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. // 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 // 02 Sep 13 Remove unneeded forward declarations; tweak private helper // function (Daryle Walker) // 30 Aug 13 Improve exception safety of "assign"; start modernizing I/O code // (Daryle Walker) // 27 Aug 13 Add cross-version constructor template, plus some private helper // functions; add constructor to exception class to take custom // messages (Daryle Walker) // 25 Aug 13 Add constexpr qualification wherever possible (Daryle Walker) // 05 May 12 Reduced use of implicit gcd (Mario Lang) // 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 <boost/config.hpp> // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC, etc #ifndef BOOST_NO_IOSTREAM #include <iomanip> // for std::setw #include <ios> // for std::noskipws, streamsize #include <istream> // for std::istream #include <ostream> // for std::ostream #include <sstream> // for std::ostringstream #endif #include <cstddef> // for NULL #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/detail/workaround.hpp> // for BOOST_WORKAROUND #include <boost/assert.hpp> // for BOOST_ASSERT #include <boost/integer/common_factor_rt.hpp> // for boost::integer::gcd, lcm #include <limits> // for std::numeric_limits #include <boost/static_assert.hpp> // for BOOST_STATIC_ASSERT #include <boost/throw_exception.hpp> // 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 integer::gcd( n, m ); } template <typename IntType> IntType lcm(IntType n, IntType m) { // Defer to the version in Boost.Math return integer::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") {} explicit bad_rational( char const *what ) : std::domain_error( what ) {} }; 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: // Component type typedef IntType int_type; BOOST_CONSTEXPR rational() : num(0), den(1) {} BOOST_CONSTEXPR rational(param_type n) : num(n), den(1) {} rational(param_type n, param_type d) : num(n), den(d) { normalize(); } #ifndef BOOST_NO_MEMBER_TEMPLATES template < typename NewType > BOOST_CONSTEXPR explicit rational(rational<NewType> const &r) : num(r.numerator()), den(is_normalized(int_type(r.numerator()), int_type(r.denominator())) ? r.denominator() : (BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){} #endif // Default copy constructor and assignment are fine // Add assignment from IntType rational& operator=(param_type i) { num = i; den = 1; return *this; } // Assign in place rational& assign(param_type n, param_type d); // Access to representation BOOST_CONSTEXPR const IntType& numerator() const { return num; } BOOST_CONSTEXPR const 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) { num += i * den; return *this; } rational& operator-= (param_type i) { num -= i * den; return *this; } rational& operator*= (param_type i); rational& operator/= (param_type i); // Increment and decrement const rational& operator++() { num += den; return *this; } const rational& operator--() { num -= den; return *this; } // Operator not BOOST_CONSTEXPR 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 BOOST_CONSTEXPR 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; BOOST_CONSTEXPR bool operator== (const rational& r) const; bool operator< (param_type i) const; bool operator> (param_type i) const; BOOST_CONSTEXPR bool operator== (param_type i) const; private: // Implementation - numerator and denominator (normalized). // Other possibilities - separate whole-part, or sign, fields? IntType num; IntType den; // Helper functions static BOOST_CONSTEXPR int_type inner_gcd( param_type a, param_type b, int_type const &zero = int_type(0) ) { return b == zero ? a : inner_gcd(b, a % b, zero); } static BOOST_CONSTEXPR int_type inner_abs( param_type x, int_type const &zero = int_type(0) ) { return x < zero ? -x : +x; } // 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(); static BOOST_CONSTEXPR bool is_normalized( param_type n, param_type d, int_type const &zero = int_type(0), int_type const &one = int_type(1) ) { return d > zero && ( n != zero || d == one ) && inner_abs( inner_gcd(n, d, zero), zero ) == one; } }; // Assign in place template <typename IntType> inline rational<IntType>& rational<IntType>::assign(param_type n, param_type d) { return *this = rational( n, d ); } // Unary plus and minus template <typename IntType> BOOST_CONSTEXPR 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 = integer::gcd(den, r_den); den /= g; // = b1 from the calculations above num = num * (r_den / g) + r_num * den; g = integer::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 = integer::gcd(den, r_den); den /= g; num = num * (r_den / g) - r_num * den; g = integer::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 = integer::gcd(num, r_den); IntType gcd2 = integer::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) BOOST_THROW_EXCEPTION(bad_rational()); if (num == zero) return *this; // Avoid overflow and preserve normalization IntType gcd1 = integer::gcd(num, r_num); IntType gcd2 = integer::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) { // Avoid overflow and preserve normalization IntType gcd = integer::gcd(i, den); num *= i / gcd; den /= gcd; return *this; } template <typename IntType> rational<IntType>& rational<IntType>::operator/= (param_type i) { // Avoid repeated construction IntType const zero(0); if(i == zero) BOOST_THROW_EXCEPTION(bad_rational()); if (num == zero) return *this; // Avoid overflow and preserve normalization IntType const gcd = integer::gcd(num, i); num /= gcd; den *= i / gcd; if (den < zero) { num = -num; den = -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, static_cast<int_type>(this->num / this->den), static_cast<int_type>(this->num % this->den) }, rs = { r.num, r.den, static_cast<int_type>(r.num / r.den), static_cast<int_type>(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 for ( ;; ) { // 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 { return operator==(i)? false: !operator<(i); } template <typename IntType> BOOST_CONSTEXPR inline bool rational<IntType>::operator== (const rational<IntType>& r) const { return ((num == r.num) && (den == r.den)); } template <typename IntType> BOOST_CONSTEXPR 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) ) && ( integer::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) BOOST_THROW_EXCEPTION(bad_rational()); // Handle the case of zero separately, to avoid division by zero if (num == zero) { den = IntType(1); return; } IntType g = integer::gcd(num, den); num /= g; den /= g; // Ensure that the denominator is positive if (den < zero) { num = -num; den = -den; } // ...But acknowledge that the previous step doesn't always work. // (Nominally, this should be done before the mutating steps, but this // member function is only called during the constructor, so we never have // to worry about zombie objects.) if (den < zero) BOOST_THROW_EXCEPTION(bad_rational("bad rational: non-zero singular denominator")); BOOST_ASSERT( this->test_invariant() ); } #ifndef BOOST_NO_IOSTREAM 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) { using std::ios; IntType n = IntType(0), d = IntType(1); char c = 0; detail::resetter sentry(is); if ( is >> n ) { if ( is.get(c) ) { if ( c == '/' ) { if ( is >> std::noskipws >> d ) try { r.assign( n, d ); } catch ( bad_rational & ) { // normalization fail try { is.setstate(ios::failbit); } catch ( ... ) {} // don't throw ios_base::failure... if ( is.exceptions() & ios::failbit ) throw; // ...but the original exception instead // ELSE: suppress the exception, use just error flags } } else is.setstate( ios::failbit ); } } return is; } // Add manipulators for output format? template <typename IntType> std::ostream& operator<< (std::ostream& os, const rational<IntType>& r) { using namespace std; // The slash directly precedes the denominator, which has no prefixes. ostringstream ss; ss.copyfmt( os ); ss.tie( NULL ); ss.exceptions( ios::goodbit ); ss.width( 0 ); ss << noshowpos << noshowbase << '/' << r.denominator(); // The numerator holds the showpos, internal, and showbase flags. string const tail = ss.str(); streamsize const w = os.width() - static_cast<streamsize>( tail.size() ); ss.clear(); ss.str( "" ); ss.flags( os.flags() ); ss << setw( w < 0 || (os.flags() & ios::adjustfield) != ios::internal ? 0 : w ) << r.numerator(); return os << ss.str() + tail; } #endif // BOOST_NO_IOSTREAM // Type conversion template <typename T, typename IntType> BOOST_CONSTEXPR inline T rational_cast(const rational<IntType>& src) { 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) { return r.numerator() >= IntType(0)? r: -r; } namespace integer { template <typename IntType> struct gcd_evaluator< rational<IntType> > { typedef rational<IntType> result_type, first_argument_type, second_argument_type; result_type operator() ( first_argument_type const &a , second_argument_type const &b ) const { return result_type(integer::gcd(a.numerator(), b.numerator()), integer::lcm(a.denominator(), b.denominator())); } }; template <typename IntType> struct lcm_evaluator< rational<IntType> > { typedef rational<IntType> result_type, first_argument_type, second_argument_type; result_type operator() ( first_argument_type const &a , second_argument_type const &b ) const { return result_type(integer::lcm(a.numerator(), b.numerator()), integer::gcd(a.denominator(), b.denominator())); } }; } // namespace integer } // namespace boost #endif // BOOST_RATIONAL_HPP