--- /dev/null
+/* mpfr_log -- natural logarithm of a floating-point number
+
+Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
+Contributed by the Arenaire and Cacao 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 2.1 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.LIB. If not, 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 log(x) is done using the formula :
+ if we want p bits of the result,
+
+ pi
+ log(x) ~ ------------ - m log 2
+ 2 AG(1,4/s)
+
+ where s = x 2^m > 2^(p/2)
+
+ More precisely, if F(x) = int(1/sqrt(1-(1-x^2)*sin(t)^2), t=0..PI/2),
+ then for s>=1.26 we have log(s) < F(4/s) < log(s)*(1+4/s^2)
+ from which we deduce pi/2/AG(1,4/s)*(1-4/s^2) < log(s) < pi/2/AG(1,4/s)
+ so the relative error 4/s^2 is < 4/2^p i.e. 4 ulps.
+*/
+
+int
+mpfr_log (mpfr_ptr r, mpfr_srcptr a, mp_rnd_t rnd_mode)
+{
+ int inexact;
+ mp_prec_t p, q;
+ mpfr_t tmp1, tmp2;
+ mp_limb_t *tmp1p, *tmp2p;
+ MPFR_SAVE_EXPO_DECL (expo);
+ MPFR_ZIV_DECL (loop);
+ MPFR_TMP_DECL(marker);
+
+ MPFR_LOG_FUNC (("a[%#R]=%R rnd=%d", a, a, rnd_mode),
+ ("r[%#R]=%R inexact=%d", r, r, inexact));
+
+ /* Special cases */
+ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (a)))
+ {
+ /* If a is NaN, the result is NaN */
+ if (MPFR_IS_NAN (a))
+ {
+ MPFR_SET_NAN (r);
+ MPFR_RET_NAN;
+ }
+ /* check for infinity before zero */
+ else if (MPFR_IS_INF (a))
+ {
+ if (MPFR_IS_NEG (a))
+ /* log(-Inf) = NaN */
+ {
+ MPFR_SET_NAN (r);
+ MPFR_RET_NAN;
+ }
+ else /* log(+Inf) = +Inf */
+ {
+ MPFR_SET_INF (r);
+ MPFR_SET_POS (r);
+ MPFR_RET (0);
+ }
+ }
+ else /* a is zero */
+ {
+ MPFR_ASSERTD (MPFR_IS_ZERO (a));
+ MPFR_SET_INF (r);
+ MPFR_SET_NEG (r);
+ MPFR_RET (0); /* log(0) is an exact -infinity */
+ }
+ }
+ /* If a is negative, the result is NaN */
+ else if (MPFR_UNLIKELY (MPFR_IS_NEG (a)))
+ {
+ MPFR_SET_NAN (r);
+ MPFR_RET_NAN;
+ }
+ /* If a is 1, the result is 0 */
+ else if (MPFR_UNLIKELY (MPFR_GET_EXP (a) == 1 && mpfr_cmp_ui (a, 1) == 0))
+ {
+ MPFR_SET_ZERO (r);
+ MPFR_SET_POS (r);
+ MPFR_RET (0); /* only "normal" case where the result is exact */
+ }
+
+ q = MPFR_PREC (r);
+
+ /* use initial precision about q+lg(q)+5 */
+ p = q + 5 + 2 * MPFR_INT_CEIL_LOG2 (q);
+ /* % ~(mp_prec_t)BITS_PER_MP_LIMB ;
+ m=q; while (m) { p++; m >>= 1; } */
+ /* if (MPFR_LIKELY(p % BITS_PER_MP_LIMB != 0))
+ p += BITS_PER_MP_LIMB - (p%BITS_PER_MP_LIMB); */
+
+ MPFR_TMP_MARK(marker);
+ MPFR_SAVE_EXPO_MARK (expo);
+
+ MPFR_ZIV_INIT (loop, p);
+ for (;;)
+ {
+ mp_size_t size;
+ long m;
+ mp_exp_t cancel;
+
+ /* Calculus of m (depends on p) */
+ m = (p + 1) / 2 - MPFR_GET_EXP (a) + 1;
+
+ /* All the mpfr_t needed have a precision of p */
+ size = (p-1)/BITS_PER_MP_LIMB+1;
+ MPFR_TMP_INIT (tmp1p, tmp1, p, size);
+ MPFR_TMP_INIT (tmp2p, tmp2, p, size);
+
+ mpfr_mul_2si (tmp2, a, m, GMP_RNDN); /* s=a*2^m, err<=1 ulp */
+ mpfr_div (tmp1, __gmpfr_four, tmp2, GMP_RNDN);/* 4/s, err<=2 ulps */
+ mpfr_agm (tmp2, __gmpfr_one, tmp1, GMP_RNDN); /* AG(1,4/s),err<=3 ulps */
+ mpfr_mul_2ui (tmp2, tmp2, 1, GMP_RNDN); /* 2*AG(1,4/s), err<=3 ulps */
+ mpfr_const_pi (tmp1, GMP_RNDN); /* compute pi, err<=1ulp */
+ mpfr_div (tmp2, tmp1, tmp2, GMP_RNDN); /* pi/2*AG(1,4/s), err<=5ulps */
+ mpfr_const_log2 (tmp1, GMP_RNDN); /* compute log(2), err<=1ulp */
+ mpfr_mul_si (tmp1, tmp1, m, GMP_RNDN); /* compute m*log(2),err<=2ulps */
+ mpfr_sub (tmp1, tmp2, tmp1, GMP_RNDN); /* log(a), err<=7ulps+cancel */
+
+ if (MPFR_LIKELY (MPFR_IS_PURE_FP (tmp1) && MPFR_IS_PURE_FP (tmp2)))
+ {
+ cancel = MPFR_GET_EXP (tmp2) - MPFR_GET_EXP (tmp1);
+ MPFR_LOG_MSG (("canceled bits=%ld\n", (long) cancel));
+ MPFR_LOG_VAR (tmp1);
+ if (MPFR_UNLIKELY (cancel < 0))
+ cancel = 0;
+
+ /* we have 7 ulps of error from the above roundings,
+ 4 ulps from the 4/s^2 second order term,
+ plus the canceled bits */
+ if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp1, p-cancel-4, q, rnd_mode)))
+ break;
+
+ /* VL: I think it is better to have an increment that it isn't
+ too low; in particular, the increment must be positive even
+ if cancel = 0 (can this occur?). */
+ p += cancel >= 8 ? cancel : 8;
+ }
+ else
+ {
+ /* TODO: find why this case can occur and what is best to do
+ with it. */
+ p += 32;
+ }
+
+ MPFR_ZIV_NEXT (loop, p);
+ }
+ MPFR_ZIV_FREE (loop);
+ inexact = mpfr_set (r, tmp1, rnd_mode);
+ /* We clean */
+ MPFR_TMP_FREE(marker);
+
+ MPFR_SAVE_EXPO_FREE (expo);
+ return mpfr_check_range (r, inexact, rnd_mode);
+}
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