//  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.

//  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

//  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 <iostream>              // for std::istream and std::ostream

#include <iomanip>               // 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

namespace boost {

// Note: We use n and m as temporaries in this function, so there is no value

// in using const IntType& as we would only need to make a copy anyway...

template <typename IntType>

IntType gcd(IntType n, IntType m)

{

    // Avoid repeated construction

    IntType zero(0);

    // This is abs() - given the existence of broken compilers with Koenig

    // lookup issues and other problems, I code this explicitly. (Remember,

    // IntType may be a user-defined type).

    if (n < zero)

        n = -n;

    if (m < zero)

        m = -m;

    // As n and m are now positive, we can be sure that %= returns a

    // positive value (the standard guarantees this for built-in types,

    // and we require it of user-defined types).

    for(;;) {

      if(m == zero)

        return n;

      n %= m;

      if(n == zero)

        return m;

      m %= n;

    }

}

template <typename IntType>

IntType lcm(IntType n, IntType m)

{

    // Avoid repeated construction

    IntType zero(0);

    if (n == zero || m == zero)

        return zero;

    n /= gcd(n, m);

    n *= m;

    if (n < zero)

        n = -n;

    return n;

}

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>

    > > > > > > > > > > > > > > > >

{

    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.

    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 = gcd(den, r_den);

    den /= g;  // = b1 from the calculations above

    num = num * (r_den / g) + r_num * den;

    g = 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 = gcd(den, r_den);

    den /= g;

    num = num * (r_den / g) - r_num * den;

    g = 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 = gcd<IntType>(num, r_den);

    IntType gcd2 = gcd<IntType>(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 = gcd<IntType>(num, r_num);

    IntType gcd2 = gcd<IntType>(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

    IntType zero(0);

    // If the two values have different signs, we don't need to do the

    // expensive calculations below. We take advantage here of the fact

    // that the denominator is always positive.

    if (num < zero && r.num >= zero) // -ve < +ve

        return true;

    if (num >= zero && r.num <= zero) // +ve or zero is not < -ve or zero

        return false;

    // Avoid overflow

    IntType gcd1 = gcd<IntType>(num, r.num);

    IntType gcd2 = gcd<IntType>(r.den, den);

    return (num/gcd1) * (r.den/gcd2) < (den/gcd2) * (r.num/gcd1);

}

template <typename IntType>

bool rational<IntType>::operator< (param_type i) const

{

    // Avoid repeated construction

    IntType zero(0);

    // If the two values have different signs, we don't need to do the

    // expensive calculations below. We take advantage here of the fact

    // that the denominator is always positive.

    if (num < zero && i >= zero) // -ve < +ve

        return true;

    if (num >= zero && i <= zero) // +ve or zero is not < -ve or zero

        return false;

    // Now, use the fact that n/d truncates towards zero as long as n and d

    // are both positive.

    // Divide instead of multiplying to avoid overflow issues. Of course,

    // division may be slower, but accuracy is more important than speed...

    if (num > zero)

        return (num/den) < i;

    else

        return -i < (-num/den);

}

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));

}

// 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 = gcd<IntType>(num, den);

    num /= g;

    den /= g;

    // Ensure that the denominator is positive

    if (den < zero) {

        num = -num;

        den = -den;

    }

}

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))