code:
////////////////////////////////////////////////////////////////////
//
// Copyright Vicente J. Botet Escriba 2010
//
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt or copy at
// http://www.boost.org/LICENSE_1_0.txt)
//
// See http://www.boost.org/libs/mpl for documentation.
//
////////////////////////////////////////////////////////////////////
#ifndef BOOST_MPL_GCD_HPP_INCLUDED
#define BOOST_MPL_GCD_HPP_INCLUDED

#include <boost/mpl/integral_c.hpp>
#include <boost/ratio/detail/mpl/abs.hpp>
#include <boost/mpl/aux_/largest_int.hpp>
#include <boost/mpl/aux_/na_spec.hpp>
#include <boost/mpl/aux_/lambda_support.hpp>
#include <boost/mpl/aux_/config/integral.hpp>
#include <boost/mpl/aux_/config/static_constant.hpp>
#include <boost/mpl/aux_/config/dependent_nttp.hpp>
#include <boost/cstdint.hpp>

#if    !defined(BOOST_MPL_CFG_NO_NESTED_VALUE_ARITHMETIC_2) \
    && !defined(BOOST_MPL_PREPROCESSING_MODE) \
    && !defined(__CUDACC__) \
    && ( defined(BOOST_MSVC) \
        || BOOST_WORKAROUND(__EDG_VERSION__, <= 238) \
        )

#   define BOOST_MPL_CFG_NO_NESTED_VALUE_ARITHMETIC_2

#endif

namespace boost { namespace mpl {

template< typename Tag1, typename Tag2 > struct gcd_impl;

template< typename T > struct gcd_tag
{
    typedef typename T::tag type;
};

template<
      typename BOOST_MPL_AUX_NA_PARAM(N1)
    , typename BOOST_MPL_AUX_NA_PARAM(N2)
    >
struct gcd
    : gcd_impl<
          typename gcd_tag<N1>::type
        , typename gcd_tag<N2>::type
        >::template apply<N1, N2>::type
{
    BOOST_MPL_AUX_LAMBDA_SUPPORT(2, gcd, (N1, N2))
};

BOOST_MPL_AUX_NA_SPEC(2, gcd)

template<
      typename T
    , T n1
    , T n2
    >
struct gcd_c
    : gcd<integral_c<T,n1>,integral_c<T,n2> >
{
};

namespace aux {

  // Workaround for error: the type of partial specialization template parameter constant "n2"
    // depends on another template parameter
    // Note: this solution could be wrong for n1 or n2 = [2**63 .. 2**64-1]
#if defined(BOOST_MPL_CFG_NO_DEPENDENT_NONTYPE_PARAMETER_IN_PARTIAL_SPEC)

  template< typename T1, boost::intmax_t n1, bool n1_is_0
                  , typename T2, boost::intmax_t n2, bool n2_is_0 >
      struct gcd_aux
          : gcd_aux<T2, n2, n2==0, T1, (n1 % n2), (n1 % n2)==0>
      {};

      template <typename T1, boost::intmax_t n1, typename T2, boost::intmax_t n2>
      struct gcd_aux<T1, n1, false, T2, n2, true> : integral_c<T1, n1>
      {};

      template <typename T1, boost::intmax_t n1, typename T2, boost::intmax_t n2, bool C>
      struct gcd_aux<T1, n1, true, T2, n2, C> : integral_c<T2, n2>
      {};

#else // defined(BOOST_MPL_CFG_NO_DEPENDENT_NONTYPE_PARAMETER_IN_PARTIAL_SPEC)

    template< typename T1, T1 n1, bool n1_is_0, typename T2, T2 n2, bool n2_is_0 >
    struct gcd_aux

        : gcd_aux<T2, n2, n2==0,
                    typename aux::largest_int<T1, T2>::type,
                    //~ T1,
                    (n1 % n2), (n1 % n2)==0>
    {};

    template <typename T1, T1 n1, typename T2, T2 n2>
    struct gcd_aux<T1, n1, false, T2, n2, true> : integral_c<T1, n1>
    {};

    template <typename T1, T1 n1, typename T2, T2 n2, bool C>
    struct gcd_aux<T1, n1, true, T2, n2, C> : integral_c<T2, n2>
    {};
#endif // defined(BOOST_MPL_CFG_NO_DEPENDENT_NONTYPE_PARAMETER_IN_PARTIAL_SPEC)
}

template<>
struct gcd_impl<integral_c_tag, integral_c_tag>
{
    template< typename N1, typename N2 > struct apply
        : abs<aux::gcd_aux< typename N1::value_type, N1::value, N1::value==0,
                        typename N2::value_type, N2::value, N2::value==0  > >
    {
    };
};

}}

#endif // BOOST_MPL_GCD_HPP_INCLUDED


code:
////////////////////////////////////////////////////////////////////
//
// Copyright Vicente J. Botet Escriba 2010
//
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt or copy at
// http://www.boost.org/LICENSE_1_0.txt)
//
// See http://www.boost.org/libs/mpl for documentation.
//
////////////////////////////////////////////////////////////////////
#ifndef BOOST_MPL_LCM_HPP_INCLUDED
#define BOOST_MPL_LCM_HPP_INCLUDED

#include <boost/mpl/integral_c.hpp>
#include <boost/ratio/detail/mpl/abs.hpp>
#include <boost/mpl/aux_/largest_int.hpp>
#include <boost/mpl/aux_/na_spec.hpp>
#include <boost/mpl/aux_/lambda_support.hpp>
#include <boost/mpl/aux_/config/integral.hpp>
#include <boost/mpl/aux_/config/static_constant.hpp>
#include <boost/mpl/aux_/config/dependent_nttp.hpp>
#include <boost/cstdint.hpp>

#if    !defined(BOOST_MPL_CFG_NO_NESTED_VALUE_ARITHMETIC_2) \
    && !defined(BOOST_MPL_PREPROCESSING_MODE) \
    && !defined(__CUDACC__) \
    && ( defined(BOOST_MSVC) \
        || BOOST_WORKAROUND(__EDG_VERSION__, <= 238) \
        )

#   define BOOST_MPL_CFG_NO_NESTED_VALUE_ARITHMETIC_2

#endif

namespace boost { namespace mpl {

template< typename Tag1, typename Tag2 > struct lcm_impl;

template< typename T > struct lcm_tag
{
    typedef typename T::tag type;
};

template<
      typename BOOST_MPL_AUX_NA_PARAM(N1)
    , typename BOOST_MPL_AUX_NA_PARAM(N2)
    >
struct lcm
    : lcm_impl<
          typename lcm_tag<N1>::type
        , typename lcm_tag<N2>::type
        >::template apply<N1, N2>::type
{
    BOOST_MPL_AUX_LAMBDA_SUPPORT(2, lcm, (N1, N2))
};

BOOST_MPL_AUX_NA_SPEC(2, lcm)

template<
      typename T
    , T n1
    , T n2
    >
struct lcm_c
    : lcm<integral_c<T,n1>,integral_c<T,n2> >
{
};

namespace aux {
  // Workaround for error: the type of partial specialization template parameter constant "n2"
  // depends on another template parameter
  // Note: this solution could be wrong for n1 or n2 = [2**63 .. 2**64-1]
  #if defined(BOOST_MPL_CFG_NO_DEPENDENT_NONTYPE_PARAMETER_IN_PARTIAL_SPEC)
  template< typename T1, boost::intmax_t n1, bool n1_is_0
                          , typename T2, boost::intmax_t n2, bool n2_is_0 >
      struct lcm_aux
          : abs<integral_c< typename aux::largest_int<T1, T2>::type,
              ( n1 / gcd<integral_c<T1,n1>, integral_c<T2,n2> >::value * n2 )
          > >
      {};

      template <typename T1, boost::intmax_t n1, typename T2, boost::intmax_t n2>
      struct lcm_aux<T1, n1, false, T2, n2, true> : integral_c<T2, 0>
      {};

      template <typename T1, boost::intmax_t n1, typename T2, boost::intmax_t n2, bool C>
      struct lcm_aux<T1, n1, true, T2, n2, C> : integral_c<T1, 0>
      {};

#else // defined(BOOST_MPL_CFG_NO_DEPENDENT_NONTYPE_PARAMETER_IN_PARTIAL_SPEC)

    template< typename T1, T1 n1, bool n1_is_0, typename T2, T2 n2, bool n2_is_0 >
    struct lcm_aux

        : abs<integral_c< typename aux::largest_int<T1, T2>::type,
            ( n1 / gcd<integral_c<T1,n1>, integral_c<T2,n2> >::value * n2 )
        > >
    {};

    template <typename T1, T1 n1, typename T2, T2 n2>
    struct lcm_aux<T1, n1, false, T2, n2, true> : integral_c<T2, 0>
    {};

    template <typename T1, T1 n1, typename T2, T2 n2, bool C>
    struct lcm_aux<T1, n1, true, T2, n2, C> : integral_c<T1, 0>
    {};
#endif // defined(BOOST_MPL_CFG_NO_DEPENDENT_NONTYPE_PARAMETER_IN_PARTIAL_SPEC)
}

template<>
struct lcm_impl<integral_c_tag, integral_c_tag>
{
    template< typename N1, typename N2 > struct apply
        : abs<aux::lcm_aux< typename N1::value_type, N1::value, N1::value==0,
                        typename N2::value_type, N2::value, N2::value==0  > >
    {
    };
};

}}

#endif // BOOST_MPL_LCM_HPP_INCLUDED

license: mit


code:
////////////////////////////////////////////////////////////////////
//
// Copyright Vicente J. Botet Escriba 2010
//
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE_1_0.txt or copy at
// http://www.boost.org/LICENSE_1_0.txt)
//
// See http://www.boost.org/libs/mpl for documentation.
//
////////////////////////////////////////////////////////////////////
#ifndef BOOST_MPL_SIGN_HPP_INCLUDED
#define BOOST_MPL_SIGN_HPP_INCLUDED

#include <boost/mpl/integral_c.hpp>
#include <boost/mpl/aux_/na_spec.hpp>
#include <boost/mpl/aux_/lambda_support.hpp>
#include <boost/mpl/aux_/config/integral.hpp>
#include <boost/mpl/aux_/config/static_constant.hpp>

#if    !defined(BOOST_MPL_CFG_NO_NESTED_VALUE_ARITHMETIC_2) \
    && !defined(BOOST_MPL_PREPROCESSING_MODE) \
    && !defined(__CUDACC__) \
    && ( defined(BOOST_MSVC) \
        || BOOST_WORKAROUND(__EDG_VERSION__, <= 238) \
        )

#   define BOOST_MPL_CFG_NO_NESTED_VALUE_ARITHMETIC_2

#endif

namespace boost { namespace mpl {

template< typename Tag > struct sign_impl;

template< typename T > struct sign_tag
{
    typedef typename T::tag type;
};

template<
      typename BOOST_MPL_AUX_NA_PARAM(N)
    >
struct sign
    : sign_impl<
          typename sign_tag<N>::type
        >::template apply<N>::type
{
    BOOST_MPL_AUX_LAMBDA_SUPPORT(1, sign, (N))
};

BOOST_MPL_AUX_NA_SPEC(1, sign)

template<
      typename T
    , T n1
    >
struct sign_c
    : sign<integral_c<T,n1> >
{
};

#if defined(BOOST_MPL_CFG_NO_NESTED_VALUE_ARITHMETIC_2)
namespace aux {
template< typename T, T n > struct sign_wknd
{
    BOOST_STATIC_CONSTANT(T, value = (n == 0 ? 0 : (n < 0 ? -1 : 1)));
    typedef integral_c<T,value> type;
};
}
#endif

template<>
struct sign_impl<integral_c_tag>
{
#if defined(BOOST_MPL_CFG_NO_NESTED_VALUE_ARITHMETIC_2)
    template< typename N > struct apply
        : aux::sign_wknd< typename N::value_type, N::value >
#else
    template< typename N > struct apply
        : integral_c< typename N::value_type, (N::value == 0 ? 0 : (N::value < 0 ? -1 : 1)) >
#endif
    {
    };
};

}}

#endif // BOOST_MPL_SIGN_HPP_INCLUDED


code:
//  ratio.hpp  ---------------------------------------------------------------//

//  Copyright 2008 Howard Hinnant
//  Copyright 2008 Beman Dawes
//  Copyright 2009 Vicente J. Botet Escriba

//  Distributed under the Boost Software License, Version 1.0.
//  See http://www.boost.org/LICENSE_1_0.txt

/*

This code was derived by Beman Dawes from Howard Hinnant's time2_demo prototype.
Many thanks to Howard for making his code available under the Boost license.
The original code was modified to conform to Boost conventions and to section
20.4 Compile-time rational arithmetic [ratio], of the C++ committee working
paper N2798.
See http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2008/n2798.pdf.

time2_demo contained this comment:

    Much thanks to Andrei Alexandrescu,
                   Walter Brown,
                   Peter Dimov,
                   Jeff Garland,
                   Terry Golubiewski,
                   Daniel Krugler,
                   Anthony Williams.
*/

// The way overflow is managed for ratio_less is taken from llvm/libcxx/include/ratio

#ifndef BOOST_RATIO_DETAIL_RATIO_OPERATIONS_HPP
#define BOOST_RATIO_DETAIL_RATIO_OPERATIONS_HPP

#include <boost/ratio/config.hpp>
#include <boost/ratio/detail/mpl/abs.hpp>
#include <boost/ratio/detail/mpl/sign.hpp>
#include <cstdlib>
#include <climits>
#include <limits>
#include <boost/cstdint.hpp>
#include <boost/type_traits/integral_constant.hpp>
#include <boost/core/enable_if.hpp>
#include <boost/integer_traits.hpp>

//
// We simply cannot include this header on gcc without getting copious warnings of the kind:
//
// boost/integer.hpp:77:30: warning: use of C99 long long integer constant
//
// And yet there is no other reasonable implementation, so we declare this a system header
// to suppress these warnings.
//
#if defined(__GNUC__) && (__GNUC__ >= 4)
#pragma GCC system_header
#endif

namespace boost
{

//----------------------------------------------------------------------------//
//                                 helpers                                    //
//----------------------------------------------------------------------------//

namespace ratio_detail
{

  template <boost::intmax_t X, boost::intmax_t Y, boost::intmax_t = mpl::sign_c<boost::intmax_t, Y>::value>
  class br_add;

  template <boost::intmax_t X, boost::intmax_t Y>
  class br_add<X, Y, 1>
  {
      static const boost::intmax_t min = boost::integer_traits<boost::intmax_t>::const_min;
      static const boost::intmax_t max = boost::integer_traits<boost::intmax_t>::const_max;

      BOOST_RATIO_STATIC_ASSERT(X <= max - Y , BOOST_RATIO_OVERFLOW_IN_ADD, ());
  public:
      static const boost::intmax_t value = X + Y;
  };

  template <boost::intmax_t X, boost::intmax_t Y>
  class br_add<X, Y, 0>
  {
  public:
      static const boost::intmax_t value = X;
  };

  template <boost::intmax_t X, boost::intmax_t Y>
  class br_add<X, Y, -1>
  {
      static const boost::intmax_t min = boost::integer_traits<boost::intmax_t>::const_min;
      static const boost::intmax_t max = boost::integer_traits<boost::intmax_t>::const_max;

      BOOST_RATIO_STATIC_ASSERT(min - Y <= X, BOOST_RATIO_OVERFLOW_IN_ADD, ());
  public:
      static const boost::intmax_t value = X + Y;
  };

  template <boost::intmax_t X, boost::intmax_t Y, boost::intmax_t = mpl::sign_c<boost::intmax_t, Y>::value>
  class br_sub;

  template <boost::intmax_t X, boost::intmax_t Y>
  class br_sub<X, Y, 1>
  {
      static const boost::intmax_t min = boost::integer_traits<boost::intmax_t>::const_min;
      static const boost::intmax_t max = boost::integer_traits<boost::intmax_t>::const_max;

      BOOST_RATIO_STATIC_ASSERT(min + Y <= X, BOOST_RATIO_OVERFLOW_IN_SUB, ());
  public:
      static const boost::intmax_t value = X - Y;
  };

  template <boost::intmax_t X, boost::intmax_t Y>
  class br_sub<X, Y, 0>
  {
  public:
      static const boost::intmax_t value = X;
  };

  template <boost::intmax_t X, boost::intmax_t Y>
  class br_sub<X, Y, -1>
  {
      static const boost::intmax_t min = boost::integer_traits<boost::intmax_t>::const_min;
      static const boost::intmax_t max = boost::integer_traits<boost::intmax_t>::const_max;

      BOOST_RATIO_STATIC_ASSERT(X <= max + Y, BOOST_RATIO_OVERFLOW_IN_SUB, ());
  public:
      static const boost::intmax_t value = X - Y;
  };

  template <boost::intmax_t X, boost::intmax_t Y>
  class br_mul
  {
      static const boost::intmax_t nan =
          boost::intmax_t(BOOST_RATIO_UINTMAX_C(1) << (sizeof(boost::intmax_t) * CHAR_BIT - 1));
      static const boost::intmax_t min = boost::integer_traits<boost::intmax_t>::const_min;
      static const boost::intmax_t max = boost::integer_traits<boost::intmax_t>::const_max;

      static const boost::intmax_t a_x = mpl::abs_c<boost::intmax_t, X>::value;
      static const boost::intmax_t a_y = mpl::abs_c<boost::intmax_t, Y>::value;

      BOOST_RATIO_STATIC_ASSERT(X != nan, BOOST_RATIO_OVERFLOW_IN_MUL, ());
      BOOST_RATIO_STATIC_ASSERT(Y != nan, BOOST_RATIO_OVERFLOW_IN_MUL, ());
      BOOST_RATIO_STATIC_ASSERT(a_x <= max / a_y, BOOST_RATIO_OVERFLOW_IN_MUL, ());
  public:
      static const boost::intmax_t value = X * Y;
  };

  template <boost::intmax_t Y>
  class br_mul<0, Y>
  {
  public:
      static const boost::intmax_t value = 0;
  };

  template <boost::intmax_t X>
  class br_mul<X, 0>
  {
  public:
      static const boost::intmax_t value = 0;
  };

  template <>
  class br_mul<0, 0>
  {
  public:
      static const boost::intmax_t value = 0;
  };

  // Not actually used but left here in case needed in future maintenance
  template <boost::intmax_t X, boost::intmax_t Y>
  class br_div
  {
      static const boost::intmax_t nan = boost::intmax_t(BOOST_RATIO_UINTMAX_C(1) << (sizeof(boost::intmax_t) * CHAR_BIT - 1));
      static const boost::intmax_t min = boost::integer_traits<boost::intmax_t>::const_min;
      static const boost::intmax_t max = boost::integer_traits<boost::intmax_t>::const_max;

      BOOST_RATIO_STATIC_ASSERT(X != nan, BOOST_RATIO_OVERFLOW_IN_DIV, ());
      BOOST_RATIO_STATIC_ASSERT(Y != nan, BOOST_RATIO_OVERFLOW_IN_DIV, ());
      BOOST_RATIO_STATIC_ASSERT(Y != 0, BOOST_RATIO_DIVIDE_BY_0, ());
  public:
      static const boost::intmax_t value = X / Y;
  };

  // ratio arithmetic
  template <class R1, class R2> struct ratio_add;
  template <class R1, class R2> struct ratio_subtract;
  template <class R1, class R2> struct ratio_multiply;
  template <class R1, class R2> struct ratio_divide;

  template <class R1, class R2>
  struct ratio_add
  {
      //The nested typedef type shall be a synonym for ratio<T1, T2>::type where T1 has the value R1::num *
      //R2::den + R2::num * R1::den and T2 has the value R1::den * R2::den.
      // As the preceding doesn't works because of overflow on boost::intmax_t we need something more elaborated.
  private:
      static const boost::intmax_t gcd_n1_n2 = mpl::gcd_c<boost::intmax_t, R1::num, R2::num>::value;
      static const boost::intmax_t gcd_d1_d2 = mpl::gcd_c<boost::intmax_t, R1::den, R2::den>::value;
  public:
      // No need to normalize as ratio_multiply is already normalized
      typedef typename ratio_multiply
         <
             ratio<gcd_n1_n2, R1::den / gcd_d1_d2>,
             ratio
             <
                 boost::ratio_detail::br_add
                 <
                     boost::ratio_detail::br_mul<R1::num / gcd_n1_n2, R2::den / gcd_d1_d2>::value,
                     boost::ratio_detail::br_mul<R2::num / gcd_n1_n2, R1::den / gcd_d1_d2>::value
                 >::value,
                 R2::den
             >
         >::type type;
  };
  template <class R, boost::intmax_t D>
  struct ratio_add<R, ratio<0,D> >
  {
    typedef R type;
  };

  template <class R1, class R2>
  struct ratio_subtract
  {
      //The nested typedef type shall be a synonym for ratio<T1, T2>::type where T1 has the value
      // R1::num *R2::den - R2::num * R1::den and T2 has the value R1::den * R2::den.
      // As the preceding doesn't works because of overflow on boost::intmax_t we need something more elaborated.
  private:
      static const boost::intmax_t gcd_n1_n2 = mpl::gcd_c<boost::intmax_t, R1::num, R2::num>::value;
      static const boost::intmax_t gcd_d1_d2 = mpl::gcd_c<boost::intmax_t, R1::den, R2::den>::value;
  public:
      // No need to normalize as ratio_multiply is already normalized
      typedef typename ratio_multiply
         <
             ratio<gcd_n1_n2, R1::den / gcd_d1_d2>,
             ratio
             <
                 boost::ratio_detail::br_sub
                 <
                     boost::ratio_detail::br_mul<R1::num / gcd_n1_n2, R2::den / gcd_d1_d2>::value,
                     boost::ratio_detail::br_mul<R2::num / gcd_n1_n2, R1::den / gcd_d1_d2>::value
                 >::value,
                 R2::den
             >
         >::type type;
  };

  template <class R, boost::intmax_t D>
  struct ratio_subtract<R, ratio<0,D> >
  {
    typedef R type;
  };

  template <class R1, class R2>
  struct ratio_multiply
  {
      // The nested typedef type  shall be a synonym for ratio<R1::num * R2::den - R2::num * R1::den, R1::den * R2::den>::type.
      // As the preceding doesn't works because of overflow on boost::intmax_t we need something more elaborated.
  private:
     static const boost::intmax_t gcd_n1_d2 = mpl::gcd_c<boost::intmax_t, R1::num, R2::den>::value;
     static const boost::intmax_t gcd_d1_n2 = mpl::gcd_c<boost::intmax_t, R1::den, R2::num>::value;
  public:
      typedef typename ratio
         <
             boost::ratio_detail::br_mul<R1::num / gcd_n1_d2, R2::num / gcd_d1_n2>::value,
             boost::ratio_detail::br_mul<R2::den / gcd_n1_d2, R1::den / gcd_d1_n2>::value
         >::type type;
  };

  template <class R1, class R2>
  struct ratio_divide
  {
      // The nested typedef type  shall be a synonym for ratio<R1::num * R2::den, R2::num * R1::den>::type.
      // As the preceding doesn't works because of overflow on boost::intmax_t we need something more elaborated.
  private:
      static const boost::intmax_t gcd_n1_n2 = mpl::gcd_c<boost::intmax_t, R1::num, R2::num>::value;
      static const boost::intmax_t gcd_d1_d2 = mpl::gcd_c<boost::intmax_t, R1::den, R2::den>::value;
  public:
      typedef typename ratio
         <
             boost::ratio_detail::br_mul<R1::num / gcd_n1_n2, R2::den / gcd_d1_d2>::value,
             boost::ratio_detail::br_mul<R2::num / gcd_n1_n2, R1::den / gcd_d1_d2>::value
         >::type type;
  };
  template <class R1, class R2>
  struct is_evenly_divisible_by
  {
  private:
      static const boost::intmax_t gcd_n1_n2 = mpl::gcd_c<boost::intmax_t, R1::num, R2::num>::value;
      static const boost::intmax_t gcd_d1_d2 = mpl::gcd_c<boost::intmax_t, R1::den, R2::den>::value;
  public:
      typedef integral_constant<bool,
             ((R2::num / gcd_n1_n2 ==1) && (R1::den / gcd_d1_d2)==1)
      > type;
  };

  template <class T>
  struct is_ratio : public boost::false_type
  {};
  template <boost::intmax_t N, boost::intmax_t D>
  struct is_ratio<ratio<N, D> > : public boost::true_type
  {};

  template <class R1, class R2,
            boost::intmax_t Q1 = R1::num / R1::den, boost::intmax_t M1 = R1::num % R1::den,
            boost::intmax_t Q2 = R2::num / R2::den, boost::intmax_t M2 = R2::num % R2::den>
  struct ratio_less1
  {
    static const bool value = Q1 < Q2;
  };

  template <class R1, class R2, boost::intmax_t Q>
  struct ratio_less1<R1, R2, Q, 0, Q, 0>
  {
    static const bool value = false;
  };

  template <class R1, class R2, boost::intmax_t Q, boost::intmax_t M2>
  struct ratio_less1<R1, R2, Q, 0, Q, M2>
  {
    static const bool value = true;
  };

  template <class R1, class R2, boost::intmax_t Q, boost::intmax_t M1>
  struct ratio_less1<R1, R2, Q, M1, Q, 0>
  {
    static const bool value = false;
  };

  template <class R1, class R2, boost::intmax_t Q, boost::intmax_t M1, boost::intmax_t M2>
  struct ratio_less1<R1, R2, Q, M1, Q, M2>
  {
    static const bool value = ratio_less1<ratio<R2::den, M2>, ratio<R1::den, M1>
                                            >::value;
  };

  template <
      class R1,
      class R2,
      boost::intmax_t S1 = mpl::sign_c<boost::intmax_t, R1::num>::value,
    boost::intmax_t S2 = mpl::sign_c<boost::intmax_t, R2::num>::value
>
  struct ratio_less
  {
      static const bool value = S1 < S2;
  };

  template <class R1, class R2>
  struct ratio_less<R1, R2, 1LL, 1LL>
  {
      static const bool value = ratio_less1<R1, R2>::value;
  };

  template <class R1, class R2>
  struct ratio_less<R1, R2, -1LL, -1LL>
  {
      static const bool value = ratio_less1<ratio<-R2::num, R2::den>,
                                            ratio<-R1::num, R1::den> >::value;
  };

}  // namespace ratio_detail

}  // namespace boost

#endif  // BOOST_RATIO_HPP


code:
//  ratio_io
//
//  (C) Copyright Howard Hinnant
//  (C) Copyright 2010 Vicente J. Botet Escriba
//  Use, modification and distribution are subject to the Boost Software License,
//  Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
//  http://www.boost.org/LICENSE_1_0.txt).
//
// This code was adapted by Vicente from Howard Hinnant's experimental work
// on chrono i/o under lvm/libc++ to Boost

#ifndef BOOST_RATIO_DETAIL_RATIO_IO_HPP
#define BOOST_RATIO_DETAIL_RATIO_IO_HPP

/*

    ratio_io synopsis

#include <ratio>
#include <string>

namespace boost
{

template <class Ratio, class CharT>
struct ratio_string
{
    static basic_string<CharT> short_name();
    static basic_string<CharT> long_name();
};

}  // boost

*/

#include <boost/config.hpp>
#include <boost/ratio/ratio.hpp>
#include <boost/type_traits/integral_constant.hpp>
#include <string>
#include <sstream>

#if defined(BOOST_NO_CXX11_UNICODE_LITERALS) || defined(BOOST_NO_CXX11_CHAR16_T) || defined(BOOST_NO_CXX11_CHAR32_T) || defined(BOOST_NO_CXX11_U16STRING) || defined(BOOST_NO_CXX11_U32STRING)
#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT
#undef BOOST_RATIO_HAS_UNICODE_SUPPORT
#endif
#else
#define BOOST_RATIO_HAS_UNICODE_SUPPORT 1
#endif

namespace boost {

//template <class Ratio>
//struct ratio_string_is_localizable : false_type {};
//template <class Ratio>
//struct ratio_string_id : integral_constant<int,0> {};

template <class Ratio, class CharT>
struct ratio_string
{
    static std::basic_string<CharT> short_name() {return long_name();}
    static std::basic_string<CharT> long_name();
    static std::basic_string<CharT> symbol() {return short_name();}
    static std::basic_string<CharT> prefix() {return long_name();}
};

template <class Ratio, class CharT>
std::basic_string<CharT>
ratio_string<Ratio, CharT>::long_name()
{
    std::basic_ostringstream<CharT> os;
    os << CharT('[') << Ratio::num << CharT('/')
                        << Ratio::den << CharT(']');
    return os.str();
}

#ifdef BOOST_RATIO_HAS_STATIC_STRING
namespace ratio_detail {
template <class Ratio, class CharT>
struct ratio_string_static
{
    static std::string short_name() {
        return std::basic_string<CharT>(
                static_string::c_str<
                        typename ratio_static_string<Ratio, CharT>::short_name
                    >::value);
    }
    static std::string long_name()  {
        return std::basic_string<CharT>(
                static_string::c_str<
                    typename ratio_static_string<Ratio, CharT>::long_name
                >::value);
    }
    static std::basic_string<CharT> symbol() {return short_name();}
    static std::basic_string<CharT> prefix() {return long_name();}
};
}
#endif
// atto
//template <>
//struct ratio_string_is_localizable<atto> : true_type {};
//
//template <>
//struct ratio_string_id<atto> : integral_constant<int,-18> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<atto, CharT> :
    ratio_detail::ratio_string_static<atto,CharT>
{};

#else
template <>
struct ratio_string<atto, char>
{
    static std::string short_name() {return std::string(1, 'a');}
    static std::string long_name()  {return std::string("atto");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<atto, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'a');}
    static std::u16string long_name()  {return std::u16string(u"atto");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<atto, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'a');}
    static std::u32string long_name()  {return std::u32string(U"atto");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<atto, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'a');}
    static std::wstring long_name()  {return std::wstring(L"atto");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// femto

//template <>
//struct ratio_string_is_localizable<femto> : true_type {};
//
//template <>
//struct ratio_string_id<femto> : integral_constant<int,-15> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<femto, CharT> :
    ratio_detail::ratio_string_static<femto,CharT>
{};

#else

template <>
struct ratio_string<femto, char>
{
    static std::string short_name() {return std::string(1, 'f');}
    static std::string long_name()  {return std::string("femto");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<femto, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'f');}
    static std::u16string long_name()  {return std::u16string(u"femto");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<femto, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'f');}
    static std::u32string long_name()  {return std::u32string(U"femto");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<femto, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'f');}
    static std::wstring long_name()  {return std::wstring(L"femto");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// pico

//template <>
//struct ratio_string_is_localizable<pico> : true_type {};
//
//template <>
//struct ratio_string_id<pico> : integral_constant<int,-12> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<pico, CharT> :
    ratio_detail::ratio_string_static<pico,CharT>
{};

#else
template <>
struct ratio_string<pico, char>
{
    static std::string short_name() {return std::string(1, 'p');}
    static std::string long_name()  {return std::string("pico");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<pico, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'p');}
    static std::u16string long_name()  {return std::u16string(u"pico");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<pico, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'p');}
    static std::u32string long_name()  {return std::u32string(U"pico");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<pico, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'p');}
    static std::wstring long_name()  {return std::wstring(L"pico");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// nano

//template <>
//struct ratio_string_is_localizable<nano> : true_type {};
//
//template <>
//struct ratio_string_id<nano> : integral_constant<int,-9> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<nano, CharT> :
    ratio_detail::ratio_string_static<nano,CharT>
{};

#else
template <>
struct ratio_string<nano, char>
{
    static std::string short_name() {return std::string(1, 'n');}
    static std::string long_name()  {return std::string("nano");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<nano, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'n');}
    static std::u16string long_name()  {return std::u16string(u"nano");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<nano, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'n');}
    static std::u32string long_name()  {return std::u32string(U"nano");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<nano, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'n');}
    static std::wstring long_name()  {return std::wstring(L"nano");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// micro

//template <>
//struct ratio_string_is_localizable<micro> : true_type {};
//
//template <>
//struct ratio_string_id<micro> : integral_constant<int,-6> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<micro, CharT> :
    ratio_detail::ratio_string_static<micro,CharT>
{};

#else
template <>
struct ratio_string<micro, char>
{
    static std::string short_name() {return std::string("\xC2\xB5");}
    static std::string long_name()  {return std::string("micro");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<micro, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'\xB5');}
    static std::u16string long_name()  {return std::u16string(u"micro");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<micro, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'\xB5');}
    static std::u32string long_name()  {return std::u32string(U"micro");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<micro, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'\xB5');}
    static std::wstring long_name()  {return std::wstring(L"micro");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// milli

//template <>
//struct ratio_string_is_localizable<milli> : true_type {};
//
//template <>
//struct ratio_string_id<milli> : integral_constant<int,-3> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<milli, CharT> :
    ratio_detail::ratio_string_static<milli,CharT>
{};

#else
template <>
struct ratio_string<milli, char>
{
    static std::string short_name() {return std::string(1, 'm');}
    static std::string long_name()  {return std::string("milli");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<milli, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'm');}
    static std::u16string long_name()  {return std::u16string(u"milli");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<milli, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'm');}
    static std::u32string long_name()  {return std::u32string(U"milli");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<milli, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'm');}
    static std::wstring long_name()  {return std::wstring(L"milli");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// centi

//template <>
//struct ratio_string_is_localizable<centi> : true_type {};
//
//template <>
//struct ratio_string_id<centi> : integral_constant<int,-2> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<centi, CharT> :
    ratio_detail::ratio_string_static<centi,CharT>
{};

#else
template <>
struct ratio_string<centi, char>
{
    static std::string short_name() {return std::string(1, 'c');}
    static std::string long_name()  {return std::string("centi");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<centi, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'c');}
    static std::u16string long_name()  {return std::u16string(u"centi");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<centi, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'c');}
    static std::u32string long_name()  {return std::u32string(U"centi");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<centi, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'c');}
    static std::wstring long_name()  {return std::wstring(L"centi");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// deci

//template <>
//struct ratio_string_is_localizable<deci> : true_type {};
//
//template <>
//struct ratio_string_id<deci> : integral_constant<int,-1> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<deci, CharT> :
    ratio_detail::ratio_string_static<deci,CharT>
{};

#else

template <>
struct ratio_string<deci, char>
{
    static std::string short_name() {return std::string(1, 'd');}
    static std::string long_name()  {return std::string("deci");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<deci, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'd');}
    static std::u16string long_name()  {return std::u16string(u"deci");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<deci, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'd');}
    static std::u32string long_name()  {return std::u32string(U"deci");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<deci, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'd');}
    static std::wstring long_name()  {return std::wstring(L"deci");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// unit

//template <>
//struct ratio_string_is_localizable<ratio<1> > : true_type {};
//
//template <>
//struct ratio_string_id<ratio<1> > : integral_constant<int,0> {};
// deca

//template <>
//struct ratio_string_is_localizable<deca> : true_type {};
//
//template <>
//struct ratio_string_id<deca> : integral_constant<int,1> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<deca, CharT> :
    ratio_detail::ratio_string_static<deca,CharT>
{};

#else
template <>
struct ratio_string<deca, char>
{
    static std::string short_name() {return std::string("da");}
    static std::string long_name()  {return std::string("deca");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<deca, char16_t>
{
    static std::u16string short_name() {return std::u16string(u"da");}
    static std::u16string long_name()  {return std::u16string(u"deca");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<deca, char32_t>
{
    static std::u32string short_name() {return std::u32string(U"da");}
    static std::u32string long_name()  {return std::u32string(U"deca");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<deca, wchar_t>
{
    static std::wstring short_name() {return std::wstring(L"da");}
    static std::wstring long_name()  {return std::wstring(L"deca");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// hecto

//template <>
//struct ratio_string_is_localizable<hecto> : true_type {};
//
//template <>
//struct ratio_string_id<hecto> : integral_constant<int,2> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<hecto, CharT> :
    ratio_detail::ratio_string_static<hecto,CharT>
{};

#else
template <>
struct ratio_string<hecto, char>
{
    static std::string short_name() {return std::string(1, 'h');}
    static std::string long_name()  {return std::string("hecto");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<hecto, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'h');}
    static std::u16string long_name()  {return std::u16string(u"hecto");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<hecto, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'h');}
    static std::u32string long_name()  {return std::u32string(U"hecto");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<hecto, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'h');}
    static std::wstring long_name()  {return std::wstring(L"hecto");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// kilo

//template <>
//struct ratio_string_is_localizable<kilo> : true_type {};
//
//template <>
//struct ratio_string_id<kilo> : integral_constant<int,3> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<kilo, CharT> :
    ratio_detail::ratio_string_static<kilo,CharT>
{};

#else
template <>
struct ratio_string<kilo, char>
{
    static std::string short_name() {return std::string(1, 'k');}
    static std::string long_name()  {return std::string("kilo");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<kilo, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'k');}
    static std::u16string long_name()  {return std::u16string(u"kilo");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<kilo, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'k');}
    static std::u32string long_name()  {return std::u32string(U"kilo");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<kilo, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'k');}
    static std::wstring long_name()  {return std::wstring(L"kilo");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// mega

//template <>
//struct ratio_string_is_localizable<mega> : true_type {};
//
//template <>
//struct ratio_string_id<mega> : integral_constant<int,6> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<mega, CharT> :
    ratio_detail::ratio_string_static<mega,CharT>
{};

#else

template <>
struct ratio_string<mega, char>
{
    static std::string short_name() {return std::string(1, 'M');}
    static std::string long_name()  {return std::string("mega");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<mega, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'M');}
    static std::u16string long_name()  {return std::u16string(u"mega");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<mega, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'M');}
    static std::u32string long_name()  {return std::u32string(U"mega");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<mega, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'M');}
    static std::wstring long_name()  {return std::wstring(L"mega");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// giga

//template <>
//struct ratio_string_is_localizable<giga> : true_type {};
//
//template <>
//struct ratio_string_id<giga> : integral_constant<int,9> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<giga, CharT> :
    ratio_detail::ratio_string_static<giga,CharT>
{};

#else

template <>
struct ratio_string<giga, char>
{
    static std::string short_name() {return std::string(1, 'G');}
    static std::string long_name()  {return std::string("giga");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<giga, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'G');}
    static std::u16string long_name()  {return std::u16string(u"giga");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<giga, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'G');}
    static std::u32string long_name()  {return std::u32string(U"giga");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<giga, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'G');}
    static std::wstring long_name()  {return std::wstring(L"giga");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// tera

//template <>
//struct ratio_string_is_localizable<tera> : true_type {};
//
//template <>
//struct ratio_string_id<tera> : integral_constant<int,12> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<tera, CharT> :
    ratio_detail::ratio_string_static<tera,CharT>
{};

#else
template <>
struct ratio_string<tera, char>
{
    static std::string short_name() {return std::string(1, 'T');}
    static std::string long_name()  {return std::string("tera");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<tera, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'T');}
    static std::u16string long_name()  {return std::u16string(u"tera");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<tera, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'T');}
    static std::u32string long_name()  {return std::u32string(U"tera");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<tera, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'T');}
    static std::wstring long_name()  {return std::wstring(L"tera");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// peta

//template <>
//struct ratio_string_is_localizable<peta> : true_type {};
//
//template <>
//struct ratio_string_id<peta> : integral_constant<int,15> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<peta, CharT> :
    ratio_detail::ratio_string_static<peta,CharT>
{};

#else
template <>
struct ratio_string<peta, char>
{
    static std::string short_name() {return std::string(1, 'P');}
    static std::string long_name()  {return std::string("peta");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<peta, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'P');}
    static std::u16string long_name()  {return std::u16string(u"peta");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<peta, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'P');}
    static std::u32string long_name()  {return std::u32string(U"peta");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<peta, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'P');}
    static std::wstring long_name()  {return std::wstring(L"peta");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

// exa

//template <>
//struct ratio_string_is_localizable<exa> : true_type {};
//
//template <>
//struct ratio_string_id<exa> : integral_constant<int,18> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<exa, CharT> :
    ratio_detail::ratio_string_static<exa,CharT>
{};

#else
template <>
struct ratio_string<exa, char>
{
    static std::string short_name() {return std::string(1, 'E');}
    static std::string long_name()  {return std::string("exa");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<exa, char16_t>
{
    static std::u16string short_name() {return std::u16string(1, u'E');}
    static std::u16string long_name()  {return std::u16string(u"exa");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<exa, char32_t>
{
    static std::u32string short_name() {return std::u32string(1, U'E');}
    static std::u32string long_name()  {return std::u32string(U"exa");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<exa, wchar_t>
{
    static std::wstring short_name() {return std::wstring(1, L'E');}
    static std::wstring long_name()  {return std::wstring(L"exa");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

#ifdef BOOST_RATIO_EXTENSIONS

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<kibi, CharT> :
    ratio_detail::ratio_string_static<kibi,CharT>
{};

#else
template <>
struct ratio_string<kibi, char>
{
    static std::string short_name() {return std::string("Ki");}
    static std::string long_name()  {return std::string("kibi");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<kibi, char16_t>
{
    static std::u16string short_name() {return std::u16string(u"Ki");}
    static std::u16string long_name()  {return std::u16string(u"kibi");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<kibi, char32_t>
{
    static std::u32string short_name() {return std::u32string(U"Ki");}
    static std::u32string long_name()  {return std::u32string(U"kibi");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<kibi, wchar_t>
{
    static std::wstring short_name() {return std::wstring(L"Ki");}
    static std::wstring long_name()  {return std::wstring(L"kibi");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<mebi, CharT> :
    ratio_detail::ratio_string_static<mebi,CharT>
{};

#else
template <>
struct ratio_string<mebi, char>
{
    static std::string short_name() {return std::string("Mi");}
    static std::string long_name()  {return std::string("mebi");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<mebi, char16_t>
{
    static std::u16string short_name() {return std::u16string(u"Mi");}
    static std::u16string long_name()  {return std::u16string(u"mebi");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<mebi, char32_t>
{
    static std::u32string short_name() {return std::u32string(U"Mi");}
    static std::u32string long_name()  {return std::u32string(U"mebi");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<mebi, wchar_t>
{
    static std::wstring short_name() {return std::wstring(L"Mi");}
    static std::wstring long_name()  {return std::wstring(L"mebi");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<gibi, CharT> :
    ratio_detail::ratio_string_static<gibi,CharT>
{};

#else
template <>
struct ratio_string<gibi, char>
{
    static std::string short_name() {return std::string("Gi");}
    static std::string long_name()  {return std::string("gibi");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<gibi, char16_t>
{
    static std::u16string short_name() {return std::u16string(u"Gi");}
    static std::u16string long_name()  {return std::u16string(u"gibi");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<gibi, char32_t>
{
    static std::u32string short_name() {return std::u32string(U"Gi");}
    static std::u32string long_name()  {return std::u32string(U"gibi");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<gibi, wchar_t>
{
    static std::wstring short_name() {return std::wstring(L"Gi");}
    static std::wstring long_name()  {return std::wstring(L"gibi");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<tebi, CharT> :
    ratio_detail::ratio_string_static<tebi,CharT>
{};

#else
template <>
struct ratio_string<tebi, char>
{
    static std::string short_name() {return std::string("Ti");}
    static std::string long_name()  {return std::string("tebi");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<tebi, char16_t>
{
    static std::u16string short_name() {return std::u16string(u"Ti");}
    static std::u16string long_name()  {return std::u16string(u"tebi");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<tebi, char32_t>
{
    static std::u32string short_name() {return std::u32string(U"Ti");}
    static std::u32string long_name()  {return std::u32string(U"tebi");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<tebi, wchar_t>
{
    static std::wstring short_name() {return std::wstring(L"Ti");}
    static std::wstring long_name()  {return std::wstring(L"tebi");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<pebi, CharT> :
    ratio_detail::ratio_string_static<pebi,CharT>
{};

#else
template <>
struct ratio_string<pebi, char>
{
    static std::string short_name() {return std::string("Pi");}
    static std::string long_name()  {return std::string("pebi");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<pebi, char16_t>
{
    static std::u16string short_name() {return std::u16string(u"Pi");}
    static std::u16string long_name()  {return std::u16string(u"pebi");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<pebi, char32_t>
{
    static std::u32string short_name() {return std::u32string(U"Pi");}
    static std::u32string long_name()  {return std::u32string(U"pebi");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<pebi, wchar_t>
{
    static std::wstring short_name() {return std::wstring(L"Pi");}
    static std::wstring long_name()  {return std::wstring(L"pebi");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<exbi, CharT> :
    ratio_detail::ratio_string_static<exbi,CharT>
{};

#else
template <>
struct ratio_string<exbi, char>
{
    static std::string short_name() {return std::string("Ei");}
    static std::string long_name()  {return std::string("exbi");}
    static std::string symbol() {return short_name();}
    static std::string prefix() {return long_name();}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<exbi, char16_t>
{
    static std::u16string short_name() {return std::u16string(u"Ei");}
    static std::u16string long_name()  {return std::u16string(u"exbi");}
    static std::u16string symbol() {return short_name();}
    static std::u16string prefix() {return long_name();}
};

template <>
struct ratio_string<exbi, char32_t>
{
    static std::u32string short_name() {return std::u32string(U"Ei");}
    static std::u32string long_name()  {return std::u32string(U"exbi");}
    static std::u32string symbol() {return short_name();}
    static std::u32string prefix() {return long_name();}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<exbi, wchar_t>
{
    static std::wstring short_name() {return std::wstring(L"Ei");}
    static std::wstring long_name()  {return std::wstring(L"exbi");}
    static std::wstring symbol() {return short_name();}
    static std::wstring prefix() {return long_name();}
};
#endif
#endif
#endif
}

#endif  // BOOST_RATIO_RATIO_IO_HPP


code:
//  abs.hpp
//
//  (C) Copyright 2011 Vicente J. Botet Escriba
//  Use, modification and distribution are subject to the Boost Software License,
//  Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
//  http://www.boost.org/LICENSE_1_0.txt).
//

#ifndef BOOST_RATIO_MPL_RATIONAL_C_TAG_HPP
#define BOOST_RATIO_MPL_RATIONAL_C_TAG_HPP

#ifdef BOOST_RATIO_EXTENSIONS

#include <boost/mpl/int.hpp>

namespace boost {
namespace mpl {

struct rational_c_tag : int_<10> {};

}
}

#endif // BOOST_RATIO_EXTENSIONS
#endif  // BOOST_RATIO_MPL_RATIONAL_C_TAG_HPP


code:
//  ratio.hpp  ---------------------------------------------------------------//

//  Copyright 2008 Howard Hinnant
//  Copyright 2008 Beman Dawes
//  Copyright 2009 Vicente J. Botet Escriba

//  Distributed under the Boost Software License, Version 1.0.
//  See http://www.boost.org/LICENSE_1_0.txt

/*

This code was derived by Beman Dawes from Howard Hinnant's time2_demo prototype.
Many thanks to Howard for making his code available under the Boost license.
The original code was modified to conform to Boost conventions and to section
20.4 Compile-time rational arithmetic [ratio], of the C++ committee working
paper N2798.
See http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2008/n2798.pdf.

time2_demo contained this comment:

    Much thanks to Andrei Alexandrescu,
                   Walter Brown,
                   Peter Dimov,
                   Jeff Garland,
                   Terry Golubiewski,
                   Daniel Krugler,
                   Anthony Williams.
*/

// The way overflow is managed for ratio_less is taken from llvm/libcxx/include/ratio

#ifndef BOOST_RATIO_RATIO_HPP
#define BOOST_RATIO_RATIO_HPP

#include <boost/ratio/config.hpp>
#include <boost/ratio/detail/mpl/abs.hpp>
#include <boost/ratio/detail/mpl/sign.hpp>
#include <boost/ratio/detail/mpl/gcd.hpp>
#include <boost/ratio/detail/mpl/lcm.hpp>
#include <cstdlib>
#include <climits>
#include <limits>
#include <boost/cstdint.hpp>
#include <boost/type_traits/integral_constant.hpp>
#include <boost/core/enable_if.hpp>
#include <boost/integer_traits.hpp>
#include <boost/ratio/ratio_fwd.hpp>
#include <boost/ratio/detail/overflow_helpers.hpp>
#ifdef BOOST_RATIO_EXTENSIONS
#include <boost/rational.hpp>
#include <boost/ratio/mpl/rational_c_tag.hpp>
#endif

//
// We simply cannot include this header on gcc without getting copious warnings of the kind:
//
// boost/integer.hpp:77:30: warning: use of C99 long long integer constant
//
// And yet there is no other reasonable implementation, so we declare this a system header
// to suppress these warnings.
//
#if defined(__GNUC__) && (__GNUC__ >= 4)
#pragma GCC system_header
#endif

namespace boost
{

//----------------------------------------------------------------------------//
//                                                                            //
//                20.6.1 Class template ratio [ratio.ratio]                   //
//                                                                            //
//----------------------------------------------------------------------------//

template <boost::intmax_t N, boost::intmax_t D>
class ratio
{
    static const boost::intmax_t ABS_N = mpl::abs_c<boost::intmax_t, N>::value;
    static const boost::intmax_t ABS_D = mpl::abs_c<boost::intmax_t, D>::value;
    BOOST_RATIO_STATIC_ASSERT(ABS_N >= 0, BOOST_RATIO_NUMERATOR_IS_OUT_OF_RANGE, ());
    BOOST_RATIO_STATIC_ASSERT(ABS_D > 0, BOOST_RATIO_DENOMINATOR_IS_OUT_OF_RANGE, ());
    BOOST_RATIO_STATIC_ASSERT(D != 0, BOOST_RATIO_DIVIDE_BY_0 , ());
    static const boost::intmax_t SIGN_N = mpl::sign_c<boost::intmax_t,N>::value
      * mpl::sign_c<boost::intmax_t,D>::value;
    static const boost::intmax_t GCD = mpl::gcd_c<boost::intmax_t, ABS_N, ABS_D>::value;
public:
    BOOST_STATIC_CONSTEXPR boost::intmax_t num = SIGN_N * ABS_N / GCD;
    BOOST_STATIC_CONSTEXPR boost::intmax_t den = ABS_D / GCD;

#ifdef BOOST_RATIO_EXTENSIONS
    typedef mpl::rational_c_tag tag;
    typedef boost::rational<boost::intmax_t> value_type;
    typedef boost::intmax_t num_type;
    typedef boost::intmax_t den_type;
    ratio()
    {}
    template <boost::intmax_t _N2, boost::intmax_t _D2>
    ratio(const ratio<_N2, _D2>&,
        typename enable_if_c
            <
                (ratio<_N2, _D2>::num == num &&
                ratio<_N2, _D2>::den == den)
            >::type* = 0)
    {}

    template <boost::intmax_t _N2, boost::intmax_t _D2>
        typename enable_if_c
        <
            (ratio<_N2, _D2>::num == num &&
            ratio<_N2, _D2>::den == den),
            ratio&
        >::type
    operator=(const ratio<_N2, _D2>&) {return *this;}

    static value_type value() {return value_type(num,den);}
    value_type operator()() const {return value();}
#endif
    typedef ratio<num, den> type;
};

#if defined(BOOST_NO_CXX11_CONSTEXPR)
template <boost::intmax_t N, boost::intmax_t D>
const    boost::intmax_t ratio<N, D>::num;
template <boost::intmax_t N, boost::intmax_t D>
const    boost::intmax_t ratio<N, D>::den;
#endif

//----------------------------------------------------------------------------//
//                                                                            //
//                20.6.2 Arithmetic on ratio types [ratio.arithmetic]         //
//                                                                            //
//----------------------------------------------------------------------------//

template <class R1, class R2>
struct ratio_add
: boost::ratio_detail::ratio_add<R1, R2>::type
{
};

template <class R1, class R2>
struct ratio_subtract
: boost::ratio_detail::ratio_subtract<R1, R2>::type
{
};

template <class R1, class R2>
struct ratio_multiply
: boost::ratio_detail::ratio_multiply<R1, R2>::type
{
};

template <class R1, class R2>
struct ratio_divide
: boost::ratio_detail::ratio_divide<R1, R2>::type
{
};

//----------------------------------------------------------------------------//
//                                                                            //
//                20.6.3 Comparision of ratio types [ratio.comparison]        //
//                                                                            //
//----------------------------------------------------------------------------//

// ratio_equal

template <class R1, class R2>
struct ratio_equal
    : public boost::integral_constant<bool,
                               (R1::num == R2::num && R1::den == R2::den)>
{};

template <class R1, class R2>
struct ratio_not_equal
    : public boost::integral_constant<bool, !ratio_equal<R1, R2>::value>
{};

// ratio_less

template <class R1, class R2>
struct ratio_less
    : boost::integral_constant<bool, boost::ratio_detail::ratio_less<R1, R2>::value>
{};

template <class R1, class R2>
struct ratio_less_equal
    : boost::integral_constant<bool, !ratio_less<R2, R1>::value>
{};

template <class R1, class R2>
struct ratio_greater
    : boost::integral_constant<bool, ratio_less<R2, R1>::value>
{};

template <class R1, class R2>
struct ratio_greater_equal
    : boost::integral_constant<bool, !ratio_less<R1, R2>::value>
{};

template <class R1, class R2>
struct ratio_gcd :
    ratio<mpl::gcd_c<boost::intmax_t, R1::num, R2::num>::value,
        mpl::lcm_c<boost::intmax_t, R1::den, R2::den>::value>::type
{
};

    //----------------------------------------------------------------------------//
    //                                                                            //
    //                More arithmetic on ratio types [ratio.arithmetic]           //
    //                                                                            //
    //----------------------------------------------------------------------------//

#ifdef BOOST_RATIO_EXTENSIONS
template <class R>
struct ratio_negate
    : ratio<-R::num, R::den>::type
{
};
template <class R>
struct ratio_abs
    : ratio<mpl::abs_c<boost::intmax_t, R::num>::value, R::den>::type
{
};
template <class R>
struct ratio_sign
    : mpl::sign_c<boost::intmax_t, R::num>
{
};

template <class R>
struct ratio_inverse
    : ratio<R::den, R::num>::type
{
};

template <class R1, class R2>
struct ratio_lcm :
    ratio<mpl::lcm_c<boost::intmax_t, R1::num, R2::num>::value,
        mpl::gcd_c<boost::intmax_t, R1::den, R2::den>::value>::type
{
};

template <class R1, class R2>
struct ratio_modulo :
    ratio<(R1::num * R2::den) % (R2::num * R1::den), R1::den * R2::den>::type
{
};

namespace detail {
  template <class R1, class R2, bool r1ltr2>
  struct ratio_min : R1 {};
  template <class R1, class R2>
  struct ratio_min<R1,R2,false> : R2 {};

  template <class R1, class R2, bool r1ltr2>
  struct ratio_max : R2 {};
  template <class R1, class R2>
  struct ratio_max<R1,R2,false> : R1 {};
}

template <class R1, class R2>
struct ratio_min : detail::ratio_min<R1, R2, ratio_less<R1,R2>::value>::type
{
};

template <class R1, class R2>
struct ratio_max : detail::ratio_max<R1, R2, ratio_less<R1,R2>::value>::type
{
};

template<typename R, int p>
struct ratio_power :
  ratio_multiply<
    typename ratio_power<R, p%2>::type,
    typename ratio_power<typename ratio_multiply<R, R>::type, p/2>::type
  >::type
{};

template<typename R>
struct ratio_power<R, 0> : ratio<1>::type {};

template<typename R>
struct ratio_power<R, 1> : R {};

template<typename R>
struct ratio_power<R, -1> : ratio_divide<ratio<1>, R>::type {};

#endif
}  // namespace boost

#endif  // BOOST_RATIO_RATIO_HPP


code:
//  ratio_fwd.hpp  ---------------------------------------------------------------//

//  Copyright 2008 Howard Hinnant
//  Copyright 2008 Beman Dawes
//  Copyright 2009 Vicente J. Botet Escriba

//  Distributed under the Boost Software License, Version 1.0.
//  See http://www.boost.org/LICENSE_1_0.txt

/*

This code was derived by Beman Dawes from Howard Hinnant's time2_demo prototype.
Many thanks to Howard for making his code available under the Boost license.
The original code was modified to conform to Boost conventions and to section
20.4 Compile-time rational arithmetic [ratio], of the C++ committee working
paper N2798.
See http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2008/n2798.pdf.

time2_demo contained this comment:

    Much thanks to Andrei Alexandrescu,
                   Walter Brown,
                   Peter Dimov,
                   Jeff Garland,
                   Terry Golubiewski,
                   Daniel Krugler,
                   Anthony Williams.
*/

// The way overflow is managed for ratio_less is taken from llvm/libcxx/include/ratio

#ifndef BOOST_RATIO_RATIO_FWD_HPP
#define BOOST_RATIO_RATIO_FWD_HPP

#include <boost/ratio/config.hpp>

#if defined(__GNUC__) && (__GNUC__ >= 4)
#pragma GCC system_header
#endif

namespace boost
{

//----------------------------------------------------------------------------//
//                                                                            //
//              20.6 Compile-time rational arithmetic [ratio]                 //
//                                                                            //
//----------------------------------------------------------------------------//

// ratio
template <boost::intmax_t N, boost::intmax_t D = 1> class ratio;

// ratio arithmetic
template <class R1, class R2> struct ratio_add;
template <class R1, class R2> struct ratio_subtract;
template <class R1, class R2> struct ratio_multiply;
template <class R1, class R2> struct ratio_divide;
#ifdef BOOST_RATIO_EXTENSIONS
template <class R1, class R2> struct ratio_gcd;
template <class R1, class R2> struct ratio_lcm;
template <class R> struct ratio_negate;
template <class R> struct ratio_abs;
template <class R> struct ratio_sign;
template <class R, int P> struct ratio_power;
#endif

// ratio comparison
template <class R1, class R2> struct ratio_equal;
template <class R1, class R2> struct ratio_not_equal;
template <class R1, class R2> struct ratio_less;
template <class R1, class R2> struct ratio_less_equal;
template <class R1, class R2> struct ratio_greater;
template <class R1, class R2> struct ratio_greater_equal;

// convenience SI typedefs
typedef ratio<BOOST_RATIO_INTMAX_C(1), BOOST_RATIO_INTMAX_C(1000000000000000000)> atto;
typedef ratio<BOOST_RATIO_INTMAX_C(1),    BOOST_RATIO_INTMAX_C(1000000000000000)> femto;
typedef ratio<BOOST_RATIO_INTMAX_C(1),       BOOST_RATIO_INTMAX_C(1000000000000)> pico;
typedef ratio<BOOST_RATIO_INTMAX_C(1),          BOOST_RATIO_INTMAX_C(1000000000)> nano;
typedef ratio<BOOST_RATIO_INTMAX_C(1),             BOOST_RATIO_INTMAX_C(1000000)> micro;
typedef ratio<BOOST_RATIO_INTMAX_C(1),                BOOST_RATIO_INTMAX_C(1000)> milli;
typedef ratio<BOOST_RATIO_INTMAX_C(1),                 BOOST_RATIO_INTMAX_C(100)> centi;
typedef ratio<BOOST_RATIO_INTMAX_C(1),                  BOOST_RATIO_INTMAX_C(10)> deci;
typedef ratio<                 BOOST_RATIO_INTMAX_C(10), BOOST_RATIO_INTMAX_C(1)> deca;
typedef ratio<                BOOST_RATIO_INTMAX_C(100), BOOST_RATIO_INTMAX_C(1)> hecto;
typedef ratio<               BOOST_RATIO_INTMAX_C(1000), BOOST_RATIO_INTMAX_C(1)> kilo;
typedef ratio<            BOOST_RATIO_INTMAX_C(1000000), BOOST_RATIO_INTMAX_C(1)> mega;
typedef ratio<         BOOST_RATIO_INTMAX_C(1000000000), BOOST_RATIO_INTMAX_C(1)> giga;
typedef ratio<      BOOST_RATIO_INTMAX_C(1000000000000), BOOST_RATIO_INTMAX_C(1)> tera;
typedef ratio<   BOOST_RATIO_INTMAX_C(1000000000000000), BOOST_RATIO_INTMAX_C(1)> peta;
typedef ratio<BOOST_RATIO_INTMAX_C(1000000000000000000), BOOST_RATIO_INTMAX_C(1)> exa;

#ifdef BOOST_RATIO_EXTENSIONS

#define BOOST_RATIO_1024 BOOST_RATIO_INTMAX_C(1024)

// convenience IEC typedefs
typedef ratio<                                                                                BOOST_RATIO_1024> kibi;
typedef ratio<                                                                    BOOST_RATIO_1024*BOOST_RATIO_1024> mebi;
typedef ratio<                                                   BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024> gibi;
typedef ratio<                                  BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024> tebi;
typedef ratio<                 BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024> pebi;
typedef ratio<BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024*BOOST_RATIO_1024> exbi;

#endif
}  // namespace boost

#endif  // BOOST_RATIO_HPP


code:
//  ratio_io
//
//  (C) Copyright Howard Hinnant
//  (C) Copyright 2010 Vicente J. Botet Escriba
//  Use, modification and distribution are subject to the Boost Software License,
//  Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
//  http://www.boost.org/LICENSE_1_0.txt).
//
// This code was adapted by Vicente from Howard Hinnant's experimental work
// on chrono i/o under lvm/libc++ to Boost

#ifndef BOOST_RATIO_RATIO_IO_HPP
#define BOOST_RATIO_RATIO_IO_HPP

/*

    ratio_io synopsis

#include <ratio>
#include <string>

namespace boost
{

template <class Ratio, class CharT>
struct ratio_string
{
    static basic_string<CharT> prefix();
    static basic_string<CharT> symbol();
};

}  // boost

*/
#include <boost/ratio/config.hpp>

#ifdef BOOST_RATIO_PROVIDES_DEPRECATED_FEATURES_SINCE_V2_0_0
#include <boost/ratio/detail/ratio_io.hpp>
#else

#include <boost/config.hpp>
#include <boost/ratio/ratio.hpp>
#include <boost/type_traits/integral_constant.hpp>
#include <string>
#include <sstream>

#if defined(BOOST_NO_CXX11_UNICODE_LITERALS) || defined(BOOST_NO_CXX11_CHAR16_T) || defined(BOOST_NO_CXX11_CHAR32_T) || defined(BOOST_NO_CXX11_U16STRING) || defined(BOOST_NO_CXX11_U32STRING)
#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT
#undef BOOST_RATIO_HAS_UNICODE_SUPPORT
#endif
#else
#define BOOST_RATIO_HAS_UNICODE_SUPPORT 1
#endif

namespace boost {

//template <class Ratio>
//struct ratio_string_is_localizable : false_type {};
//template <class Ratio>
//struct ratio_string_id : integral_constant<int,0> {};

template <class Ratio, class CharT>
struct ratio_string
{
    static std::basic_string<CharT> symbol() {return prefix();}
    static std::basic_string<CharT> prefix();
};

template <class Ratio, class CharT>
std::basic_string<CharT>
ratio_string<Ratio, CharT>::prefix()
{
    std::basic_ostringstream<CharT> os;
    os << CharT('[') << Ratio::num << CharT('/')
                        << Ratio::den << CharT(']');
    return os.str();
}

#ifdef BOOST_RATIO_HAS_STATIC_STRING
namespace ratio_detail {
template <class Ratio, class CharT>
struct ratio_string_static
{
    static std::string symbol() {
        return std::basic_string<CharT>(
                static_string::c_str<
                        typename ratio_static_string<Ratio, CharT>::symbol
                    >::value);
    }
    static std::string prefix()  {
        return std::basic_string<CharT>(
                static_string::c_str<
                    typename ratio_static_string<Ratio, CharT>::prefix
                >::value);
    }
};
}
#endif
// atto
//template <>
//struct ratio_string_is_localizable<atto> : true_type {};
//
//template <>
//struct ratio_string_id<atto> : integral_constant<int,-18> {};

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<atto, CharT> :
    ratio_detail::ratio_string_static<atto,CharT>
{};

#else
template <>
struct ratio_string<atto, char>
{
    static std::string symbol() {return std::string(1, 'a');}
    static std::string prefix()  {return std::string("atto");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<atto, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'a');}
    static std::u16string prefix()  {return std::u16string(u"atto");}
};

template <>
struct ratio_string<atto, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'a');}
    static std::u32string prefix()  {return std::u32string(U"atto");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<atto, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'a');}
    static std::wstring prefix()  {return std::wstring(L"atto");}
};
#endif
#endif

// femto

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<femto, CharT> :
    ratio_detail::ratio_string_static<femto,CharT>
{};

#else

template <>
struct ratio_string<femto, char>
{
    static std::string symbol() {return std::string(1, 'f');}
    static std::string prefix()  {return std::string("femto");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<femto, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'f');}
    static std::u16string prefix()  {return std::u16string(u"femto");}
};

template <>
struct ratio_string<femto, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'f');}
    static std::u32string prefix()  {return std::u32string(U"femto");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<femto, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'f');}
    static std::wstring prefix()  {return std::wstring(L"femto");}
};
#endif
#endif

// pico

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<pico, CharT> :
    ratio_detail::ratio_string_static<pico,CharT>
{};

#else
template <>
struct ratio_string<pico, char>
{
    static std::string symbol() {return std::string(1, 'p');}
    static std::string prefix()  {return std::string("pico");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<pico, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'p');}
    static std::u16string prefix()  {return std::u16string(u"pico");}
};

template <>
struct ratio_string<pico, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'p');}
    static std::u32string prefix()  {return std::u32string(U"pico");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<pico, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'p');}
    static std::wstring prefix()  {return std::wstring(L"pico");}
};
#endif
#endif

// nano

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<nano, CharT> :
    ratio_detail::ratio_string_static<nano,CharT>
{};

#else
template <>
struct ratio_string<nano, char>
{
    static std::string symbol() {return std::string(1, 'n');}
    static std::string prefix()  {return std::string("nano");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<nano, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'n');}
    static std::u16string prefix()  {return std::u16string(u"nano");}
};

template <>
struct ratio_string<nano, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'n');}
    static std::u32string prefix()  {return std::u32string(U"nano");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<nano, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'n');}
    static std::wstring prefix()  {return std::wstring(L"nano");}
};
#endif
#endif

// micro

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<micro, CharT> :
    ratio_detail::ratio_string_static<micro,CharT>
{};

#else
template <>
struct ratio_string<micro, char>
{
    static std::string symbol() {return std::string("\xC2\xB5");}
    static std::string prefix()  {return std::string("micro");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<micro, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'\xB5');}
    static std::u16string prefix()  {return std::u16string(u"micro");}
};

template <>
struct ratio_string<micro, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'\xB5');}
    static std::u32string prefix()  {return std::u32string(U"micro");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<micro, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'\xB5');}
    static std::wstring prefix()  {return std::wstring(L"micro");}
};
#endif
#endif

// milli

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<milli, CharT> :
    ratio_detail::ratio_string_static<milli,CharT>
{};

#else
template <>
struct ratio_string<milli, char>
{
    static std::string symbol() {return std::string(1, 'm');}
    static std::string prefix()  {return std::string("milli");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<milli, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'm');}
    static std::u16string prefix()  {return std::u16string(u"milli");}
};

template <>
struct ratio_string<milli, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'm');}
    static std::u32string prefix()  {return std::u32string(U"milli");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<milli, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'm');}
    static std::wstring prefix()  {return std::wstring(L"milli");}
};
#endif
#endif

// centi

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<centi, CharT> :
    ratio_detail::ratio_string_static<centi,CharT>
{};

#else
template <>
struct ratio_string<centi, char>
{
    static std::string symbol() {return std::string(1, 'c');}
    static std::string prefix()  {return std::string("centi");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<centi, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'c');}
    static std::u16string prefix()  {return std::u16string(u"centi");}
};

template <>
struct ratio_string<centi, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'c');}
    static std::u32string prefix()  {return std::u32string(U"centi");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<centi, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'c');}
    static std::wstring prefix()  {return std::wstring(L"centi");}
};
#endif
#endif

// deci

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<deci, CharT> :
    ratio_detail::ratio_string_static<deci,CharT>
{};

#else

template <>
struct ratio_string<deci, char>
{
    static std::string symbol() {return std::string(1, 'd');}
    static std::string prefix()  {return std::string("deci");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<deci, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'd');}
    static std::u16string prefix()  {return std::u16string(u"deci");}
};

template <>
struct ratio_string<deci, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'd');}
    static std::u32string prefix()  {return std::u32string(U"deci");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<deci, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'd');}
    static std::wstring prefix()  {return std::wstring(L"deci");}
};
#endif
#endif

// unit

// deca

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<deca, CharT> :
    ratio_detail::ratio_string_static<deca,CharT>
{};

#else
template <>
struct ratio_string<deca, char>
{
    static std::string symbol() {return std::string("da");}
    static std::string prefix()  {return std::string("deca");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<deca, char16_t>
{
    static std::u16string symbol() {return std::u16string(u"da");}
    static std::u16string prefix()  {return std::u16string(u"deca");}
};

template <>
struct ratio_string<deca, char32_t>
{
    static std::u32string symbol() {return std::u32string(U"da");}
    static std::u32string prefix()  {return std::u32string(U"deca");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<deca, wchar_t>
{
    static std::wstring symbol() {return std::wstring(L"da");}
    static std::wstring prefix()  {return std::wstring(L"deca");}
};
#endif
#endif

// hecto

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<hecto, CharT> :
    ratio_detail::ratio_string_static<hecto,CharT>
{};

#else
template <>
struct ratio_string<hecto, char>
{
    static std::string symbol() {return std::string(1, 'h');}
    static std::string prefix()  {return std::string("hecto");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<hecto, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'h');}
    static std::u16string prefix()  {return std::u16string(u"hecto");}
};

template <>
struct ratio_string<hecto, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'h');}
    static std::u32string prefix()  {return std::u32string(U"hecto");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<hecto, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'h');}
    static std::wstring prefix()  {return std::wstring(L"hecto");}
};
#endif
#endif

// kilo

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<kilo, CharT> :
    ratio_detail::ratio_string_static<kilo,CharT>
{};

#else
template <>
struct ratio_string<kilo, char>
{
    static std::string symbol() {return std::string(1, 'k');}
    static std::string prefix()  {return std::string("kilo");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<kilo, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'k');}
    static std::u16string prefix()  {return std::u16string(u"kilo");}
};

template <>
struct ratio_string<kilo, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'k');}
    static std::u32string prefix()  {return std::u32string(U"kilo");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<kilo, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'k');}
    static std::wstring prefix()  {return std::wstring(L"kilo");}
};
#endif
#endif

// mega

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<mega, CharT> :
    ratio_detail::ratio_string_static<mega,CharT>
{};

#else

template <>
struct ratio_string<mega, char>
{
    static std::string symbol() {return std::string(1, 'M');}
    static std::string prefix()  {return std::string("mega");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<mega, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'M');}
    static std::u16string prefix()  {return std::u16string(u"mega");}
};

template <>
struct ratio_string<mega, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'M');}
    static std::u32string prefix()  {return std::u32string(U"mega");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<mega, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'M');}
    static std::wstring prefix()  {return std::wstring(L"mega");}
};
#endif
#endif

// giga

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<giga, CharT> :
    ratio_detail::ratio_string_static<giga,CharT>
{};

#else

template <>
struct ratio_string<giga, char>
{
    static std::string symbol() {return std::string(1, 'G');}
    static std::string prefix()  {return std::string("giga");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<giga, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'G');}
    static std::u16string prefix()  {return std::u16string(u"giga");}
};

template <>
struct ratio_string<giga, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'G');}
    static std::u32string prefix()  {return std::u32string(U"giga");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<giga, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'G');}
    static std::wstring prefix()  {return std::wstring(L"giga");}
};
#endif
#endif

// tera

//template <>
#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<tera, CharT> :
    ratio_detail::ratio_string_static<tera,CharT>
{};

#else
template <>
struct ratio_string<tera, char>
{
    static std::string symbol() {return std::string(1, 'T');}
    static std::string prefix()  {return std::string("tera");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<tera, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'T');}
    static std::u16string prefix()  {return std::u16string(u"tera");}
};

template <>
struct ratio_string<tera, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'T');}
    static std::u32string prefix()  {return std::u32string(U"tera");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<tera, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'T');}
    static std::wstring prefix()  {return std::wstring(L"tera");}
};
#endif
#endif

// peta

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<peta, CharT> :
    ratio_detail::ratio_string_static<peta,CharT>
{};

#else
template <>
struct ratio_string<peta, char>
{
    static std::string symbol() {return std::string(1, 'P');}
    static std::string prefix()  {return std::string("peta");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<peta, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'P');}
    static std::u16string prefix()  {return std::u16string(u"peta");}
};

template <>
struct ratio_string<peta, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'P');}
    static std::u32string prefix()  {return std::u32string(U"peta");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<peta, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'P');}
    static std::wstring prefix()  {return std::wstring(L"peta");}
};
#endif
#endif

// exa

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<exa, CharT> :
    ratio_detail::ratio_string_static<exa,CharT>
{};

#else
template <>
struct ratio_string<exa, char>
{
    static std::string symbol() {return std::string(1, 'E');}
    static std::string prefix()  {return std::string("exa");}
};

#if defined BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<exa, char16_t>
{
    static std::u16string symbol() {return std::u16string(1, u'E');}
    static std::u16string prefix()  {return std::u16string(u"exa");}
};

template <>
struct ratio_string<exa, char32_t>
{
    static std::u32string symbol() {return std::u32string(1, U'E');}
    static std::u32string prefix()  {return std::u32string(U"exa");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<exa, wchar_t>
{
    static std::wstring symbol() {return std::wstring(1, L'E');}
    static std::wstring prefix()  {return std::wstring(L"exa");}
};
#endif
#endif

#ifdef BOOST_RATIO_EXTENSIONS

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<kibi, CharT> :
    ratio_detail::ratio_string_static<kibi,CharT>
{};

#else
template <>
struct ratio_string<kibi, char>
{
    static std::string symbol() {return std::string("Ki");}
    static std::string prefix()  {return std::string("kibi");}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<kibi, char16_t>
{
    static std::u16string symbol() {return std::u16string(u"Ki");}
    static std::u16string prefix()  {return std::u16string(u"kibi");}
};

template <>
struct ratio_string<kibi, char32_t>
{
    static std::u32string symbol() {return std::u32string(U"Ki");}
    static std::u32string prefix()  {return std::u32string(U"kibi");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<kibi, wchar_t>
{
    static std::wstring symbol() {return std::wstring(L"Ki");}
    static std::wstring prefix()  {return std::wstring(L"kibi");}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<mebi, CharT> :
    ratio_detail::ratio_string_static<mebi,CharT>
{};

#else
template <>
struct ratio_string<mebi, char>
{
    static std::string symbol() {return std::string("Mi");}
    static std::string prefix()  {return std::string("mebi");}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<mebi, char16_t>
{
    static std::u16string symbol() {return std::u16string(u"Mi");}
    static std::u16string prefix()  {return std::u16string(u"mebi");}
};

template <>
struct ratio_string<mebi, char32_t>
{
    static std::u32string symbol() {return std::u32string(U"Mi");}
    static std::u32string prefix()  {return std::u32string(U"mebi");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<mebi, wchar_t>
{
    static std::wstring symbol() {return std::wstring(L"Mi");}
    static std::wstring prefix()  {return std::wstring(L"mebi");}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<gibi, CharT> :
    ratio_detail::ratio_string_static<gibi,CharT>
{};

#else
template <>
struct ratio_string<gibi, char>
{
    static std::string symbol() {return std::string("Gi");}
    static std::string prefix()  {return std::string("gibi");}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<gibi, char16_t>
{
    static std::u16string symbol() {return std::u16string(u"Gi");}
    static std::u16string prefix()  {return std::u16string(u"gibi");}
};

template <>
struct ratio_string<gibi, char32_t>
{
    static std::u32string symbol() {return std::u32string(U"Gi");}
    static std::u32string prefix()  {return std::u32string(U"gibi");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<gibi, wchar_t>
{
    static std::wstring symbol() {return std::wstring(L"Gi");}
    static std::wstring prefix()  {return std::wstring(L"gibi");}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<tebi, CharT> :
    ratio_detail::ratio_string_static<tebi,CharT>
{};

#else
template <>
struct ratio_string<tebi, char>
{
    static std::string symbol() {return std::string("Ti");}
    static std::string prefix()  {return std::string("tebi");}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<tebi, char16_t>
{
    static std::u16string symbol() {return std::u16string(u"Ti");}
    static std::u16string prefix()  {return std::u16string(u"tebi");}
};

template <>
struct ratio_string<tebi, char32_t>
{
    static std::u32string symbol() {return std::u32string(U"Ti");}
    static std::u32string prefix()  {return std::u32string(U"tebi");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<tebi, wchar_t>
{
    static std::wstring symbol() {return std::wstring(L"Ti");}
    static std::wstring prefix()  {return std::wstring(L"tebi");}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<pebi, CharT> :
    ratio_detail::ratio_string_static<pebi,CharT>
{};

#else
template <>
struct ratio_string<pebi, char>
{
    static std::string symbol() {return std::string("Pi");}
    static std::string prefix()  {return std::string("pebi");}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<pebi, char16_t>
{
    static std::u16string symbol() {return std::u16string(u"Pi");}
    static std::u16string prefix()  {return std::u16string(u"pebi");}
};

template <>
struct ratio_string<pebi, char32_t>
{
    static std::u32string symbol() {return std::u32string(U"Pi");}
    static std::u32string prefix()  {return std::u32string(U"pebi");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<pebi, wchar_t>
{
    static std::wstring symbol() {return std::wstring(L"Pi");}
    static std::wstring prefix()  {return std::wstring(L"pebi");}
};
#endif
#endif

#ifdef BOOST_RATIO_HAS_STATIC_STRING
template <typename CharT>
struct ratio_string<exbi, CharT> :
    ratio_detail::ratio_string_static<exbi,CharT>
{};

#else
template <>
struct ratio_string<exbi, char>
{
    static std::string symbol() {return std::string("Ei");}
    static std::string prefix()  {return std::string("exbi");}
};

#if BOOST_RATIO_HAS_UNICODE_SUPPORT

template <>
struct ratio_string<exbi, char16_t>
{
    static std::u16string symbol() {return std::u16string(u"Ei");}
    static std::u16string prefix()  {return std::u16string(u"exbi");}
};

template <>
struct ratio_string<exbi, char32_t>
{
    static std::u32string symbol() {return std::u32string(U"Ei");}
    static std::u32string prefix()  {return std::u32string(U"exbi");}
};

#endif

#ifndef BOOST_NO_STD_WSTRING
template <>
struct ratio_string<exbi, wchar_t>
{
    static std::wstring symbol() {return std::wstring(L"Ei");}
    static std::wstring prefix()  {return std::wstring(L"exbi");}
};
#endif
#endif
#endif

}

#endif  // BOOST_RATIO_PROVIDES_DEPRECATED_FEATURES_SINCE_V2_0_0
#endif  // BOOST_RATIO_RATIO_IO_HPP