// 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 | |