Edit: /usr/include/boost/rational.hpp (39720B)
// 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
(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 // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC, etc
#ifndef BOOST_NO_IOSTREAM
#include // for std::setw
#include // for std::noskipws, streamsize
#include // for std::istream
#include // for std::ostream
#include // for std::ostringstream
#endif
#include // for NULL
#include // for std::domain_error
#include // for std::string implicit constructor
#include // for boost::addable etc
#include // for std::abs
#include // for boost::call_traits
#include // for BOOST_WORKAROUND
#include // for BOOST_ASSERT
#include // for boost::integer::gcd, lcm
#include // for std::numeric_limits
#include // for BOOST_STATIC_ASSERT
#include
#include
#include
#include
#include
#include
// 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
IntType gcd(IntType n, IntType m)
{
// Defer to the version in Boost.Integer
return integer::gcd( n, m );
}
template
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
struct is_compatible_integer;
template
struct is_compatible_integer::value>::type>
{
BOOST_STATIC_CONSTANT(bool, value = ((std::numeric_limits::is_specialized && std::numeric_limits::is_integer
&& (std::numeric_limits::digits <= std::numeric_limits::digits)
&& (std::numeric_limits::radix == std::numeric_limits::radix)
&& ((std::numeric_limits::is_signed == false) || (std::numeric_limits::is_signed == true))
&& is_convertible::value)
|| is_same::value)
|| (is_class::value && is_class::value && is_convertible::value));
};
template
struct is_compatible_integer::value>::type>
{
BOOST_STATIC_CONSTANT(bool, value = false);
};
template
struct is_backward_compatible_integer;
template
struct is_backward_compatible_integer::value>::type>
{
BOOST_STATIC_CONSTANT(bool, value = (std::numeric_limits::is_specialized && std::numeric_limits::is_integer
&& !is_compatible_integer::value
&& (std::numeric_limits::radix == std::numeric_limits::radix)
&& is_convertible::value));
};
template
struct is_backward_compatible_integer::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
class rational
{
// Class-wide pre-conditions
BOOST_STATIC_ASSERT( ::std::numeric_limits::is_specialized );
// Helper types
typedef typename boost::call_traits::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 //, typename enable_if_c::value>::type>
BOOST_CONSTEXPR rational(const T& n, typename enable_if_c<
rational_detail::is_compatible_integer::value
>::type const* = 0) : num(n), den(1) {}
template
BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
rational_detail::is_compatible_integer::value && rational_detail::is_compatible_integer::value
>::type const* = 0) : num(n), den(d) {
normalize();
}
template < typename NewType >
BOOST_CONSTEXPR explicit
rational(rational const &r, typename enable_if_c::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 const &r, typename disable_if_c::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
BOOST_CXX14_CONSTEXPR typename enable_if_c<
rational_detail::is_compatible_integer::value, rational &
>::type operator=(const T& n) { return assign(static_cast(n), static_cast(1)); }
// Assign in place
template
BOOST_CXX14_CONSTEXPR typename enable_if_c<
rational_detail::is_compatible_integer::value && rational_detail::is_compatible_integer::value, rational &
>::type assign(const T& n, const U& d)
{
return *this = rational(static_cast(n), static_cast(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 //, typename enable_if_c::value>::type>
BOOST_CXX14_CONSTEXPR rational(const T& n, typename enable_if_c<
rational_detail::is_backward_compatible_integer::value
>::type const* = 0)
{
assign(n, static_cast(1));
}
template
BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
(!rational_detail::is_compatible_integer::value
|| !rational_detail::is_compatible_integer::value)
&& std::numeric_limits::is_specialized && std::numeric_limits::is_integer
&& (std::numeric_limits::radix == std::numeric_limits::radix)
&& is_convertible::value &&
std::numeric_limits::is_specialized && std::numeric_limits::is_integer
&& (std::numeric_limits::radix == std::numeric_limits::radix)
&& is_convertible::value
>::type const* = 0)
{
assign(n, d);
}
template
BOOST_CXX14_CONSTEXPR typename enable_if_c<
std::numeric_limits::is_specialized && std::numeric_limits::is_integer
&& !rational_detail::is_compatible_integer::value
&& (std::numeric_limits::radix == std::numeric_limits::radix)
&& is_convertible::value,
rational &
>::type operator=(const T& n) { return assign(n, static_cast(1)); }
template
BOOST_CXX14_CONSTEXPR typename enable_if_c<
(!rational_detail::is_compatible_integer::value
|| !rational_detail::is_compatible_integer::value)
&& std::numeric_limits::is_specialized && std::numeric_limits::is_integer
&& (std::numeric_limits::radix == std::numeric_limits::radix)
&& is_convertible::value &&
std::numeric_limits::is_specialized && std::numeric_limits::is_integer
&& (std::numeric_limits::radix == std::numeric_limits::radix)
&& is_convertible::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(static_cast(n), static_cast(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
BOOST_CXX14_CONSTEXPR typename boost::enable_if_c::value, rational&>::type operator+= (const T& i)
{
num += i * den;
return *this;
}
template
BOOST_CXX14_CONSTEXPR typename boost::enable_if_c::value, rational&>::type operator-= (const T& i)
{
num -= i * den;
return *this;
}
template
BOOST_CXX14_CONSTEXPR typename boost::enable_if_c::value, rational&>::type operator*= (const T& i)
{
// Avoid overflow and preserve normalization
IntType gcd = integer::gcd(static_cast(i), den);
num *= i / gcd;
den /= gcd;
return *this;
}
template
BOOST_CXX14_CONSTEXPR typename boost::enable_if_c::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(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
BOOST_CXX14_CONSTEXPR typename boost::enable_if_c::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
BOOST_CXX14_CONSTEXPR typename boost::enable_if_c::value, bool>::type operator>(const T& i) const
{
return operator==(i) ? false : !operator<(i);
}
template
BOOST_CONSTEXPR typename boost::enable_if_c::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
BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits::digits > std::numeric_limits::digits) && (std::numeric_limits::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
{
return val < (T(1) << std::numeric_limits::digits);
}
//
// (2) From a signed type with more digits than IntType, and IntType also signed:
//
template
BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits::digits > std::numeric_limits::digits) && (std::numeric_limits::is_signed == true) && (std::numeric_limits::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::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::digits)) && (val >= -(T(1) << std::numeric_limits::digits));
}
//
// (3) From a signed type with more digits than IntType, and IntType unsigned:
//
template
BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits::digits > std::numeric_limits::digits) && (std::numeric_limits::is_signed == true) && (std::numeric_limits::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
{
return (val < (T(1) << std::numeric_limits::digits)) && (val >= 0);
}
//
// (4) From a signed type with fewer digits than IntType, and IntType unsigned:
//
template
BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::is_signed == true) && (std::numeric_limits::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
BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::is_signed == false) && (std::numeric_limits::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
BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::is_signed == false) && (std::numeric_limits::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
BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::is_signed == true) && (std::numeric_limits::is_signed == true), bool>::type is_safe_narrowing_conversion(const T&)
{
return true;
}
};
// Unary plus and minus
template
BOOST_CONSTEXPR
inline rational operator+ (const rational& r)
{
return r;
}
template
BOOST_CXX14_CONSTEXPR
inline rational operator- (const rational& r)
{
return rational(static_cast(-r.numerator()), r.denominator());
}
// Arithmetic assignment operators
template
BOOST_CXX14_CONSTEXPR rational& rational::operator+= (const rational& 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
BOOST_CXX14_CONSTEXPR rational& rational::operator-= (const rational& 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
BOOST_CXX14_CONSTEXPR rational& rational::operator*= (const rational& 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
BOOST_CXX14_CONSTEXPR rational& rational::operator/= (const rational& 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 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
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value || is_same, Arg>::value, rational >::type
operator + (const rational& a, const Arg& b)
{
rational t(a);
return t += b;
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value, rational >::type
operator + (const Arg& b, const rational& a)
{
rational t(a);
return t += b;
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value || is_same, Arg>::value, rational >::type
operator - (const rational& a, const Arg& b)
{
rational t(a);
return t -= b;
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value, rational >::type
operator - (const Arg& b, const rational& a)
{
rational t(a);
return -(t -= b);
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value || is_same, Arg>::value, rational >::type
operator * (const rational& a, const Arg& b)
{
rational t(a);
return t *= b;
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value, rational >::type
operator * (const Arg& b, const rational& a)
{
rational t(a);
return t *= b;
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value || is_same, Arg>::value, rational >::type
operator / (const rational& a, const Arg& b)
{
rational t(a);
return t /= b;
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value, rational >::type
operator / (const Arg& b, const rational& a)
{
rational t(b);
return t /= a;
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value || is_same, Arg>::value, bool>::type
operator <= (const rational& a, const Arg& b)
{
return !(a > b);
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value, bool>::type
operator <= (const Arg& b, const rational& a)
{
return a >= b;
}
template
BOOST_CXX14_CONSTEXPR
inline typename boost::enable_if_c <
rational_detail::is_compatible_integer::value || is_same, Arg>::value, bool>::type
operator >= (const rational& a, const Arg& b)
{
return !(a < b);
}
template