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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
// 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.Math; 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 (Joaqu�n M L�pez Mu�oz)
// 18 Oct 06 Use EXPLICIT_TEMPLATE_TYPE helper macros from Boost.Config
// (Joaqu�n M L�pez Mu�oz)
// 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 <iostream> // for std::istream and std::ostream
#include <ios> // for std::noskipws
#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/config.hpp> // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC
#include <boost/detail/workaround.hpp> // for BOOST_WORKAROUND
#include <boost/assert.hpp> // for BOOST_ASSERT
#include <boost/math/common_factor_rt.hpp> // for boost::math::gcd, lcm
#include <limits> // for std::numeric_limits
#include <boost/static_assert.hpp> // for BOOST_STATIC_ASSERT
// 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.Math
return math::gcd( n, m );
}
template <typename IntType>
IntType lcm(IntType n, IntType m)
{
// Defer to the version in Boost.Math
return math::lcm( n, m );
}
#endif // BOOST_CONTROL_RATIONAL_HAS_GCD
class bad_rational : public std::domain_error
{
public:
explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}
};
template <typename IntType>
class rational;
template <typename IntType>
rational<IntType> abs(const rational<IntType>& r);
template <typename IntType>
class rational :
less_than_comparable < rational<IntType>,
equality_comparable < rational<IntType>,
less_than_comparable2 < rational<IntType>, IntType,
equality_comparable2 < rational<IntType>, IntType,
addable < rational<IntType>,
subtractable < rational<IntType>,
multipliable < rational<IntType>,
dividable < rational<IntType>,
addable2 < rational<IntType>, IntType,
subtractable2 < rational<IntType>, IntType,
subtractable2_left < rational<IntType>, IntType,
multipliable2 < rational<IntType>, IntType,
dividable2 < rational<IntType>, IntType,
dividable2_left < rational<IntType>, IntType,
incrementable < rational<IntType>,
decrementable < rational<IntType>
> > > > > > > > > > > > > > > >
{
// 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:
typedef IntType int_type;
rational() : num(0), den(1) {}
rational(param_type n) : num(n), den(1) {}
rational(param_type n, param_type d) : num(n), den(d) { normalize(); }
// Default copy constructor and assignment are fine
// Add assignment from IntType
rational& operator=(param_type n) { return assign(n, 1); }
// Assign in place
rational& assign(param_type n, param_type d);
// Access to representation
IntType numerator() const { return num; }
IntType denominator() const { return den; }
// Arithmetic assignment operators
rational& operator+= (const rational& r);
rational& operator-= (const rational& r);
rational& operator*= (const rational& r);
rational& operator/= (const rational& r);
rational& operator+= (param_type i);
rational& operator-= (param_type i);
rational& operator*= (param_type i);
rational& operator/= (param_type i);
// Increment and decrement
const rational& operator++();
const rational& operator--();
// Operator not
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
operator bool_type() const { return operator !() ? 0 : &helper::parts; }
#if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
#pragma parse_mfunc_templ reset
#endif
// Comparison operators
bool operator< (const rational& r) const;
bool operator== (const rational& r) const;
bool operator< (param_type i) const;
bool operator> (param_type i) const;
bool operator== (param_type i) const;
private:
// Implementation - numerator and denominator (normalized).
// Other possibilities - separate whole-part, or sign, fields?
IntType num;
IntType den;
// 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.
bool test_invariant() const;
void normalize();
};
// Assign in place
template <typename IntType>
inline rational<IntType>& rational<IntType>::assign(param_type n, param_type d)
{
num = n;
den = d;
normalize();
return *this;
}
// Unary plus and minus
template <typename IntType>
inline rational<IntType> operator+ (const rational<IntType>& r)
{
return r;
}
template <typename IntType>
inline rational<IntType> operator- (const rational<IntType>& r)
{
return rational<IntType>(-r.numerator(), r.denominator());
}
// Arithmetic assignment operators
template <typename IntType>
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 = math::gcd(den, r_den);
den /= g; // = b1 from the calculations above
num = num * (r_den / g) + r_num * den;
g = math::gcd(num, g);
num /= g;
den *= r_den/g;
return *this;
}
template <typename IntType>
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 = math::gcd(den, r_den);
den /= g;
num = num * (r_den / g) - r_num * den;
g = math::gcd(num, g);
num /= g;
den *= r_den/g;
return *this;
}
template <typename IntType>
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 = math::gcd(num, r_den);
IntType gcd2 = math::gcd(r_num, den);
num = (num/gcd1) * (r_num/gcd2);
den = (den/gcd2) * (r_den/gcd1);
return *this;
}
template <typename IntType>
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)
throw bad_rational();
if (num == zero)
return *this;
// Avoid overflow and preserve normalization
IntType gcd1 = math::gcd(num, r_num);
IntType gcd2 = math::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;
}
// Mixed-mode operators
template <typename IntType>
inline rational<IntType>&
rational<IntType>::operator+= (param_type i)
{
return operator+= (rational<IntType>(i));
}
template <typename IntType>
inline rational<IntType>&
rational<IntType>::operator-= (param_type i)
{
return operator-= (rational<IntType>(i));
}
template <typename IntType>
inline rational<IntType>&
rational<IntType>::operator*= (param_type i)
{
return operator*= (rational<IntType>(i));
}
template <typename IntType>
inline rational<IntType>&
rational<IntType>::operator/= (param_type i)
{
return operator/= (rational<IntType>(i));
}
// Increment and decrement
template <typename IntType>
inline const rational<IntType>& rational<IntType>::operator++()
{
// This can never denormalise the fraction
num += den;
return *this;
}
template <typename IntType>
inline const rational<IntType>& rational<IntType>::operator--()
{
// This can never denormalise the fraction
num -= den;
return *this;
}
// Comparison operators
template <typename IntType>
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, this->num /
this->den, this->num % this->den }, rs = { r.num, r.den, r.num / r.den,
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
while ( true )
{
// 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>
bool rational<IntType>::operator< (param_type i) const
{
// Avoid repeated construction
int_type const zero( 0 );
// Break value into mixed-fraction form, w/ always-nonnegative remainder
BOOST_ASSERT( this->den > zero );
int_type q = this->num / this->den, r = this->num % this->den;
while ( r < zero ) { r += this->den; --q; }
// Compare with just the quotient, since the remainder always bumps the
// value up. [Since q = floor(n/d), and if n/d < i then q < i, if n/d == i
// then q == i, if n/d == i + r/d then q == i, and if n/d >= i + 1 then
// q >= i + 1 > i; therefore n/d < i iff q < i.]
return q < i;
}
template <typename IntType>
bool rational<IntType>::operator> (param_type i) const
{
// Trap equality first
if (num == i && den == IntType(1))
return false;
// Otherwise, we can use operator<
return !operator<(i);
}
template <typename IntType>
inline bool rational<IntType>::operator== (const rational<IntType>& r) const
{
return ((num == r.num) && (den == r.den));
}
template <typename IntType>
inline bool rational<IntType>::operator== (param_type i) const
{
return ((den == IntType(1)) && (num == i));
}
// Invariant check
template <typename IntType>
inline bool rational<IntType>::test_invariant() const
{
return ( this->den > int_type(0) ) && ( math::gcd(this->num, this->den) ==
int_type(1) );
}
// Normalisation
template <typename IntType>
void rational<IntType>::normalize()
{
// Avoid repeated construction
IntType zero(0);
if (den == zero)
throw bad_rational();
// Handle the case of zero separately, to avoid division by zero
if (num == zero) {
den = IntType(1);
return;
}
IntType g = math::gcd(num, den);
num /= g;
den /= g;
// Ensure that the denominator is positive
if (den < zero) {
num = -num;
den = -den;
}
BOOST_ASSERT( this->test_invariant() );
}
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)
{
IntType n = IntType(0), d = IntType(1);
char c = 0;
detail::resetter sentry(is);
is >> n;
c = is.get();
if (c != '/')
is.clear(std::istream::badbit); // old GNU c++ lib has no ios_base
#if !defined(__GNUC__) || (defined(__GNUC__) && (__GNUC__ >= 3)) || defined __SGI_STL_PORT
is >> std::noskipws;
#else
is.unsetf(ios::skipws); // compiles, but seems to have no effect.
#endif
is >> d;
if (is)
r.assign(n, d);
return is;
}
// Add manipulators for output format?
template <typename IntType>
std::ostream& operator<< (std::ostream& os, const rational<IntType>& r)
{
os << r.numerator() << '/' << r.denominator();
return os;
}
// Type conversion
template <typename T, typename IntType>
inline T rational_cast(
const rational<IntType>& src BOOST_APPEND_EXPLICIT_TEMPLATE_TYPE(T))
{
return static_cast<T>(src.numerator())/static_cast<T>(src.denominator());
}
// Do not use any abs() defined on IntType - it isn't worth it, given the
// difficulties involved (Koenig lookup required, there may not *be* an abs()
// defined, etc etc).
template <typename IntType>
inline rational<IntType> abs(const rational<IntType>& r)
{
if (r.numerator() >= IntType(0))
return r;
return rational<IntType>(-r.numerator(), r.denominator());
}
} // namespace boost
#endif // BOOST_RATIONAL_HPP
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