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/* mpfr_cbrt -- cube root function.
Copyright 2002-2020 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"
/* The computation of y = x^(1/3) is done as follows.
Let n = PREC(y), or PREC(y) + 1 if the rounding mode is MPFR_RNDN.
We seek to compute an integer cube root in precision n and the
associated inexact bit (non-zero iff the remainder is non-zero).
Let x = sign * m * 2^(3*e) where m is an integer >= 2^(3n-3), i.e.
m has at least 3n-2 bits.
Let s be the integer cube root of m, i.e. the maximum integer such that
m = s^3 + t with t >= 0.
TODO: Couldn't the size of m be fixed between 3n-2 and 3n? In the case
where the initial size of m is > 3n, if a discarded bit was non-zero,
this could be remembered for the inexact bit. Said otherwise, discard
3k bits of the mpz_root argument instead of discarding k bits of its
result (integer cube root).
The constraint m >= 2^(3n-3) allows one to have sufficient precision
for s: s >= 2^(n-1), i.e. s has at least n bits.
Let s' be s shifted to the right so that s' has exactly n bits.
Then |x|^(1/3) = s * 2^e or (s+1) * 2^e depending on the rounding mode,
the sign, and whether s' is inexact (t > 0 or some discarded bit in the
shift of s is non-zero).
*/
int
mpfr_cbrt (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
mpz_t m;
mpfr_exp_t e, sh;
mpfr_prec_t n, size_m, tmp;
int inexact, inexact2, negative, r;
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));
/* special values */
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))
{
MPFR_SET_INF (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
/* case 0: cbrt(+/- 0) = +/- 0 */
else /* x is necessarily 0 */
{
MPFR_ASSERTD (MPFR_IS_ZERO (x));
MPFR_SET_ZERO (y);
MPFR_SET_SAME_SIGN (y, x);
MPFR_RET (0);
}
}
/* General case */
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 % 3;
if (r < 0)
r += 3;
MPFR_ASSERTD (r >= 0 && r < 3 && (e - r) % 3 == 0);
/* x = (m*2^r) * 2^(e-r) = (m*2^r) * 2^(3*q) */
MPFR_LOG_MSG (("e=%" MPFR_EXP_FSPEC "d r=%d\n", (mpfr_eexp_t) e, r));
MPFR_MPZ_SIZEINBASE2 (size_m, m);
n = MPFR_PREC (y) + (rnd_mode == MPFR_RNDN);
/* We will need to shift m by r' bits to the left and subtract r' from e
so that m has at least 3n-2 bits and e becomes a multiple of 3.
Since r = e % 3, we write r' = 3 * sh + r.
If m already has at least 3n-2 bits, then we will use r' = r, so that
let us focus on the case size_m < 3 * n - 2.
We want 3 * n - 2 <= size_m + 3 * sh + r <= 3 * n.
Let d = 3 * n - size_m - r > 0. Thus we want 0 <= d - 3 * sh <= 2,
i.e. sh = floor(d/3) = trunc(d/3).
If size_m >= 3 * n - 2, then d <= 2, so that sh <= 0, whether a trunc
(ISO C99 and later) or a floor (possible before C99) is done with the
integer division; and the code will use r' = r as wanted. */
sh = (3 * (mpfr_exp_t) n - (mpfr_exp_t) size_m - r) / 3;
if (sh > 0)
r += 3 * sh; /* denoted r' above */
if (r > 0)
{
mpz_mul_2exp (m, m, r);
e -= r;
}
MPFR_ASSERTD (e % 3 == 0);
e /= 3;
/* invariant: x = m*2^(3*e) */
/* 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, 3) == 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 || (mpz_scan1 (m, 0) < sh);
mpz_fdiv_q_2exp (m, m, sh);
e += 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 */
inexact2 = mpfr_set_z (y, m, MPFR_RNDN);
MPFR_ASSERTD (inexact == 0 || inexact2 == 0);
inexact += inexact2;
MPFR_SET_EXP (y, MPFR_GET_EXP (y) + e);
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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