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 #include <boost/type_traits/is_array.hpp>
  91
  92 // Control whether depreciated GCD and LCM functions are included (default: yes)
  93 #ifndef BOOST_CONTROL_RATIONAL_HAS_GCD
  94 #define BOOST_CONTROL_RATIONAL_HAS_GCD  1
  95 #endif
  96
  97 namespace boost {
  98
  99 #if BOOST_CONTROL_RATIONAL_HAS_GCD
 100 template <typename IntType>
 101 IntType gcd(IntType n, IntType m)
 102 {
 103     // Defer to the version in Boost.Integer
 104     return integer::gcd( n, m );
 105 }
 106
 107 template <typename IntType>
 108 IntType lcm(IntType n, IntType m)
 109 {
 110     // Defer to the version in Boost.Integer
 111     return integer::lcm( n, m );
 112 }
 113 #endif  // BOOST_CONTROL_RATIONAL_HAS_GCD
 114
 115 namespace rational_detail{
 116
 117    template <class FromInt, class ToInt, typename Enable = void>
 118    struct is_compatible_integer;
 119
 120    template <class FromInt, class ToInt>
 121    struct is_compatible_integer<FromInt, ToInt, typename enable_if_c<!is_array<FromInt>::value>::type>
 122    {
 123       BOOST_STATIC_CONSTANT(bool, value = ((std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
 124          && (std::numeric_limits<FromInt>::digits <= std::numeric_limits<ToInt>::digits)
 125          && (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
 126          && ((std::numeric_limits<FromInt>::is_signed == false) || (std::numeric_limits<ToInt>::is_signed == true))
 127          && is_convertible<FromInt, ToInt>::value)
 128          || is_same<FromInt, ToInt>::value)
 129          || (is_class<ToInt>::value && is_class<FromInt>::value && is_convertible<FromInt, ToInt>::value));
 130    };
 131
 132    template <class FromInt, class ToInt>
 133    struct is_compatible_integer<FromInt, ToInt, typename enable_if_c<is_array<FromInt>::value>::type>
 134    {
 135       BOOST_STATIC_CONSTANT(bool, value = false);
 136    };
 137
 138    template <class FromInt, class ToInt, typename Enable = void>
 139    struct is_backward_compatible_integer;
 140
 141    template <class FromInt, class ToInt>
 142    struct is_backward_compatible_integer<FromInt, ToInt, typename enable_if_c<!is_array<FromInt>::value>::type>
 143    {
 144       BOOST_STATIC_CONSTANT(bool, value = (std::numeric_limits<FromInt>::is_specialized && std::numeric_limits<FromInt>::is_integer
 145          && !is_compatible_integer<FromInt, ToInt>::value
 146          && (std::numeric_limits<FromInt>::radix == std::numeric_limits<ToInt>::radix)
 147          && is_convertible<FromInt, ToInt>::value));
 148    };
 149
 150    template <class FromInt, class ToInt>
 151    struct is_backward_compatible_integer<FromInt, ToInt, typename enable_if_c<is_array<FromInt>::value>::type>
 152    {
 153       BOOST_STATIC_CONSTANT(bool, value = false);
 154    };
 155 }
 156
 157 class bad_rational : public std::domain_error
 158 {
 159 public:
 160     explicit bad_rational() : std::domain_error("bad rational: zero denominator") {}
 161     explicit bad_rational( char const *what ) : std::domain_error( what ) {}
 162 };
 163
 164 template <typename IntType>
 165 class rational
 166 {
 167     // Class-wide pre-conditions
 168     BOOST_STATIC_ASSERT( ::std::numeric_limits<IntType>::is_specialized );
 169
 170     // Helper types
 171     typedef typename boost::call_traits<IntType>::param_type param_type;
 172
 173     struct helper { IntType parts[2]; };
 174     typedef IntType (helper::* bool_type)[2];
 175
 176 public:
 177     // Component type
 178     typedef IntType int_type;
 179
 180     BOOST_CONSTEXPR
 181     rational() : num(0), den(1) {}
 182
 183     template <class T>//, typename enable_if_c<!is_array<T>::value>::type>
 184     BOOST_CONSTEXPR rational(const T& n, typename enable_if_c<
 185        rational_detail::is_compatible_integer<T, IntType>::value
 186     >::type const* = 0) : num(n), den(1) {}
 187
 188     template <class T, class U>
 189     BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
 190        rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value
 191     >::type const* = 0) : num(n), den(d) {
 192        normalize();
 193     }
 194
 195     template < typename NewType >
 196     BOOST_CONSTEXPR explicit
 197        rational(rational<NewType> const &r, typename enable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
 198        : num(r.numerator()), den(is_normalized(int_type(r.numerator()),
 199        int_type(r.denominator())) ? r.denominator() :
 200        (BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
 201
 202     template < typename NewType >
 203     BOOST_CONSTEXPR explicit
 204        rational(rational<NewType> const &r, typename disable_if_c<rational_detail::is_compatible_integer<NewType, IntType>::value>::type const* = 0)
 205        : num(r.numerator()), den(is_normalized(int_type(r.numerator()),
 206        int_type(r.denominator())) && is_safe_narrowing_conversion(r.denominator()) && is_safe_narrowing_conversion(r.numerator()) ? r.denominator() :
 207        (BOOST_THROW_EXCEPTION(bad_rational("bad rational: denormalized conversion")), 0)){}
 208     // Default copy constructor and assignment are fine
 209
 210     // Add assignment from IntType
 211     template <class T>
 212     BOOST_CXX14_CONSTEXPR typename enable_if_c<
 213        rational_detail::is_compatible_integer<T, IntType>::value, rational &
 214     >::type operator=(const T& n) { return assign(static_cast<IntType>(n), static_cast<IntType>(1)); }
 215
 216     // Assign in place
 217     template <class T, class U>
 218     BOOST_CXX14_CONSTEXPR typename enable_if_c<
 219        rational_detail::is_compatible_integer<T, IntType>::value && rational_detail::is_compatible_integer<U, IntType>::value, rational &
 220     >::type assign(const T& n, const U& d)
 221     {
 222        return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
 223     }
 224     //
 225     // The following overloads should probably *not* be provided -
 226     // but are provided for backwards compatibity reasons only.
 227     // These allow for construction/assignment from types that
 228     // are wider than IntType only if there is an implicit
 229     // conversion from T to IntType, they will throw a bad_rational
 230     // if the conversion results in loss of precision or undefined behaviour.
 231     //
 232     template <class T>//, typename enable_if_c<!is_array<T>::value>::type>
 233     BOOST_CXX14_CONSTEXPR rational(const T& n, typename enable_if_c<
 234        rational_detail::is_backward_compatible_integer<T, IntType>::value
 235     >::type const* = 0)
 236     {
 237        assign(n, static_cast<T>(1));
 238     }
 239     template <class T, class U>
 240     BOOST_CXX14_CONSTEXPR rational(const T& n, const U& d, typename enable_if_c<
 241        (!rational_detail::is_compatible_integer<T, IntType>::value
 242        || !rational_detail::is_compatible_integer<U, IntType>::value)
 243        && std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
 244        && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
 245        && is_convertible<T, IntType>::value &&
 246        std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
 247        && (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
 248        && is_convertible<U, IntType>::value
 249     >::type const* = 0)
 250     {
 251        assign(n, d);
 252     }
 253     template <class T>
 254     BOOST_CXX14_CONSTEXPR typename enable_if_c<
 255        std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
 256        && !rational_detail::is_compatible_integer<T, IntType>::value
 257        && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
 258        && is_convertible<T, IntType>::value,
 259        rational &
 260     >::type operator=(const T& n) { return assign(n, static_cast<T>(1)); }
 261
 262     template <class T, class U>
 263     BOOST_CXX14_CONSTEXPR typename enable_if_c<
 264        (!rational_detail::is_compatible_integer<T, IntType>::value
 265           || !rational_detail::is_compatible_integer<U, IntType>::value)
 266        && std::numeric_limits<T>::is_specialized && std::numeric_limits<T>::is_integer
 267        && (std::numeric_limits<T>::radix == std::numeric_limits<IntType>::radix)
 268        && is_convertible<T, IntType>::value &&
 269        std::numeric_limits<U>::is_specialized && std::numeric_limits<U>::is_integer
 270        && (std::numeric_limits<U>::radix == std::numeric_limits<IntType>::radix)
 271        && is_convertible<U, IntType>::value,
 272        rational &
 273     >::type assign(const T& n, const U& d)
 274     {
 275        if(!is_safe_narrowing_conversion(n) || !is_safe_narrowing_conversion(d))
 276           BOOST_THROW_EXCEPTION(bad_rational());
 277        return *this = rational<IntType>(static_cast<IntType>(n), static_cast<IntType>(d));
 278     }
 279
 280     // Access to representation
 281     BOOST_CONSTEXPR
 282     const IntType& numerator() const { return num; }
 283     BOOST_CONSTEXPR
 284     const IntType& denominator() const { return den; }
 285
 286     // Arithmetic assignment operators
 287     BOOST_CXX14_CONSTEXPR rational& operator+= (const rational& r);
 288     BOOST_CXX14_CONSTEXPR rational& operator-= (const rational& r);
 289     BOOST_CXX14_CONSTEXPR rational& operator*= (const rational& r);
 290     BOOST_CXX14_CONSTEXPR rational& operator/= (const rational& r);
 291
 292     template <class T>
 293     BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator+= (const T& i)
 294     {
 295        num += i * den;
 296        return *this;
 297     }
 298     template <class T>
 299     BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator-= (const T& i)
 300     {
 301        num -= i * den;
 302        return *this;
 303     }
 304     template <class T>
 305     BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator*= (const T& i)
 306     {
 307        // Avoid overflow and preserve normalization
 308        IntType gcd = integer::gcd(static_cast<IntType>(i), den);
 309        num *= i / gcd;
 310        den /= gcd;
 311        return *this;
 312     }
 313     template <class T>
 314     BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, rational&>::type operator/= (const T& i)
 315     {
 316        // Avoid repeated construction
 317        IntType const zero(0);
 318
 319        if(i == zero) BOOST_THROW_EXCEPTION(bad_rational());
 320        if(num == zero) return *this;
 321
 322        // Avoid overflow and preserve normalization
 323        IntType const gcd = integer::gcd(num, static_cast<IntType>(i));
 324        num /= gcd;
 325        den *= i / gcd;
 326
 327        if(den < zero) {
 328           num = -num;
 329           den = -den;
 330        }
 331
 332        return *this;
 333     }
 334
 335     // Increment and decrement
 336     BOOST_CXX14_CONSTEXPR const rational& operator++() { num += den; return *this; }
 337     BOOST_CXX14_CONSTEXPR const rational& operator--() { num -= den; return *this; }
 338
 339     BOOST_CXX14_CONSTEXPR rational operator++(int)
 340     {
 341        rational t(*this);
 342        ++(*this);
 343        return t;
 344     }
 345     BOOST_CXX14_CONSTEXPR rational operator--(int)
 346     {
 347        rational t(*this);
 348        --(*this);
 349        return t;
 350     }
 351
 352     // Operator not
 353     BOOST_CONSTEXPR
 354     bool operator!() const { return !num; }
 355
 356     // Boolean conversion
 357
 358 #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
 359     // The "ISO C++ Template Parser" option in CW 8.3 chokes on the
 360     // following, hence we selectively disable that option for the
 361     // offending memfun.
 362 #pragma parse_mfunc_templ off
 363 #endif
 364
 365     BOOST_CONSTEXPR
 366     operator bool_type() const { return operator !() ? 0 : &helper::parts; }
 367
 368 #if BOOST_WORKAROUND(__MWERKS__,<=0x3003)
 369 #pragma parse_mfunc_templ reset
 370 #endif
 371
 372     // Comparison operators
 373     BOOST_CXX14_CONSTEXPR bool operator< (const rational& r) const;
 374     BOOST_CXX14_CONSTEXPR bool operator> (const rational& r) const { return r < *this; }
 375     BOOST_CONSTEXPR
 376     bool operator== (const rational& r) const;
 377
 378     template <class T>
 379     BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator< (const T& i) const
 380     {
 381        // Avoid repeated construction
 382        int_type const  zero(0);
 383
 384        // Break value into mixed-fraction form, w/ always-nonnegative remainder
 385        BOOST_ASSERT(this->den > zero);
 386        int_type  q = this->num / this->den, r = this->num % this->den;
 387        while(r < zero)  { r += this->den; --q; }
 388
 389        // Compare with just the quotient, since the remainder always bumps the
 390        // value up.  [Since q = floor(n/d), and if n/d < i then q < i, if n/d == i
 391        // then q == i, if n/d == i + r/d then q == i, and if n/d >= i + 1 then
 392        // q >= i + 1 > i; therefore n/d < i iff q < i.]
 393        return q < i;
 394     }
 395     template <class T>
 396     BOOST_CXX14_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator>(const T& i) const
 397     {
 398        return operator==(i) ? false : !operator<(i);
 399     }
 400     template <class T>
 401     BOOST_CONSTEXPR typename boost::enable_if_c<rational_detail::is_compatible_integer<T, IntType>::value, bool>::type operator== (const T& i) const
 402     {
 403        return ((den == IntType(1)) && (num == i));
 404     }
 405
 406 private:
 407     // Implementation - numerator and denominator (normalized).
 408     // Other possibilities - separate whole-part, or sign, fields?
 409     IntType num;
 410     IntType den;
 411
 412     // Helper functions
 413     static BOOST_CONSTEXPR
 414     int_type inner_gcd( param_type a, param_type b, int_type const &zero =
 415      int_type(0) )
 416     { return b == zero ? a : inner_gcd(b, a % b, zero); }
 417
 418     static BOOST_CONSTEXPR
 419     int_type inner_abs( param_type x, int_type const &zero = int_type(0) )
 420     { return x < zero ? -x : +x; }
 421
 422     // Representation note: Fractions are kept in normalized form at all
 423     // times. normalized form is defined as gcd(num,den) == 1 and den > 0.
 424     // In particular, note that the implementation of abs() below relies
 425     // on den always being positive.
 426     BOOST_CXX14_CONSTEXPR bool test_invariant() const;
 427     BOOST_CXX14_CONSTEXPR void normalize();
 428
 429     static BOOST_CONSTEXPR
 430     bool is_normalized( param_type n, param_type d, int_type const &zero =
 431      int_type(0), int_type const &one = int_type(1) )
 432     {
 433         return d > zero && ( n != zero || d == one ) && inner_abs( inner_gcd(n,
 434          d, zero), zero ) == one;
 435     }
 436     //
 437     // Conversion checks:
 438     //
 439     // (1) From an unsigned type with more digits than IntType:
 440     //
 441     template <class T>
 442     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)
 443     {
 444        return val < (T(1) << std::numeric_limits<IntType>::digits);
 445     }
 446     //
 447     // (2) From a signed type with more digits than IntType, and IntType also signed:
 448     //
 449     template <class T>
 450     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)
 451     {
 452        // Note that this check assumes IntType has a 2's complement representation,
 453        // we don't want to try to convert a std::numeric_limits<IntType>::min() to
 454        // a T because that conversion may not be allowed (this happens when IntType
 455        // is from Boost.Multiprecision).
 456        return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= -(T(1) << std::numeric_limits<IntType>::digits));
 457     }
 458     //
 459     // (3) From a signed type with more digits than IntType, and IntType unsigned:
 460     //
 461     template <class T>
 462     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)
 463     {
 464        return (val < (T(1) << std::numeric_limits<IntType>::digits)) && (val >= 0);
 465     }
 466     //
 467     // (4) From a signed type with fewer digits than IntType, and IntType unsigned:
 468     //
 469     template <class T>
 470     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)
 471     {
 472        return val >= 0;
 473     }
 474     //
 475     // (5) From an unsigned type with fewer digits than IntType, and IntType signed:
 476     //
 477     template <class T>
 478     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&)
 479     {
 480        return true;
 481     }
 482     //
 483     // (6) From an unsigned type with fewer digits than IntType, and IntType unsigned:
 484     //
 485     template <class T>
 486     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&)
 487     {
 488        return true;
 489     }
 490     //
 491     // (7) From an signed type with fewer digits than IntType, and IntType signed:
 492     //
 493     template <class T>
 494     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&)
 495     {
 496        return true;
 497     }
 498 };
 499
 500 // Unary plus and minus
 501 template <typename IntType>
 502 BOOST_CONSTEXPR
 503 inline rational<IntType> operator+ (const rational<IntType>& r)
 504 {
 505     return r;
 506 }
 507
 508 template <typename IntType>
 509 BOOST_CXX14_CONSTEXPR
 510 inline rational<IntType> operator- (const rational<IntType>& r)
 511 {
 512     return rational<IntType>(static_cast<IntType>(-r.numerator()), r.denominator());
 513 }
 514
 515 // Arithmetic assignment operators
 516 template <typename IntType>
 517 BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator+= (const rational<IntType>& r)
 518 {
 519     // This calculation avoids overflow, and minimises the number of expensive
 520     // calculations. Thanks to Nickolay Mladenov for this algorithm.
 521     //
 522     // Proof:
 523     // We have to compute a/b + c/d, where gcd(a,b)=1 and gcd(b,c)=1.
 524     // Let g = gcd(b,d), and b = b1*g, d=d1*g. Then gcd(b1,d1)=1
 525     //
 526     // The result is (a*d1 + c*b1) / (b1*d1*g).
 527     // Now we have to normalize this ratio.
 528     // Let's assume h | gcd((a*d1 + c*b1), (b1*d1*g)), and h > 1
 529     // If h | b1 then gcd(h,d1)=1 and hence h|(a*d1+c*b1) => h|a.
 530     // But since gcd(a,b1)=1 we have h=1.
 531     // Similarly h|d1 leads to h=1.
 532     // So we have that h | gcd((a*d1 + c*b1) , (b1*d1*g)) => h|g
 533     // Finally we have gcd((a*d1 + c*b1), (b1*d1*g)) = gcd((a*d1 + c*b1), g)
 534     // Which proves that instead of normalizing the result, it is better to
 535     // divide num and den by gcd((a*d1 + c*b1), g)
 536
 537     // Protect against self-modification
 538     IntType r_num = r.num;
 539     IntType r_den = r.den;
 540
 541     IntType g = integer::gcd(den, r_den);
 542     den /= g;  // = b1 from the calculations above
 543     num = num * (r_den / g) + r_num * den;
 544     g = integer::gcd(num, g);
 545     num /= g;
 546     den *= r_den/g;
 547
 548     return *this;
 549 }
 550
 551 template <typename IntType>
 552 BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator-= (const rational<IntType>& r)
 553 {
 554     // Protect against self-modification
 555     IntType r_num = r.num;
 556     IntType r_den = r.den;
 557
 558     // This calculation avoids overflow, and minimises the number of expensive
 559     // calculations. It corresponds exactly to the += case above
 560     IntType g = integer::gcd(den, r_den);
 561     den /= g;
 562     num = num * (r_den / g) - r_num * den;
 563     g = integer::gcd(num, g);
 564     num /= g;
 565     den *= r_den/g;
 566
 567     return *this;
 568 }
 569
 570 template <typename IntType>
 571 BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator*= (const rational<IntType>& r)
 572 {
 573     // Protect against self-modification
 574     IntType r_num = r.num;
 575     IntType r_den = r.den;
 576
 577     // Avoid overflow and preserve normalization
 578     IntType gcd1 = integer::gcd(num, r_den);
 579     IntType gcd2 = integer::gcd(r_num, den);
 580     num = (num/gcd1) * (r_num/gcd2);
 581     den = (den/gcd2) * (r_den/gcd1);
 582     return *this;
 583 }
 584
 585 template <typename IntType>
 586 BOOST_CXX14_CONSTEXPR rational<IntType>& rational<IntType>::operator/= (const rational<IntType>& r)
 587 {
 588     // Protect against self-modification
 589     IntType r_num = r.num;
 590     IntType r_den = r.den;
 591
 592     // Avoid repeated construction
 593     IntType zero(0);
 594
 595     // Trap division by zero
 596     if (r_num == zero)
 597         BOOST_THROW_EXCEPTION(bad_rational());
 598     if (num == zero)
 599         return *this;
 600
 601     // Avoid overflow and preserve normalization
 602     IntType gcd1 = integer::gcd(num, r_num);
 603     IntType gcd2 = integer::gcd(r_den, den);
 604     num = (num/gcd1) * (r_den/gcd2);
 605     den = (den/gcd2) * (r_num/gcd1);
 606
 607     if (den < zero) {
 608         num = -num;
 609         den = -den;
 610     }
 611     return *this;
 612 }
 613
 614
 615 //
 616 // Non-member operators: previously these were provided by Boost.Operator, but these had a number of
 617 // drawbacks, most notably, that in order to allow inter-operability with IntType code such as this:
 618 //
 619 // rational<int> r(3);
 620 // assert(r == 3.5); // compiles and passes!!
 621 //
 622 // Happens to be allowed as well :-(
 623 //
 624 // There are three possible cases for each operator:
 625 // 1) rational op rational.
 626 // 2) rational op integer
 627 // 3) integer op rational
 628 // Cases (1) and (2) are folded into the one function.
 629 //
 630 template <class IntType, class Arg>
 631 BOOST_CXX14_CONSTEXPR
 632 inline typename boost::enable_if_c <
 633    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
 634    operator + (const rational<IntType>& a, const Arg& b)
 635 {
 636       rational<IntType> t(a);
 637       return t += b;
 638 }
 639 template <class Arg, class IntType>
 640 BOOST_CXX14_CONSTEXPR
 641 inline typename boost::enable_if_c <
 642    rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
 643    operator + (const Arg& b, const rational<IntType>& a)
 644 {
 645       rational<IntType> t(a);
 646       return t += b;
 647 }
 648
 649 template <class IntType, class Arg>
 650 BOOST_CXX14_CONSTEXPR
 651 inline typename boost::enable_if_c <
 652    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
 653    operator - (const rational<IntType>& a, const Arg& b)
 654 {
 655       rational<IntType> t(a);
 656       return t -= b;
 657 }
 658 template <class Arg, class IntType>
 659 BOOST_CXX14_CONSTEXPR
 660 inline typename boost::enable_if_c <
 661    rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
 662    operator - (const Arg& b, const rational<IntType>& a)
 663 {
 664       rational<IntType> t(a);
 665       return -(t -= b);
 666 }
 667
 668 template <class IntType, class Arg>
 669 BOOST_CXX14_CONSTEXPR
 670 inline typename boost::enable_if_c <
 671    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
 672    operator * (const rational<IntType>& a, const Arg& b)
 673 {
 674       rational<IntType> t(a);
 675       return t *= b;
 676 }
 677 template <class Arg, class IntType>
 678 BOOST_CXX14_CONSTEXPR
 679 inline typename boost::enable_if_c <
 680    rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
 681    operator * (const Arg& b, const rational<IntType>& a)
 682 {
 683       rational<IntType> t(a);
 684       return t *= b;
 685 }
 686
 687 template <class IntType, class Arg>
 688 BOOST_CXX14_CONSTEXPR
 689 inline typename boost::enable_if_c <
 690    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, rational<IntType> >::type
 691    operator / (const rational<IntType>& a, const Arg& b)
 692 {
 693       rational<IntType> t(a);
 694       return t /= b;
 695 }
 696 template <class Arg, class IntType>
 697 BOOST_CXX14_CONSTEXPR
 698 inline typename boost::enable_if_c <
 699    rational_detail::is_compatible_integer<Arg, IntType>::value, rational<IntType> >::type
 700    operator / (const Arg& b, const rational<IntType>& a)
 701 {
 702       rational<IntType> t(b);
 703       return t /= a;
 704 }
 705
 706 template <class IntType, class Arg>
 707 BOOST_CXX14_CONSTEXPR
 708 inline typename boost::enable_if_c <
 709    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
 710    operator <= (const rational<IntType>& a, const Arg& b)
 711 {
 712       return !(a > b);
 713 }
 714 template <class Arg, class IntType>
 715 BOOST_CXX14_CONSTEXPR
 716 inline typename boost::enable_if_c <
 717    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 718    operator <= (const Arg& b, const rational<IntType>& a)
 719 {
 720       return a >= b;
 721 }
 722
 723 template <class IntType, class Arg>
 724 BOOST_CXX14_CONSTEXPR
 725 inline typename boost::enable_if_c <
 726    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
 727    operator >= (const rational<IntType>& a, const Arg& b)
 728 {
 729       return !(a < b);
 730 }
 731 template <class Arg, class IntType>
 732 BOOST_CXX14_CONSTEXPR
 733 inline typename boost::enable_if_c <
 734    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 735    operator >= (const Arg& b, const rational<IntType>& a)
 736 {
 737       return a <= b;
 738 }
 739
 740 template <class IntType, class Arg>
 741 BOOST_CONSTEXPR
 742 inline typename boost::enable_if_c <
 743    rational_detail::is_compatible_integer<Arg, IntType>::value || is_same<rational<IntType>, Arg>::value, bool>::type
 744    operator != (const rational<IntType>& a, const Arg& b)
 745 {
 746       return !(a == b);
 747 }
 748 template <class Arg, class IntType>
 749 BOOST_CONSTEXPR
 750 inline typename boost::enable_if_c <
 751    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 752    operator != (const Arg& b, const rational<IntType>& a)
 753 {
 754       return !(b == a);
 755 }
 756
 757 template <class Arg, class IntType>
 758 BOOST_CXX14_CONSTEXPR
 759 inline typename boost::enable_if_c <
 760    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 761    operator < (const Arg& b, const rational<IntType>& a)
 762 {
 763       return a > b;
 764 }
 765 template <class Arg, class IntType>
 766 BOOST_CXX14_CONSTEXPR
 767 inline typename boost::enable_if_c <
 768    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 769    operator > (const Arg& b, const rational<IntType>& a)
 770 {
 771       return a < b;
 772 }
 773 template <class Arg, class IntType>
 774 BOOST_CONSTEXPR
 775 inline typename boost::enable_if_c <
 776    rational_detail::is_compatible_integer<Arg, IntType>::value, bool>::type
 777    operator == (const Arg& b, const rational<IntType>& a)
 778 {
 779       return a == b;
 780 }
 781
 782 // Comparison operators
 783 template <typename IntType>
 784 BOOST_CXX14_CONSTEXPR
 785 bool rational<IntType>::operator< (const rational<IntType>& r) const
 786 {
 787     // Avoid repeated construction
 788     int_type const  zero( 0 );
 789
 790     // This should really be a class-wide invariant.  The reason for these
 791     // checks is that for 2's complement systems, INT_MIN has no corresponding
 792     // positive, so negating it during normalization keeps it INT_MIN, which
 793     // is bad for later calculations that assume a positive denominator.
 794     BOOST_ASSERT( this->den > zero );
 795     BOOST_ASSERT( r.den > zero );
 796
 797     // Determine relative order by expanding each value to its simple continued
 798     // fraction representation using the Euclidian GCD algorithm.
 799     struct { int_type  n, d, q, r; }
 800      ts = { this->num, this->den, static_cast<int_type>(this->num / this->den),
 801      static_cast<int_type>(this->num % this->den) },
 802      rs = { r.num, r.den, static_cast<int_type>(r.num / r.den),
 803      static_cast<int_type>(r.num % r.den) };
 804     unsigned  reverse = 0u;
 805
 806     // Normalize negative moduli by repeatedly adding the (positive) denominator
 807     // and decrementing the quotient.  Later cycles should have all positive
 808     // values, so this only has to be done for the first cycle.  (The rules of
 809     // C++ require a nonnegative quotient & remainder for a nonnegative dividend
 810     // & positive divisor.)
 811     while ( ts.r < zero )  { ts.r += ts.d; --ts.q; }
 812     while ( rs.r < zero )  { rs.r += rs.d; --rs.q; }
 813
 814     // Loop through and compare each variable's continued-fraction components
 815     for ( ;; )
 816     {
 817         // The quotients of the current cycle are the continued-fraction
 818         // components.  Comparing two c.f. is comparing their sequences,
 819         // stopping at the first difference.
 820         if ( ts.q != rs.q )
 821         {
 822             // Since reciprocation changes the relative order of two variables,
 823             // and c.f. use reciprocals, the less/greater-than test reverses
 824             // after each index.  (Start w/ non-reversed @ whole-number place.)
 825             return reverse ? ts.q > rs.q : ts.q < rs.q;
 826         }
 827
 828         // Prepare the next cycle
 829         reverse ^= 1u;
 830
 831         if ( (ts.r == zero) || (rs.r == zero) )
 832         {
 833             // At least one variable's c.f. expansion has ended
 834             break;
 835         }
 836
 837         ts.n = ts.d;         ts.d = ts.r;
 838         ts.q = ts.n / ts.d;  ts.r = ts.n % ts.d;
 839         rs.n = rs.d;         rs.d = rs.r;
 840         rs.q = rs.n / rs.d;  rs.r = rs.n % rs.d;
 841     }
 842
 843     // Compare infinity-valued components for otherwise equal sequences
 844     if ( ts.r == rs.r )
 845     {
 846         // Both remainders are zero, so the next (and subsequent) c.f.
 847         // components for both sequences are infinity.  Therefore, the sequences
 848         // and their corresponding values are equal.
 849         return false;
 850     }
 851     else
 852     {
 853 #ifdef BOOST_MSVC
 854 #pragma warning(push)
 855 #pragma warning(disable:4800)
 856 #endif
 857         // Exactly one of the remainders is zero, so all following c.f.
 858         // components of that variable are infinity, while the other variable
 859         // has a finite next c.f. component.  So that other variable has the
 860         // lesser value (modulo the reversal flag!).
 861         return ( ts.r != zero ) != static_cast<bool>( reverse );
 862 #ifdef BOOST_MSVC
 863 #pragma warning(pop)
 864 #endif
 865     }
 866 }
 867
 868 template <typename IntType>
 869 BOOST_CONSTEXPR
 870 inline bool rational<IntType>::operator== (const rational<IntType>& r) const
 871 {
 872     return ((num == r.num) && (den == r.den));
 873 }
 874
 875 // Invariant check
 876 template <typename IntType>
 877 BOOST_CXX14_CONSTEXPR
 878 inline bool rational<IntType>::test_invariant() const
 879 {
 880     return ( this->den > int_type(0) ) && ( integer::gcd(this->num, this->den) ==
 881      int_type(1) );
 882 }
 883
 884 // Normalisation
 885 template <typename IntType>
 886 BOOST_CXX14_CONSTEXPR void rational<IntType>::normalize()
 887 {
 888     // Avoid repeated construction
 889     IntType zero(0);
 890
 891     if (den == zero)
 892        BOOST_THROW_EXCEPTION(bad_rational());
 893
 894     // Handle the case of zero separately, to avoid division by zero
 895     if (num == zero) {
 896         den = IntType(1);
 897         return;
 898     }
 899
 900     IntType g = integer::gcd(num, den);
 901
 902     num /= g;
 903     den /= g;
 904
 905     if (den < -(std::numeric_limits<IntType>::max)()) {
 906         BOOST_THROW_EXCEPTION(bad_rational("bad rational: non-zero singular denominator"));
 907     }
 908
 909     // Ensure that the denominator is positive
 910     if (den < zero) {
 911         num = -num;
 912         den = -den;
 913     }
 914
 915     BOOST_ASSERT( this->test_invariant() );
 916 }
 917
 918 #ifndef BOOST_NO_IOSTREAM
 919 namespace detail {
 920
 921     // A utility class to reset the format flags for an istream at end
 922     // of scope, even in case of exceptions
 923     struct resetter {
 924         resetter(std::istream& is) : is_(is), f_(is.flags()) {}
 925         ~resetter() { is_.flags(f_); }
 926         std::istream& is_;
 927         std::istream::fmtflags f_;      // old GNU c++ lib has no ios_base
 928     };
 929
 930 }
 931
 932 // Input and output
 933 template <typename IntType>
 934 std::istream& operator>> (std::istream& is, rational<IntType>& r)
 935 {
 936     using std::ios;
 937
 938     IntType n = IntType(0), d = IntType(1);
 939     char c = 0;
 940     detail::resetter sentry(is);
 941
 942     if ( is >> n )
 943     {
 944         if ( is.get(c) )
 945         {
 946             if ( c == '/' )
 947             {
 948                 if ( is >> std::noskipws >> d )
 949                     try {
 950                         r.assign( n, d );
 951                     } catch ( bad_rational & ) {        // normalization fail
 952                         try { is.setstate(ios::failbit); }
 953                         catch ( ... ) {}  // don't throw ios_base::failure...
 954                         if ( is.exceptions() & ios::failbit )
 955                             throw;   // ...but the original exception instead
 956                         // ELSE: suppress the exception, use just error flags
 957                     }
 958             }
 959             else
 960                 is.setstate( ios::failbit );
 961         }
 962     }
 963
 964     return is;
 965 }
 966
 967 // Add manipulators for output format?
 968 template <typename IntType>
 969 std::ostream& operator<< (std::ostream& os, const rational<IntType>& r)
 970 {
 971     // The slash directly precedes the denominator, which has no prefixes.
 972     std::ostringstream  ss;
 973
 974     ss.copyfmt( os );
 975     ss.tie( NULL );
 976     ss.exceptions( std::ios::goodbit );
 977     ss.width( 0 );
 978     ss << std::noshowpos << std::noshowbase << '/' << r.denominator();
 979
 980     // The numerator holds the showpos, internal, and showbase flags.
 981     std::string const   tail = ss.str();
 982     std::streamsize const  w =
 983         os.width() - static_cast<std::streamsize>( tail.size() );
 984
 985     ss.clear();
 986     ss.str( "" );
 987     ss.flags( os.flags() );
 988     ss << std::setw( w < 0 || (os.flags() & std::ios::adjustfield) !=
 989                      std::ios::internal ? 0 : w ) << r.numerator();
 990     return os << ss.str() + tail;
 991 }
 992 #endif  // BOOST_NO_IOSTREAM
 993
 994 // Type conversion
 995 template <typename T, typename IntType>
 996 BOOST_CONSTEXPR
 997 inline T rational_cast(const rational<IntType>& src)
 998 {
 999     return static_cast<T>(src.numerator())/static_cast<T>(src.denominator());
1000 }
1001
1002 // Do not use any abs() defined on IntType - it isn't worth it, given the
1003 // difficulties involved (Koenig lookup required, there may not *be* an abs()
1004 // defined, etc etc).
1005 template <typename IntType>
1006 BOOST_CXX14_CONSTEXPR
1007 inline rational<IntType> abs(const rational<IntType>& r)
1008 {
1009     return r.numerator() >= IntType(0)? r: -r;
1010 }
1011
1012 namespace integer {
1013
1014 template <typename IntType>
1015 struct gcd_evaluator< rational<IntType> >
1016 {
1017     typedef rational<IntType> result_type,
1018                               first_argument_type, second_argument_type;
1019     result_type operator() (  first_argument_type const &a
1020                            , second_argument_type const &b
1021                            ) const
1022     {
1023         return result_type(integer::gcd(a.numerator(), b.numerator()),
1024                            integer::lcm(a.denominator(), b.denominator()));
1025     }
1026 };
1027
1028 template <typename IntType>
1029 struct lcm_evaluator< rational<IntType> >
1030 {
1031     typedef rational<IntType> result_type,
1032                               first_argument_type, second_argument_type;
1033     result_type operator() (  first_argument_type const &a
1034                            , second_argument_type const &b
1035                            ) const
1036     {
1037         return result_type(integer::lcm(a.numerator(), b.numerator()),
1038                            integer::gcd(a.denominator(), b.denominator()));
1039     }
1040 };
1041
1042 } // namespace integer
1043
1044 } // namespace boost
1045
1046 #endif  // BOOST_RATIONAL_HPP