efee5258bc
The MPFR library is a C library for multiple-precision floating-point computations with exact rounding (also called correct rounding). It is based on the GMP multiple-precision library and should replace the MPF class in further releases of GMP. GCC >= 4.2 requires MPFR.
334 lines
11 KiB
C
334 lines
11 KiB
C
/* mpfr_exp -- exponential of a floating-point number
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Copyright 1999, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 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 /* for MPFR_MPZ_SIZEINBASE2 */
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#include "mpfr-impl.h"
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/* y <- exp(p/2^r) within 1 ulp, using 2^m terms from the series
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Assume |p/2^r| < 1.
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We use the following binary splitting formula:
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P(a,b) = p if a+1=b, P(a,c)*P(c,b) otherwise
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Q(a,b) = a*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise
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T(a,b) = P(a,b) if a+1=b, Q(c,b)*T(a,c)+P(a,c)*T(c,b) otherwise
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Then exp(p/2^r) ~ T(0,i)/Q(0,i) for i so that (p/2^r)^i/i! is small enough.
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Since P(a,b) = p^(b-a), and we consider only values of b-a of the form 2^j,
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we don't need to compute P(), we only precompute p^(2^j) in the ptoj[] array
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below.
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Since Q(a,b) is divisible by 2^(r*(b-a-1)), we don't compute the power of
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two part.
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*/
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static void
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mpfr_exp_rational (mpfr_ptr y, mpz_ptr p, long r, int m,
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mpz_t *Q, mpfr_prec_t *mult)
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{
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unsigned long n, i, j;
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mpz_t *S, *ptoj;
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mpfr_prec_t *log2_nb_terms;
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mpfr_exp_t diff, expo;
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mpfr_prec_t precy = MPFR_PREC(y), prec_i_have, prec_ptoj;
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int k, l;
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MPFR_ASSERTN ((size_t) m < sizeof (long) * CHAR_BIT - 1);
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S = Q + (m+1);
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ptoj = Q + 2*(m+1); /* ptoj[i] = mantissa^(2^i) */
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log2_nb_terms = mult + (m+1);
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/* Normalize p */
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MPFR_ASSERTD (mpz_cmp_ui (p, 0) != 0);
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n = mpz_scan1 (p, 0); /* number of trailing zeros in p */
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mpz_tdiv_q_2exp (p, p, n);
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r -= n; /* since |p/2^r| < 1 and p >= 1, r >= 1 */
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/* Set initial var */
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mpz_set (ptoj[0], p);
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for (k = 1; k < m; k++)
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mpz_mul (ptoj[k], ptoj[k-1], ptoj[k-1]); /* ptoj[k] = p^(2^k) */
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mpz_set_ui (Q[0], 1);
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mpz_set_ui (S[0], 1);
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k = 0;
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mult[0] = 0; /* the multiplier P[k]/Q[k] for the remaining terms
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satisfies P[k]/Q[k] <= 2^(-mult[k]) */
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log2_nb_terms[0] = 0; /* log2(#terms) [exact in 1st loop where 2^k] */
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prec_i_have = 0;
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/* Main Loop */
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n = 1UL << m;
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for (i = 1; (prec_i_have < precy) && (i < n); i++)
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{
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/* invariant: Q[0]*Q[1]*...*Q[k] equals i! */
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k++;
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log2_nb_terms[k] = 0; /* 1 term */
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mpz_set_ui (Q[k], i + 1);
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mpz_set_ui (S[k], i + 1);
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j = i + 1; /* we have computed j = i+1 terms so far */
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l = 0;
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while ((j & 1) == 0) /* combine and reduce */
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{
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/* invariant: S[k] corresponds to 2^l consecutive terms */
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mpz_mul (S[k], S[k], ptoj[l]);
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mpz_mul (S[k-1], S[k-1], Q[k]);
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/* Q[k] corresponds to 2^l consecutive terms too.
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Since it does not contains the factor 2^(r*2^l),
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when going from l to l+1 we need to multiply
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by 2^(r*2^(l+1))/2^(r*2^l) = 2^(r*2^l) */
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mpz_mul_2exp (S[k-1], S[k-1], r << l);
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mpz_add (S[k-1], S[k-1], S[k]);
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mpz_mul (Q[k-1], Q[k-1], Q[k]);
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log2_nb_terms[k-1] ++; /* number of terms in S[k-1]
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is a power of 2 by construction */
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MPFR_MPZ_SIZEINBASE2 (prec_i_have, Q[k]);
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MPFR_MPZ_SIZEINBASE2 (prec_ptoj, ptoj[l]);
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mult[k-1] += prec_i_have + (r << l) - prec_ptoj - 1;
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prec_i_have = mult[k] = mult[k-1];
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/* since mult[k] >= mult[k-1] + nbits(Q[k]),
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we have Q[0]*...*Q[k] <= 2^mult[k] = 2^prec_i_have */
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l ++;
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j >>= 1;
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k --;
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}
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}
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/* accumulate all products in S[0] and Q[0]. Warning: contrary to above,
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here we do not have log2_nb_terms[k-1] = log2_nb_terms[k]+1. */
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l = 0; /* number of accumulated terms in the right part S[k]/Q[k] */
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while (k > 0)
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{
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j = log2_nb_terms[k-1];
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mpz_mul (S[k], S[k], ptoj[j]);
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mpz_mul (S[k-1], S[k-1], Q[k]);
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l += 1 << log2_nb_terms[k];
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mpz_mul_2exp (S[k-1], S[k-1], r * l);
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mpz_add (S[k-1], S[k-1], S[k]);
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mpz_mul (Q[k-1], Q[k-1], Q[k]);
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k--;
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}
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/* Q[0] now equals i! */
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MPFR_MPZ_SIZEINBASE2 (prec_i_have, S[0]);
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diff = (mpfr_exp_t) prec_i_have - 2 * (mpfr_exp_t) precy;
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expo = diff;
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if (diff >= 0)
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mpz_fdiv_q_2exp (S[0], S[0], diff);
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else
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mpz_mul_2exp (S[0], S[0], -diff);
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MPFR_MPZ_SIZEINBASE2 (prec_i_have, Q[0]);
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diff = (mpfr_exp_t) prec_i_have - (mpfr_prec_t) precy;
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expo -= diff;
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if (diff > 0)
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mpz_fdiv_q_2exp (Q[0], Q[0], diff);
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else
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mpz_mul_2exp (Q[0], Q[0], -diff);
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mpz_tdiv_q (S[0], S[0], Q[0]);
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mpfr_set_z (y, S[0], MPFR_RNDD);
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MPFR_SET_EXP (y, MPFR_GET_EXP (y) + expo - r * (i - 1) );
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}
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#define shift (GMP_NUMB_BITS/2)
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int
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mpfr_exp_3 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
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{
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mpfr_t t, x_copy, tmp;
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mpz_t uk;
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mpfr_exp_t ttt, shift_x;
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unsigned long twopoweri;
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mpz_t *P;
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mpfr_prec_t *mult;
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int i, k, loop;
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int prec_x;
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mpfr_prec_t realprec, Prec;
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int iter;
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int inexact = 0;
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MPFR_SAVE_EXPO_DECL (expo);
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MPFR_ZIV_DECL (ziv_loop);
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MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
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("y[%#R]=%R inexact=%d", y, y, inexact));
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MPFR_SAVE_EXPO_MARK (expo);
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/* decompose x */
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/* we first write x = 1.xxxxxxxxxxxxx
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----- k bits -- */
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prec_x = MPFR_INT_CEIL_LOG2 (MPFR_PREC (x)) - MPFR_LOG2_GMP_NUMB_BITS;
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if (prec_x < 0)
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prec_x = 0;
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ttt = MPFR_GET_EXP (x);
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mpfr_init2 (x_copy, MPFR_PREC(x));
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mpfr_set (x_copy, x, MPFR_RNDD);
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/* we shift to get a number less than 1 */
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if (ttt > 0)
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{
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shift_x = ttt;
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mpfr_div_2ui (x_copy, x, ttt, MPFR_RNDN);
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ttt = MPFR_GET_EXP (x_copy);
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}
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else
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shift_x = 0;
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MPFR_ASSERTD (ttt <= 0);
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/* Init prec and vars */
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realprec = MPFR_PREC (y) + MPFR_INT_CEIL_LOG2 (prec_x + MPFR_PREC (y));
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Prec = realprec + shift + 2 + shift_x;
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mpfr_init2 (t, Prec);
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mpfr_init2 (tmp, Prec);
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mpz_init (uk);
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/* Main loop */
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MPFR_ZIV_INIT (ziv_loop, realprec);
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for (;;)
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{
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int scaled = 0;
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MPFR_BLOCK_DECL (flags);
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k = MPFR_INT_CEIL_LOG2 (Prec) - MPFR_LOG2_GMP_NUMB_BITS;
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/* now we have to extract */
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twopoweri = GMP_NUMB_BITS;
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/* Allocate tables */
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P = (mpz_t*) (*__gmp_allocate_func) (3*(k+2)*sizeof(mpz_t));
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for (i = 0; i < 3*(k+2); i++)
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mpz_init (P[i]);
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mult = (mpfr_prec_t*) (*__gmp_allocate_func) (2*(k+2)*sizeof(mpfr_prec_t));
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/* Particular case for i==0 */
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mpfr_extract (uk, x_copy, 0);
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MPFR_ASSERTD (mpz_cmp_ui (uk, 0) != 0);
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mpfr_exp_rational (tmp, uk, shift + twopoweri - ttt, k + 1, P, mult);
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for (loop = 0; loop < shift; loop++)
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mpfr_sqr (tmp, tmp, MPFR_RNDD);
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twopoweri *= 2;
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/* General case */
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iter = (k <= prec_x) ? k : prec_x;
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for (i = 1; i <= iter; i++)
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{
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mpfr_extract (uk, x_copy, i);
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if (MPFR_LIKELY (mpz_cmp_ui (uk, 0) != 0))
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{
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mpfr_exp_rational (t, uk, twopoweri - ttt, k - i + 1, P, mult);
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mpfr_mul (tmp, tmp, t, MPFR_RNDD);
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}
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MPFR_ASSERTN (twopoweri <= LONG_MAX/2);
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twopoweri *=2;
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}
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/* Clear tables */
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for (i = 0; i < 3*(k+2); i++)
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mpz_clear (P[i]);
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(*__gmp_free_func) (P, 3*(k+2)*sizeof(mpz_t));
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(*__gmp_free_func) (mult, 2*(k+2)*sizeof(mpfr_prec_t));
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if (shift_x > 0)
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{
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MPFR_BLOCK (flags, {
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for (loop = 0; loop < shift_x - 1; loop++)
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mpfr_sqr (tmp, tmp, MPFR_RNDD);
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mpfr_sqr (t, tmp, MPFR_RNDD);
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} );
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if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
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{
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/* tmp <= exact result, so that it is a real overflow. */
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inexact = mpfr_overflow (y, rnd_mode, 1);
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MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
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break;
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}
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if (MPFR_UNLIKELY (MPFR_UNDERFLOW (flags)))
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{
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/* This may be a spurious underflow. So, let's scale
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the result. */
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mpfr_mul_2ui (tmp, tmp, 1, MPFR_RNDD); /* no overflow, exact */
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mpfr_sqr (t, tmp, MPFR_RNDD);
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if (MPFR_IS_ZERO (t))
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{
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/* approximate result < 2^(emin - 3), thus
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exact result < 2^(emin - 2). */
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inexact = mpfr_underflow (y, (rnd_mode == MPFR_RNDN) ?
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MPFR_RNDZ : rnd_mode, 1);
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MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
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break;
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}
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scaled = 1;
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}
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}
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if (mpfr_can_round (shift_x > 0 ? t : tmp, realprec, MPFR_RNDD, MPFR_RNDZ,
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MPFR_PREC(y) + (rnd_mode == MPFR_RNDN)))
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{
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inexact = mpfr_set (y, shift_x > 0 ? t : tmp, rnd_mode);
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if (MPFR_UNLIKELY (scaled && MPFR_IS_PURE_FP (y)))
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{
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int inex2;
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mpfr_exp_t ey;
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/* The result has been scaled and needs to be corrected. */
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ey = MPFR_GET_EXP (y);
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inex2 = mpfr_mul_2si (y, y, -2, rnd_mode);
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if (inex2) /* underflow */
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{
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if (rnd_mode == MPFR_RNDN && inexact < 0 &&
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MPFR_IS_ZERO (y) && ey == __gmpfr_emin + 1)
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{
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/* Double rounding case: in MPFR_RNDN, the scaled
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result has been rounded downward to 2^emin.
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As the exact result is > 2^(emin - 2), correct
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rounding must be done upward. */
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/* TODO: make sure in coverage tests that this line
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is reached. */
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inexact = mpfr_underflow (y, MPFR_RNDU, 1);
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}
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else
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{
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/* No double rounding. */
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inexact = inex2;
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}
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MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
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}
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}
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break;
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}
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MPFR_ZIV_NEXT (ziv_loop, realprec);
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Prec = realprec + shift + 2 + shift_x;
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mpfr_set_prec (t, Prec);
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mpfr_set_prec (tmp, Prec);
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}
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MPFR_ZIV_FREE (ziv_loop);
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mpz_clear (uk);
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mpfr_clear (tmp);
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mpfr_clear (t);
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mpfr_clear (x_copy);
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MPFR_SAVE_EXPO_FREE (expo);
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return inexact;
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}
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