dist/src/ai.c

/* mpfr_ai -- Airy function Ai

Copyright 2010-2020 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
https://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"

/* Reminder and notations:
   -----------------------

   Ai is the solution of:
        / y'' - x*y = 0
       {    Ai(0)   = 1/ ( 9^(1/3)*Gamma(2/3) )
        \  Ai'(0)   = -1/ ( 3^(1/3)*Gamma(1/3) )

   Series development:
       Ai(x) = sum (a_i*x^i)
             = sum (t_i)

   Recurrences:
       a_(i+3) = a_i / ((i+2)*(i+3))
       t_(i+3) = t_i * x^3 / ((i+2)*(i+3))

   Values:
       a_0 = Ai(0)  ~  0.355
       a_1 = Ai'(0) ~ -0.259
*/


/* Airy function Ai evaluated by the most naive algorithm.
   Assume that x is a finite number. */
static int
mpfr_ai1 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd)
{
  MPFR_ZIV_DECL (loop);
  MPFR_SAVE_EXPO_DECL (expo);
  mpfr_prec_t wprec;             /* working precision */
  mpfr_prec_t prec;              /* target precision */
  mpfr_prec_t err;               /* used to estimate the evaluation error */
  mpfr_prec_t correct_bits;      /* estimates the number of correct bits*/
  unsigned long int k;
  unsigned long int cond;        /* condition number of the series */
  unsigned long int assumed_exponent; /* used as a lowerbound of |EXP(Ai(x))| */
  int r;
  mpfr_t s;                      /* used to store the partial sum */
  mpfr_t ti, tip1;   /* used to store successive values of t_i */
  mpfr_t x3;                     /* used to store x^3 */
  mpfr_t tmp_sp, tmp2_sp;        /* small precision variables */
  unsigned long int x3u;         /* used to store ceil(x^3) */
  mpfr_t temp1, temp2;
  int test1, test2;

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

  /* Save current exponents range */
  MPFR_SAVE_EXPO_MARK (expo);

  if (MPFR_UNLIKELY (MPFR_IS_ZERO (x)))
    {
      mpfr_t y1, y2;
      prec = MPFR_ADD_PREC (MPFR_PREC (y), 3);
      mpfr_init2 (y1, prec);
      mpfr_init2 (y2, prec);
      MPFR_ZIV_INIT (loop, prec);

      /* ZIV loop */
      for (;;)
        {
          mpfr_gamma_one_and_two_third (y1, y2, prec); /* y2 = Gamma(2/3)(1 + delta1), |delta1| <= 2^{1-prec}. */

          r = mpfr_set_ui (y1, 9, MPFR_RNDN);
          MPFR_ASSERTD (r == 0);
          mpfr_cbrt (y1, y1, MPFR_RNDN); /* y1 = cbrt(9)(1 + delta2), |delta2| <= 2^{-prec}. */
          mpfr_mul (y1, y1, y2, MPFR_RNDN);
          mpfr_ui_div (y1, 1, y1, MPFR_RNDN);
          if (MPFR_LIKELY (MPFR_CAN_ROUND (y1, prec - 3, MPFR_PREC (y), rnd)))
            break;
          MPFR_ZIV_NEXT (loop, prec);
        }
      r = mpfr_set (y, y1, rnd);
      MPFR_ZIV_FREE (loop);
      MPFR_SAVE_EXPO_FREE (expo);
      mpfr_clear (y1);
      mpfr_clear (y2);
      return mpfr_check_range (y, r, rnd);
    }

  /* now x is not zero */
  MPFR_ASSERTD(!MPFR_IS_ZERO(x));

  /* FIXME: underflow for large values of |x| ? */


  /* Set initial precision */
  /* If we compute sum(i=0, N-1, t_i), the relative error is bounded by  */
  /*       2*(4N)*2^(1-wprec)*C(|x|)/Ai(x)                               */
  /* where C(|x|) = 1 if 0<=x<=1                                         */
  /*   and C(|x|) = (1/2)*x^(-1/4)*exp(2/3 x^(3/2))  if x >= 1           */

  /* A priori, we do not know N, so we estimate it to ~ prec             */
  /* If 0<=x<=1, we estimate Ai(x) ~ 1/8                                 */
  /* if 1<=x,    we estimate Ai(x) ~ (1/4)*x^(-1/4)*exp(-2/3 * x^(3/2))  */
  /* if x<=0,    ?????                                                   */

  /* We begin with 11 guard bits */
  prec = MPFR_ADD_PREC (MPFR_PREC (y), 11);
  MPFR_ZIV_INIT (loop, prec);

  /* The working precision is heuristically chosen in order to obtain  */
  /* approximately prec correct bits in the sum. To sum up: the sum    */
  /* is stopped when the *exact* sum gives ~ prec correct bit. And     */
  /* when it is stopped, the accuracy of the computed sum, with respect*/
  /* to the exact one should be ~prec bits.                            */
  mpfr_init2 (tmp_sp, MPFR_SMALL_PRECISION);
  mpfr_init2 (tmp2_sp, MPFR_SMALL_PRECISION);
  mpfr_abs (tmp_sp, x, MPFR_RNDU);
  mpfr_pow_ui (tmp_sp, tmp_sp, 3, MPFR_RNDU);
  mpfr_sqrt (tmp_sp, tmp_sp, MPFR_RNDU); /* tmp_sp ~ |x|^(3/2) */

  /* 0.96179669392597567 >~ 2/3 * log2(e). See algorithms.tex */
  mpfr_set_str (tmp2_sp, "0.96179669392597567", 10, MPFR_RNDU);
  mpfr_mul (tmp2_sp, tmp_sp, tmp2_sp, MPFR_RNDU);

  /* cond represents the number of lost bits in the evaluation of the sum */
  if (MPFR_GET_EXP (x) <= 0)
    cond = 0;
  else
    {
      MPFR_BLOCK_DECL (flags);

      MPFR_BLOCK (flags, cond = mpfr_get_ui (tmp2_sp, MPFR_RNDU));
      MPFR_ASSERTN (! MPFR_ERANGEFLAG (flags));
      cond -= (MPFR_GET_EXP (x) - 1) / 4 + 1;
    }

  /* The variable assumed_exponent is used to store the maximal assumed */
  /* exponent of Ai(x). More precisely, we assume that |Ai(x)| will be  */
  /* greater than 2^{-assumed_exponent}.                                */
  if (MPFR_IS_POS (x))
    {
      if (MPFR_GET_EXP (x) <= 0)
        assumed_exponent = 3;
      else
        {
          unsigned long int t;
          MPFR_BLOCK_DECL (flags);

          MPFR_BLOCK (flags, t = mpfr_get_ui (tmp2_sp, MPFR_RNDU));
          MPFR_ASSERTN (! MPFR_ERANGEFLAG (flags));
          assumed_exponent = t + 2 + (MPFR_GET_EXP (x) / 4 + 1);
          MPFR_ASSERTN (assumed_exponent > t);
        }
    }
  /* We do not know Ai (x) yet */
  /* We cover the case when EXP (Ai (x))>=-10 */
  else
    assumed_exponent = 10;

  {
    unsigned long int t, u;

    t = assumed_exponent + cond;
    MPFR_ASSERTN (t >= cond);
    u = MPFR_INT_CEIL_LOG2 (prec) + 5;
    t += u;
    MPFR_ASSERTN (t >= u);
    wprec = MPFR_ADD_PREC (prec, t);
  }

  mpfr_init (ti);
  mpfr_init (tip1);
  mpfr_init (temp1);
  mpfr_init (temp2);
  mpfr_init (x3);
  mpfr_init (s);

  /* ZIV loop */
  for (;;)
    {
      MPFR_LOG_MSG (("Working precision: %Pu\n", wprec));
      mpfr_set_prec (ti, wprec);
      mpfr_set_prec (tip1, wprec);
      mpfr_set_prec (x3, wprec);
      mpfr_set_prec (s, wprec);

      mpfr_sqr (x3, x, MPFR_RNDU);
      mpfr_mul (x3, x3, x, (MPFR_IS_POS (x)?MPFR_RNDU:MPFR_RNDD));  /* x3=x^3 */
      if (MPFR_IS_NEG (x))
        MPFR_CHANGE_SIGN (x3);
      x3u = mpfr_get_ui (x3, MPFR_RNDU);   /* x3u >= ceil(x^3) */
      if (MPFR_IS_NEG (x))
        MPFR_CHANGE_SIGN (x3);

      mpfr_gamma_one_and_two_third (temp1, temp2, wprec);
      mpfr_set_ui (ti, 9, MPFR_RNDN);
      mpfr_cbrt (ti, ti, MPFR_RNDN);
      mpfr_mul (ti, ti, temp2, MPFR_RNDN);
      mpfr_ui_div (ti, 1, ti , MPFR_RNDN); /* ti = 1/( Gamma (2/3)*9^(1/3) ) */

      mpfr_set_ui (tip1, 3, MPFR_RNDN);
      mpfr_cbrt (tip1, tip1, MPFR_RNDN);
      mpfr_mul (tip1, tip1, temp1, MPFR_RNDN);
      mpfr_neg (tip1, tip1, MPFR_RNDN);
      mpfr_div (tip1, x, tip1, MPFR_RNDN); /* tip1 = -x/(Gamma (1/3)*3^(1/3)) */

      mpfr_add (s, ti, tip1, MPFR_RNDN);


      /* Evaluation of the series */
      k = 2;
      for (;;)
        {
          mpfr_mul (ti, ti, x3, MPFR_RNDN);
          mpfr_mul (tip1, tip1, x3, MPFR_RNDN);

          mpfr_div_ui2 (ti, ti, k, (k+1), MPFR_RNDN);
          mpfr_div_ui2 (tip1, tip1, (k+1), (k+2), MPFR_RNDN);

          k += 3;
          mpfr_add (s, s, ti, MPFR_RNDN);
          mpfr_add (s, s, tip1, MPFR_RNDN);

          /* FIXME: if s==0 */
          test1 = MPFR_IS_ZERO (ti)
            || (MPFR_GET_EXP (ti) + (mpfr_exp_t)prec + 3 <= MPFR_GET_EXP (s));
          test2 = MPFR_IS_ZERO (tip1)
            || (MPFR_GET_EXP (tip1) + (mpfr_exp_t)prec + 3 <= MPFR_GET_EXP (s));

          if ( test1 && test2 && (x3u <= k*(k+1)/2) )
            break; /* FIXME: if k*(k+1) overflows */
        }

      MPFR_LOG_MSG (("Truncation rank: %lu\n", k));

      err = 4 + MPFR_INT_CEIL_LOG2 (k) + cond - MPFR_GET_EXP (s);

      /* err is the number of bits lost due to the evaluation error */
      /* wprec-(prec+1): number of bits lost due to the approximation error */
      MPFR_LOG_MSG (("Roundoff error: %Pu\n", err));
      MPFR_LOG_MSG (("Approxim error: %Pu\n", wprec-prec-1));

      if (wprec < err + 1)
        correct_bits = 0;
      else if (wprec < err + prec +1)
        correct_bits =  wprec - err - 1; /* since wprec > err + 1,
                                            correct_bits > 0 */
      else
        correct_bits = prec;

      if (MPFR_LIKELY (MPFR_CAN_ROUND (s, correct_bits, MPFR_PREC (y), rnd)))
        break;

      if (correct_bits == 0)
        {
          assumed_exponent *= 2;
          MPFR_LOG_MSG (("Not a single bit correct (assumed_exponent=%lu)\n",
                         assumed_exponent));
          wprec = prec + 5 + MPFR_INT_CEIL_LOG2 (prec) + cond +
            assumed_exponent;
        }
      else if (correct_bits < prec)
        { /* The precision was badly chosen */
          MPFR_LOG_MSG (("Bad assumption on the exponent of Ai(x)"
                         " (E=%" MPFR_EXP_FSPEC "d)\n",
                         (mpfr_eexp_t) MPFR_GET_EXP (s)));
          wprec = prec + err + 1;
        }
      else
        { /* We are really in a bad case of the TMD */
          MPFR_ZIV_NEXT (loop, prec);

          /* We update wprec */
          /* We assume that K will not be multiplied by more than 4 */
          wprec = prec + (MPFR_INT_CEIL_LOG2 (k) + 2) + 5 + cond
            - MPFR_GET_EXP (s);
        }

    } /* End of ZIV loop */

  MPFR_ZIV_FREE (loop);

  r = mpfr_set (y, s, rnd);

  mpfr_clear (ti);
  mpfr_clear (tip1);
  mpfr_clear (temp1);
  mpfr_clear (temp2);
  mpfr_clear (x3);
  mpfr_clear (s);
  mpfr_clear (tmp_sp);
  mpfr_clear (tmp2_sp);

  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, r, rnd);
}


/* Airy function Ai evaluated by Smith algorithm.
   Assume that x is a finite non-zero number. */
static int
mpfr_ai2 (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd)
{
  MPFR_ZIV_DECL (loop);
  MPFR_SAVE_EXPO_DECL (expo);
  mpfr_prec_t wprec;             /* working precision */
  mpfr_prec_t prec;              /* target precision */
  mpfr_prec_t err;               /* used to estimate the evaluation error */
  mpfr_prec_t correctBits;       /* estimates the number of correct bits*/
  unsigned long int i, j, L, t;
  unsigned long int cond;        /* condition number of the series */
  unsigned long int assumed_exponent; /* used as a lowerbound of |EXP(Ai(x))| */
  int r;                         /* returned ternary value */
  mpfr_t s;                      /* used to store the partial sum */
  mpfr_t u0, u1;
  mpfr_t *z;                     /* used to store the (x^3j) */
  mpfr_t result;
  mpfr_t tmp_sp, tmp2_sp;        /* small precision variables */
  unsigned long int x3u;         /* used to store ceil (x^3) */
  mpfr_t temp1, temp2;
  int test0, test1;

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

  /* Save current exponents range */
  MPFR_SAVE_EXPO_MARK (expo);

  /* FIXME: underflow for large values of |x| */

  /* Set initial precision */
  /* See the analysis for the naive evaluation */

  /* We begin with 11 guard bits */
  prec = MPFR_PREC (y) + 11;
  MPFR_ZIV_INIT (loop, prec);

  mpfr_init2 (tmp_sp, MPFR_SMALL_PRECISION);
  mpfr_init2 (tmp2_sp, MPFR_SMALL_PRECISION);
  mpfr_abs (tmp_sp, x, MPFR_RNDU);
  mpfr_pow_ui (tmp_sp, tmp_sp, 3, MPFR_RNDU);
  mpfr_sqrt (tmp_sp, tmp_sp, MPFR_RNDU); /* tmp_sp ~ |x|^(3/2) */

  /* 0.96179669392597567 >~ 2/3 * log2(e). See algorithms.tex */
  mpfr_set_str (tmp2_sp, "0.96179669392597567", 10, MPFR_RNDU);
  mpfr_mul (tmp2_sp, tmp_sp, tmp2_sp, MPFR_RNDU);

  /* cond represents the number of lost bits in the evaluation of the sum */
  if (MPFR_GET_EXP (x) <= 0)
    cond = 0;
  else
    {
      MPFR_BLOCK_DECL (flags);

      MPFR_BLOCK (flags, cond = mpfr_get_ui (tmp2_sp, MPFR_RNDU));
      MPFR_ASSERTN (! MPFR_ERANGEFLAG (flags));
      cond -= (MPFR_GET_EXP (x) - 1) / 4 + 1;
    }

  /* This variable is used to store the maximal assumed exponent of       */
  /* Ai(x). More precisely, we assume that |Ai(x)| will be greater than   */
  /* 2^{-assumed_exponent}.                                               */
  if (MPFR_IS_POS (x))
    {
      if (MPFR_GET_EXP (x) <= 0)
        assumed_exponent = 3;
      else
        {
          unsigned long int t;
          MPFR_BLOCK_DECL (flags);

          MPFR_BLOCK (flags, t = mpfr_get_ui (tmp2_sp, MPFR_RNDU));
          MPFR_ASSERTN (! MPFR_ERANGEFLAG (flags));
          assumed_exponent = t + 2 + (MPFR_GET_EXP (x) / 4 + 1);
          MPFR_ASSERTN (assumed_exponent > t);
        }
    }
  /* We do not know Ai(x) yet */
  /* We cover the case when EXP(Ai(x))>=-10 */
  else
    assumed_exponent = 10;

  {
    unsigned long int t, u;

    t = assumed_exponent + cond;
    MPFR_ASSERTN (t >= cond);
    u = MPFR_INT_CEIL_LOG2 (prec) + 6;
    t += u;
    MPFR_ASSERTN (t >= u);
    wprec = MPFR_ADD_PREC (prec, t);
  }

  /* We assume that the truncation rank will be ~ prec */
  L = __gmpfr_isqrt (prec);
  MPFR_LOG_MSG (("size of blocks L = %lu\n", L));

  z = (mpfr_t *) mpfr_allocate_func ( (L + 1) * sizeof (mpfr_t) );
  MPFR_ASSERTN (z != NULL);
  for (j=0; j<=L; j++)
    mpfr_init (z[j]);

  mpfr_init (s);
  mpfr_init (u0); mpfr_init (u1);
  mpfr_init (result);
  mpfr_init (temp1);
  mpfr_init (temp2);

  /* ZIV loop */
  for (;;)
    {
      MPFR_LOG_MSG (("working precision: %Pu\n", wprec));

      for (j=0; j<=L; j++)
        mpfr_set_prec (z[j], wprec);
      mpfr_set_prec (s, wprec);
      mpfr_set_prec (u0, wprec); mpfr_set_prec (u1, wprec);
      mpfr_set_prec (result, wprec);

      mpfr_set_ui (u0, 1, MPFR_RNDN);
      mpfr_set (u1, x, MPFR_RNDN);

      mpfr_set_ui (z[0], 1, MPFR_RNDU);
      mpfr_sqr (z[1], u1, MPFR_RNDU);
      mpfr_mul (z[1], z[1], x, (MPFR_IS_POS (x) ? MPFR_RNDU : MPFR_RNDD) );

      if (MPFR_IS_NEG (x))
        MPFR_CHANGE_SIGN (z[1]);
      x3u = mpfr_get_ui (z[1], MPFR_RNDU);   /* x3u >= ceil (x^3) */
      if (MPFR_IS_NEG (x))
        MPFR_CHANGE_SIGN (z[1]);

      for (j=2; j<=L ;j++)
        {
          if (j%2 == 0)
            mpfr_sqr (z[j], z[j/2], MPFR_RNDN);
          else
            mpfr_mul (z[j], z[j-1], z[1], MPFR_RNDN);
        }

      mpfr_gamma_one_and_two_third (temp1, temp2, wprec);
      mpfr_set_ui (u0, 9, MPFR_RNDN);
      mpfr_cbrt (u0, u0, MPFR_RNDN);
      mpfr_mul (u0, u0, temp2, MPFR_RNDN);
      mpfr_ui_div (u0, 1, u0 , MPFR_RNDN); /* u0 = 1/( Gamma (2/3)*9^(1/3) ) */

      mpfr_set_ui (u1, 3, MPFR_RNDN);
      mpfr_cbrt (u1, u1, MPFR_RNDN);
      mpfr_mul (u1, u1, temp1, MPFR_RNDN);
      mpfr_neg (u1, u1, MPFR_RNDN);
      mpfr_div (u1, x, u1, MPFR_RNDN); /* u1 = -x/(Gamma (1/3)*3^(1/3)) */

      mpfr_set_ui (result, 0, MPFR_RNDN);
      t = 0;

      /* Evaluation of the series by Smith' method    */
      for (i=0; ; i++)
        {
          t += 3 * L;

          /* k = 0 */
          t -= 3;
          mpfr_set (s, z[L-1], MPFR_RNDN);
          for (j=L-2; ; j--)
            {
              t -= 3;
              mpfr_div_ui2 (s, s, (t+2), (t+3), MPFR_RNDN);
              mpfr_add (s, s, z[j], MPFR_RNDN);
              if (j==0)
                break;
            }
          mpfr_mul (s, s, u0, MPFR_RNDN);
          mpfr_add (result, result, s, MPFR_RNDN);

          mpfr_mul (u0, u0, z[L], MPFR_RNDN);
          for (j=0; j<=L-1; j++)
            {
              mpfr_div_ui2 (u0, u0, (t + 2), (t + 3), MPFR_RNDN);
              t += 3;
            }

          t++;

          /* k = 1 */
          t -= 3;
          mpfr_set (s, z[L-1], MPFR_RNDN);
          for (j=L-2; ; j--)
            {
              t -= 3;
              mpfr_div_ui2 (s, s, (t + 2), (t + 3), MPFR_RNDN);
              mpfr_add (s, s, z[j], MPFR_RNDN);
              if (j==0)
                break;
            }
          mpfr_mul (s, s, u1, MPFR_RNDN);
          mpfr_add (result, result, s, MPFR_RNDN);

          mpfr_mul (u1, u1, z[L], MPFR_RNDN);
          for (j=0; j<=L-1; j++)
            {
              mpfr_div_ui2 (u1, u1, (t + 2), (t + 3), MPFR_RNDN);
              t += 3;
            }

          t++;

          /* k = 2 */
          t++;

          /* End of the loop over k */
          t -= 3;

          test0 = MPFR_IS_ZERO (u0) ||
            MPFR_GET_EXP (u0) + (mpfr_exp_t)prec + 4 <= MPFR_GET_EXP (result);
          test1 = MPFR_IS_ZERO (u1) ||
            MPFR_GET_EXP (u1) + (mpfr_exp_t)prec + 4 <= MPFR_GET_EXP (result);

          if ( test0 && test1 && (x3u <= (t + 2) * (t + 3) / 2) )
            break;
        }

      MPFR_LOG_MSG (("Truncation rank: %lu\n", t));

      err = (5 + MPFR_INT_CEIL_LOG2 (L+1) + MPFR_INT_CEIL_LOG2 (i+1)
             + cond - MPFR_GET_EXP (result));

      /* err is the number of bits lost due to the evaluation error */
      /* wprec-(prec+1): number of bits lost due to the approximation error */
      MPFR_LOG_MSG (("Roundoff error: %Pu\n", err));
      MPFR_LOG_MSG (("Approxim error: %Pu\n", wprec - prec - 1));

      if (wprec < err+1)
        correctBits = 0;
      else
        {
          if (wprec < err+prec+1)
            correctBits = wprec - err - 1;
          else
            correctBits = prec;
        }

      if (MPFR_LIKELY (MPFR_CAN_ROUND (result, correctBits,
                                       MPFR_PREC (y), rnd)))
        break;

      for (j=0; j<=L; j++)
        mpfr_clear (z[j]);
      mpfr_free_func (z, (L + 1) * sizeof (mpfr_t));
      L = __gmpfr_isqrt (t);
      MPFR_LOG_MSG (("size of blocks L = %lu\n", L));
      z = (mpfr_t *) mpfr_allocate_func ( (L + 1) * sizeof (mpfr_t));
      MPFR_ASSERTN (z != NULL);
      for (j=0; j<=L; j++)
        mpfr_init (z[j]);

      if (correctBits == 0)
        {
          assumed_exponent *= 2;
          MPFR_LOG_MSG (("Not a single bit correct (assumed_exponent=%lu)\n",
                         assumed_exponent));
          wprec = prec + 6 + MPFR_INT_CEIL_LOG2 (t) + cond + assumed_exponent;
        }
    else
      {
        if (correctBits < prec)
          { /* The precision was badly chosen */
            MPFR_LOG_MSG (("Bad assumption on the exponent of Ai(x)"
                           " (E=%" MPFR_EXP_FSPEC "d)\n",
                           (mpfr_eexp_t) MPFR_GET_EXP (result)));
            wprec = prec + err + 1;
          }
        else
          { /* We are really in a bad case of the TMD */
            MPFR_ZIV_NEXT (loop, prec);

            /* We update wprec */
            /* We assume that t will not be multiplied by more than 4 */
            wprec = (prec + (MPFR_INT_CEIL_LOG2 (t) + 2) + 6 + cond
                     - MPFR_GET_EXP (result));
          }
      }
    } /* End of ZIV loop */

  MPFR_ZIV_FREE (loop);

  r = mpfr_set (y, result, rnd);

  mpfr_clear (tmp_sp);
  mpfr_clear (tmp2_sp);
  for (j=0; j<=L; j++)
    mpfr_clear (z[j]);
  mpfr_free_func (z, (L + 1) * sizeof (mpfr_t));

  mpfr_clear (s);
  mpfr_clear (u0); mpfr_clear (u1);
  mpfr_clear (result);
  mpfr_clear (temp1);
  mpfr_clear (temp2);

  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, r, rnd);
}

/* We consider that the boundary between the area where the naive method
   should preferably be used and the area where Smith' method should preferably
   be used has the following form:
   it is a triangle defined by two lines (one for the negative values of x, and
   one for the positive values of x) crossing at x=0.

   More precisely,

   * If x<0 and MPFR_AI_THRESHOLD1*x + MPFR_AI_THRESHOLD2*prec > MPFR_AI_SCALE,
   use Smith' algorithm;
   * If x>0 and MPFR_AI_THRESHOLD3*x + MPFR_AI_THRESHOLD2*prec > MPFR_AI_SCALE,
   use Smith' algorithm;
   * otherwise, use the naive method.
*/

#define MPFR_AI_SCALE 1048576

int
mpfr_ai (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd)
{
  mpfr_t temp1, temp2;
  int use_ai2;
  MPFR_SAVE_EXPO_DECL (expo);

  /* Special cases */
  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (x))
        return mpfr_set_ui (y, 0, rnd);
      /* the cases x = +0 or -0 will be treated below */
    }

  /* The exponent range must be large enough for the computation of temp1. */
  MPFR_SAVE_EXPO_MARK (expo);

  mpfr_init2 (temp1, MPFR_SMALL_PRECISION);
  mpfr_init2 (temp2, MPFR_SMALL_PRECISION);

  mpfr_set (temp1, x, MPFR_RNDN);
  mpfr_set_si (temp2, MPFR_AI_THRESHOLD2, MPFR_RNDN);
  mpfr_mul_ui (temp2, temp2, MPFR_PREC (y) > ULONG_MAX ?
               ULONG_MAX : (unsigned long) MPFR_PREC (y), MPFR_RNDN);

  if (MPFR_IS_NEG (x))
      mpfr_mul_si (temp1, temp1, MPFR_AI_THRESHOLD1, MPFR_RNDN);
  else
      mpfr_mul_si (temp1, temp1, MPFR_AI_THRESHOLD3, MPFR_RNDN);

  mpfr_add (temp1, temp1, temp2, MPFR_RNDN);
  mpfr_clear (temp2);

  use_ai2 = mpfr_cmp_si (temp1, MPFR_AI_SCALE) > 0;
  mpfr_clear (temp1);

  MPFR_SAVE_EXPO_FREE (expo); /* Ignore all previous exceptions. */

  /* we use ai2 if |x|*AI_THRESHOLD1/3 + PREC(y)*AI_THRESHOLD2 > AI_SCALE,
     which means x cannot be zero in mpfr_ai2 */
  return use_ai2 ? mpfr_ai2 (y, x, rnd) : mpfr_ai1 (y, x, rnd);
}