gcc/mpfr/expm1.c

/* mpfr_expm1 -- Compute exp(x)-1

Copyright 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 expm1 is done by
    expm1(x)=exp(x)-1
 */

int
mpfr_expm1 (mpfr_ptr y, mpfr_srcptr x , mp_rnd_t rnd_mode)
{
  int inexact;
  mp_exp_t ex;
  MPFR_SAVE_EXPO_DECL (expo);

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      /* check for inf or -inf (expm1(-inf)=-1) */
      else if (MPFR_IS_INF (x))
        {
          if (MPFR_IS_POS (x))
            {
              MPFR_SET_INF (y);
              MPFR_SET_POS (y);
              MPFR_RET (0);
            }
          else
            return mpfr_set_si (y, -1, rnd_mode);
        }
      else
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (x));
          MPFR_SET_ZERO (y);   /* expm1(+/- 0) = +/- 0 */
          MPFR_SET_SAME_SIGN (y, x);
          MPFR_RET (0);
        }
    }

  ex = MPFR_GET_EXP (x);
  if (ex < 0)
    {
      /* For -1 < x < 0, abs(expm1(x)-x) < x^2/2.
         For 0 < x < 1,  abs(expm1(x)-x) < x^2. */
      if (MPFR_IS_POS (x))
        MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 0, 1, rnd_mode, {});
      else
        MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, - ex, 1, 0, rnd_mode, {});
    }

  MPFR_SAVE_EXPO_MARK (expo);

  if (MPFR_IS_NEG (x) && ex > 5)  /* x <= -32 */
    {
      mpfr_t minus_one, t;
      mp_exp_t err;

      mpfr_init2 (minus_one, 2);
      mpfr_init2 (t, 64);
      mpfr_set_si (minus_one, -1, GMP_RNDN);
      mpfr_const_log2 (t, GMP_RNDU); /* round upward since x is negative */
      mpfr_div (t, x, t, GMP_RNDU); /* > x / ln(2) */
      err = mpfr_cmp_si (t, MPFR_EMIN_MIN >= -LONG_MAX ?
                         MPFR_EMIN_MIN : -LONG_MAX) <= 0 ?
        - (MPFR_EMIN_MIN >= -LONG_MAX ? MPFR_EMIN_MIN : -LONG_MAX) :
        - mpfr_get_si (t, GMP_RNDU);
      /* exp(x) = 2^(x/ln(2))
               <= 2^max(MPFR_EMIN_MIN,-LONG_MAX,ceil(x/ln(2)+epsilon))
         with epsilon > 0 */
      mpfr_clear (t);
      MPFR_SMALL_INPUT_AFTER_SAVE_EXPO (y, minus_one, err, 0, 0, rnd_mode,
                                        expo, { mpfr_clear (minus_one); });
      mpfr_clear (minus_one);
    }

  /* General case */
  {
    /* Declaration of the intermediary variable */
    mpfr_t t;
    /* Declaration of the size variable */
    mp_prec_t Ny = MPFR_PREC(y);   /* target precision */
    mp_prec_t Nt;                  /* working precision */
    mp_exp_t err, exp_te;          /* error */
    MPFR_ZIV_DECL (loop);

    /* compute the precision of intermediary variable */
    /* the optimal number of bits : see algorithms.tex */
    Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 6;

    /* if |x| is smaller than 2^(-e), we will loose about e bits in the
       subtraction exp(x) - 1 */
    if (ex < 0)
      Nt += - ex;

    /* initialize auxiliary variable */
    mpfr_init2 (t, Nt);

    /* First computation of expm1 */
    MPFR_ZIV_INIT (loop, Nt);
    for (;;)
      {
        MPFR_BLOCK_DECL (flags);

        /* exp(x) may overflow and underflow */
        MPFR_BLOCK (flags, mpfr_exp (t, x, GMP_RNDN));
        if (MPFR_OVERFLOW (flags))
          {
            inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN_POS);
            MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
            break;
          }
        else if (MPFR_UNDERFLOW (flags))
          {
            inexact = mpfr_set_si (y, -1, rnd_mode);
            MPFR_ASSERTD (inexact == 0);
            inexact = -1;
            if (MPFR_IS_LIKE_RNDZ (rnd_mode, 1))
              {
                inexact = 1;
                mpfr_nexttozero (y);
              }
            break;
          }

        exp_te = MPFR_GET_EXP (t);         /* FIXME: exp(x) may overflow! */
        mpfr_sub_ui (t, t, 1, GMP_RNDN);   /* exp(x)-1 */

        /* error estimate */
        /*err=Nt-(__gmpfr_ceil_log2(1+pow(2,MPFR_EXP(te)-MPFR_EXP(t))));*/
        err = Nt - (MAX (exp_te - MPFR_GET_EXP (t), 0) + 1);

        if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
          {
            inexact = mpfr_set (y, t, rnd_mode);
            break;
          }

        /* increase the precision */
        MPFR_ZIV_NEXT (loop, Nt);
        mpfr_set_prec (t, Nt);
      }
    MPFR_ZIV_FREE (loop);

    mpfr_clear (t);
  }

  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, inexact, rnd_mode);
}