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b58ddff026
* merged mpfr 3.0.0 and gmp 5.0.1 in buildtools trunk git-svn-id: file:///srv/svn/repos/haiku/buildtools/trunk@37378 a95241bf-73f2-0310-859d-f6bbb57e9c96
251 lines
10 KiB
C
251 lines
10 KiB
C
/* mpfr_pow_si -- power function x^y with y a signed int
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Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free Software Foundation, Inc.
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Contributed by the Arenaire and Cacao projects, INRIA.
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This file is part of the GNU MPFR Library.
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The GNU MPFR Library is free software; you can redistribute it and/or modify
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it under the terms of the GNU Lesser General Public License as published by
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the Free Software Foundation; either version 3 of the License, or (at your
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option) any later version.
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The GNU MPFR Library is distributed in the hope that it will be useful, but
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WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
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or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
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License for more details.
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You should have received a copy of the GNU Lesser General Public License
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along with the GNU MPFR Library; see the file COPYING.LESSER. If not, see
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http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
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51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
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#define MPFR_NEED_LONGLONG_H
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#include "mpfr-impl.h"
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/* The computation of y = pow_si(x,n) is done by
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* y = pow_ui(x,n) if n >= 0
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* y = 1 / pow_ui(x,-n) if n < 0
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*/
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int
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mpfr_pow_si (mpfr_ptr y, mpfr_srcptr x, long int n, mpfr_rnd_t rnd)
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{
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MPFR_LOG_FUNC (("x[%#R]=%R n=%ld rnd=%d", x, x, n, rnd),
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("y[%#R]=%R", y, y));
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if (n >= 0)
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return mpfr_pow_ui (y, x, n, rnd);
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else
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{
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if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
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{
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if (MPFR_IS_NAN (x))
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{
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MPFR_SET_NAN (y);
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MPFR_RET_NAN;
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}
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else if (MPFR_IS_INF (x))
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{
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MPFR_SET_ZERO (y);
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if (MPFR_IS_POS (x) || ((unsigned) n & 1) == 0)
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MPFR_SET_POS (y);
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else
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MPFR_SET_NEG (y);
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MPFR_RET (0);
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}
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else /* x is zero */
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{
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MPFR_ASSERTD (MPFR_IS_ZERO (x));
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MPFR_SET_INF(y);
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if (MPFR_IS_POS (x) || ((unsigned) n & 1) == 0)
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MPFR_SET_POS (y);
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else
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MPFR_SET_NEG (y);
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MPFR_RET(0);
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}
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}
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/* detect exact powers: x^(-n) is exact iff x is a power of 2 */
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if (mpfr_cmp_si_2exp (x, MPFR_SIGN(x), MPFR_EXP(x) - 1) == 0)
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{
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mpfr_exp_t expx = MPFR_EXP (x) - 1, expy;
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MPFR_ASSERTD (n < 0);
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/* Warning: n * expx may overflow!
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*
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* Some systems (apparently alpha-freebsd) abort with
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* LONG_MIN / 1, and LONG_MIN / -1 is undefined.
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* http://www.freebsd.org/cgi/query-pr.cgi?pr=72024
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*
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* Proof of the overflow checking. The expressions below are
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* assumed to be on the rational numbers, but the word "overflow"
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* still has its own meaning in the C context. / still denotes
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* the integer (truncated) division, and // denotes the exact
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* division.
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* - First, (__gmpfr_emin - 1) / n and (__gmpfr_emax - 1) / n
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* cannot overflow due to the constraints on the exponents of
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* MPFR numbers.
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* - If n = -1, then n * expx = - expx, which is representable
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* because of the constraints on the exponents of MPFR numbers.
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* - If expx = 0, then n * expx = 0, which is representable.
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* - If n < -1 and expx > 0:
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* + If expx > (__gmpfr_emin - 1) / n, then
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* expx >= (__gmpfr_emin - 1) / n + 1
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* > (__gmpfr_emin - 1) // n,
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* and
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* n * expx < __gmpfr_emin - 1,
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* i.e.
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* n * expx <= __gmpfr_emin - 2.
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* This corresponds to an underflow, with a null result in
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* the rounding-to-nearest mode.
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* + If expx <= (__gmpfr_emin - 1) / n, then n * expx cannot
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* overflow since 0 < expx <= (__gmpfr_emin - 1) / n and
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* 0 > n * expx >= n * ((__gmpfr_emin - 1) / n)
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* >= __gmpfr_emin - 1.
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* - If n < -1 and expx < 0:
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* + If expx < (__gmpfr_emax - 1) / n, then
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* expx <= (__gmpfr_emax - 1) / n - 1
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* < (__gmpfr_emax - 1) // n,
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* and
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* n * expx > __gmpfr_emax - 1,
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* i.e.
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* n * expx >= __gmpfr_emax.
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* This corresponds to an overflow (2^(n * expx) has an
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* exponent > __gmpfr_emax).
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* + If expx >= (__gmpfr_emax - 1) / n, then n * expx cannot
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* overflow since 0 > expx >= (__gmpfr_emax - 1) / n and
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* 0 < n * expx <= n * ((__gmpfr_emax - 1) / n)
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* <= __gmpfr_emax - 1.
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* Note: one could use expx bounds based on MPFR_EXP_MIN and
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* MPFR_EXP_MAX instead of __gmpfr_emin and __gmpfr_emax. The
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* current bounds do not lead to noticeably slower code and
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* allow us to avoid a bug in Sun's compiler for Solaris/x86
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* (when optimizations are enabled); known affected versions:
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* cc: Sun C 5.8 2005/10/13
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* cc: Sun C 5.8 Patch 121016-02 2006/03/31
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* cc: Sun C 5.8 Patch 121016-04 2006/10/18
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*/
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expy =
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n != -1 && expx > 0 && expx > (__gmpfr_emin - 1) / n ?
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MPFR_EMIN_MIN - 2 /* Underflow */ :
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n != -1 && expx < 0 && expx < (__gmpfr_emax - 1) / n ?
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MPFR_EMAX_MAX /* Overflow */ : n * expx;
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return mpfr_set_si_2exp (y, n % 2 ? MPFR_INT_SIGN (x) : 1,
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expy, rnd);
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}
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/* General case */
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{
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/* Declaration of the intermediary variable */
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mpfr_t t;
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/* Declaration of the size variable */
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mpfr_prec_t Ny; /* target precision */
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mpfr_prec_t Nt; /* working precision */
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mpfr_rnd_t rnd1;
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int size_n;
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int inexact;
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unsigned long abs_n;
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MPFR_SAVE_EXPO_DECL (expo);
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MPFR_ZIV_DECL (loop);
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abs_n = - (unsigned long) n;
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count_leading_zeros (size_n, (mp_limb_t) abs_n);
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size_n = GMP_NUMB_BITS - size_n;
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/* initial working precision */
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Ny = MPFR_PREC (y);
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Nt = Ny + size_n + 3 + MPFR_INT_CEIL_LOG2 (Ny);
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MPFR_SAVE_EXPO_MARK (expo);
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/* initialise of intermediary variable */
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mpfr_init2 (t, Nt);
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/* We will compute rnd(rnd1(1/x) ^ |n|), where rnd1 is the rounding
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toward sign(x), to avoid spurious overflow or underflow, as in
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mpfr_pow_z. */
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rnd1 = MPFR_EXP (x) < 1 ? MPFR_RNDZ :
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(MPFR_SIGN (x) > 0 ? MPFR_RNDU : MPFR_RNDD);
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MPFR_ZIV_INIT (loop, Nt);
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for (;;)
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{
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MPFR_BLOCK_DECL (flags);
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/* compute (1/x)^|n| */
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MPFR_BLOCK (flags, mpfr_ui_div (t, 1, x, rnd1));
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MPFR_ASSERTD (! MPFR_UNDERFLOW (flags));
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/* t = (1/x)*(1+theta) where |theta| <= 2^(-Nt) */
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if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
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goto overflow;
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MPFR_BLOCK (flags, mpfr_pow_ui (t, t, abs_n, rnd));
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/* t = (1/x)^|n|*(1+theta')^(|n|+1) where |theta'| <= 2^(-Nt).
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If (|n|+1)*2^(-Nt) <= 1/2, which is satisfied as soon as
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Nt >= bits(n)+2, then we can use Lemma \ref{lemma_graillat}
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from algorithms.tex, which yields x^n*(1+theta) with
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|theta| <= 2(|n|+1)*2^(-Nt), thus the error is bounded by
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2(|n|+1) ulps <= 2^(bits(n)+2) ulps. */
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if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
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{
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overflow:
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MPFR_ZIV_FREE (loop);
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mpfr_clear (t);
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MPFR_SAVE_EXPO_FREE (expo);
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MPFR_LOG_MSG (("overflow\n", 0));
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return mpfr_overflow (y, rnd, abs_n & 1 ?
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MPFR_SIGN (x) : MPFR_SIGN_POS);
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}
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if (MPFR_UNLIKELY (MPFR_UNDERFLOW (flags)))
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{
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MPFR_ZIV_FREE (loop);
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mpfr_clear (t);
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MPFR_LOG_MSG (("underflow\n", 0));
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if (rnd == MPFR_RNDN)
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{
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mpfr_t y2, nn;
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/* We cannot decide now whether the result should be
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rounded toward zero or away from zero. So, like
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in mpfr_pow_pos_z, let's use the general case of
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mpfr_pow in precision 2. */
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MPFR_ASSERTD (mpfr_cmp_si_2exp (x, MPFR_SIGN (x),
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MPFR_EXP (x) - 1) != 0);
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mpfr_init2 (y2, 2);
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mpfr_init2 (nn, sizeof (long) * CHAR_BIT);
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inexact = mpfr_set_si (nn, n, MPFR_RNDN);
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MPFR_ASSERTN (inexact == 0);
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inexact = mpfr_pow_general (y2, x, nn, rnd, 1,
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(mpfr_save_expo_t *) NULL);
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mpfr_clear (nn);
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mpfr_set (y, y2, MPFR_RNDN);
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mpfr_clear (y2);
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MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
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goto end;
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}
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else
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{
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MPFR_SAVE_EXPO_FREE (expo);
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return mpfr_underflow (y, rnd, abs_n & 1 ?
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MPFR_SIGN (x) : MPFR_SIGN_POS);
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}
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}
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/* error estimate -- see pow function in algorithms.ps */
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if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - size_n - 2, Ny, rnd)))
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break;
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/* actualisation of the precision */
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MPFR_ZIV_NEXT (loop, Nt);
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mpfr_set_prec (t, Nt);
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}
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MPFR_ZIV_FREE (loop);
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inexact = mpfr_set (y, t, rnd);
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mpfr_clear (t);
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end:
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MPFR_SAVE_EXPO_FREE (expo);
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return mpfr_check_range (y, inexact, rnd);
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}
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}
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}
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