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/* mpfr_log -- natural logarithm of a floating-point number
Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005 Free Software Foundation.
This file is part of the MPFR Library.
The MPFR 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 MPFR 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 MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin Place, Fifth Floor, Boston,
MA 02110-1301, USA. */
/*#define DEBUG */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* The computation of log(a) is done using the formula :
if we want p bits of the result,
pi
log(a) ~ ------------ - m log 2
2 AG(1,4/s)
where s = x 2^m > 2^(p/2)
More precisely, if F(x) = int(1/sqrt(1-(1-x^2)*sin(t)^2), t=0..PI/2),
then for s>=1.26 we have log(s) < F(4/s) < log(s)*(1+4/s^2)
from which we deduce pi/2/AG(1,4/s)*(1-4/s^2) < log(s) < pi/2/AG(1,4/s)
so the relative error 4/s^2 is < 4/2^p i.e. 4 ulps.
*/
int
mpfr_log (mpfr_ptr r, mpfr_srcptr a, mp_rnd_t rnd_mode)
{
int inexact;
mp_prec_t p, q;
mpfr_t tmp1, tmp2;
mp_limb_t *tmp1p, *tmp2p;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_TMP_DECL(marker);
MPFR_LOG_FUNC (("a[%#R]=%R rnd=%d", a, a, rnd_mode),
("r[%#R]=%R inexact=%d", r, r, inexact));
/* Special cases */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (a)))
{
/* If a is NaN, the result is NaN */
if (MPFR_IS_NAN (a))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* check for infinity before zero */
else if (MPFR_IS_INF (a))
{
if (MPFR_IS_NEG (a))
/* log(-Inf) = NaN */
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
else /* log(+Inf) = +Inf */
{
MPFR_SET_INF (r);
MPFR_SET_POS (r);
MPFR_RET (0);
}
}
else /* a is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (a));
MPFR_SET_INF (r);
MPFR_SET_NEG (r);
MPFR_RET (0); /* log(0) is an exact -infinity */
}
}
/* If a is negative, the result is NaN */
else if (MPFR_UNLIKELY (MPFR_IS_NEG (a)))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
/* If a is 1, the result is 0 */
else if (MPFR_UNLIKELY (MPFR_GET_EXP (a) == 1 && mpfr_cmp_ui (a, 1) == 0))
{
MPFR_SET_ZERO (r);
MPFR_SET_POS (r);
MPFR_RET (0); /* only "normal" case where the result is exact */
}
q = MPFR_PREC (r);
/* use initial precision about q+lg(q)+5 */
p = q + 5 + 2*MPFR_INT_CEIL_LOG2 (q);
/* % ~(mp_prec_t)BITS_PER_MP_LIMB ;
m=q; while (m) { p++; m >>= 1; } */
/* if (MPFR_LIKELY(p % BITS_PER_MP_LIMB != 0))
p += BITS_PER_MP_LIMB - (p%BITS_PER_MP_LIMB); */
MPFR_TMP_MARK(marker);
MPFR_SAVE_EXPO_MARK (expo);
MPFR_ZIV_INIT (loop, p);
for (;;)
{
mp_size_t size;
long m;
mp_exp_t cancel;
/* Calculus of m (depends on p) */
m = (p + 1) / 2 - MPFR_GET_EXP (a) + 1;
/* All the mpfr_t needed have a precision of p */
size = (p-1)/BITS_PER_MP_LIMB+1;
MPFR_TMP_INIT (tmp1p, tmp1, p, size);
MPFR_TMP_INIT (tmp2p, tmp2, p, size);
mpfr_mul_2si (tmp2, a, m, GMP_RNDN); /* s=a*2^m, err<=1 ulp */
mpfr_div (tmp1, __gmpfr_four, tmp2, GMP_RNDN);/* 4/s, err<=2 ulps */
mpfr_agm (tmp2, __gmpfr_one, tmp1, GMP_RNDN); /* AG(1,4/s),err<=3 ulps */
mpfr_mul_2ui (tmp2, tmp2, 1, GMP_RNDN); /* 2*AG(1,4/s), err<=3 ulps */
mpfr_const_pi (tmp1, GMP_RNDN); /* compute pi, err<=1ulp */
mpfr_div (tmp2, tmp1, tmp2, GMP_RNDN); /* pi/2*AG(1,4/s), err<=5ulps */
mpfr_const_log2 (tmp1, GMP_RNDN); /* compute log(2), err<=1ulp */
mpfr_mul_si (tmp1, tmp1, m, GMP_RNDN); /* compute m*log(2),err<=2ulps */
mpfr_sub (tmp1, tmp2, tmp1, GMP_RNDN); /* log(a), err<=7ulps+cancel */
cancel = MPFR_GET_EXP (tmp2) - MPFR_GET_EXP (tmp1);
MPFR_LOG_MSG (("canceled bits=%ld\n", cancel));
MPFR_LOG_VAR (tmp1);
if (MPFR_UNLIKELY (cancel < 0))
cancel = 0;
/* we have 7 ulps of error from the above roundings,
4 ulps from the 4/s^2 second order term,
plus the canceled bits */
if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp1, p-cancel-4, q, rnd_mode)))
break;
p += cancel;
MPFR_ZIV_NEXT (loop, p);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (r, tmp1, rnd_mode);
/* We clean */
MPFR_TMP_FREE(marker);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (r, inexact, rnd_mode);
}
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