| // 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 (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 <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 |
| |