| /* 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); |
| } |