/* mpc_div -- Divide two complex numbers.
Copyright (C) INRIA, 2002, 2003, 2004, 2005, 2008, 2009, 2010
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"
static int
mpc_div_zero (mpc_ptr a, mpc_srcptr z, mpc_srcptr w, mpc_rnd_t rnd)
{
/* Assumes w==0, implementation according to C99 G.5.1.8 */
int sign = MPFR_SIGNBIT (MPC_RE (w));
mpfr_t infty;
mpfr_set_inf (infty, sign);
mpfr_mul (MPC_RE (a), infty, MPC_RE (z), MPC_RND_RE (rnd));
mpfr_mul (MPC_IM (a), infty, MPC_IM (z), MPC_RND_IM (rnd));
return MPC_INEX (0, 0); /* exact */
}
static int
mpc_div_inf_fin (mpc_ptr rop, mpc_srcptr z, mpc_srcptr w)
{
/* Assumes w finite and non-zero and z infinite; implementation
according to C99 G.5.1.8 */
int a, b, x, y;
a = (mpfr_inf_p (MPC_RE (z)) ? MPFR_SIGNBIT (MPC_RE (z)) : 0);
b = (mpfr_inf_p (MPC_IM (z)) ? MPFR_SIGNBIT (MPC_IM (z)) : 0);
/* x = MPC_MPFR_SIGN (a * MPC_RE (w) + b * MPC_IM (w)) */
/* y = MPC_MPFR_SIGN (b * MPC_RE (w) - a * MPC_IM (w)) */
if (a == 0 || b == 0) {
x = a * MPC_MPFR_SIGN (MPC_RE (w)) + b * MPC_MPFR_SIGN (MPC_IM (w));
y = b * MPC_MPFR_SIGN (MPC_RE (w)) - a * MPC_MPFR_SIGN (MPC_IM (w));
}
else {
/* Both parts of z are infinite; x could be determined by sign
considerations and comparisons. Since operations with non-finite
numbers are not considered time-critical, we let mpfr do the work. */
mpfr_t sign;
mpfr_init2 (sign, 2);
/* This is enough to determine the sign of sums and differences. */
if (a == 1)
if (b == 1) {
mpfr_add (sign, MPC_RE (w), MPC_IM (w), GMP_RNDN);
x = MPC_MPFR_SIGN (sign);
mpfr_sub (sign, MPC_RE (w), MPC_IM (w), GMP_RNDN);
y = MPC_MPFR_SIGN (sign);
}
else { /* b == -1 */
mpfr_sub (sign, MPC_RE (w), MPC_IM (w), GMP_RNDN);
x = MPC_MPFR_SIGN (sign);
mpfr_add (sign, MPC_RE (w), MPC_IM (w), GMP_RNDN);
y = -MPC_MPFR_SIGN (sign);
}
else /* a == -1 */
if (b == 1) {
mpfr_sub (sign, MPC_IM (w), MPC_RE (w), GMP_RNDN);
x = MPC_MPFR_SIGN (sign);
mpfr_add (sign, MPC_RE (w), MPC_IM (w), GMP_RNDN);
y = MPC_MPFR_SIGN (sign);
}
else { /* b == -1 */
mpfr_add (sign, MPC_RE (w), MPC_IM (w), GMP_RNDN);
x = -MPC_MPFR_SIGN (sign);
mpfr_sub (sign, MPC_IM (w), MPC_RE (w), GMP_RNDN);
y = MPC_MPFR_SIGN (sign);
}
mpfr_clear (sign);
}
if (x == 0)
mpfr_set_nan (MPC_RE (rop));
else
mpfr_set_inf (MPC_RE (rop), x);
if (y == 0)
mpfr_set_nan (MPC_IM (rop));
else
mpfr_set_inf (MPC_IM (rop), y);
return MPC_INEX (0, 0); /* exact */
}
static int
mpc_div_fin_inf (mpc_ptr rop, mpc_srcptr z, mpc_srcptr w)
{
/* Assumes z finite and w infinite; implementation according to
C99 G.5.1.8 */
mpfr_t c, d, a, b, x, y, zero;
mpfr_init2 (c, 2); /* needed to hold a signed zero, +1 or -1 */
mpfr_init2 (d, 2);
mpfr_init2 (x, 2);
mpfr_init2 (y, 2);
mpfr_init2 (zero, 2);
mpfr_set_ui (zero, 0ul, GMP_RNDN);
mpfr_init2 (a, mpfr_get_prec (MPC_RE (z)));
mpfr_init2 (b, mpfr_get_prec (MPC_IM (z)));
mpfr_set_ui (c, (mpfr_inf_p (MPC_RE (w)) ? 1 : 0), GMP_RNDN);
MPFR_COPYSIGN (c, c, MPC_RE (w), GMP_RNDN);
mpfr_set_ui (d, (mpfr_inf_p (MPC_IM (w)) ? 1 : 0), GMP_RNDN);
MPFR_COPYSIGN (d, d, MPC_IM (w), GMP_RNDN);
mpfr_mul (a, MPC_RE (z), c, GMP_RNDN); /* exact */
mpfr_mul (b, MPC_IM (z), d, GMP_RNDN);
mpfr_add (x, a, b, GMP_RNDN);
mpfr_mul (b, MPC_IM (z), c, GMP_RNDN);
mpfr_mul (a, MPC_RE (z), d, GMP_RNDN);
mpfr_sub (y, b, a, GMP_RNDN);
MPFR_COPYSIGN (MPC_RE (rop), zero, x, GMP_RNDN);
MPFR_COPYSIGN (MPC_IM (rop), zero, y, GMP_RNDN);
mpfr_clear (c);
mpfr_clear (d);
mpfr_clear (x);
mpfr_clear (y);
mpfr_clear (zero);
mpfr_clear (a);
mpfr_clear (b);
return MPC_INEX (0, 0); /* exact */
}
int
mpc_div (mpc_ptr a, mpc_srcptr b, mpc_srcptr c, mpc_rnd_t rnd)
{
int ok_re = 0, ok_im = 0;
mpc_t res, c_conj;
mpfr_t q;
mpfr_prec_t prec;
int inexact_prod, inexact_norm, inexact_re, inexact_im, loops = 0;
/* save signs of operands in case there are overlaps */
int brs = MPFR_SIGNBIT (MPC_RE (b));
int bis = MPFR_SIGNBIT (MPC_IM (b));
int crs = MPFR_SIGNBIT (MPC_RE (c));
int cis = MPFR_SIGNBIT (MPC_IM (c));
/* According to the C standard G.3, there are three types of numbers: */
/* finite (both parts are usual real numbers; contains 0), infinite */
/* (at least one part is a real infinity) and all others; the latter */
/* are numbers containing a nan, but no infinity, and could reasonably */
/* be called nan. */
/* By G.5.1.4, infinite/finite=infinite; finite/infinite=0; */
/* all other divisions that are not finite/finite return nan+i*nan. */
/* Division by 0 could be handled by the following case of division by */
/* a real; we handle it separately instead. */
if (mpc_zero_p (c))
return mpc_div_zero (a, b, c, rnd);
else {
if (mpc_inf_p (b) && mpc_fin_p (c))
return mpc_div_inf_fin (a, b, c);
else if (mpc_fin_p (b) && mpc_inf_p (c))
return mpc_div_fin_inf (a, b, c);
else if (!mpc_fin_p (b) || !mpc_fin_p (c)) {
mpfr_set_nan (MPC_RE (a));
mpfr_set_nan (MPC_IM (a));
return MPC_INEX (0, 0);
}
}
/* check for real divisor */
if (mpfr_zero_p(MPC_IM(c))) /* (re_b+i*im_b)/c = re_b/c + i * (im_b/c) */
{
/* warning: a may overlap with b,c so treat the imaginary part first */
inexact_im = mpfr_div (MPC_IM(a), MPC_IM(b), MPC_RE(c), MPC_RND_IM(rnd));
inexact_re = mpfr_div (MPC_RE(a), MPC_RE(b), MPC_RE(c), MPC_RND_RE(rnd));
/* correct signs of zeroes if necessary, which does not affect the
inexact flags */
if (mpfr_zero_p (MPC_RE (a)))
mpfr_setsign (MPC_RE (a), MPC_RE (a), (brs != crs && bis != cis),
GMP_RNDN); /* exact */
if (mpfr_zero_p (MPC_IM (a)))
mpfr_setsign (MPC_IM (a), MPC_IM (a), (bis != crs && brs == cis),
GMP_RNDN);
return MPC_INEX(inexact_re, inexact_im);
}
/* check for purely imaginary divisor */
if (mpfr_zero_p(MPC_RE(c)))
{
/* (re_b+i*im_b)/(i*c) = im_b/c - i * (re_b/c) */
int overlap = (a == b) || (a == c);
int imag_b = mpfr_zero_p (MPC_RE (b));
mpfr_t cloc;
mpc_t tmpa;
mpc_ptr dest = (overlap) ? tmpa : a;
if (overlap)
mpc_init3 (tmpa, MPC_PREC_RE (a), MPC_PREC_IM (a));
cloc[0] = MPC_IM(c)[0]; /* copies mpfr struct IM(c) into cloc */
inexact_re = mpfr_div (MPC_RE(dest), MPC_IM(b), cloc, MPC_RND_RE(rnd));
mpfr_neg (cloc, cloc, GMP_RNDN);
/* changes the sign only in cloc, not in c; no need to correct later */
inexact_im = mpfr_div (MPC_IM(dest), MPC_RE(b), cloc, MPC_RND_IM(rnd));
if (overlap)
{
/* Note: we could use mpc_swap here, but this might cause problems
if a and tmpa have been allocated using different methods, since
it will swap the significands of a and tmpa. See http://
lists.gforge.inria.fr/pipermail/mpc-discuss/2009-August/000504.html */
mpc_set (a, tmpa, MPC_RNDNN); /* exact */
mpc_clear (tmpa);
}
/* correct signs of zeroes if necessary, which does not affect the
inexact flags */
if (mpfr_zero_p (MPC_RE (a)))
mpfr_setsign (MPC_RE (a), MPC_RE (a), (brs != crs && bis != cis),
GMP_RNDN); /* exact */
if (imag_b)
mpfr_setsign (MPC_IM (a), MPC_IM (a), (bis != crs && brs == cis),
GMP_RNDN);
return MPC_INEX(inexact_re, inexact_im);
}
prec = MPC_MAX_PREC(a);
mpc_init2 (res, 2);
mpfr_init (q);
/* create the conjugate of c in c_conj without allocating new memory */
MPC_RE (c_conj)[0] = MPC_RE (c)[0];
MPC_IM (c_conj)[0] = MPC_IM (c)[0];
MPFR_CHANGE_SIGN (MPC_IM (c_conj));
do
{
loops ++;
prec += loops <= 2 ? mpc_ceil_log2 (prec) + 5 : prec / 2;
mpc_set_prec (res, prec);
mpfr_set_prec (q, prec);
/* first compute norm(c)^2 */
inexact_norm = mpc_norm (q, c, GMP_RNDD);
/* now compute b*conjugate(c) */
/* We need rounding away from zero for both the real and the imagin- */
/* ary part; then the final result is also rounded away from zero. */
/* The error is less than 1 ulp. Since this is not implemented in */
/* mpc, we round towards zero and add 1 ulp to the absolute values */
/* if they are not exact. */
inexact_prod = mpc_mul (res, b, c_conj, MPC_RNDZZ);
inexact_re = MPC_INEX_RE (inexact_prod);
inexact_im = MPC_INEX_IM (inexact_prod);
if (inexact_re != 0)
MPFR_ADD_ONE_ULP (MPC_RE (res));
if (inexact_im != 0)
MPFR_ADD_ONE_ULP (MPC_IM (res));
/* divide the product by the norm */
if (inexact_norm == 0 && (inexact_re == 0 || inexact_im == 0))
{
/* The division has good chances to be exact in at least one part. */
/* Since this can cause problems when not rounding to the nearest, */
/* we use the division code of mpfr, which handles the situation. */
if (MPFR_SIGN (MPC_RE (res)) > 0)
{
inexact_re |= mpfr_div (MPC_RE (res), MPC_RE (res), q, GMP_RNDU);
ok_re = mpfr_inf_p (MPC_RE (res)) || mpfr_zero_p (MPC_RE (res)) ||
mpfr_can_round (MPC_RE (res), prec - 4, GMP_RNDU,
MPC_RND_RE(rnd), MPC_PREC_RE(a));
}
else
{
inexact_re |= mpfr_div (MPC_RE (res), MPC_RE (res), q, GMP_RNDD);
ok_re = mpfr_inf_p (MPC_RE (res)) || mpfr_zero_p (MPC_RE (res)) ||
mpfr_can_round (MPC_RE (res), prec - 4, GMP_RNDD,
MPC_RND_RE(rnd), MPC_PREC_RE(a));
}
if (ok_re || !inexact_re) /* compute imaginary part */
{
if (MPFR_SIGN (MPC_IM (res)) > 0)
{
inexact_im |= mpfr_div (MPC_IM (res), MPC_IM (res), q, GMP_RNDU);
ok_im = mpfr_can_round (MPC_IM (res), prec - 4, GMP_RNDU,
MPC_RND_IM(rnd), MPC_PREC_IM(a));
}
else
{
inexact_im |= mpfr_div (MPC_IM (res), MPC_IM (res), q, GMP_RNDD);
ok_im = mpfr_can_round (MPC_IM (res), prec - 4, GMP_RNDD,
MPC_RND_IM(rnd), MPC_PREC_IM(a));
}
}
}
else
{
/* The division is inexact, so for efficiency reasons we invert q */
/* only once and multiply by the inverse. */
/* We do not decide about the sign of the difference. */
if (mpfr_ui_div (q, 1, q, GMP_RNDU) || inexact_norm)
{
/* if 1/q is inexact, the approximations of the real and
imaginary part below will be inexact, unless RE(res)
or IM(res) is zero */
inexact_re = inexact_re || !mpfr_zero_p (MPC_RE (res));
inexact_im = inexact_im || !mpfr_zero_p (MPC_IM (res));
}
if (MPFR_SIGN (MPC_RE (res)) > 0)
{
inexact_re = mpfr_mul (MPC_RE (res), MPC_RE (res), q, GMP_RNDU)
|| inexact_re;
ok_re = mpfr_inf_p (MPC_RE (res)) || mpfr_zero_p (MPC_RE (res)) ||
mpfr_can_round (MPC_RE (res), prec - 4, GMP_RNDU,
MPC_RND_RE(rnd), MPC_PREC_RE(a));
}
else
{
inexact_re = mpfr_mul (MPC_RE (res), MPC_RE (res), q, GMP_RNDD)
|| inexact_re;
ok_re = mpfr_inf_p (MPC_RE (res)) || mpfr_zero_p (MPC_RE (res)) ||
mpfr_can_round (MPC_RE (res), prec - 4, GMP_RNDD,
MPC_RND_RE(rnd), MPC_PREC_RE(a));
}
if (ok_re) /* compute imaginary part */
{
if (MPFR_SIGN (MPC_IM (res)) > 0)
{
inexact_im = mpfr_mul (MPC_IM (res), MPC_IM (res), q, GMP_RNDU)
|| inexact_im;
ok_im = mpfr_can_round (MPC_IM (res), prec - 4, GMP_RNDU,
MPC_RND_IM(rnd), MPC_PREC_IM(a));
}
else
{
inexact_im = mpfr_mul (MPC_IM (res), MPC_IM (res), q, GMP_RNDD)
|| inexact_im;
ok_im = mpfr_can_round (MPC_IM (res), prec - 4, GMP_RNDD,
MPC_RND_IM(rnd), MPC_PREC_IM(a));
}
}
}
/* check for overflow or underflow on the imaginary part */
if (ok_im == 0 &&
(mpfr_inf_p (MPC_IM (res)) || mpfr_zero_p (MPC_IM (res))))
ok_im = 1;
}
while ((!ok_re && inexact_re) || (!ok_im && inexact_im));
mpc_set (a, res, rnd);
/* fix inexact flags in case of overflow/underflow */
if (mpfr_inf_p (MPC_RE (res)))
inexact_re = mpfr_sgn (MPC_RE (res));
else if (mpfr_zero_p (MPC_RE (res)))
inexact_re = -mpfr_sgn (MPC_RE (res));
if (mpfr_inf_p (MPC_IM (res)))
inexact_im = mpfr_sgn (MPC_IM (res));
else if (mpfr_zero_p (MPC_IM (res)))
inexact_im = -mpfr_sgn (MPC_IM (res));
mpc_clear (res);
mpfr_clear (q);
return (MPC_INEX (inexact_re, inexact_im));
/* Only exactness vs. inexactness is tested, not the sign. */
}