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708 lines
23 KiB
C++
708 lines
23 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.Math; 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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// 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.Math
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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.Math
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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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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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less_than_comparable < rational<IntType>,
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equality_comparable < rational<IntType>,
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less_than_comparable2 < rational<IntType>, IntType,
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equality_comparable2 < rational<IntType>, IntType,
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addable < rational<IntType>,
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subtractable < rational<IntType>,
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multipliable < rational<IntType>,
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dividable < rational<IntType>,
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addable2 < rational<IntType>, IntType,
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subtractable2 < rational<IntType>, IntType,
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subtractable2_left < rational<IntType>, IntType,
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multipliable2 < rational<IntType>, IntType,
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dividable2 < rational<IntType>, IntType,
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dividable2_left < rational<IntType>, IntType,
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incrementable < rational<IntType>,
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decrementable < rational<IntType>
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> > > > > > > > > > > > > > > >
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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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BOOST_CONSTEXPR
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rational(param_type n) : num(n), den(1) {}
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rational(param_type n, param_type d) : num(n), den(d) { normalize(); }
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#ifndef BOOST_NO_MEMBER_TEMPLATES
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template < typename NewType >
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BOOST_CONSTEXPR explicit
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rational(rational<NewType> const &r)
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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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#endif
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// Default copy constructor and assignment are fine
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// Add assignment from IntType
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rational& operator=(param_type i) { num = i; den = 1; return *this; }
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// Assign in place
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rational& assign(param_type n, param_type d);
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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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rational& operator+= (const rational& r);
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rational& operator-= (const rational& r);
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rational& operator*= (const rational& r);
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rational& operator/= (const rational& r);
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rational& operator+= (param_type i) { num += i * den; return *this; }
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rational& operator-= (param_type i) { num -= i * den; return *this; }
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rational& operator*= (param_type i);
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rational& operator/= (param_type i);
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// Increment and decrement
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const rational& operator++() { num += den; return *this; }
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const rational& operator--() { num -= den; return *this; }
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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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bool operator< (const rational& r) const;
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BOOST_CONSTEXPR
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bool operator== (const rational& r) const;
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bool operator< (param_type i) const;
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bool operator> (param_type i) const;
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BOOST_CONSTEXPR
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bool operator== (param_type i) const;
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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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bool test_invariant() const;
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void normalize();
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static BOOST_CONSTEXPR
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bool is_normalized( param_type n, param_type d, int_type const &zero =
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int_type(0), int_type const &one = int_type(1) )
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{
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return d > zero && ( n != zero || d == one ) && inner_abs( inner_gcd(n,
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d, zero), zero ) == one;
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}
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};
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// Assign in place
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template <typename IntType>
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inline rational<IntType>& rational<IntType>::assign(param_type n, param_type d)
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{
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return *this = rational( n, d );
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}
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// Unary plus and minus
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template <typename IntType>
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BOOST_CONSTEXPR
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inline rational<IntType> operator+ (const rational<IntType>& r)
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{
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return r;
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}
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template <typename IntType>
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inline rational<IntType> operator- (const rational<IntType>& r)
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{
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return rational<IntType>(-r.numerator(), r.denominator());
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}
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// Arithmetic assignment operators
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template <typename IntType>
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rational<IntType>& rational<IntType>::operator+= (const rational<IntType>& r)
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{
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// This calculation avoids overflow, and minimises the number of expensive
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// calculations. Thanks to Nickolay Mladenov for this algorithm.
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//
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// Proof:
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// We have to compute a/b + c/d, where gcd(a,b)=1 and gcd(b,c)=1.
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// Let g = gcd(b,d), and b = b1*g, d=d1*g. Then gcd(b1,d1)=1
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//
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// The result is (a*d1 + c*b1) / (b1*d1*g).
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// Now we have to normalize this ratio.
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// Let's assume h | gcd((a*d1 + c*b1), (b1*d1*g)), and h > 1
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// If h | b1 then gcd(h,d1)=1 and hence h|(a*d1+c*b1) => h|a.
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// But since gcd(a,b1)=1 we have h=1.
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// Similarly h|d1 leads to h=1.
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// So we have that h | gcd((a*d1 + c*b1) , (b1*d1*g)) => h|g
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// Finally we have gcd((a*d1 + c*b1), (b1*d1*g)) = gcd((a*d1 + c*b1), g)
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// Which proves that instead of normalizing the result, it is better to
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// divide num and den by gcd((a*d1 + c*b1), g)
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// Protect against self-modification
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IntType r_num = r.num;
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IntType r_den = r.den;
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IntType g = integer::gcd(den, r_den);
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den /= g; // = b1 from the calculations above
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num = num * (r_den / g) + r_num * den;
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g = integer::gcd(num, g);
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num /= g;
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den *= r_den/g;
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return *this;
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}
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template <typename IntType>
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rational<IntType>& rational<IntType>::operator-= (const rational<IntType>& r)
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{
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// Protect against self-modification
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IntType r_num = r.num;
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IntType r_den = r.den;
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// This calculation avoids overflow, and minimises the number of expensive
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// calculations. It corresponds exactly to the += case above
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IntType g = integer::gcd(den, r_den);
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den /= g;
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num = num * (r_den / g) - r_num * den;
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g = integer::gcd(num, g);
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num /= g;
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den *= r_den/g;
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return *this;
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}
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template <typename IntType>
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rational<IntType>& rational<IntType>::operator*= (const rational<IntType>& r)
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{
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// Protect against self-modification
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IntType r_num = r.num;
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IntType r_den = r.den;
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// Avoid overflow and preserve normalization
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IntType gcd1 = integer::gcd(num, r_den);
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IntType gcd2 = integer::gcd(r_num, den);
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num = (num/gcd1) * (r_num/gcd2);
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den = (den/gcd2) * (r_den/gcd1);
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return *this;
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}
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template <typename IntType>
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rational<IntType>& rational<IntType>::operator/= (const rational<IntType>& r)
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{
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// Protect against self-modification
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IntType r_num = r.num;
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IntType r_den = r.den;
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// Avoid repeated construction
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IntType zero(0);
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// Trap division by zero
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if (r_num == zero)
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BOOST_THROW_EXCEPTION(bad_rational());
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if (num == zero)
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return *this;
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// Avoid overflow and preserve normalization
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IntType gcd1 = integer::gcd(num, r_num);
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IntType gcd2 = integer::gcd(r_den, den);
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num = (num/gcd1) * (r_den/gcd2);
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den = (den/gcd2) * (r_num/gcd1);
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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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// Mixed-mode operators
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template <typename IntType>
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inline rational<IntType>&
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rational<IntType>::operator*= (param_type i)
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{
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// Avoid overflow and preserve normalization
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IntType gcd = integer::gcd(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 <typename IntType>
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rational<IntType>&
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rational<IntType>::operator/= (param_type 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, 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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// Comparison operators
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template <typename IntType>
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bool rational<IntType>::operator< (const rational<IntType>& r) 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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// This should really be a class-wide invariant. The reason for these
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// checks is that for 2's complement systems, INT_MIN has no corresponding
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// positive, so negating it during normalization keeps it INT_MIN, which
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// is bad for later calculations that assume a positive denominator.
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BOOST_ASSERT( this->den > zero );
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BOOST_ASSERT( r.den > zero );
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// Determine relative order by expanding each value to its simple continued
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// fraction representation using the Euclidian GCD algorithm.
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struct { int_type n, d, q, r; }
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ts = { this->num, this->den, static_cast<int_type>(this->num / this->den),
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static_cast<int_type>(this->num % this->den) },
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rs = { r.num, r.den, static_cast<int_type>(r.num / r.den),
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static_cast<int_type>(r.num % r.den) };
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unsigned reverse = 0u;
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// Normalize negative moduli by repeatedly adding the (positive) denominator
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// and decrementing the quotient. Later cycles should have all positive
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// values, so this only has to be done for the first cycle. (The rules of
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// C++ require a nonnegative quotient & remainder for a nonnegative dividend
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// & positive divisor.)
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while ( ts.r < zero ) { ts.r += ts.d; --ts.q; }
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while ( rs.r < zero ) { rs.r += rs.d; --rs.q; }
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// Loop through and compare each variable's continued-fraction components
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for ( ;; )
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{
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// The quotients of the current cycle are the continued-fraction
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// components. Comparing two c.f. is comparing their sequences,
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// stopping at the first difference.
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if ( ts.q != rs.q )
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{
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// Since reciprocation changes the relative order of two variables,
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// and c.f. use reciprocals, the less/greater-than test reverses
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// after each index. (Start w/ non-reversed @ whole-number place.)
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return reverse ? ts.q > rs.q : ts.q < rs.q;
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}
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// Prepare the next cycle
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reverse ^= 1u;
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if ( (ts.r == zero) || (rs.r == zero) )
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{
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// At least one variable's c.f. expansion has ended
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break;
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}
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ts.n = ts.d; ts.d = ts.r;
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ts.q = ts.n / ts.d; ts.r = ts.n % ts.d;
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rs.n = rs.d; rs.d = rs.r;
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rs.q = rs.n / rs.d; rs.r = rs.n % rs.d;
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}
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// Compare infinity-valued components for otherwise equal sequences
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if ( ts.r == rs.r )
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{
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// 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>
|
|
bool rational<IntType>::operator< (param_type 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 <typename IntType>
|
|
bool rational<IntType>::operator> (param_type i) const
|
|
{
|
|
return operator==(i)? false: !operator<(i);
|
|
}
|
|
|
|
template <typename IntType>
|
|
BOOST_CONSTEXPR
|
|
inline bool rational<IntType>::operator== (const rational<IntType>& r) const
|
|
{
|
|
return ((num == r.num) && (den == r.den));
|
|
}
|
|
|
|
template <typename IntType>
|
|
BOOST_CONSTEXPR
|
|
inline bool rational<IntType>::operator== (param_type i) const
|
|
{
|
|
return ((den == IntType(1)) && (num == i));
|
|
}
|
|
|
|
// Invariant check
|
|
template <typename IntType>
|
|
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>
|
|
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;
|
|
|
|
// Ensure that the denominator is positive
|
|
if (den < zero) {
|
|
num = -num;
|
|
den = -den;
|
|
}
|
|
|
|
// ...But acknowledge that the previous step doesn't always work.
|
|
// (Nominally, this should be done before the mutating steps, but this
|
|
// member function is only called during the constructor, so we never have
|
|
// to worry about zombie objects.)
|
|
if (den < zero)
|
|
BOOST_THROW_EXCEPTION(bad_rational("bad rational: non-zero singular denominator"));
|
|
|
|
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>
|
|
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
|