random.h

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// random number generation -*- C++ -*-
 
// Copyright (C) 2009-2025 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library.  This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
 
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.
 
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.
 
// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
// <http://www.gnu.org/licenses/>.
 
/**
 * @file bits/random.h
 *  This is an internal header file, included by other library headers.
 *  Do not attempt to use it directly. @headername{random}
 */
 
#ifndef _RANDOM_H
#define _RANDOM_H 1
 
#include <vector>
#include <bits/uniform_int_dist.h>
 
namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
 
  // [26.4] Random number generation
 
  /**
   * @defgroup random Random Number Generation
   * @ingroup numerics
   *
   * A facility for generating random numbers on selected distributions.
   * @{
   */
 
  // std::uniform_random_bit_generator is defined in <bits/uniform_int_dist.h>
 
  /**
   * @brief A function template for converting the output of a (integral)
   * uniform random number generator to a floatng point result in the range
   * [0-1).
   */
  template<typename _RealType, size_t __bits,
           typename _UniformRandomNumberGenerator>
    _RealType
    generate_canonical(_UniformRandomNumberGenerator& __g);
 
  /// @cond undocumented
  // Implementation-space details.
  namespace __detail
  {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++17-extensions"
 
    template<typename _UIntType, size_t __w,
             bool = __w < static_cast<size_t>
                          (std::numeric_limits<_UIntType>::digits)>
      struct _Shift
      { static constexpr _UIntType __value = 0; };
 
    template<typename _UIntType, size_t __w>
      struct _Shift<_UIntType, __w, true>
      { static constexpr _UIntType __value = _UIntType(1) << __w; };
 
    template<int __s,
             int __which = ((__s <= __CHAR_BIT__ * sizeof (int))
                            + (__s <= __CHAR_BIT__ * sizeof (long))
                            + (__s <= __CHAR_BIT__ * sizeof (long long))
                            /* assume long long no bigger than __int128 */
                            + (__s <= 128))>
      struct _Select_uint_least_t
      {
        static_assert(__which < 0, /* needs to be dependent */
                      "sorry, would be too much trouble for a slow result");
      };
 
    template<int __s>
      struct _Select_uint_least_t<__s, 4>
      { using type = unsigned int; };
 
    template<int __s>
      struct _Select_uint_least_t<__s, 3>
      { using type = unsigned long; };
 
    template<int __s>
      struct _Select_uint_least_t<__s, 2>
      { using type = unsigned long long; };
 
#if __SIZEOF_INT128__ > __SIZEOF_LONG_LONG__
    template<int __s>
      struct _Select_uint_least_t<__s, 1>
      { __extension__ using type = unsigned __int128; };
#elif __has_builtin(__builtin_add_overflow) \
    && __has_builtin(__builtin_sub_overflow) \
    && defined __UINT64_TYPE__
    template<int __s>
      struct _Select_uint_least_t<__s, 1>
      {
        // This is NOT a general-purpose 128-bit integer type.
        // It only supports (type(a) * x + c) % m as needed by __mod.
        struct type
        {
          explicit
          type(uint64_t __a) noexcept : _M_lo(__a), _M_hi(0) { }
 
          // pre: __l._M_hi == 0
          friend type
          operator*(type __l, uint64_t __x) noexcept
          {
            // Split 64-bit values __l._M_lo and __x into high and low 32-bit
            // limbs and multiply those individually.
            // l * x = (l0 + l1) * (x0 + x1) = l0x0 + l0x1 + l1x0 + l1x1
 
            constexpr uint64_t __mask = 0xffffffff;
            uint64_t __ll[2] = { __l._M_lo >> 32, __l._M_lo & __mask };
            uint64_t __xx[2] = { __x >> 32, __x & __mask };
            uint64_t __l0x0 = __ll[0] * __xx[0];
            uint64_t __l0x1 = __ll[0] * __xx[1];
            uint64_t __l1x0 = __ll[1] * __xx[0];
            uint64_t __l1x1 = __ll[1] * __xx[1];
            // These bits are the low half of __l._M_hi
            // and the high half of __l._M_lo.
            uint64_t __mid
              = (__l0x1 & __mask) + (__l1x0 & __mask) + (__l1x1 >> 32);
            __l._M_hi = __l0x0 + (__l0x1 >> 32) + (__l1x0 >> 32) + (__mid >> 32);
            __l._M_lo = (__mid << 32) + (__l1x1 & __mask);
            return __l;
          }
 
          friend type
          operator+(type __l, uint64_t __c) noexcept
          {
            __l._M_hi += __builtin_add_overflow(__l._M_lo, __c, &__l._M_lo);
            return __l;
          }
 
          friend type
          operator%(type __l, uint64_t __m) noexcept
          {
            if (__builtin_expect(__l._M_hi == 0, 0))
              {
                __l._M_lo %= __m;
                return __l;
              }
 
            int __shift = __builtin_clzll(__m) + 64
                            - __builtin_clzll(__l._M_hi);
            type __x(0);
            if (__shift >= 64)
              {
                __x._M_hi = __m << (__shift - 64);
                __x._M_lo = 0;
              }
            else
              {
                __x._M_hi = __m >> (64 - __shift);
                __x._M_lo = __m << __shift;
              }
 
            while (__l._M_hi != 0 || __l._M_lo >= __m)
              {
                if (__x <= __l)
                  {
                    __l._M_hi -= __x._M_hi;
                    __l._M_hi -= __builtin_sub_overflow(__l._M_lo, __x._M_lo,
                                                        &__l._M_lo);
                  }
                __x._M_lo = (__x._M_lo >> 1) | (__x._M_hi << 63);
                __x._M_hi >>= 1;
              }
            return __l;
          }
 
          // pre: __l._M_hi == 0
          explicit operator uint64_t() const noexcept
          { return _M_lo; }
 
          friend bool operator<(const type& __l, const type& __r) noexcept
          {
            if (__l._M_hi < __r._M_hi)
              return true;
            else if (__l._M_hi == __r._M_hi)
              return __l._M_lo < __r._M_lo;
            else
              return false;
          }
 
          friend bool operator<=(const type& __l, const type& __r) noexcept
          { return !(__r < __l); }
 
          uint64_t _M_lo;
          uint64_t _M_hi;
        };
      };
#endif
 
    // Assume a != 0, a < m, c < m, x < m.
    template<typename _Tp, _Tp __m, _Tp __a, _Tp __c,
             bool __big_enough = (!(__m & (__m - 1))
                                  || (_Tp(-1) - __c) / __a >= __m - 1),
             bool __schrage_ok = __m % __a < __m / __a>
      struct _Mod
      {
        static _Tp
        __calc(_Tp __x)
        {
          using _Tp2
            = typename _Select_uint_least_t<std::__lg(__a)
                                            + std::__lg(__m) + 2>::type;
          return static_cast<_Tp>((_Tp2(__a) * __x + __c) % __m);
        }
      };
 
    // Schrage.
    template<typename _Tp, _Tp __m, _Tp __a, _Tp __c>
      struct _Mod<_Tp, __m, __a, __c, false, true>
      {
        static _Tp
        __calc(_Tp __x);
      };
 
    // Special cases:
    // - for m == 2^n or m == 0, unsigned integer overflow is safe.
    // - a * (m - 1) + c fits in _Tp, there is no overflow.
    template<typename _Tp, _Tp __m, _Tp __a, _Tp __c, bool __s>
      struct _Mod<_Tp, __m, __a, __c, true, __s>
      {
        static _Tp
        __calc(_Tp __x)
        {
          _Tp __res = __a * __x + __c;
          if (__m)
            __res %= __m;
          return __res;
        }
      };
 
    template<typename _Tp, _Tp __m, _Tp __a = 1, _Tp __c = 0>
      inline _Tp
      __mod(_Tp __x)
      {
        if constexpr (__a == 0)
          return __c;
        else // N.B. _Mod must not be instantiated with a == 0
          return _Mod<_Tp, __m, __a, __c>::__calc(__x);
      }
 
    /*
     * An adaptor class for converting the output of any Generator into
     * the input for a specific Distribution.
     */
    template<typename _Engine, typename _DInputType>
      struct _Adaptor
      {
        static_assert(std::is_floating_point<_DInputType>::value,
                      "template argument must be a floating point type");
 
      public:
        _Adaptor(_Engine& __g)
        : _M_g(__g) { }
 
        _DInputType
        min() const
        { return _DInputType(0); }
 
        _DInputType
        max() const
        { return _DInputType(1); }
 
        /*
         * Converts a value generated by the adapted random number generator
         * into a value in the input domain for the dependent random number
         * distribution.
         */
        _DInputType
        operator()()
        {
          return std::generate_canonical<_DInputType,
                                    std::numeric_limits<_DInputType>::digits,
                                    _Engine>(_M_g);
        }
 
      private:
        _Engine& _M_g;
      };
 
    // Detect whether a template argument _Sseq is a valid seed sequence for
    // a random number engine _Engine with result type _Res.
    // Used to constrain _Engine::_Engine(_Sseq&) and _Engine::seed(_Sseq&)
    // as required by [rand.eng.general].
 
    template<typename _Sseq>
      using __seed_seq_generate_t = decltype(
          std::declval<_Sseq&>().generate(std::declval<uint_least32_t*>(),
                                          std::declval<uint_least32_t*>()));
 
    template<typename _Sseq, typename _Engine, typename _Res,
             typename _GenerateCheck = __seed_seq_generate_t<_Sseq>>
      using _If_seed_seq_for = _Require<
        __not_<is_same<__remove_cvref_t<_Sseq>, _Engine>>,
        is_unsigned<typename _Sseq::result_type>,
        __not_<is_convertible<_Sseq, _Res>>
      >;
 
#pragma GCC diagnostic pop
  } // namespace __detail
  /// @endcond
 
  /**
   * @addtogroup random_generators Random Number Generators
   * @ingroup random
   *
   * These classes define objects which provide random or pseudorandom
   * numbers, either from a discrete or a continuous interval.  The
   * random number generator supplied as a part of this library are
   * all uniform random number generators which provide a sequence of
   * random number uniformly distributed over their range.
   *
   * A number generator is a function object with an operator() that
   * takes zero arguments and returns a number.
   *
   * A compliant random number generator must satisfy the following
   * requirements.  <table border=1 cellpadding=10 cellspacing=0>
   * <caption align=top>Random Number Generator Requirements</caption>
   * <tr><td>To be documented.</td></tr> </table>
   *
   * @{
   */
 
  /**
   * @brief A model of a linear congruential random number generator.
   *
   * A random number generator that produces pseudorandom numbers via
   * linear function:
   * @f[
   *     x_{i+1}\leftarrow(ax_{i} + c) \bmod m
   * @f]
   *
   * The template parameter @p _UIntType must be an unsigned integral type
   * large enough to store values up to (__m-1). If the template parameter
   * @p __m is 0, the modulus @p __m used is
   * std::numeric_limits<_UIntType>::max() plus 1. Otherwise, the template
   * parameters @p __a and @p __c must be less than @p __m.
   *
   * The size of the state is @f$1@f$.
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
    class linear_congruential_engine
    {
      static_assert(std::is_unsigned<_UIntType>::value,
                    "result_type must be an unsigned integral type");
      static_assert(__m == 0u || (__a < __m && __c < __m),
                    "template argument substituting __m out of bounds");
 
      template<typename _Sseq>
        using _If_seed_seq
          = __detail::_If_seed_seq_for<_Sseq, linear_congruential_engine,
                                       _UIntType>;
 
    public:
      /** The type of the generated random value. */
      typedef _UIntType result_type;
 
      /** The multiplier. */
      static constexpr result_type multiplier   = __a;
      /** An increment. */
      static constexpr result_type increment    = __c;
      /** The modulus. */
      static constexpr result_type modulus      = __m;
      static constexpr result_type default_seed = 1u;
 
      /**
       * @brief Constructs a %linear_congruential_engine random number
       *        generator engine with seed 1.
       */
      linear_congruential_engine() : linear_congruential_engine(default_seed)
      { }
 
      /**
       * @brief Constructs a %linear_congruential_engine random number
       *        generator engine with seed @p __s.  The default seed value
       *        is 1.
       *
       * @param __s The initial seed value.
       */
      explicit
      linear_congruential_engine(result_type __s)
      { seed(__s); }
 
      /**
       * @brief Constructs a %linear_congruential_engine random number
       *        generator engine seeded from the seed sequence @p __q.
       *
       * @param __q the seed sequence.
       */
      template<typename _Sseq, typename = _If_seed_seq<_Sseq>>
        explicit
        linear_congruential_engine(_Sseq& __q)
        { seed(__q); }
 
      /**
       * @brief Reseeds the %linear_congruential_engine random number generator
       *        engine sequence to the seed @p __s.
       *
       * @param __s The new seed.
       */
      void
      seed(result_type __s = default_seed);
 
      /**
       * @brief Reseeds the %linear_congruential_engine random number generator
       *        engine
       * sequence using values from the seed sequence @p __q.
       *
       * @param __q the seed sequence.
       */
      template<typename _Sseq>
        _If_seed_seq<_Sseq>
        seed(_Sseq& __q);
 
      /**
       * @brief Gets the smallest possible value in the output range.
       *
       * The minimum depends on the @p __c parameter: if it is zero, the
       * minimum generated must be > 0, otherwise 0 is allowed.
       */
      static constexpr result_type
      min()
      { return __c == 0u ? 1u : 0u; }
 
      /**
       * @brief Gets the largest possible value in the output range.
       */
      static constexpr result_type
      max()
      { return __m - 1u; }
 
      /**
       * @brief Discard a sequence of random numbers.
       */
      void
      discard(unsigned long long __z)
      {
        for (; __z != 0ULL; --__z)
          (*this)();
      }
 
      /**
       * @brief Gets the next random number in the sequence.
       */
      result_type
      operator()()
      {
        _M_x = __detail::__mod<_UIntType, __m, __a, __c>(_M_x);
        return _M_x;
      }
 
      /**
       * @brief Compares two linear congruential random number generator
       * objects of the same type for equality.
       *
       * @param __lhs A linear congruential random number generator object.
       * @param __rhs Another linear congruential random number generator
       *              object.
       *
       * @returns true if the infinite sequences of generated values
       *          would be equal, false otherwise.
       */
      friend bool
      operator==(const linear_congruential_engine& __lhs,
                 const linear_congruential_engine& __rhs)
      { return __lhs._M_x == __rhs._M_x; }
 
      /**
       * @brief Writes the textual representation of the state x(i) of x to
       *        @p __os.
       *
       * @param __os  The output stream.
       * @param __lcr A % linear_congruential_engine random number generator.
       * @returns __os.
       */
      template<typename _UIntType1, _UIntType1 __a1, _UIntType1 __c1,
               _UIntType1 __m1, typename _CharT, typename _Traits>
        friend std::basic_ostream<_CharT, _Traits>&
        operator<<(std::basic_ostream<_CharT, _Traits>& __os,
                   const std::linear_congruential_engine<_UIntType1,
                   __a1, __c1, __m1>& __lcr);
 
      /**
       * @brief Sets the state of the engine by reading its textual
       *        representation from @p __is.
       *
       * The textual representation must have been previously written using
       * an output stream whose imbued locale and whose type's template
       * specialization arguments _CharT and _Traits were the same as those
       * of @p __is.
       *
       * @param __is  The input stream.
       * @param __lcr A % linear_congruential_engine random number generator.
       * @returns __is.
       */
      template<typename _UIntType1, _UIntType1 __a1, _UIntType1 __c1,
               _UIntType1 __m1, typename _CharT, typename _Traits>
        friend std::basic_istream<_CharT, _Traits>&
        operator>>(std::basic_istream<_CharT, _Traits>& __is,
                   std::linear_congruential_engine<_UIntType1, __a1,
                   __c1, __m1>& __lcr);
 
    private:
      _UIntType _M_x;
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Compares two linear congruential random number generator
   * objects of the same type for inequality.
   *
   * @param __lhs A linear congruential random number generator object.
   * @param __rhs Another linear congruential random number generator
   *              object.
   *
   * @returns true if the infinite sequences of generated values
   *          would be different, false otherwise.
   */
  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
    inline bool
    operator!=(const std::linear_congruential_engine<_UIntType, __a,
               __c, __m>& __lhs,
               const std::linear_congruential_engine<_UIntType, __a,
               __c, __m>& __rhs)
    { return !(__lhs == __rhs); }
#endif
 
  /**
   * A generalized feedback shift register discrete random number generator.
   *
   * This algorithm avoids multiplication and division and is designed to be
   * friendly to a pipelined architecture.  If the parameters are chosen
   * correctly, this generator will produce numbers with a very long period and
   * fairly good apparent entropy, although still not cryptographically strong.
   *
   * The best way to use this generator is with the predefined mt19937 class.
   *
   * This algorithm was originally invented by Makoto Matsumoto and
   * Takuji Nishimura.
   *
   * @tparam __w  Word size, the number of bits in each element of
   *              the state vector.
   * @tparam __n  The degree of recursion.
   * @tparam __m  The period parameter.
   * @tparam __r  The separation point bit index.
   * @tparam __a  The last row of the twist matrix.
   * @tparam __u  The first right-shift tempering matrix parameter.
   * @tparam __d  The first right-shift tempering matrix mask.
   * @tparam __s  The first left-shift tempering matrix parameter.
   * @tparam __b  The first left-shift tempering matrix mask.
   * @tparam __t  The second left-shift tempering matrix parameter.
   * @tparam __c  The second left-shift tempering matrix mask.
   * @tparam __l  The second right-shift tempering matrix parameter.
   * @tparam __f  Initialization multiplier.
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _UIntType, size_t __w,
           size_t __n, size_t __m, size_t __r,
           _UIntType __a, size_t __u, _UIntType __d, size_t __s,
           _UIntType __b, size_t __t,
           _UIntType __c, size_t __l, _UIntType __f>
    class mersenne_twister_engine
    {
      static_assert(std::is_unsigned<_UIntType>::value,
                    "result_type must be an unsigned integral type");
      static_assert(1u <= __m && __m <= __n,
                    "template argument substituting __m out of bounds");
      static_assert(__r <= __w, "template argument substituting "
                    "__r out of bound");
      static_assert(__u <= __w, "template argument substituting "
                    "__u out of bound");
      static_assert(__s <= __w, "template argument substituting "
                    "__s out of bound");
      static_assert(__t <= __w, "template argument substituting "
                    "__t out of bound");
      static_assert(__l <= __w, "template argument substituting "
                    "__l out of bound");
      static_assert(__w <= std::numeric_limits<_UIntType>::digits,
                    "template argument substituting __w out of bound");
      static_assert(__a <= (__detail::_Shift<_UIntType, __w>::__value - 1),
                    "template argument substituting __a out of bound");
      static_assert(__b <= (__detail::_Shift<_UIntType, __w>::__value - 1),
                    "template argument substituting __b out of bound");
      static_assert(__c <= (__detail::_Shift<_UIntType, __w>::__value - 1),
                    "template argument substituting __c out of bound");
      static_assert(__d <= (__detail::_Shift<_UIntType, __w>::__value - 1),
                    "template argument substituting __d out of bound");
      static_assert(__f <= (__detail::_Shift<_UIntType, __w>::__value - 1),
                    "template argument substituting __f out of bound");
 
      template<typename _Sseq>
        using _If_seed_seq
          = __detail::_If_seed_seq_for<_Sseq, mersenne_twister_engine,
                                       _UIntType>;
 
    public:
      /** The type of the generated random value. */
      typedef _UIntType result_type;
 
      // parameter values
      static constexpr size_t      word_size                 = __w;
      static constexpr size_t      state_size                = __n;
      static constexpr size_t      shift_size                = __m;
      static constexpr size_t      mask_bits                 = __r;
      static constexpr result_type xor_mask                  = __a;
      static constexpr size_t      tempering_u               = __u;
      static constexpr result_type tempering_d               = __d;
      static constexpr size_t      tempering_s               = __s;
      static constexpr result_type tempering_b               = __b;
      static constexpr size_t      tempering_t               = __t;
      static constexpr result_type tempering_c               = __c;
      static constexpr size_t      tempering_l               = __l;
      static constexpr result_type initialization_multiplier = __f;
      static constexpr result_type default_seed = 5489u;
 
      // constructors and member functions
 
      mersenne_twister_engine() : mersenne_twister_engine(default_seed) { }
 
      explicit
      mersenne_twister_engine(result_type __sd)
      { seed(__sd); }
 
      /**
       * @brief Constructs a %mersenne_twister_engine random number generator
       *        engine seeded from the seed sequence @p __q.
       *
       * @param __q the seed sequence.
       */
      template<typename _Sseq, typename = _If_seed_seq<_Sseq>>
        explicit
        mersenne_twister_engine(_Sseq& __q)
        { seed(__q); }
 
      void
      seed(result_type __sd = default_seed);
 
      template<typename _Sseq>
        _If_seed_seq<_Sseq>
        seed(_Sseq& __q);
 
      /**
       * @brief Gets the smallest possible value in the output range.
       */
      static constexpr result_type
      min()
      { return 0; }
 
      /**
       * @brief Gets the largest possible value in the output range.
       */
      static constexpr result_type
      max()
      { return __detail::_Shift<_UIntType, __w>::__value - 1; }
 
      /**
       * @brief Discard a sequence of random numbers.
       */
      void
      discard(unsigned long long __z);
 
      result_type
      operator()();
 
      /**
       * @brief Compares two % mersenne_twister_engine random number generator
       *        objects of the same type for equality.
       *
       * @param __lhs A % mersenne_twister_engine random number generator
       *              object.
       * @param __rhs Another % mersenne_twister_engine random number
       *              generator object.
       *
       * @returns true if the infinite sequences of generated values
       *          would be equal, false otherwise.
       */
      friend bool
      operator==(const mersenne_twister_engine& __lhs,
                 const mersenne_twister_engine& __rhs)
      { return (std::equal(__lhs._M_x, __lhs._M_x + state_size, __rhs._M_x)
                && __lhs._M_p == __rhs._M_p); }
 
      /**
       * @brief Inserts the current state of a % mersenne_twister_engine
       *        random number generator engine @p __x into the output stream
       *        @p __os.
       *
       * @param __os An output stream.
       * @param __x  A % mersenne_twister_engine random number generator
       *             engine.
       *
       * @returns The output stream with the state of @p __x inserted or in
       * an error state.
       */
      template<typename _UIntType1,
               size_t __w1, size_t __n1,
               size_t __m1, size_t __r1,
               _UIntType1 __a1, size_t __u1,
               _UIntType1 __d1, size_t __s1,
               _UIntType1 __b1, size_t __t1,
               _UIntType1 __c1, size_t __l1, _UIntType1 __f1,
               typename _CharT, typename _Traits>
        friend std::basic_ostream<_CharT, _Traits>&
        operator<<(std::basic_ostream<_CharT, _Traits>& __os,
                   const std::mersenne_twister_engine<_UIntType1, __w1, __n1,
                   __m1, __r1, __a1, __u1, __d1, __s1, __b1, __t1, __c1,
                   __l1, __f1>& __x);
 
      /**
       * @brief Extracts the current state of a % mersenne_twister_engine
       *        random number generator engine @p __x from the input stream
       *        @p __is.
       *
       * @param __is An input stream.
       * @param __x  A % mersenne_twister_engine random number generator
       *             engine.
       *
       * @returns The input stream with the state of @p __x extracted or in
       * an error state.
       */
      template<typename _UIntType1,
               size_t __w1, size_t __n1,
               size_t __m1, size_t __r1,
               _UIntType1 __a1, size_t __u1,
               _UIntType1 __d1, size_t __s1,
               _UIntType1 __b1, size_t __t1,
               _UIntType1 __c1, size_t __l1, _UIntType1 __f1,
               typename _CharT, typename _Traits>
        friend std::basic_istream<_CharT, _Traits>&
        operator>>(std::basic_istream<_CharT, _Traits>& __is,
                   std::mersenne_twister_engine<_UIntType1, __w1, __n1, __m1,
                   __r1, __a1, __u1, __d1, __s1, __b1, __t1, __c1,
                   __l1, __f1>& __x);
 
    private:
      void _M_gen_rand();
 
      _UIntType _M_x[state_size];
      size_t    _M_p;
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Compares two % mersenne_twister_engine random number generator
   *        objects of the same type for inequality.
   *
   * @param __lhs A % mersenne_twister_engine random number generator
   *              object.
   * @param __rhs Another % mersenne_twister_engine random number
   *              generator object.
   *
   * @returns true if the infinite sequences of generated values
   *          would be different, false otherwise.
   */
  template<typename _UIntType, size_t __w,
           size_t __n, size_t __m, size_t __r,
           _UIntType __a, size_t __u, _UIntType __d, size_t __s,
           _UIntType __b, size_t __t,
           _UIntType __c, size_t __l, _UIntType __f>
    inline bool
    operator!=(const std::mersenne_twister_engine<_UIntType, __w, __n, __m,
               __r, __a, __u, __d, __s, __b, __t, __c, __l, __f>& __lhs,
               const std::mersenne_twister_engine<_UIntType, __w, __n, __m,
               __r, __a, __u, __d, __s, __b, __t, __c, __l, __f>& __rhs)
    { return !(__lhs == __rhs); }
#endif
 
  /**
   * @brief The Marsaglia-Zaman generator.
   *
   * This is a model of a Generalized Fibonacci discrete random number
   * generator, sometimes referred to as the SWC generator.
   *
   * A discrete random number generator that produces pseudorandom
   * numbers using:
   * @f[
   *     x_{i}\leftarrow(x_{i - s} - x_{i - r} - carry_{i-1}) \bmod m
   * @f]
   *
   * The size of the state is @f$r@f$
   * and the maximum period of the generator is @f$(m^r - m^s - 1)@f$.
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _UIntType, size_t __w, size_t __s, size_t __r>
    class subtract_with_carry_engine
    {
      static_assert(std::is_unsigned<_UIntType>::value,
                    "result_type must be an unsigned integral type");
      static_assert(0u < __s && __s < __r,
                    "0 < s < r");
      static_assert(0u < __w && __w <= std::numeric_limits<_UIntType>::digits,
                    "template argument substituting __w out of bounds");
 
      template<typename _Sseq>
        using _If_seed_seq
          = __detail::_If_seed_seq_for<_Sseq, subtract_with_carry_engine,
                                       _UIntType>;
 
    public:
      /** The type of the generated random value. */
      typedef _UIntType result_type;
 
      // parameter values
      static constexpr size_t      word_size    = __w;
      static constexpr size_t      short_lag    = __s;
      static constexpr size_t      long_lag     = __r;
      static constexpr uint_least32_t default_seed = 19780503u;
 
      subtract_with_carry_engine() : subtract_with_carry_engine(0u)
      { }
 
      /**
       * @brief Constructs an explicitly seeded %subtract_with_carry_engine
       *        random number generator.
       */
      explicit
      subtract_with_carry_engine(result_type __sd)
      { seed(__sd); }
 
      /**
       * @brief Constructs a %subtract_with_carry_engine random number engine
       *        seeded from the seed sequence @p __q.
       *
       * @param __q the seed sequence.
       */
      template<typename _Sseq, typename = _If_seed_seq<_Sseq>>
        explicit
        subtract_with_carry_engine(_Sseq& __q)
        { seed(__q); }
 
      /**
       * @brief Seeds the initial state @f$x_0@f$ of the random number
       *        generator.
       *
       * N1688[4.19] modifies this as follows.  If @p __value == 0,
       * sets value to 19780503.  In any case, with a linear
       * congruential generator lcg(i) having parameters @f$ m_{lcg} =
       * 2147483563, a_{lcg} = 40014, c_{lcg} = 0, and lcg(0) = value
       * @f$, sets @f$ x_{-r} \dots x_{-1} @f$ to @f$ lcg(1) \bmod m
       * \dots lcg(r) \bmod m @f$ respectively.  If @f$ x_{-1} = 0 @f$
       * set carry to 1, otherwise sets carry to 0.
       */
      void
      seed(result_type __sd = 0u);
 
      /**
       * @brief Seeds the initial state @f$x_0@f$ of the
       * % subtract_with_carry_engine random number generator.
       */
      template<typename _Sseq>
        _If_seed_seq<_Sseq>
        seed(_Sseq& __q);
 
      /**
       * @brief Gets the inclusive minimum value of the range of random
       * integers returned by this generator.
       */
      static constexpr result_type
      min()
      { return 0; }
 
      /**
       * @brief Gets the inclusive maximum value of the range of random
       * integers returned by this generator.
       */
      static constexpr result_type
      max()
      { return __detail::_Shift<_UIntType, __w>::__value - 1; }
 
      /**
       * @brief Discard a sequence of random numbers.
       */
      void
      discard(unsigned long long __z)
      {
        for (; __z != 0ULL; --__z)
          (*this)();
      }
 
      /**
       * @brief Gets the next random number in the sequence.
       */
      result_type
      operator()();
 
      /**
       * @brief Compares two % subtract_with_carry_engine random number
       *        generator objects of the same type for equality.
       *
       * @param __lhs A % subtract_with_carry_engine random number generator
       *              object.
       * @param __rhs Another % subtract_with_carry_engine random number
       *              generator object.
       *
       * @returns true if the infinite sequences of generated values
       *          would be equal, false otherwise.
      */
      friend bool
      operator==(const subtract_with_carry_engine& __lhs,
                 const subtract_with_carry_engine& __rhs)
      { return (std::equal(__lhs._M_x, __lhs._M_x + long_lag, __rhs._M_x)
                && __lhs._M_carry == __rhs._M_carry
                && __lhs._M_p == __rhs._M_p); }
 
      /**
       * @brief Inserts the current state of a % subtract_with_carry_engine
       *        random number generator engine @p __x into the output stream
       *        @p __os.
       *
       * @param __os An output stream.
       * @param __x  A % subtract_with_carry_engine random number generator
       *             engine.
       *
       * @returns The output stream with the state of @p __x inserted or in
       * an error state.
       */
      template<typename _UIntType1, size_t __w1, size_t __s1, size_t __r1,
               typename _CharT, typename _Traits>
        friend std::basic_ostream<_CharT, _Traits>&
        operator<<(std::basic_ostream<_CharT, _Traits>& __os,
                   const std::subtract_with_carry_engine<_UIntType1, __w1,
                   __s1, __r1>& __x);
 
      /**
       * @brief Extracts the current state of a % subtract_with_carry_engine
       *        random number generator engine @p __x from the input stream
       *        @p __is.
       *
       * @param __is An input stream.
       * @param __x  A % subtract_with_carry_engine random number generator
       *             engine.
       *
       * @returns The input stream with the state of @p __x extracted or in
       * an error state.
       */
      template<typename _UIntType1, size_t __w1, size_t __s1, size_t __r1,
               typename _CharT, typename _Traits>
        friend std::basic_istream<_CharT, _Traits>&
        operator>>(std::basic_istream<_CharT, _Traits>& __is,
                   std::subtract_with_carry_engine<_UIntType1, __w1,
                   __s1, __r1>& __x);
 
    private:
      /// The state of the generator.  This is a ring buffer.
      _UIntType  _M_x[long_lag];
      _UIntType  _M_carry;              ///< The carry
      size_t     _M_p;                  ///< Current index of x(i - r).
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Compares two % subtract_with_carry_engine random number
   *        generator objects of the same type for inequality.
   *
   * @param __lhs A % subtract_with_carry_engine random number generator
   *              object.
   * @param __rhs Another % subtract_with_carry_engine random number
   *              generator object.
   *
   * @returns true if the infinite sequences of generated values
   *          would be different, false otherwise.
   */
  template<typename _UIntType, size_t __w, size_t __s, size_t __r>
    inline bool
    operator!=(const std::subtract_with_carry_engine<_UIntType, __w,
               __s, __r>& __lhs,
               const std::subtract_with_carry_engine<_UIntType, __w,
               __s, __r>& __rhs)
    { return !(__lhs == __rhs); }
#endif
 
  /**
   * Produces random numbers from some base engine by discarding blocks of
   * data.
   *
   * @pre @f$ 0 \leq r \leq p @f$
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _RandomNumberEngine, size_t __p, size_t __r>
    class discard_block_engine
    {
      static_assert(1 <= __r && __r <= __p,
                    "template argument substituting __r out of bounds");
 
    public:
      /** The type of the generated random value. */
      typedef typename _RandomNumberEngine::result_type result_type;
 
      template<typename _Sseq>
        using _If_seed_seq
          = __detail::_If_seed_seq_for<_Sseq, discard_block_engine,
                                       result_type>;
 
      // parameter values
      static constexpr size_t block_size = __p;
      static constexpr size_t used_block = __r;
 
      /**
       * @brief Constructs a default %discard_block_engine engine.
       *
       * The underlying engine is default constructed as well.
       */
      discard_block_engine()
      : _M_b(), _M_n(0) { }
 
      /**
       * @brief Copy constructs a %discard_block_engine engine.
       *
       * Copies an existing base class random number generator.
       * @param __rng An existing (base class) engine object.
       */
      explicit
      discard_block_engine(const _RandomNumberEngine& __rng)
      : _M_b(__rng), _M_n(0) { }
 
      /**
       * @brief Move constructs a %discard_block_engine engine.
       *
       * Copies an existing base class random number generator.
       * @param __rng An existing (base class) engine object.
       */
      explicit
      discard_block_engine(_RandomNumberEngine&& __rng)
      : _M_b(std::move(__rng)), _M_n(0) { }
 
      /**
       * @brief Seed constructs a %discard_block_engine engine.
       *
       * Constructs the underlying generator engine seeded with @p __s.
       * @param __s A seed value for the base class engine.
       */
      explicit
      discard_block_engine(result_type __s)
      : _M_b(__s), _M_n(0) { }
 
      /**
       * @brief Generator construct a %discard_block_engine engine.
       *
       * @param __q A seed sequence.
       */
      template<typename _Sseq, typename = _If_seed_seq<_Sseq>>
        explicit
        discard_block_engine(_Sseq& __q)
        : _M_b(__q), _M_n(0)
        { }
 
      /**
       * @brief Reseeds the %discard_block_engine object with the default
       *        seed for the underlying base class generator engine.
       */
      void
      seed()
      {
        _M_b.seed();
        _M_n = 0;
      }
 
      /**
       * @brief Reseeds the %discard_block_engine object with the default
       *        seed for the underlying base class generator engine.
       */
      void
      seed(result_type __s)
      {
        _M_b.seed(__s);
        _M_n = 0;
      }
 
      /**
       * @brief Reseeds the %discard_block_engine object with the given seed
       *        sequence.
       * @param __q A seed generator function.
       */
      template<typename _Sseq>
        _If_seed_seq<_Sseq>
        seed(_Sseq& __q)
        {
          _M_b.seed(__q);
          _M_n = 0;
        }
 
      /**
       * @brief Gets a const reference to the underlying generator engine
       *        object.
       */
      const _RandomNumberEngine&
      base() const noexcept
      { return _M_b; }
 
      /**
       * @brief Gets the minimum value in the generated random number range.
       */
      static constexpr result_type
      min()
      { return _RandomNumberEngine::min(); }
 
      /**
       * @brief Gets the maximum value in the generated random number range.
       */
      static constexpr result_type
      max()
      { return _RandomNumberEngine::max(); }
 
      /**
       * @brief Discard a sequence of random numbers.
       */
      void
      discard(unsigned long long __z)
      {
        for (; __z != 0ULL; --__z)
          (*this)();
      }
 
      /**
       * @brief Gets the next value in the generated random number sequence.
       */
      result_type
      operator()();
 
      /**
       * @brief Compares two %discard_block_engine random number generator
       *        objects of the same type for equality.
       *
       * @param __lhs A %discard_block_engine random number generator object.
       * @param __rhs Another %discard_block_engine random number generator
       *              object.
       *
       * @returns true if the infinite sequences of generated values
       *          would be equal, false otherwise.
       */
      friend bool
      operator==(const discard_block_engine& __lhs,
                 const discard_block_engine& __rhs)
      { return __lhs._M_b == __rhs._M_b && __lhs._M_n == __rhs._M_n; }
 
      /**
       * @brief Inserts the current state of a %discard_block_engine random
       *        number generator engine @p __x into the output stream
       *        @p __os.
       *
       * @param __os An output stream.
       * @param __x  A %discard_block_engine random number generator engine.
       *
       * @returns The output stream with the state of @p __x inserted or in
       * an error state.
       */
      template<typename _RandomNumberEngine1, size_t __p1, size_t __r1,
               typename _CharT, typename _Traits>
        friend std::basic_ostream<_CharT, _Traits>&
        operator<<(std::basic_ostream<_CharT, _Traits>& __os,
                   const std::discard_block_engine<_RandomNumberEngine1,
                   __p1, __r1>& __x);
 
      /**
       * @brief Extracts the current state of a % subtract_with_carry_engine
       *        random number generator engine @p __x from the input stream
       *        @p __is.
       *
       * @param __is An input stream.
       * @param __x  A %discard_block_engine random number generator engine.
       *
       * @returns The input stream with the state of @p __x extracted or in
       * an error state.
       */
      template<typename _RandomNumberEngine1, size_t __p1, size_t __r1,
               typename _CharT, typename _Traits>
        friend std::basic_istream<_CharT, _Traits>&
        operator>>(std::basic_istream<_CharT, _Traits>& __is,
                   std::discard_block_engine<_RandomNumberEngine1,
                   __p1, __r1>& __x);
 
    private:
      _RandomNumberEngine _M_b;
      size_t _M_n;
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Compares two %discard_block_engine random number generator
   *        objects of the same type for inequality.
   *
   * @param __lhs A %discard_block_engine random number generator object.
   * @param __rhs Another %discard_block_engine random number generator
   *              object.
   *
   * @returns true if the infinite sequences of generated values
   *          would be different, false otherwise.
   */
  template<typename _RandomNumberEngine, size_t __p, size_t __r>
    inline bool
    operator!=(const std::discard_block_engine<_RandomNumberEngine, __p,
               __r>& __lhs,
               const std::discard_block_engine<_RandomNumberEngine, __p,
               __r>& __rhs)
    { return !(__lhs == __rhs); }
#endif
 
  /**
   * Produces random numbers by combining random numbers from some base
   * engine to produce random numbers with a specified number of bits @p __w.
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _RandomNumberEngine, size_t __w, typename _UIntType>
    class independent_bits_engine
    {
      static_assert(std::is_unsigned<_UIntType>::value,
                    "result_type must be an unsigned integral type");
      static_assert(0u < __w && __w <= std::numeric_limits<_UIntType>::digits,
                    "template argument substituting __w out of bounds");
 
      template<typename _Sseq>
        using _If_seed_seq
          = __detail::_If_seed_seq_for<_Sseq, independent_bits_engine,
                                       _UIntType>;
 
    public:
      /** The type of the generated random value. */
      typedef _UIntType result_type;
 
      /**
       * @brief Constructs a default %independent_bits_engine engine.
       *
       * The underlying engine is default constructed as well.
       */
      independent_bits_engine()
      : _M_b() { }
 
      /**
       * @brief Copy constructs a %independent_bits_engine engine.
       *
       * Copies an existing base class random number generator.
       * @param __rng An existing (base class) engine object.
       */
      explicit
      independent_bits_engine(const _RandomNumberEngine& __rng)
      : _M_b(__rng) { }
 
      /**
       * @brief Move constructs a %independent_bits_engine engine.
       *
       * Copies an existing base class random number generator.
       * @param __rng An existing (base class) engine object.
       */
      explicit
      independent_bits_engine(_RandomNumberEngine&& __rng)
      : _M_b(std::move(__rng)) { }
 
      /**
       * @brief Seed constructs a %independent_bits_engine engine.
       *
       * Constructs the underlying generator engine seeded with @p __s.
       * @param __s A seed value for the base class engine.
       */
      explicit
      independent_bits_engine(result_type __s)
      : _M_b(__s) { }
 
      /**
       * @brief Generator construct a %independent_bits_engine engine.
       *
       * @param __q A seed sequence.
       */
      template<typename _Sseq, typename = _If_seed_seq<_Sseq>>
        explicit
        independent_bits_engine(_Sseq& __q)
        : _M_b(__q)
        { }
 
      /**
       * @brief Reseeds the %independent_bits_engine object with the default
       *        seed for the underlying base class generator engine.
       */
      void
      seed()
      { _M_b.seed(); }
 
      /**
       * @brief Reseeds the %independent_bits_engine object with the default
       *        seed for the underlying base class generator engine.
       */
      void
      seed(result_type __s)
      { _M_b.seed(__s); }
 
      /**
       * @brief Reseeds the %independent_bits_engine object with the given
       *        seed sequence.
       * @param __q A seed generator function.
       */
      template<typename _Sseq>
        _If_seed_seq<_Sseq>
        seed(_Sseq& __q)
        { _M_b.seed(__q); }
 
      /**
       * @brief Gets a const reference to the underlying generator engine
       *        object.
       */
      const _RandomNumberEngine&
      base() const noexcept
      { return _M_b; }
 
      /**
       * @brief Gets the minimum value in the generated random number range.
       */
      static constexpr result_type
      min()
      { return 0U; }
 
      /**
       * @brief Gets the maximum value in the generated random number range.
       */
      static constexpr result_type
      max()
      { return __detail::_Shift<_UIntType, __w>::__value - 1; }
 
      /**
       * @brief Discard a sequence of random numbers.
       */
      void
      discard(unsigned long long __z)
      {
        for (; __z != 0ULL; --__z)
          (*this)();
      }
 
      /**
       * @brief Gets the next value in the generated random number sequence.
       */
      result_type
      operator()();
 
      /**
       * @brief Compares two %independent_bits_engine random number generator
       * objects of the same type for equality.
       *
       * @param __lhs A %independent_bits_engine random number generator
       *              object.
       * @param __rhs Another %independent_bits_engine random number generator
       *              object.
       *
       * @returns true if the infinite sequences of generated values
       *          would be equal, false otherwise.
       */
      friend bool
      operator==(const independent_bits_engine& __lhs,
                 const independent_bits_engine& __rhs)
      { return __lhs._M_b == __rhs._M_b; }
 
      /**
       * @brief Extracts the current state of a % subtract_with_carry_engine
       *        random number generator engine @p __x from the input stream
       *        @p __is.
       *
       * @param __is An input stream.
       * @param __x  A %independent_bits_engine random number generator
       *             engine.
       *
       * @returns The input stream with the state of @p __x extracted or in
       *          an error state.
       */
      template<typename _CharT, typename _Traits>
        friend std::basic_istream<_CharT, _Traits>&
        operator>>(std::basic_istream<_CharT, _Traits>& __is,
                   std::independent_bits_engine<_RandomNumberEngine,
                   __w, _UIntType>& __x)
        {
          __is >> __x._M_b;
          return __is;
        }
 
    private:
      _RandomNumberEngine _M_b;
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Compares two %independent_bits_engine random number generator
   * objects of the same type for inequality.
   *
   * @param __lhs A %independent_bits_engine random number generator
   *              object.
   * @param __rhs Another %independent_bits_engine random number generator
   *              object.
   *
   * @returns true if the infinite sequences of generated values
   *          would be different, false otherwise.
   */
  template<typename _RandomNumberEngine, size_t __w, typename _UIntType>
    inline bool
    operator!=(const std::independent_bits_engine<_RandomNumberEngine, __w,
               _UIntType>& __lhs,
               const std::independent_bits_engine<_RandomNumberEngine, __w,
               _UIntType>& __rhs)
    { return !(__lhs == __rhs); }
#endif
 
  /**
   * @brief Inserts the current state of a %independent_bits_engine random
   *        number generator engine @p __x into the output stream @p __os.
   *
   * @param __os An output stream.
   * @param __x  A %independent_bits_engine random number generator engine.
   *
   * @returns The output stream with the state of @p __x inserted or in
   *          an error state.
   */
  template<typename _RandomNumberEngine, size_t __w, typename _UIntType,
           typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
               const std::independent_bits_engine<_RandomNumberEngine,
               __w, _UIntType>& __x)
    {
      __os << __x.base();
      return __os;
    }
 
 
  /**
   * @brief Produces random numbers by reordering random numbers from some
   * base engine.
   *
   * The values from the base engine are stored in a sequence of size @p __k
   * and shuffled by an algorithm that depends on those values.
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _RandomNumberEngine, size_t __k>
    class shuffle_order_engine
    {
      static_assert(1u <= __k, "template argument substituting "
                    "__k out of bound");
 
    public:
      /** The type of the generated random value. */
      typedef typename _RandomNumberEngine::result_type result_type;
 
      template<typename _Sseq>
        using _If_seed_seq
          = __detail::_If_seed_seq_for<_Sseq, shuffle_order_engine,
                                       result_type>;
 
      static constexpr size_t table_size = __k;
 
      /**
       * @brief Constructs a default %shuffle_order_engine engine.
       *
       * The underlying engine is default constructed as well.
       */
      shuffle_order_engine()
      : _M_b()
      { _M_initialize(); }
 
      /**
       * @brief Copy constructs a %shuffle_order_engine engine.
       *
       * Copies an existing base class random number generator.
       * @param __rng An existing (base class) engine object.
       */
      explicit
      shuffle_order_engine(const _RandomNumberEngine& __rng)
      : _M_b(__rng)
      { _M_initialize(); }
 
      /**
       * @brief Move constructs a %shuffle_order_engine engine.
       *
       * Copies an existing base class random number generator.
       * @param __rng An existing (base class) engine object.
       */
      explicit
      shuffle_order_engine(_RandomNumberEngine&& __rng)
      : _M_b(std::move(__rng))
      { _M_initialize(); }
 
      /**
       * @brief Seed constructs a %shuffle_order_engine engine.
       *
       * Constructs the underlying generator engine seeded with @p __s.
       * @param __s A seed value for the base class engine.
       */
      explicit
      shuffle_order_engine(result_type __s)
      : _M_b(__s)
      { _M_initialize(); }
 
      /**
       * @brief Generator construct a %shuffle_order_engine engine.
       *
       * @param __q A seed sequence.
       */
      template<typename _Sseq, typename = _If_seed_seq<_Sseq>>
        explicit
        shuffle_order_engine(_Sseq& __q)
        : _M_b(__q)
        { _M_initialize(); }
 
      /**
       * @brief Reseeds the %shuffle_order_engine object with the default seed
                for the underlying base class generator engine.
       */
      void
      seed()
      {
        _M_b.seed();
        _M_initialize();
      }
 
      /**
       * @brief Reseeds the %shuffle_order_engine object with the default seed
       *        for the underlying base class generator engine.
       */
      void
      seed(result_type __s)
      {
        _M_b.seed(__s);
        _M_initialize();
      }
 
      /**
       * @brief Reseeds the %shuffle_order_engine object with the given seed
       *        sequence.
       * @param __q A seed generator function.
       */
      template<typename _Sseq>
        _If_seed_seq<_Sseq>
        seed(_Sseq& __q)
        {
          _M_b.seed(__q);
          _M_initialize();
        }
 
      /**
       * Gets a const reference to the underlying generator engine object.
       */
      const _RandomNumberEngine&
      base() const noexcept
      { return _M_b; }
 
      /**
       * Gets the minimum value in the generated random number range.
       */
      static constexpr result_type
      min()
      { return _RandomNumberEngine::min(); }
 
      /**
       * Gets the maximum value in the generated random number range.
       */
      static constexpr result_type
      max()
      { return _RandomNumberEngine::max(); }
 
      /**
       * Discard a sequence of random numbers.
       */
      void
      discard(unsigned long long __z)
      {
        for (; __z != 0ULL; --__z)
          (*this)();
      }
 
      /**
       * Gets the next value in the generated random number sequence.
       */
      result_type
      operator()();
 
      /**
       * Compares two %shuffle_order_engine random number generator objects
       * of the same type for equality.
       *
       * @param __lhs A %shuffle_order_engine random number generator object.
       * @param __rhs Another %shuffle_order_engine random number generator
       *              object.
       *
       * @returns true if the infinite sequences of generated values
       *          would be equal, false otherwise.
      */
      friend bool
      operator==(const shuffle_order_engine& __lhs,
                 const shuffle_order_engine& __rhs)
      { return (__lhs._M_b == __rhs._M_b
                && std::equal(__lhs._M_v, __lhs._M_v + __k, __rhs._M_v)
                && __lhs._M_y == __rhs._M_y); }
 
      /**
       * @brief Inserts the current state of a %shuffle_order_engine random
       *        number generator engine @p __x into the output stream
        @p __os.
       *
       * @param __os An output stream.
       * @param __x  A %shuffle_order_engine random number generator engine.
       *
       * @returns The output stream with the state of @p __x inserted or in
       * an error state.
       */
      template<typename _RandomNumberEngine1, size_t __k1,
               typename _CharT, typename _Traits>
        friend std::basic_ostream<_CharT, _Traits>&
        operator<<(std::basic_ostream<_CharT, _Traits>& __os,
                   const std::shuffle_order_engine<_RandomNumberEngine1,
                   __k1>& __x);
 
      /**
       * @brief Extracts the current state of a % subtract_with_carry_engine
       *        random number generator engine @p __x from the input stream
       *        @p __is.
       *
       * @param __is An input stream.
       * @param __x  A %shuffle_order_engine random number generator engine.
       *
       * @returns The input stream with the state of @p __x extracted or in
       * an error state.
       */
      template<typename _RandomNumberEngine1, size_t __k1,
               typename _CharT, typename _Traits>
        friend std::basic_istream<_CharT, _Traits>&
        operator>>(std::basic_istream<_CharT, _Traits>& __is,
                   std::shuffle_order_engine<_RandomNumberEngine1, __k1>& __x);
 
    private:
      void _M_initialize()
      {
        for (size_t __i = 0; __i < __k; ++__i)
          _M_v[__i] = _M_b();
        _M_y = _M_b();
      }
 
      _RandomNumberEngine _M_b;
      result_type _M_v[__k];
      result_type _M_y;
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * Compares two %shuffle_order_engine random number generator objects
   * of the same type for inequality.
   *
   * @param __lhs A %shuffle_order_engine random number generator object.
   * @param __rhs Another %shuffle_order_engine random number generator
   *              object.
   *
   * @returns true if the infinite sequences of generated values
   *          would be different, false otherwise.
   */
  template<typename _RandomNumberEngine, size_t __k>
    inline bool
    operator!=(const std::shuffle_order_engine<_RandomNumberEngine,
               __k>& __lhs,
               const std::shuffle_order_engine<_RandomNumberEngine,
               __k>& __rhs)
    { return !(__lhs == __rhs); }
#endif
 
  /**
   * The classic Minimum Standard rand0 of Lewis, Goodman, and Miller.
   */
  typedef linear_congruential_engine<uint_fast32_t, 16807UL, 0UL, 2147483647UL>
  minstd_rand0;
 
  /**
   * An alternative LCR (Lehmer Generator function).
   */
  typedef linear_congruential_engine<uint_fast32_t, 48271UL, 0UL, 2147483647UL>
  minstd_rand;
 
  /**
   * The classic Mersenne Twister.
   *
   * Reference:
   * M. Matsumoto and T. Nishimura, Mersenne Twister: A 623-Dimensionally
   * Equidistributed Uniform Pseudo-Random Number Generator, ACM Transactions
   * on Modeling and Computer Simulation, Vol. 8, No. 1, January 1998, pp 3-30.
   */
  typedef mersenne_twister_engine<
    uint_fast32_t,
    32, 624, 397, 31,
    0x9908b0dfUL, 11,
    0xffffffffUL, 7,
    0x9d2c5680UL, 15,
    0xefc60000UL, 18, 1812433253UL> mt19937;
 
  /**
   * An alternative Mersenne Twister.
   */
  typedef mersenne_twister_engine<
    uint_fast64_t,
    64, 312, 156, 31,
    0xb5026f5aa96619e9ULL, 29,
    0x5555555555555555ULL, 17,
    0x71d67fffeda60000ULL, 37,
    0xfff7eee000000000ULL, 43,
    6364136223846793005ULL> mt19937_64;
 
  typedef subtract_with_carry_engine<uint_fast32_t, 24, 10, 24>
    ranlux24_base;
 
  typedef subtract_with_carry_engine<uint_fast64_t, 48, 5, 12>
    ranlux48_base;
 
  typedef discard_block_engine<ranlux24_base, 223, 23> ranlux24;
 
  typedef discard_block_engine<ranlux48_base, 389, 11> ranlux48;
 
  typedef shuffle_order_engine<minstd_rand0, 256> knuth_b;
 
  typedef minstd_rand0 default_random_engine;
 
  /**
   * A standard interface to a platform-specific non-deterministic
   * random number generator (if any are available).
   *
   * @headerfile random
   * @since C++11
   */
  class random_device
  {
  public:
    /** The type of the generated random value. */
    typedef unsigned int result_type;
 
    // constructors, destructors and member functions
 
    random_device() { _M_init("default"); }
 
    explicit
    random_device(const std::string& __token) { _M_init(__token); }
 
    ~random_device()
    { _M_fini(); }
 
    static constexpr result_type
    min()
    { return std::numeric_limits<result_type>::min(); }
 
    static constexpr result_type
    max()
    { return std::numeric_limits<result_type>::max(); }
 
    double
    entropy() const noexcept
    { return this->_M_getentropy(); }
 
    result_type
    operator()()
    { return this->_M_getval(); }
 
    // No copy functions.
    random_device(const random_device&) = delete;
    void operator=(const random_device&) = delete;
 
  private:
 
    void _M_init(const std::string& __token);
    void _M_init_pretr1(const std::string& __token);
    void _M_fini();
 
    result_type _M_getval();
    result_type _M_getval_pretr1();
    double _M_getentropy() const noexcept;
 
    void _M_init(const char*, size_t); // not exported from the shared library
 
    __extension__ union
    {
      struct
      {
        void*      _M_file;
        result_type (*_M_func)(void*);
        int _M_fd;
      };
      mt19937    _M_mt;
    };
  };
 
  /// @} group random_generators
 
  /**
   * @addtogroup random_distributions Random Number Distributions
   * @ingroup random
   * @{
   */
 
  /**
   * @addtogroup random_distributions_uniform Uniform Distributions
   * @ingroup random_distributions
   * @{
   */
 
  // std::uniform_int_distribution is defined in <bits/uniform_int_dist.h>
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Return true if two uniform integer distributions have
   *        different parameters.
   */
  template<typename _IntType>
    inline bool
    operator!=(const std::uniform_int_distribution<_IntType>& __d1,
               const std::uniform_int_distribution<_IntType>& __d2)
    { return !(__d1 == __d2); }
#endif
 
  /**
   * @brief Inserts a %uniform_int_distribution random number
   *        distribution @p __x into the output stream @p os.
   *
   * @param __os An output stream.
   * @param __x  A %uniform_int_distribution random number distribution.
   *
   * @returns The output stream with the state of @p __x inserted or in
   * an error state.
   */
  template<typename _IntType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>&,
               const std::uniform_int_distribution<_IntType>&);
 
  /**
   * @brief Extracts a %uniform_int_distribution random number distribution
   * @p __x from the input stream @p __is.
   *
   * @param __is An input stream.
   * @param __x  A %uniform_int_distribution random number generator engine.
   *
   * @returns The input stream with @p __x extracted or in an error state.
   */
  template<typename _IntType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>&,
               std::uniform_int_distribution<_IntType>&);
 
 
  /**
   * @brief Uniform continuous distribution for random numbers.
   *
   * A continuous random distribution on the range [min, max) with equal
   * probability throughout the range.  The URNG should be real-valued and
   * deliver number in the range [0, 1).
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _RealType = double>
    class uniform_real_distribution
    {
      static_assert(std::is_floating_point<_RealType>::value,
                    "result_type must be a floating point type");
 
    public:
      /** The type of the range of the distribution. */
      typedef _RealType result_type;
 
      /** Parameter type. */
      struct param_type
      {
        typedef uniform_real_distribution<_RealType> distribution_type;
 
        param_type() : param_type(0) { }
 
        explicit
        param_type(_RealType __a, _RealType __b = _RealType(1))
        : _M_a(__a), _M_b(__b)
        {
          __glibcxx_assert(_M_a <= _M_b);
        }
 
        result_type
        a() const
        { return _M_a; }
 
        result_type
        b() const
        { return _M_b; }
 
        friend bool
        operator==(const param_type& __p1, const param_type& __p2)
        { return __p1._M_a == __p2._M_a && __p1._M_b == __p2._M_b; }
 
#if __cpp_impl_three_way_comparison < 201907L
        friend bool
        operator!=(const param_type& __p1, const param_type& __p2)
        { return !(__p1 == __p2); }
#endif
 
      private:
        _RealType _M_a;
        _RealType _M_b;
      };
 
    public:
      /**
       * @brief Constructs a uniform_real_distribution object.
       *
       * The lower bound is set to 0.0 and the upper bound to 1.0
       */
      uniform_real_distribution() : uniform_real_distribution(0.0) { }
 
      /**
       * @brief Constructs a uniform_real_distribution object.
       *
       * @param __a [IN]  The lower bound of the distribution.
       * @param __b [IN]  The upper bound of the distribution.
       */
      explicit
      uniform_real_distribution(_RealType __a, _RealType __b = _RealType(1))
      : _M_param(__a, __b)
      { }
 
      explicit
      uniform_real_distribution(const param_type& __p)
      : _M_param(__p)
      { }
 
      /**
       * @brief Resets the distribution state.
       *
       * Does nothing for the uniform real distribution.
       */
      void
      reset() { }
 
      result_type
      a() const
      { return _M_param.a(); }
 
      result_type
      b() const
      { return _M_param.b(); }
 
      /**
       * @brief Returns the parameter set of the distribution.
       */
      param_type
      param() const
      { return _M_param; }
 
      /**
       * @brief Sets the parameter set of the distribution.
       * @param __param The new parameter set of the distribution.
       */
      void
      param(const param_type& __param)
      { _M_param = __param; }
 
      /**
       * @brief Returns the inclusive lower bound of the distribution range.
       */
      result_type
      min() const
      { return this->a(); }
 
      /**
       * @brief Returns the inclusive upper bound of the distribution range.
       */
      result_type
      max() const
      { return this->b(); }
 
      /**
       * @brief Generating functions.
       */
      template<typename _UniformRandomNumberGenerator>
        result_type
        operator()(_UniformRandomNumberGenerator& __urng)
        { return this->operator()(__urng, _M_param); }
 
      template<typename _UniformRandomNumberGenerator>
        result_type
        operator()(_UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        {
          __detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
            __aurng(__urng);
          return (__aurng() * (__p.b() - __p.a())) + __p.a();
        }
 
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate(_ForwardIterator __f, _ForwardIterator __t,
                   _UniformRandomNumberGenerator& __urng)
        { this->__generate(__f, __t, __urng, _M_param); }
 
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate(_ForwardIterator __f, _ForwardIterator __t,
                   _UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        { this->__generate_impl(__f, __t, __urng, __p); }
 
      template<typename _UniformRandomNumberGenerator>
        void
        __generate(result_type* __f, result_type* __t,
                   _UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        { this->__generate_impl(__f, __t, __urng, __p); }
 
      /**
       * @brief Return true if two uniform real distributions have
       *        the same parameters.
       */
      friend bool
      operator==(const uniform_real_distribution& __d1,
                 const uniform_real_distribution& __d2)
      { return __d1._M_param == __d2._M_param; }
 
    private:
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
                        _UniformRandomNumberGenerator& __urng,
                        const param_type& __p);
 
      param_type _M_param;
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Return true if two uniform real distributions have
   *        different parameters.
   */
  template<typename _IntType>
    inline bool
    operator!=(const std::uniform_real_distribution<_IntType>& __d1,
               const std::uniform_real_distribution<_IntType>& __d2)
    { return !(__d1 == __d2); }
#endif
 
  /**
   * @brief Inserts a %uniform_real_distribution random number
   *        distribution @p __x into the output stream @p __os.
   *
   * @param __os An output stream.
   * @param __x  A %uniform_real_distribution random number distribution.
   *
   * @returns The output stream with the state of @p __x inserted or in
   *          an error state.
   */
  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>&,
               const std::uniform_real_distribution<_RealType>&);
 
  /**
   * @brief Extracts a %uniform_real_distribution random number distribution
   * @p __x from the input stream @p __is.
   *
   * @param __is An input stream.
   * @param __x  A %uniform_real_distribution random number generator engine.
   *
   * @returns The input stream with @p __x extracted or in an error state.
   */
  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>&,
               std::uniform_real_distribution<_RealType>&);
 
  /// @} group random_distributions_uniform
 
  /**
   * @addtogroup random_distributions_normal Normal Distributions
   * @ingroup random_distributions
   * @{
   */
 
  /**
   * @brief A normal continuous distribution for random numbers.
   *
   * The formula for the normal probability density function is
   * @f[
   *     p(x|\mu,\sigma) = \frac{1}{\sigma \sqrt{2 \pi}}
   *            e^{- \frac{{x - \mu}^ {2}}{2 \sigma ^ {2}} }
   * @f]
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _RealType = double>
    class normal_distribution
    {
      static_assert(std::is_floating_point<_RealType>::value,
                    "result_type must be a floating point type");
 
    public:
      /** The type of the range of the distribution. */
      typedef _RealType result_type;
 
      /** Parameter type. */
      struct param_type
      {
        typedef normal_distribution<_RealType> distribution_type;
 
        param_type() : param_type(0.0) { }
 
        explicit
        param_type(_RealType __mean, _RealType __stddev = _RealType(1))
        : _M_mean(__mean), _M_stddev(__stddev)
        {
          __glibcxx_assert(_M_stddev > _RealType(0));
        }
 
        _RealType
        mean() const
        { return _M_mean; }
 
        _RealType
        stddev() const
        { return _M_stddev; }
 
        friend bool
        operator==(const param_type& __p1, const param_type& __p2)
        { return (__p1._M_mean == __p2._M_mean
                  && __p1._M_stddev == __p2._M_stddev); }
 
#if __cpp_impl_three_way_comparison < 201907L
        friend bool
        operator!=(const param_type& __p1, const param_type& __p2)
        { return !(__p1 == __p2); }
#endif
 
      private:
        _RealType _M_mean;
        _RealType _M_stddev;
      };
 
    public:
      normal_distribution() : normal_distribution(0.0) { }
 
      /**
       * Constructs a normal distribution with parameters @f$mean@f$ and
       * standard deviation.
       */
      explicit
      normal_distribution(result_type __mean,
                          result_type __stddev = result_type(1))
      : _M_param(__mean, __stddev)
      { }
 
      explicit
      normal_distribution(const param_type& __p)
      : _M_param(__p)
      { }
 
      /**
       * @brief Resets the distribution state.
       */
      void
      reset()
      { _M_saved_available = false; }
 
      /**
       * @brief Returns the mean of the distribution.
       */
      _RealType
      mean() const
      { return _M_param.mean(); }
 
      /**
       * @brief Returns the standard deviation of the distribution.
       */
      _RealType
      stddev() const
      { return _M_param.stddev(); }
 
      /**
       * @brief Returns the parameter set of the distribution.
       */
      param_type
      param() const
      { return _M_param; }
 
      /**
       * @brief Sets the parameter set of the distribution.
       * @param __param The new parameter set of the distribution.
       */
      void
      param(const param_type& __param)
      { _M_param = __param; }
 
      /**
       * @brief Returns the greatest lower bound value of the distribution.
       */
      result_type
      min() const
      { return std::numeric_limits<result_type>::lowest(); }
 
      /**
       * @brief Returns the least upper bound value of the distribution.
       */
      result_type
      max() const
      { return std::numeric_limits<result_type>::max(); }
 
      /**
       * @brief Generating functions.
       */
      template<typename _UniformRandomNumberGenerator>
        result_type
        operator()(_UniformRandomNumberGenerator& __urng)
        { return this->operator()(__urng, _M_param); }
 
      template<typename _UniformRandomNumberGenerator>
        result_type
        operator()(_UniformRandomNumberGenerator& __urng,
                   const param_type& __p);
 
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate(_ForwardIterator __f, _ForwardIterator __t,
                   _UniformRandomNumberGenerator& __urng)
        { this->__generate(__f, __t, __urng, _M_param); }
 
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate(_ForwardIterator __f, _ForwardIterator __t,
                   _UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        { this->__generate_impl(__f, __t, __urng, __p); }
 
      template<typename _UniformRandomNumberGenerator>
        void
        __generate(result_type* __f, result_type* __t,
                   _UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        { this->__generate_impl(__f, __t, __urng, __p); }
 
      /**
       * @brief Return true if two normal distributions have
       *        the same parameters and the sequences that would
       *        be generated are equal.
       */
      template<typename _RealType1>
        friend bool
        operator==(const std::normal_distribution<_RealType1>& __d1,
                   const std::normal_distribution<_RealType1>& __d2);
 
      /**
       * @brief Inserts a %normal_distribution random number distribution
       * @p __x into the output stream @p __os.
       *
       * @param __os An output stream.
       * @param __x  A %normal_distribution random number distribution.
       *
       * @returns The output stream with the state of @p __x inserted or in
       * an error state.
       */
      template<typename _RealType1, typename _CharT, typename _Traits>
        friend std::basic_ostream<_CharT, _Traits>&
        operator<<(std::basic_ostream<_CharT, _Traits>& __os,
                   const std::normal_distribution<_RealType1>& __x);
 
      /**
       * @brief Extracts a %normal_distribution random number distribution
       * @p __x from the input stream @p __is.
       *
       * @param __is An input stream.
       * @param __x  A %normal_distribution random number generator engine.
       *
       * @returns The input stream with @p __x extracted or in an error
       *          state.
       */
      template<typename _RealType1, typename _CharT, typename _Traits>
        friend std::basic_istream<_CharT, _Traits>&
        operator>>(std::basic_istream<_CharT, _Traits>& __is,
                   std::normal_distribution<_RealType1>& __x);
 
    private:
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
                        _UniformRandomNumberGenerator& __urng,
                        const param_type& __p);
 
      param_type  _M_param;
      result_type _M_saved = 0;
      bool        _M_saved_available = false;
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Return true if two normal distributions are different.
   */
  template<typename _RealType>
    inline bool
    operator!=(const std::normal_distribution<_RealType>& __d1,
               const std::normal_distribution<_RealType>& __d2)
    { return !(__d1 == __d2); }
#endif
 
  /**
   * @brief A lognormal_distribution random number distribution.
   *
   * The formula for the normal probability mass function is
   * @f[
   *     p(x|m,s) = \frac{1}{sx\sqrt{2\pi}}
   *                \exp{-\frac{(\ln{x} - m)^2}{2s^2}}
   * @f]
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _RealType = double>
    class lognormal_distribution
    {
      static_assert(std::is_floating_point<_RealType>::value,
                    "result_type must be a floating point type");
 
    public:
      /** The type of the range of the distribution. */
      typedef _RealType result_type;
 
      /** Parameter type. */
      struct param_type
      {
        typedef lognormal_distribution<_RealType> distribution_type;
 
        param_type() : param_type(0.0) { }
 
        explicit
        param_type(_RealType __m, _RealType __s = _RealType(1))
        : _M_m(__m), _M_s(__s)
        { }
 
        _RealType
        m() const
        { return _M_m; }
 
        _RealType
        s() const
        { return _M_s; }
 
        friend bool
        operator==(const param_type& __p1, const param_type& __p2)
        { return __p1._M_m == __p2._M_m && __p1._M_s == __p2._M_s; }
 
#if __cpp_impl_three_way_comparison < 201907L
        friend bool
        operator!=(const param_type& __p1, const param_type& __p2)
        { return !(__p1 == __p2); }
#endif
 
      private:
        _RealType _M_m;
        _RealType _M_s;
      };
 
      lognormal_distribution() : lognormal_distribution(0.0) { }
 
      explicit
      lognormal_distribution(_RealType __m, _RealType __s = _RealType(1))
      : _M_param(__m, __s), _M_nd()
      { }
 
      explicit
      lognormal_distribution(const param_type& __p)
      : _M_param(__p), _M_nd()
      { }
 
      /**
       * Resets the distribution state.
       */
      void
      reset()
      { _M_nd.reset(); }
 
      /**
       *
       */
      _RealType
      m() const
      { return _M_param.m(); }
 
      _RealType
      s() const
      { return _M_param.s(); }
 
      /**
       * @brief Returns the parameter set of the distribution.
       */
      param_type
      param() const
      { return _M_param; }
 
      /**
       * @brief Sets the parameter set of the distribution.
       * @param __param The new parameter set of the distribution.
       */
      void
      param(const param_type& __param)
      { _M_param = __param; }
 
      /**
       * @brief Returns the greatest lower bound value of the distribution.
       */
      result_type
      min() const
      { return result_type(0); }
 
      /**
       * @brief Returns the least upper bound value of the distribution.
       */
      result_type
      max() const
      { return std::numeric_limits<result_type>::max(); }
 
      /**
       * @brief Generating functions.
       */
      template<typename _UniformRandomNumberGenerator>
        result_type
        operator()(_UniformRandomNumberGenerator& __urng)
        { return this->operator()(__urng, _M_param); }
 
      template<typename _UniformRandomNumberGenerator>
        result_type
        operator()(_UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        { return std::exp(__p.s() * _M_nd(__urng) + __p.m()); }
 
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate(_ForwardIterator __f, _ForwardIterator __t,
                   _UniformRandomNumberGenerator& __urng)
        { this->__generate(__f, __t, __urng, _M_param); }
 
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate(_ForwardIterator __f, _ForwardIterator __t,
                   _UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        { this->__generate_impl(__f, __t, __urng, __p); }
 
      template<typename _UniformRandomNumberGenerator>
        void
        __generate(result_type* __f, result_type* __t,
                   _UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        { this->__generate_impl(__f, __t, __urng, __p); }
 
      /**
       * @brief Return true if two lognormal distributions have
       *        the same parameters and the sequences that would
       *        be generated are equal.
       */
      friend bool
      operator==(const lognormal_distribution& __d1,
                 const lognormal_distribution& __d2)
      { return (__d1._M_param == __d2._M_param
                && __d1._M_nd == __d2._M_nd); }
 
      /**
       * @brief Inserts a %lognormal_distribution random number distribution
       * @p __x into the output stream @p __os.
       *
       * @param __os An output stream.
       * @param __x  A %lognormal_distribution random number distribution.
       *
       * @returns The output stream with the state of @p __x inserted or in
       * an error state.
       */
      template<typename _RealType1, typename _CharT, typename _Traits>
        friend std::basic_ostream<_CharT, _Traits>&
        operator<<(std::basic_ostream<_CharT, _Traits>& __os,
                   const std::lognormal_distribution<_RealType1>& __x);
 
      /**
       * @brief Extracts a %lognormal_distribution random number distribution
       * @p __x from the input stream @p __is.
       *
       * @param __is An input stream.
       * @param __x A %lognormal_distribution random number
       *            generator engine.
       *
       * @returns The input stream with @p __x extracted or in an error state.
       */
      template<typename _RealType1, typename _CharT, typename _Traits>
        friend std::basic_istream<_CharT, _Traits>&
        operator>>(std::basic_istream<_CharT, _Traits>& __is,
                   std::lognormal_distribution<_RealType1>& __x);
 
    private:
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
                        _UniformRandomNumberGenerator& __urng,
                        const param_type& __p);
 
      param_type _M_param;
 
      std::normal_distribution<result_type> _M_nd;
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Return true if two lognormal distributions are different.
   */
  template<typename _RealType>
    inline bool
    operator!=(const std::lognormal_distribution<_RealType>& __d1,
               const std::lognormal_distribution<_RealType>& __d2)
    { return !(__d1 == __d2); }
#endif
 
  /// @} group random_distributions_normal
 
  /**
   * @addtogroup random_distributions_poisson Poisson Distributions
   * @ingroup random_distributions
   * @{
   */
 
  /**
   * @brief A gamma continuous distribution for random numbers.
   *
   * The formula for the gamma probability density function is:
   * @f[
   *     p(x|\alpha,\beta) = \frac{1}{\beta\Gamma(\alpha)}
   *                         (x/\beta)^{\alpha - 1} e^{-x/\beta}
   * @f]
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _RealType = double>
    class gamma_distribution
    {
      static_assert(std::is_floating_point<_RealType>::value,
                    "result_type must be a floating point type");
 
    public:
      /** The type of the range of the distribution. */
      typedef _RealType result_type;
 
      /** Parameter type. */
      struct param_type
      {
        typedef gamma_distribution<_RealType> distribution_type;
        friend class gamma_distribution<_RealType>;
 
        param_type() : param_type(1.0) { }
 
        explicit
        param_type(_RealType __alpha_val, _RealType __beta_val = _RealType(1))
        : _M_alpha(__alpha_val), _M_beta(__beta_val)
        {
          __glibcxx_assert(_M_alpha > _RealType(0));
          _M_initialize();
        }
 
        _RealType
        alpha() const
        { return _M_alpha; }
 
        _RealType
        beta() const
        { return _M_beta; }
 
        friend bool
        operator==(const param_type& __p1, const param_type& __p2)
        { return (__p1._M_alpha == __p2._M_alpha
                  && __p1._M_beta == __p2._M_beta); }
 
#if __cpp_impl_three_way_comparison < 201907L
        friend bool
        operator!=(const param_type& __p1, const param_type& __p2)
        { return !(__p1 == __p2); }
#endif
 
      private:
        void
        _M_initialize();
 
        _RealType _M_alpha;
        _RealType _M_beta;
 
        _RealType _M_malpha, _M_a2;
      };
 
    public:
      /**
       * @brief Constructs a gamma distribution with parameters 1 and 1.
       */
      gamma_distribution() : gamma_distribution(1.0) { }
 
      /**
       * @brief Constructs a gamma distribution with parameters
       * @f$\alpha@f$ and @f$\beta@f$.
       */
      explicit
      gamma_distribution(_RealType __alpha_val,
                         _RealType __beta_val = _RealType(1))
      : _M_param(__alpha_val, __beta_val), _M_nd()
      { }
 
      explicit
      gamma_distribution(const param_type& __p)
      : _M_param(__p), _M_nd()
      { }
 
      /**
       * @brief Resets the distribution state.
       */
      void
      reset()
      { _M_nd.reset(); }
 
      /**
       * @brief Returns the @f$\alpha@f$ of the distribution.
       */
      _RealType
      alpha() const
      { return _M_param.alpha(); }
 
      /**
       * @brief Returns the @f$\beta@f$ of the distribution.
       */
      _RealType
      beta() const
      { return _M_param.beta(); }
 
      /**
       * @brief Returns the parameter set of the distribution.
       */
      param_type
      param() const
      { return _M_param; }
 
      /**
       * @brief Sets the parameter set of the distribution.
       * @param __param The new parameter set of the distribution.
       */
      void
      param(const param_type& __param)
      { _M_param = __param; }
 
      /**
       * @brief Returns the greatest lower bound value of the distribution.
       */
      result_type
      min() const
      { return result_type(0); }
 
      /**
       * @brief Returns the least upper bound value of the distribution.
       */
      result_type
      max() const
      { return std::numeric_limits<result_type>::max(); }
 
      /**
       * @brief Generating functions.
       */
      template<typename _UniformRandomNumberGenerator>
        result_type
        operator()(_UniformRandomNumberGenerator& __urng)
        { return this->operator()(__urng, _M_param); }
 
      template<typename _UniformRandomNumberGenerator>
        result_type
        operator()(_UniformRandomNumberGenerator& __urng,
                   const param_type& __p);
 
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate(_ForwardIterator __f, _ForwardIterator __t,
                   _UniformRandomNumberGenerator& __urng)
        { this->__generate(__f, __t, __urng, _M_param); }
 
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate(_ForwardIterator __f, _ForwardIterator __t,
                   _UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        { this->__generate_impl(__f, __t, __urng, __p); }
 
      template<typename _UniformRandomNumberGenerator>
        void
        __generate(result_type* __f, result_type* __t,
                   _UniformRandomNumberGenerator& __urng,
                   const param_type& __p)
        { this->__generate_impl(__f, __t, __urng, __p); }
 
      /**
       * @brief Return true if two gamma distributions have the same
       *        parameters and the sequences that would be generated
       *        are equal.
       */
      friend bool
      operator==(const gamma_distribution& __d1,
                 const gamma_distribution& __d2)
      { return (__d1._M_param == __d2._M_param
                && __d1._M_nd == __d2._M_nd); }
 
      /**
       * @brief Inserts a %gamma_distribution random number distribution
       * @p __x into the output stream @p __os.
       *
       * @param __os An output stream.
       * @param __x  A %gamma_distribution random number distribution.
       *
       * @returns The output stream with the state of @p __x inserted or in
       * an error state.
       */
      template<typename _RealType1, typename _CharT, typename _Traits>
        friend std::basic_ostream<_CharT, _Traits>&
        operator<<(std::basic_ostream<_CharT, _Traits>& __os,
                   const std::gamma_distribution<_RealType1>& __x);
 
      /**
       * @brief Extracts a %gamma_distribution random number distribution
       * @p __x from the input stream @p __is.
       *
       * @param __is An input stream.
       * @param __x  A %gamma_distribution random number generator engine.
       *
       * @returns The input stream with @p __x extracted or in an error state.
       */
      template<typename _RealType1, typename _CharT, typename _Traits>
        friend std::basic_istream<_CharT, _Traits>&
        operator>>(std::basic_istream<_CharT, _Traits>& __is,
                   std::gamma_distribution<_RealType1>& __x);
 
    private:
      template<typename _ForwardIterator,
               typename _UniformRandomNumberGenerator>
        void
        __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
                        _UniformRandomNumberGenerator& __urng,
                        const param_type& __p);
 
      param_type _M_param;
 
      std::normal_distribution<result_type> _M_nd;
    };
 
#if __cpp_impl_three_way_comparison < 201907L
  /**
   * @brief Return true if two gamma distributions are different.
   */
   template<typename _RealType>
     inline bool
     operator!=(const std::gamma_distribution<_RealType>& __d1,
                const std::gamma_distribution<_RealType>& __d2)
     { return !(__d1 == __d2); }
#endif
 
  /// @} group random_distributions_poisson
 
  /**
   * @addtogroup random_distributions_normal Normal Distributions
   * @ingroup random_distributions
   * @{
   */
 
  /**
   * @brief A chi_squared_distribution random number distribution.
   *
   * The formula for the normal probability mass function is
   * @f$p(x|n) = \frac{x^{(n/2) - 1}e^{-x/2}}{\Gamma(n/2) 2^{n/2}}@f$
   *
   * @headerfile random
   * @since C++11
   */
  template<typename _RealType = double>
    class chi_squared_distribution
    {
      static_assert(std::is_floating_point<_RealType>::value,
                    "result_type must be a floating point type");
 
    public:
      /** The type of the range of the distribution. */
      typedef _RealType result_type;
 
      /** Parameter type. */
      struct param_type
      {
        typedef chi_squared_distribution<_RealType> distribution_type;
 
        param_type() : param_type(1) { }
 
        explicit
        param_type(_RealType __n)
        : _M_n(__n)
        { }
 
        _RealType
        n() const
        { return _M_n; }
 
        friend bool
        operator==(const param_type& __p1, const param_type& __p2)
        { return __p1._M_n == __p2._M_n; }
 
#if __cpp_impl_three_way_comparison < 201907L
        friend bool
        operator!=(const param_type& __p1, const param_type& __p2)
        { return !(__p1 == __p2); }
#endif
 
      private:
        _RealType _M_n;
      };
 
      chi_squared_distribution() : chi_squared_distribution(1) { }
 
      explicit
      chi_squared_distribution(_RealType __n)
      : _M_param(__n), _M_gd(__n / 2)
      { }
 
      explicit
      chi_squared_distribution(const param_type& __p)
      : _M_param(__p), _M_gd(__p.n() / 2)
      { }
 
      /**
       * @brief Resets the distribution state.
       */
      void
      reset()
      { _M_gd.reset(); }
 
      /**
       *
       */
      _RealType
      n() const
      { return _M_param.n(); }
 
      /**
       * @brief Returns the parameter set of the distribution.
       */
      param_type
      param() const
      { return _M_param; }
 
      /**
       * @brief Sets the parameter set of the distribution.
       * @param __param The new parameter set of the distribution.
       */
      void
      param(const param_type& __param)
      {
        _M_param = __param;
        typedef typename std::gamma_distribution<result_type>::param_type
          param_type;
        _M_gd.param(param_type{__param.n() / 2});
      }
 
      /**
       * @brief Returns the greatest lower bound value of the distribution.
       */
      result_type
      min() const
      { return result_type(0); }
 
      /**
       * @brief Returns the least upper bound value of the distribution.
       */
      result_type