1047 lines
39 KiB
C++
1047 lines
39 KiB
C++
// Boost rational.hpp header file ------------------------------------------//
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// (C) Copyright Paul Moore 1999. Permission to copy, use, modify, sell and
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// distribute this software is granted provided this copyright notice appears
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// in all copies. This software is provided "as is" without express or
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// implied warranty, and with no claim as to its suitability for any purpose.
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// boostinspect:nolicense (don't complain about the lack of a Boost license)
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// (Paul Moore hasn't been in contact for years, so there's no way to change the
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// license.)
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// See http://www.boost.org/libs/rational for documentation.
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// Credits:
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// Thanks to the boost mailing list in general for useful comments.
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// Particular contributions included:
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// Andrew D Jewell, for reminding me to take care to avoid overflow
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// Ed Brey, for many comments, including picking up on some dreadful typos
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// Stephen Silver contributed the test suite and comments on user-defined
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// IntType
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// Nickolay Mladenov, for the implementation of operator+=
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// Revision History
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// 02 Sep 13 Remove unneeded forward declarations; tweak private helper
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// function (Daryle Walker)
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// 30 Aug 13 Improve exception safety of "assign"; start modernizing I/O code
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// (Daryle Walker)
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// 27 Aug 13 Add cross-version constructor template, plus some private helper
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// functions; add constructor to exception class to take custom
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// messages (Daryle Walker)
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// 25 Aug 13 Add constexpr qualification wherever possible (Daryle Walker)
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// 05 May 12 Reduced use of implicit gcd (Mario Lang)
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// 05 Nov 06 Change rational_cast to not depend on division between different
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// types (Daryle Walker)
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// 04 Nov 06 Off-load GCD and LCM to Boost.Integer; add some invariant checks;
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// add std::numeric_limits<> requirement to help GCD (Daryle Walker)
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// 31 Oct 06 Recoded both operator< to use round-to-negative-infinity
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// divisions; the rational-value version now uses continued fraction
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// expansion to avoid overflows, for bug #798357 (Daryle Walker)
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// 20 Oct 06 Fix operator bool_type for CW 8.3 (Joaquín M López Muñoz)
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// 18 Oct 06 Use EXPLICIT_TEMPLATE_TYPE helper macros from Boost.Config
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// (Joaquín M López Muñoz)
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// 27 Dec 05 Add Boolean conversion operator (Daryle Walker)
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// 28 Sep 02 Use _left versions of operators from operators.hpp
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// 05 Jul 01 Recode gcd(), avoiding std::swap (Helmut Zeisel)
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// 03 Mar 01 Workarounds for Intel C++ 5.0 (David Abrahams)
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// 05 Feb 01 Update operator>> to tighten up input syntax
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// 05 Feb 01 Final tidy up of gcd code prior to the new release
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// 27 Jan 01 Recode abs() without relying on abs(IntType)
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// 21 Jan 01 Include Nickolay Mladenov's operator+= algorithm,
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// tidy up a number of areas, use newer features of operators.hpp
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// (reduces space overhead to zero), add operator!,
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// introduce explicit mixed-mode arithmetic operations
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// 12 Jan 01 Include fixes to handle a user-defined IntType better
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// 19 Nov 00 Throw on divide by zero in operator /= (John (EBo) David)
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// 23 Jun 00 Incorporate changes from Mark Rodgers for Borland C++
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// 22 Jun 00 Change _MSC_VER to BOOST_MSVC so other compilers are not
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// affected (Beman Dawes)
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// 6 Mar 00 Fix operator-= normalization, #include <string> (Jens Maurer)
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// 14 Dec 99 Modifications based on comments from the boost list
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// 09 Dec 99 Initial Version (Paul Moore)
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#ifndef BOOST_RATIONAL_HPP
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#define BOOST_RATIONAL_HPP
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#include <boost/config.hpp> // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC, etc
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#ifndef BOOST_NO_IOSTREAM
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#include <iomanip> // for std::setw
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#include <ios> // for std::noskipws, streamsize
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#include <istream> // for std::istream
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#include <ostream> // for std::ostream
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#include <sstream> // for std::ostringstream
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#endif
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#include <cstddef> // for NULL
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#include <stdexcept> // for std::domain_error
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#include <string> // for std::string implicit constructor
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#include <boost/operators.hpp> // for boost::addable etc
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#include <cstdlib> // for std::abs
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#include <boost/call_traits.hpp> // for boost::call_traits
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#include <boost/detail/workaround.hpp> // for BOOST_WORKAROUND
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#include <boost/assert.hpp> // for BOOST_ASSERT
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#include <boost/integer/common_factor_rt.hpp> // for boost::integer::gcd, lcm
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#include <limits> // for std::numeric_limits
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#include <boost/static_assert.hpp> // for BOOST_STATIC_ASSERT
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#include <boost/throw_exception.hpp>
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#include <boost/utility/enable_if.hpp>
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#include <boost/type_traits/is_convertible.hpp>
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#include <boost/type_traits/is_class.hpp>
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#include <boost/type_traits/is_same.hpp>
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#include <boost/type_traits/is_array.hpp>
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// Control whether depreciated GCD and LCM functions are included (default: yes)
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#ifndef BOOST_CONTROL_RATIONAL_HAS_GCD
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#define BOOST_CONTROL_RATIONAL_HAS_GCD 1
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#endif
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namespace boost {
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#if BOOST_CONTROL_RATIONAL_HAS_GCD
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template <typename IntType>
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IntType gcd(IntType n, IntType m)
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{
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// Defer to the version in Boost.Integer
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return integer::gcd( n, m );
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}
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template <typename IntType>
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IntType lcm(IntType n, IntType m)
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{
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// Defer to the version in Boost.Integer
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return integer::lcm( n, m );
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}
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#endif // BOOST_CONTROL_RATIONAL_HAS_GCD
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namespace rational_detail{
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template <class FromInt, class ToInt, typename Enable = void>
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struct is_compatible_integer;
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template <class FromInt, class ToInt>
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struct is_compatible_integer<FromInt, ToInt, typename enable_if_c<!is_array<FromInt>::value>::type>
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{
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BOOST_STATIC_CONSTANT(bool, value = ((std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
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&& (std::numeric_limits<FromInt>::digits <= std::numeric_limits<ToInt>::digits)
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&& (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
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&& ((std::numeric_limits<FromInt>::is_signed == false) || (std::numeric_limits<ToInt>::is_signed == true))
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&& is_convertible<FromInt, ToInt>::value)
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|| is_same<FromInt, ToInt>::value)
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|| (is_class<ToInt>::value && is_class<FromInt>::value && is_convertible<FromInt, ToInt>::value));
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};
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template <class FromInt, class ToInt>
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struct is_compatible_integer<FromInt, ToInt, typename enable_if_c<is_array<FromInt>::value>::type>
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{
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BOOST_STATIC_CONSTANT(bool, value = false);
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};
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template <class FromInt, class ToInt, typename Enable = void>
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struct is_backward_compatible_integer;
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template <class FromInt, class ToInt>
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struct is_backward_compatible_integer<FromInt, ToInt, typename enable_if_c<!is_array<FromInt>::value>::type>
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{
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BOOST_STATIC_CONSTANT(bool, value = (std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
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&& !is_compatible_integer<FromInt, ToInt>::value
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&& (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
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&& is_convertible<FromInt, ToInt>::value));
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};
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template <class FromInt, class ToInt>
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struct is_backward_compatible_integer<FromInt, ToInt, typename enable_if_c<is_array<FromInt>::value>::type>
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{
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BOOST_STATIC_CONSTANT(bool, value = false);
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};
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}
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class bad_rational : public std::domain_error
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{
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public:
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explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}
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explicit bad_rational( char const *what ) : std::domain_error( what ) {}
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};
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template <typename IntType>
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class rational
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{
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// Class-wide pre-conditions
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BOOST_STATIC_ASSERT( ::std::numeric_limits<IntType>::is_specialized );
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// Helper types
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typedef typename boost::call_traits<IntType>::param_type param_type;
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struct helper { IntType parts[2]; };
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typedef IntType (helper::* bool_type)[2];
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public:
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// Component type
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typedef IntType int_type;
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BOOST_CONSTEXPR
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rational() : num(0), den(1) {}
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template <class T>//, typename enable_if_c<!is_array<T>::value>::type>
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BOOST_CONSTEXPR rational(const T& n, typename enable_if_c<
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rational_detail::is_compatible_integer<T, IntType>::value
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>::type const* = 0) : num(n), den(1) {}
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template <class T, class U>
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BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
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rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value
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>::type const* = 0) : num(n), den(d) {
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normalize();
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}
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template < typename NewType >
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BOOST_CONSTEXPR explicit
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rational(rational<NewType> const &r, typename enable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
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: num(r.numerator()), den(is_normalized(int_type(r.numerator()),
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int_type(r.denominator())) ? r.denominator() :
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(BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
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template < typename NewType >
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BOOST_CONSTEXPR explicit
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rational(rational<NewType> const &r, typename disable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
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: num(r.numerator()), den(is_normalized(int_type(r.numerator()),
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int_type(r.denominator())) && is_safe_narrowing_conversion(r.denominator()) && is_safe_narrowing_conversion(r.numerator()) ? r.denominator() :
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(BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
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// Default copy constructor and assignment are fine
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// Add assignment from IntType
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template <class T>
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BOOST_CXX14_CONSTEXPR typename enable_if_c<
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rational_detail::is_compatible_integer<T, IntType>::value, rational &
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>::type operator=(const T& n) { return assign(static_cast<IntType>(n), static_cast<IntType>(1)); }
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// Assign in place
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template <class T, class U>
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BOOST_CXX14_CONSTEXPR typename enable_if_c<
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rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value, rational &
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>::type assign(const T& n, const U& d)
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{
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return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
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}
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//
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// The following overloads should probably *not* be provided -
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// but are provided for backwards compatibity reasons only.
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// These allow for construction/assignment from types that
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// are wider than IntType only if there is an implicit
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// conversion from T to IntType, they will throw a bad_rational
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// if the conversion results in loss of precision or undefined behaviour.
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//
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template <class T>//, typename enable_if_c<!is_array<T>::value>::type>
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BOOST_CXX14_CONSTEXPR rational(const T& n, typename enable_if_c<
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rational_detail::is_backward_compatible_integer<T, IntType>::value
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>::type const* = 0)
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{
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assign(n, static_cast<T>(1));
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}
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template <class T, class U>
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BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
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(!rational_detail::is_compatible_integer<T, IntType>::value
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|| !rational_detail::is_compatible_integer<U, IntType>::value)
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&& std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
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&& (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
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&& is_convertible<T, IntType>::value &&
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std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
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&& (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
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&& is_convertible<U, IntType>::value
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>::type const* = 0)
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{
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assign(n, d);
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}
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template <class T>
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BOOST_CXX14_CONSTEXPR typename enable_if_c<
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std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
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&& !rational_detail::is_compatible_integer<T, IntType>::value
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&& (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
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&& is_convertible<T, IntType>::value,
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rational &
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>::type operator=(const T& n) { return assign(n, static_cast<T>(1)); }
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template <class T, class U>
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BOOST_CXX14_CONSTEXPR typename enable_if_c<
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(!rational_detail::is_compatible_integer<T, IntType>::value
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|| !rational_detail::is_compatible_integer<U, IntType>::value)
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&& std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
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&& (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
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&& is_convertible<T, IntType>::value &&
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std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
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&& (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
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&& is_convertible<U, IntType>::value,
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rational &
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>::type assign(const T& n, const U& d)
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{
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if(!is_safe_narrowing_conversion(n) || !is_safe_narrowing_conversion(d))
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BOOST_THROW_EXCEPTION(bad_rational());
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return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
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}
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// Access to representation
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BOOST_CONSTEXPR
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const IntType& numerator() const { return num; }
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BOOST_CONSTEXPR
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const IntType& denominator() const { return den; }
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// Arithmetic assignment operators
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BOOST_CXX14_CONSTEXPR rational& operator+= (const rational& r);
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BOOST_CXX14_CONSTEXPR rational& operator-= (const rational& r);
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BOOST_CXX14_CONSTEXPR rational& operator*= (const rational& r);
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BOOST_CXX14_CONSTEXPR rational& operator/= (const rational& r);
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template <class T>
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BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator+= (const T& i)
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{
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num += i * den;
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return *this;
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}
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template <class T>
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BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator-= (const T& i)
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{
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num -= i * den;
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return *this;
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}
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template <class T>
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BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator*= (const T& i)
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{
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// Avoid overflow and preserve normalization
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IntType gcd = integer::gcd(static_cast<IntType>(i), den);
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num *= i / gcd;
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den /= gcd;
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return *this;
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}
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template <class T>
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BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator/= (const T& i)
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{
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// Avoid repeated construction
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IntType const zero(0);
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if(i == zero) BOOST_THROW_EXCEPTION(bad_rational());
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if(num == zero) return *this;
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// Avoid overflow and preserve normalization
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IntType const gcd = integer::gcd(num, static_cast<IntType>(i));
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num /= gcd;
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den *= i / gcd;
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if(den < zero) {
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num = -num;
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den = -den;
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}
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return *this;
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}
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// Increment and decrement
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BOOST_CXX14_CONSTEXPR const rational& operator++() { num += den; return *this; }
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BOOST_CXX14_CONSTEXPR const rational& operator--() { num -= den; return *this; }
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BOOST_CXX14_CONSTEXPR rational operator++(int)
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{
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rational t(*this);
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++(*this);
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return t;
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}
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BOOST_CXX14_CONSTEXPR rational operator--(int)
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{
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rational t(*this);
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--(*this);
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return t;
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}
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// Operator not
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BOOST_CONSTEXPR
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bool operator!() const { return !num; }
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// Boolean conversion
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#if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
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// The "ISO C++ Template Parser" option in CW 8.3 chokes on the
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// following, hence we selectively disable that option for the
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// offending memfun.
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#pragma parse_mfunc_templ off
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#endif
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BOOST_CONSTEXPR
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operator bool_type() const { return operator !() ? 0 : &helper::parts; }
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#if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
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#pragma parse_mfunc_templ reset
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#endif
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// Comparison operators
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BOOST_CXX14_CONSTEXPR bool operator< (const rational& r) const;
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BOOST_CXX14_CONSTEXPR bool operator> (const rational& r) const { return r < *this; }
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BOOST_CONSTEXPR
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bool operator== (const rational& r) const;
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template <class T>
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BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator< (const T& i) const
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{
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// Avoid repeated construction
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int_type const zero(0);
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// Break value into mixed-fraction form, w/ always-nonnegative remainder
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BOOST_ASSERT(this->den > zero);
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int_type q = this->num / this->den, r = this->num % this->den;
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while(r < zero) { r += this->den; --q; }
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// Compare with just the quotient, since the remainder always bumps the
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// value up. [Since q = floor(n/d), and if n/d < i then q < i, if n/d == i
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// then q == i, if n/d == i + r/d then q == i, and if n/d >= i + 1 then
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// q >= i + 1 > i; therefore n/d < i iff q < i.]
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return q < i;
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}
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template <class T>
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BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator>(const T& i) const
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{
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return operator==(i) ? false : !operator<(i);
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}
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template <class T>
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BOOST_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator== (const T& i) const
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{
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return ((den == IntType(1)) && (num == i));
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}
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private:
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// Implementation - numerator and denominator (normalized).
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// Other possibilities - separate whole-part, or sign, fields?
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IntType num;
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IntType den;
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// Helper functions
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static BOOST_CONSTEXPR
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int_type inner_gcd( param_type a, param_type b, int_type const &zero =
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int_type(0) )
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{ return b == zero ? a : inner_gcd(b, a % b, zero); }
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static BOOST_CONSTEXPR
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int_type inner_abs( param_type x, int_type const &zero = int_type(0) )
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{ return x < zero ? -x : +x; }
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// Representation note: Fractions are kept in normalized form at all
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// times. normalized form is defined as gcd(num,den) == 1 and den > 0.
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// In particular, note that the implementation of abs() below relies
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// on den always being positive.
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BOOST_CXX14_CONSTEXPR bool test_invariant() const;
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BOOST_CXX14_CONSTEXPR void normalize();
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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 > 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 < 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 == 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 > 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 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 == 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
|