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/Src/Dependencies/Boost/boost/rational.hpp

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  1//  Boost rational.hpp header file  ------------------------------------------//
  2
  3//  (C) Copyright Paul Moore 1999. Permission to copy, use, modify, sell and
  4//  distribute this software is granted provided this copyright notice appears
  5//  in all copies. This software is provided "as is" without express or
  6//  implied warranty, and with no claim as to its suitability for any purpose.
  7
  8// boostinspect:nolicense (don't complain about the lack of a Boost license)
  9// (Paul Moore hasn't been in contact for years, so there's no way to change the
 10// license.)
 11
 12//  See http://www.boost.org/libs/rational for documentation.
 13
 14//  Credits:
 15//  Thanks to the boost mailing list in general for useful comments.
 16//  Particular contributions included:
 17//    Andrew D Jewell, for reminding me to take care to avoid overflow
 18//    Ed Brey, for many comments, including picking up on some dreadful typos
 19//    Stephen Silver contributed the test suite and comments on user-defined
 20//    IntType
 21//    Nickolay Mladenov, for the implementation of operator+=
 22
 23//  Revision History
 24//  05 Nov 06  Change rational_cast to not depend on division between different
 25//             types (Daryle Walker)
 26//  04 Nov 06  Off-load GCD and LCM to Boost.Math; add some invariant checks;
 27//             add std::numeric_limits<> requirement to help GCD (Daryle Walker)
 28//  31 Oct 06  Recoded both operator< to use round-to-negative-infinity
 29//             divisions; the rational-value version now uses continued fraction
 30//             expansion to avoid overflows, for bug #798357 (Daryle Walker)
 31//  20 Oct 06  Fix operator bool_type for CW 8.3 (Joaquín M López Muńoz)
 32//  18 Oct 06  Use EXPLICIT_TEMPLATE_TYPE helper macros from Boost.Config
 33//             (Joaquín M López Muńoz)
 34//  27 Dec 05  Add Boolean conversion operator (Daryle Walker)
 35//  28 Sep 02  Use _left versions of operators from operators.hpp
 36//  05 Jul 01  Recode gcd(), avoiding std::swap (Helmut Zeisel)
 37//  03 Mar 01  Workarounds for Intel C++ 5.0 (David Abrahams)
 38//  05 Feb 01  Update operator>> to tighten up input syntax
 39//  05 Feb 01  Final tidy up of gcd code prior to the new release
 40//  27 Jan 01  Recode abs() without relying on abs(IntType)
 41//  21 Jan 01  Include Nickolay Mladenov's operator+= algorithm,
 42//             tidy up a number of areas, use newer features of operators.hpp
 43//             (reduces space overhead to zero), add operator!,
 44//             introduce explicit mixed-mode arithmetic operations
 45//  12 Jan 01  Include fixes to handle a user-defined IntType better
 46//  19 Nov 00  Throw on divide by zero in operator /= (John (EBo) David)
 47//  23 Jun 00  Incorporate changes from Mark Rodgers for Borland C++
 48//  22 Jun 00  Change _MSC_VER to BOOST_MSVC so other compilers are not
 49//             affected (Beman Dawes)
 50//   6 Mar 00  Fix operator-= normalization, #include <string> (Jens Maurer)
 51//  14 Dec 99  Modifications based on comments from the boost list
 52//  09 Dec 99  Initial Version (Paul Moore)
 53
 54#ifndef BOOST_RATIONAL_HPP
 55#define BOOST_RATIONAL_HPP
 56
 57#include <iostream>              // for std::istream and std::ostream
 58#include <ios>                   // for std::noskipws
 59#include <stdexcept>             // for std::domain_error
 60#include <string>                // for std::string implicit constructor
 61#include <boost/operators.hpp>   // for boost::addable etc
 62#include <cstdlib>               // for std::abs
 63#include <boost/call_traits.hpp> // for boost::call_traits
 64#include <boost/config.hpp>      // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC
 65#include <boost/detail/workaround.hpp> // for BOOST_WORKAROUND
 66#include <boost/assert.hpp>      // for BOOST_ASSERT
 67#include <boost/math/common_factor_rt.hpp>  // for boost::math::gcd, lcm
 68#include <limits>                // for std::numeric_limits
 69#include <boost/static_assert.hpp>  // for BOOST_STATIC_ASSERT
 70
 71// Control whether depreciated GCD and LCM functions are included (default: yes)
 72#ifndef BOOST_CONTROL_RATIONAL_HAS_GCD
 73#define BOOST_CONTROL_RATIONAL_HAS_GCD  1
 74#endif
 75
 76namespace boost {
 77
 78#if BOOST_CONTROL_RATIONAL_HAS_GCD
 79template <typename IntType>
 80IntType gcd(IntType n, IntType m)
 81{
 82    // Defer to the version in Boost.Math
 83    return math::gcd( n, m );
 84}
 85
 86template <typename IntType>
 87IntType lcm(IntType n, IntType m)
 88{
 89    // Defer to the version in Boost.Math
 90    return math::lcm( n, m );
 91}
 92#endif  // BOOST_CONTROL_RATIONAL_HAS_GCD
 93
 94class bad_rational : public std::domain_error
 95{
 96public:
 97    explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}
 98};
 99
100template <typename IntType>
101class rational;
102
103template <typename IntType>
104rational<IntType> abs(const rational<IntType>& r);
105
106template <typename IntType>
107class rational :
108    less_than_comparable < rational<IntType>,
109    equality_comparable < rational<IntType>,
110    less_than_comparable2 < rational<IntType>, IntType,
111    equality_comparable2 < rational<IntType>, IntType,
112    addable < rational<IntType>,
113    subtractable < rational<IntType>,
114    multipliable < rational<IntType>,
115    dividable < rational<IntType>,
116    addable2 < rational<IntType>, IntType,
117    subtractable2 < rational<IntType>, IntType,
118    subtractable2_left < rational<IntType>, IntType,
119    multipliable2 < rational<IntType>, IntType,
120    dividable2 < rational<IntType>, IntType,
121    dividable2_left < rational<IntType>, IntType,
122    incrementable < rational<IntType>,
123    decrementable < rational<IntType>
124    > > > > > > > > > > > > > > > >
125{
126    // Class-wide pre-conditions
127    BOOST_STATIC_ASSERT( ::std::numeric_limits<IntType>::is_specialized );
128
129    // Helper types
130    typedef typename boost::call_traits<IntType>::param_type param_type;
131
132    struct helper { IntType parts[2]; };
133    typedef IntType (helper::* bool_type)[2];
134
135public:
136    typedef IntType int_type;
137    rational() : num(0), den(1) {}
138    rational(param_type n) : num(n), den(1) {}
139    rational(param_type n, param_type d) : num(n), den(d) { normalize(); }
140
141    // Default copy constructor and assignment are fine
142
143    // Add assignment from IntType
144    rational& operator=(param_type n) { return assign(n, 1); }
145
146    // Assign in place
147    rational& assign(param_type n, param_type d);
148
149    // Access to representation
150    IntType numerator() const { return num; }
151    IntType denominator() const { return den; }
152
153    // Arithmetic assignment operators
154    rational& operator+= (const rational& r);
155    rational& operator-= (const rational& r);
156    rational& operator*= (const rational& r);
157    rational& operator/= (const rational& r);
158
159    rational& operator+= (param_type i);
160    rational& operator-= (param_type i);
161    rational& operator*= (param_type i);
162    rational& operator/= (param_type i);
163
164    // Increment and decrement
165    const rational& operator++();
166    const rational& operator--();
167
168    // Operator not
169    bool operator!() const { return !num; }
170
171    // Boolean conversion
172    
173#if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
174    // The "ISO C++ Template Parser" option in CW 8.3 chokes on the
175    // following, hence we selectively disable that option for the
176    // offending memfun.
177#pragma parse_mfunc_templ off
178#endif
179
180    operator bool_type() const { return operator !() ? 0 : &helper::parts; }
181
182#if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
183#pragma parse_mfunc_templ reset
184#endif
185
186    // Comparison operators
187    bool operator< (const rational& r) const;
188    bool operator== (const rational& r) const;
189
190    bool operator< (param_type i) const;
191    bool operator> (param_type i) const;
192    bool operator== (param_type i) const;
193
194private:
195    // Implementation - numerator and denominator (normalized).
196    // Other possibilities - separate whole-part, or sign, fields?
197    IntType num;
198    IntType den;
199
200    // Representation note: Fractions are kept in normalized form at all
201    // times. normalized form is defined as gcd(num,den) == 1 and den > 0.
202    // In particular, note that the implementation of abs() below relies
203    // on den always being positive.
204    bool test_invariant() const;
205    void normalize();
206};
207
208// Assign in place
209template <typename IntType>
210inline rational<IntType>& rational<IntType>::assign(param_type n, param_type d)
211{
212    num = n;
213    den = d;
214    normalize();
215    return *this;
216}
217
218// Unary plus and minus
219template <typename IntType>
220inline rational<IntType> operator+ (const rational<IntType>& r)
221{
222    return r;
223}
224
225template <typename IntType>
226inline rational<IntType> operator- (const rational<IntType>& r)
227{
228    return rational<IntType>(-r.numerator(), r.denominator());
229}
230
231// Arithmetic assignment operators
232template <typename IntType>
233rational<IntType>& rational<IntType>::operator+= (const rational<IntType>& r)
234{
235    // This calculation avoids overflow, and minimises the number of expensive
236    // calculations. Thanks to Nickolay Mladenov for this algorithm.
237    //
238    // Proof:
239    // We have to compute a/b + c/d, where gcd(a,b)=1 and gcd(b,c)=1.
240    // Let g = gcd(b,d), and b = b1*g, d=d1*g. Then gcd(b1,d1)=1
241    //
242    // The result is (a*d1 + c*b1) / (b1*d1*g).
243    // Now we have to normalize this ratio.
244    // Let's assume h | gcd((a*d1 + c*b1), (b1*d1*g)), and h > 1
245    // If h | b1 then gcd(h,d1)=1 and hence h|(a*d1+c*b1) => h|a.
246    // But since gcd(a,b1)=1 we have h=1.
247    // Similarly h|d1 leads to h=1.
248    // So we have that h | gcd((a*d1 + c*b1) , (b1*d1*g)) => h|g
249    // Finally we have gcd((a*d1 + c*b1), (b1*d1*g)) = gcd((a*d1 + c*b1), g)
250    // Which proves that instead of normalizing the result, it is better to
251    // divide num and den by gcd((a*d1 + c*b1), g)
252
253    // Protect against self-modification
254    IntType r_num = r.num;
255    IntType r_den = r.den;
256
257    IntType g = math::gcd(den, r_den);
258    den /= g;  // = b1 from the calculations above
259    num = num * (r_den / g) + r_num * den;
260    g = math::gcd(num, g);
261    num /= g;
262    den *= r_den/g;
263
264    return *this;
265}
266
267template <typename IntType>
268rational<IntType>& rational<IntType>::operator-= (const rational<IntType>& r)
269{
270    // Protect against self-modification
271    IntType r_num = r.num;
272    IntType r_den = r.den;
273
274    // This calculation avoids overflow, and minimises the number of expensive
275    // calculations. It corresponds exactly to the += case above
276    IntType g = math::gcd(den, r_den);
277    den /= g;
278    num = num * (r_den / g) - r_num * den;
279    g = math::gcd(num, g);
280    num /= g;
281    den *= r_den/g;
282
283    return *this;
284}
285
286template <typename IntType>
287rational<IntType>& rational<IntType>::operator*= (const rational<IntType>& r)
288{
289    // Protect against self-modification
290    IntType r_num = r.num;
291    IntType r_den = r.den;
292
293    // Avoid overflow and preserve normalization
294    IntType gcd1 = math::gcd(num, r_den);
295    IntType gcd2 = math::gcd(r_num, den);
296    num = (num/gcd1) * (r_num/gcd2);
297    den = (den/gcd2) * (r_den/gcd1);
298    return *this;
299}
300
301template <typename IntType>
302rational<IntType>& rational<IntType>::operator/= (const rational<IntType>& r)
303{
304    // Protect against self-modification
305    IntType r_num = r.num;
306    IntType r_den = r.den;
307
308    // Avoid repeated construction
309    IntType zero(0);
310
311    // Trap division by zero
312    if (r_num == zero)
313        throw bad_rational();
314    if (num == zero)
315        return *this;
316
317    // Avoid overflow and preserve normalization
318    IntType gcd1 = math::gcd(num, r_num);
319    IntType gcd2 = math::gcd(r_den, den);
320    num = (num/gcd1) * (r_den/gcd2);
321    den = (den/gcd2) * (r_num/gcd1);
322
323    if (den < zero) {
324        num = -num;
325        den = -den;
326    }
327    return *this;
328}
329
330// Mixed-mode operators
331template <typename IntType>
332inline rational<IntType>&
333rational<IntType>::operator+= (param_type i)
334{
335    return operator+= (rational<IntType>(i));
336}
337
338template <typename IntType>
339inline rational<IntType>&
340rational<IntType>::operator-= (param_type i)
341{
342    return operator-= (rational<IntType>(i));
343}
344
345template <typename IntType>
346inline rational<IntType>&
347rational<IntType>::operator*= (param_type i)
348{
349    return operator*= (rational<IntType>(i));
350}
351
352template <typename IntType>
353inline rational<IntType>&
354rational<IntType>::operator/= (param_type i)
355{
356    return operator/= (rational<IntType>(i));
357}
358
359// Increment and decrement
360template <typename IntType>
361inline const rational<IntType>& rational<IntType>::operator++()
362{
363    // This can never denormalise the fraction
364    num += den;
365    return *this;
366}
367
368template <typename IntType>
369inline const rational<IntType>& rational<IntType>::operator--()
370{
371    // This can never denormalise the fraction
372    num -= den;
373    return *this;
374}
375
376// Comparison operators
377template <typename IntType>
378bool rational<IntType>::operator< (const rational<IntType>& r) const
379{
380    // Avoid repeated construction
381    int_type const  zero( 0 );
382
383    // This should really be a class-wide invariant.  The reason for these
384    // checks is that for 2's complement systems, INT_MIN has no corresponding
385    // positive, so negating it during normalization keeps it INT_MIN, which
386    // is bad for later calculations that assume a positive denominator.
387    BOOST_ASSERT( this->den > zero );
388    BOOST_ASSERT( r.den > zero );
389
390    // Determine relative order by expanding each value to its simple continued
391    // fraction representation using the Euclidian GCD algorithm.
392    struct { int_type  n, d, q, r; }  ts = { this->num, this->den, this->num /
393     this->den, this->num % this->den }, rs = { r.num, r.den, r.num / r.den,
394     r.num % r.den };
395    unsigned  reverse = 0u;
396
397    // Normalize negative moduli by repeatedly adding the (positive) denominator
398    // and decrementing the quotient.  Later cycles should have all positive
399    // values, so this only has to be done for the first cycle.  (The rules of
400    // C++ require a nonnegative quotient & remainder for a nonnegative dividend
401    // & positive divisor.)
402    while ( ts.r < zero )  { ts.r += ts.d; --ts.q; }
403    while ( rs.r < zero )  { rs.r += rs.d; --rs.q; }
404
405    // Loop through and compare each variable's continued-fraction components
406    while ( true )
407    {
408        // The quotients of the current cycle are the continued-fraction
409        // components.  Comparing two c.f. is comparing their sequences,
410        // stopping at the first difference.
411        if ( ts.q != rs.q )
412        {
413            // Since reciprocation changes the relative order of two variables,
414            // and c.f. use reciprocals, the less/greater-than test reverses
415            // after each index.  (Start w/ non-reversed @ whole-number place.)
416            return reverse ? ts.q > rs.q : ts.q < rs.q;
417        }
418
419        // Prepare the next cycle
420        reverse ^= 1u;
421
422        if ( (ts.r == zero) || (rs.r == zero) )
423        {
424            // At least one variable's c.f. expansion has ended
425            break;
426        }
427
428        ts.n = ts.d;         ts.d = ts.r;
429        ts.q = ts.n / ts.d;  ts.r = ts.n % ts.d;
430        rs.n = rs.d;         rs.d = rs.r;
431        rs.q = rs.n / rs.d;  rs.r = rs.n % rs.d;
432    }
433
434    // Compare infinity-valued components for otherwise equal sequences
435    if ( ts.r == rs.r )
436    {
437        // Both remainders are zero, so the next (and subsequent) c.f.
438        // components for both sequences are infinity.  Therefore, the sequences
439        // and their corresponding values are equal.
440        return false;
441    }
442    else
443    {
444#ifdef BOOST_MSVC
445#pragma warning(push)
446#pragma warning(disable:4800)
447#endif
448        // Exactly one of the remainders is zero, so all following c.f.
449        // components of that variable are infinity, while the other variable
450        // has a finite next c.f. component.  So that other variable has the
451        // lesser value (modulo the reversal flag!).
452        return ( ts.r != zero ) != static_cast<bool>( reverse );
453#ifdef BOOST_MSVC
454#pragma warning(pop)
455#endif
456    }
457}
458
459template <typename IntType>
460bool rational<IntType>::operator< (param_type i) const
461{
462    // Avoid repeated construction
463    int_type const  zero( 0 );
464
465    // Break value into mixed-fraction form, w/ always-nonnegative remainder
466    BOOST_ASSERT( this->den > zero );
467    int_type  q = this->num / this->den, r = this->num % this->den;
468    while ( r < zero )  { r += this->den; --q; }
469
470    // Compare with just the quotient, since the remainder always bumps the
471    // value up.  [Since q = floor(n/d), and if n/d < i then q < i, if n/d == i
472    // then q == i, if n/d == i + r/d then q == i, and if n/d >= i + 1 then
473    // q >= i + 1 > i; therefore n/d < i iff q < i.]
474    return q < i;
475}
476
477template <typename IntType>
478bool rational<IntType>::operator> (param_type i) const
479{
480    // Trap equality first
481    if (num == i && den == IntType(1))
482        return false;
483
484    // Otherwise, we can use operator<
485    return !operator<(i);
486}
487
488template <typename IntType>
489inline bool rational<IntType>::operator== (const rational<IntType>& r) const
490{
491    return ((num == r.num) && (den == r.den));
492}
493
494template <typename IntType>
495inline bool rational<IntType>::operator== (param_type i) const
496{
497    return ((den == IntType(1)) && (num == i));
498}
499
500// Invariant check
501template <typename IntType>
502inline bool rational<IntType>::test_invariant() const
503{
504    return ( this->den > int_type(0) ) && ( math::gcd(this->num, this->den) ==
505     int_type(1) );
506}
507
508// Normalisation
509template <typename IntType>
510void rational<IntType>::normalize()
511{
512    // Avoid repeated construction
513    IntType zero(0);
514
515    if (den == zero)
516        throw bad_rational();
517
518    // Handle the case of zero separately, to avoid division by zero
519    if (num == zero) {
520        den = IntType(1);
521        return;
522    }
523
524    IntType g = math::gcd(num, den);
525
526    num /= g;
527    den /= g;
528
529    // Ensure that the denominator is positive
530    if (den < zero) {
531        num = -num;
532        den = -den;
533    }
534
535    BOOST_ASSERT( this->test_invariant() );
536}
537
538namespace detail {
539
540    // A utility class to reset the format flags for an istream at end
541    // of scope, even in case of exceptions
542    struct resetter {
543        resetter(std::istream& is) : is_(is), f_(is.flags()) {}
544        ~resetter() { is_.flags(f_); }
545        std::istream& is_;
546        std::istream::fmtflags f_;      // old GNU c++ lib has no ios_base
547    };
548
549}
550
551// Input and output
552template <typename IntType>
553std::istream& operator>> (std::istream& is, rational<IntType>& r)
554{
555    IntType n = IntType(0), d = IntType(1);
556    char c = 0;
557    detail::resetter sentry(is);
558
559    is >> n;
560    c = is.get();
561
562    if (c != '/')
563        is.clear(std::istream::badbit);  // old GNU c++ lib has no ios_base
564
565#if !defined(__GNUC__) || (defined(__GNUC__) && (__GNUC__ >= 3)) || defined __SGI_STL_PORT
566    is >> std::noskipws;
567#else
568    is.unsetf(ios::skipws); // compiles, but seems to have no effect.
569#endif
570    is >> d;
571
572    if (is)
573        r.assign(n, d);
574
575    return is;
576}
577
578// Add manipulators for output format?
579template <typename IntType>
580std::ostream& operator<< (std::ostream& os, const rational<IntType>& r)
581{
582    os << r.numerator() << '/' << r.denominator();
583    return os;
584}
585
586// Type conversion
587template <typename T, typename IntType>
588inline T rational_cast(
589    const rational<IntType>& src BOOST_APPEND_EXPLICIT_TEMPLATE_TYPE(T))
590{
591    return static_cast<T>(src.numerator())/static_cast<T>(src.denominator());
592}
593
594// Do not use any abs() defined on IntType - it isn't worth it, given the
595// difficulties involved (Koenig lookup required, there may not *be* an abs()
596// defined, etc etc).
597template <typename IntType>
598inline rational<IntType> abs(const rational<IntType>& r)
599{
600    if (r.numerator() >= IntType(0))
601        return r;
602
603    return rational<IntType>(-r.numerator(), r.denominator());
604}
605
606} // namespace boost
607
608#endif  // BOOST_RATIONAL_HPP
609