ceph/src/boost/boost/rational.hpp

   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 //  02 Sep 13  Remove unneeded forward declarations; tweak private helper
  25 //             function (Daryle Walker)
  26 //  30 Aug 13  Improve exception safety of "assign"; start modernizing I/O code
  27 //             (Daryle Walker)
  28 //  27 Aug 13  Add cross-version constructor template, plus some private helper
  29 //             functions; add constructor to exception class to take custom
  30 //             messages (Daryle Walker)
  31 //  25 Aug 13  Add constexpr qualification wherever possible (Daryle Walker)
  32 //  05 May 12  Reduced use of implicit gcd (Mario Lang)
  33 //  05 Nov 06  Change rational_cast to not depend on division between different
  34 //             types (Daryle Walker)
  35 //  04 Nov 06  Off-load GCD and LCM to Boost.Integer; add some invariant checks;
  36 //             add std::numeric_limits<> requirement to help GCD (Daryle Walker)
  37 //  31 Oct 06  Recoded both operator< to use round-to-negative-infinity
  38 //             divisions; the rational-value version now uses continued fraction
  39 //             expansion to avoid overflows, for bug #798357 (Daryle Walker)
  40 //  20 Oct 06  Fix operator bool_type for CW 8.3 (Joaquín M López Muñoz)
  41 //  18 Oct 06  Use EXPLICIT_TEMPLATE_TYPE helper macros from Boost.Config
  42 //             (Joaquín M López Muñoz)
  43 //  27 Dec 05  Add Boolean conversion operator (Daryle Walker)
  44 //  28 Sep 02  Use _left versions of operators from operators.hpp
  45 //  05 Jul 01  Recode gcd(), avoiding std::swap (Helmut Zeisel)
  46 //  03 Mar 01  Workarounds for Intel C++ 5.0 (David Abrahams)
  47 //  05 Feb 01  Update operator>> to tighten up input syntax
  48 //  05 Feb 01  Final tidy up of gcd code prior to the new release
  49 //  27 Jan 01  Recode abs() without relying on abs(IntType)
  50 //  21 Jan 01  Include Nickolay Mladenov's operator+= algorithm,
  51 //             tidy up a number of areas, use newer features of operators.hpp
  52 //             (reduces space overhead to zero), add operator!,
  53 //             introduce explicit mixed-mode arithmetic operations
  54 //  12 Jan 01  Include fixes to handle a user-defined IntType better
  55 //  19 Nov 00  Throw on divide by zero in operator /= (John (EBo) David)
  56 //  23 Jun 00  Incorporate changes from Mark Rodgers for Borland C++
  57 //  22 Jun 00  Change _MSC_VER to BOOST_MSVC so other compilers are not
  58 //             affected (Beman Dawes)
  59 //   6 Mar 00  Fix operator-= normalization, #include <string> (Jens Maurer)
  60 //  14 Dec 99  Modifications based on comments from the boost list
  61 //  09 Dec 99  Initial Version (Paul Moore)
  62
  63 #ifndef BOOST_RATIONAL_HPP
  64 #define BOOST_RATIONAL_HPP
  65
  66 #include <boost/config.hpp>      // for BOOST_NO_STDC_NAMESPACE, BOOST_MSVC, etc
  67 #ifndef BOOST_NO_IOSTREAM
  68 #include <iomanip>               // for std::setw
  69 #include <ios>                   // for std::noskipws, streamsize
  70 #include <istream>               // for std::istream
  71 #include <ostream>               // for std::ostream
  72 #include <sstream>               // for std::ostringstream
  73 #endif
  74 #include <cstddef>               // for NULL
  75 #include <stdexcept>             // for std::domain_error
  76 #include <string>                // for std::string implicit constructor
  77 #include <boost/operators.hpp>   // for boost::addable etc
  78 #include <cstdlib>               // for std::abs
  79 #include <boost/call_traits.hpp> // for boost::call_traits
  80 #include <boost/detail/workaround.hpp> // for BOOST_WORKAROUND
  81 #include <boost/assert.hpp>      // for BOOST_ASSERT
  82 #include <boost/integer/common_factor_rt.hpp> // for boost::integer::gcd, lcm
  83 #include <limits>                // for std::numeric_limits
  84 #include <boost/static_assert.hpp>  // for BOOST_STATIC_ASSERT
  85 #include <boost/throw_exception.hpp>
  86 #include <boost/utility/enable_if.hpp>
  87 #include <boost/type_traits/is_convertible.hpp>
  88 #include <boost/type_traits/is_class.hpp>
  89 #include <boost/type_traits/is_same.hpp>
  90
  91 // Control whether depreciated GCD and LCM functions are included (default: yes)
  92 #ifndef BOOST_CONTROL_RATIONAL_HAS_GCD
  93 #define BOOST_CONTROL_RATIONAL_HAS_GCD  1
  94 #endif
  95
  96 namespace boost {
  97
  98 #if BOOST_CONTROL_RATIONAL_HAS_GCD
  99 template <typename IntType>
 100 IntType gcd(IntType n, IntType m)
 101 {
 102     // Defer to the version in Boost.Integer
 103     return integer::gcd( n, m );
 104 }
 105
 106 template <typename IntType>
 107 IntType lcm(IntType n, IntType m)
 108 {
 109     // Defer to the version in Boost.Integer
 110     return integer::lcm( n, m );
 111 }
 112 #endif  // BOOST_CONTROL_RATIONAL_HAS_GCD
 113
 114 namespace rational_detail{
 115
 116    template <class FromInt, class ToInt>
 117    struct is_compatible_integer
 118    {
 119       BOOST_STATIC_CONSTANT(bool, value = ((std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
 120          && (std::numeric_limits<FromInt>::digits <= std::numeric_limits<ToInt>::digits)
 121          && (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
 122          && ((std::numeric_limits<FromInt>::is_signed == false) || (std::numeric_limits<ToInt>::is_signed == true))
 123          && is_convertible<FromInt, ToInt>::value)
 124          || is_same<FromInt, ToInt>::value)
 125          || (is_class<ToInt>::value && is_class<FromInt>::value && is_convertible<FromInt, ToInt>::value));
 126    };
 127
 128 }
 129
 130 class bad_rational : public std::domain_error
 131 {
 132 public:
 133     explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}
 134     explicit bad_rational( char const *what ) : std::domain_error( what ) {}
 135 };
 136
 137 template <typename IntType>
 138 class rational
 139 {
 140     // Class-wide pre-conditions
 141     BOOST_STATIC_ASSERT( ::std::numeric_limits<IntType>::is_specialized );
 142
 143     // Helper types
 144     typedef typename boost::call_traits<IntType>::param_type param_type;
 145
 146     struct helper { IntType parts[2]; };
 147     typedef IntType (helper::* bool_type)[2];
 148
 149 public:
 150     // Component type
 151     typedef IntType int_type;
 152
 153     BOOST_CONSTEXPR
 154     rational() : num(0), den(1) {}
 155     template <class T>
 156     BOOST_CONSTEXPR rational(const T& n, typename enable_if_c<
 157        rational_detail::is_compatible_integer<T, IntType>::value
 158     >::type const* = 0) : num(n), den(1) {}
 159     template <class T, class U>
 160     rational(const T& n, const U& d, typename enable_if_c<
 161        rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value
 162     >::type const* = 0) : num(n), den(d) {
 163        normalize();
 164     }
 165
 166     template < typename NewType >
 167     BOOST_CONSTEXPR explicit
 168        rational(rational<NewType> const &r, typename enable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
 169        : num(r.numerator()), den(is_normalized(int_type(r.numerator()),
 170        int_type(r.denominator())) ? r.denominator() :
 171        (BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
 172
 173     template < typename NewType >
 174     BOOST_CONSTEXPR explicit
 175        rational(rational<NewType> const &r, typename disable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
 176        : num(r.numerator()), den(is_normalized(int_type(r.numerator()),
 177        int_type(r.denominator())) && is_safe_narrowing_conversion(r.denominator()) && is_safe_narrowing_conversion(r.numerator()) ? r.denominator() :
 178        (BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
 179     // Default copy constructor and assignment are fine
 180
 181     // Add assignment from IntType
 182     template <class T>
 183     typename enable_if_c<
 184        rational_detail::is_compatible_integer<T, IntType>::value, rational &
 185     >::type operator=(const T& n) { return assign(static_cast<IntType>(n), static_cast<IntType>(1)); }
 186
 187     // Assign in place
 188     template <class T, class U>
 189     typename enable_if_c<
 190        rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value, rational &
 191     >::type assign(const T& n, const U& d)
 192     {
 193        return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
 194     }
 195     //
 196     // The following overloads should probably *not* be provided -
 197     // but are provided for backwards compatibity reasons only.
 198     // These allow for construction/assignment from types that
 199     // are wider than IntType only if there is an implicit
 200     // conversion from T to IntType, they will throw a bad_rational
 201     // if the conversion results in loss of precision or undefined behaviour.
 202     //
 203     template <class T>
 204     rational(const T& n, typename enable_if_c<
 205        std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
 206        && !rational_detail::is_compatible_integer<T, IntType>::value
 207        && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
 208        && is_convertible<T, IntType>::value
 209     >::type const* = 0)
 210     {
 211        assign(n, static_cast<T>(1));
 212     }
 213     template <class T, class U>
 214     rational(const T& n, const U& d, typename enable_if_c<
 215        (!rational_detail::is_compatible_integer<T, IntType>::value
 216        || !rational_detail::is_compatible_integer<U, IntType>::value)
 217        && std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
 218        && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
 219        && is_convertible<T, IntType>::value &&
 220        std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
 221        && (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
 222        && is_convertible<U, IntType>::value
 223     >::type const* = 0)
 224     {
 225        assign(n, d);
 226     }
 227     template <class T>
 228     typename enable_if_c<
 229        std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
 230        && !rational_detail::is_compatible_integer<T, IntType>::value
 231        && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
 232        && is_convertible<T, IntType>::value,
 233        rational &
 234     >::type operator=(const T& n) { return assign(n, static_cast<T>(1)); }
 235
 236     template <class T, class U>
 237     typename enable_if_c<
 238        (!rational_detail::is_compatible_integer<T, IntType>::value
 239           || !rational_detail::is_compatible_integer<U, IntType>::value)
 240        && std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
 241        && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
 242        && is_convertible<T, IntType>::value &&
 243        std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
 244        && (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
 245        && is_convertible<U, IntType>::value,
 246        rational &
 247     >::type assign(const T& n, const U& d)
 248     {
 249        if(!is_safe_narrowing_conversion(n) || !is_safe_narrowing_conversion(d))
 250           BOOST_THROW_EXCEPTION(bad_rational());
 251        return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
 252     }
 253
 254     // Access to representation
 255     BOOST_CONSTEXPR
 256     const IntType& numerator() const { return num; }
 257     BOOST_CONSTEXPR
 258     const IntType& denominator() const { return den; }
 259
 260     // Arithmetic assignment operators
 261     rational& operator+= (const rational& r);
 262     rational& operator-= (const rational& r);
 263     rational& operator*= (const rational& r);
 264     rational& operator/= (const rational& r);
 265
 266     template <class T>
 267     typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator+= (const T& i)
 268     {
 269        num += i * den;
 270        return *this;
 271     }
 272     template <class T>
 273     typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator-= (const T& i)
 274     {
 275        num -= i * den;
 276        return *this;
 277     }
 278     template <class T>
 279     typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator*= (const T& i)
 280     {
 281        // Avoid overflow and preserve normalization
 282        IntType gcd = integer::gcd(static_cast<IntType>(i), den);
 283        num *= i / gcd;
 284        den /= gcd;
 285        return *this;
 286     }
 287     template <class T>
 288     typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator/= (const T& i)
 289     {
 290        // Avoid repeated construction
 291        IntType const zero(0);
 292
 293        if(i == zero) BOOST_THROW_EXCEPTION(bad_rational());
 294        if(num == zero) return *this;
 295
 296        // Avoid overflow and preserve normalization
 297        IntType const gcd = integer::gcd(num, static_cast<IntType>(i));
 298        num /= gcd;
 299        den *= i / gcd;
 300
 301        if(den < zero) {
 302           num = -num;
 303           den = -den;
 304        }
 305
 306        return *this;
 307     }
 308
 309     // Increment and decrement
 310     const rational& operator++() { num += den; return *this; }
 311     const rational& operator--() { num -= den; return *this; }
 312
 313     rational operator++(int)
 314     {
 315        rational t(*this);
 316        ++(*this);
 317        return t;
 318     }
 319     rational operator--(int)
 320     {
 321        rational t(*this);
 322        --(*this);
 323        return t;
 324     }
 325
 326     // Operator not
 327     BOOST_CONSTEXPR
 328     bool operator!() const { return !num; }
 329
 330     // Boolean conversion
 331
 332 #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
 333     // The "ISO C++ Template Parser" option in CW 8.3 chokes on the
 334     // following, hence we selectively disable that option for the
 335     // offending memfun.
 336 #pragma parse_mfunc_templ off
 337 #endif
 338
 339     BOOST_CONSTEXPR
 340     operator bool_type() const { return operator !() ? 0 : &helper::parts; }
 341
 342 #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
 343 #pragma parse_mfunc_templ reset
 344 #endif
 345
 346     // Comparison operators
 347     bool operator< (const rational& r) const;
 348     bool operator> (const rational& r) const { return r < *this; }
 349     BOOST_CONSTEXPR
 350     bool operator== (const rational& r) const;
 351
 352     template <class T>
 353     typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator< (const T& i) const
 354     {
 355        // Avoid repeated construction
 356        int_type const  zero(0);
 357
 358        // Break value into mixed-fraction form, w/ always-nonnegative remainder
 359        BOOST_ASSERT(this->den > zero);
 360        int_type  q = this->num / this->den, r = this->num % this->den;
 361        while(r < zero)  { r += this->den; --q; }
 362
 363        // Compare with just the quotient, since the remainder always bumps the
 364        // value up.  [Since q = floor(n/d), and if n/d < i then q < i, if n/d == i
 365        // then q == i, if n/d == i + r/d then q == i, and if n/d >= i + 1 then
 366        // q >= i + 1 > i; therefore n/d < i iff q < i.]
 367        return q < i;
 368     }
 369     template <class T>
 370     typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator>(const T& i) const
 371     {
 372        return operator==(i) ? false : !operator<(i);
 373     }
 374     template <class T>
 375     BOOST_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator== (const T& i) const
 376     {
 377        return ((den == IntType(1)) && (num == i));
 378     }
 379
 380 private:
 381     // Implementation - numerator and denominator (normalized).
 382     // Other possibilities - separate whole-part, or sign, fields?
 383     IntType num;
 384     IntType den;
 385
 386     // Helper functions
 387     static BOOST_CONSTEXPR
 388     int_type inner_gcd( param_type a, param_type b, int_type const &zero =
 389      int_type(0) )
 390     { return b == zero ? a : inner_gcd(b, a % b, zero); }
 391
 392     static BOOST_CONSTEXPR
 393     int_type inner_abs( param_type x, int_type const &zero = int_type(0) )
 394     { return x < zero ? -x : +x; }
 395
 396     // Representation note: Fractions are kept in normalized form at all
 397     // times. normalized form is defined as gcd(num,den) == 1 and den > 0.
 398     // In particular, note that the implementation of abs() below relies
 399     // on den always being positive.
 400     bool test_invariant() const;
 401     void normalize();
 402
 403     static BOOST_CONSTEXPR
 404     bool is_normalized( param_type n, param_type d, int_type const &zero =
 405      int_type(0), int_type const &one = int_type(1) )
 406     {
 407         return d > zero && ( n != zero || d == one ) && inner_abs( inner_gcd(n,
 408          d, zero), zero ) == one;
 409     }
 410     //
 411     // Conversion checks:
 412     //
 413     // (1) From an unsigned type with more digits than IntType:
 414     //
 415     template <class T>
 416     BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
 417     {
 418        return val < (T(1) << std::numeric_limits<IntType>::digits);
 419     }
 420     //
 421     // (2) From a signed type with more digits than IntType, and IntType also signed:
 422     //
 423     template <class T>
 424     BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T& val)
 425     {
 426        // Note that this check assumes IntType has a 2's complement representation,
 427        // we don't want to try to convert a std::numeric_limits<IntType>::min() to
 428        // a T because that conversion may not be allowed (this happens when IntType
 429        // is from Boost.Multiprecision).
 430        return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= -(T(1) << std::numeric_limits<IntType>::digits));
 431     }
 432     //
 433     // (3) From a signed type with more digits than IntType, and IntType unsigned:
 434     //
 435     template <class T>
 436     BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits > std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
 437     {
 438        return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= 0);
 439     }
 440     //
 441     // (4) From a signed type with fewer digits than IntType, and IntType unsigned:
 442     //
 443     template <class T>
 444     BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T& val)
 445     {
 446        return val >= 0;
 447     }
 448     //
 449     // (5) From an unsigned type with fewer digits than IntType, and IntType signed:
 450     //
 451     template <class T>
 452     BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T&)
 453     {
 454        return true;
 455     }
 456     //
 457     // (6) From an unsigned type with fewer digits than IntType, and IntType unsigned:
 458     //
 459     template <class T>
 460     BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == false) && (std::numeric_limits<IntType>::is_signed == false), bool>::type is_safe_narrowing_conversion(const T&)
 461     {
 462        return true;
 463     }
 464     //
 465     // (7) From an signed type with fewer digits than IntType, and IntType signed:
 466     //
 467     template <class T>
 468     BOOST_CONSTEXPR static typename boost::enable_if_c<(std::numeric_limits<T>::digits <= std::numeric_limits<IntType>::digits) && (std::numeric_limits<T>::is_signed == true) && (std::numeric_limits<IntType>::is_signed == true), bool>::type is_safe_narrowing_conversion(const T&)
 469     {
 470        return true;
 471     }
 472 };
 473
 474 // Unary plus and minus
 475 template <typename IntType>
 476 BOOST_CONSTEXPR
 477 inline rational<IntType> operator+ (const rational<IntType>& r)
 478 {
 479     return r;
 480 }
 481
 482 template <typename IntType>
 483 inline rational<IntType> operator- (const rational<IntType>& r)
 484 {
 485     return rational<IntType>(static_cast<IntType>(-r.numerator()), r.denominator());
 486 }
 487
 488 // Arithmetic assignment operators
 489 template <typename IntType>
 490 rational<IntType>& rational<IntType>::operator+= (const rational<IntType>& r)
 491 {
 492     // This calculation avoids overflow, and minimises the number of expensive
 493     // calculations. Thanks to Nickolay Mladenov for this algorithm.
 494     //
 495     // Proof:
 496     // We have to compute a/b + c/d, where gcd(a,b)=1 and gcd(b,c)=1.
 497     // Let g = gcd(b,d), and b = b1*g, d=d1*g. Then gcd(b1,d1)=1
 498     //
 499     // The result is (a*d1 + c*b1) / (b1*d1*g).
 500     // Now we have to normalize this ratio.
 501     // Let's assume h | gcd((a*d1 + c*b1), (b1*d1*g)), and h > 1
 502     // If h | b1 then gcd(h,d1)=1 and hence h|(a*d1+c*b1) => h|a.
 503     // But since gcd(a,b1)=1 we have h=1.
 504     // Similarly h|d1 leads to h=1.
 505     // So we have that h | gcd((a*d1 + c*b1) , (b1*d1*g)) => h|g
 506     // Finally we have gcd((a*d1 + c*b1), (b1*d1*g)) = gcd((a*d1 + c*b1), g)
 507     // Which proves that instead of normalizing the result, it is better to
 508     // divide num and den by gcd((a*d1 + c*b1), g)
 509
 510     // Protect against self-modification
 511     IntType r_num = r.num;
 512     IntType r_den = r.den;
 513
 514     IntType g = integer::gcd(den, r_den);
 515     den /= g;  // = b1 from the calculations above
 516     num = num * (r_den / g) + r_num * den;
 517     g = integer::gcd(num, g);
 518     num /= g;
 519     den *= r_den/g;
 520
 521     return *this;
 522 }
 523
 524 template <typename IntType>
 525 rational<IntType>& rational<IntType>::operator-= (const rational<IntType>& r)
 526 {
 527     // Protect against self-modification
 528     IntType r_num = r.num;
 529     IntType r_den = r.den;
 530
 531     // This calculation avoids overflow, and minimises the number of expensive
 532     // calculations. It corresponds exactly to the += case above
 533     IntType g = integer::gcd(den, r_den);
 534     den /= g;
 535     num = num * (r_den / g) - r_num * den;
 536     g = integer::gcd(num, g);
 537     num /= g;
 538     den *= r_den/g;
 539
 540     return *this;
 541 }
 542
 543 template <typename IntType>
 544 rational<IntType>& rational<IntType>::operator*= (const rational<IntType>& r)
 545 {
 546     // Protect against self-modification
 547     IntType r_num = r.num;
 548     IntType r_den = r.den;
 549
 550     // Avoid overflow and preserve normalization
 551     IntType gcd1 = integer::gcd(num, r_den);
 552     IntType gcd2 = integer::gcd(r_num, den);
 553     num = (num/gcd1) * (r_num/gcd2);
 554     den = (den/gcd2) * (r_den/gcd1);
 555     return *this;
 556 }
 557
 558 template <typename IntType>
 559 rational<IntType>& rational<IntType>::operator/= (const rational<IntType>& r)
 560 {
 561     // Protect against self-modification
 562     IntType r_num = r.num;
 563     IntType r_den = r.den;
 564
 565     // Avoid repeated construction
 566     IntType zero(0);
 567
 568     // Trap division by zero
 569     if (r_num == zero)
 570         BOOST_THROW_EXCEPTION(bad_rational());
 571     if (num == zero)
 572         return *this;
 573
 574     // Avoid overflow and preserve normalization
 575     IntType gcd1 = integer::gcd(num, r_num);
 576     IntType gcd2 = integer::gcd(r_den, den);
 577     num = (num/gcd1) * (r_den/gcd2);
 578     den = (den/gcd2) * (r_num/gcd1);
 579
 580     if (den < zero) {
 581         num = -num;
 582         den = -den;
 583     }
 584     return *this;
 585 }
 586
 587
 588 //
 589 // Non-member operators: previously these were provided by Boost.Operator, but these had a number of
 590 // drawbacks, most notably, that in order to allow inter-operability with IntType code such as this:
 591 //
 592 // rational<int> r(3);
 593 // assert(r == 3.5); // compiles and passes!!
 594 //
 595 // Happens to be allowed as well :-(
 596 //
 597 // There are three possible cases for each operator:
 598 // 1) rational op rational.
 599 // 2) rational op integer
 600 // 3) integer op rational
 601 // Cases (1) and (2) are folded into the one function.
 602 //
 603 template <class IntType, class Arg>
 604 inline typename boost::enable_if_c <
 605    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
 606    operator + (const rational<IntType>& a, const Arg& b)
 607 {
 608       rational<IntType> t(a);
 609       return t += b;
 610 }
 611 template <class Arg, class IntType>
 612 inline typename boost::enable_if_c <
 613    rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
 614    operator + (const Arg& b, const rational<IntType>& a)
 615 {
 616       rational<IntType> t(a);
 617       return t += b;
 618 }
 619
 620 template <class IntType, class Arg>
 621 inline typename boost::enable_if_c <
 622    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
 623    operator - (const rational<IntType>& a, const Arg& b)
 624 {
 625       rational<IntType> t(a);
 626       return t -= b;
 627 }
 628 template <class Arg, class IntType>
 629 inline typename boost::enable_if_c <
 630    rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
 631    operator - (const Arg& b, const rational<IntType>& a)
 632 {
 633       rational<IntType> t(a);
 634       return -(t -= b);
 635 }
 636
 637 template <class IntType, class Arg>
 638 inline typename boost::enable_if_c <
 639    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
 640    operator * (const rational<IntType>& a, const Arg& b)
 641 {
 642       rational<IntType> t(a);
 643       return t *= b;
 644 }
 645 template <class Arg, class IntType>
 646 inline typename boost::enable_if_c <
 647    rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
 648    operator * (const Arg& b, const rational<IntType>& a)
 649 {
 650       rational<IntType> t(a);
 651       return t *= b;
 652 }
 653
 654 template <class IntType, class Arg>
 655 inline typename boost::enable_if_c <
 656    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
 657    operator / (const rational<IntType>& a, const Arg& b)
 658 {
 659       rational<IntType> t(a);
 660       return t /= b;
 661 }
 662 template <class Arg, class IntType>
 663 inline typename boost::enable_if_c <
 664    rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
 665    operator / (const Arg& b, const rational<IntType>& a)
 666 {
 667       rational<IntType> t(b);
 668       return t /= a;
 669 }
 670
 671 template <class IntType, class Arg>
 672 inline typename boost::enable_if_c <
 673    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
 674    operator <= (const rational<IntType>& a, const Arg& b)
 675 {
 676       return !(a > b);
 677 }
 678 template <class Arg, class IntType>
 679 inline typename boost::enable_if_c <
 680    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 681    operator <= (const Arg& b, const rational<IntType>& a)
 682 {
 683       return a >= b;
 684 }
 685
 686 template <class IntType, class Arg>
 687 inline typename boost::enable_if_c <
 688    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
 689    operator >= (const rational<IntType>& a, const Arg& b)
 690 {
 691       return !(a < b);
 692 }
 693 template <class Arg, class IntType>
 694 inline typename boost::enable_if_c <
 695    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 696    operator >= (const Arg& b, const rational<IntType>& a)
 697 {
 698       return a <= b;
 699 }
 700
 701 template <class IntType, class Arg>
 702 inline typename boost::enable_if_c <
 703    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
 704    operator != (const rational<IntType>& a, const Arg& b)
 705 {
 706       return !(a == b);
 707 }
 708 template <class Arg, class IntType>
 709 inline typename boost::enable_if_c <
 710    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 711    operator != (const Arg& b, const rational<IntType>& a)
 712 {
 713       return !(b == a);
 714 }
 715
 716 template <class Arg, class IntType>
 717 inline typename boost::enable_if_c <
 718    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 719    operator < (const Arg& b, const rational<IntType>& a)
 720 {
 721       return a > b;
 722 }
 723 template <class Arg, class IntType>
 724 inline typename boost::enable_if_c <
 725    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 726    operator > (const Arg& b, const rational<IntType>& a)
 727 {
 728       return a < b;
 729 }
 730 template <class Arg, class IntType>
 731 inline typename boost::enable_if_c <
 732    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 733    operator == (const Arg& b, const rational<IntType>& a)
 734 {
 735       return a == b;
 736 }
 737
 738 // Comparison operators
 739 template <typename IntType>
 740 bool rational<IntType>::operator< (const rational<IntType>& r) const
 741 {
 742     // Avoid repeated construction
 743     int_type const  zero( 0 );
 744
 745     // This should really be a class-wide invariant.  The reason for these
 746     // checks is that for 2's complement systems, INT_MIN has no corresponding
 747     // positive, so negating it during normalization keeps it INT_MIN, which
 748     // is bad for later calculations that assume a positive denominator.
 749     BOOST_ASSERT( this->den > zero );
 750     BOOST_ASSERT( r.den > zero );
 751
 752     // Determine relative order by expanding each value to its simple continued
 753     // fraction representation using the Euclidian GCD algorithm.
 754     struct { int_type  n, d, q, r; }
 755      ts = { this->num, this->den, static_cast<int_type>(this->num / this->den),
 756      static_cast<int_type>(this->num % this->den) },
 757      rs = { r.num, r.den, static_cast<int_type>(r.num / r.den),
 758      static_cast<int_type>(r.num % r.den) };
 759     unsigned  reverse = 0u;
 760
 761     // Normalize negative moduli by repeatedly adding the (positive) denominator
 762     // and decrementing the quotient.  Later cycles should have all positive
 763     // values, so this only has to be done for the first cycle.  (The rules of
 764     // C++ require a nonnegative quotient & remainder for a nonnegative dividend
 765     // & positive divisor.)
 766     while ( ts.r < zero )  { ts.r += ts.d; --ts.q; }
 767     while ( rs.r < zero )  { rs.r += rs.d; --rs.q; }
 768
 769     // Loop through and compare each variable's continued-fraction components
 770     for ( ;; )
 771     {
 772         // The quotients of the current cycle are the continued-fraction
 773         // components.  Comparing two c.f. is comparing their sequences,
 774         // stopping at the first difference.
 775         if ( ts.q != rs.q )
 776         {
 777             // Since reciprocation changes the relative order of two variables,
 778             // and c.f. use reciprocals, the less/greater-than test reverses
 779             // after each index.  (Start w/ non-reversed @ whole-number place.)
 780             return reverse ? ts.q > rs.q : ts.q < rs.q;
 781         }
 782
 783         // Prepare the next cycle
 784         reverse ^= 1u;
 785
 786         if ( (ts.r == zero) || (rs.r == zero) )
 787         {
 788             // At least one variable's c.f. expansion has ended
 789             break;
 790         }
 791
 792         ts.n = ts.d;         ts.d = ts.r;
 793         ts.q = ts.n / ts.d;  ts.r = ts.n % ts.d;
 794         rs.n = rs.d;         rs.d = rs.r;
 795         rs.q = rs.n / rs.d;  rs.r = rs.n % rs.d;
 796     }
 797
 798     // Compare infinity-valued components for otherwise equal sequences
 799     if ( ts.r == rs.r )
 800     {
 801         // Both remainders are zero, so the next (and subsequent) c.f.
 802         // components for both sequences are infinity.  Therefore, the sequences
 803         // and their corresponding values are equal.
 804         return false;
 805     }
 806     else
 807     {
 808 #ifdef BOOST_MSVC
 809 #pragma warning(push)
 810 #pragma warning(disable:4800)
 811 #endif
 812         // Exactly one of the remainders is zero, so all following c.f.
 813         // components of that variable are infinity, while the other variable
 814         // has a finite next c.f. component.  So that other variable has the
 815         // lesser value (modulo the reversal flag!).
 816         return ( ts.r != zero ) != static_cast<bool>( reverse );
 817 #ifdef BOOST_MSVC
 818 #pragma warning(pop)
 819 #endif
 820     }
 821 }
 822
 823 template <typename IntType>
 824 BOOST_CONSTEXPR
 825 inline bool rational<IntType>::operator== (const rational<IntType>& r) const
 826 {
 827     return ((num == r.num) && (den == r.den));
 828 }
 829
 830 // Invariant check
 831 template <typename IntType>
 832 inline bool rational<IntType>::test_invariant() const
 833 {
 834     return ( this->den > int_type(0) ) && ( integer::gcd(this->num, this->den) ==
 835      int_type(1) );
 836 }
 837
 838 // Normalisation
 839 template <typename IntType>
 840 void rational<IntType>::normalize()
 841 {
 842     // Avoid repeated construction
 843     IntType zero(0);
 844
 845     if (den == zero)
 846        BOOST_THROW_EXCEPTION(bad_rational());
 847
 848     // Handle the case of zero separately, to avoid division by zero
 849     if (num == zero) {
 850         den = IntType(1);
 851         return;
 852     }
 853
 854     IntType g = integer::gcd(num, den);
 855
 856     num /= g;
 857     den /= g;
 858
 859     // Ensure that the denominator is positive
 860     if (den < zero) {
 861         num = -num;
 862         den = -den;
 863     }
 864
 865     // ...But acknowledge that the previous step doesn't always work.
 866     // (Nominally, this should be done before the mutating steps, but this
 867     // member function is only called during the constructor, so we never have
 868     // to worry about zombie objects.)
 869     if (den < zero)
 870        BOOST_THROW_EXCEPTION(bad_rational("bad rational: non-zero singular denominator"));
 871
 872     BOOST_ASSERT( this->test_invariant() );
 873 }
 874
 875 #ifndef BOOST_NO_IOSTREAM
 876 namespace detail {
 877
 878     // A utility class to reset the format flags for an istream at end
 879     // of scope, even in case of exceptions
 880     struct resetter {
 881         resetter(std::istream& is) : is_(is), f_(is.flags()) {}
 882         ~resetter() { is_.flags(f_); }
 883         std::istream& is_;
 884         std::istream::fmtflags f_;      // old GNU c++ lib has no ios_base
 885     };
 886
 887 }
 888
 889 // Input and output
 890 template <typename IntType>
 891 std::istream& operator>> (std::istream& is, rational<IntType>& r)
 892 {
 893     using std::ios;
 894
 895     IntType n = IntType(0), d = IntType(1);
 896     char c = 0;
 897     detail::resetter sentry(is);
 898
 899     if ( is >> n )
 900     {
 901         if ( is.get(c) )
 902         {
 903             if ( c == '/' )
 904             {
 905                 if ( is >> std::noskipws >> d )
 906                     try {
 907                         r.assign( n, d );
 908                     } catch ( bad_rational & ) {        // normalization fail
 909                         try { is.setstate(ios::failbit); }
 910                         catch ( ... ) {}  // don't throw ios_base::failure...
 911                         if ( is.exceptions() & ios::failbit )
 912                             throw;   // ...but the original exception instead
 913                         // ELSE: suppress the exception, use just error flags
 914                     }
 915             }
 916             else
 917                 is.setstate( ios::failbit );
 918         }
 919     }
 920
 921     return is;
 922 }
 923
 924 // Add manipulators for output format?
 925 template <typename IntType>
 926 std::ostream& operator<< (std::ostream& os, const rational<IntType>& r)
 927 {
 928     // The slash directly precedes the denominator, which has no prefixes.
 929     std::ostringstream  ss;
 930
 931     ss.copyfmt( os );
 932     ss.tie( NULL );
 933     ss.exceptions( std::ios::goodbit );
 934     ss.width( 0 );
 935     ss << std::noshowpos << std::noshowbase << '/' << r.denominator();
 936
 937     // The numerator holds the showpos, internal, and showbase flags.
 938     std::string const   tail = ss.str();
 939     std::streamsize const  w =
 940         os.width() - static_cast<std::streamsize>( tail.size() );
 941
 942     ss.clear();
 943     ss.str( "" );
 944     ss.flags( os.flags() );
 945     ss << std::setw( w < 0 || (os.flags() & std::ios::adjustfield) !=
 946                      std::ios::internal ? 0 : w ) << r.numerator();
 947     return os << ss.str() + tail;
 948 }
 949 #endif  // BOOST_NO_IOSTREAM
 950
 951 // Type conversion
 952 template <typename T, typename IntType>
 953 BOOST_CONSTEXPR
 954 inline T rational_cast(const rational<IntType>& src)
 955 {
 956     return static_cast<T>(src.numerator())/static_cast<T>(src.denominator());
 957 }
 958
 959 // Do not use any abs() defined on IntType - it isn't worth it, given the
 960 // difficulties involved (Koenig lookup required, there may not *be* an abs()
 961 // defined, etc etc).
 962 template <typename IntType>
 963 inline rational<IntType> abs(const rational<IntType>& r)
 964 {
 965     return r.numerator() >= IntType(0)? r: -r;
 966 }
 967
 968 namespace integer {
 969
 970 template <typename IntType>
 971 struct gcd_evaluator< rational<IntType> >
 972 {
 973     typedef rational<IntType> result_type,
 974                               first_argument_type, second_argument_type;
 975     result_type operator() (  first_argument_type const &a
 976                            , second_argument_type const &b
 977                            ) const
 978     {
 979         return result_type(integer::gcd(a.numerator(), b.numerator()),
 980                            integer::lcm(a.denominator(), b.denominator()));
 981     }
 982 };
 983
 984 template <typename IntType>
 985 struct lcm_evaluator< rational<IntType> >
 986 {
 987     typedef rational<IntType> result_type,
 988                               first_argument_type, second_argument_type;
 989     result_type operator() (  first_argument_type const &a
 990                            , second_argument_type const &b
 991                            ) const
 992     {
 993         return result_type(integer::lcm(a.numerator(), b.numerator()),
 994                            integer::gcd(a.denominator(), b.denominator()));
 995     }
 996 };
 997
 998 } // namespace integer
 999
1000 } // namespace boost
1001
1002 #endif  // BOOST_RATIONAL_HPP