/* mpc_log10 -- Take the base-10 logarithm of a complex number.
Copyright (C) 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 <limits.h> /* for CHAR_BIT */
#include "mpc-impl.h"
/* Auxiliary functions which implement Ziv's strategy for special cases.
if flag = 0: compute only real part
if flag = 1: compute only imaginary
Exact cases should be dealt with separately. */
static int
mpc_log10_aux (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd, int flag, int nb)
{
mp_prec_t prec = (MPFR_PREC_MIN > 4) ? MPFR_PREC_MIN : 4;
mpc_t tmp;
mpfr_t log10;
int ok = 0, ret;
prec = mpfr_get_prec ((flag == 0) ? mpc_realref (rop) : mpc_imagref (rop));
prec += 10;
mpc_init2 (tmp, prec);
mpfr_init2 (log10, prec);
while (ok == 0)
{
mpfr_set_ui (log10, 10, GMP_RNDN); /* exact since prec >= 4 */
mpfr_log (log10, log10, GMP_RNDN);
/* In each case we have two roundings, thus the final value is
x * (1+u)^2 where x is the exact value, and |u| <= 2^(-prec-1).
Thus the error is always less than 3 ulps. */
switch (nb)
{
case 0: /* imag <- atan2(y/x) */
mpfr_atan2 (mpc_imagref (tmp), mpc_imagref (op), mpc_realref (op),
MPC_RND_IM (rnd));
mpfr_div (mpc_imagref (tmp), mpc_imagref (tmp), log10, GMP_RNDN);
ok = mpfr_can_round (mpc_imagref (tmp), prec - 2, GMP_RNDN,
GMP_RNDZ, MPC_PREC_IM(rop) +
(MPC_RND_IM (rnd) == GMP_RNDN));
if (ok)
ret = mpfr_set (mpc_imagref (rop), mpc_imagref (tmp),
MPC_RND_IM (rnd));
break;
case 1: /* real <- log(x) */
mpfr_log (mpc_realref (tmp), mpc_realref (op), MPC_RND_RE (rnd));
mpfr_div (mpc_realref (tmp), mpc_realref (tmp), log10, GMP_RNDN);
ok = mpfr_can_round (mpc_realref (tmp), prec - 2, GMP_RNDN,
GMP_RNDZ, MPC_PREC_RE(rop) +
(MPC_RND_RE (rnd) == GMP_RNDN));
if (ok)
ret = mpfr_set (mpc_realref (rop), mpc_realref (tmp),
MPC_RND_RE (rnd));
break;
case 2: /* imag <- pi */
mpfr_const_pi (mpc_imagref (tmp), MPC_RND_IM (rnd));
mpfr_div (mpc_imagref (tmp), mpc_imagref (tmp), log10, GMP_RNDN);
ok = mpfr_can_round (mpc_imagref (tmp), prec - 2, GMP_RNDN,
GMP_RNDZ, MPC_PREC_IM(rop) +
(MPC_RND_IM (rnd) == GMP_RNDN));
if (ok)
ret = mpfr_set (mpc_imagref (rop), mpc_imagref (tmp),
MPC_RND_IM (rnd));
break;
}
prec += prec / 2;
mpc_set_prec (tmp, prec);
mpfr_set_prec (log10, prec);
}
mpc_clear (tmp);
mpfr_clear (log10);
return ret;
}
int
mpc_log10 (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
{
int ok = 0, loops = 0, re_cmp, im_cmp, inex_re, inex_im, negative_zero;
mpfr_t w;
mpfr_prec_t prec;
mpfr_rnd_t rnd_im;
mpc_t ww;
mpc_rnd_t invrnd;
/* special values: NaN and infinities: same as mpc_log */
if (!mpc_fin_p (op)) /* real or imaginary parts are NaN or Inf */
{
if (mpfr_nan_p (mpc_realref (op)))
{
if (mpfr_inf_p (mpc_imagref (op)))
/* (NaN, Inf) -> (+Inf, NaN) */
mpfr_set_inf (mpc_realref (rop), +1);
else
/* (NaN, xxx) -> (NaN, NaN) */
mpfr_set_nan (mpc_realref (rop));
mpfr_set_nan (mpc_imagref (rop));
inex_im = 0; /* Inf/NaN is exact */
}
else if (mpfr_nan_p (mpc_imagref (op)))
{
if (mpfr_inf_p (mpc_realref (op)))
/* (Inf, NaN) -> (+Inf, NaN) */
mpfr_set_inf (mpc_realref (rop), +1);
else
/* (xxx, NaN) -> (NaN, NaN) */
mpfr_set_nan (mpc_realref (rop));
mpfr_set_nan (mpc_imagref (rop));
inex_im = 0; /* Inf/NaN is exact */
}
else /* We have an infinity in at least one part. */
{
/* (+Inf, y) -> (+Inf, 0) for finite positive-signed y */
if (mpfr_inf_p (mpc_realref (op)) && mpfr_signbit (mpc_realref (op))
== 0 && mpfr_number_p (mpc_imagref (op)))
inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op),
mpc_realref (op), MPC_RND_IM (rnd));
else
/* (xxx, Inf) -> (+Inf, atan2(Inf/xxx))
(Inf, yyy) -> (+Inf, atan2(yyy/Inf)) */
inex_im = mpc_log10_aux (rop, op, rnd, 1, 0);
mpfr_set_inf (mpc_realref (rop), +1);
}
return MPC_INEX(0, inex_im);
}
/* special cases: real and purely imaginary numbers */
re_cmp = mpfr_cmp_ui (mpc_realref (op), 0);
im_cmp = mpfr_cmp_ui (mpc_imagref (op), 0);
if (im_cmp == 0) /* Im(op) = 0 */
{
if (re_cmp == 0) /* Re(op) = 0 */
{
if (mpfr_signbit (mpc_realref (op)) == 0)
inex_im = mpfr_atan2 (mpc_imagref (rop), mpc_imagref (op),
mpc_realref (op), MPC_RND_IM (rnd));
else
inex_im = mpc_log10_aux (rop, op, rnd, 1, 0);
mpfr_set_inf (mpc_realref (rop), -1);
inex_re = 0; /* -Inf is exact */
}
else if (re_cmp > 0)
{
inex_re = mpfr_log10 (mpc_realref (rop), mpc_realref (op),
MPC_RND_RE (rnd));
inex_im = mpfr_set (mpc_imagref (rop), mpc_imagref (op),
MPC_RND_IM (rnd));
}
else /* log10(x + 0*i) for negative x */
{ /* op = x + 0*i; let w = -x = |x| */
negative_zero = mpfr_signbit (mpc_imagref (op));
if (negative_zero)
rnd_im = INV_RND (MPC_RND_IM (rnd));
else
rnd_im = MPC_RND_IM (rnd);
ww->re[0] = *mpc_realref (op);
MPFR_CHANGE_SIGN (ww->re);
ww->im[0] = *mpc_imagref (op);
if (mpfr_cmp_ui (ww->re, 1) == 0)
inex_re = mpfr_set_ui (mpc_realref (rop), 0, MPC_RND_RE (rnd));
else
inex_re = mpc_log10_aux (rop, ww, rnd, 0, 1);
inex_im = mpc_log10_aux (rop, op, MPC_RND (0,rnd_im), 1, 2);
if (negative_zero)
{
mpc_conj (rop, rop, MPC_RNDNN);
inex_im = -inex_im;
}
}
return MPC_INEX(inex_re, inex_im);
}
else if (re_cmp == 0)
{
if (im_cmp > 0)
{
inex_re = mpfr_log10 (mpc_realref (rop), mpc_imagref (op), MPC_RND_RE (rnd));
inex_im = mpc_log10_aux (rop, op, rnd, 1, 2);
/* division by 2 does not change the ternary flag */
mpfr_div_2ui (mpc_imagref (rop), mpc_imagref (rop), 1, GMP_RNDN);
}
else
{
w [0] = *mpc_imagref (op);
MPFR_CHANGE_SIGN (w);
inex_re = mpfr_log10 (mpc_realref (rop), w, MPC_RND_RE (rnd));
invrnd = MPC_RND (0, INV_RND (MPC_RND_IM (rnd)));
inex_im = mpc_log10_aux (rop, op, invrnd, 1, 2);
/* division by 2 does not change the ternary flag */
mpfr_div_2ui (mpc_imagref (rop), mpc_imagref (rop), 1, GMP_RNDN);
mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), GMP_RNDN);
inex_im = -inex_im; /* negate the ternary flag */
}
return MPC_INEX(inex_re, inex_im);
}
/* generic case: neither Re(op) nor Im(op) is NaN, Inf or zero */
prec = MPC_PREC_RE(rop);
mpfr_init2 (w, prec);
mpc_init2 (ww, prec);
/* let op = x + iy; compute log(op)/log(10) */
while (ok == 0)
{
loops ++;
prec += (loops <= 2) ? mpc_ceil_log2 (prec) + 4 : prec / 2;
mpfr_set_prec (w, prec);
mpc_set_prec (ww, prec);
mpc_log (ww, op, MPC_RNDNN);
mpfr_set_ui (w, 10, GMP_RNDN); /* exact since prec >= 4 */
mpfr_log (w, w, GMP_RNDN);
mpc_div_fr (ww, ww, w, MPC_RNDNN);
ok = mpfr_can_round (mpc_realref (ww), prec - 2, GMP_RNDN, GMP_RNDZ,
MPC_PREC_RE(rop) + (MPC_RND_RE (rnd) == GMP_RNDN));
/* Special code to deal with cases where the real part of log10(x+i*y)
is exact, like x=3 and y=1. Since Re(log10(x+i*y)) = log10(x^2+y^2)/2
this happens whenever x^2+y^2 is a nonnegative power of 10.
Indeed x^2+y^2 cannot equal 10^(a/2^b) for a, b integers, a odd, b>0,
since x^2+y^2 is rational, and 10^(a/2^b) is irrational.
Similarly, for b=0, x^2+y^2 cannot equal 10^a for a < 0 since x^2+y^2
is a rational with denominator a power of 2.
Now let x^2+y^2 = 10^s. Without loss of generality we can assume
x = u/2^e and y = v/2^e with u, v, e integers: u^2+v^2 = 10^s*2^(2e)
thus u^2+v^2 = 0 mod 2^(2e). By recurrence on e, necessarily
u = v = 0 mod 2^e, thus x and y are necessarily integers.
*/
if ((ok == 0) && (loops == 1) && mpfr_integer_p (mpc_realref (op)) &&
mpfr_integer_p (mpc_imagref (op)))
{
mpz_t x, y;
unsigned long s, v;
mpz_init (x);
mpz_init (y);
mpfr_get_z (x, mpc_realref (op), GMP_RNDN); /* exact */
mpfr_get_z (y, mpc_imagref (op), GMP_RNDN); /* exact */
mpz_mul (x, x, x);
mpz_mul (y, y, y);
mpz_add (x, x, y); /* x^2+y^2 */
v = mpz_scan1 (x, 0);
/* if x = 10^s then necessarily s = v */
s = mpz_sizeinbase (x, 10);
/* since s is either the number of digits of x or one more,
then x = 10^(s-1) or 10^(s-2) */
if (s == v + 1 || s == v + 2)
{
mpz_div_2exp (x, x, v);
mpz_ui_pow_ui (y, 5, v);
if (mpz_cmp (y, x) == 0) /* Re(log10(x+i*y)) is exactly v/2 */
{
/* we reset the precision of Re(ww) so that v can be
represented exactly */
mpfr_set_prec (mpc_realref (ww), sizeof(unsigned long)*CHAR_BIT);
mpfr_set_ui_2exp (mpc_realref (ww), v, -1, GMP_RNDN); /* exact */
ok = 1;
}
}
mpz_clear (x);
mpz_clear (y);
}
ok = ok && mpfr_can_round (mpc_imagref (ww), prec-2, GMP_RNDN, GMP_RNDZ,
MPC_PREC_IM(rop) + (MPC_RND_IM (rnd) == GMP_RNDN));
}
inex_re = mpfr_set (mpc_realref(rop), mpc_realref (ww), MPC_RND_RE (rnd));
inex_im = mpfr_set (mpc_imagref(rop), mpc_imagref (ww), MPC_RND_IM (rnd));
mpfr_clear (w);
mpc_clear (ww);
return MPC_INEX(inex_re, inex_im);
}