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 //  See http://www.boost.org/libs/rational for documentation.

     9 

    10 //  Credits:

    11 //  Thanks to the boost mailing list in general for useful comments.

    12 //  Particular contributions included:

    13 //    Andrew D Jewell, for reminding me to take care to avoid overflow

    14 //    Ed Brey, for many comments, including picking up on some dreadful typos

    15 //    Stephen Silver contributed the test suite and comments on user-defined

    16 //    IntType

    17 //    Nickolay Mladenov, for the implementation of operator+=

    18 

    19 //  Revision History

    20 //  20 Oct 06  Fix operator bool_type for CW 8.3 (Joaquín M López Muñoz)

    21 //  18 Oct 06  Use EXPLICIT_TEMPLATE_TYPE helper macros from Boost.Config

    22 //             (Joaquín M López Muñoz)

    23 //  27 Dec 05  Add Boolean conversion operator (Daryle Walker)

    24 //  28 Sep 02  Use _left versions of operators from operators.hpp

    25 //  05 Jul 01  Recode gcd(), avoiding std::swap (Helmut Zeisel)

    26 //  03 Mar 01  Workarounds for Intel C++ 5.0 (David Abrahams)

    27 //  05 Feb 01  Update operator>> to tighten up input syntax

    28 //  05 Feb 01  Final tidy up of gcd code prior to the new release

    29 //  27 Jan 01  Recode abs() without relying on abs(IntType)

    30 //  21 Jan 01  Include Nickolay Mladenov's operator+= algorithm,

    31 //             tidy up a number of areas, use newer features of operators.hpp

    32 //             (reduces space overhead to zero), add operator!,

    33 //             introduce explicit mixed-mode arithmetic operations

    34 //  12 Jan 01  Include fixes to handle a user-defined IntType better

    35 //  19 Nov 00  Throw on divide by zero in operator /= (John (EBo) David)

    36 //  23 Jun 00  Incorporate changes from Mark Rodgers for Borland C++

    37 //  22 Jun 00  Change _MSC_VER to BOOST_MSVC so other compilers are not

    38 //             affected (Beman Dawes)

    39 //   6 Mar 00  Fix operator-= normalization, #include <string> (Jens Maurer)

    40 //  14 Dec 99  Modifications based on comments from the boost list

    41 //  09 Dec 99  Initial Version (Paul Moore)

    42 

    43 #ifndef BOOST_RATIONAL_HPP

    44 #define BOOST_RATIONAL_HPP

    45 

    46 #include <iostream>              // for std::istream and std::ostream

    47 #include <iomanip>               // for std::noskipws

    48 #include <stdexcept>             // for std::domain_error

    49 #include <string>                // for std::string implicit constructor

    50 #include <boost/operators.hpp>   // for boost::addable etc

    51 #include <cstdlib>               // for std::abs

    52 #include <boost/call_traits.hpp> // for boost::call_traits

    53 #include <boost/config.hpp>      // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC

    54 #include <boost/detail/workaround.hpp> // for BOOST_WORKAROUND

    55 

    56 namespace boost {

    57 

    58 // Note: We use n and m as temporaries in this function, so there is no value

    59 // in using const IntType& as we would only need to make a copy anyway...

    60 template <typename IntType>

    61 IntType gcd(IntType n, IntType m)

    62 {

    63     // Avoid repeated construction

    64     IntType zero(0);

    65 

    66     // This is abs() - given the existence of broken compilers with Koenig

    67     // lookup issues and other problems, I code this explicitly. (Remember,

    68     // IntType may be a user-defined type).

    69     if (n < zero)

    70         n = -n;

    71     if (m < zero)

    72         m = -m;

    73 

    74     // As n and m are now positive, we can be sure that %= returns a

    75     // positive value (the standard guarantees this for built-in types,

    76     // and we require it of user-defined types).

    77     for(;;) {

    78       if(m == zero)

    79         return n;

    80       n %= m;

    81       if(n == zero)

    82         return m;

    83       m %= n;

    84     }

    85 }

    86 

    87 template <typename IntType>

    88 IntType lcm(IntType n, IntType m)

    89 {

    90     // Avoid repeated construction

    91     IntType zero(0);

    92 

    93     if (n == zero || m == zero)

    94         return zero;

    95 

    96     n /= gcd(n, m);

    97     n *= m;

    98 

    99     if (n < zero)

   100         n = -n;

   101     return n;

   102 }

   103 

   104 class bad_rational : public std::domain_error

   105 {

   106 public:

   107     explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}

   108 };

   109 

   110 template <typename IntType>

   111 class rational;

   112 

   113 template <typename IntType>

   114 rational<IntType> abs(const rational<IntType>& r);

   115 

   116 template <typename IntType>

   117 class rational :

   118     less_than_comparable < rational<IntType>,

   119     equality_comparable < rational<IntType>,

   120     less_than_comparable2 < rational<IntType>, IntType,

   121     equality_comparable2 < rational<IntType>, IntType,

   122     addable < rational<IntType>,

   123     subtractable < rational<IntType>,

   124     multipliable < rational<IntType>,

   125     dividable < rational<IntType>,

   126     addable2 < rational<IntType>, IntType,

   127     subtractable2 < rational<IntType>, IntType,

   128     subtractable2_left < rational<IntType>, IntType,

   129     multipliable2 < rational<IntType>, IntType,

   130     dividable2 < rational<IntType>, IntType,

   131     dividable2_left < rational<IntType>, IntType,

   132     incrementable < rational<IntType>,

   133     decrementable < rational<IntType>

   134     > > > > > > > > > > > > > > > >

   135 {

   136     typedef typename boost::call_traits<IntType>::param_type param_type;

   137 

   138     struct helper { IntType parts[2]; };

   139     typedef IntType (helper::* bool_type)[2];

   140 

   141 public:

   142     typedef IntType int_type;

   143     rational() : num(0), den(1) {}

   144     rational(param_type n) : num(n), den(1) {}

   145     rational(param_type n, param_type d) : num(n), den(d) { normalize(); }

   146 

   147     // Default copy constructor and assignment are fine

   148 

   149     // Add assignment from IntType

   150     rational& operator=(param_type n) { return assign(n, 1); }

   151 

   152     // Assign in place

   153     rational& assign(param_type n, param_type d);

   154 

   155     // Access to representation

   156     IntType numerator() const { return num; }

   157     IntType denominator() const { return den; }

   158 

   159     // Arithmetic assignment operators

   160     rational& operator+= (const rational& r);

   161     rational& operator-= (const rational& r);

   162     rational& operator*= (const rational& r);

   163     rational& operator/= (const rational& r);

   164 

   165     rational& operator+= (param_type i);

   166     rational& operator-= (param_type i);

   167     rational& operator*= (param_type i);

   168     rational& operator/= (param_type i);

   169 

   170     // Increment and decrement

   171     const rational& operator++();

   172     const rational& operator--();

   173 

   174     // Operator not

   175     bool operator!() const { return !num; }

   176 

   177     // Boolean conversion

   178     

   179 #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)

   180     // The "ISO C++ Template Parser" option in CW 8.3 chokes on the

   181     // following, hence we selectively disable that option for the

   182     // offending memfun.

   183 #pragma parse_mfunc_templ off

   184 #endif

   185 

   186     operator bool_type() const { return operator !() ? 0 : &helper::parts; }

   187 

   188 #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)

   189 #pragma parse_mfunc_templ reset

   190 #endif

   191 

   192     // Comparison operators

   193     bool operator< (const rational& r) const;

   194     bool operator== (const rational& r) const;

   195 

   196     bool operator< (param_type i) const;

   197     bool operator> (param_type i) const;

   198     bool operator== (param_type i) const;

   199 

   200 private:

   201     // Implementation - numerator and denominator (normalized).

   202     // Other possibilities - separate whole-part, or sign, fields?

   203     IntType num;

   204     IntType den;

   205 

   206     // Representation note: Fractions are kept in normalized form at all

   207     // times. normalized form is defined as gcd(num,den) == 1 and den > 0.

   208     // In particular, note that the implementation of abs() below relies

   209     // on den always being positive.

   210     void normalize();

   211 };

   212 

   213 // Assign in place

   214 template <typename IntType>

   215 inline rational<IntType>& rational<IntType>::assign(param_type n, param_type d)

   216 {

   217     num = n;

   218     den = d;

   219     normalize();

   220     return *this;

   221 }

   222 

   223 // Unary plus and minus

   224 template <typename IntType>

   225 inline rational<IntType> operator+ (const rational<IntType>& r)

   226 {

   227     return r;

   228 }

   229 

   230 template <typename IntType>

   231 inline rational<IntType> operator- (const rational<IntType>& r)

   232 {

   233     return rational<IntType>(-r.numerator(), r.denominator());

   234 }

   235 

   236 // Arithmetic assignment operators

   237 template <typename IntType>

   238 rational<IntType>& rational<IntType>::operator+= (const rational<IntType>& r)

   239 {

   240     // This calculation avoids overflow, and minimises the number of expensive

   241     // calculations. Thanks to Nickolay Mladenov for this algorithm.

   242     //

   243     // Proof:

   244     // We have to compute a/b + c/d, where gcd(a,b)=1 and gcd(b,c)=1.

   245     // Let g = gcd(b,d), and b = b1*g, d=d1*g. Then gcd(b1,d1)=1

   246     //

   247     // The result is (a*d1 + c*b1) / (b1*d1*g).

   248     // Now we have to normalize this ratio.

   249     // Let's assume h | gcd((a*d1 + c*b1), (b1*d1*g)), and h > 1

   250     // If h | b1 then gcd(h,d1)=1 and hence h|(a*d1+c*b1) => h|a.

   251     // But since gcd(a,b1)=1 we have h=1.

   252     // Similarly h|d1 leads to h=1.

   253     // So we have that h | gcd((a*d1 + c*b1) , (b1*d1*g)) => h|g

   254     // Finally we have gcd((a*d1 + c*b1), (b1*d1*g)) = gcd((a*d1 + c*b1), g)

   255     // Which proves that instead of normalizing the result, it is better to

   256     // divide num and den by gcd((a*d1 + c*b1), g)

   257 

   258     // Protect against self-modification

   259     IntType r_num = r.num;

   260     IntType r_den = r.den;

   261 

   262     IntType g = gcd(den, r_den);

   263     den /= g;  // = b1 from the calculations above

   264     num = num * (r_den / g) + r_num * den;

   265     g = gcd(num, g);

   266     num /= g;

   267     den *= r_den/g;

   268 

   269     return *this;

   270 }

   271 

   272 template <typename IntType>

   273 rational<IntType>& rational<IntType>::operator-= (const rational<IntType>& r)

   274 {

   275     // Protect against self-modification

   276     IntType r_num = r.num;

   277     IntType r_den = r.den;

   278 

   279     // This calculation avoids overflow, and minimises the number of expensive

   280     // calculations. It corresponds exactly to the += case above

   281     IntType g = gcd(den, r_den);

   282     den /= g;

   283     num = num * (r_den / g) - r_num * den;

   284     g = gcd(num, g);

   285     num /= g;

   286     den *= r_den/g;

   287 

   288     return *this;

   289 }

   290 

   291 template <typename IntType>

   292 rational<IntType>& rational<IntType>::operator*= (const rational<IntType>& r)

   293 {

   294     // Protect against self-modification

   295     IntType r_num = r.num;

   296     IntType r_den = r.den;

   297 

   298     // Avoid overflow and preserve normalization

   299     IntType gcd1 = gcd<IntType>(num, r_den);

   300     IntType gcd2 = gcd<IntType>(r_num, den);

   301     num = (num/gcd1) * (r_num/gcd2);

   302     den = (den/gcd2) * (r_den/gcd1);

   303     return *this;

   304 }

   305 

   306 template <typename IntType>

   307 rational<IntType>& rational<IntType>::operator/= (const rational<IntType>& r)

   308 {

   309     // Protect against self-modification

   310     IntType r_num = r.num;

   311     IntType r_den = r.den;

   312 

   313     // Avoid repeated construction

   314     IntType zero(0);

   315 

   316     // Trap division by zero

   317     if (r_num == zero)

   318         throw bad_rational();

   319     if (num == zero)

   320         return *this;

   321 

   322     // Avoid overflow and preserve normalization

   323     IntType gcd1 = gcd<IntType>(num, r_num);

   324     IntType gcd2 = gcd<IntType>(r_den, den);

   325     num = (num/gcd1) * (r_den/gcd2);

   326     den = (den/gcd2) * (r_num/gcd1);

   327 

   328     if (den < zero) {

   329         num = -num;

   330         den = -den;

   331     }

   332     return *this;

   333 }

   334 

   335 // Mixed-mode operators

   336 template <typename IntType>

   337 inline rational<IntType>&

   338 rational<IntType>::operator+= (param_type i)

   339 {

   340     return operator+= (rational<IntType>(i));

   341 }

   342 

   343 template <typename IntType>

   344 inline rational<IntType>&

   345 rational<IntType>::operator-= (param_type i)

   346 {

   347     return operator-= (rational<IntType>(i));

   348 }

   349 

   350 template <typename IntType>

   351 inline rational<IntType>&

   352 rational<IntType>::operator*= (param_type i)

   353 {

   354     return operator*= (rational<IntType>(i));

   355 }

   356 

   357 template <typename IntType>

   358 inline rational<IntType>&

   359 rational<IntType>::operator/= (param_type i)

   360 {

   361     return operator/= (rational<IntType>(i));

   362 }

   363 

   364 // Increment and decrement

   365 template <typename IntType>

   366 inline const rational<IntType>& rational<IntType>::operator++()

   367 {

   368     // This can never denormalise the fraction

   369     num += den;

   370     return *this;

   371 }

   372 

   373 template <typename IntType>

   374 inline const rational<IntType>& rational<IntType>::operator--()

   375 {

   376     // This can never denormalise the fraction

   377     num -= den;

   378     return *this;

   379 }

   380 

   381 // Comparison operators

   382 template <typename IntType>

   383 bool rational<IntType>::operator< (const rational<IntType>& r) const

   384 {

   385     // Avoid repeated construction

   386     IntType zero(0);

   387 

   388     // If the two values have different signs, we don't need to do the

   389     // expensive calculations below. We take advantage here of the fact

   390     // that the denominator is always positive.

   391     if (num < zero && r.num >= zero) // -ve < +ve

   392         return true;

   393     if (num >= zero && r.num <= zero) // +ve or zero is not < -ve or zero

   394         return false;

   395 

   396     // Avoid overflow

   397     IntType gcd1 = gcd<IntType>(num, r.num);

   398     IntType gcd2 = gcd<IntType>(r.den, den);

   399     return (num/gcd1) * (r.den/gcd2) < (den/gcd2) * (r.num/gcd1);

   400 }

   401 

   402 template <typename IntType>

   403 bool rational<IntType>::operator< (param_type i) const

   404 {

   405     // Avoid repeated construction

   406     IntType zero(0);

   407 

   408     // If the two values have different signs, we don't need to do the

   409     // expensive calculations below. We take advantage here of the fact

   410     // that the denominator is always positive.

   411     if (num < zero && i >= zero) // -ve < +ve

   412         return true;

   413     if (num >= zero && i <= zero) // +ve or zero is not < -ve or zero

   414         return false;

   415 

   416     // Now, use the fact that n/d truncates towards zero as long as n and d

   417     // are both positive.

   418     // Divide instead of multiplying to avoid overflow issues. Of course,

   419     // division may be slower, but accuracy is more important than speed...

   420     if (num > zero)

   421         return (num/den) < i;

   422     else

   423         return -i < (-num/den);

   424 }

   425 

   426 template <typename IntType>

   427 bool rational<IntType>::operator> (param_type i) const

   428 {

   429     // Trap equality first

   430     if (num == i && den == IntType(1))

   431         return false;

   432 

   433     // Otherwise, we can use operator<

   434     return !operator<(i);

   435 }

   436 

   437 template <typename IntType>

   438 inline bool rational<IntType>::operator== (const rational<IntType>& r) const

   439 {

   440     return ((num == r.num) && (den == r.den));

   441 }

   442 

   443 template <typename IntType>

   444 inline bool rational<IntType>::operator== (param_type i) const

   445 {

   446     return ((den == IntType(1)) && (num == i));

   447 }

   448 

   449 // Normalisation

   450 template <typename IntType>

   451 void rational<IntType>::normalize()

   452 {

   453     // Avoid repeated construction

   454     IntType zero(0);

   455 

   456     if (den == zero)

   457         throw bad_rational();

   458 

   459     // Handle the case of zero separately, to avoid division by zero

   460     if (num == zero) {

   461         den = IntType(1);

   462         return;

   463     }

   464 

   465     IntType g = gcd<IntType>(num, den);

   466 

   467     num /= g;

   468     den /= g;

   469 

   470     // Ensure that the denominator is positive

   471     if (den < zero) {

   472         num = -num;

   473         den = -den;

   474     }

   475 }

   476 

   477 namespace detail {

   478 

   479     // A utility class to reset the format flags for an istream at end

   480     // of scope, even in case of exceptions

   481     struct resetter {

   482         resetter(std::istream& is) : is_(is), f_(is.flags()) {}

   483         ~resetter() { is_.flags(f_); }

   484         std::istream& is_;

   485         std::istream::fmtflags f_;      // old GNU c++ lib has no ios_base

   486     };

   487 

   488 }

   489 

   490 // Input and output

   491 template <typename IntType>

   492 std::istream& operator>> (std::istream& is, rational<IntType>& r)

   493 {

   494     IntType n = IntType(0), d = IntType(1);

   495     char c = 0;

   496     detail::resetter sentry(is);

   497 

   498     is >> n;

   499     c = is.get();

   500 

   501     if (c != '/')

   502         is.clear(std::istream::badbit);  // old GNU c++ lib has no ios_base

   503 

   504 #if !defined(__GNUC__) || (defined(__GNUC__) && (__GNUC__ >= 3)) || defined __SGI_STL_PORT

   505     is >> std::noskipws;

   506 #else

   507     is.unsetf(ios::skipws); // compiles, but seems to have no effect.

   508 #endif

   509     is >> d;

   510 

   511     if (is)

   512         r.assign(n, d);

   513 

   514     return is;

   515 }

   516 

   517 // Add manipulators for output format?

   518 template <typename IntType>

   519 std::ostream& operator<< (std::ostream& os, const rational<IntType>& r)

   520 {

   521     os << r.numerator() << '/' << r.denominator();

   522     return os;

   523 }

   524 

   525 // Type conversion

   526 template <typename T, typename IntType>

   527 inline T rational_cast(

   528     const rational<IntType>& src BOOST_APPEND_EXPLICIT_TEMPLATE_TYPE(T))

   529 {

   530     return static_cast<T>(src.numerator())/src.denominator();

   531 }

   532 

   533 // Do not use any abs() defined on IntType - it isn't worth it, given the

   534 // difficulties involved (Koenig lookup required, there may not *be* an abs()

   535 // defined, etc etc).

   536 template <typename IntType>

   537 inline rational<IntType> abs(const rational<IntType>& r)

   538 {

   539     if (r.numerator() >= IntType(0))

   540         return r;

   541 

   542     return rational<IntType>(-r.numerator(), r.denominator());

   543 }

   544 

   545 } // namespace boost

   546 

   547 #endif  // BOOST_RATIONAL_HPP

   548