boost/rational.hpp
// 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 // 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 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 IntType int_type; typedef typename boost::call_traits<IntType>::param_type param_type; public: 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; } // 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) { 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