/* mpfr_log -- natural logarithm of a floating-point number Copyright 1999-2017 Free Software Foundation, Inc. Contributed by the AriC and Caramba 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 3 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.LESSER. If not, see http://www.gnu.org/licenses/ or 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, mpfr_rnd_t rnd_mode) { int inexact; mpfr_prec_t p, q; mpfr_t tmp1, tmp2; mpfr_exp_t exp_a; MPFR_SAVE_EXPO_DECL (expo); MPFR_ZIV_DECL (loop); MPFR_GROUP_DECL(group); MPFR_LOG_FUNC (("a[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (a), mpfr_log_prec, a, rnd_mode), ("r[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (r), mpfr_log_prec, 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_SET_DIVBY0 (); MPFR_RET (0); /* log(0) is an exact -infinity */ } } /* If a is negative, the result is NaN */ if (MPFR_UNLIKELY (MPFR_IS_NEG (a))) { MPFR_SET_NAN (r); MPFR_RET_NAN; } exp_a = MPFR_GET_EXP (a); /* If a is 1, the result is +0 */ if (MPFR_UNLIKELY (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+2*lg(q)+cte */ p = q + 2 * MPFR_INT_CEIL_LOG2 (q) + 10; /* % ~(mpfr_prec_t)GMP_NUMB_BITS ; m=q; while (m) { p++; m >>= 1; } */ /* if (MPFR_LIKELY(p % GMP_NUMB_BITS != 0)) p += GMP_NUMB_BITS - (p%GMP_NUMB_BITS); */ MPFR_SAVE_EXPO_MARK (expo); MPFR_GROUP_INIT_2 (group, p, tmp1, tmp2); MPFR_ZIV_INIT (loop, p); for (;;) { mpfr_exp_t m; mpfr_exp_t cancel; /* Calculus of m (depends on p) If mpfr_exp_t has N bits, then both (p + 3) / 2 and |exp_a| fit on N-2 bits, so that there cannot be an overflow. */ m = (p + 3) / 2 - exp_a; /* In standard configuration (_MPFR_EXP_FORMAT <= 3), one has mpfr_exp_t <= long, so that the following assertion is always true. */ MPFR_ASSERTN (m >= LONG_MIN && m <= LONG_MAX); /* FIXME: Why 1 ulp and not 1/2 ulp? Ditto with some other ones below. The error concerning the AGM should be explained since 4/s is inexact (one needs a bound on its derivative). */ mpfr_mul_2si (tmp2, a, m, MPFR_RNDN); /* s=a*2^m, err<=1 ulp */ MPFR_ASSERTD (MPFR_EXP (tmp2) >= (p + 3) / 2); /* [FIXME] and one can have the equality, even if p is even. This means that if a is a power of 2 and p is even, then s = (1/2) * 2^((p+2)/2) = 2^(p/2), so that the condition s > 2^(p/2) from algorithms.tex is not satisfied. */ mpfr_div (tmp1, __gmpfr_four, tmp2, MPFR_RNDN);/* 4/s, err<=2 ulps */ mpfr_agm (tmp2, __gmpfr_one, tmp1, MPFR_RNDN); /* AG(1,4/s),err<=3 ulps */ mpfr_mul_2ui (tmp2, tmp2, 1, MPFR_RNDN); /* 2*AG(1,4/s), err<=3 ulps */ mpfr_const_pi (tmp1, MPFR_RNDN); /* compute pi, err<=1ulp */ mpfr_div (tmp2, tmp1, tmp2, MPFR_RNDN); /* pi/2*AG(1,4/s), err<=5ulps */ mpfr_const_log2 (tmp1, MPFR_RNDN); /* compute log(2), err<=1ulp */ mpfr_mul_si (tmp1, tmp1, m, MPFR_RNDN); /* compute m*log(2),err<=2ulps */ mpfr_sub (tmp1, tmp2, tmp1, MPFR_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 + MPFR_INT_CEIL_LOG2 (p); } else { /* TODO: find why this case can occur and what is best to do with it. */ p += MPFR_INT_CEIL_LOG2 (p); } MPFR_ZIV_NEXT (loop, p); MPFR_GROUP_REPREC_2 (group, p, tmp1, tmp2); } MPFR_ZIV_FREE (loop); inexact = mpfr_set (r, tmp1, rnd_mode); /* We clean */ MPFR_GROUP_CLEAR (group); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (r, inexact, rnd_mode); }