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/* mpfr_root -- kth root.
Copyright 2005 Free Software Foundation.
Contributed by the Spaces project, INRIA Lorraine.
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"
/* The computation of y = x^(1/k) is done as follows:
Let x = sign * m * 2^(k*e) where m is an integer
with 2^(k*(n-1)) <= m < 2^(k*n) where n = PREC(y)
and m = s^k + r where 0 <= r and m < (s+1)^k
we want that s has n bits i.e. s >= 2^(n-1), or m >= 2^(k*(n-1))
i.e. m must have at least k*(n-1)+1 bits
then, not taking into account the sign, the result will be
x^(1/k) = s * 2^e or (s+1) * 2^e according to the rounding mode.
*/
int
mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mp_rnd_t rnd_mode)
{
mpz_t m;
mp_exp_t e, r, sh;
mp_prec_t n, size_m, tmp;
int inexact, negative;
MPFR_SAVE_EXPO_DECL (expo);
if (MPFR_UNLIKELY (k <= 1))
{
if (k < 1) /* k==0 => y=x^(1/0)=x^(+Inf) */
#if 0
/* For 0 <= x < 1 => +0.
For x = 1 => 1.
For x > 1, => +Inf.
For x < 0 => NaN.
*/
{
if (MPFR_IS_NEG (x) && !MPFR_IS_ZERO (x))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
inexact = mpfr_cmp (x, __gmpfr_one);
if (inexact == 0)
return mpfr_set_ui (y, 1, rnd_mode); /* 1 may be Out of Range */
else if (inexact < 0)
return mpfr_set_ui (y, 0, rnd_mode); /* 0+ */
else
{
mpfr_set_inf (y, 1);
return 0;
}
}
#endif
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* y =x^(1/1)=x */
return mpfr_set (y, x, rnd_mode);
}
/* Singular values */
else if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */
MPFR_RET_NAN;
}
else if (MPFR_IS_INF (x)) /* +Inf^(1/k) = +Inf
-Inf^(1/k) = -Inf if k odd
-Inf^(1/k) = NaN if k even */
{
if (MPFR_IS_NEG(x) && (k % 2 == 0))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
MPFR_SET_INF (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
else /* x is necessarily 0: (+0)^(1/k) = +0
(-0)^(1/k) = -0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
/* Returns NAN for x < 0 and k even */
else if (MPFR_IS_NEG (x) && (k % 2 == 0))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* General case */
MPFR_SAVE_EXPO_MARK (expo);
mpz_init (m);
e = mpfr_get_z_exp (m, x); /* x = m * 2^e */
if ((negative = MPFR_IS_NEG(x)))
mpz_neg (m, m);
r = e % (mp_exp_t) k;
if (r < 0)
r += k; /* now r = e (mod k) with 0 <= e < r */
/* x = (m*2^r) * 2^(e-r) where e-r is a multiple of k */
MPFR_MPZ_SIZEINBASE2 (size_m, m);
/* for rounding to nearest, we want the round bit to be in the root */
n = MPFR_PREC (y) + (rnd_mode == GMP_RNDN);
/* we now multiply m by 2^(r+k*sh) so that root(m,k) will give
exactly n bits: we want k*(n-1)+1 <= size_m + k*sh + r <= k*n
i.e. sh = floor ((kn-size_m-r)/k) */
if ((mp_exp_t) size_m + r > k * (mp_exp_t) n)
sh = 0; /* we already have too many bits */
else
sh = (k * (mp_exp_t) n - (mp_exp_t) size_m - r) / k;
sh = k * sh + r;
if (sh >= 0)
{
mpz_mul_2exp (m, m, sh);
e = e - sh;
}
else if (r > 0)
{
mpz_mul_2exp (m, m, r);
e = e - r;
}
/* invariant: x = m*2^e, with e divisible by k */
/* we reuse the variable m to store the cube root, since it is not needed
any more: we just need to know if the root is exact */
inexact = mpz_root (m, m, k) == 0;
MPFR_MPZ_SIZEINBASE2 (tmp, m);
sh = tmp - n;
if (sh > 0) /* we have to flush to 0 the last sh bits from m */
{
inexact = inexact || ((mp_exp_t) mpz_scan1 (m, 0) < sh);
mpz_div_2exp (m, m, sh);
e += k * sh;
}
if (inexact)
{
if (negative)
rnd_mode = MPFR_INVERT_RND (rnd_mode);
if (rnd_mode == GMP_RNDU
|| (rnd_mode == GMP_RNDN && mpz_tstbit (m, 0)))
inexact = 1, mpz_add_ui (m, m, 1);
else
inexact = -1;
}
/* either inexact is not zero, and the conversion is exact, i.e. inexact
is not changed; or inexact=0, and inexact is set only when
rnd_mode=GMP_RNDN and bit (n+1) from m is 1 */
inexact += mpfr_set_z (y, m, GMP_RNDN);
MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / (mp_exp_t) k);
if (negative)
{
MPFR_CHANGE_SIGN (y);
inexact = -inexact;
}
mpz_clear (m);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}
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