/* mpc_asin -- arcsine of a complex number.
Copyright (C) 2009, 2010, 2011, 2012, 2013, 2014 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"
/* Special case op = 1 + i*y for tiny y (see algorithms.tex).
Return 0 if special formula fails, otherwise put in z1 the approximate
value which needs to be converted to rop.
z1 is a temporary variable with enough precision.
*/
static int
mpc_asin_special (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd, mpc_ptr z1)
{
mpfr_exp_t ey = mpfr_get_exp (mpc_imagref (op));
mpfr_t abs_y;
mpfr_prec_t p;
int inex;
/* |Re(asin(1+i*y)) - pi/2| <= y^(1/2) */
if (ey >= 0 || ((-ey) / 2 < mpfr_get_prec (mpc_realref (z1))))
return 0;
mpfr_const_pi (mpc_realref (z1), MPFR_RNDN);
mpfr_div_2exp (mpc_realref (z1), mpc_realref (z1), 1, MPFR_RNDN); /* exact */
p = mpfr_get_prec (mpc_realref (z1));
/* if z1 has precision p, the error on z1 is 1/2*ulp(z1) = 2^(-p) so far,
and since ey <= -2p, then y^(1/2) <= 1/2*ulp(z1) too, thus the total
error is bounded by ulp(z1) */
if (!mpfr_can_round (mpc_realref(z1), p, MPFR_RNDN, MPFR_RNDZ,
mpfr_get_prec (mpc_realref(rop)) +
(MPC_RND_RE(rnd) == MPFR_RNDN)))
return 0;
/* |Im(asin(1+i*y)) - y^(1/2)| <= (1/12) * y^(3/2) for y >= 0 (err >= 0)
|Im(asin(1-i*y)) + y^(1/2)| <= (1/12) * y^(3/2) for y >= 0 (err <= 0) */
abs_y[0] = mpc_imagref (op)[0];
if (mpfr_signbit (mpc_imagref (op)))
MPFR_CHANGE_SIGN (abs_y);
inex = mpfr_sqrt (mpc_imagref (z1), abs_y, MPFR_RNDN); /* error <= 1/2 ulp */
if (mpfr_signbit (mpc_imagref (op)))
MPFR_CHANGE_SIGN (mpc_imagref (z1));
/* If z1 has precision p, the error on z1 is 1/2*ulp(z1) = 2^(-p) so far,
and (1/12) * y^(3/2) <= (1/8) * y * y^(1/2) <= 2^(ey-3)*2^p*ulp(y^(1/2))
thus for p+ey-3 <= -1 we have (1/12) * y^(3/2) <= (1/2) * ulp(y^(1/2)),
and the total error is bounded by ulp(z1).
Note: if y^(1/2) is exactly representable, and ends with many zeroes,
then mpfr_can_round below will fail; however in that case we know that
Im(asin(1+i*y)) is away from +/-y^(1/2) wrt zero. */
if (inex == 0) /* enlarge im(z1) so that the inexact flag is correct */
{
if (mpfr_signbit (mpc_imagref (op)))
mpfr_nextbelow (mpc_imagref (z1));
else
mpfr_nextabove (mpc_imagref (z1));
return 1;
}
p = mpfr_get_prec (mpc_imagref (z1));
if (!mpfr_can_round (mpc_imagref(z1), p - 1, MPFR_RNDA, MPFR_RNDZ,
mpfr_get_prec (mpc_imagref(rop)) +
(MPC_RND_IM(rnd) == MPFR_RNDN)))
return 0;
return 1;
}
int
mpc_asin (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
{
mpfr_prec_t p, p_re, p_im;
mpfr_rnd_t rnd_re, rnd_im;
mpc_t z1;
int inex, loop = 0;
/* special values */
if (mpfr_nan_p (mpc_realref (op)) || mpfr_nan_p (mpc_imagref (op)))
{
if (mpfr_inf_p (mpc_realref (op)) || mpfr_inf_p (mpc_imagref (op)))
{
mpfr_set_nan (mpc_realref (rop));
mpfr_set_inf (mpc_imagref (rop), mpfr_signbit (mpc_imagref (op)) ? -1 : +1);
}
else if (mpfr_zero_p (mpc_realref (op)))
{
mpfr_set (mpc_realref (rop), mpc_realref (op), MPFR_RNDN);
mpfr_set_nan (mpc_imagref (rop));
}
else
{
mpfr_set_nan (mpc_realref (rop));
mpfr_set_nan (mpc_imagref (rop));
}
return 0;
}
if (mpfr_inf_p (mpc_realref (op)) || mpfr_inf_p (mpc_imagref (op)))
{
int inex_re;
if (mpfr_inf_p (mpc_realref (op)))
{
int inf_im = mpfr_inf_p (mpc_imagref (op));
inex_re = set_pi_over_2 (mpc_realref (rop),
(mpfr_signbit (mpc_realref (op)) ? -1 : 1), MPC_RND_RE (rnd));
mpfr_set_inf (mpc_imagref (rop), (mpfr_signbit (mpc_imagref (op)) ? -1 : 1));
if (inf_im)
mpfr_div_2ui (mpc_realref (rop), mpc_realref (rop), 1, MPFR_RNDN);
}
else
{
mpfr_set_zero (mpc_realref (rop), (mpfr_signbit (mpc_realref (op)) ? -1 : 1));
inex_re = 0;
mpfr_set_inf (mpc_imagref (rop), (mpfr_signbit (mpc_imagref (op)) ? -1 : 1));
}
return MPC_INEX (inex_re, 0);
}
/* pure real argument */
if (mpfr_zero_p (mpc_imagref (op)))
{
int inex_re;
int inex_im;
int s_im;
s_im = mpfr_signbit (mpc_imagref (op));
if (mpfr_cmp_ui (mpc_realref (op), 1) > 0)
{
if (s_im)
inex_im = -mpfr_acosh (mpc_imagref (rop), mpc_realref (op),
INV_RND (MPC_RND_IM (rnd)));
else
inex_im = mpfr_acosh (mpc_imagref (rop), mpc_realref (op),
MPC_RND_IM (rnd));
inex_re = set_pi_over_2 (mpc_realref (rop),
(mpfr_signbit (mpc_realref (op)) ? -1 : 1), MPC_RND_RE (rnd));
if (s_im)
mpc_conj (rop, rop, MPC_RNDNN);
}
else if (mpfr_cmp_si (mpc_realref (op), -1) < 0)
{
mpfr_t minus_op_re;
minus_op_re[0] = mpc_realref (op)[0];
MPFR_CHANGE_SIGN (minus_op_re);
if (s_im)
inex_im = -mpfr_acosh (mpc_imagref (rop), minus_op_re,
INV_RND (MPC_RND_IM (rnd)));
else
inex_im = mpfr_acosh (mpc_imagref (rop), minus_op_re,
MPC_RND_IM (rnd));
inex_re = set_pi_over_2 (mpc_realref (rop),
(mpfr_signbit (mpc_realref (op)) ? -1 : 1), MPC_RND_RE (rnd));
if (s_im)
mpc_conj (rop, rop, MPC_RNDNN);
}
else
{
inex_im = mpfr_set_ui (mpc_imagref (rop), 0, MPC_RND_IM (rnd));
if (s_im)
mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), MPFR_RNDN);
inex_re = mpfr_asin (mpc_realref (rop), mpc_realref (op), MPC_RND_RE (rnd));
}
return MPC_INEX (inex_re, inex_im);
}
/* pure imaginary argument */
if (mpfr_zero_p (mpc_realref (op)))
{
int inex_im;
int s;
s = mpfr_signbit (mpc_realref (op));
mpfr_set_ui (mpc_realref (rop), 0, MPFR_RNDN);
if (s)
mpfr_neg (mpc_realref (rop), mpc_realref (rop), MPFR_RNDN);
inex_im = mpfr_asinh (mpc_imagref (rop), mpc_imagref (op), MPC_RND_IM (rnd));
return MPC_INEX (0, inex_im);
}
/* regular complex: asin(z) = -i*log(i*z+sqrt(1-z^2)) */
p_re = mpfr_get_prec (mpc_realref(rop));
p_im = mpfr_get_prec (mpc_imagref(rop));
rnd_re = MPC_RND_RE(rnd);
rnd_im = MPC_RND_IM(rnd);
p = p_re >= p_im ? p_re : p_im;
mpc_init2 (z1, p);
while (1)
{
mpfr_exp_t ex, ey, err;
loop ++;
p += (loop <= 2) ? mpc_ceil_log2 (p) + 3 : p / 2;
mpfr_set_prec (mpc_realref(z1), p);
mpfr_set_prec (mpc_imagref(z1), p);
/* try special code for 1+i*y with tiny y */
if (loop == 1 && mpc_asin_special (rop, op, rnd, z1))
break;
/* z1 <- z^2 */
mpc_sqr (z1, op, MPC_RNDNN);
/* err(x) <= 1/2 ulp(x), err(y) <= 1/2 ulp(y) */
/* z1 <- 1-z1 */
ex = mpfr_get_exp (mpc_realref(z1));
mpfr_ui_sub (mpc_realref(z1), 1, mpc_realref(z1), MPFR_RNDN);
mpfr_neg (mpc_imagref(z1), mpc_imagref(z1), MPFR_RNDN);
ex = ex - mpfr_get_exp (mpc_realref(z1));
ex = (ex <= 0) ? 0 : ex;
/* err(x) <= 2^ex * ulp(x) */
ex = ex + mpfr_get_exp (mpc_realref(z1)) - p;
/* err(x) <= 2^ex */
ey = mpfr_get_exp (mpc_imagref(z1)) - p - 1;
/* err(y) <= 2^ey */
ex = (ex >= ey) ? ex : ey; /* err(x), err(y) <= 2^ex, i.e., the norm
of the error is bounded by |h|<=2^(ex+1/2) */
/* z1 <- sqrt(z1): if z1 = z + h, then sqrt(z1) = sqrt(z) + h/2/sqrt(t) */
ey = mpfr_get_exp (mpc_realref(z1)) >= mpfr_get_exp (mpc_imagref(z1))
? mpfr_get_exp (mpc_realref(z1)) : mpfr_get_exp (mpc_imagref(z1));
/* we have |z1| >= 2^(ey-1) thus 1/|z1| <= 2^(1-ey) */
mpc_sqrt (z1, z1, MPC_RNDNN);
ex = (2 * ex + 1) - 2 - (ey - 1); /* |h^2/4/|t| <= 2^ex */
ex = (ex + 1) / 2; /* ceil(ex/2) */
/* express ex in terms of ulp(z1) */
ey = mpfr_get_exp (mpc_realref(z1)) <= mpfr_get_exp (mpc_imagref(z1))
? mpfr_get_exp (mpc_realref(z1)) : mpfr_get_exp (mpc_imagref(z1));
ex = ex - ey + p;
/* take into account the rounding error in the mpc_sqrt call */
err = (ex <= 0) ? 1 : ex + 1;
/* err(x) <= 2^err * ulp(x), err(y) <= 2^err * ulp(y) */
/* z1 <- i*z + z1 */
ex = mpfr_get_exp (mpc_realref(z1));
ey = mpfr_get_exp (mpc_imagref(z1));
mpfr_sub (mpc_realref(z1), mpc_realref(z1), mpc_imagref(op), MPFR_RNDN);
mpfr_add (mpc_imagref(z1), mpc_imagref(z1), mpc_realref(op), MPFR_RNDN);
if (mpfr_cmp_ui (mpc_realref(z1), 0) == 0 || mpfr_cmp_ui (mpc_imagref(z1), 0) == 0)
continue;
ex -= mpfr_get_exp (mpc_realref(z1)); /* cancellation in x */
ey -= mpfr_get_exp (mpc_imagref(z1)); /* cancellation in y */
ex = (ex >= ey) ? ex : ey; /* maximum cancellation */
err += ex;
err = (err <= 0) ? 1 : err + 1; /* rounding error in sub/add */
/* z1 <- log(z1): if z1 = z + h, then log(z1) = log(z) + h/t with
|t| >= min(|z1|,|z|) */
ex = mpfr_get_exp (mpc_realref(z1));
ey = mpfr_get_exp (mpc_imagref(z1));
ex = (ex >= ey) ? ex : ey;
err += ex - p; /* revert to absolute error <= 2^err */
mpc_log (z1, z1, MPFR_RNDN);
err -= ex - 1; /* 1/|t| <= 1/|z| <= 2^(1-ex) */
/* express err in terms of ulp(z1) */
ey = mpfr_get_exp (mpc_realref(z1)) <= mpfr_get_exp (mpc_imagref(z1))
? mpfr_get_exp (mpc_realref(z1)) : mpfr_get_exp (mpc_imagref(z1));
err = err - ey + p;
/* take into account the rounding error in the mpc_log call */
err = (err <= 0) ? 1 : err + 1;
/* z1 <- -i*z1 */
mpfr_swap (mpc_realref(z1), mpc_imagref(z1));
mpfr_neg (mpc_imagref(z1), mpc_imagref(z1), MPFR_RNDN);
if (mpfr_can_round (mpc_realref(z1), p - err, MPFR_RNDN, MPFR_RNDZ,
p_re + (rnd_re == MPFR_RNDN)) &&
mpfr_can_round (mpc_imagref(z1), p - err, MPFR_RNDN, MPFR_RNDZ,
p_im + (rnd_im == MPFR_RNDN)))
break;
}
inex = mpc_set (rop, z1, rnd);
mpc_clear (z1);
return inex;
}