[article Boost.Integer
    [quickbook 1.6]
    [compatibility-mode 1.5]
    [copyright 2001-2009 Beman Dawes, Daryle Walker, Gennaro Prota, John Maddock]
    [license
        Distributed under the Boost Software License, Version 1.0.
        (See accompanying file LICENSE_1_0.txt or copy at
        [@https://www.boost.org/LICENSE_1_0.txt])
    ]
    [authors [Dawes, Beman], [Walker, Daryle], [Prota, Gennaro], [Maddock, John]]
    [/last-revision $Date: 2008-02-21 12:58:15 +0000 (Thu, 21 Feb 2008) $]
]

[template super[x]'''<superscript>'''[x]'''</superscript>''']

[section:overview Overview]

Boost.Integer provides integer type support, particularly helpful in generic programming.
It provides the means to select an integer type based upon its properties, like the number of bits or
the maximum supported value, as well as compile-time bit mask selection.  There is a derivative of
std::numeric_limits that provides integral constant expressions for `min` and `max`.
Finally, it provides two compile-time algorithms: determining the highest power of two in a
compile-time value; and computing min and max of constant expressions.

[table
   [[Component][Header][Purpose]]
   [
      [Forward Declarations.]
      [[^[@../../../../boost/integer_fwd.hpp <boost/integer_fwd.hpp>]]]
      [Forward declarations of classes and class templates - for use when just the name of a class is needed.]
   ]
   [
      [[link boost_integer.traits Integer Traits].]
      [[^[@../../../../boost/integer_traits.hpp <boost/integer_traits.hpp>]]]
      [Class template [^boost::integer_traits], derives from [^std::numeric_limits] and adds [^const_min] and [^const_max] members.]
   ]
   [
      [[link boost_integer.integer Integer Type Selection].]
      [[^[@../../../../boost/integer.hpp <boost/integer.hpp>]]]
      [Templates for integer type selection based on properties such as maximum value or number of bits:
      Use to select the type of an integer when some property such as maximum value or number of bits is known.
      Useful for generic programming. ]
   ]
   [
      [[link boost_integer.gcd_lcm Greatest Common Divisor and Least Common Multiple].]
      [[^[@../../../../boost/integer/common_factor_rt.hpp <boost/integer/common_factor_rt.hpp>]] and [^[@../../../../boost/integer/common_factor_ct.hpp <boost/integer/common_factor_ct.hpp>]]]
      [Functions `gcd` and `lcm` plus function objects and compile time versions.]
   ]
   [
      [[link boost_integer.mask Integer Masks].]
      [[^[@../../../../boost/integer/integer_mask.hpp <boost/integer/integer_mask.hpp>]]]
      [Templates for the selection of integer masks, single or lowest group, based on the number of bits:
      Use to select a particular mask when the bit position(s) are based on a compile-time variable. Useful for generic programming. ]
   ]
   [
      [[link boost_integer.log2 Compile time log2 Calculation].]
      [[^[@../../../../boost/integer/static_log2.hpp <boost/integer/static_log2.hpp>]]]
      [Template for finding the highest power of two in a number:
      Use to find the bit-size/range based on a maximum value. Useful for generic programming. ]
   ]
   [
      [[link boost_integer.minmax Compile time min/max calculation].]
      [[^[@../../../../boost/integer/static_min_max.hpp <boost/integer/static_min_max.hpp>]]]
      [Templates for finding the extrema of two numbers:
      Use to find a bound based on a minimum or maximum value. Useful for generic programming. ]
   ]
   [
      [[link boost_integer.extended_euclidean Extended Euclidean algorithm].]
      [[^[@../../../../boost/integer/extended_euclidean.hpp <boost/integer/extended_euclidean.hpp>]]]
      [Solves /mx + ny = gcd(x,y)/ for /x/ and /y/.]
   ]
   [
      [[link boost_integer.mod_inverse Modular multiplicative inverse].]
      [[^[@../../../../boost/integer/mod_inverse.hpp <boost/integer/mod_inverse.hpp>]]]
      [Given /a/ and /m/, solves /ax/ = 1 mod /m/ for /x/.]
   ]


]

[endsect]

[section:traits Integer Traits]

[section Motivation]

The C++ Standard Library <limits> header supplies a class template `numeric_limits<>` with specializations for each fundamental type.

For integer types, the interesting members of `std::numeric_limits<>` are:

   static const bool is_specialized;      // Will be true for integer types.
   static T min() throw();                // Smallest representable value.
   static T max() throw();                // Largest representable value.
   static const int digits;               // For integers, the number of value bits.
   static const int digits10;             // The number of base 10 digits that can be represented.
   static const bool is_signed;           // True if the type is signed.
   static const bool is_integer;          // Will be true for all integer types.

For many uses, these are sufficient.
But min() and max() are problematical because they are not constant expressions (std::5.19),
yet some usages require constant expressions.

The template class [^integer_traits] addresses this problem.

[endsect]

[section Synopsis]

   namespace boost {
     template<class T>
     class integer_traits : public std::numeric_limits<T>
     {
     public:
        static const bool is_integral = false;
        //
        // These members are defined only if T is a built-in
        // integal type:
        //
        static const T const_min = ``['implementation-defined]``;
        static const T const_max = ``['implementation-defined]``;
     };
   }

[endsect]

[section Description]

Template class [^integer_traits] is derived from [^std::numeric_limits]. The primary specialization adds the single
[^bool] member [^is_integral] with the compile-time constant value [^false].
However, for all integral types [^T] (std::3.9.1/7 [basic.fundamental]), there are specializations
provided with the following compile-time constants defined:

[table
   [[member][type][value]]
   [[[^is_integral]][bool][[^true]]]
   [[[^const_min]][[^T]][equivalent to [^std::numeric_limits<T>::min()]]]
   [[[^const_max]][[^T]][equivalent to [^std::numeric_limits<T>::max()]]]
]

Note: The /is_integral/ flag is provided, because a user-defined integer class should specialize
[^std::numeric_limits<>::is_integer = true], while compile-time constants
[^const_min] and [^const_max] are not provided for that user-defined class, unless boost::integer_traits is also specialized.

[endsect]

[section Test Program]

The program [^[@../../test/integer_traits_test.cpp integer_traits_test.cpp]] exercises the [^integer_traits] class.

[endsect]

[section Acknowledgements]

Beman Dawes, Ed Brey, Steve Cleary, and Nathan Myers discussed the integer traits idea on the boost mailing list in August 1999.

[endsect]
[endsect]

[section:integer Integer Type Selection]

The [@../../../../boost/integer.hpp <boost/integer.hpp>] type selection templates allow
integer types to be selected based on desired characteristics such as number of bits or maximum value.
This facility is particularly useful for solving generic programming problems.

[section:synopsis Synopsis]

   namespace boost
   {
     //  fast integers from least integers
     template<typename LeastInt>
     struct int_fast_t
     {
         typedef ``['implementation-defined-type]``  type;
     };

     //  signed
     template<int Bits>
     struct int_t
     {
         /* Member exact may or may not be defined depending upon Bits */
         typedef ``['implementation-defined-type]``  exact;
         typedef ``['implementation-defined-type]``  least;
         typedef int_fast_t<least>::fast      fast;
     };

     //  unsigned
     template<int Bits>
     struct uint_t
     {
         /* Member exact may or may not be defined depending upon Bits */
         typedef ``['implementation-defined-type]``  exact;
         typedef ``['implementation-defined-type]``  least;
         typedef int_fast_t<least>::fast      fast;
     };

     //  signed
     template<long long MaxValue>
     struct int_max_value_t
     {
         typedef ``['implementation-defined-type]``  least;
         typedef int_fast_t<least>::fast      fast;
     };

     template<long long MinValue>
     struct int_min_value_t
     {
         typedef ``['implementation-defined-type]``  least;
         typedef int_fast_t<least>::fast      fast;
     };

     //  unsigned
     template<unsigned long long Value>
     struct uint_value_t
     {
         typedef ``['implementation-defined-type]``  least;
         typedef int_fast_t<least>::fast      fast;
     };
   } // namespace boost

[endsect]

[section:easiest Easiest-to-Manipulate Types]

The [^int_fast_t] class template maps its input type to the next-largest type that the processor
can manipulate the easiest, or to itself if the input type is already an easy-to-manipulate type.
For instance, processing a bunch of [^char] objects may go faster if they were converted to [^int] objects before processing.
The input type, passed as the only template parameter, must be a built-in integral type, except [^bool].
Unsigned integral types can be used, as well as signed integral types.
The output type is given as the nested type [^fast].

[*Implementation Notes:]
By default, the output type is identical to the input type. Eventually, this code's implementation should
be customized for each platform to give accurate mappings between the built-in types and the easiest-to-manipulate
built-in types. Also, there is no guarantee that the output type actually is easier to manipulate than the input type.

[endsect]

[section:sized Sized Types]

The [^int_t], [^uint_t], [^int_max_value_t], [^int_min_value_t], and [^uint_value_t] class templates find
the most appropiate built-in integral type for the given template parameter. This type is given by the
nested type [^least]. The easiest-to-manipulate version of that type is given by the nested type [^fast].
The following table describes each template's criteria.

[table Criteria for the Sized Type Class Templates
   [
      [Class Template][Template Parameter Mapping]
   ]
   [
      [[^boost::int_t<N>::least]]
      [The smallest, built-in, signed integral type with at least /N/ bits, including the sign bit.
      The parameter should be a positive number. A compile-time error results if the parameter is
      larger than the number of bits in the largest integer type.]
   ]
   [
      [[^boost::int_t<N>::fast]]
      [The easiest-to-manipulate, built-in, signed integral type with at least /N/ bits, including the sign bit.
      The parameter should be a positive number. A compile-time error results if the parameter is
      larger than the number of bits in the largest integer type.]
   ]
   [
      [[^boost::int_t<N>::exact]]
      [A built-in, signed integral type with exactly /N/ bits, including the sign bit.
      The parameter should be a positive number.  Note that the member /exact/ is defined
      [*only] if there exists a type with exactly /N/ bits.]
   ]
   [
      [[^boost::uint_t<N>::least]]
      [The smallest, built-in, unsigned integral type with at least /N/ bits.
      The parameter should be a positive number. A compile-time error results if the
      parameter is larger than the number of bits in the largest integer type.]
   ]
   [
      [[^boost::uint_t<N>::fast]]
      [The easiest-to-manipulate, built-in, unsigned integral type with at least /N/ bits.
      The parameter should be a positive number. A compile-time error results if the
      parameter is larger than the number of bits in the largest integer type.]
   ]
   [
      [[^boost::uint_t<N>::exact]]
      [A built-in, unsigned integral type with exactly /N/ bits.
      The parameter should be a positive number. A compile-time error results if the
      parameter is larger than the number of bits in the largest integer type.
      Note that the member /exact/ is defined
      [*only] if there exists a type with exactly N bits.]
   ]
   [
      [[^boost::int_max_value_t<V>::least]]
      [The smallest, built-in, signed integral type that can hold all the values in the inclusive range ['0 - V].
      The parameter should be a positive number.]
   ]
   [
      [[^boost::int_max_value_t<V>::fast]]
      [The easiest-to-manipulate, built-in, signed integral type that can hold all the values in the inclusive range ['0 - V].
      The parameter should be a positive number.]
   ]
   [
      [[^boost::int_min_value_t<V>::least]]
      [The smallest, built-in, signed integral type that can hold all the values in the inclusive range ['V - 0].
      The parameter should be a negative number.]
   ]
   [
      [[^boost::int_min_value_t<V>::fast]]
      [The easiest-to-manipulate, built-in, signed integral type that can hold all the values in the inclusive range ['V - 0].
      The parameter should be a negative number.]
   ]
   [
      [[^boost::uint_value_t<V>::least]]
      [The smallest, built-in, unsigned integral type that can hold all positive values
      up to and including /V/. The parameter should be a positive number.]
   ]
   [
      [[^boost::uint_value_t<V>::fast]]
      [The easiest-to-manipulate, built-in, unsigned integral type that can hold all positive values
      up to and including /V/. The parameter should be a positive number.]
   ]
]

[endsect]

[section Example]

   #include <boost/integer.hpp>

   //...

   int main()
   {
       boost::int_t<24>::least my_var;  // my_var has at least 24-bits
       //...
       // This one is guaranteed not to be truncated:
       boost::int_max_value_t<1000>::least my1000 = 1000;
       //...
       // This one is guaranteed not to be truncated, and as fast
       // to manipulate as possible, its size may be greater than
       // that of my1000:
       boost::int_max_value_t<1000>::fast my_fast1000 = 1000;
   }

[endsect]

[section Demonstration Program]

The program [@../../test/integer_test.cpp integer_test.cpp] is a simplistic demonstration of the results from instantiating
various examples of the sized type class templates.

[endsect]

[section Rationale]

The rationale for the design of the templates in this header includes:

* Avoid recursion because of concern about C++'s limited guaranteed recursion depth (17).
* Avoid macros on general principles.
* Try to keep the design as simple as possible.

[endsect]

[section Alternative]

If the number of bits required is known beforehand, it may be more appropriate to use the types supplied
in [@../../../../boost/cstdint.hpp <boost/cstdint.hpp>].

[endsect]

[section Credits]

The author of most of the Boost integer type choosing templates is
[@https://www.boost.org/people/beman_dawes.html Beman Dawes].
He gives thanks to Valentin Bonnard and [@https://www.boost.org/people/kevlin_henney.htm Kevlin Henney]
for sharing their designs for similar templates.
[@https://www.boost.org/people/daryle_walker.html Daryle Walker] designed the value-based sized templates.

[endsect]
[endsect]

[include gcd/math-gcd.qbk]
[include modular_arithmetic/extended_euclidean.qbk]
[include modular_arithmetic/mod_inverse.qbk]

[section:mask Integer Masks]

[section Overview]

The class templates in [@../../../../boost/integer/integer_mask.hpp <boost/integer/integer_mask.hpp>]
provide bit masks for a certain bit position or a contiguous-bit pack of a certain size.
The types of the masking constants come from the [link boost_integer.integer integer type selection templates] header.

[endsect]

[section Synopsis]

   #include <cstddef>  // for std::size_t

   namespace boost
   {

   template <std::size_t Bit>
   struct high_bit_mask_t
   {
       typedef ``['implementation-defined-type]``  least;
       typedef ``['implementation-defined-type]``  fast;

       static const least       high_bit       = ``['implementation-defined]``;
       static const fast        high_bit_fast  = ``['implementation-defined]``;

       static const std::size_t bit_position   = Bit;
   };

   template <std::size_t Bits>
   struct low_bits_mask_t
   {
       typedef ``['implementation-defined-type]``  least;
       typedef ``['implementation-defined-type]``  fast;

       static const least       sig_bits       = ``['implementation-defined]``;
       static const fast        sig_bits_fast  = ``['implementation-defined]``;

       static const std::size_t bit_count      = Bits;
   };

   // Specializations for low_bits_mask_t exist for certain bit counts.

   }  // namespace boost

[endsect]

[section Single Bit-Mask Class Template]

The [^boost::high_bit_mask_t] class template provides constants for bit masks representing the bit at a
certain position. The masks are equivalent to the value 2[super Bit], where [^Bit] is the template parameter.
The bit position must be a nonnegative number from zero to ['Max], where Max is one less than the
number of bits supported by the largest unsigned built-in integral type. The following table describes
the members of an instantiation of [^high_bit_mask_t].

[table Members of the `boost::high_bit_mask_t` Class Template
   [[Member][Meaning]]
   [[[^least]][The smallest, unsigned, built-in type that supports the given bit position.]]
   [[[^fast]][The easiest-to-manipulate analog of [^least].]]
   [[[^high_bit]][A [^least] constant of the value 2[super Bit].]]
   [[[^high_bit_fast]][A [^fast] analog of [^high_bit].]]
   [[[^bit_position]][The value of the template parameter, in case its needed from a renamed instantiation of the class template.]]
]

[endsect]

[section Group Bit-Mask Class Template]

The [^boost::low_bits_mask_t] class template provides constants for bit masks
equivalent to the value (2[super Bits] - 1), where [^Bits] is the template parameter.
The parameter [^Bits] must be a non-negative integer from
zero to ['Max], where Max is the number of bits supported by the largest, unsigned, built-in integral type.
The following table describes the members of [^low_bits_mask_t].

[table Members of the [^boost::low_bits_mask_t] Class Template
[[Member][Meaning]]
[[[^least]][The smallest, unsigned built-in type that supports the given bit count.]]
[[[^fast]][The easiest-to-manipulate analog of [^least].]]
[[[^sig_bits]][A [^least] constant of the desired bit-masking value.]]
[[[^sig_bits_fast]][A [^fast] analog of [^sig_bits].]]
[[[^bit_count]][The value of the template parameter, in case its needed from a renamed instantiation of the class template.]]
]

[endsect]

[section Implementation Notes]

When [^Bits] is the exact size of a built-in unsigned type, the implementation has to change to
prevent undefined behavior. Therefore, there are specializations of [^low_bits_mask_t] at those bit counts.

[endsect]

[section Example]

   #include <boost/integer/integer_mask.hpp>

   //...

   int main()
   {
       typedef boost::high_bit_mask_t<29>  mask1_type;
       typedef boost::low_bits_mask_t<15>  mask2_type;

       mask1_type::least  my_var1;
       mask2_type::fast   my_var2;
       //...

       my_var1 |= mask1_type::high_bit;
       my_var2 &= mask2_type::sig_bits_fast;

       //...
   }

[endsect]

[section Demonstration Program]

The program [@../../test/integer_mask_test.cpp integer_mask_test.cpp] is a simplistic demonstration of the
results from instantiating various examples of the bit mask class templates.

[endsect]

[section Rationale]

The class templates in this header are an extension of the [link boost_integer.integer integer type selection class templates].
The new class templates provide the same sized types, but also convenient masks to use when extracting the
highest or all the significant bits when the containing built-in type contains more bits.
This prevents contamination of values by the higher, unused bits.

[endsect]

[section Credits]

The author of the Boost bit mask class templates is [@https://www.boost.org/people/daryle_walker.html Daryle Walker].

[endsect]
[endsect]

[section:log2 Compile Time log2 Calculation]

The class template in [@../../../../boost/integer/static_log2.hpp <boost/integer/static_log2.hpp>]
determines the position of the highest bit in a given value. This facility is useful for solving generic programming problems.

[section Synopsis]

   namespace boost
   {

     typedef ``['implementation-defined]`` static_log2_argument_type;
     typedef ``['implementation-defined]`` static_log2_result_type;

     template <static_log2_argument_type arg>
     struct static_log2
     {
       static const static_log2_result_type value = ``['implementation-defined]``;
     };


     template < >
     struct static_log2< 0 >
     {
       // The logarithm of zero is undefined.
     };


   }  // namespace boost

[endsect]

[section Usage]

The [^boost::static_log2] class template takes one template parameter, a value of type
[^static_log2_argument_type]. The template only defines one member, [^value], which gives the
truncated, base-two logarithm of the template argument.

Since the logarithm of zero, for any base, is undefined, there is a specialization of [^static_log2]
for a template argument of zero. This specialization has no members, so an attempt to use the base-two
logarithm of zero results in a compile-time error.

Note:

* [^static_log2_argument_type] is an ['unsigned integer type] (C++ standard, 3.9.1p3).
* [^static_log2_result_type] is an ['integer type] (C++ standard, 3.9.1p7).

[endsect]

[section Demonstration Program]

The program [@../../test/static_log2_test.cpp static_log2_test.cpp] is a simplistic
demonstration of the results from instantiating various examples of the binary logarithm class template.

[endsect]

[section Rationale]

The base-two (binary) logarithm, abbreviated lb, function is occasionally used to give order-estimates
of computer algorithms. The truncated logarithm can be considered the highest power-of-two in a value,
which corresponds to the value's highest set bit (for binary integers). Sometimes the highest-bit position
could be used in generic programming, which requires the position to be available statically (['i.e.] at compile-time).

[endsect]

[section Credits]

The original version of the Boost binary logarithm class template was
written by [@https://www.boost.org/people/daryle_walker.html Daryle Walker] and then
enhanced by Giovanni Bajo with support for compilers without partial template specialization.
The current version was suggested, together with a reference implementation, by Vesa Karvonen.
Gennaro Prota wrote the actual source file.

[endsect]
[endsect]

[section:minmax Compile time min/max calculation]

The class templates in [@../../../../boost/integer/static_min_max.hpp <boost/integer/static_min_max.hpp>]
provide a compile-time evaluation of the minimum or maximum of two integers. These facilities are useful
for generic programming problems.

[section Synopsis]

   namespace boost
   {

   typedef ``['implementation-defined]`` static_min_max_signed_type;
   typedef ``['implementation-defined]`` static_min_max_unsigned_type;

   template <static_min_max_signed_type Value1, static_min_max_signed_type Value2 >
       struct static_signed_min;

   template <static_min_max_signed_type Value1, static_min_max_signed_type Value2>
       struct static_signed_max;

   template <static_min_max_unsigned_type Value1, static_min_max_unsigned_type Value2>
       struct static_unsigned_min;

   template <static_min_max_unsigned_type Value1, static_min_max_unsigned_type Value2>
       struct static_unsigned_max;

   }

[endsect]

[section Usage]

The four class templates provide the combinations for finding the minimum or maximum of two [^signed] or
[^unsigned] ([^long]) parameters, /Value1/ and /Value2/, at compile-time. Each template has a single static data member,
[^value], which is set to the respective minimum or maximum of the template's parameters.

[endsect]

[section Example]

   #include <boost/integer/static_min_max.hpp>

   template < unsigned long AddendSize1, unsigned long AddendSize2 >
   class adder
   {
   public:
       static  unsigned long  const  addend1_size = AddendSize1;
       static  unsigned long  const  addend2_size = AddendSize2;
       static  unsigned long  const  sum_size = boost::static_unsigned_max<AddendSize1, AddendSize2>::value + 1;

       typedef int  addend1_type[ addend1_size ];
       typedef int  addend2_type[ addend2_size ];
       typedef int  sum_type[ sum_size ];

       void  operator ()( addend1_type const &a1, addend2_type const &a2, sum_type &s ) const;
   };

   //...

   int main()
   {
       int const   a1[] = { 0, 4, 3 };  // 340
       int const   a2[] = { 9, 8 };     //  89
       int         s[ 4 ];
       adder<3,2>  obj;

       obj( a1, a2, s );  // 's' should be 429 or { 9, 2, 4, 0 }
       //...
   }

[endsect]

[section Demonstration Program]

The program [@../../test/static_min_max_test.cpp static_min_max_test.cpp] is a simplistic demonstration of
various comparisons using the compile-time extrema class templates.

[endsect]

[section Rationale]

Sometimes the minimum or maximum of several values needs to be found for later compile-time processing,
['e.g.] for a bound for another class template.

[endsect]

[section Credits]

The author of the Boost compile-time extrema class templates is [@https://www.boost.org/people/daryle_walker.html Daryle Walker].

[endsect]
[endsect]

[section:history History]

[h4 1.56.0]

* Moved `<boost/cstdint.hpp>` into [@boost:/libs/config/index.html
  Boost.Config].

[h4 1.42.0]

* Reverted Trunk to release branch state (i.e. a "known good state").
* Fixed issues: [@https://svn.boost.org/trac/boost/ticket/653 653],
[@https://svn.boost.org/trac/boost/ticket/3084 3084],
[@https://svn.boost.org/trac/boost/ticket/3177 3177],
[@https://svn.boost.org/trac/boost/ticket/3180 3180],
[@https://svn.boost.org/trac/boost/ticket/3548 3568],
[@https://svn.boost.org/trac/boost/ticket/3657 3657],
[@https://svn.boost.org/trac/boost/ticket/2134 2134].
* Added long long support to [^boost::static_log2], [^boost::static_signed_min], [^boost::static_signed_max],
[^boost::static_unsigned_min][^boost::static_unsigned_max], when available.
* The argument type and the result type of [^boost::static_signed_min] etc are now typedef'd.
Formerly, they were hardcoded as [^unsigned long] and [^int] respectively. Please, use the
provided typedefs in new code (and update old code as soon as possible).

[h4 1.32.0]

* The argument type and the result type of [^boost::static_log2] are now typedef'd.
Formerly, they were hardcoded as [^unsigned long] and [^int] respectively. Please, use the
provided typedefs in new code (and update old code as soon as possible).

[endsect]

[section:cstdint Removed from library: Standard Integer Types]

The [@boost:/libs/config/doc/html/boost_config/cstdint.html Boost.Config] module provides
the typedefs useful for writing portable code that requires certain
integer widths.

[endsect]
