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/* mpfr_exp10m1 -- Compute 10^x-1
Copyright 2001-2021 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 exp10m1 is done by expm1(x) = 10^x-1 */
/* In case x is small in absolute value, 10^x - 1 ~ x*log(10).
If this is enough to deduce correct rounding, put in the auxiliary variable
t the approximation that will be rounded to get y, and return non-zero.
If we put 0 in t, it means underflow.
Otherwise return 0. */
static int
mpfr_exp10m1_small (mpfr_ptr y, mpfr_srcptr x , mpfr_rnd_t rnd_mode,
mpfr_ptr t)
{
mpfr_prec_t prec;
mpfr_exp_t e;
/* for |x| < 0.25, we have |10^x-1-x*log(10)| < 4*x^2 */
if (MPFR_EXP(x) > -2)
return 0;
/* now EXP(x) <= -2, thus x < 0.25 */
prec = MPFR_PREC(t);
mpfr_log_ui (t, 10, MPFR_RNDN);
/* t = log(10)*(1 + theta) with |theta| <= 2^(-prec) */
mpfr_mul (t, t, x, MPFR_RNDN);
/* no underflow can occur, since log(10) > 1 */
/* t = x*log(10)*(1 + theta)^2 with |theta| <= 2^(-prec) */
/* |t - x*log(10)| <= ((1 + theta)^2 - 1) * |t| <= 3*2^(-prec)*|t| */
/* |t - x*log(10)| < 3*2^(EXP(t)-prec) */
e = 2 * MPFR_GET_EXP (x) + 2 + prec - MPFR_GET_EXP(t);
/* |4*x^2| < 2^e*2^(EXP(t)-prec) thus
|t - exp10m1(x)| < (3+2^e)*2^(EXP(t)-prec) */
e = (e <= 1) ? 2 + (e == 1) : e + 1;
/* now |t - exp10m1(x)| < 2^e*2^(EXP(t)-prec) */
return MPFR_CAN_ROUND (t, prec - e, MPFR_PREC(y), rnd_mode);
}
int
mpfr_exp10m1 (mpfr_ptr y, mpfr_srcptr x , mpfr_rnd_t rnd_mode)
{
int inexact, nloop;
mpfr_t t;
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);
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_IS_SINGULAR (x))
return mpfr_expm1 (y, x, rnd_mode); /* singular cases are identical */
MPFR_ASSERTN(!MPFR_IS_ZERO(x));
MPFR_SAVE_EXPO_MARK (expo);
/* Check case where result is -1 or nextabove(-1) because x is a huge
negative number. */
if (MPFR_IS_NEG(x) && mpfr_cmpabs_ui (x, 2 + (MPFR_PREC(y) - 1) / 3) > 0)
{
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_INEXACT);
/* 1/2*ulp(-1) = 2^(-PREC(y)):
since |x| >= PREC(y)/3 + 1, then 3*|x| >= PREC(y) + 3,
thus 10^x < 8^x <= 2^(-PREC(y)-3) <= 1/2*ulp(-1), thus the
result is -1 for RNDA,RNDD,RNDN, and nextabove(-1) for RNDZ,RNDU */
mpfr_set_si (y, -1, MPFR_RNDZ);
if (!MPFR_IS_LIKE_RNDZ(rnd_mode,1))
inexact = -1;
else
{
mpfr_nextabove (y);
inexact = 1;
}
goto end;
}
Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
mpfr_init2 (t, Nt);
MPFR_ZIV_INIT (loop, Nt);
for (nloop = 0;; nloop++)
{
int inex1;
MPFR_BLOCK_DECL (flags);
/* 10^x may overflow and underflow */
MPFR_BLOCK (flags, inex1 = mpfr_exp10 (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);
goto clear;
}
/* integer case */
if (inex1 == 0)
{
inexact = mpfr_sub_ui (y, t, 1, rnd_mode);
goto clear;
}
/* The case of underflow in 10^x (huge negative x)
was already detected before Ziv's loop. */
MPFR_ASSERTD(!MPFR_UNDERFLOW (flags));
MPFR_ASSERTN(!MPFR_IS_ZERO(t));
exp_te = MPFR_GET_EXP (t);
mpfr_sub_ui (t, t, 1, MPFR_RNDN); /* 10^x-1 */
/* error estimate */
/* err = __gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))) */
if (!MPFR_IS_ZERO(t))
{
err = MAX (exp_te - MPFR_GET_EXP (t), 0) + 1;
/* if inex1=0, this means that t=o(10^x) is exact, thus the correct
rounding is simply o(t-1) */
if (inex1 == 0 ||
MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - err, Ny, rnd_mode)))
break;
}
/* check small case: we need to do it at each step of Ziv's loop,
since the multiplication x*log(10) might not enable correct
rounding at the first loop */
if (mpfr_exp10m1_small (y, x, rnd_mode, t))
{
if (MPFR_IS_ZERO(t)) /* underflow */
{
mpfr_clear (t);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_underflow (y, (rnd_mode == MPFR_RNDN) ? MPFR_RNDZ
: rnd_mode, 1);
}
break;
}
/* increase the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
inexact = mpfr_set (y, t, rnd_mode);
clear:
MPFR_ZIV_FREE (loop);
mpfr_clear (t);
end:
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
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