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/* mpfr_pow_ui -- compute the power of a floating-point by a machine integer
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"
#ifndef POW_U
#define POW_U mpfr_pow_ui
#define MPZ_SET_U mpz_set_ui
#define UTYPE unsigned long int
#define FSPEC "l"
#endif
/* sets y to x^n, and return 0 if exact, non-zero otherwise */
int
POW_U (mpfr_ptr y, mpfr_srcptr x, UTYPE n, mpfr_rnd_t rnd)
{
UTYPE m;
mpfr_t res;
mpfr_prec_t prec, err;
int inexact;
mpfr_rnd_t rnd1;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_ZIV_DECL (loop);
MPFR_BLOCK_DECL (flags);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg n=%" FSPEC "u rnd=%d",
mpfr_get_prec (x), mpfr_log_prec, x, n, rnd),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, inexact));
/* x^0 = 1 for any x, even a NaN */
if (MPFR_UNLIKELY (n == 0))
return mpfr_set_ui (y, 1, rnd);
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))
{
/* Inf^n = Inf, (-Inf)^n = Inf for n even, -Inf for n odd */
if (MPFR_IS_NEG (x) && (n & 1) == 1)
MPFR_SET_NEG (y);
else
MPFR_SET_POS (y);
MPFR_SET_INF (y);
MPFR_RET (0);
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
/* 0^n = 0 for any n */
MPFR_SET_ZERO (y);
if (MPFR_IS_POS (x) || (n & 1) == 0)
MPFR_SET_POS (y);
else
MPFR_SET_NEG (y);
MPFR_RET (0);
}
}
else if (MPFR_UNLIKELY (n <= 2))
{
if (n < 2)
/* x^1 = x */
return mpfr_set (y, x, rnd);
else
/* x^2 = sqr(x) */
return mpfr_sqr (y, x, rnd);
}
/* Augment exponent range */
MPFR_SAVE_EXPO_MARK (expo);
/* setup initial precision */
prec = MPFR_PREC (y) + 3 + GMP_NUMB_BITS
+ MPFR_INT_CEIL_LOG2 (MPFR_PREC (y));
mpfr_init2 (res, prec);
rnd1 = MPFR_IS_POS (x) ? MPFR_RNDU : MPFR_RNDD; /* away */
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
int i;
for (m = n, i = 0; m; i++, m >>= 1)
;
/* now 2^(i-1) <= n < 2^i */
MPFR_ASSERTD (prec > (mpfr_prec_t) i);
err = prec - 1 - (mpfr_prec_t) i;
/* First step: compute square from x */
MPFR_BLOCK (flags,
inexact = mpfr_sqr (res, x, MPFR_RNDU);
MPFR_ASSERTD (i >= 2);
if (n & ((UTYPE) 1 << (i-2)))
inexact |= mpfr_mul (res, res, x, rnd1);
for (i -= 3; i >= 0 && !MPFR_BLOCK_EXCEP; i--)
{
inexact |= mpfr_sqr (res, res, MPFR_RNDU);
if (n & ((UTYPE) 1 << i))
inexact |= mpfr_mul (res, res, x, rnd1);
});
/* let r(n) be the number of roundings: we have r(2)=1, r(3)=2,
and r(2n)=2r(n)+1, r(2n+1)=2r(n)+2, thus r(n)=n-1.
Using Higham's method, to each rounding corresponds a factor
(1-theta) with 0 <= theta <= 2^(1-p), thus at the end the
absolute error is bounded by (n-1)*2^(1-p)*res <= 2*(n-1)*ulp(res)
since 2^(-p)*x <= ulp(x). Since n < 2^i, this gives a maximal
error of 2^(1+i)*ulp(res).
*/
if (MPFR_LIKELY (inexact == 0
|| MPFR_OVERFLOW (flags) || MPFR_UNDERFLOW (flags)
|| MPFR_CAN_ROUND (res, err, MPFR_PREC (y), rnd)))
break;
/* Actualisation of the precision */
MPFR_ZIV_NEXT (loop, prec);
mpfr_set_prec (res, prec);
}
MPFR_ZIV_FREE (loop);
if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags) || MPFR_UNDERFLOW (flags)))
{
mpz_t z;
/* Internal overflow or underflow. However, the approximation error has
* not been taken into account. So, let's solve this problem by using
* mpfr_pow_z, which can handle it. This case could be improved in the
* future, without having to use mpfr_pow_z.
*/
MPFR_LOG_MSG (("Internal overflow or underflow,"
" let's use mpfr_pow_z.\n", 0));
mpfr_clear (res);
MPFR_SAVE_EXPO_FREE (expo);
mpz_init (z);
MPZ_SET_U (z, n);
inexact = mpfr_pow_z (y, x, z, rnd);
mpz_clear (z);
return inexact;
}
inexact = mpfr_set (y, res, rnd);
mpfr_clear (res);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd);
}
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