int
mpfr_exp (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
  mpfr_exp_t expx;
  mpfr_prec_t precy;
  int inexact;
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
                 ("y[%#R]=%R inexact=%d", y, y, inexact));

  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))
        {
          if (MPFR_IS_POS(x))
            MPFR_SET_INF(y);
          else
            MPFR_SET_ZERO(y);
          MPFR_SET_POS(y);
          MPFR_RET(0);
        }
      else
        {
          MPFR_ASSERTD(MPFR_IS_ZERO(x));
          return mpfr_set_ui (y, 1, rnd_mode);
        }
    }

  /* First, let's detect most overflow and underflow cases. */
  {
    mpfr_t e, bound;

    /* We must extended the exponent range and save the flags now. */
    MPFR_SAVE_EXPO_MARK (expo);

    mpfr_init2 (e, sizeof (mpfr_exp_t) * CHAR_BIT);
    mpfr_init2 (bound, 32);

    inexact = mpfr_set_exp_t (e, expo.saved_emax, MPFR_RNDN);
    MPFR_ASSERTD (inexact == 0);
    mpfr_const_log2 (bound, expo.saved_emax < 0 ? MPFR_RNDD : MPFR_RNDU);
    mpfr_mul (bound, bound, e, MPFR_RNDU);
    if (MPFR_UNLIKELY (mpfr_cmp (x, bound) >= 0))
      {
        /* x > log(2^emax), thus exp(x) > 2^emax */
        mpfr_clears (e, bound, (mpfr_ptr) 0);
        MPFR_SAVE_EXPO_FREE (expo);
        return mpfr_overflow (y, rnd_mode, 1);
      }

    inexact = mpfr_set_exp_t (e, expo.saved_emin, MPFR_RNDN);
    MPFR_ASSERTD (inexact == 0);
    inexact = mpfr_sub_ui (e, e, 2, MPFR_RNDN);
    MPFR_ASSERTD (inexact == 0);
    mpfr_const_log2 (bound, expo.saved_emin < 0 ? MPFR_RNDU : MPFR_RNDD);
    mpfr_mul (bound, bound, e, MPFR_RNDD);
    if (MPFR_UNLIKELY (mpfr_cmp (x, bound) <= 0))
      {
        /* x < log(2^(emin - 2)), thus exp(x) < 2^(emin - 2) */
        mpfr_clears (e, bound, (mpfr_ptr) 0);
        MPFR_SAVE_EXPO_FREE (expo);
        return mpfr_underflow (y, rnd_mode == MPFR_RNDN ? MPFR_RNDZ : rnd_mode,
                               1);
      }

    /* Other overflow/underflow cases must be detected
       by the generic routines. */
    mpfr_clears (e, bound, (mpfr_ptr) 0);
    MPFR_SAVE_EXPO_FREE (expo);
  }

  expx  = MPFR_GET_EXP (x);
  precy = MPFR_PREC (y);

  /* if x < 2^(-precy), then exp(x) i.e. gives 1 +/- 1 ulp(1) */
  if (MPFR_UNLIKELY (expx < 0 && (mpfr_uexp_t) (-expx) > precy))
    {
      mpfr_exp_t emin = __gmpfr_emin;
      mpfr_exp_t emax = __gmpfr_emax;
      int signx = MPFR_SIGN (x);

      MPFR_SET_POS (y);
      if (MPFR_IS_NEG_SIGN (signx) && (rnd_mode == MPFR_RNDD ||
                                       rnd_mode == MPFR_RNDZ))
        {
          __gmpfr_emin = 0;
          __gmpfr_emax = 0;
          mpfr_setmax (y, 0);  /* y = 1 - epsilon */
          inexact = -1;
        }
      else
        {
          __gmpfr_emin = 1;
          __gmpfr_emax = 1;
          mpfr_setmin (y, 1);  /* y = 1 */
          if (MPFR_IS_POS_SIGN (signx) && (rnd_mode == MPFR_RNDU ||
                                           rnd_mode == MPFR_RNDA))
            {
              mp_size_t yn;
              int sh;

              yn = 1 + (MPFR_PREC(y) - 1) / GMP_NUMB_BITS;
              sh = (mpfr_prec_t) yn * GMP_NUMB_BITS - MPFR_PREC(y);
              MPFR_MANT(y)[0] += MPFR_LIMB_ONE << sh;
              inexact = 1;
            }
          else
            inexact = -MPFR_FROM_SIGN_TO_INT(signx);
        }

      __gmpfr_emin = emin;
      __gmpfr_emax = emax;
    }
  else  /* General case */
    {
      if (MPFR_UNLIKELY (precy >= MPFR_EXP_THRESHOLD))
        /* mpfr_exp_3 saves the exponent range and flags itself, otherwise
           the flag changes in mpfr_exp_3 are lost */
        inexact = mpfr_exp_3 (y, x, rnd_mode); /* O(M(n) log(n)^2) */
      else
        {
          MPFR_SAVE_EXPO_MARK (expo);
          inexact = mpfr_exp_2 (y, x, rnd_mode); /* O(n^(1/3) M(n)) */
          MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, __gmpfr_flags);
          MPFR_SAVE_EXPO_FREE (expo);
        }
    }

  return mpfr_check_range (y, inexact, rnd_mode);
}
Example #2
0
File: mul.c Project: gnooth/xcl
static int
mpfr_mul3 (mpfr_ptr a, mpfr_srcptr b, mpfr_srcptr c, mpfr_rnd_t rnd_mode)
{
    /* Old implementation */
    int sign_product, cc, inexact;
    mpfr_exp_t ax;
    mp_limb_t *tmp;
    mp_limb_t b1;
    mpfr_prec_t bq, cq;
    mp_size_t bn, cn, tn, k;
    MPFR_TMP_DECL(marker);

    /* deal with special cases */
    if (MPFR_ARE_SINGULAR(b,c))
    {
        if (MPFR_IS_NAN(b) || MPFR_IS_NAN(c))
        {
            MPFR_SET_NAN(a);
            MPFR_RET_NAN;
        }
        sign_product = MPFR_MULT_SIGN( MPFR_SIGN(b) , MPFR_SIGN(c) );
        if (MPFR_IS_INF(b))
        {
            if (MPFR_IS_INF(c) || MPFR_NOTZERO(c))
            {
                MPFR_SET_SIGN(a,sign_product);
                MPFR_SET_INF(a);
                MPFR_RET(0); /* exact */
            }
            else
            {
                MPFR_SET_NAN(a);
                MPFR_RET_NAN;
            }
        }
        else if (MPFR_IS_INF(c))
        {
            if (MPFR_NOTZERO(b))
            {
                MPFR_SET_SIGN(a, sign_product);
                MPFR_SET_INF(a);
                MPFR_RET(0); /* exact */
            }
            else
            {
                MPFR_SET_NAN(a);
                MPFR_RET_NAN;
            }
        }
        else
        {
            MPFR_ASSERTD(MPFR_IS_ZERO(b) || MPFR_IS_ZERO(c));
            MPFR_SET_SIGN(a, sign_product);
            MPFR_SET_ZERO(a);
            MPFR_RET(0); /* 0 * 0 is exact */
        }
    }
    sign_product = MPFR_MULT_SIGN( MPFR_SIGN(b) , MPFR_SIGN(c) );

    ax = MPFR_GET_EXP (b) + MPFR_GET_EXP (c);

    bq = MPFR_PREC(b);
    cq = MPFR_PREC(c);

    MPFR_ASSERTD(bq+cq > bq); /* PREC_MAX is /2 so no integer overflow */

    bn = (bq+GMP_NUMB_BITS-1)/GMP_NUMB_BITS; /* number of limbs of b */
    cn = (cq+GMP_NUMB_BITS-1)/GMP_NUMB_BITS; /* number of limbs of c */
    k = bn + cn; /* effective nb of limbs used by b*c (= tn or tn+1) below */
    tn = (bq + cq + GMP_NUMB_BITS - 1) / GMP_NUMB_BITS;
    /* <= k, thus no int overflow */
    MPFR_ASSERTD(tn <= k);

    /* Check for no size_t overflow*/
    MPFR_ASSERTD((size_t) k <= ((size_t) -1) / BYTES_PER_MP_LIMB);
    MPFR_TMP_MARK(marker);
    tmp = (mp_limb_t *) MPFR_TMP_ALLOC((size_t) k * BYTES_PER_MP_LIMB);

    /* multiplies two mantissa in temporary allocated space */
    b1 = (MPFR_LIKELY(bn >= cn)) ?
         mpn_mul (tmp, MPFR_MANT(b), bn, MPFR_MANT(c), cn)
         : mpn_mul (tmp, MPFR_MANT(c), cn, MPFR_MANT(b), bn);

    /* now tmp[0]..tmp[k-1] contains the product of both mantissa,
       with tmp[k-1]>=2^(GMP_NUMB_BITS-2) */
    b1 >>= GMP_NUMB_BITS - 1; /* msb from the product */

    /* if the mantissas of b and c are uniformly distributed in ]1/2, 1],
       then their product is in ]1/4, 1/2] with probability 2*ln(2)-1 ~ 0.386
       and in [1/2, 1] with probability 2-2*ln(2) ~ 0.614 */
    tmp += k - tn;
    if (MPFR_UNLIKELY(b1 == 0))
        mpn_lshift (tmp, tmp, tn, 1); /* tn <= k, so no stack corruption */
    cc = mpfr_round_raw (MPFR_MANT (a), tmp, bq + cq,
                         MPFR_IS_NEG_SIGN(sign_product),
                         MPFR_PREC (a), rnd_mode, &inexact);

    /* cc = 1 ==> result is a power of two */
    if (MPFR_UNLIKELY(cc))
        MPFR_MANT(a)[MPFR_LIMB_SIZE(a)-1] = MPFR_LIMB_HIGHBIT;

    MPFR_TMP_FREE(marker);

    {
        mpfr_exp_t ax2 = ax + (mpfr_exp_t) (b1 - 1 + cc);
        if (MPFR_UNLIKELY( ax2 > __gmpfr_emax))
            return mpfr_overflow (a, rnd_mode, sign_product);
        if (MPFR_UNLIKELY( ax2 < __gmpfr_emin))
        {
            /* In the rounding to the nearest mode, if the exponent of the exact
               result (i.e. before rounding, i.e. without taking cc into account)
               is < __gmpfr_emin - 1 or the exact result is a power of 2 (i.e. if
               both arguments are powers of 2), then round to zero. */
            if (rnd_mode == MPFR_RNDN &&
                    (ax + (mpfr_exp_t) b1 < __gmpfr_emin ||
                     (mpfr_powerof2_raw (b) && mpfr_powerof2_raw (c))))
                rnd_mode = MPFR_RNDZ;
            return mpfr_underflow (a, rnd_mode, sign_product);
        }
        MPFR_SET_EXP (a, ax2);
        MPFR_SET_SIGN(a, sign_product);
    }
    MPFR_RET (inexact);
}
Example #3
0
int
mpfr_tanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
{
  /****** Declaration ******/
  mpfr_t x;
  int inexact;
  MPFR_SAVE_EXPO_DECL (expo);

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

  /* Special value checking */
  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
    {
      if (MPFR_IS_NAN (xt))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (xt))
        {
          /* tanh(inf) = 1 && tanh(-inf) = -1 */
          return mpfr_set_si (y, MPFR_INT_SIGN (xt), rnd_mode);
        }
      else /* tanh (0) = 0 and xt is zero */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO(xt));
          MPFR_SET_ZERO (y);
          MPFR_SET_SAME_SIGN (y, xt);
          MPFR_RET (0);
        }
    }

  /* tanh(x) = x - x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */
  MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 0,
                                    rnd_mode, {});

  MPFR_TMP_INIT_ABS (x, xt);

  MPFR_SAVE_EXPO_MARK (expo);

  /* General case */
  {
    /* Declaration of the intermediary variable */
    mpfr_t t, te;
    mpfr_exp_t d;

    /* Declaration of the size variable */
    mpfr_prec_t Ny = MPFR_PREC(y);   /* target precision */
    mpfr_prec_t Nt;                  /* working precision */
    long int err;                  /* error */
    int sign = MPFR_SIGN (xt);
    MPFR_ZIV_DECL (loop);
    MPFR_GROUP_DECL (group);

    /* First check for BIG overflow of exp(2*x):
       For x > 0, exp(2*x) > 2^(2*x)
       If 2 ^(2*x) > 2^emax or x>emax/2, there is an overflow */
    if (MPFR_UNLIKELY (mpfr_cmp_si (x, __gmpfr_emax/2) >= 0)) {
      /* initialise of intermediary variables
         since 'set_one' label assumes the variables have been
         initialize */
      MPFR_GROUP_INIT_2 (group, MPFR_PREC_MIN, t, te);
      goto set_one;
    }

    /* Compute the precision of intermediary variable */
    /* The optimal number of bits: see algorithms.tex */
    Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 4;
    /* if x is small, there will be a cancellation in exp(2x)-1 */
    if (MPFR_GET_EXP (x) < 0)
      Nt += -MPFR_GET_EXP (x);

    /* initialise of intermediary variable */
    MPFR_GROUP_INIT_2 (group, Nt, t, te);

    MPFR_ZIV_INIT (loop, Nt);
    for (;;) {
      /* tanh = (exp(2x)-1)/(exp(2x)+1) */
      mpfr_mul_2ui (te, x, 1, MPFR_RNDN);  /* 2x */
      /* since x > 0, we can only have an overflow */
      mpfr_exp (te, te, MPFR_RNDN);        /* exp(2x) */
      if (MPFR_UNLIKELY (MPFR_IS_INF (te))) {
      set_one:
        inexact = MPFR_FROM_SIGN_TO_INT (sign);
        mpfr_set4 (y, __gmpfr_one, MPFR_RNDN, sign);
        if (MPFR_IS_LIKE_RNDZ (rnd_mode, MPFR_IS_NEG_SIGN (sign)))
          {
            inexact = -inexact;
            mpfr_nexttozero (y);
          }
        break;
      }
      d = MPFR_GET_EXP (te);              /* For Error calculation */
      mpfr_add_ui (t, te, 1, MPFR_RNDD);   /* exp(2x) + 1*/
      mpfr_sub_ui (te, te, 1, MPFR_RNDU);  /* exp(2x) - 1*/
      d = d - MPFR_GET_EXP (te);
      mpfr_div (t, te, t, MPFR_RNDN);      /* (exp(2x)-1)/(exp(2x)+1)*/

      /* Calculation of the error */
      d = MAX(3, d + 1);
      err = Nt - (d + 1);

      if (MPFR_LIKELY ((d <= Nt / 2) && MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
        {
          inexact = mpfr_set4 (y, t, rnd_mode, sign);
          break;
        }

      /* if t=1, we still can round since |sinh(x)| < 1 */
      if (MPFR_GET_EXP (t) == 1)
        goto set_one;

      /* Actualisation of the precision */
      MPFR_ZIV_NEXT (loop, Nt);
      MPFR_GROUP_REPREC_2 (group, Nt, t, te);
    }
    MPFR_ZIV_FREE (loop);
    MPFR_GROUP_CLEAR (group);
  }
  MPFR_SAVE_EXPO_FREE (expo);
  inexact = mpfr_check_range (y, inexact, rnd_mode);

  return inexact;
}
Example #4
0
int
mpfr_sin (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode)
{
  mpfr_t c, xr;
  mpfr_srcptr xx;
  mpfr_exp_t expx, err;
  mpfr_prec_t precy, m;
  int inexact, sign, reduce;
  MPFR_ZIV_DECL (loop);
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC (("x[%#R]=%R rnd=%d", x, x, rnd_mode),
                  ("y[%#R]=%R inexact=%d", y, y, inexact));

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x)))
    {
      if (MPFR_IS_NAN (x) || MPFR_IS_INF (x))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;

        }
      else /* x is zero */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (x));
          MPFR_SET_ZERO (y);
          MPFR_SET_SAME_SIGN (y, x);
          MPFR_RET (0);
        }
    }

  /* sin(x) = x - x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
  MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, x, -2 * MPFR_GET_EXP (x), 2, 0,
                                    rnd_mode, {});

  MPFR_SAVE_EXPO_MARK (expo);

  /* Compute initial precision */
  precy = MPFR_PREC (y);

  if (precy >= MPFR_SINCOS_THRESHOLD)
    return mpfr_sin_fast (y, x, rnd_mode);

  m = precy + MPFR_INT_CEIL_LOG2 (precy) + 13;
  expx = MPFR_GET_EXP (x);

  mpfr_init (c);
  mpfr_init (xr);

  MPFR_ZIV_INIT (loop, m);
  for (;;)
    {
      /* first perform argument reduction modulo 2*Pi (if needed),
         also helps to determine the sign of sin(x) */
      if (expx >= 2) /* If Pi < x < 4, we need to reduce too, to determine
                        the sign of sin(x). For 2 <= |x| < Pi, we could avoid
                        the reduction. */
        {
          reduce = 1;
          /* As expx + m - 1 will silently be converted into mpfr_prec_t
             in the mpfr_set_prec call, the assert below may be useful to
             avoid undefined behavior. */
          MPFR_ASSERTN (expx + m - 1 <= MPFR_PREC_MAX);
          mpfr_set_prec (c, expx + m - 1);
          mpfr_set_prec (xr, m);
          mpfr_const_pi (c, MPFR_RNDN);
          mpfr_mul_2ui (c, c, 1, MPFR_RNDN);
          mpfr_remainder (xr, x, c, MPFR_RNDN);
          /* The analysis is similar to that of cos.c:
             |xr - x - 2kPi| <= 2^(2-m). Thus we can decide the sign
             of sin(x) if xr is at distance at least 2^(2-m) of both
             0 and +/-Pi. */
          mpfr_div_2ui (c, c, 1, MPFR_RNDN);
          /* Since c approximates Pi with an error <= 2^(2-expx-m) <= 2^(-m),
             it suffices to check that c - |xr| >= 2^(2-m). */
          if (MPFR_SIGN (xr) > 0)
            mpfr_sub (c, c, xr, MPFR_RNDZ);
          else
            mpfr_add (c, c, xr, MPFR_RNDZ);
          if (MPFR_IS_ZERO(xr)
              || MPFR_EXP(xr) < (mpfr_exp_t) 3 - (mpfr_exp_t) m
              || MPFR_EXP(c) < (mpfr_exp_t) 3 - (mpfr_exp_t) m)
            goto ziv_next;

          /* |xr - x - 2kPi| <= 2^(2-m), thus |sin(xr) - sin(x)| <= 2^(2-m) */
          xx = xr;
        }
      else /* the input argument is already reduced */
        {
          reduce = 0;
          xx = x;
        }

      sign = MPFR_SIGN(xx);
      /* now that the argument is reduced, precision m is enough */
      mpfr_set_prec (c, m);
      mpfr_cos (c, xx, MPFR_RNDZ);    /* can't be exact */
      mpfr_nexttoinf (c);           /* now c = cos(x) rounded away */
      mpfr_mul (c, c, c, MPFR_RNDU); /* away */
      mpfr_ui_sub (c, 1, c, MPFR_RNDZ);
      mpfr_sqrt (c, c, MPFR_RNDZ);
      if (MPFR_IS_NEG_SIGN(sign))
        MPFR_CHANGE_SIGN(c);

      /* Warning: c may be 0! */
      if (MPFR_UNLIKELY (MPFR_IS_ZERO (c)))
        {
          /* Huge cancellation: increase prec a lot! */
          m = MAX (m, MPFR_PREC (x));
          m = 2 * m;
        }
      else
        {
          /* the absolute error on c is at most 2^(3-m-EXP(c)),
             plus 2^(2-m) if there was an argument reduction.
             Since EXP(c) <= 1, 3-m-EXP(c) >= 2-m, thus the error
             is at most 2^(3-m-EXP(c)) in case of argument reduction. */
          err = 2 * MPFR_GET_EXP (c) + (mpfr_exp_t) m - 3 - (reduce != 0);
          if (MPFR_CAN_ROUND (c, err, precy, rnd_mode))
            break;

          /* check for huge cancellation (Near 0) */
          if (err < (mpfr_exp_t) MPFR_PREC (y))
            m += MPFR_PREC (y) - err;
          /* Check if near 1 */
          if (MPFR_GET_EXP (c) == 1)
            m += m;
        }

    ziv_next:
      /* Else generic increase */
      MPFR_ZIV_NEXT (loop, m);
    }
  MPFR_ZIV_FREE (loop);

  inexact = mpfr_set (y, c, rnd_mode);
  /* inexact cannot be 0, since this would mean that c was representable
     within the target precision, but in that case mpfr_can_round will fail */

  mpfr_clear (c);
  mpfr_clear (xr);

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