path: root/src/exp3.c

/* mpfr_exp -- exponential of a floating-point number

Copyright 1999, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012 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 /* for MPFR_MPZ_SIZEINBASE2 */
#include "mpfr-impl.h"

/* y <- exp(p/2^r) within 1 ulp, using 2^m terms from the series
   Assume |p/2^r| < 1.
   We use the following binary splitting formula:
   P(a,b) = p if a+1=b, P(a,c)*P(c,b) otherwise
   Q(a,b) = a*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise
   T(a,b) = P(a,b) if a+1=b, Q(c,b)*T(a,c)+P(a,c)*T(c,b) otherwise
   Then exp(p/2^r) ~ T(0,i)/Q(0,i) for i so that (p/2^r)^i/i! is small enough.

   Since P(a,b) = p^(b-a), and we consider only values of b-a of the form 2^j,
   we don't need to compute P(), we only precompute p^(2^j) in the ptoj[] array
   below.

   Since Q(a,b) is divisible by 2^(r*(b-a-1)), we don't compute the power of
   two part.
*/
static void
mpfr_exp_rational (mpfr_ptr y, mpz_ptr p, long r, int m,
                   mpz_t *Q, mpfr_prec_t *mult)
{
  unsigned long n, i, j;
  mpz_t *S, *ptoj;
  mpfr_prec_t *log2_nb_terms;
  mpfr_exp_t diff, expo;
  mpfr_prec_t precy = MPFR_PREC(y), prec_i_have, prec_ptoj;
  int k, l;

  MPFR_ASSERTN ((size_t) m < sizeof (long) * CHAR_BIT - 1);

  S    = Q + (m+1);
  ptoj = Q + 2*(m+1);                     /* ptoj[i] = mantissa^(2^i) */
  log2_nb_terms = mult + (m+1);

  /* Normalize p */
  MPFR_ASSERTD (mpz_cmp_ui (p, 0) != 0);
  n = mpz_scan1 (p, 0); /* number of trailing zeros in p */
  mpz_tdiv_q_2exp (p, p, n);
  r -= n; /* since |p/2^r| < 1 and p >= 1, r >= 1 */

  /* Set initial var */
  mpz_set (ptoj[0], p);
  for (k = 1; k < m; k++)
    mpz_mul (ptoj[k], ptoj[k-1], ptoj[k-1]); /* ptoj[k] = p^(2^k) */
  mpz_set_ui (Q[0], 1);
  mpz_set_ui (S[0], 1);
  k = 0;
  mult[0] = 0; /* the multiplier P[k]/Q[k] for the remaining terms
                  satisfies P[k]/Q[k] <= 2^(-mult[k]) */
  log2_nb_terms[0] = 0; /* log2(#terms) [exact in 1st loop where 2^k] */
  prec_i_have = 0;

  /* Main Loop */
  n = 1UL << m;
  for (i = 1; (prec_i_have < precy) && (i < n); i++)
    {
      /* invariant: Q[0]*Q[1]*...*Q[k] equals i! */
      k++;
      log2_nb_terms[k] = 0; /* 1 term */
      mpz_set_ui (Q[k], i + 1);
      mpz_set_ui (S[k], i + 1);
      j = i + 1; /* we have computed j = i+1 terms so far */
      l = 0;
      while ((j & 1) == 0) /* combine and reduce */
        {
          /* invariant: S[k] corresponds to 2^l consecutive terms */
          mpz_mul (S[k], S[k], ptoj[l]);
          mpz_mul (S[k-1], S[k-1], Q[k]);
          /* Q[k] corresponds to 2^l consecutive terms too.
             Since it does not contains the factor 2^(r*2^l),
             when going from l to l+1 we need to multiply
             by 2^(r*2^(l+1))/2^(r*2^l) = 2^(r*2^l) */
          mpz_mul_2exp (S[k-1], S[k-1], r << l);
          mpz_add (S[k-1], S[k-1], S[k]);
          mpz_mul (Q[k-1], Q[k-1], Q[k]);
          log2_nb_terms[k-1] ++; /* number of terms in S[k-1]
                                    is a power of 2 by construction */
          MPFR_MPZ_SIZEINBASE2 (prec_i_have, Q[k]);
          MPFR_MPZ_SIZEINBASE2 (prec_ptoj, ptoj[l]);
          mult[k-1] += prec_i_have + (r << l) - prec_ptoj - 1;
          prec_i_have = mult[k] = mult[k-1];
          /* since mult[k] >= mult[k-1] + nbits(Q[k]),
             we have Q[0]*...*Q[k] <= 2^mult[k] = 2^prec_i_have */
          l ++;
          j >>= 1;
          k --;
        }
    }

  /* accumulate all products in S[0] and Q[0]. Warning: contrary to above,
     here we do not have log2_nb_terms[k-1] = log2_nb_terms[k]+1. */
  l = 0; /* number of accumulated terms in the right part S[k]/Q[k] */
  while (k > 0)
    {
      j = log2_nb_terms[k-1];
      mpz_mul (S[k], S[k], ptoj[j]);
      mpz_mul (S[k-1], S[k-1], Q[k]);
      l += 1 << log2_nb_terms[k];
      mpz_mul_2exp (S[k-1], S[k-1], r * l);
      mpz_add (S[k-1], S[k-1], S[k]);
      mpz_mul (Q[k-1], Q[k-1], Q[k]);
      k--;
    }

  /* Q[0] now equals i! */
  MPFR_MPZ_SIZEINBASE2 (prec_i_have, S[0]);
  diff = (mpfr_exp_t) prec_i_have - 2 * (mpfr_exp_t) precy;
  expo = diff;
  if (diff >= 0)
    mpz_fdiv_q_2exp (S[0], S[0], diff);
  else
    mpz_mul_2exp (S[0], S[0], -diff);

  MPFR_MPZ_SIZEINBASE2 (prec_i_have, Q[0]);
  diff = (mpfr_exp_t) prec_i_have - (mpfr_prec_t) precy;
  expo -= diff;
  if (diff > 0)
    mpz_fdiv_q_2exp (Q[0], Q[0], diff);
  else
    mpz_mul_2exp (Q[0], Q[0], -diff);

  mpz_tdiv_q (S[0], S[0], Q[0]);
  mpfr_set_z (y, S[0], MPFR_RNDD);
  MPFR_SET_EXP (y, MPFR_GET_EXP (y) + expo - r * (i - 1) );
}

#define shift (GMP_NUMB_BITS/2)

int
mpfr_exp_3 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
  mpfr_t t, x_copy, tmp;
  mpz_t uk;
  mpfr_exp_t ttt, shift_x;
  unsigned long twopoweri;
  mpz_t *P;
  mpfr_prec_t *mult;
  int i, k, loop;
  int prec_x;
  mpfr_prec_t realprec, Prec;
  int iter;
  int inexact = 0;
  MPFR_SAVE_EXPO_DECL (expo);
  MPFR_ZIV_DECL (ziv_loop);

  MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, rnd_mode),
     ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec(y), mpfr_log_prec, y,
      inexact));

  MPFR_SAVE_EXPO_MARK (expo);

  /* decompose x */
  /* we first write x = 1.xxxxxxxxxxxxx
     ----- k bits -- */
  prec_x = MPFR_INT_CEIL_LOG2 (MPFR_PREC (x)) - MPFR_LOG2_GMP_NUMB_BITS;
  if (prec_x < 0)
    prec_x = 0;

  ttt = MPFR_GET_EXP (x);
  mpfr_init2 (x_copy, MPFR_PREC(x));
  mpfr_set (x_copy, x, MPFR_RNDD);

  /* we shift to get a number less than 1 */
  if (ttt > 0)
    {
      shift_x = ttt;
      mpfr_div_2ui (x_copy, x, ttt, MPFR_RNDN);
      ttt = MPFR_GET_EXP (x_copy);
    }
  else
    shift_x = 0;
  MPFR_ASSERTD (ttt <= 0);

  /* Init prec and vars */
  realprec = MPFR_PREC (y) + MPFR_INT_CEIL_LOG2 (prec_x + MPFR_PREC (y));
  Prec = realprec + shift + 2 + shift_x;
  mpfr_init2 (t, Prec);
  mpfr_init2 (tmp, Prec);
  mpz_init (uk);

  /* Main loop */
  MPFR_ZIV_INIT (ziv_loop, realprec);
  for (;;)
    {
      int scaled = 0;
      MPFR_BLOCK_DECL (flags);

      k = MPFR_INT_CEIL_LOG2 (Prec) - MPFR_LOG2_GMP_NUMB_BITS;

      /* now we have to extract */
      twopoweri = GMP_NUMB_BITS;

      /* Allocate tables */
      P    = (mpz_t*) (*__gmp_allocate_func) (3*(k+2)*sizeof(mpz_t));
      for (i = 0; i < 3*(k+2); i++)
        mpz_init (P[i]);
      mult = (mpfr_prec_t*) (*__gmp_allocate_func) (2*(k+2)*sizeof(mpfr_prec_t));

      /* Particular case for i==0 */
      mpfr_extract (uk, x_copy, 0);
      MPFR_ASSERTD (mpz_cmp_ui (uk, 0) != 0);
      mpfr_exp_rational (tmp, uk, shift + twopoweri - ttt, k + 1, P, mult);
      for (loop = 0; loop < shift; loop++)
        mpfr_sqr (tmp, tmp, MPFR_RNDD);
      twopoweri *= 2;

      /* General case */
      iter = (k <= prec_x) ? k : prec_x;
      for (i = 1; i <= iter; i++)
        {
          mpfr_extract (uk, x_copy, i);
          if (MPFR_LIKELY (mpz_cmp_ui (uk, 0) != 0))
            {
              mpfr_exp_rational (t, uk, twopoweri - ttt, k  - i + 1, P, mult);
              mpfr_mul (tmp, tmp, t, MPFR_RNDD);
            }
          MPFR_ASSERTN (twopoweri <= LONG_MAX/2);
          twopoweri *=2;
        }

      /* Clear tables */
      for (i = 0; i < 3*(k+2); i++)
        mpz_clear (P[i]);
      (*__gmp_free_func) (P, 3*(k+2)*sizeof(mpz_t));
      (*__gmp_free_func) (mult, 2*(k+2)*sizeof(mpfr_prec_t));

      if (shift_x > 0)
        {
          MPFR_BLOCK (flags, {
              for (loop = 0; loop < shift_x - 1; loop++)
                mpfr_sqr (tmp, tmp, MPFR_RNDD);
              mpfr_sqr (t, tmp, MPFR_RNDD);
            } );

          if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags)))
            {
              /* tmp <= exact result, so that it is a real overflow. */
              inexact = mpfr_overflow (y, rnd_mode, 1);
              MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
              break;
            }

          if (MPFR_UNLIKELY (MPFR_UNDERFLOW (flags)))
            {
              /* This may be a spurious underflow. So, let's scale
                 the result. */
              mpfr_mul_2ui (tmp, tmp, 1, MPFR_RNDD);  /* no overflow, exact */
              mpfr_sqr (t, tmp, MPFR_RNDD);
              if (MPFR_IS_ZERO (t))
                {
                  /* approximate result < 2^(emin - 3), thus
                     exact result < 2^(emin - 2). */
                  inexact = mpfr_underflow (y, (rnd_mode == MPFR_RNDN) ?
                                            MPFR_RNDZ : rnd_mode, 1);
                  MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
                  break;
                }
              scaled = 1;
            }
        }

      if (mpfr_can_round (shift_x > 0 ? t : tmp, realprec, MPFR_RNDD, MPFR_RNDZ,
                          MPFR_PREC(y) + (rnd_mode == MPFR_RNDN)))
        {
          inexact = mpfr_set (y, shift_x > 0 ? t : tmp, rnd_mode);
          if (MPFR_UNLIKELY (scaled && MPFR_IS_PURE_FP (y)))
            {
              int inex2;
              mpfr_exp_t ey;

              /* The result has been scaled and needs to be corrected. */
              ey = MPFR_GET_EXP (y);
              inex2 = mpfr_mul_2si (y, y, -2, rnd_mode);
              if (inex2)  /* underflow */
                {
                  if (rnd_mode == MPFR_RNDN && inexact < 0 &&
                      MPFR_IS_ZERO (y) && ey == __gmpfr_emin + 1)
                    {
                      /* Double rounding case: in MPFR_RNDN, the scaled
                         result has been rounded downward to 2^emin.
                         As the exact result is > 2^(emin - 2), correct
                         rounding must be done upward. */
                      /* TODO: make sure in coverage tests that this line
                         is reached. */
                      inexact = mpfr_underflow (y, MPFR_RNDU, 1);
                    }
                  else
                    {
                      /* No double rounding. */
                      inexact = inex2;
                    }
                  MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_UNDERFLOW);
                }
            }
          break;
        }

      MPFR_ZIV_NEXT (ziv_loop, realprec);
      Prec = realprec + shift + 2 + shift_x;
      mpfr_set_prec (t, Prec);
      mpfr_set_prec (tmp, Prec);
    }
  MPFR_ZIV_FREE (ziv_loop);

  mpz_clear (uk);
  mpfr_clear (tmp);
  mpfr_clear (t);
  mpfr_clear (x_copy);
  MPFR_SAVE_EXPO_FREE (expo);
  return inexact;
}