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/* mpfr_pow_z -- power function x^z with z a MPZ
Copyright 2005 Free Software Foundation, Inc.
This file is part of the MPFR Library.
The 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 2.1 of the License, or (at your
option) any later version.
The 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 MPFR Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin Place, Fifth Floor, Boston,
MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
static int
mpfr_pow_pos_z (mpfr_ptr y, mpfr_srcptr x, mpz_srcptr z, mp_rnd_t rnd)
{
mpfr_t res;
mp_prec_t prec, err;
int inexact;
mp_rnd_t rnd1;
mpz_t absz;
mp_size_t size_z;
MPFR_ZIV_DECL (loop);
MPFR_ASSERTD (mpz_sgn (z) != 0);
if (MPFR_UNLIKELY (mpz_cmpabs_ui (z, 1) == 0))
return mpfr_set (y, x, rnd);
rnd1 = MPFR_IS_POS (x) ? GMP_RNDU : GMP_RNDD; /* away */
absz[0] = z[0];
SIZ (absz) = ABS(SIZ(absz)); /* Hack to get abs(z) */
MPFR_MPZ_SIZEINBASE2 (size_z, z);
prec = MPFR_PREC (y) + 3 + size_z + MPFR_INT_CEIL_LOG2 (MPFR_PREC (y));
mpfr_init2 (res, prec);
MPFR_ZIV_INIT (loop, prec);
for (;;)
{
mp_size_t i = size_z;
/* now 2^(i-1) <= z < 2^i */
/* see pow_ui.c for the error analusis, which is identical */
MPFR_ASSERTD (prec > (mpfr_prec_t) i);
err = prec - 1 - (mpfr_prec_t) i;
MPFR_ASSERTD (i >= 2);
mpfr_clear_overflow ();
mpfr_clear_underflow ();
/* First step: compute square from y */
inexact = mpfr_mul (res, x, x, GMP_RNDU);
if (mpz_tstbit (absz, i-2))
inexact |= mpfr_mul (res, res, x, rnd1);
for (i -= 3; i >= 0 && !mpfr_underflow_p() && !mpfr_overflow_p (); i--)
{
inexact |= mpfr_mul (res, res, res, GMP_RNDU);
if (mpz_tstbit (absz, i))
inexact |= mpfr_mul (res, res, x, rnd1);
}
/* printf ("pow_z ");
mpfr_dump_mant (MPFR_MANT (res), prec, MPFR_PREC (x), err); */
if (MPFR_LIKELY (inexact == 0
|| mpfr_overflow_p () || mpfr_underflow_p ()
|| 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);
inexact = mpfr_set (y, res, rnd);
mpfr_clear (res);
/* Check Overflow */
if (MPFR_UNLIKELY (mpfr_overflow_p ()))
return mpfr_overflow (y, rnd,
mpz_odd_p (absz) ? MPFR_SIGN (x) : MPFR_SIGN_POS);
/* Check Underflow */
else if (MPFR_UNLIKELY (mpfr_underflow_p ()))
{
if (rnd == GMP_RNDN)
rnd = GMP_RNDZ;
return mpfr_underflow (y, rnd,
mpz_odd_p (absz) ? MPFR_SIGN (x) : MPFR_SIGN_POS);
}
return inexact;
}
/* The computation of y=pow(x,z) is done by
* y=pow_ui(x,z) if z>0
* else
* y=1/pow_ui(x,z) if z<0
*/
int
mpfr_pow_z (mpfr_ptr y, mpfr_srcptr x, mpz_srcptr z, mp_rnd_t rnd)
{
int inexact;
mpz_t tmp;
MPFR_SAVE_EXPO_DECL (expo);
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else if (mpz_sgn (z) == 0) /* y^0 = 1 for any y except NAN */
return mpfr_set_ui (y, 1, rnd);
else if (MPFR_IS_INF (x))
{
/* Inf^n = Inf, (-Inf)^n = Inf for n even, -Inf for n odd */
/* Inf ^(-n) = 0, sign = + if x>0 or z even */
if (mpz_sgn (z) > 0)
MPFR_SET_INF (y);
else
MPFR_SET_ZERO (y);
if (MPFR_UNLIKELY (MPFR_IS_NEG (x) && mpz_odd_p (z)))
MPFR_SET_NEG (y);
else
MPFR_SET_POS (y);
MPFR_RET (0);
}
else /* x is zero */
{
MPFR_ASSERTD (MPFR_IS_ZERO(x));
if (mpz_sgn (z) > 0)
/* 0^n = +/-0 for any n */
MPFR_SET_ZERO (y);
else
/* 0^(-n) if +/- INF */
MPFR_SET_INF (y);
if (MPFR_LIKELY (MPFR_IS_POS (x) || mpz_even_p (z)))
MPFR_SET_POS (y);
else
MPFR_SET_NEG (y);
MPFR_RET(0);
}
}
if (MPFR_UNLIKELY (mpz_sgn (z) == 0))
/* y^0 = 1 for any y except NAN */
return mpfr_set_ui (y, 1, rnd);
/* detect exact powers: x^-n is exact iff x is a power of 2
Do it if n > 0 too (faster). */
if (MPFR_UNLIKELY (mpfr_cmp_si_2exp (x, MPFR_SIGN (x),
MPFR_EXP (x) - 1) == 0))
{
mp_exp_t expx = MPFR_EXP (x); /* warning: x and y may be the same
variable */
mpfr_set_si (y, mpz_odd_p (z) ? MPFR_INT_SIGN(x) : 1, rnd);
MPFR_ASSERTD (MPFR_IS_FP (y));
mpz_init (tmp);
mpz_mul_si (tmp, z, expx-1);
MPFR_ASSERTD (MPFR_GET_EXP (y) == 1);
mpz_add_ui (tmp, tmp, 1);
inexact = 0;
if (MPFR_UNLIKELY (mpz_cmp_si (tmp, __gmpfr_emin) < 0))
{
/* The following test is necessary because in the rounding to the
* nearest mode, mpfr_underflow always rounds away from 0. In
* this rounding mode, we need to round to 0 if:
* _ |y| < 2^(emin-2), or
* _ |y| = 2^(emin-2) and the absolute value of the exact
* result is <= 2^(emin-2).
* NOTE: y is a power of 2 and inexact = 0!
*/
if (rnd == GMP_RNDN && mpz_cmp_si (tmp, __gmpfr_emin-1) < 0)
rnd = GMP_RNDZ;
inexact = mpfr_underflow (y, rnd, MPFR_SIGN (y));
}
else if (MPFR_UNLIKELY (mpz_cmp_si (tmp, __gmpfr_emax) > 0))
inexact = mpfr_overflow (y, rnd, MPFR_SIGN(x));
else
MPFR_SET_EXP (y, mpz_get_si (tmp));
mpz_clear (tmp);
MPFR_RET (inexact);
}
MPFR_SAVE_EXPO_MARK (expo);
__gmpfr_emin -= 3; /* So that we can check for underflow properly */
if (mpz_sgn (z) > 0)
inexact = mpfr_pow_pos_z (y, x, z, rnd);
else
{
/* Declaration of the intermediary variable */
mpfr_t t;
mp_prec_t Nt; /* Precision of the intermediary variable */
MPFR_ZIV_DECL (loop);
/* compute the precision of intermediary variable */
Nt = MAX (MPFR_PREC (x), MPFR_PREC (y));
/* the optimal number of bits : see algorithms.ps */
Nt = Nt + 3 + MPFR_INT_CEIL_LOG2 (Nt);
/* initialise of intermediary variable */
mpfr_init2 (t, Nt);
MPFR_ZIV_INIT (loop, Nt);
for (;;)
{
/* compute 1/(x^n) n>0 */
mpfr_pow_pos_z (t, x, z, GMP_RNDN);
inexact = MPFR_IS_ZERO (t) || MPFR_IS_INF (t);
mpfr_ui_div (t, 1, t, GMP_RNDN);
inexact = inexact || MPFR_IS_ZERO (t) || MPFR_IS_INF (t);
if (MPFR_LIKELY (inexact != 0
|| MPFR_CAN_ROUND (t, Nt-3, MPFR_PREC (y), rnd)))
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);
mpfr_clear (t);
}
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd);
}
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