//  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))

 {

     return static_cast<T>(src.numerator())/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