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243 lines
7.8 KiB
C
243 lines
7.8 KiB
C
/* mpfr_log1p -- Compute log(1+x)
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Copyright 2001-2025 Free Software Foundation, Inc.
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Contributed by the Pascaline and Caramba 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.
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If not, see <https://www.gnu.org/licenses/>. */
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#define MPFR_NEED_LONGLONG_H
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#include "mpfr-impl.h"
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/* Put in y an approximation of log(1+x) for x small.
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We assume |x| < 1/2, in which case:
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|x/2| <= |log(1+x)| <= |2x|.
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Return k such that the error is bounded by 2^k*ulp(y).
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*/
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static int
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mpfr_log1p_small (mpfr_ptr y, mpfr_srcptr x)
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{
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mpfr_prec_t p = MPFR_PREC(y), err;
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mpfr_t t, u;
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unsigned long i;
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int k;
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MPFR_ASSERTD(MPFR_GET_EXP (x) <= -1); /* ensures |x| < 1/2 */
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/* in the following, theta represents a value with |theta| <= 2^(1-p)
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(might be a different value each time) */
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mpfr_init2 (t, p);
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mpfr_init2 (u, p);
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mpfr_set (t, x, MPFR_RNDF); /* t = x * (1 + theta) */
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mpfr_set (y, t, MPFR_RNDF); /* exact */
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for (i = 2; ; i++)
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{
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mpfr_mul (t, t, x, MPFR_RNDF); /* t = x^i * (1 + theta)^i */
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mpfr_div_ui (u, t, i, MPFR_RNDF); /* u = x^i/i * (1 + theta)^(i+1) */
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if (MPFR_GET_EXP (u) <= MPFR_GET_EXP (y) - p) /* |u| < ulp(y) */
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break;
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if (i & 1)
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mpfr_add (y, y, u, MPFR_RNDF); /* error <= ulp(y) */
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else
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mpfr_sub (y, y, u, MPFR_RNDF); /* error <= ulp(y) */
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}
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/* We assume |(1 + theta)^(i+1)| <= 2.
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The neglected part is at most |u| + |u|/2 + ... <= 2|u| < 2 ulp(y)
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which has to be multiplied by |(1 + theta)^(i+1)| <= 2, thus at most
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4 ulp(y).
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The rounding error on y is bounded by:
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* for the (i-2) add/sub, each error is bounded by ulp(y),
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and since |y| <= |x|, this yields (i-2)*ulp(x)
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* from Lemma 3.1 from [Higham02] (see algorithms.tex),
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the relative error on u at step i is bounded by:
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(i+1)*epsilon/(1-(i+1)*epsilon) where epsilon = 2^(1-p).
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If (i+1)*epsilon <= 1/2, then the relative error on u at
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step i is bounded by 2*(i+1)*epsilon, and since |u| <= 1/2^(i+1)
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at step i, this gives an absolute error bound of;
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2*epsilon*x*(3/2^3 + 4/2^4 + 5/2^5 + ...) <= 2*2^(1-p)*x =
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4*2^(-p)*x <= 4*ulp(x).
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If (i+1)*epsilon <= 1/2, then the relative error on u at step i
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is bounded by (i+1)*epsilon/(1-(i+1)*epsilon) <= 1, thus it follows
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|(1 + theta)^(i+1)| <= 2.
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Finally the total error is bounded by 4*ulp(y) + (i-2)*ulp(x) + 4*ulp(x)
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= 4*ulp(y) + (i+2)*ulp(x).
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Since x/2 <= y, we have ulp(x) <= 2*ulp(y), thus the error is bounded by:
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(2*i+8)*ulp(y).
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*/
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err = 2 * i + 8;
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k = __gmpfr_int_ceil_log2 (err);
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MPFR_ASSERTN(k < p);
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/* if k < p, since k = ceil(log2(err)), we have err <= 2^k <= 2^(p-1),
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thus i+4 = err/2 <= 2^(p-2), thus (i+4)*epsilon <= 1/2, which implies
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our assumption (i+1)*epsilon <= 1/2. */
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mpfr_clear (t);
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mpfr_clear (u);
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return k;
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}
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/* The computation of log1p is done by
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log1p(x) = log(1+x)
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except when x is very small, in which case log1p(x) = x + tiny error,
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or when x is small, where we use directly the Taylor expansion.
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*/
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int
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mpfr_log1p (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
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{
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int comp, inexact;
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mpfr_exp_t ex;
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MPFR_SAVE_EXPO_DECL (expo);
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MPFR_LOG_FUNC
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(("x[%Pd]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode),
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("y[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y,
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inexact));
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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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/* check for inf or -inf (result is not defined) */
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else if (MPFR_IS_INF (x))
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{
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if (MPFR_IS_POS (x))
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{
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MPFR_SET_INF (y);
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MPFR_SET_POS (y);
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MPFR_RET (0);
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}
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else
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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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}
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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_ZERO (y); /* log1p(+/- 0) = +/- 0 */
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MPFR_SET_SAME_SIGN (y, x);
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MPFR_RET (0);
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}
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}
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ex = MPFR_GET_EXP (x);
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if (ex < 0) /* -0.5 < x < 0.5 */
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{
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/* For x > 0, abs(log(1+x)-x) < x^2/2.
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For x > -0.5, abs(log(1+x)-x) < x^2. */
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if (MPFR_IS_POS (x))
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MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex - 1, 0, 0, rnd_mode, {});
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else
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MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
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}
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comp = mpfr_cmp_si (x, -1);
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/* log1p(x) is undefined for x < -1 */
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if (MPFR_UNLIKELY(comp <= 0))
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{
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if (comp == 0)
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/* x=0: log1p(-1)=-inf (divide-by-zero exception) */
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{
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MPFR_SET_INF (y);
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MPFR_SET_NEG (y);
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MPFR_SET_DIVBY0 ();
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MPFR_RET (0);
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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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MPFR_SAVE_EXPO_MARK (expo);
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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 = MPFR_PREC(y); /* target precision */
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mpfr_prec_t Nt; /* working precision */
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mpfr_exp_t err; /* error */
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MPFR_ZIV_DECL (loop);
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/* compute the precision of intermediary variable */
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/* the optimal number of bits : see algorithms.tex */
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Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;
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/* if |x| is smaller than 2^(-e), we will loose about e bits
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in log(1+x) */
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if (MPFR_EXP(x) < 0)
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Nt += -MPFR_EXP(x);
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/* initialize of intermediary variable */
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mpfr_init2 (t, Nt);
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/* First computation of log1p */
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MPFR_ZIV_INIT (loop, Nt);
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for (;;)
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{
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int k;
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/* small case: assuming the AGM algorithm used by mpfr_log uses
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log2(p) steps for a precision of p bits, we try the special
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variant whenever EXP(x) <= -p/log2(p). */
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k = 1 + __gmpfr_int_ceil_log2 (Ny); /* the +1 avoids a division by 0
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when Ny=1 */
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if (MPFR_GET_EXP (x) + 1 <= - (mpfr_exp_t) (Ny / k))
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/* this implies EXP(x) <= -1 thus x < 1/2 */
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err = Nt - mpfr_log1p_small (t, x);
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else
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{
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/* compute log1p */
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inexact = mpfr_add_ui (t, x, 1, MPFR_RNDN); /* 1+x */
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/* if inexact = 0, then t = x+1, and the result is simply log(t) */
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if (inexact == 0)
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{
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inexact = mpfr_log (y, t, rnd_mode);
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goto end;
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}
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mpfr_log (t, t, MPFR_RNDN); /* log(1+x) */
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/* the error is bounded by (1/2+2^(1-EXP(t))*ulp(t)
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(cf algorithms.tex)
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if EXP(t)>=2, then error <= ulp(t)
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if EXP(t)<=1, then error <= 2^(2-EXP(t))*ulp(t) */
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err = Nt - MAX (0, 2 - MPFR_GET_EXP (t));
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}
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if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
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break;
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/* increase 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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inexact = mpfr_set (y, t, rnd_mode);
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end:
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MPFR_ZIV_FREE (loop);
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mpfr_clear (t);
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}
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MPFR_SAVE_EXPO_FREE (expo);
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return mpfr_check_range (y, inexact, rnd_mode);
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}
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