コード例 #1
0
ファイル: get_d.c プロジェクト: MiKTeX/miktex
double
mpfr_get_d_2exp (long *expptr, mpfr_srcptr src, mpfr_rnd_t rnd_mode)
{
  double ret;
  mpfr_exp_t exp;
  mpfr_t tmp;

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (src)))
    {
      int negative;
      *expptr = 0;
      if (MPFR_IS_NAN (src))
        return MPFR_DBL_NAN;
      negative = MPFR_IS_NEG (src);
      if (MPFR_IS_INF (src))
        return negative ? MPFR_DBL_INFM : MPFR_DBL_INFP;
      MPFR_ASSERTD (MPFR_IS_ZERO(src));
      return negative ? DBL_NEG_ZERO : 0.0;
    }

  MPFR_ALIAS (tmp, src, MPFR_SIGN (src), 0);
  ret = mpfr_get_d (tmp, rnd_mode);

  if (MPFR_IS_PURE_FP(src))
    {
      exp = MPFR_GET_EXP (src);

      /* rounding can give 1.0, adjust back to 0.5 <= abs(ret) < 1.0 */
      if (ret == 1.0)
        {
          ret = 0.5;
          exp++;
        }
      else if (ret == -1.0)
        {
          ret = -0.5;
          exp++;
        }

      MPFR_ASSERTN ((ret >= 0.5 && ret < 1.0)
                    || (ret <= -0.5 && ret > -1.0));
      MPFR_ASSERTN (exp >= LONG_MIN && exp <= LONG_MAX);
    }
  else
    exp = 0;

  *expptr = exp;
  return ret;
}
コード例 #2
0
ファイル: pow.c プロジェクト: Distrotech/mpfr
/* Assumes that the exponent range has already been extended and if y is
   an integer, then the result is not exact in unbounded exponent range. */
int
mpfr_pow_general (mpfr_ptr z, mpfr_srcptr x, mpfr_srcptr y,
                  mpfr_rnd_t rnd_mode, int y_is_integer, mpfr_save_expo_t *expo)
{
  mpfr_t t, u, k, absx;
  int neg_result = 0;
  int k_non_zero = 0;
  int check_exact_case = 0;
  int inexact;
  /* Declaration of the size variable */
  mpfr_prec_t Nz = MPFR_PREC(z);               /* target precision */
  mpfr_prec_t Nt;                              /* working precision */
  mpfr_exp_t err;                              /* error */
  MPFR_ZIV_DECL (ziv_loop);


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

  /* We put the absolute value of x in absx, pointing to the significand
     of x to avoid allocating memory for the significand of absx. */
  MPFR_ALIAS(absx, x, /*sign=*/ 1, /*EXP=*/ MPFR_EXP(x));

  /* We will compute the absolute value of the result. So, let's
     invert the rounding mode if the result is negative. */
  if (MPFR_IS_NEG (x) && is_odd (y))
    {
      neg_result = 1;
      rnd_mode = MPFR_INVERT_RND (rnd_mode);
    }

  /* compute the precision of intermediary variable */
  /* the optimal number of bits : see algorithms.tex */
  Nt = Nz + 5 + MPFR_INT_CEIL_LOG2 (Nz);

  /* initialise of intermediary variable */
  mpfr_init2 (t, Nt);

  MPFR_ZIV_INIT (ziv_loop, Nt);
  for (;;)
    {
      MPFR_BLOCK_DECL (flags1);

      /* compute exp(y*ln|x|), using MPFR_RNDU to get an upper bound, so
         that we can detect underflows. */
      mpfr_log (t, absx, MPFR_IS_NEG (y) ? MPFR_RNDD : MPFR_RNDU); /* ln|x| */
      mpfr_mul (t, y, t, MPFR_RNDU);                              /* y*ln|x| */
      if (k_non_zero)
        {
          MPFR_LOG_MSG (("subtract k * ln(2)\n", 0));
          mpfr_const_log2 (u, MPFR_RNDD);
          mpfr_mul (u, u, k, MPFR_RNDD);
          /* Error on u = k * log(2): < k * 2^(-Nt) < 1. */
          mpfr_sub (t, t, u, MPFR_RNDU);
          MPFR_LOG_MSG (("t = y * ln|x| - k * ln(2)\n", 0));
          MPFR_LOG_VAR (t);
        }
      /* estimate of the error -- see pow function in algorithms.tex.
         The error on t is at most 1/2 + 3*2^(EXP(t)+1) ulps, which is
         <= 2^(EXP(t)+3) for EXP(t) >= -1, and <= 2 ulps for EXP(t) <= -2.
         Additional error if k_no_zero: treal = t * errk, with
         1 - |k| * 2^(-Nt) <= exp(-|k| * 2^(-Nt)) <= errk <= 1,
         i.e., additional absolute error <= 2^(EXP(k)+EXP(t)-Nt).
         Total error <= 2^err1 + 2^err2 <= 2^(max(err1,err2)+1). */
      err = MPFR_NOTZERO (t) && MPFR_GET_EXP (t) >= -1 ?
        MPFR_GET_EXP (t) + 3 : 1;
      if (k_non_zero)
        {
          if (MPFR_GET_EXP (k) > err)
            err = MPFR_GET_EXP (k);
          err++;
        }
      MPFR_BLOCK (flags1, mpfr_exp (t, t, MPFR_RNDN));  /* exp(y*ln|x|)*/
      /* We need to test */
      if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (t) || MPFR_UNDERFLOW (flags1)))
        {
          mpfr_prec_t Ntmin;
          MPFR_BLOCK_DECL (flags2);

          MPFR_ASSERTN (!k_non_zero);
          MPFR_ASSERTN (!MPFR_IS_NAN (t));

          /* Real underflow? */
          if (MPFR_IS_ZERO (t))
            {
              /* Underflow. We computed rndn(exp(t)), where t >= y*ln|x|.
                 Therefore rndn(|x|^y) = 0, and we have a real underflow on
                 |x|^y. */
              inexact = mpfr_underflow (z, rnd_mode == MPFR_RNDN ? MPFR_RNDZ
                                        : rnd_mode, MPFR_SIGN_POS);
              if (expo != NULL)
                MPFR_SAVE_EXPO_UPDATE_FLAGS (*expo, MPFR_FLAGS_INEXACT
                                             | MPFR_FLAGS_UNDERFLOW);
              break;
            }

          /* Real overflow? */
          if (MPFR_IS_INF (t))
            {
              /* Note: we can probably use a low precision for this test. */
              mpfr_log (t, absx, MPFR_IS_NEG (y) ? MPFR_RNDU : MPFR_RNDD);
              mpfr_mul (t, y, t, MPFR_RNDD);            /* y * ln|x| */
              MPFR_BLOCK (flags2, mpfr_exp (t, t, MPFR_RNDD));
              /* t = lower bound on exp(y * ln|x|) */
              if (MPFR_OVERFLOW (flags2))
                {
                  /* We have computed a lower bound on |x|^y, and it
                     overflowed. Therefore we have a real overflow
                     on |x|^y. */
                  inexact = mpfr_overflow (z, rnd_mode, MPFR_SIGN_POS);
                  if (expo != NULL)
                    MPFR_SAVE_EXPO_UPDATE_FLAGS (*expo, MPFR_FLAGS_INEXACT
                                                 | MPFR_FLAGS_OVERFLOW);
                  break;
                }
            }

          k_non_zero = 1;
          Ntmin = sizeof(mpfr_exp_t) * CHAR_BIT;
          if (Ntmin > Nt)
            {
              Nt = Ntmin;
              mpfr_set_prec (t, Nt);
            }
          mpfr_init2 (u, Nt);
          mpfr_init2 (k, Ntmin);
          mpfr_log2 (k, absx, MPFR_RNDN);
          mpfr_mul (k, y, k, MPFR_RNDN);
          mpfr_round (k, k);
          MPFR_LOG_VAR (k);
          /* |y| < 2^Ntmin, therefore |k| < 2^Nt. */
          continue;
        }
      if (MPFR_LIKELY (MPFR_CAN_ROUND (t, Nt - err, Nz, rnd_mode)))
        {
          inexact = mpfr_set (z, t, rnd_mode);
          break;
        }

      /* check exact power, except when y is an integer (since the
         exact cases for y integer have already been filtered out) */
      if (check_exact_case == 0 && ! y_is_integer)
        {
          if (mpfr_pow_is_exact (z, absx, y, rnd_mode, &inexact))
            break;
          check_exact_case = 1;
        }

      /* reactualisation of the precision */
      MPFR_ZIV_NEXT (ziv_loop, Nt);
      mpfr_set_prec (t, Nt);
      if (k_non_zero)
        mpfr_set_prec (u, Nt);
    }
  MPFR_ZIV_FREE (ziv_loop);

  if (k_non_zero)
    {
      int inex2;
      long lk;

      /* The rounded result in an unbounded exponent range is z * 2^k. As
       * MPFR chooses underflow after rounding, the mpfr_mul_2si below will
       * correctly detect underflows and overflows. However, in rounding to
       * nearest, if z * 2^k = 2^(emin - 2), then the double rounding may
       * affect the result. We need to cope with that before overwriting z.
       * This can occur only if k < 0 (this test is necessary to avoid a
       * potential integer overflow).
       * If inexact >= 0, then the real result is <= 2^(emin - 2), so that
       * o(2^(emin - 2)) = +0 is correct. If inexact < 0, then the real
       * result is > 2^(emin - 2) and we need to round to 2^(emin - 1).
       */
      MPFR_ASSERTN (MPFR_EXP_MAX <= LONG_MAX);
      lk = mpfr_get_si (k, MPFR_RNDN);
      /* Due to early overflow detection, |k| should not be much larger than
       * MPFR_EMAX_MAX, and as MPFR_EMAX_MAX <= MPFR_EXP_MAX/2 <= LONG_MAX/2,
       * an overflow should not be possible in mpfr_get_si (and lk is exact).
       * And one even has the following assertion. TODO: complete proof.
       */
      MPFR_ASSERTD (lk > LONG_MIN && lk < LONG_MAX);
      /* Note: even in case of overflow (lk inexact), the code is correct.
       * Indeed, for the 3 occurrences of lk:
       *   - The test lk < 0 is correct as sign(lk) = sign(k).
       *   - In the test MPFR_GET_EXP (z) == __gmpfr_emin - 1 - lk,
       *     if lk is inexact, then lk = LONG_MIN <= MPFR_EXP_MIN
       *     (the minimum value of the mpfr_exp_t type), and
       *     __gmpfr_emin - 1 - lk >= MPFR_EMIN_MIN - 1 - 2 * MPFR_EMIN_MIN
       *     >= - MPFR_EMIN_MIN - 1 = MPFR_EMAX_MAX - 1. However, from the
       *     choice of k, z has been chosen to be around 1, so that the
       *     result of the test is false, as if lk were exact.
       *   - In the mpfr_mul_2si (z, z, lk, rnd_mode), if lk is inexact,
       *     then |lk| >= LONG_MAX >= MPFR_EXP_MAX, and as z is around 1,
       *     mpfr_mul_2si underflows or overflows in the same way as if
       *     lk were exact.
       * TODO: give a bound on |t|, then on |EXP(z)|.
       */
      if (rnd_mode == MPFR_RNDN && inexact < 0 && lk < 0 &&
          MPFR_GET_EXP (z) == __gmpfr_emin - 1 - lk && mpfr_powerof2_raw (z))
        {
          /* Rounding to nearest, real result > z * 2^k = 2^(emin - 2),
           * underflow case: as the minimum precision is > 1, we will
           * obtain the correct result and exceptions by replacing z by
           * nextabove(z).
           */
          MPFR_ASSERTN (MPFR_PREC_MIN > 1);
          mpfr_nextabove (z);
        }
      MPFR_CLEAR_FLAGS ();
      inex2 = mpfr_mul_2si (z, z, lk, rnd_mode);
      if (inex2)  /* underflow or overflow */
        {
          inexact = inex2;
          if (expo != NULL)
            MPFR_SAVE_EXPO_UPDATE_FLAGS (*expo, __gmpfr_flags);
        }
      mpfr_clears (u, k, (mpfr_ptr) 0);
    }
  mpfr_clear (t);

  /* update the sign of the result if x was negative */
  if (neg_result)
    {
      MPFR_SET_NEG(z);
      inexact = -inexact;
    }

  return inexact;
}