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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 )
BOOST_TRY {
r.assign( n, d );
} BOOST_CATCH ( bad_rational & ) { // normalization fail
BOOST_TRY { is.setstate(ios::failbit); }
BOOST_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
BOOST_CATCH_END
}
BOOST_CATCH_END
}
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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