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/* mpc_asin -- arcsine of a complex number.
Copyright (C) 2009, 2010, 2011, 2012, 2013, 2014, 2020 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 <limits.h> /* for ULONG_MAX */
#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;
}
/* Put in s an approximation of asin(z) using:
asin z = z + 1/2*z^3/3 + (1*3)/(2*4)*z^5/5 + ...
Assume |Re(z)|, |Im(z)| < 1/2.
Return non-zero if we can get the correct result by rounding s:
mpc_set (rop, s, ...) */
static int
mpc_asin_series (mpc_srcptr rop, mpc_ptr s, mpc_srcptr z, mpc_rnd_t rnd)
{
mpc_t w, t;
unsigned long k, err, kx, ky;
mpfr_prec_t p;
mpfr_exp_t ex, ey, e;
/* assume z = (x,y) with |x|,|y| < 2^(-e) with e >= 1, see the error
analysis in algorithms.tex */
ex = mpfr_get_exp (mpc_realref (z));
MPC_ASSERT(ex <= -1);
ey = mpfr_get_exp (mpc_imagref (z));
MPC_ASSERT(ey <= -1);
e = (ex >= ey) ? ex : ey;
e = -e;
/* now e >= 1 */
p = mpfr_get_prec (mpc_realref (s)); /* working precision */
MPC_ASSERT(mpfr_get_prec (mpc_imagref (s)) == p);
mpc_init2 (w, p);
mpc_init2 (t, p);
mpc_set (s, z, MPC_RNDNN);
mpc_sqr (w, z, MPC_RNDNN);
mpc_set (t, z, MPC_RNDNN);
for (k = 1; ;k++)
{
mpfr_exp_t exp_x, exp_y;
mpc_mul (t, t, w, MPC_RNDNN);
mpc_mul_ui (t, t, (2 * k - 1) * (2 * k - 1), MPC_RNDNN);
mpc_div_ui (t, t, (2 * k) * (2 * k + 1), MPC_RNDNN);
exp_x = mpfr_get_exp (mpc_realref (s));
exp_y = mpfr_get_exp (mpc_imagref (s));
if (mpfr_get_exp (mpc_realref (t)) < exp_x - p &&
mpfr_get_exp (mpc_imagref (t)) < exp_y - p)
/* Re(t) < 1/2 ulp(Re(s)) and Im(t) < 1/2 ulp(Im(s)),
thus adding t to s will not change s */
break;
mpc_add (s, s, t, MPC_RNDNN);
}
mpc_clear (w);
mpc_clear (t);
/* check (2k-1)^2 is exactly representable */
MPC_ASSERT(2 * k - 1 <= ULONG_MAX / (2 * k - 1));
/* maximal absolute error on Re(s),Im(s) is:
(5k-3)k/2*2^(-1-p) for e=1
5k/2*2^(-e-p) for e >= 2 */
if (e == 1)
{
MPC_ASSERT(5 * k - 3 <= ULONG_MAX / k);
kx = (5 * k - 3) * k;
}
else
kx = 5 * k;
kx = (kx + 1) / 2; /* takes into account the 1/2 factor in both cases */
/* now (5k-3)k/2 <= kx for e=1, and 5k/2 <= kx for e >= 2, thus
the maximal absolute error on Re(s),Im(s) is bounded by kx*2^(-e-p) */
e = -e;
ky = kx;
/* for the real part, convert the maximal absolute error kx*2^(e-p) into
relative error */
ex = mpfr_get_exp (mpc_realref (s));
/* ulp(Re(s)) = 2^(ex+1-p) */
err = 0;
/* invariant: the error will be kx*2^err */
if (ex+1 > e) /* divide kx by 2^(ex+1-e) */
while (ex+1 > e)
{
kx = (kx + 1) / 2;
ex --;
}
else /* multiply the error by 2^(e-(ex+1)), thus add e-(ex+1) to err */
err += e - (ex+1);
/* now the rounding error is bounded by kx*2^err*ulp(Re(s)), add the
mathematical error which is bounded by ulp(Re(s)): the first neglected
term is less than 1/2*ulp(Re(s)), and each term decreases by at least
a factor 2, since |z^2| <= 1/2. */
kx ++;
for (; kx > 2; err ++, kx = (kx + 1) / 2);
/* can we round Re(s) with error less than 2^(EXP(Re(s))-err) ? */
if (!mpfr_can_round (mpc_realref (s), p - err, MPFR_RNDN, MPFR_RNDZ,
mpfr_get_prec (mpc_realref (rop)) +
(MPC_RND_RE(rnd) == MPFR_RNDN)))
return 0;
/* same for the imaginary part */
ey = mpfr_get_exp (mpc_imagref (s));
/* we take for e the exponent of Im(z), which amounts to divide the error by
2^delta where delta is the exponent difference between Re(z) and Im(z)
(see algorithms.tex) */
e = mpfr_get_exp (mpc_imagref (z));
/* ulp(Im(s)) = 2^(ey+1-p) */
if (ey+1 > e) /* divide ky by 2^(ey+1-e) */
while (ey+1 > e)
{
ky = (ky + 1) / 2;
ey --;
}
else /* multiply ky by 2^(e-(ey+1)) */
ky <<= e - (ey+1);
/* now the rounding error is bounded by ky*ulp(Im(s)), add the
mathematical error which is bounded by ulp(Im(s)): the first neglected
term is less than 1/2*ulp(Im(s)), and each term decreases by at least
a factor 2, since |z^2| <= 1/2. */
ky ++;
for (err = 0; ky > 2; err ++, ky = (ky + 1) / 2);
/* can we round Im(s) with error less than 2^(EXP(Im(s))-err) ? */
return mpfr_can_round (mpc_imagref (s), p - err, MPFR_RNDN, MPFR_RNDZ,
mpfr_get_prec (mpc_imagref (rop)) +
(MPC_RND_IM(rnd) == MPFR_RNDN));
}
/* Put in s an approximation of asin(z) for |Re(z)| < 1 and tiny Im(y)
(see algorithms.tex). Assume z = x + i*y:
|Re(asin(z)) - asin(x)| <= Pi*beta^2/(1-beta)
|Im(asin(z)) - y/sqrt(1-x^2)| <= Pi/2*beta^2/(1-beta)
where beta = |y|/(1-|x|).
We assume |x| >= 1/2 > |y| here, thus beta < 1, and the bounds
simplify to 16y^2.
Assume Re(s) and Im(s) have the same precision.
Return non-zero if we can get the correct result by rounding s:
mpc_set (rop, s, ...) */
static int
mpc_asin_tiny (mpc_srcptr rop, mpc_ptr s, mpc_srcptr z, mpc_rnd_t rnd)
{
mpfr_exp_t ey, e1, e2, es;
mpfr_prec_t p = mpfr_get_prec (mpc_realref (s)), err;
ey = mpfr_get_exp (mpc_imagref (z));
MPC_ASSERT(mpfr_get_exp (mpc_realref (z)) >= 0); /* |x| >= 1/2 */
MPC_ASSERT(ey < 0); /* |y| < 1/2 */
e2 = 2 * ey + 4;
/* for |x| >= 0.5, |asin x| >= 0.5, thus no need to compute asin(x)
if 16y^2 >= 1/2 ulp(1) for the target variable */
if (e2 >= - (mpfr_exp_t) mpfr_get_prec (mpc_realref (rop)))
return 0;
/* real part */
mpfr_asin (mpc_realref (s), mpc_realref (z), MPFR_RNDN);
/* check that we can round with error < 16y^2 */
e1 = mpfr_get_exp (mpc_realref (s)) - p;
err = (e1 >= e2) ? 0 : e2 - e1;
if (!mpfr_can_round (mpc_realref (s), p - err, MPFR_RNDN, MPFR_RNDZ,
mpfr_get_prec (mpc_realref (rop)) +
(MPC_RND_RE(rnd) == MPFR_RNDN)))
return 0;
/* now compute the approximate imaginary part y/sqrt(1-x^2) */
mpfr_sqr (mpc_imagref (s), mpc_realref (z), MPFR_RNDN);
/* now Im(s) approximates x^2, with 1/4 <= Im(s) <= 1 and absolute error
less than 2^-p */
mpfr_ui_sub (mpc_imagref (s), 1, mpc_imagref (s), MPFR_RNDN);
/* now Im(s) approximates 1-x^2, with 0 <= Im(s) <= 3/4, assuming p >= 2,
and absolute error less than 2^(1-p) */
mpfr_sqrt (mpc_imagref (s), mpc_imagref (s), MPFR_RNDN); /* sqrt(1-x^2) */
es = mpfr_get_exp (mpc_imagref (s));
/* now Im(s) approximates sqrt(1-x^2), with 0 <= Im(s) <= 7/8,
assuming p >= 3, and absolute error less than 2^-p + 2^(1-p)/s
<= 3*2^-p/s, thus relative error less than 3*2^-p/s^2.
Since |s| >= 2^(es-1), the relative error is less than 3*2^(-p+2-2*es) */
mpfr_div (mpc_imagref (s), mpc_imagref (z), mpc_imagref (s), MPFR_RNDN);
/* now Im(s) approximates y/sqrt(1-x^2), with relative error less than
(3*2^(2-2*es)+2)*2^-p. Since es <= 0, 2-2*es >= 2, thus the relative
error is less than 2^(4-2*es-p) */
err = 4 - 2 * es;
return mpfr_can_round (mpc_imagref (s), p - err, MPFR_RNDN, MPFR_RNDZ,
mpfr_get_prec (mpc_imagref (rop)) +
(MPC_RND_IM(rnd) == MPFR_RNDN));
}
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, inex_re, inex_im, loop = 0;
mpfr_exp_t saved_emin, saved_emax, err, olderr;
/* 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)))
{
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 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 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);
}
saved_emin = mpfr_get_emin ();
saved_emax = mpfr_get_emax ();
mpfr_set_emin (mpfr_get_emin_min ());
mpfr_set_emax (mpfr_get_emax_max ());
/* 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);
olderr = err = 0; /* number of lost bits */
while (1)
{
mpfr_exp_t ex, ey;
loop ++;
p += err - olderr; /* add extra number of lost bits in previous loop */
olderr = err;
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 && mpfr_cmp_ui (mpc_realref(op), 1) == 0 &&
mpc_asin_special (rop, op, rnd, z1))
break;
/* try special code for small z */
if (mpfr_get_exp (mpc_realref (op)) <= -1 &&
mpfr_get_exp (mpc_imagref (op)) <= -1 &&
mpc_asin_series (rop, z1, op, rnd))
break;
/* try special code for 1/2 <= |x| < 1 and |y| < 1/2 */
if (mpfr_get_exp (mpc_realref (op)) == 0 &&
mpfr_get_exp (mpc_imagref (op)) <= -1 &&
mpc_asin_tiny (rop, z1, op, rnd))
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);
/* if Re(z1) = 0, we can't determine the relative error */
if (mpfr_zero_p (mpc_realref(z1)))
continue;
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_zero_p (mpc_realref(z1)) || mpfr_zero_p (mpc_imagref(z1)))
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);
/* restore the exponent range, and check the range of results */
mpfr_set_emin (saved_emin);
mpfr_set_emax (saved_emax);
inex_re = mpfr_check_range (mpc_realref (rop), MPC_INEX_RE (inex),
MPC_RND_RE (rnd));
inex_im = mpfr_check_range (mpc_imagref (rop), MPC_INEX_IM (inex),
MPC_RND_IM (rnd));
return MPC_INEX (inex_re, inex_im);
}
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