add contrib/restricted/boost/rational/include to fix warning

commit_hash:804663433eb5a988140a4afdf332c597ad933686

1 files changed, 1046 insertions, 0 deletions

diff --git a/contrib/restricted/boost/rational/include/boost/rational.hpp b/contrib/restricted/boost/rational/include/boost/rational.hpp
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@@ -0,0 +1,1046 @@
+//  Boost rational.hpp header file  ------------------------------------------//
+
+//  (C) Copyright Paul Moore 1999. Permission to copy, use, modify, sell and
+//  distribute this software is granted provided this copyright notice appears
+//  in all copies. This software is provided "as is" without express or
+//  implied warranty, and with no claim as to its suitability for any purpose.
+
+// boostinspect:nolicense (don't complain about the lack of a Boost license)
+// (Paul Moore hasn't been in contact for years, so there's no way to change the
+// license.)
+
+//  See http://www.boost.org/libs/rational for documentation.
+
+//  Credits:
+//  Thanks to the boost mailing list in general for useful comments.
+//  Particular contributions included:
+//    Andrew D Jewell, for reminding me to take care to avoid overflow
+//    Ed Brey, for many comments, including picking up on some dreadful typos
+//    Stephen Silver contributed the test suite and comments on user-defined
+//    IntType
+//    Nickolay Mladenov, for the implementation of operator+=
+
+//  Revision History
+//  12 Nov 20  Fix operators to work with C++20 rules (Glen Joseph Fernandes)
+//  02 Sep 13  Remove unneeded forward declarations; tweak private helper
+//             function (Daryle Walker)
+//  30 Aug 13  Improve exception safety of "assign"; start modernizing I/O code
+//             (Daryle Walker)
+//  27 Aug 13  Add cross-version constructor template, plus some private helper
+//             functions; add constructor to exception class to take custom
+//             messages (Daryle Walker)
+//  25 Aug 13  Add constexpr qualification wherever possible (Daryle Walker)
+//  05 May 12  Reduced use of implicit gcd (Mario Lang)
+//  05 Nov 06  Change rational_cast to not depend on division between different
+//             types (Daryle Walker)
+//  04 Nov 06  Off-load GCD and LCM to Boost.Integer; add some invariant checks;
+//             add std::numeric_limits<> requirement to help GCD (Daryle Walker)
+//  31 Oct 06  Recoded both operator< to use round-to-negative-infinity
+//             divisions; the rational-value version now uses continued fraction
+//             expansion to avoid overflows, for bug #798357 (Daryle Walker)
+//  20 Oct 06  Fix operator bool_type for CW 8.3 (Joaquín M López Muñoz)
+//  18 Oct 06  Use EXPLICIT_TEMPLATE_TYPE helper macros from Boost.Config
+//             (Joaquín M López Muñoz)
+//  27 Dec 05  Add Boolean conversion operator (Daryle Walker)
+//  28 Sep 02  Use _left versions of operators from operators.hpp
+//  05 Jul 01  Recode gcd(), avoiding std::swap (Helmut Zeisel)
+//  03 Mar 01  Workarounds for Intel C++ 5.0 (David Abrahams)
+//  05 Feb 01  Update operator>> to tighten up input syntax
+//  05 Feb 01  Final tidy up of gcd code prior to the new release
+//  27 Jan 01  Recode abs() without relying on abs(IntType)
+//  21 Jan 01  Include Nickolay Mladenov's operator+= algorithm,
+//             tidy up a number of areas, use newer features of operators.hpp
+//             (reduces space overhead to zero), add operator!,
+//             introduce explicit mixed-mode arithmetic operations
+//  12 Jan 01  Include fixes to handle a user-defined IntType better
+//  19 Nov 00  Throw on divide by zero in operator /= (John (EBo) David)
+//  23 Jun 00  Incorporate changes from Mark Rodgers for Borland C++
+//  22 Jun 00  Change _MSC_VER to BOOST_MSVC so other compilers are not
+//             affected (Beman Dawes)
+//   6 Mar 00  Fix operator-= normalization, #include <string> (Jens Maurer)
+//  14 Dec 99  Modifications based on comments from the boost list
+//  09 Dec 99  Initial Version (Paul Moore)
+
+#ifndef BOOST_RATIONAL_HPP
+#define BOOST_RATIONAL_HPP
+
+#include <boost/config.hpp>      // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC, etc
+#ifndef BOOST_NO_IOSTREAM
+#include <iomanip>               // for std::setw
+#include <ios>                   // for std::noskipws, streamsize
+#include <istream>               // for std::istream
+#include <ostream>               // for std::ostream
+#include <sstream>               // for std::ostringstream
+#endif
+#include <cstddef>               // for NULL
+#include <stdexcept>             // for std::domain_error
+#include <string>                // for std::string implicit constructor
+#include <cstdlib>               // for std::abs
+#include <boost/call_traits.hpp> // for boost::call_traits
+#include <boost/detail/workaround.hpp> // for BOOST_WORKAROUND
+#include <boost/assert.hpp>      // for BOOST_ASSERT
+#include <boost/integer/common_factor_rt.hpp> // for boost::integer::gcd, lcm
+#include <limits>                // for std::numeric_limits
+#include <boost/static_assert.hpp>  // for BOOST_STATIC_ASSERT
+#include <boost/throw_exception.hpp>
+#include <boost/utility/enable_if.hpp>
+#include <boost/type_traits/is_convertible.hpp>
+#include <boost/type_traits/is_class.hpp>
+#include <boost/type_traits/is_same.hpp>
+#include <boost/type_traits/is_array.hpp>
+
+// Control whether depreciated GCD and LCM functions are included (default: yes)
+#ifndef BOOST_CONTROL_RATIONAL_HAS_GCD
+#define BOOST_CONTROL_RATIONAL_HAS_GCD  1
+#endif
+
+namespace boost {
+
+#if BOOST_CONTROL_RATIONAL_HAS_GCD
+template <typename IntType>
+IntType gcd(IntType n, IntType m)
+{
+    // Defer to the version in Boost.Integer
+    return integer::gcd( n, m );
+}
+
+template <typename IntType>
+IntType lcm(IntType n, IntType m)
+{
+    // Defer to the version in Boost.Integer
+    return integer::lcm( n, m );
+}
+#endif  // BOOST_CONTROL_RATIONAL_HAS_GCD
+
+namespace rational_detail{
+
+   template <class FromInt, class ToInt, typename Enable = void>
+   struct is_compatible_integer;
+
+   template <class FromInt, class ToInt>
+   struct is_compatible_integer<FromInt, ToInt, typename enable_if_c<!is_array<FromInt>::value>::type>
+   {
+      BOOST_STATIC_CONSTANT(bool, value = ((std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
+         && (std::numeric_limits<FromInt>::digits <= std::numeric_limits<ToInt>::digits)
+         && (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
+         && ((std::numeric_limits<FromInt>::is_signed == false) || (std::numeric_limits<ToInt>::is_signed == true))
+         && is_convertible<FromInt, ToInt>::value)
+         || is_same<FromInt, ToInt>::value)
+         || (is_class<ToInt>::value && is_class<FromInt>::value && is_convertible<FromInt, ToInt>::value));
+   };
+
+   template <class FromInt, class ToInt>
+   struct is_compatible_integer<FromInt, ToInt, typename enable_if_c<is_array<FromInt>::value>::type>
+   {
+      BOOST_STATIC_CONSTANT(bool, value = false);
+   };
+
+   template <class FromInt, class ToInt, typename Enable = void>
+   struct is_backward_compatible_integer;
+
+   template <class FromInt, class ToInt>
+   struct is_backward_compatible_integer<FromInt, ToInt, typename enable_if_c<!is_array<FromInt>::value>::type>
+   {
+      BOOST_STATIC_CONSTANT(bool, value = (std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
+         && !is_compatible_integer<FromInt, ToInt>::value
+         && (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
+         && is_convertible<FromInt, ToInt>::value));
+   };
+
+   template <class FromInt, class ToInt>
+   struct is_backward_compatible_integer<FromInt, ToInt, typename enable_if_c<is_array<FromInt>::value>::type>
+   {
+      BOOST_STATIC_CONSTANT(bool, value = false);
+   };
+}
+
+class bad_rational : public std::domain_error
+{
+public:
+    explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}
+    explicit bad_rational( char const *what ) : std::domain_error( what ) {}
+};
+
+template <typename IntType>
+class rational
+{
+    // Class-wide pre-conditions
+    BOOST_STATIC_ASSERT( ::std::numeric_limits<IntType>::is_specialized );
+
+    // Helper types
+    typedef typename boost::call_traits<IntType>::param_type param_type;
+
+    struct helper { IntType parts[2]; };
+    typedef IntType (helper::* bool_type)[2];
+
+public:
+    // Component type
+    typedef IntType int_type;
+
+    BOOST_CONSTEXPR
+    rational() : num(0), den(1) {}
+
+    template <class T>//, typename enable_if_c<!is_array<T>::value>::type>
+    BOOST_CONSTEXPR rational(const T& n, typename enable_if_c<
+       rational_detail::is_compatible_integer<T, IntType>::value
+    >::type const* = 0) : num(n), den(1) {}
+
+    template <class T, class U>
+    BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
+       rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value
+    >::type const* = 0) : num(n), den(d) {
+       normalize();
+    }
+
+    template < typename NewType >
+    BOOST_CONSTEXPR explicit
+       rational(rational<NewType> const &r, typename enable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
+       : num(r.numerator()), den(is_normalized(int_type(r.numerator()),
+       int_type(r.denominator())) ? r.denominator() :
+       (BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
+
+    template < typename NewType >
+    BOOST_CONSTEXPR explicit
+       rational(rational<NewType> const &r, typename disable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
+       : num(r.numerator()), den(is_normalized(int_type(r.numerator()),
+       int_type(r.denominator())) && is_safe_narrowing_conversion(r.denominator()) && is_safe_narrowing_conversion(r.numerator()) ? r.denominator() :
+       (BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
+    // Default copy constructor and assignment are fine
+
+    // Add assignment from IntType
+    template <class T>
+    BOOST_CXX14_CONSTEXPR typename enable_if_c<
+       rational_detail::is_compatible_integer<T, IntType>::value, rational &
+    >::type operator=(const T& n) { return assign(static_cast<IntType>(n), static_cast<IntType>(1)); }
+
+    // Assign in place
+    template <class T, class U>
+    BOOST_CXX14_CONSTEXPR typename enable_if_c<
+       rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value, rational &
+    >::type assign(const T& n, const U& d)
+    {
+       return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
+    }
+    //
+    // The following overloads should probably *not* be provided - 
+    // but are provided for backwards compatibity reasons only.
+    // These allow for construction/assignment from types that
+    // are wider than IntType only if there is an implicit
+    // conversion from T to IntType, they will throw a bad_rational
+    // if the conversion results in loss of precision or undefined behaviour.
+    //
+    template <class T>//, typename enable_if_c<!is_array<T>::value>::type>
+    BOOST_CXX14_CONSTEXPR rational(const T& n, typename enable_if_c<
+       rational_detail::is_backward_compatible_integer<T, IntType>::value
+    >::type const* = 0)
+    {
+       assign(n, static_cast<T>(1));
+    }
+    template <class T, class U>
+    BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
+       (!rational_detail::is_compatible_integer<T, IntType>::value
+       || !rational_detail::is_compatible_integer<U, IntType>::value)
+       && std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
+       && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
+       && is_convertible<T, IntType>::value &&
+       std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
+       && (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
+       && is_convertible<U, IntType>::value
+    >::type const* = 0)
+    {
+       assign(n, d);
+    }
+    template <class T>
+    BOOST_CXX14_CONSTEXPR typename enable_if_c<
+       std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
+       && !rational_detail::is_compatible_integer<T, IntType>::value
+       && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
+       && is_convertible<T, IntType>::value,
+       rational &
+    >::type operator=(const T& n) { return assign(n, static_cast<T>(1)); }
+
+    template <class T, class U>
+    BOOST_CXX14_CONSTEXPR typename enable_if_c<
+       (!rational_detail::is_compatible_integer<T, IntType>::value
+          || !rational_detail::is_compatible_integer<U, IntType>::value)
+       && std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
+       && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
+       && is_convertible<T, IntType>::value &&
+       std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
+       && (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
+       && is_convertible<U, IntType>::value,
+       rational &
+    >::type assign(const T& n, const U& d)
+    {
+       if(!is_safe_narrowing_conversion(n) || !is_safe_narrowing_conversion(d))
+          BOOST_THROW_EXCEPTION(bad_rational());
+       return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
+    }
+
+    // Access to representation
+    BOOST_CONSTEXPR
+    const IntType& numerator() const { return num; }
+    BOOST_CONSTEXPR
+    const IntType& denominator() const { return den; }
+
+    // Arithmetic assignment operators
+    BOOST_CXX14_CONSTEXPR rational& operator+= (const rational& r);
+    BOOST_CXX14_CONSTEXPR rational& operator-= (const rational& r);
+    BOOST_CXX14_CONSTEXPR rational& operator*= (const rational& r);
+    BOOST_CXX14_CONSTEXPR rational& operator/= (const rational& r);
+
+    template <class T>
+    BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator+= (const T& i)
+    {
+       num += i * den;
+       return *this;
+    }
+    template <class T>
+    BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator-= (const T& i)
+    {
+       num -= i * den;
+       return *this;
+    }
+    template <class T>
+    BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator*= (const T& i)
+    {
+       // Avoid overflow and preserve normalization
+       IntType gcd = integer::gcd(static_cast<IntType>(i), den);
+       num *= i / gcd;
+       den /= gcd;
+       return *this;
+    }
+    template <class T>
+    BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator/= (const T& i)
+    {
+       // Avoid repeated construction
+       IntType const zero(0);
+
+       if(i == zero) BOOST_THROW_EXCEPTION(bad_rational());
+       if(num == zero) return *this;
+
+       // Avoid overflow and preserve normalization
+       IntType const gcd = integer::gcd(num, static_cast<IntType>(i));
+       num /= gcd;
+       den *= i / gcd;
+
+       if(den < zero) {
+          num = -num;
+          den = -den;
+       }
+
+       return *this;
+    }
+
+    // Increment and decrement
+    BOOST_CXX14_CONSTEXPR const rational& operator++() { num += den; return *this; }
+    BOOST_CXX14_CONSTEXPR const rational& operator--() { num -= den; return *this; }
+
+    BOOST_CXX14_CONSTEXPR rational operator++(int)
+    {
+       rational t(*this);
+       ++(*this);
+       return t;
+    }
+    BOOST_CXX14_CONSTEXPR rational operator--(int)
+    {
+       rational t(*this);
+       --(*this);
+       return t;
+    }
+
+    // Operator not
+    BOOST_CONSTEXPR
+    bool operator!() const { return !num; }
+
+    // Boolean conversion
+    
+#if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
+    // The "ISO C++ Template Parser" option in CW 8.3 chokes on the
+    // following, hence we selectively disable that option for the
+    // offending memfun.
+#pragma parse_mfunc_templ off
+#endif
+
+    BOOST_CONSTEXPR
+    operator bool_type() const { return operator !() ? 0 : &helper::parts; }
+
+#if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
+#pragma parse_mfunc_templ reset
+#endif
+
+    // Comparison operators
+    BOOST_CXX14_CONSTEXPR bool operator< (const rational& r) const;
+    BOOST_CXX14_CONSTEXPR bool operator> (const rational& r) const { return r < *this; }
+    BOOST_CONSTEXPR
+    bool operator== (const rational& r) const;
+
+    template <class T>
+    BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator< (const T& i) const
+    {
+       // Avoid repeated construction
+       int_type const  zero(0);
+
+       // Break value into mixed-fraction form, w/ always-nonnegative remainder
+       BOOST_ASSERT(this->den > zero);
+       int_type  q = this->num / this->den, r = this->num % this->den;
+       while(r < zero)  { r += this->den; --q; }
+
+       // Compare with just the quotient, since the remainder always bumps the
+       // value up.  [Since q = floor(n/d), and if n/d < i then q < i, if n/d == i
+       // then q == i, if n/d == i + r/d then q == i, and if n/d >= i + 1 then
+       // q >= i + 1 > i; therefore n/d < i iff q < i.]
+       return q < i;
+    }
+    template <class T>
+    BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator>(const T& i) const
+    {
+       return operator==(i) ? false : !operator<(i);
+    }
+    template <class T>
+    BOOST_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator== (const T& i) const
+    {
+       return ((den == IntType(1)) && (num == i));
+    }
+
+private:
+    // Implementation - numerator and denominator (normalized).
+    // Other possibilities - separate whole-part, or sign, fields?
+    IntType num;
+    IntType den;
+
+    // Helper functions
+    static BOOST_CONSTEXPR
+    int_type inner_gcd( param_type a, param_type b, int_type const &zero =
+     int_type(0) )
+    { return b == zero ? a : inner_gcd(b, a % b, zero); }
+
+    static BOOST_CONSTEXPR
+    int_type inner_abs( param_type x, int_type const &zero = int_type(0) )
+    { return x < zero ? -x : +x; }
+
+    // Representation note: Fractions are kept in normalized form at all
+    // times. normalized form is defined as gcd(num,den) == 1 and den > 0.
+    // In particular, note that the implementation of abs() below relies
+    // on den always being positive.
+    BOOST_CXX14_CONSTEXPR bool test_invariant() const;
+    BOOST_CXX14_CONSTEXPR void normalize();
+
+    static BOOST_CONSTEXPR
+    bool is_normalized( param_type n, param_type d, int_type const &zero =
+     int_type(0), int_type const &one = int_type(1) )
+    {
+        return d > zero && ( n != zero || d == one ) && inner_abs( inner_gcd(n,
+         d, zero), zero ) == one;
+    }
+    //
+    // Conversion checks:
+    //
+    // (1) From an unsigned type with more digits than IntType:
+    //
+    template <class T>
+    BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
+    {
+       return val < (T(1) << std::numeric_limits<IntType>::digits);
+    }
+    //
+    // (2) From a signed type with more digits than IntType, and IntType also signed:
+    //
+    template <class T>
+    BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T& val)
+    {
+       // Note that this check assumes IntType has a 2's complement representation,
+       // we don't want to try to convert a std::numeric_limits<IntType>::min() to
+       // a T because that conversion may not be allowed (this happens when IntType
+       // is from Boost.Multiprecision).
+       return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= -(T(1) << std::numeric_limits<IntType>::digits));
+    }
+    //
+    // (3) From a signed type with more digits than IntType, and IntType unsigned:
+    //
+    template <class T>
+    BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
+    {
+       return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= 0);
+    }
+    //
+    // (4) From a signed type with fewer digits than IntType, and IntType unsigned:
+    //
+    template <class T>
+    BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
+    {
+       return val >= 0;
+    }
+    //
+    // (5) From an unsigned type with fewer digits than IntType, and IntType signed:
+    //
+    template <class T>
+    BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T&)
+    {
+       return true;
+    }
+    //
+    // (6) From an unsigned type with fewer digits than IntType, and IntType unsigned:
+    //
+    template <class T>
+    BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T&)
+    {
+       return true;
+    }
+    //
+    // (7) From an signed type with fewer digits than IntType, and IntType signed:
+    //
+    template <class T>
+    BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T&)
+    {
+       return true;
+    }
+};
+
+// Unary plus and minus
+template <typename IntType>
+BOOST_CONSTEXPR
+inline rational<IntType> operator+ (const rational<IntType>& r)
+{
+    return r;
+}
+
+template <typename IntType>
+BOOST_CXX14_CONSTEXPR
+inline rational<IntType> operator- (const rational<IntType>& r)
+{
+    return rational<IntType>(static_cast<IntType>(-r.numerator()), r.denominator());
+}
+
+// Arithmetic assignment operators
+template <typename IntType>
+BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator+= (const rational<IntType>& r)
+{
+    // This calculation avoids overflow, and minimises the number of expensive
+    // calculations. Thanks to Nickolay Mladenov for this algorithm.
+    //
+    // Proof:
+    // We have to compute a/b + c/d, where gcd(a,b)=1 and gcd(b,c)=1.
+    // Let g = gcd(b,d), and b = b1*g, d=d1*g. Then gcd(b1,d1)=1
+    //
+    // The result is (a*d1 + c*b1) / (b1*d1*g).
+    // Now we have to normalize this ratio.
+    // Let's assume h | gcd((a*d1 + c*b1), (b1*d1*g)), and h > 1
+    // If h | b1 then gcd(h,d1)=1 and hence h|(a*d1+c*b1) => h|a.
+    // But since gcd(a,b1)=1 we have h=1.
+    // Similarly h|d1 leads to h=1.
+    // So we have that h | gcd((a*d1 + c*b1) , (b1*d1*g)) => h|g
+    // Finally we have gcd((a*d1 + c*b1), (b1*d1*g)) = gcd((a*d1 + c*b1), g)
+    // Which proves that instead of normalizing the result, it is better to
+    // divide num and den by gcd((a*d1 + c*b1), g)
+
+    // Protect against self-modification
+    IntType r_num = r.num;
+    IntType r_den = r.den;
+
+    IntType g = integer::gcd(den, r_den);
+    den /= g;  // = b1 from the calculations above
+    num = num * (r_den / g) + r_num * den;
+    g = integer::gcd(num, g);
+    num /= g;
+    den *= r_den/g;
+
+    return *this;
+}
+
+template <typename IntType>
+BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator-= (const rational<IntType>& r)
+{
+    // Protect against self-modification
+    IntType r_num = r.num;
+    IntType r_den = r.den;
+
+    // This calculation avoids overflow, and minimises the number of expensive
+    // calculations. It corresponds exactly to the += case above
+    IntType g = integer::gcd(den, r_den);
+    den /= g;
+    num = num * (r_den / g) - r_num * den;
+    g = integer::gcd(num, g);
+    num /= g;
+    den *= r_den/g;
+
+    return *this;
+}
+
+template <typename IntType>
+BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator*= (const rational<IntType>& r)
+{
+    // Protect against self-modification
+    IntType r_num = r.num;
+    IntType r_den = r.den;
+
+    // Avoid overflow and preserve normalization
+    IntType gcd1 = integer::gcd(num, r_den);
+    IntType gcd2 = integer::gcd(r_num, den);
+    num = (num/gcd1) * (r_num/gcd2);
+    den = (den/gcd2) * (r_den/gcd1);
+    return *this;
+}
+
+template <typename IntType>
+BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator/= (const rational<IntType>& r)
+{
+    // Protect against self-modification
+    IntType r_num = r.num;
+    IntType r_den = r.den;
+
+    // Avoid repeated construction
+    IntType zero(0);
+
+    // Trap division by zero
+    if (r_num == zero)
+        BOOST_THROW_EXCEPTION(bad_rational());
+    if (num == zero)
+        return *this;
+
+    // Avoid overflow and preserve normalization
+    IntType gcd1 = integer::gcd(num, r_num);
+    IntType gcd2 = integer::gcd(r_den, den);
+    num = (num/gcd1) * (r_den/gcd2);
+    den = (den/gcd2) * (r_num/gcd1);
+
+    if (den < zero) {
+        num = -num;
+        den = -den;
+    }
+    return *this;
+}
+
+
+//
+// Non-member operators: previously these were provided by Boost.Operator, but these had a number of
+// drawbacks, most notably, that in order to allow inter-operability with IntType code such as this:
+//
+// rational<int> r(3);
+// assert(r == 3.5); // compiles and passes!!
+//
+// Happens to be allowed as well :-(
+//
+// There are three possible cases for each operator:
+// 1) rational op rational.
+// 2) rational op integer
+// 3) integer op rational
+// Cases (1) and (2) are folded into the one function.
+//
+template <class IntType, class Arg>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
+   operator + (const rational<IntType>& a, const Arg& b)
+{
+      rational<IntType> t(a);
+      return t += b;
+}
+template <class Arg, class IntType>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
+   operator + (const Arg& b, const rational<IntType>& a)
+{
+      rational<IntType> t(a);
+      return t += b;
+}
+
+template <class IntType, class Arg>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
+   operator - (const rational<IntType>& a, const Arg& b)
+{
+      rational<IntType> t(a);
+      return t -= b;
+}
+template <class Arg, class IntType>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
+   operator - (const Arg& b, const rational<IntType>& a)
+{
+      rational<IntType> t(a);
+      return -(t -= b);
+}
+
+template <class IntType, class Arg>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
+   operator * (const rational<IntType>& a, const Arg& b)
+{
+      rational<IntType> t(a);
+      return t *= b;
+}
+template <class Arg, class IntType>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
+   operator * (const Arg& b, const rational<IntType>& a)
+{
+      rational<IntType> t(a);
+      return t *= b;
+}
+
+template <class IntType, class Arg>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
+   operator / (const rational<IntType>& a, const Arg& b)
+{
+      rational<IntType> t(a);
+      return t /= b;
+}
+template <class Arg, class IntType>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
+   operator / (const Arg& b, const rational<IntType>& a)
+{
+      rational<IntType> t(b);
+      return t /= a;
+}
+
+template <class IntType, class Arg>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
+   operator <= (const rational<IntType>& a, const Arg& b)
+{
+      return !a.operator>(b);
+}
+template <class Arg, class IntType>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
+   operator <= (const Arg& b, const rational<IntType>& a)
+{
+      return a >= b;
+}
+
+template <class IntType, class Arg>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
+   operator >= (const rational<IntType>& a, const Arg& b)
+{
+      return !a.operator<(b);
+}
+template <class Arg, class IntType>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
+   operator >= (const Arg& b, const rational<IntType>& a)
+{
+      return a <= b;
+}
+
+template <class IntType, class Arg>
+BOOST_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
+   operator != (const rational<IntType>& a, const Arg& b)
+{
+      return !a.operator==(b);
+}
+template <class Arg, class IntType>
+BOOST_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
+   operator != (const Arg& b, const rational<IntType>& a)
+{
+      return !(b == a);
+}
+
+template <class Arg, class IntType>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
+   operator < (const Arg& b, const rational<IntType>& a)
+{
+      return a.operator>(b);
+}
+template <class Arg, class IntType>
+BOOST_CXX14_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
+   operator > (const Arg& b, const rational<IntType>& a)
+{
+      return a.operator<(b);
+}
+template <class Arg, class IntType>
+BOOST_CONSTEXPR
+inline typename boost::enable_if_c <
+   rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
+   operator == (const Arg& b, const rational<IntType>& a)
+{
+      return a.operator==(b);
+}
+
+// Comparison operators
+template <typename IntType>
+BOOST_CXX14_CONSTEXPR
+bool rational<IntType>::operator< (const rational<IntType>& r) const
+{
+    // Avoid repeated construction
+    int_type const  zero( 0 );
+
+    // This should really be a class-wide invariant.  The reason for these
+    // checks is that for 2's complement systems, INT_MIN has no corresponding
+    // positive, so negating it during normalization keeps it INT_MIN, which
+    // is bad for later calculations that assume a positive denominator.
+    BOOST_ASSERT( this->den > zero );
+    BOOST_ASSERT( r.den > zero );
+
+    // Determine relative order by expanding each value to its simple continued
+    // fraction representation using the Euclidian GCD algorithm.
+    struct { int_type  n, d, q, r; }
+     ts = { this->num, this->den, static_cast<int_type>(this->num / this->den),
+     static_cast<int_type>(this->num % this->den) },
+     rs = { r.num, r.den, static_cast<int_type>(r.num / r.den),
+     static_cast<int_type>(r.num % r.den) };
+    unsigned  reverse = 0u;
+
+    // Normalize negative moduli by repeatedly adding the (positive) denominator
+    // and decrementing the quotient.  Later cycles should have all positive
+    // values, so this only has to be done for the first cycle.  (The rules of
+    // C++ require a nonnegative quotient & remainder for a nonnegative dividend
+    // & positive divisor.)
+    while ( ts.r < zero )  { ts.r += ts.d; --ts.q; }
+    while ( rs.r < zero )  { rs.r += rs.d; --rs.q; }
+
+    // Loop through and compare each variable's continued-fraction components
+    for ( ;; )
+    {
+        // The quotients of the current cycle are the continued-fraction
+        // components.  Comparing two c.f. is comparing their sequences,
+        // stopping at the first difference.
+        if ( ts.q != rs.q )
+        {
+            // Since reciprocation changes the relative order of two variables,
+            // and c.f. use reciprocals, the less/greater-than test reverses
+            // after each index.  (Start w/ non-reversed @ whole-number place.)
+            return reverse ? ts.q > rs.q : ts.q < rs.q;
+        }
+
+        // Prepare the next cycle
+        reverse ^= 1u;
+
+        if ( (ts.r == zero) || (rs.r == zero) )
+        {
+            // At least one variable's c.f. expansion has ended
+            break;
+        }
+
+        ts.n = ts.d;         ts.d = ts.r;
+        ts.q = ts.n / ts.d;  ts.r = ts.n % ts.d;
+        rs.n = rs.d;         rs.d = rs.r;
+        rs.q = rs.n / rs.d;  rs.r = rs.n % rs.d;
+    }
+
+    // Compare infinity-valued components for otherwise equal sequences
+    if ( ts.r == rs.r )
+    {
+        // Both remainders are zero, so the next (and subsequent) c.f.
+        // components for both sequences are infinity.  Therefore, the sequences
+        // and their corresponding values are equal.
+        return false;
+    }
+    else
+    {
+#ifdef BOOST_MSVC
+#pragma warning(push)
+#pragma warning(disable:4800)
+#endif
+        // Exactly one of the remainders is zero, so all following c.f.
+        // components of that variable are infinity, while the other variable
+        // has a finite next c.f. component.  So that other variable has the
+        // lesser value (modulo the reversal flag!).
+        return ( ts.r != zero ) != static_cast<bool>( reverse );
+#ifdef BOOST_MSVC
+#pragma warning(pop)
+#endif
+    }
+}
+
+template <typename IntType>
+BOOST_CONSTEXPR
+inline bool rational<IntType>::operator== (const rational<IntType>& r) const
+{
+    return ((num == r.num) && (den == r.den));
+}
+
+// Invariant check
+template <typename IntType>
+BOOST_CXX14_CONSTEXPR
+inline bool rational<IntType>::test_invariant() const
+{
+    return ( this->den > int_type(0) ) && ( integer::gcd(this->num, this->den) ==
+     int_type(1) );
+}
+
+// Normalisation
+template <typename IntType>
+BOOST_CXX14_CONSTEXPR void rational<IntType>::normalize()
+{
+    // Avoid repeated construction
+    IntType zero(0);
+
+    if (den == zero)
+       BOOST_THROW_EXCEPTION(bad_rational());
+
+    // Handle the case of zero separately, to avoid division by zero
+    if (num == zero) {
+        den = IntType(1);
+        return;
+    }
+
+    IntType g = integer::gcd(num, den);
+
+    num /= g;
+    den /= g;
+
+    if (den < -(std::numeric_limits<IntType>::max)()) {
+        BOOST_THROW_EXCEPTION(bad_rational("bad rational: non-zero singular denominator"));
+    }
+
+    // Ensure that the denominator is positive
+    if (den < zero) {
+        num = -num;
+        den = -den;
+    }
+
+    BOOST_ASSERT( this->test_invariant() );
+}
+
+#ifndef BOOST_NO_IOSTREAM
+namespace detail {
+
+    // A utility class to reset the format flags for an istream at end
+    // of scope, even in case of exceptions
+    struct resetter {
+        resetter(std::istream& is) : is_(is), f_(is.flags()) {}
+        ~resetter() { is_.flags(f_); }
+        std::istream& is_;
+        std::istream::fmtflags f_;      // old GNU c++ lib has no ios_base
+    };
+
+}
+
+// Input and output
+template <typename IntType>
+std::istream& operator>> (std::istream& is, rational<IntType>& r)
+{
+    using std::ios;
+
+    IntType n = IntType(0), d = IntType(1);
+    char c = 0;
+    detail::resetter sentry(is);
+
+    if ( is >> n )
+    {
+        if ( is.get(c) )
+        {
+            if ( c == '/' )
+            {
+                if ( is >> std::noskipws >> d )
+                    try {
+                        r.assign( n, d );
+                    } catch ( bad_rational & ) {        // normalization fail
+                        try { is.setstate(ios::failbit); }
+                        catch ( ... ) {}  // don't throw ios_base::failure...
+                        if ( is.exceptions() & ios::failbit )
+                            throw;   // ...but the original exception instead
+                        // ELSE: suppress the exception, use just error flags
+                    }
+            }
+            else
+                is.setstate( ios::failbit );
+        }
+    }
+
+    return is;
+}
+
+// Add manipulators for output format?
+template <typename IntType>
+std::ostream& operator<< (std::ostream& os, const rational<IntType>& r)
+{
+    // The slash directly precedes the denominator, which has no prefixes.
+    std::ostringstream  ss;
+
+    ss.copyfmt( os );
+    ss.tie( NULL );
+    ss.exceptions( std::ios::goodbit );
+    ss.width( 0 );
+    ss << std::noshowpos << std::noshowbase << '/' << r.denominator();
+
+    // The numerator holds the showpos, internal, and showbase flags.
+    std::string const   tail = ss.str();
+    std::streamsize const  w =
+        os.width() - static_cast<std::streamsize>( tail.size() );
+
+    ss.clear();
+    ss.str( "" );
+    ss.flags( os.flags() );
+    ss << std::setw( w < 0 || (os.flags() & std::ios::adjustfield) !=
+                     std::ios::internal ? 0 : w ) << r.numerator();
+    return os << ss.str() + tail;
+}
+#endif  // BOOST_NO_IOSTREAM
+
+// Type conversion
+template <typename T, typename IntType>
+BOOST_CONSTEXPR
+inline T rational_cast(const rational<IntType>& src)
+{
+    return static_cast<T>(src.numerator())/static_cast<T>(src.denominator());
+}
+
+// Do not use any abs() defined on IntType - it isn't worth it, given the
+// difficulties involved (Koenig lookup required, there may not *be* an abs()
+// defined, etc etc).
+template <typename IntType>
+BOOST_CXX14_CONSTEXPR
+inline rational<IntType> abs(const rational<IntType>& r)
+{
+    return r.numerator() >= IntType(0)? r: -r;
+}
+
+namespace integer {
+
+template <typename IntType>
+struct gcd_evaluator< rational<IntType> >
+{
+    typedef rational<IntType> result_type,
+                              first_argument_type, second_argument_type;
+    result_type operator() (  first_argument_type const &a
+                           , second_argument_type const &b
+                           ) const
+    {
+        return result_type(integer::gcd(a.numerator(), b.numerator()),
+                           integer::lcm(a.denominator(), b.denominator()));
+    }
+};
+
+template <typename IntType>
+struct lcm_evaluator< rational<IntType> >
+{
+    typedef rational<IntType> result_type,
+                              first_argument_type, second_argument_type;
+    result_type operator() (  first_argument_type const &a
+                           , second_argument_type const &b
+                           ) const
+    {
+        return result_type(integer::lcm(a.numerator(), b.numerator()),
+                           integer::gcd(a.denominator(), b.denominator()));
+    }
+};
+
+} // namespace integer
+
+} // namespace boost
+
+#endif  // BOOST_RATIONAL_HPP