/* mpfr_log -- natural logarithm of a floating-point number

Copyright 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc.
Contributed by the AriC and Caramel 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_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 */
  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);
  /* % ~(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 (;;)
    {
      long m;
      mpfr_exp_t cancel;

      /* Calculus of m (depends on p) */
      m = (p + 1) / 2 - MPFR_GET_EXP (a) + 1;

      mpfr_mul_2si (tmp2, a, m, MPFR_RNDN);    /* s=a*2^m,        err<=1 ulp  */
      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 >= 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_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);
}
