_imagref (rop_cos),
(mpfr_sgn (mpc_imagref (op_loc)) == mpfr_sgn (s) ? -1 : +1));
mpfr_clear (s);
mpfr_clear (c);
}
}
if (overlap)
mpc_clear (op_loc);
return MPC_INEX12 (MPC_INEX (0,0), MPC_INEX (0,0));
/* everything is exact */
}
static int
mpc_sin_cos_real (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op,
mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos)
/* assumes that op is real */
{
int inex_sin_re = 0, inex_cos_re = 0;
/* Until further notice, assume computations exact; in particular,
by definition, for not computed values. */
mpfr_t s, c;
int inex_s, inex_c;
int sign_im = mpfr_signbit (mpc_imagref (op));
/* sin(x +-0*i) = sin(x) +-0*i*sign(cos(x)) */
/* cos(x +-i*0) = cos(x) -+i*0*sign(sin(x)) */
if (rop_sin != 0)
mpfr_init2 (s, MPC_PREC_RE (rop_sin));
else
mpfr_init2 (s, 2); /* We need only the sign. */
if (rop_cos != NULL)
mpfr_init2 (c, MPC_PREC_RE (rop_cos));
else
mpfr_init2 (c, 2);
inex_s = mpfr_sin (s, mpc_realref (op), MPC_RND_RE (rnd_sin));
inex_c = mpfr_cos (c, mpc_realref (op), MPC_RND_RE (rnd_cos));
/* We cannot use mpfr_sin_cos since we may need two distinct rounding
modes and the exact return values. If we need only the sign, an
arbitrary rounding mode will work. */
if (rop_sin != NULL) {
mpfr_set (mpc_realref (rop_sin), s, MPFR_RNDN); /* exact */
inex_sin_re = inex_s;
mpfr_set_zero (mpc_imagref (rop_sin),
( ( sign_im && !mpfr_signbit(c))
|| (!sign_im && mpfr_signbit(c)) ? -1 : 1));
}
if (rop_cos != NULL) {
mpfr_set (mpc_realref (rop_cos), c, MPFR_RNDN); /* exact */
inex_cos_re = inex_c;
mpfr_set_zero (mpc_imagref (rop_cos),
( ( sign_im && mpfr_signbit(s))
|| (!sign_im && !mpfr_signbit(s)) ? -1 : 1));
}
mpfr_clear (s);
mpfr_clear (c);
return MPC_INEX12 (MPC_INEX (inex_sin_re, 0), MPC_INEX (inex_cos_re, 0));
}
static int
mpc_sin_cos_imag (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op,
mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos)
/* assumes that op is purely imaginary, but not zero */
{
int inex_sin_im = 0, inex_cos_re = 0;
/* assume exact if not computed */
int overlap;
mpc_t op_loc;
overlap = (rop_sin == op || rop_cos == op);
if (overlap) {
mpc_init3 (op_loc, MPC_PREC_RE (op), MPC_PREC_IM (op));
mpc_set (op_loc, op, MPC_RNDNN);
}
else
op_loc [0] = op [0];
if (rop_sin != NULL) {
/* sin(+-O +i*y) = +-0 +i*sinh(y) */
mpfr_set (mpc_realref(rop_sin), mpc_realref(op_loc), MPFR_RNDN);
inex_sin_im = mpfr_sinh (mpc_imagref(rop_sin), mpc_imagref(op_loc), MPC_RND_IM(rnd_sin));
}
if (rop_cos != NULL) {
/* cos(-0 - i * y) = cos(+0 + i * y) = cosh(y) - i * 0,
cos(-0 + i * y) = cos(+0 - i * y) = cosh(y) + i * 0,
where y > 0 */
inex_cos_re = mpfr_cosh (mpc_realref (rop_cos), mpc_imagref (op_loc), MPC_RND_RE (rnd_cos));
mpfr_set_ui (mpc_imagref (rop_cos), 0ul, MPC_RND_IM (rnd_cos));
if (mpfr_signbit (mpc_realref (op_loc)) == mpfr_signbit (mpc_imagref (op_loc)))
MPFR_CHANGE_SIGN (mpc_imagref (rop_cos));
}
if (overlap)
mpc_clear (op_loc);
return MPC_INEX12 (MPC_INEX (0, inex_sin_im), MPC_INEX (inex_cos_re, 0));
}
/* Fix an inexact overflow, when x is +Inf or -Inf:
When rnd is towards zero, change x into the largest (in absolute value)
floating-point number.
Return the inexact flag. */
int
mpc_fix_inf (mpfr_t x, mpfr_rnd_t rnd)
{
MPC_ASSERT (mpfr_inf_p (x));
if (!MPC_IS_LIKE_RNDZ(rnd, MPFR_SIGNBIT(x)))
return mpfr_sgn (x);
else
{
if (mpfr_sgn (x) < 0)
mpfr_nextabove (x);
else
mpfr_nextbelow (x);
return -mpfr_sgn (x);
}
}
/* Fix an inexact underflow, when x is +0 or -0:
When rnd is away from zero, change x into the closest floating-point number.
Return the inexact flag. */
int
mpc_fix_zero (mpfr_t x, mpfr_rnd_t rnd)
{
if (!MPC_IS_LIKE_RNDA(rnd, MPFR_SIGNBIT(x)))
return mpfr_signbit (x) == 0 ? -1 : 1;
else
{
if (mpfr_signbit (x) == 0)
{
mpfr_nextabove (x);
return 1;
}
else
{
mpfr_nextbelow (x);
return -1;
}
}
}
int
mpc_sin_cos (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op,
mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos)
/* Feature not documented in the texinfo file: One of rop_sin or
rop_cos may be NULL, in which case it is not computed, and the
corresponding ternary inexact value is set to 0 (exact). */
{
if (!mpc_fin_p (op))
return mpc_sin_cos_nonfinite (rop_sin, rop_cos, op, rnd_sin, rnd_cos);
else if (mpfr_zero_p (mpc_imagref (op)))
return mpc_sin_cos_real (rop_sin, rop_cos, op, rnd_sin, rnd_cos);
else if (mpfr_zero_p (mpc_realref (op)))
return mpc_sin_cos_imag (rop_sin, rop_cos, op, rnd_sin, rnd_cos);
else {
/* let op = a + i*b, then sin(op) = sin(a)*cosh(b) + i*cos(a)*sinh(b)
and cos(op) = cos(a)*cosh(b) - i*sin(a)*sinh(b).
For Re(sin(op)) (and analogously, the other parts), we use the
following algorithm, 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).
*/
mpfr_t s, c, sh, ch, sch, csh;
mpfr_prec_t prec;
int ok;
int inex_re, inex_im, inex_sin, inex_cos, loop = 0;
prec = 2;
if (rop_sin != NULL)
{
mp_prec_t er, ei;
prec = MPC_MAX (prec, MPC_MAX_PREC (rop_sin));
/* since the Taylor expansion of sin(x) at x=0 is x - x^3/6 + O(x^5),
if x <= 2^(-p), then the second term/x is about 2^(-2p)/6, thus we
need at least 2p+3 bits of precision. This is true only when x is
exactly representable in the target precision. */
if (MPC_MAX_PREC (op) <= prec)
{
er = mpfr_get_exp (mpc_realref (op));
ei = mpfr_get_exp (mpc_imagref (op));
/* consider the maximal exponent only */
er = (er < ei) ? ei : er;
if (er < 0)
if (prec < 2 * (mp_prec_t) (-er) + 3)
prec = 2 * (mp_prec_t) (-er) + 3;
}
}
if (rop_cos != NULL)
prec = MPC_MAX (prec, MPC_MAX_PREC (rop_cos));
mpfr_init2 (s, 2);
mpfr_init2 (c, 2);
mpfr_init2 (sh, 2);
mpfr_init2 (ch, 2);
mpfr_init2 (sch, 2);
mpfr_init2 (csh, 2);
do {
loop ++;
ok = 1;
prec += (loop <= 2) ? mpc_ceil_log2 (prec) + 5 : prec / 2;
mpfr_set_prec (s, prec);
mpfr_set_prec (c, prec);
mpfr_set_prec (sh, prec);
mpfr_set_prec (ch, prec);
mpfr_set_prec (sch, prec);
mpfr_set_prec (csh, prec);
mpfr_sin_cos (s, c, mpc_realref(op), MPFR_RNDN);
mpfr_sinh_cosh (sh, ch, mpc_imagref(op), MPFR_RNDN);
if (rop_sin != NULL) {
/* real part of sine */
mpfr_mul (sch, s, ch, MPFR_RNDN);
ok = (!mpfr_number_p (sch))
|| mpfr_can_round (sch, prec - 2, MPFR_RNDN, MPFR_RNDZ,
MPC_PREC_RE (rop_sin)
+ (MPC_RND_RE (rnd_sin) == MPFR_RNDN));
if (ok) {
/* imaginary part of sine */
mpfr_mul (csh, c, sh, MPFR_RNDN);
ok = (!mpfr_number_p (csh))
|| mpfr_can_round (csh, prec - 2, MPFR_RNDN, MPFR_RNDZ,
MPC_PREC_IM (rop_sin)
+ (MPC_RND_IM (rnd_sin) == MPFR_RNDN));
}
}
if (rop_cos != NULL && ok) {
/* real part of cosine */
mpfr_mul (c, c, ch, MPFR_RNDN);
ok = (!mpfr_number_p (c))
|| mpfr_can_round (c, prec - 2, MPFR_RNDN, MPFR_RNDZ,
MPC_PREC_RE (rop_cos)
+ (MPC_RND_RE (rnd_cos) == MPFR_RNDN));
if (ok) {
/* imaginary part of cosine */
mpfr_mul (s, s, sh, MPFR_RNDN);
mpfr_neg (s, s, MPFR_RNDN);
ok = (!mpfr_number_p (s))
|| mpfr_can_round (s, prec - 2, MPFR_RNDN, MPFR_RNDZ,
MPC_PREC_IM (rop_cos)
+ (MPC_RND_IM (rnd_cos) == MPFR_RNDN));
}
}
} while (ok == 0);
if (rop_sin != NULL) {
inex_re = mpfr_set (mpc_realref (rop_sin), sch, MPC_RND_RE (rnd_sin));
if (mpfr_inf_p (sch))
inex_re = mpc_fix_inf (mpc_realref (rop_sin), MPC_RND_RE (rnd_sin));
inex_im = mpfr_set (mpc_imagref (rop_sin), csh, MPC_RND_IM (rnd_sin));
if (mpfr_inf_p (csh))
inex_im = mpc_fix_inf (mpc_imagref (rop_sin), MPC_RND_IM (rnd_sin));
inex_sin = MPC_INEX (inex_re, inex_im);
}
else
inex_sin = MPC_INEX (0,0); /* return exact if not computed */
if (rop_cos != NULL) {
inex_re = mpfr_set (mpc_realref (rop_cos), c, MPC_RND_RE (rnd_cos));
if (mpfr_inf_p (c))
inex_re = mpc_fix_inf (mpc_realref (rop_cos), MPC_RND_RE (rnd_cos));
inex_im = mpfr_set (mpc_imagref (rop_cos), s, MPC_RND_IM (rnd_cos));
if (mpfr_inf_p (s))
inex_im = mpc_fix_inf (mpc_imagref (rop_cos), MPC_RND_IM (rnd_cos));
inex_cos = MPC_INEX (inex_re, inex_im);
}
else
inex_cos = MPC_INEX (0,0); /* return exact if not computed */
mpfr_clear (s);
mpfr_clear (c);
mpfr_clear (sh);
mpfr_clear (ch);
mpfr_clear (sch);
mpfr_clear (csh);
return (MPC_INEX12 (inex_sin, inex_cos));
}
}