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- // Boost rational.hpp header file ------------------------------------------//
- // (C) Copyright Paul Moore 1999. Permission to copy, use, modify, sell and
- // distribute this software is granted provided this copyright notice appears
- // in all copies. This software is provided "as is" without express or
- // implied warranty, and with no claim as to its suitability for any purpose.
- // boostinspect:nolicense (don't complain about the lack of a Boost license)
- // (Paul Moore hasn't been in contact for years, so there's no way to change the
- // license.)
- // See http://www.boost.org/libs/rational for documentation.
- // Credits:
- // Thanks to the boost mailing list in general for useful comments.
- // Particular contributions included:
- // Andrew D Jewell, for reminding me to take care to avoid overflow
- // Ed Brey, for many comments, including picking up on some dreadful typos
- // Stephen Silver contributed the test suite and comments on user-defined
- // IntType
- // Nickolay Mladenov, for the implementation of operator+=
- // Revision History
- // 02 Sep 13 Remove unneeded forward declarations; tweak private helper
- // function (Daryle Walker)
- // 30 Aug 13 Improve exception safety of "assign"; start modernizing I/O code
- // (Daryle Walker)
- // 27 Aug 13 Add cross-version constructor template, plus some private helper
- // functions; add constructor to exception class to take custom
- // messages (Daryle Walker)
- // 25 Aug 13 Add constexpr qualification wherever possible (Daryle Walker)
- // 05 May 12 Reduced use of implicit gcd (Mario Lang)
- // 05 Nov 06 Change rational_cast to not depend on division between different
- // types (Daryle Walker)
- // 04 Nov 06 Off-load GCD and LCM to Boost.Integer; add some invariant checks;
- // add std::numeric_limits<> requirement to help GCD (Daryle Walker)
- // 31 Oct 06 Recoded both operator< to use round-to-negative-infinity
- // divisions; the rational-value version now uses continued fraction
- // expansion to avoid overflows, for bug #798357 (Daryle Walker)
- // 20 Oct 06 Fix operator bool_type for CW 8.3 (Joaquín M López Muñoz)
- // 18 Oct 06 Use EXPLICIT_TEMPLATE_TYPE helper macros from Boost.Config
- // (Joaquín M López Muñoz)
- // 27 Dec 05 Add Boolean conversion operator (Daryle Walker)
- // 28 Sep 02 Use _left versions of operators from operators.hpp
- // 05 Jul 01 Recode gcd(), avoiding std::swap (Helmut Zeisel)
- // 03 Mar 01 Workarounds for Intel C++ 5.0 (David Abrahams)
- // 05 Feb 01 Update operator>> to tighten up input syntax
- // 05 Feb 01 Final tidy up of gcd code prior to the new release
- // 27 Jan 01 Recode abs() without relying on abs(IntType)
- // 21 Jan 01 Include Nickolay Mladenov's operator+= algorithm,
- // tidy up a number of areas, use newer features of operators.hpp
- // (reduces space overhead to zero), add operator!,
- // introduce explicit mixed-mode arithmetic operations
- // 12 Jan 01 Include fixes to handle a user-defined IntType better
- // 19 Nov 00 Throw on divide by zero in operator /= (John (EBo) David)
- // 23 Jun 00 Incorporate changes from Mark Rodgers for Borland C++
- // 22 Jun 00 Change _MSC_VER to BOOST_MSVC so other compilers are not
- // affected (Beman Dawes)
- // 6 Mar 00 Fix operator-= normalization, #include <string> (Jens Maurer)
- // 14 Dec 99 Modifications based on comments from the boost list
- // 09 Dec 99 Initial Version (Paul Moore)
- #ifndef BOOST_RATIONAL_HPP
- #define BOOST_RATIONAL_HPP
- #include <boost/config.hpp> // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC, etc
- #ifndef BOOST_NO_IOSTREAM
- #include <iomanip> // for std::setw
- #include <ios> // for std::noskipws, streamsize
- #include <istream> // for std::istream
- #include <ostream> // for std::ostream
- #include <sstream> // for std::ostringstream
- #endif
- #include <cstddef> // for NULL
- #include <stdexcept> // for std::domain_error
- #include <string> // for std::string implicit constructor
- #include <boost/operators.hpp> // for boost::addable etc
- #include <cstdlib> // for std::abs
- #include <boost/call_traits.hpp> // for boost::call_traits
- #include <boost/detail/workaround.hpp> // for BOOST_WORKAROUND
- #include <boost/assert.hpp> // for BOOST_ASSERT
- #include <boost/integer/common_factor_rt.hpp> // for boost::integer::gcd, lcm
- #include <limits> // for std::numeric_limits
- #include <boost/static_assert.hpp> // for BOOST_STATIC_ASSERT
- #include <boost/throw_exception.hpp>
- #include <boost/utility/enable_if.hpp>
- #include <boost/type_traits/is_convertible.hpp>
- #include <boost/type_traits/is_class.hpp>
- #include <boost/type_traits/is_same.hpp>
- #include <boost/type_traits/is_array.hpp>
- // Control whether depreciated GCD and LCM functions are included (default: yes)
- #ifndef BOOST_CONTROL_RATIONAL_HAS_GCD
- #define BOOST_CONTROL_RATIONAL_HAS_GCD 1
- #endif
- namespace boost {
- #if BOOST_CONTROL_RATIONAL_HAS_GCD
- template <typename IntType>
- IntType gcd(IntType n, IntType m)
- {
- // Defer to the version in Boost.Integer
- return integer::gcd( n, m );
- }
- template <typename IntType>
- IntType lcm(IntType n, IntType m)
- {
- // Defer to the version in Boost.Integer
- return integer::lcm( n, m );
- }
- #endif // BOOST_CONTROL_RATIONAL_HAS_GCD
- namespace rational_detail{
- template <class FromInt, class ToInt, typename Enable = void>
- struct is_compatible_integer;
- template <class FromInt, class ToInt>
- struct is_compatible_integer<FromInt, ToInt, typename enable_if_c<!is_array<FromInt>::value>::type>
- {
- BOOST_STATIC_CONSTANT(bool, value = ((std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
- && (std::numeric_limits<FromInt>::digits <= std::numeric_limits<ToInt>::digits)
- && (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
- && ((std::numeric_limits<FromInt>::is_signed == false) || (std::numeric_limits<ToInt>::is_signed == true))
- && is_convertible<FromInt, ToInt>::value)
- || is_same<FromInt, ToInt>::value)
- || (is_class<ToInt>::value && is_class<FromInt>::value && is_convertible<FromInt, ToInt>::value));
- };
- template <class FromInt, class ToInt>
- struct is_compatible_integer<FromInt, ToInt, typename enable_if_c<is_array<FromInt>::value>::type>
- {
- BOOST_STATIC_CONSTANT(bool, value = false);
- };
- template <class FromInt, class ToInt, typename Enable = void>
- struct is_backward_compatible_integer;
- template <class FromInt, class ToInt>
- struct is_backward_compatible_integer<FromInt, ToInt, typename enable_if_c<!is_array<FromInt>::value>::type>
- {
- BOOST_STATIC_CONSTANT(bool, value = (std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
- && !is_compatible_integer<FromInt, ToInt>::value
- && (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
- && is_convertible<FromInt, ToInt>::value));
- };
- template <class FromInt, class ToInt>
- struct is_backward_compatible_integer<FromInt, ToInt, typename enable_if_c<is_array<FromInt>::value>::type>
- {
- BOOST_STATIC_CONSTANT(bool, value = false);
- };
- }
- class bad_rational : public std::domain_error
- {
- public:
- explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}
- explicit bad_rational( char const *what ) : std::domain_error( what ) {}
- };
- template <typename IntType>
- class rational
- {
- // Class-wide pre-conditions
- BOOST_STATIC_ASSERT( ::std::numeric_limits<IntType>::is_specialized );
- // Helper types
- typedef typename boost::call_traits<IntType>::param_type param_type;
- struct helper { IntType parts[2]; };
- typedef IntType (helper::* bool_type)[2];
- public:
- // Component type
- typedef IntType int_type;
- BOOST_CONSTEXPR
- rational() : num(0), den(1) {}
- template <class T>//, typename enable_if_c<!is_array<T>::value>::type>
- BOOST_CONSTEXPR rational(const T& n, typename enable_if_c<
- rational_detail::is_compatible_integer<T, IntType>::value
- >::type const* = 0) : num(n), den(1) {}
- template <class T, class U>
- BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
- rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value
- >::type const* = 0) : num(n), den(d) {
- normalize();
- }
- template < typename NewType >
- BOOST_CONSTEXPR explicit
- rational(rational<NewType> const &r, typename enable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
- : num(r.numerator()), den(is_normalized(int_type(r.numerator()),
- int_type(r.denominator())) ? r.denominator() :
- (BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
- template < typename NewType >
- BOOST_CONSTEXPR explicit
- rational(rational<NewType> const &r, typename disable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
- : num(r.numerator()), den(is_normalized(int_type(r.numerator()),
- int_type(r.denominator())) && is_safe_narrowing_conversion(r.denominator()) && is_safe_narrowing_conversion(r.numerator()) ? r.denominator() :
- (BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
- // Default copy constructor and assignment are fine
- // Add assignment from IntType
- template <class T>
- BOOST_CXX14_CONSTEXPR typename enable_if_c<
- rational_detail::is_compatible_integer<T, IntType>::value, rational &
- >::type operator=(const T& n) { return assign(static_cast<IntType>(n), static_cast<IntType>(1)); }
- // Assign in place
- template <class T, class U>
- BOOST_CXX14_CONSTEXPR typename enable_if_c<
- rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value, rational &
- >::type assign(const T& n, const U& d)
- {
- return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
- }
- //
- // The following overloads should probably *not* be provided -
- // but are provided for backwards compatibity reasons only.
- // These allow for construction/assignment from types that
- // are wider than IntType only if there is an implicit
- // conversion from T to IntType, they will throw a bad_rational
- // if the conversion results in loss of precision or undefined behaviour.
- //
- template <class T>//, typename enable_if_c<!is_array<T>::value>::type>
- BOOST_CXX14_CONSTEXPR rational(const T& n, typename enable_if_c<
- rational_detail::is_backward_compatible_integer<T, IntType>::value
- >::type const* = 0)
- {
- assign(n, static_cast<T>(1));
- }
- template <class T, class U>
- BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
- (!rational_detail::is_compatible_integer<T, IntType>::value
- || !rational_detail::is_compatible_integer<U, IntType>::value)
- && std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
- && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
- && is_convertible<T, IntType>::value &&
- std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
- && (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
- && is_convertible<U, IntType>::value
- >::type const* = 0)
- {
- assign(n, d);
- }
- template <class T>
- BOOST_CXX14_CONSTEXPR typename enable_if_c<
- std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
- && !rational_detail::is_compatible_integer<T, IntType>::value
- && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
- && is_convertible<T, IntType>::value,
- rational &
- >::type operator=(const T& n) { return assign(n, static_cast<T>(1)); }
- template <class T, class U>
- BOOST_CXX14_CONSTEXPR typename enable_if_c<
- (!rational_detail::is_compatible_integer<T, IntType>::value
- || !rational_detail::is_compatible_integer<U, IntType>::value)
- && std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
- && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
- && is_convertible<T, IntType>::value &&
- std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
- && (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
- && is_convertible<U, IntType>::value,
- rational &
- >::type assign(const T& n, const U& d)
- {
- if(!is_safe_narrowing_conversion(n) || !is_safe_narrowing_conversion(d))
- BOOST_THROW_EXCEPTION(bad_rational());
- return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
- }
- // Access to representation
- BOOST_CONSTEXPR
- const IntType& numerator() const { return num; }
- BOOST_CONSTEXPR
- const IntType& denominator() const { return den; }
- // Arithmetic assignment operators
- BOOST_CXX14_CONSTEXPR rational& operator+= (const rational& r);
- BOOST_CXX14_CONSTEXPR rational& operator-= (const rational& r);
- BOOST_CXX14_CONSTEXPR rational& operator*= (const rational& r);
- BOOST_CXX14_CONSTEXPR rational& operator/= (const rational& r);
- template <class T>
- BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator+= (const T& i)
- {
- num += i * den;
- return *this;
- }
- template <class T>
- BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator-= (const T& i)
- {
- num -= i * den;
- return *this;
- }
- template <class T>
- BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator*= (const T& i)
- {
- // Avoid overflow and preserve normalization
- IntType gcd = integer::gcd(static_cast<IntType>(i), den);
- num *= i / gcd;
- den /= gcd;
- return *this;
- }
- template <class T>
- BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator/= (const T& i)
- {
- // Avoid repeated construction
- IntType const zero(0);
- if(i == zero) BOOST_THROW_EXCEPTION(bad_rational());
- if(num == zero) return *this;
- // Avoid overflow and preserve normalization
- IntType const gcd = integer::gcd(num, static_cast<IntType>(i));
- num /= gcd;
- den *= i / gcd;
- if(den < zero) {
- num = -num;
- den = -den;
- }
- return *this;
- }
- // Increment and decrement
- BOOST_CXX14_CONSTEXPR const rational& operator++() { num += den; return *this; }
- BOOST_CXX14_CONSTEXPR const rational& operator--() { num -= den; return *this; }
- BOOST_CXX14_CONSTEXPR rational operator++(int)
- {
- rational t(*this);
- ++(*this);
- return t;
- }
- BOOST_CXX14_CONSTEXPR rational operator--(int)
- {
- rational t(*this);
- --(*this);
- return t;
- }
- // Operator not
- BOOST_CONSTEXPR
- bool operator!() const { return !num; }
- // Boolean conversion
-
- #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
- // The "ISO C++ Template Parser" option in CW 8.3 chokes on the
- // following, hence we selectively disable that option for the
- // offending memfun.
- #pragma parse_mfunc_templ off
- #endif
- BOOST_CONSTEXPR
- operator bool_type() const { return operator !() ? 0 : &helper::parts; }
- #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
- #pragma parse_mfunc_templ reset
- #endif
- // Comparison operators
- BOOST_CXX14_CONSTEXPR bool operator< (const rational& r) const;
- BOOST_CXX14_CONSTEXPR bool operator> (const rational& r) const { return r < *this; }
- BOOST_CONSTEXPR
- bool operator== (const rational& r) const;
- template <class T>
- BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator< (const T& i) const
- {
- // Avoid repeated construction
- int_type const zero(0);
- // Break value into mixed-fraction form, w/ always-nonnegative remainder
- BOOST_ASSERT(this->den > zero);
- int_type q = this->num / this->den, r = this->num % this->den;
- while(r < zero) { r += this->den; --q; }
- // Compare with just the quotient, since the remainder always bumps the
- // value up. [Since q = floor(n/d), and if n/d < i then q < i, if n/d == i
- // then q == i, if n/d == i + r/d then q == i, and if n/d >= i + 1 then
- // q >= i + 1 > i; therefore n/d < i iff q < i.]
- return q < i;
- }
- template <class T>
- BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator>(const T& i) const
- {
- return operator==(i) ? false : !operator<(i);
- }
- template <class T>
- BOOST_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator== (const T& i) const
- {
- return ((den == IntType(1)) && (num == i));
- }
- private:
- // Implementation - numerator and denominator (normalized).
- // Other possibilities - separate whole-part, or sign, fields?
- IntType num;
- IntType den;
- // Helper functions
- static BOOST_CONSTEXPR
- int_type inner_gcd( param_type a, param_type b, int_type const &zero =
- int_type(0) )
- { return b == zero ? a : inner_gcd(b, a % b, zero); }
- static BOOST_CONSTEXPR
- int_type inner_abs( param_type x, int_type const &zero = int_type(0) )
- { return x < zero ? -x : +x; }
- // Representation note: Fractions are kept in normalized form at all
- // times. normalized form is defined as gcd(num,den) == 1 and den > 0.
- // In particular, note that the implementation of abs() below relies
- // on den always being positive.
- BOOST_CXX14_CONSTEXPR bool test_invariant() const;
- BOOST_CXX14_CONSTEXPR void normalize();
- static BOOST_CONSTEXPR
- bool is_normalized( param_type n, param_type d, int_type const &zero =
- int_type(0), int_type const &one = int_type(1) )
- {
- return d > zero && ( n != zero || d == one ) && inner_abs( inner_gcd(n,
- d, zero), zero ) == one;
- }
- //
- // Conversion checks:
- //
- // (1) From an unsigned type with more digits than IntType:
- //
- template <class T>
- BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
- {
- return val < (T(1) << std::numeric_limits<IntType>::digits);
- }
- //
- // (2) From a signed type with more digits than IntType, and IntType also signed:
- //
- template <class T>
- BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T& val)
- {
- // Note that this check assumes IntType has a 2's complement representation,
- // we don't want to try to convert a std::numeric_limits<IntType>::min() to
- // a T because that conversion may not be allowed (this happens when IntType
- // is from Boost.Multiprecision).
- return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= -(T(1) << std::numeric_limits<IntType>::digits));
- }
- //
- // (3) From a signed type with more digits than IntType, and IntType unsigned:
- //
- template <class T>
- BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
- {
- return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= 0);
- }
- //
- // (4) From a signed type with fewer digits than IntType, and IntType unsigned:
- //
- template <class T>
- BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
- {
- return val >= 0;
- }
- //
- // (5) From an unsigned type with fewer digits than IntType, and IntType signed:
- //
- template <class T>
- BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T&)
- {
- return true;
- }
- //
- // (6) From an unsigned type with fewer digits than IntType, and IntType unsigned:
- //
- template <class T>
- BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T&)
- {
- return true;
- }
- //
- // (7) From an signed type with fewer digits than IntType, and IntType signed:
- //
- template <class T>
- BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T&)
- {
- return true;
- }
- };
- // Unary plus and minus
- template <typename IntType>
- BOOST_CONSTEXPR
- inline rational<IntType> operator+ (const rational<IntType>& r)
- {
- return r;
- }
- template <typename IntType>
- BOOST_CXX14_CONSTEXPR
- inline rational<IntType> operator- (const rational<IntType>& r)
- {
- return rational<IntType>(static_cast<IntType>(-r.numerator()), r.denominator());
- }
- // Arithmetic assignment operators
- template <typename IntType>
- BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator+= (const rational<IntType>& r)
- {
- // This calculation avoids overflow, and minimises the number of expensive
- // calculations. Thanks to Nickolay Mladenov for this algorithm.
- //
- // Proof:
- // We have to compute a/b + c/d, where gcd(a,b)=1 and gcd(b,c)=1.
- // Let g = gcd(b,d), and b = b1*g, d=d1*g. Then gcd(b1,d1)=1
- //
- // The result is (a*d1 + c*b1) / (b1*d1*g).
- // Now we have to normalize this ratio.
- // Let's assume h | gcd((a*d1 + c*b1), (b1*d1*g)), and h > 1
- // If h | b1 then gcd(h,d1)=1 and hence h|(a*d1+c*b1) => h|a.
- // But since gcd(a,b1)=1 we have h=1.
- // Similarly h|d1 leads to h=1.
- // So we have that h | gcd((a*d1 + c*b1) , (b1*d1*g)) => h|g
- // Finally we have gcd((a*d1 + c*b1), (b1*d1*g)) = gcd((a*d1 + c*b1), g)
- // Which proves that instead of normalizing the result, it is better to
- // divide num and den by gcd((a*d1 + c*b1), g)
- // Protect against self-modification
- IntType r_num = r.num;
- IntType r_den = r.den;
- IntType g = integer::gcd(den, r_den);
- den /= g; // = b1 from the calculations above
- num = num * (r_den / g) + r_num * den;
- g = integer::gcd(num, g);
- num /= g;
- den *= r_den/g;
- return *this;
- }
- template <typename IntType>
- BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator-= (const rational<IntType>& r)
- {
- // Protect against self-modification
- IntType r_num = r.num;
- IntType r_den = r.den;
- // This calculation avoids overflow, and minimises the number of expensive
- // calculations. It corresponds exactly to the += case above
- IntType g = integer::gcd(den, r_den);
- den /= g;
- num = num * (r_den / g) - r_num * den;
- g = integer::gcd(num, g);
- num /= g;
- den *= r_den/g;
- return *this;
- }
- template <typename IntType>
- BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator*= (const rational<IntType>& r)
- {
- // Protect against self-modification
- IntType r_num = r.num;
- IntType r_den = r.den;
- // Avoid overflow and preserve normalization
- IntType gcd1 = integer::gcd(num, r_den);
- IntType gcd2 = integer::gcd(r_num, den);
- num = (num/gcd1) * (r_num/gcd2);
- den = (den/gcd2) * (r_den/gcd1);
- return *this;
- }
- template <typename IntType>
- BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator/= (const rational<IntType>& r)
- {
- // Protect against self-modification
- IntType r_num = r.num;
- IntType r_den = r.den;
- // Avoid repeated construction
- IntType zero(0);
- // Trap division by zero
- if (r_num == zero)
- BOOST_THROW_EXCEPTION(bad_rational());
- if (num == zero)
- return *this;
- // Avoid overflow and preserve normalization
- IntType gcd1 = integer::gcd(num, r_num);
- IntType gcd2 = integer::gcd(r_den, den);
- num = (num/gcd1) * (r_den/gcd2);
- den = (den/gcd2) * (r_num/gcd1);
- if (den < zero) {
- num = -num;
- den = -den;
- }
- return *this;
- }
- //
- // Non-member operators: previously these were provided by Boost.Operator, but these had a number of
- // drawbacks, most notably, that in order to allow inter-operability with IntType code such as this:
- //
- // rational<int> r(3);
- // assert(r == 3.5); // compiles and passes!!
- //
- // Happens to be allowed as well :-(
- //
- // There are three possible cases for each operator:
- // 1) rational op rational.
- // 2) rational op integer
- // 3) integer op rational
- // Cases (1) and (2) are folded into the one function.
- //
- template <class IntType, class Arg>
- BOOST_CXX14_CONSTEXPR
- inline typename boost::enable_if_c <
- rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
- operator + (const rational<IntType>& a, const Arg& b)
- {
- rational<IntType> t(a);
- return t += b;
- }
- template <class Arg, class IntType>
- BOOST_CXX14_CONSTEXPR
- inline typename boost::enable_if_c <
- rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
- operator + (const Arg& b, const rational<IntType>& a)
- {
- rational<IntType> t(a);
- return t += b;
- }
- template <class IntType, class Arg>
- BOOST_CXX14_CONSTEXPR
- inline typename boost::enable_if_c <
- rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
- operator - (const rational<IntType>& a, const Arg& b)
- {
- rational<IntType> t(a);
- return t -= b;
- }
- template <class Arg, class IntType>
- BOOST_CXX14_CONSTEXPR
- inline typename boost::enable_if_c <
- rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
- operator - (const Arg& b, const rational<IntType>& a)
- {
- rational<IntType> t(a);
- return -(t -= b);
- }
- template <class IntType, class Arg>
- BOOST_CXX14_CONSTEXPR
- inline typename boost::enable_if_c <
- rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
- operator * (const rational<IntType>& a, const Arg& b)
- {
- rational<IntType> t(a);
- return t *= b;
- }
- template <class Arg, class IntType>
- BOOST_CXX14_CONSTEXPR
- inline typename boost::enable_if_c <
- rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
- operator * (const Arg& b, const rational<IntType>& a)
- {
- rational<IntType> t(a);
- return t *= b;
- }
- template <class IntType, class Arg>
- BOOST_CXX14_CONSTEXPR
- inline typename boost::enable_if_c <
- rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
- operator / (const rational<IntType>& a, const Arg& b)
- {
- rational<IntType> t(a);
- return t /= b;
- }
- template <class Arg, class IntType>
- BOOST_CXX14_CONSTEXPR
- inline typename boost::enable_if_c <
- rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
- operator / (const Arg& b, const rational<IntType>& a)
- {
- rational<IntType> t(b);
- return t /= a;
- }
- template <class IntType, class Arg>
- BOOST_CXX14_CONSTEXPR
- inline typename boost::enable_if_c <
- rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
- operator <= (const rational<IntType>& a, const Arg& b)
- {
- return !(a > 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
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