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/* mpfr_log -- natural logarithm of a floating-point number
Copyright 1999-2021 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU 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 3 of the License, or (at your
option) any later version.
The GNU 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 GNU MPFR Library; see the file COPYING.LESSER. If not, see
https://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* The computation of log(x) is done using the formula :
if we want p bits of the result,
pi
log(x) ~ ------------ - 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, mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_prec_t p, q;
mpfr_t tmp1, tmp2;
mpfr_exp_t exp_a;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_GROUP_DECL(group);
MPFR_LOG_FUNC
(("a[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (a), mpfr_log_prec, a, rnd_mode),
("r[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (r), mpfr_log_prec, 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_SET_DIVBY0 ();
MPFR_RET (0); /* log(0) is an exact -infinity */
}
}
/* If a is negative, the result is NaN */
if (MPFR_UNLIKELY (MPFR_IS_NEG (a)))
{
MPFR_SET_NAN (r);
MPFR_RET_NAN;
}
exp_a = MPFR_GET_EXP (a);
/* If a is 1, the result is +0 */
if (MPFR_UNLIKELY (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+2*lg(q)+cte */
p = q + 2 * MPFR_INT_CEIL_LOG2 (q) + 10;
/* % ~(mpfr_prec_t)GMP_NUMB_BITS ;
m=q; while (m) { p++; m >>= 1; } */
/* if (MPFR_LIKELY(p % GMP_NUMB_BITS != 0))
p += GMP_NUMB_BITS - (p%GMP_NUMB_BITS); */
MPFR_SAVE_EXPO_MARK (expo);
MPFR_GROUP_INIT_2 (group, p, tmp1, tmp2);
MPFR_ZIV_INIT (loop, p);
for (;;)
{
mpfr_t scaled_a;
mpfr_exp_t m;
mpfr_exp_t cancel;
/* Calculus of m (depends on p)
If mpfr_exp_t has N bits, then both (p + 3) / 2 and |exp_a| fit
on N-2 bits, so that there cannot be an overflow. */
m = (p + 3) / 2 - exp_a;
/* In standard configuration (_MPFR_EXP_FORMAT <= 3), one has
mpfr_exp_t <= long, so that the following assertion is always
true. This assertion is needed for the mpfr_mul_si below. */
MPFR_ASSERTN (m >= LONG_MIN && m <= LONG_MAX);
/* FIXME: Redo the error analysis. The error concerning the AGM
should be explained since 4/s is inexact (one needs a bound
on its derivative). */
MPFR_ALIAS (scaled_a, a, MPFR_SIGN_POS, (p + 3) / 2); /* s=a*2^m */
/* [FIXME] and one can have the equality, even if p is even.
This means that if a is a power of 2 and p is even, then
s = (1/2) * 2^((p+2)/2) = 2^(p/2), so that the condition
s > 2^(p/2) from algorithms.tex is not satisfied. */
mpfr_div (tmp1, __gmpfr_four, scaled_a, MPFR_RNDF); /* 4/s, err<=2 ulps */
mpfr_agm (tmp2, __gmpfr_one, tmp1, MPFR_RNDN); /* AG(1,4/s),err<=3 ulps */
mpfr_mul_2ui (tmp2, tmp2, 1, MPFR_RNDN); /* 2*AG(1,4/s), err<=3 ulps */
mpfr_const_pi (tmp1, MPFR_RNDN); /* compute pi, err<=1ulp */
mpfr_div (tmp2, tmp1, tmp2, MPFR_RNDN); /* pi/2*AG(1,4/s), err<=5ulps */
mpfr_const_log2 (tmp1, MPFR_RNDN); /* compute log(2), err<=1ulp */
mpfr_mul_si (tmp1, tmp1, m, MPFR_RNDN); /* compute m*log(2),err<=2ulps */
mpfr_sub (tmp1, tmp2, tmp1, MPFR_RNDN); /* log(a), err<=7ulps+cancel */
if (MPFR_LIKELY (MPFR_IS_PURE_FP (tmp1) && MPFR_IS_PURE_FP (tmp2)))
{
cancel = MPFR_GET_EXP (tmp2) - MPFR_GET_EXP (tmp1);
MPFR_LOG_MSG (("canceled bits=%ld\n", (long) 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;
/* VL: I think it is better to have an increment that it isn't
too low; in particular, the increment must be positive even
if cancel = 0 (can this occur?). */
p += cancel + MPFR_INT_CEIL_LOG2 (p);
}
else
{
/* TODO: find why this case can occur and what is best to do
with it. */
p += MPFR_INT_CEIL_LOG2 (p);
}
MPFR_ZIV_NEXT (loop, p);
MPFR_GROUP_REPREC_2 (group, p, tmp1, tmp2);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (r, tmp1, rnd_mode);
/* We clean */
MPFR_GROUP_CLEAR (group);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (r, inexact, rnd_mode);
}
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