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/* mpfr_exp2 -- power of 2 function 2^y
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"
/* TODO: mpfr_get_exp_t is called 3 times, with 3 different directed
rounding modes. One could reduce it to only one call thanks to the
inexact flag, but is it worth? */
/* Convert x to an mpfr_eexp_t integer, with saturation at the minimum
and maximum values. Flags are unchanged. */
static mpfr_eexp_t
round_to_eexp_t (mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpfr_flags_t flags = __gmpfr_flags;
mpfr_eexp_t e;
e = mpfr_get_exp_t (x, rnd_mode);
__gmpfr_flags = flags;
return e;
}
/* The computation of y = 2^z is done by *
* y = exp(z*log(2)). The result is exact iff z is an integer. */
int
mpfr_exp2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
int inexact;
mpfr_eexp_t xint; /* note: will fit in mpfr_exp_t */
mpfr_t xfrac;
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;
}
else if (MPFR_IS_INF (x))
{
if (MPFR_IS_POS (x))
MPFR_SET_INF (y);
else
MPFR_SET_ZERO (y);
MPFR_SET_POS (y);
MPFR_RET (0);
}
else /* 2^0 = 1 */
{
MPFR_ASSERTD (MPFR_IS_ZERO(x));
return mpfr_set_ui (y, 1, rnd_mode);
}
}
/* Since the smallest representable non-zero float is 1/2 * 2^emin,
if x <= emin - 2, the result is either 1/2 * 2^emin or 0.
Warning, for emin - 2 < x < emin - 1, we cannot conclude, since 2^x
might round to 2^(emin - 1) for rounding away or to nearest, and there
might be no underflow, since we consider underflow "after rounding". */
if (MPFR_UNLIKELY (round_to_eexp_t (x, MPFR_RNDU) <= __gmpfr_emin - 2))
return mpfr_underflow (y, rnd_mode == MPFR_RNDN ? MPFR_RNDZ : rnd_mode, 1);
if (MPFR_UNLIKELY (round_to_eexp_t (x, MPFR_RNDD) >= __gmpfr_emax))
return mpfr_overflow (y, rnd_mode, 1);
/* We now know that emin - 2 < x < emax. Note that an underflow or
overflow is still possible (we have eliminated only easy cases). */
MPFR_SAVE_EXPO_MARK (expo);
/* 2^x = 1 + x*log(2) + O(x^2) for x near zero, and for |x| <= 1 we have
|2^x - 1| <= x < 2^EXP(x). If x > 0 we must round away from 0 (dir=1);
if x < 0 we must round toward 0 (dir=0). */
MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, __gmpfr_one, - MPFR_GET_EXP (x), 0,
MPFR_IS_POS (x), rnd_mode, expo, {});
xint = mpfr_get_exp_t (x, MPFR_RNDZ);
MPFR_ASSERTD (__gmpfr_emin - 2 < xint && xint < __gmpfr_emax);
mpfr_init2 (xfrac, MPFR_PREC (x));
MPFR_DBGRES (inexact = mpfr_frac (xfrac, x, MPFR_RNDN));
MPFR_ASSERTD (inexact == 0);
if (MPFR_IS_ZERO (xfrac))
{
/* Here, emin - 1 <= x <= emax - 1, so that an underflow or overflow
will not be possible. */
mpfr_set_ui (y, 1, MPFR_RNDN);
inexact = 0;
}
else
{
/* 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; /* error */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
/* the optimal number of bits : see algorithms.tex */
Nt = Ny + 5 + MPFR_INT_CEIL_LOG2 (Ny);
/* initialize of intermediary variable */
mpfr_init2 (t, Nt);
/* First computation */
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute exp(x*ln(2))*/
mpfr_const_log2 (t, MPFR_RNDU); /* ln(2) */
mpfr_mul (t, xfrac, t, MPFR_RNDU); /* xfrac * ln(2) */
err = Nt - (MPFR_GET_EXP (t) + 2); /* Estimate of the error */
mpfr_exp (t, t, MPFR_RNDN); /* exp(xfrac * ln(2)) */
if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
break;
/* Actualisation of the precision */
MPFR_ZIV_NEXT (loop, Nt);
mpfr_set_prec (t, Nt);
}
MPFR_ZIV_FREE (loop);
inexact = mpfr_set (y, t, rnd_mode);
mpfr_clear (t);
}
mpfr_clear (xfrac);
if (MPFR_UNLIKELY (rnd_mode == MPFR_RNDN && xint == __gmpfr_emin - 1 &&
MPFR_GET_EXP (y) == 0 && mpfr_powerof2_raw (y)))
{
/* y was rounded down to 1/2 and the rounded value with an unbounded
exponent range would be 2^(emin-2), i.e. the midpoint between 0
and the smallest positive FP number. This is a double rounding
problem: we should not round to 0, but to (1/2) * 2^emin. */
MPFR_SET_EXP (y, __gmpfr_emin);
inexact = 1;
MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
}
else
{
/* The following is OK due to early overflow/underflow checking.
the exponent may be slightly out-of-range, but this will be
handled by mpfr_check_range. */
MPFR_EXP (y) += xint;
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
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