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/* mpfr_expm1 -- Compute exp(x)-1
Copyright 2001-2020 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 expm1 is done by
expm1(x)=exp(x)-1
*/
int
mpfr_expm1 (mpfr_ptr y, mpfr_srcptr x , mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_exp_t ex;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
inexact));
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* check for inf or -inf (expm1(-inf)=-1) */
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_POS (x))
{
MPFR_SET_INF (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
else
return mpfr_set_si (y, -1, rnd_mode);
}
else
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y); /* expm1(+/- 0) = +/- 0 */
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
ex = MPFR_GET_EXP (x);
if (ex < 0)
{
/* For -1 < x < 0, abs(expm1(x)-x) < x^2/2.
For 0 < x < 1, abs(expm1(x)-x) < x^2. */
if (MPFR_IS_POS (x))
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
else
MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 1, 0, rnd_mode, {});
}
MPFR_SAVE_EXPO_MARK (expo);
if (MPFR_IS_NEG (x) && ex > 5) /* x <= -32 */
{
mp_limb_t t_limb[(64 - 1) / GMP_NUMB_BITS + 1];
mpfr_t t;
mpfr_eexp_t err;
MPFR_TMP_INIT1(t_limb, t, 64);
/* since x < 0, to get an upper bound on x/log(2), we need to divide
by an upper bound on log(2) */
mpfr_div (t, x, __gmpfr_const_log2_RNDU, MPFR_RNDU); /* > x / ln(2) */
err = mpfr_get_exp_t (t, MPFR_RNDU);
MPFR_ASSERTD (err < 0);
err = err < - MPFR_EXP_MAX ? MPFR_EXP_MAX : - err;
/* err = -max(ceil(t),-MPFR_EXP_MAX). */
MPFR_LOG_MSG (("err=%" MPFR_EXP_FSPEC "d\n", err));
/* exp(x) = 2^(x/ln(2))
<= 2^max(-MPFR_EXP_MAX,ceil(x/ln(2)+epsilon))
with epsilon > 0 */
MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_mone, (mpfr_exp_t) err,
0, 0, rnd_mode, expo, {});
}
/* General case */
{
/* Declaration of the intermediary variable */
mpfr_t t;
/* Declaration of the size variable */
mpfr_prec_t Ny = MPFR_PREC(y); /* target precision */
mpfr_prec_t Nt; /* working precision */
mpfr_exp_t err, exp_te; /* error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
/* if |x| is smaller than 2^(-e), we will loose about e bits in the
subtraction exp(x) - 1 */
if (ex < 0)
Nt += - ex;
/* initialize auxiliary variable */
mpfr_init2 (t, Nt);
/* First computation of expm1 */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
MPFR_BLOCK_DECL (flags);
/* exp(x) may overflow and underflow */
MPFR_BLOCK (flags, mpfr_exp (t, x, MPFR_RNDN));
if (MPFR_OVERFLOW (flags)) /* overflow case */
{
inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN_POS);
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
break;
}
/* To get an underflow in exp(x), we need exp(x) < 0.5*2^MPFR_EMIN_MIN
thus x/log(2) < MPFR_EMIN_MIN-1. But in that case the above
MPFR_SMALL_INPUT_AFTER_SAVE_EXPO() will return the result. */
MPFR_ASSERTD(!MPFR_UNDERFLOW (flags));
exp_te = MPFR_GET_EXP (t);
mpfr_sub_ui (t, t, 1, MPFR_RNDN); /* exp(x)-1 */
/* error estimate */
/*err=Nt-(__gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))));*/
err = Nt - (MAX (exp_te - MPFR_GET_EXP (t), 0) + 1);
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
{
inexact = mpfr_set (y, t, rnd_mode);
break;
}
/* increase the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
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