Exemple #1
0
void eval_exp(T& result, const T& x)
{
   BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The exp function is only valid for floating point types.");
   if(&x == &result)
   {
      T temp;
      eval_exp(temp, x);
      result = temp;
      return;
   }
   typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
   typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
   typedef typename T::exponent_type exp_type;
   typedef typename boost::multiprecision::detail::canonical<exp_type, T>::type canonical_exp_type;
   typedef typename boost::multiprecision::detail::canonical<float, T>::type float_type;

   // Handle special arguments.
   int type = eval_fpclassify(x);
   bool isneg = eval_get_sign(x) < 0;
   if(type == FP_NAN)
   {
      result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
      return;
   }
   else if(type == FP_INFINITE)
   {
      result = x;
      if(isneg)
         result = ui_type(0u);
      else 
         result = x;
      return;
   }
   else if(type == FP_ZERO)
   {
      result = ui_type(1);
      return;
   }

   // Get local copy of argument and force it to be positive.
   T xx = x;
   T exp_series;
   if(isneg)
      xx.negate();

   // Check the range of the argument.
   static const canonical_exp_type maximum_arg_for_exp = std::numeric_limits<number<T, et_on> >::max_exponent == 0 ? (std::numeric_limits<long>::max)() : std::numeric_limits<number<T, et_on> >::max_exponent;

   if(xx.compare(maximum_arg_for_exp) >= 0)
   {
      // Overflow / underflow
      if(isneg)
         result = ui_type(0);
      else
         result = std::numeric_limits<number<T, et_on> >::has_infinity ? std::numeric_limits<number<T, et_on> >::infinity().backend() : (std::numeric_limits<number<T, et_on> >::max)().backend();
      return;
   }
   if(xx.compare(si_type(1)) <= 0)
   {
      //
      // Use series for exp(x) - 1:
      //
      T lim = std::numeric_limits<number<T, et_on> >::epsilon().backend();
      unsigned k = 2;
      exp_series = xx;
      result = si_type(1);
      if(isneg)
         eval_subtract(result, exp_series);
      else
         eval_add(result, exp_series);
      eval_multiply(exp_series, xx);
      eval_divide(exp_series, ui_type(k));
      eval_add(result, exp_series);
      while(exp_series.compare(lim) > 0)
      {
         ++k;
         eval_multiply(exp_series, xx);
         eval_divide(exp_series, ui_type(k));
         if(isneg && (k&1))
            eval_subtract(result, exp_series);
         else
            eval_add(result, exp_series);
      }
      return;
   }

   // Check for pure-integer arguments which can be either signed or unsigned.
   typename boost::multiprecision::detail::canonical<boost::intmax_t, T>::type ll;
   eval_trunc(exp_series, x);
   eval_convert_to(&ll, exp_series);
   if(x.compare(ll) == 0)
   {
      detail::pow_imp(result, get_constant_e<T>(), ll, mpl::true_());
      return;
   }

   // The algorithm for exp has been taken from MPFUN.
   // exp(t) = [ (1 + r + r^2/2! + r^3/3! + r^4/4! ...)^p2 ] * 2^n
   // where p2 is a power of 2 such as 2048, r = t_prime / p2, and
   // t_prime = t - n*ln2, with n chosen to minimize the absolute
   // value of t_prime. In the resulting Taylor series, which is
   // implemented as a hypergeometric function, |r| is bounded by
   // ln2 / p2. For small arguments, no scaling is done.

   // Compute the exponential series of the (possibly) scaled argument.

   eval_divide(result, xx, get_constant_ln2<T>());
   exp_type n;
   eval_convert_to(&n, result);

   // The scaling is 2^11 = 2048.
   static const si_type p2 = static_cast<si_type>(si_type(1) << 11);

   eval_multiply(exp_series, get_constant_ln2<T>(), static_cast<canonical_exp_type>(n));
   eval_subtract(exp_series, xx);
   eval_divide(exp_series, p2);
   exp_series.negate();
   hyp0F0(result, exp_series);

   detail::pow_imp(exp_series, result, p2, mpl::true_());
   result = ui_type(1);
   eval_ldexp(result, result, n);
   eval_multiply(exp_series, result);

   if(isneg)
      eval_divide(result, ui_type(1), exp_series);
   else
      result = exp_series;
}
Exemple #2
0
inline void eval_pow(T& result, const T& x, const T& a)
{
   BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The pow function is only valid for floating point types.");
   typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
   typedef typename boost::multiprecision::detail::canonical<unsigned, T>::type ui_type;
   typedef typename T::exponent_type exp_type;
   typedef typename boost::multiprecision::detail::canonical<exp_type, T>::type canonical_exp_type;
   typedef typename mpl::front<typename T::float_types>::type fp_type;

   if((&result == &x) || (&result == &a))
   {
      T t;
      eval_pow(t, x, a);
      result = t;
      return;
   }

   if(a.compare(si_type(1)) == 0)
   {
      result = x;
      return;
   }

   int type = eval_fpclassify(x);

   switch(type)
   {
   case FP_INFINITE:
      result = x;
      return;
   case FP_ZERO:
      result = si_type(1);
      return;
   case FP_NAN:
      result = x;
      return;
   default: ;
   }

   if(eval_get_sign(a) == 0)
   {
      result = si_type(1);
      return;
   }

   if(a.compare(si_type(-1)) < 0)
   {
      T t, da;
      t = a;
      t.negate();
      eval_pow(da, x, t);
      eval_divide(result, si_type(1), da);
      return;
   }
   
   bool bo_a_isint = false;
   typename boost::multiprecision::detail::canonical<boost::intmax_t, T>::type an;
   T fa;
   try
   {
      eval_convert_to(&an, a);
      if(a.compare(an) == 0)
      {
         detail::pow_imp(result, x, an, mpl::true_());
         return;
      }
   }
   catch(const std::exception&)
   {
      // conversion failed, just fall through, value is not an integer.
      an = (std::numeric_limits<boost::intmax_t>::max)();
   }

   if((eval_get_sign(x) < 0) && !bo_a_isint)
   {
      result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
   }

   T t, da;

   eval_subtract(da, a, an);

   if((x.compare(fp_type(0.5)) >= 0) && (x.compare(fp_type(0.9)) < 0))
   {
      if(a.compare(fp_type(1e-5f)) <= 0)
      {
         // Series expansion for small a.
         eval_log(t, x);
         eval_multiply(t, a);
         hyp0F0(result, t);
         return;
      }
      else
      {
         // Series expansion for moderately sized x. Note that for large power of a,
         // the power of the integer part of a is calculated using the pown function.
         if(an)
         {
            da.negate();
            t = si_type(1);
            eval_subtract(t, x);
            hyp1F0(result, da, t);
            detail::pow_imp(t, x, an, mpl::true_());
            eval_multiply(result, t);
         }
         else
         {
            da = a;
            da.negate();
            t = si_type(1);
            eval_subtract(t, x);
            hyp1F0(result, da, t);
         }
      }
   }
   else
   {
      // Series expansion for pow(x, a). Note that for large power of a, the power
      // of the integer part of a is calculated using the pown function.
      if(an)
      {
         eval_log(t, x);
         eval_multiply(t, da);
         eval_exp(result, t);
         detail::pow_imp(t, x, an, mpl::true_());
         eval_multiply(result, t);
      }
      else
      {
         eval_log(t, x);
         eval_multiply(t, a);
         eval_exp(result, t);
      }
   }
}
Exemple #3
0
inline void eval_pow(T& result, const T& x, const T& a)
{
   BOOST_STATIC_ASSERT_MSG(number_category<T>::value == number_kind_floating_point, "The pow function is only valid for floating point types.");
   typedef typename boost::multiprecision::detail::canonical<int, T>::type si_type;
   typedef typename mpl::front<typename T::float_types>::type fp_type;

   if((&result == &x) || (&result == &a))
   {
      T t;
      eval_pow(t, x, a);
      result = t;
      return;
   }

   if(a.compare(si_type(1)) == 0)
   {
      result = x;
      return;
   }

   int type = eval_fpclassify(x);

   switch(type)
   {
   case FP_INFINITE:
      result = x;
      return;
   case FP_ZERO:
      switch(eval_fpclassify(a))
      {
      case FP_ZERO:
         result = si_type(1);
         break;
      case FP_NAN:
         result = a;
         break;
      default:
         result = x;
         break;
      }
      return;
   case FP_NAN:
      result = x;
      return;
   default: ;
   }

   int s = eval_get_sign(a);
   if(s == 0)
   {
      result = si_type(1);
      return;
   }

   if(s < 0)
   {
      T t, da;
      t = a;
      t.negate();
      eval_pow(da, x, t);
      eval_divide(result, si_type(1), da);
      return;
   }
   
   typename boost::multiprecision::detail::canonical<boost::intmax_t, T>::type an;
   T fa;
#ifndef BOOST_NO_EXCEPTIONS
   try
   {
#endif
      eval_convert_to(&an, a);
      if(a.compare(an) == 0)
      {
         detail::pow_imp(result, x, an, mpl::true_());
         return;
      }
#ifndef BOOST_NO_EXCEPTIONS
   }
   catch(const std::exception&)
   {
      // conversion failed, just fall through, value is not an integer.
      an = (std::numeric_limits<boost::intmax_t>::max)();
   }
#endif
   if((eval_get_sign(x) < 0))
   {
      typename boost::multiprecision::detail::canonical<boost::uintmax_t, T>::type aun;
#ifndef BOOST_NO_EXCEPTIONS
      try
      {
#endif
         eval_convert_to(&aun, a);
         if(a.compare(aun) == 0)
         {
            fa = x;
            fa.negate();
            eval_pow(result, fa, a);
            if(aun & 1u)
               result.negate();
            return;
         }
#ifndef BOOST_NO_EXCEPTIONS
      }
      catch(const std::exception&)
      {
         // conversion failed, just fall through, value is not an integer.
      }
#endif
      if(std::numeric_limits<number<T, et_on> >::has_quiet_NaN)
         result = std::numeric_limits<number<T, et_on> >::quiet_NaN().backend();
      else
      {
         BOOST_THROW_EXCEPTION(std::domain_error("Result of pow is undefined or non-real and there is no NaN for this number type."));
      }
      return;
   }

   T t, da;

   eval_subtract(da, a, an);

   if((x.compare(fp_type(0.5)) >= 0) && (x.compare(fp_type(0.9)) < 0))
   {
      if(a.compare(fp_type(1e-5f)) <= 0)
      {
         // Series expansion for small a.
         eval_log(t, x);
         eval_multiply(t, a);
         hyp0F0(result, t);
         return;
      }
      else
      {
         // Series expansion for moderately sized x. Note that for large power of a,
         // the power of the integer part of a is calculated using the pown function.
         if(an)
         {
            da.negate();
            t = si_type(1);
            eval_subtract(t, x);
            hyp1F0(result, da, t);
            detail::pow_imp(t, x, an, mpl::true_());
            eval_multiply(result, t);
         }
         else
         {
            da = a;
            da.negate();
            t = si_type(1);
            eval_subtract(t, x);
            hyp1F0(result, da, t);
         }
      }
   }
   else
   {
      // Series expansion for pow(x, a). Note that for large power of a, the power
      // of the integer part of a is calculated using the pown function.
      if(an)
      {
         eval_log(t, x);
         eval_multiply(t, da);
         eval_exp(result, t);
         detail::pow_imp(t, x, an, mpl::true_());
         eval_multiply(result, t);
      }
      else
      {
         eval_log(t, x);
         eval_multiply(t, a);
         eval_exp(result, t);
      }
   }
}