/* mpfr_root -- kth root.
Copyright 2005-2017 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
http://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 y = x^(1/k) is done as follows, except for large
values of k, for which this would be inefficient or yield internal
integer overflows:
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 + t where 0 <= t 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.
*/
static int
mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k,
mpfr_rnd_t rnd_mode);
int
mpfr_root (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
{
mpz_t m;
mpfr_exp_t e, r, sh, f;
mpfr_prec_t n, size_m, tmp;
int inexact, negative;
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("x[%Pu]=%.*Rg k=%lu rnd=%d",
mpfr_get_prec (x), mpfr_log_prec, x, k, rnd_mode),
("y[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (y), mpfr_log_prec, y, inexact));
if (MPFR_UNLIKELY (k <= 1))
{
if (k == 0)
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
else /* y = x^(1/1) = x */
return mpfr_set (y, x, rnd_mode);
}
/* Singular values */
if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
{
if (MPFR_IS_NAN (x))
{
MPFR_SET_NAN (y); /* NaN^(1/k) = NaN */
MPFR_RET_NAN;
}
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);
}
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 */
if (MPFR_UNLIKELY (MPFR_IS_NEG (x) && (k % 2 == 0)))
{
MPFR_SET_NAN (y);
MPFR_RET_NAN;
}
/* Special case |x| = 1. Note that if x = -1, then k is odd
(NaN results have already been filtered), so that y = -1. */
if (mpfr_cmpabs (x, __gmpfr_one) == 0)
return mpfr_set (y, x, rnd_mode);
/* General case */
/* For large k, use exp(log(x)/k). The threshold of 100 seems to be quite
good when the precision goes to infinity. */
if (k > 100)
return mpfr_root_aux (y, x, k, rnd_mode);
MPFR_SAVE_EXPO_MARK (expo);
mpz_init (m);
e = mpfr_get_z_2exp (m, x); /* x = m * 2^e */
if ((negative = MPFR_IS_NEG(x)))
mpz_neg (m, m);
r = e % (mpfr_exp_t) k;
if (r < 0)
r += k; /* now r = e (mod k) with 0 <= r < k */
MPFR_ASSERTD (0 <= r && r < k);
/* 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 == MPFR_RNDN);
/* we now multiply m by 2^sh so that root(m,k) will give
exactly n bits: we want k*(n-1)+1 <= size_m + sh <= k*n
i.e. sh = k*f + r with f = max(floor((k*n-size_m-r)/k),0) */
if ((mpfr_exp_t) size_m + r >= k * (mpfr_exp_t) n)
f = 0; /* we already have too many bits */
else
f = (k * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / k;
sh = k * f + r;
mpz_mul_2exp (m, m, sh);
e = e - sh;
/* invariant: x = m*2^e, with e divisible by k */
/* we reuse the variable m to store the kth 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 || ((mpfr_exp_t) mpz_scan1 (m, 0) < sh);
mpz_fdiv_q_2exp (m, m, sh);
e += k * sh;
}
if (inexact)
{
if (negative)
rnd_mode = MPFR_INVERT_RND (rnd_mode);
if (rnd_mode == MPFR_RNDU || rnd_mode == MPFR_RNDA
|| (rnd_mode == MPFR_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=MPFR_RNDN and bit (n+1) from m is 1 */
inexact += mpfr_set_z (y, m, MPFR_RNDN);
MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e / (mpfr_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);
}
/* Compute y <- x^(1/k) using exp(log(x)/k).
Assume all special cases have been eliminated before.
In the extended exponent range, overflows/underflows are not possible.
Assume x > 0, or x < 0 and k odd.
Also assume |x| <> 1 because log(1) = 0, which does not have an exponent
and would yield a failure in the error bound computation. A priori, this
constraint is quite artificial because if |x| is close enough to 1, then
the exponent of log|x| does not need to be used (in the code, err would
be 1 in such a domain). So this constraint |x| <> 1 could be avoided in
the code. However, this is an exact case easy to detect, so that such a
change would be useless. Values very close to 1 are not an issue, since
an underflow is not possible before the MPFR_GET_EXP.
*/
static int
mpfr_root_aux (mpfr_ptr y, mpfr_srcptr x, unsigned long k, mpfr_rnd_t rnd_mode)
{
int inexact, exact_root = 0;
mpfr_prec_t w; /* working precision */
mpfr_t absx, t;
MPFR_GROUP_DECL(group);
MPFR_TMP_DECL(marker);
MPFR_ZIV_DECL(loop);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_TMP_INIT_ABS (absx, x);
MPFR_TMP_MARK(marker);
w = MPFR_PREC(y) + 10;
/* Take some guard bits to prepare for the 'expt' lost bits below.
If |x| < 2^k, then log|x| < k, thus taking log2(k) bits should be fine. */
if (MPFR_GET_EXP(x) > 0)
w += MPFR_INT_CEIL_LOG2 (MPFR_GET_EXP(x));
MPFR_GROUP_INIT_1(group, w, t);
MPFR_SAVE_EXPO_MARK (expo);
MPFR_ZIV_INIT (loop, w);
for (;;)
{
mpfr_exp_t expt;
unsigned int err;
mpfr_log (t, absx, MPFR_RNDN);
/* t = log|x| * (1 + theta) with |theta| <= 2^(-w) */
mpfr_div_ui (t, t, k, MPFR_RNDN);
/* No possible underflow in mpfr_log and mpfr_div_ui. */
expt = MPFR_GET_EXP (t); /* assumes t <> 0 */
/* t = log|x|/k * (1 + theta) + eps with |theta| <= 2^(-w)
and |eps| <= 1/2 ulp(t), thus the total error is bounded
by 1.5 * 2^(expt - w) */
mpfr_exp (t, t, MPFR_RNDN);
/* t = |x|^(1/k) * exp(tau) * (1 + theta1) with
|tau| <= 1.5 * 2^(expt - w) and |theta1| <= 2^(-w).
For |tau| <= 0.5 we have |exp(tau)-1| < 4/3*tau, thus
for w >= expt + 2 we have:
t = |x|^(1/k) * (1 + 2^(expt+2)*theta2) * (1 + theta1) with
|theta1|, |theta2| <= 2^(-w).
If expt+2 > 0, as long as w >= 1, we have:
t = |x|^(1/k) * (1 + 2^(expt+3)*theta3) with |theta3| < 2^(-w).
For expt+2 = 0, we have:
t = |x|^(1/k) * (1 + 2^2*theta3) with |theta3| < 2^(-w).
Finally for expt+2 < 0 we have:
t = |x|^(1/k) * (1 + 2*theta3) with |theta3| < 2^(-w).
*/
err = (expt + 2 > 0) ? expt + 3
: (expt + 2 == 0) ? 2 : 1;
/* now t = |x|^(1/k) * (1 + 2^(err-w)) thus the error is at most
2^(EXP(t) - w + err) */
if (MPFR_LIKELY (MPFR_CAN_ROUND(t, w - err, MPFR_PREC(y), rnd_mode)))
break;
/* If we fail to round correctly, check for an exact result or a
midpoint result with MPFR_RNDN (regarded as hard-to-round in
all precisions in order to determine the ternary value). */
{
mpfr_t z, zk;
mpfr_init2 (z, MPFR_PREC(y) + (rnd_mode == MPFR_RNDN));
mpfr_init2 (zk, MPFR_PREC(x));
mpfr_set (z, t, MPFR_RNDN);
inexact = mpfr_pow_ui (zk, z, k, MPFR_RNDN);
exact_root = !inexact && mpfr_equal_p (zk, absx);
if (exact_root) /* z is the exact root, thus round z directly */
inexact = mpfr_set4 (y, z, rnd_mode, MPFR_SIGN (x));
mpfr_clear (zk);
mpfr_clear (z);
if (exact_root)
break;
}
MPFR_ZIV_NEXT (loop, w);
MPFR_GROUP_REPREC_1(group, w, t);
}
MPFR_ZIV_FREE (loop);
if (!exact_root)
inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (x));
MPFR_GROUP_CLEAR(group);
MPFR_TMP_FREE(marker);
MPFR_SAVE_EXPO_FREE (expo);
return mpfr_check_range (y, inexact, rnd_mode);
}