Chris@16: //  Boost rational.hpp header file  ------------------------------------------//
Chris@16: 
Chris@16: //  (C) Copyright Paul Moore 1999. Permission to copy, use, modify, sell and
Chris@16: //  distribute this software is granted provided this copyright notice appears
Chris@16: //  in all copies. This software is provided "as is" without express or
Chris@16: //  implied warranty, and with no claim as to its suitability for any purpose.
Chris@16: 
Chris@16: // boostinspect:nolicense (don't complain about the lack of a Boost license)
Chris@16: // (Paul Moore hasn't been in contact for years, so there's no way to change the
Chris@16: // license.)
Chris@16: 
Chris@16: //  See http://www.boost.org/libs/rational for documentation.
Chris@16: 
Chris@16: //  Credits:
Chris@16: //  Thanks to the boost mailing list in general for useful comments.
Chris@16: //  Particular contributions included:
Chris@16: //    Andrew D Jewell, for reminding me to take care to avoid overflow
Chris@16: //    Ed Brey, for many comments, including picking up on some dreadful typos
Chris@16: //    Stephen Silver contributed the test suite and comments on user-defined
Chris@16: //    IntType
Chris@16: //    Nickolay Mladenov, for the implementation of operator+=
Chris@16: 
Chris@16: //  Revision History
Chris@101: //  02 Sep 13  Remove unneeded forward declarations; tweak private helper
Chris@101: //             function (Daryle Walker)
Chris@101: //  30 Aug 13  Improve exception safety of "assign"; start modernizing I/O code
Chris@101: //             (Daryle Walker)
Chris@101: //  27 Aug 13  Add cross-version constructor template, plus some private helper
Chris@101: //             functions; add constructor to exception class to take custom
Chris@101: //             messages (Daryle Walker)
Chris@101: //  25 Aug 13  Add constexpr qualification wherever possible (Daryle Walker)
Chris@101: //  05 May 12  Reduced use of implicit gcd (Mario Lang)
Chris@16: //  05 Nov 06  Change rational_cast to not depend on division between different
Chris@16: //             types (Daryle Walker)
Chris@16: //  04 Nov 06  Off-load GCD and LCM to Boost.Math; add some invariant checks;
Chris@16: //             add std::numeric_limits<> requirement to help GCD (Daryle Walker)
Chris@16: //  31 Oct 06  Recoded both operator< to use round-to-negative-infinity
Chris@16: //             divisions; the rational-value version now uses continued fraction
Chris@16: //             expansion to avoid overflows, for bug #798357 (Daryle Walker)
Chris@16: //  20 Oct 06  Fix operator bool_type for CW 8.3 (Joaquín M López Muñoz)
Chris@16: //  18 Oct 06  Use EXPLICIT_TEMPLATE_TYPE helper macros from Boost.Config
Chris@16: //             (Joaquín M López Muñoz)
Chris@16: //  27 Dec 05  Add Boolean conversion operator (Daryle Walker)
Chris@16: //  28 Sep 02  Use _left versions of operators from operators.hpp
Chris@16: //  05 Jul 01  Recode gcd(), avoiding std::swap (Helmut Zeisel)
Chris@16: //  03 Mar 01  Workarounds for Intel C++ 5.0 (David Abrahams)
Chris@16: //  05 Feb 01  Update operator>> to tighten up input syntax
Chris@16: //  05 Feb 01  Final tidy up of gcd code prior to the new release
Chris@16: //  27 Jan 01  Recode abs() without relying on abs(IntType)
Chris@16: //  21 Jan 01  Include Nickolay Mladenov's operator+= algorithm,
Chris@16: //             tidy up a number of areas, use newer features of operators.hpp
Chris@16: //             (reduces space overhead to zero), add operator!,
Chris@16: //             introduce explicit mixed-mode arithmetic operations
Chris@16: //  12 Jan 01  Include fixes to handle a user-defined IntType better
Chris@16: //  19 Nov 00  Throw on divide by zero in operator /= (John (EBo) David)
Chris@16: //  23 Jun 00  Incorporate changes from Mark Rodgers for Borland C++
Chris@16: //  22 Jun 00  Change _MSC_VER to BOOST_MSVC so other compilers are not
Chris@16: //             affected (Beman Dawes)
Chris@16: //   6 Mar 00  Fix operator-= normalization, #include <string> (Jens Maurer)
Chris@16: //  14 Dec 99  Modifications based on comments from the boost list
Chris@16: //  09 Dec 99  Initial Version (Paul Moore)
Chris@16: 
Chris@16: #ifndef BOOST_RATIONAL_HPP
Chris@16: #define BOOST_RATIONAL_HPP
Chris@16: 
Chris@101: #include <boost/config.hpp>      // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC, etc
Chris@101: #ifndef BOOST_NO_IOSTREAM
Chris@101: #include <iomanip>               // for std::setw
Chris@101: #include <ios>                   // for std::noskipws, streamsize
Chris@101: #include <istream>               // for std::istream
Chris@101: #include <ostream>               // for std::ostream
Chris@101: #include <sstream>               // for std::ostringstream
Chris@101: #endif
Chris@101: #include <cstddef>               // for NULL
Chris@16: #include <stdexcept>             // for std::domain_error
Chris@16: #include <string>                // for std::string implicit constructor
Chris@16: #include <boost/operators.hpp>   // for boost::addable etc
Chris@16: #include <cstdlib>               // for std::abs
Chris@16: #include <boost/call_traits.hpp> // for boost::call_traits
Chris@16: #include <boost/detail/workaround.hpp> // for BOOST_WORKAROUND
Chris@16: #include <boost/assert.hpp>      // for BOOST_ASSERT
Chris@101: #include <boost/integer/common_factor_rt.hpp> // for boost::integer::gcd, lcm
Chris@16: #include <limits>                // for std::numeric_limits
Chris@16: #include <boost/static_assert.hpp>  // for BOOST_STATIC_ASSERT
Chris@16: 
Chris@16: // Control whether depreciated GCD and LCM functions are included (default: yes)
Chris@16: #ifndef BOOST_CONTROL_RATIONAL_HAS_GCD
Chris@16: #define BOOST_CONTROL_RATIONAL_HAS_GCD  1
Chris@16: #endif
Chris@16: 
Chris@16: namespace boost {
Chris@16: 
Chris@16: #if BOOST_CONTROL_RATIONAL_HAS_GCD
Chris@16: template <typename IntType>
Chris@16: IntType gcd(IntType n, IntType m)
Chris@16: {
Chris@16:     // Defer to the version in Boost.Math
Chris@101:     return integer::gcd( n, m );
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@16: IntType lcm(IntType n, IntType m)
Chris@16: {
Chris@16:     // Defer to the version in Boost.Math
Chris@101:     return integer::lcm( n, m );
Chris@16: }
Chris@16: #endif  // BOOST_CONTROL_RATIONAL_HAS_GCD
Chris@16: 
Chris@16: class bad_rational : public std::domain_error
Chris@16: {
Chris@16: public:
Chris@16:     explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}
Chris@101:     explicit bad_rational( char const *what ) : std::domain_error( what ) {}
Chris@16: };
Chris@16: 
Chris@16: template <typename IntType>
Chris@16: class rational :
Chris@16:     less_than_comparable < rational<IntType>,
Chris@16:     equality_comparable < rational<IntType>,
Chris@16:     less_than_comparable2 < rational<IntType>, IntType,
Chris@16:     equality_comparable2 < rational<IntType>, IntType,
Chris@16:     addable < rational<IntType>,
Chris@16:     subtractable < rational<IntType>,
Chris@16:     multipliable < rational<IntType>,
Chris@16:     dividable < rational<IntType>,
Chris@16:     addable2 < rational<IntType>, IntType,
Chris@16:     subtractable2 < rational<IntType>, IntType,
Chris@16:     subtractable2_left < rational<IntType>, IntType,
Chris@16:     multipliable2 < rational<IntType>, IntType,
Chris@16:     dividable2 < rational<IntType>, IntType,
Chris@16:     dividable2_left < rational<IntType>, IntType,
Chris@16:     incrementable < rational<IntType>,
Chris@16:     decrementable < rational<IntType>
Chris@16:     > > > > > > > > > > > > > > > >
Chris@16: {
Chris@16:     // Class-wide pre-conditions
Chris@16:     BOOST_STATIC_ASSERT( ::std::numeric_limits<IntType>::is_specialized );
Chris@16: 
Chris@16:     // Helper types
Chris@16:     typedef typename boost::call_traits<IntType>::param_type param_type;
Chris@16: 
Chris@16:     struct helper { IntType parts[2]; };
Chris@16:     typedef IntType (helper::* bool_type)[2];
Chris@16: 
Chris@16: public:
Chris@101:     // Component type
Chris@16:     typedef IntType int_type;
Chris@101: 
Chris@101:     BOOST_CONSTEXPR
Chris@16:     rational() : num(0), den(1) {}
Chris@101:     BOOST_CONSTEXPR
Chris@16:     rational(param_type n) : num(n), den(1) {}
Chris@16:     rational(param_type n, param_type d) : num(n), den(d) { normalize(); }
Chris@16: 
Chris@101: #ifndef BOOST_NO_MEMBER_TEMPLATES
Chris@101:     template < typename NewType >
Chris@101:     BOOST_CONSTEXPR explicit
Chris@101:     rational( rational<NewType> const &r )
Chris@101:         : num( r.numerator() ), den( is_normalized(int_type( r.numerator() ),
Chris@101:           int_type( r.denominator() )) ? r.denominator() :
Chris@101:           throw bad_rational("bad rational: denormalized conversion") )
Chris@101:     {}
Chris@101: #endif
Chris@101: 
Chris@16:     // Default copy constructor and assignment are fine
Chris@16: 
Chris@16:     // Add assignment from IntType
Chris@101:     rational& operator=(param_type i) { num = i; den = 1; return *this; }
Chris@16: 
Chris@16:     // Assign in place
Chris@16:     rational& assign(param_type n, param_type d);
Chris@16: 
Chris@16:     // Access to representation
Chris@101:     BOOST_CONSTEXPR
Chris@16:     IntType numerator() const { return num; }
Chris@101:     BOOST_CONSTEXPR
Chris@16:     IntType denominator() const { return den; }
Chris@16: 
Chris@16:     // Arithmetic assignment operators
Chris@16:     rational& operator+= (const rational& r);
Chris@16:     rational& operator-= (const rational& r);
Chris@16:     rational& operator*= (const rational& r);
Chris@16:     rational& operator/= (const rational& r);
Chris@16: 
Chris@101:     rational& operator+= (param_type i) { num += i * den; return *this; }
Chris@101:     rational& operator-= (param_type i) { num -= i * den; return *this; }
Chris@16:     rational& operator*= (param_type i);
Chris@16:     rational& operator/= (param_type i);
Chris@16: 
Chris@16:     // Increment and decrement
Chris@101:     const rational& operator++() { num += den; return *this; }
Chris@101:     const rational& operator--() { num -= den; return *this; }
Chris@16: 
Chris@16:     // Operator not
Chris@101:     BOOST_CONSTEXPR
Chris@16:     bool operator!() const { return !num; }
Chris@16: 
Chris@16:     // Boolean conversion
Chris@16:     
Chris@16: #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
Chris@16:     // The "ISO C++ Template Parser" option in CW 8.3 chokes on the
Chris@16:     // following, hence we selectively disable that option for the
Chris@16:     // offending memfun.
Chris@16: #pragma parse_mfunc_templ off
Chris@16: #endif
Chris@16: 
Chris@101:     BOOST_CONSTEXPR
Chris@16:     operator bool_type() const { return operator !() ? 0 : &helper::parts; }
Chris@16: 
Chris@16: #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
Chris@16: #pragma parse_mfunc_templ reset
Chris@16: #endif
Chris@16: 
Chris@16:     // Comparison operators
Chris@16:     bool operator< (const rational& r) const;
Chris@101:     BOOST_CONSTEXPR
Chris@16:     bool operator== (const rational& r) const;
Chris@16: 
Chris@16:     bool operator< (param_type i) const;
Chris@16:     bool operator> (param_type i) const;
Chris@101:     BOOST_CONSTEXPR
Chris@16:     bool operator== (param_type i) const;
Chris@16: 
Chris@16: private:
Chris@16:     // Implementation - numerator and denominator (normalized).
Chris@16:     // Other possibilities - separate whole-part, or sign, fields?
Chris@16:     IntType num;
Chris@16:     IntType den;
Chris@16: 
Chris@101:     // Helper functions
Chris@101:     static BOOST_CONSTEXPR
Chris@101:     int_type inner_gcd( param_type a, param_type b, int_type const &zero =
Chris@101:      int_type(0) )
Chris@101:     { return b == zero ? a : inner_gcd(b, a % b, zero); }
Chris@101: 
Chris@101:     static BOOST_CONSTEXPR
Chris@101:     int_type inner_abs( param_type x, int_type const &zero = int_type(0) )
Chris@101:     { return x < zero ? -x : +x; }
Chris@101: 
Chris@16:     // Representation note: Fractions are kept in normalized form at all
Chris@16:     // times. normalized form is defined as gcd(num,den) == 1 and den > 0.
Chris@16:     // In particular, note that the implementation of abs() below relies
Chris@16:     // on den always being positive.
Chris@16:     bool test_invariant() const;
Chris@16:     void normalize();
Chris@101: 
Chris@101:     static BOOST_CONSTEXPR
Chris@101:     bool is_normalized( param_type n, param_type d, int_type const &zero =
Chris@101:      int_type(0), int_type const &one = int_type(1) )
Chris@101:     {
Chris@101:         return d > zero && ( n != zero || d == one ) && inner_abs( inner_gcd(n,
Chris@101:          d, zero), zero ) == one;
Chris@101:     }
Chris@16: };
Chris@16: 
Chris@16: // Assign in place
Chris@16: template <typename IntType>
Chris@16: inline rational<IntType>& rational<IntType>::assign(param_type n, param_type d)
Chris@16: {
Chris@101:     return *this = rational( n, d );
Chris@16: }
Chris@16: 
Chris@16: // Unary plus and minus
Chris@16: template <typename IntType>
Chris@101: BOOST_CONSTEXPR
Chris@16: inline rational<IntType> operator+ (const rational<IntType>& r)
Chris@16: {
Chris@16:     return r;
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@16: inline rational<IntType> operator- (const rational<IntType>& r)
Chris@16: {
Chris@16:     return rational<IntType>(-r.numerator(), r.denominator());
Chris@16: }
Chris@16: 
Chris@16: // Arithmetic assignment operators
Chris@16: template <typename IntType>
Chris@16: rational<IntType>& rational<IntType>::operator+= (const rational<IntType>& r)
Chris@16: {
Chris@16:     // This calculation avoids overflow, and minimises the number of expensive
Chris@16:     // calculations. Thanks to Nickolay Mladenov for this algorithm.
Chris@16:     //
Chris@16:     // Proof:
Chris@16:     // We have to compute a/b + c/d, where gcd(a,b)=1 and gcd(b,c)=1.
Chris@16:     // Let g = gcd(b,d), and b = b1*g, d=d1*g. Then gcd(b1,d1)=1
Chris@16:     //
Chris@16:     // The result is (a*d1 + c*b1) / (b1*d1*g).
Chris@16:     // Now we have to normalize this ratio.
Chris@16:     // Let's assume h | gcd((a*d1 + c*b1), (b1*d1*g)), and h > 1
Chris@16:     // If h | b1 then gcd(h,d1)=1 and hence h|(a*d1+c*b1) => h|a.
Chris@16:     // But since gcd(a,b1)=1 we have h=1.
Chris@16:     // Similarly h|d1 leads to h=1.
Chris@16:     // So we have that h | gcd((a*d1 + c*b1) , (b1*d1*g)) => h|g
Chris@16:     // Finally we have gcd((a*d1 + c*b1), (b1*d1*g)) = gcd((a*d1 + c*b1), g)
Chris@16:     // Which proves that instead of normalizing the result, it is better to
Chris@16:     // divide num and den by gcd((a*d1 + c*b1), g)
Chris@16: 
Chris@16:     // Protect against self-modification
Chris@16:     IntType r_num = r.num;
Chris@16:     IntType r_den = r.den;
Chris@16: 
Chris@101:     IntType g = integer::gcd(den, r_den);
Chris@16:     den /= g;  // = b1 from the calculations above
Chris@16:     num = num * (r_den / g) + r_num * den;
Chris@101:     g = integer::gcd(num, g);
Chris@16:     num /= g;
Chris@16:     den *= r_den/g;
Chris@16: 
Chris@16:     return *this;
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@16: rational<IntType>& rational<IntType>::operator-= (const rational<IntType>& r)
Chris@16: {
Chris@16:     // Protect against self-modification
Chris@16:     IntType r_num = r.num;
Chris@16:     IntType r_den = r.den;
Chris@16: 
Chris@16:     // This calculation avoids overflow, and minimises the number of expensive
Chris@16:     // calculations. It corresponds exactly to the += case above
Chris@101:     IntType g = integer::gcd(den, r_den);
Chris@16:     den /= g;
Chris@16:     num = num * (r_den / g) - r_num * den;
Chris@101:     g = integer::gcd(num, g);
Chris@16:     num /= g;
Chris@16:     den *= r_den/g;
Chris@16: 
Chris@16:     return *this;
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@16: rational<IntType>& rational<IntType>::operator*= (const rational<IntType>& r)
Chris@16: {
Chris@16:     // Protect against self-modification
Chris@16:     IntType r_num = r.num;
Chris@16:     IntType r_den = r.den;
Chris@16: 
Chris@16:     // Avoid overflow and preserve normalization
Chris@101:     IntType gcd1 = integer::gcd(num, r_den);
Chris@101:     IntType gcd2 = integer::gcd(r_num, den);
Chris@16:     num = (num/gcd1) * (r_num/gcd2);
Chris@16:     den = (den/gcd2) * (r_den/gcd1);
Chris@16:     return *this;
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@16: rational<IntType>& rational<IntType>::operator/= (const rational<IntType>& r)
Chris@16: {
Chris@16:     // Protect against self-modification
Chris@16:     IntType r_num = r.num;
Chris@16:     IntType r_den = r.den;
Chris@16: 
Chris@16:     // Avoid repeated construction
Chris@16:     IntType zero(0);
Chris@16: 
Chris@16:     // Trap division by zero
Chris@16:     if (r_num == zero)
Chris@16:         throw bad_rational();
Chris@16:     if (num == zero)
Chris@16:         return *this;
Chris@16: 
Chris@16:     // Avoid overflow and preserve normalization
Chris@101:     IntType gcd1 = integer::gcd(num, r_num);
Chris@101:     IntType gcd2 = integer::gcd(r_den, den);
Chris@16:     num = (num/gcd1) * (r_den/gcd2);
Chris@16:     den = (den/gcd2) * (r_num/gcd1);
Chris@16: 
Chris@16:     if (den < zero) {
Chris@16:         num = -num;
Chris@16:         den = -den;
Chris@16:     }
Chris@16:     return *this;
Chris@16: }
Chris@16: 
Chris@16: // Mixed-mode operators
Chris@16: template <typename IntType>
Chris@16: inline rational<IntType>&
Chris@16: rational<IntType>::operator*= (param_type i)
Chris@16: {
Chris@101:     // Avoid overflow and preserve normalization
Chris@101:     IntType gcd = integer::gcd(i, den);
Chris@101:     num *= i / gcd;
Chris@101:     den /= gcd;
Chris@16: 
Chris@16:     return *this;
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@101: rational<IntType>&
Chris@101: rational<IntType>::operator/= (param_type i)
Chris@16: {
Chris@101:     // Avoid repeated construction
Chris@101:     IntType const zero(0);
Chris@101: 
Chris@101:     if (i == zero) throw bad_rational();
Chris@101:     if (num == zero) return *this;
Chris@101: 
Chris@101:     // Avoid overflow and preserve normalization
Chris@101:     IntType const gcd = integer::gcd(num, i);
Chris@101:     num /= gcd;
Chris@101:     den *= i / gcd;
Chris@101: 
Chris@101:     if (den < zero) {
Chris@101:         num = -num;
Chris@101:         den = -den;
Chris@101:     }
Chris@101: 
Chris@16:     return *this;
Chris@16: }
Chris@16: 
Chris@16: // Comparison operators
Chris@16: template <typename IntType>
Chris@16: bool rational<IntType>::operator< (const rational<IntType>& r) const
Chris@16: {
Chris@16:     // Avoid repeated construction
Chris@16:     int_type const  zero( 0 );
Chris@16: 
Chris@16:     // This should really be a class-wide invariant.  The reason for these
Chris@16:     // checks is that for 2's complement systems, INT_MIN has no corresponding
Chris@16:     // positive, so negating it during normalization keeps it INT_MIN, which
Chris@16:     // is bad for later calculations that assume a positive denominator.
Chris@16:     BOOST_ASSERT( this->den > zero );
Chris@16:     BOOST_ASSERT( r.den > zero );
Chris@16: 
Chris@16:     // Determine relative order by expanding each value to its simple continued
Chris@16:     // fraction representation using the Euclidian GCD algorithm.
Chris@16:     struct { int_type  n, d, q, r; }
Chris@16:      ts = { this->num, this->den, static_cast<int_type>(this->num / this->den),
Chris@16:      static_cast<int_type>(this->num % this->den) },
Chris@16:      rs = { r.num, r.den, static_cast<int_type>(r.num / r.den),
Chris@16:      static_cast<int_type>(r.num % r.den) };
Chris@16:     unsigned  reverse = 0u;
Chris@16: 
Chris@16:     // Normalize negative moduli by repeatedly adding the (positive) denominator
Chris@16:     // and decrementing the quotient.  Later cycles should have all positive
Chris@16:     // values, so this only has to be done for the first cycle.  (The rules of
Chris@16:     // C++ require a nonnegative quotient & remainder for a nonnegative dividend
Chris@16:     // & positive divisor.)
Chris@16:     while ( ts.r < zero )  { ts.r += ts.d; --ts.q; }
Chris@16:     while ( rs.r < zero )  { rs.r += rs.d; --rs.q; }
Chris@16: 
Chris@16:     // Loop through and compare each variable's continued-fraction components
Chris@101:     for ( ;; )
Chris@16:     {
Chris@16:         // The quotients of the current cycle are the continued-fraction
Chris@16:         // components.  Comparing two c.f. is comparing their sequences,
Chris@16:         // stopping at the first difference.
Chris@16:         if ( ts.q != rs.q )
Chris@16:         {
Chris@16:             // Since reciprocation changes the relative order of two variables,
Chris@16:             // and c.f. use reciprocals, the less/greater-than test reverses
Chris@16:             // after each index.  (Start w/ non-reversed @ whole-number place.)
Chris@16:             return reverse ? ts.q > rs.q : ts.q < rs.q;
Chris@16:         }
Chris@16: 
Chris@16:         // Prepare the next cycle
Chris@16:         reverse ^= 1u;
Chris@16: 
Chris@16:         if ( (ts.r == zero) || (rs.r == zero) )
Chris@16:         {
Chris@16:             // At least one variable's c.f. expansion has ended
Chris@16:             break;
Chris@16:         }
Chris@16: 
Chris@16:         ts.n = ts.d;         ts.d = ts.r;
Chris@16:         ts.q = ts.n / ts.d;  ts.r = ts.n % ts.d;
Chris@16:         rs.n = rs.d;         rs.d = rs.r;
Chris@16:         rs.q = rs.n / rs.d;  rs.r = rs.n % rs.d;
Chris@16:     }
Chris@16: 
Chris@16:     // Compare infinity-valued components for otherwise equal sequences
Chris@16:     if ( ts.r == rs.r )
Chris@16:     {
Chris@16:         // Both remainders are zero, so the next (and subsequent) c.f.
Chris@16:         // components for both sequences are infinity.  Therefore, the sequences
Chris@16:         // and their corresponding values are equal.
Chris@16:         return false;
Chris@16:     }
Chris@16:     else
Chris@16:     {
Chris@16: #ifdef BOOST_MSVC
Chris@16: #pragma warning(push)
Chris@16: #pragma warning(disable:4800)
Chris@16: #endif
Chris@16:         // Exactly one of the remainders is zero, so all following c.f.
Chris@16:         // components of that variable are infinity, while the other variable
Chris@16:         // has a finite next c.f. component.  So that other variable has the
Chris@16:         // lesser value (modulo the reversal flag!).
Chris@16:         return ( ts.r != zero ) != static_cast<bool>( reverse );
Chris@16: #ifdef BOOST_MSVC
Chris@16: #pragma warning(pop)
Chris@16: #endif
Chris@16:     }
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@16: bool rational<IntType>::operator< (param_type i) const
Chris@16: {
Chris@16:     // Avoid repeated construction
Chris@16:     int_type const  zero( 0 );
Chris@16: 
Chris@16:     // Break value into mixed-fraction form, w/ always-nonnegative remainder
Chris@16:     BOOST_ASSERT( this->den > zero );
Chris@16:     int_type  q = this->num / this->den, r = this->num % this->den;
Chris@16:     while ( r < zero )  { r += this->den; --q; }
Chris@16: 
Chris@16:     // Compare with just the quotient, since the remainder always bumps the
Chris@16:     // value up.  [Since q = floor(n/d), and if n/d < i then q < i, if n/d == i
Chris@16:     // then q == i, if n/d == i + r/d then q == i, and if n/d >= i + 1 then
Chris@16:     // q >= i + 1 > i; therefore n/d < i iff q < i.]
Chris@16:     return q < i;
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@16: bool rational<IntType>::operator> (param_type i) const
Chris@16: {
Chris@101:     return operator==(i)? false: !operator<(i);
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@101: BOOST_CONSTEXPR
Chris@16: inline bool rational<IntType>::operator== (const rational<IntType>& r) const
Chris@16: {
Chris@16:     return ((num == r.num) && (den == r.den));
Chris@16: }
Chris@16: 
Chris@16: template <typename IntType>
Chris@101: BOOST_CONSTEXPR
Chris@16: inline bool rational<IntType>::operator== (param_type i) const
Chris@16: {
Chris@16:     return ((den == IntType(1)) && (num == i));
Chris@16: }
Chris@16: 
Chris@16: // Invariant check
Chris@16: template <typename IntType>
Chris@16: inline bool rational<IntType>::test_invariant() const
Chris@16: {
Chris@101:     return ( this->den > int_type(0) ) && ( integer::gcd(this->num, this->den) ==
Chris@16:      int_type(1) );
Chris@16: }
Chris@16: 
Chris@16: // Normalisation
Chris@16: template <typename IntType>
Chris@16: void rational<IntType>::normalize()
Chris@16: {
Chris@16:     // Avoid repeated construction
Chris@16:     IntType zero(0);
Chris@16: 
Chris@16:     if (den == zero)
Chris@16:         throw bad_rational();
Chris@16: 
Chris@16:     // Handle the case of zero separately, to avoid division by zero
Chris@16:     if (num == zero) {
Chris@16:         den = IntType(1);
Chris@16:         return;
Chris@16:     }
Chris@16: 
Chris@101:     IntType g = integer::gcd(num, den);
Chris@16: 
Chris@16:     num /= g;
Chris@16:     den /= g;
Chris@16: 
Chris@16:     // Ensure that the denominator is positive
Chris@16:     if (den < zero) {
Chris@16:         num = -num;
Chris@16:         den = -den;
Chris@16:     }
Chris@16: 
Chris@101:     // ...But acknowledge that the previous step doesn't always work.
Chris@101:     // (Nominally, this should be done before the mutating steps, but this
Chris@101:     // member function is only called during the constructor, so we never have
Chris@101:     // to worry about zombie objects.)
Chris@101:     if (den < zero)
Chris@101:         throw bad_rational( "bad rational: non-zero singular denominator" );
Chris@101: 
Chris@16:     BOOST_ASSERT( this->test_invariant() );
Chris@16: }
Chris@16: 
Chris@101: #ifndef BOOST_NO_IOSTREAM
Chris@16: namespace detail {
Chris@16: 
Chris@16:     // A utility class to reset the format flags for an istream at end
Chris@16:     // of scope, even in case of exceptions
Chris@16:     struct resetter {
Chris@16:         resetter(std::istream& is) : is_(is), f_(is.flags()) {}
Chris@16:         ~resetter() { is_.flags(f_); }
Chris@16:         std::istream& is_;
Chris@16:         std::istream::fmtflags f_;      // old GNU c++ lib has no ios_base
Chris@16:     };
Chris@16: 
Chris@16: }
Chris@16: 
Chris@16: // Input and output
Chris@16: template <typename IntType>
Chris@16: std::istream& operator>> (std::istream& is, rational<IntType>& r)
Chris@16: {
Chris@101:     using std::ios;
Chris@101: 
Chris@16:     IntType n = IntType(0), d = IntType(1);
Chris@16:     char c = 0;
Chris@16:     detail::resetter sentry(is);
Chris@16: 
Chris@101:     if ( is >> n )
Chris@101:     {
Chris@101:         if ( is.get(c) )
Chris@101:         {
Chris@101:             if ( c == '/' )
Chris@101:             {
Chris@101:                 if ( is >> std::noskipws >> d )
Chris@101:                     try {
Chris@101:                         r.assign( n, d );
Chris@101:                     } catch ( bad_rational & ) {        // normalization fail
Chris@101:                         try { is.setstate(ios::failbit); }
Chris@101:                         catch ( ... ) {}  // don't throw ios_base::failure...
Chris@101:                         if ( is.exceptions() & ios::failbit )
Chris@101:                             throw;   // ...but the original exception instead
Chris@101:                         // ELSE: suppress the exception, use just error flags
Chris@101:                     }
Chris@101:             }
Chris@101:             else
Chris@101:                 is.setstate( ios::failbit );
Chris@101:         }
Chris@101:     }
Chris@16: 
Chris@16:     return is;
Chris@16: }
Chris@16: 
Chris@16: // Add manipulators for output format?
Chris@16: template <typename IntType>
Chris@16: std::ostream& operator<< (std::ostream& os, const rational<IntType>& r)
Chris@16: {
Chris@101:     using namespace std;
Chris@101: 
Chris@101:     // The slash directly precedes the denominator, which has no prefixes.
Chris@101:     ostringstream  ss;
Chris@101: 
Chris@101:     ss.copyfmt( os );
Chris@101:     ss.tie( NULL );
Chris@101:     ss.exceptions( ios::goodbit );
Chris@101:     ss.width( 0 );
Chris@101:     ss << noshowpos << noshowbase << '/' << r.denominator();
Chris@101: 
Chris@101:     // The numerator holds the showpos, internal, and showbase flags.
Chris@101:     string const   tail = ss.str();
Chris@101:     streamsize const  w = os.width() - static_cast<streamsize>( tail.size() );
Chris@101: 
Chris@101:     ss.clear();
Chris@101:     ss.str( "" );
Chris@101:     ss.flags( os.flags() );
Chris@101:     ss << setw( w < 0 || (os.flags() & ios::adjustfield) != ios::internal ? 0 :
Chris@101:      w ) << r.numerator();
Chris@101:     return os << ss.str() + tail;
Chris@16: }
Chris@101: #endif  // BOOST_NO_IOSTREAM
Chris@16: 
Chris@16: // Type conversion
Chris@16: template <typename T, typename IntType>
Chris@101: BOOST_CONSTEXPR
Chris@101: inline T rational_cast(const rational<IntType>& src)
Chris@16: {
Chris@16:     return static_cast<T>(src.numerator())/static_cast<T>(src.denominator());
Chris@16: }
Chris@16: 
Chris@16: // Do not use any abs() defined on IntType - it isn't worth it, given the
Chris@16: // difficulties involved (Koenig lookup required, there may not *be* an abs()
Chris@16: // defined, etc etc).
Chris@16: template <typename IntType>
Chris@16: inline rational<IntType> abs(const rational<IntType>& r)
Chris@16: {
Chris@101:     return r.numerator() >= IntType(0)? r: -r;
Chris@16: }
Chris@16: 
Chris@16: } // namespace boost
Chris@16: 
Chris@16: #endif  // BOOST_RATIONAL_HPP
Chris@16: