// random number generation (out of line) -*- C++ -*-

// Copyright (C) 2009-2026 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.tcc
 *  This is an internal header file, included by other library headers.
 *  Do not attempt to use it directly. @headername{random}
 */

#ifndef _RANDOM_TCC
#define _RANDOM_TCC 1

#include <numeric> // std::accumulate and std::partial_sum

namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION

  /// @cond undocumented
  // (Further) implementation-space details.
  namespace __detail
  {
    // General case for x = (ax + c) mod m -- use Schrage's algorithm
    // to avoid integer overflow.
    //
    // Preconditions:  a > 0, m > 0.
    //
    // Note: only works correctly for __m % __a < __m / __a.
    template<typename _Tp, _Tp __m, _Tp __a, _Tp __c>
      _Tp
      _Mod<_Tp, __m, __a, __c, false, true>::
      __calc(_Tp __x)
      {
	if (__a == 1)
	  __x %= __m;
	else
	  {
	    static const _Tp __q = __m / __a;
	    static const _Tp __r = __m % __a;

	    _Tp __t1 = __a * (__x % __q);
	    _Tp __t2 = __r * (__x / __q);
	    if (__t1 >= __t2)
	      __x = __t1 - __t2;
	    else
	      __x = __m - __t2 + __t1;
	  }

	if (__c != 0)
	  {
	    const _Tp __d = __m - __x;
	    if (__d > __c)
	      __x += __c;
	    else
	      __x = __c - __d;
	  }
	return __x;
      }

    template<typename _InputIterator, typename _OutputIterator,
	     typename _Tp>
      _OutputIterator
      __normalize(_InputIterator __first, _InputIterator __last,
		  _OutputIterator __result, const _Tp& __factor)
      {
	for (; __first != __last; ++__first, (void) ++__result)
	  *__result = *__first / __factor;
	return __result;
      }

  } // namespace __detail
  /// @endcond

#if ! __cpp_inline_variables
  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
    constexpr _UIntType
    linear_congruential_engine<_UIntType, __a, __c, __m>::multiplier;

  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
    constexpr _UIntType
    linear_congruential_engine<_UIntType, __a, __c, __m>::increment;

  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
    constexpr _UIntType
    linear_congruential_engine<_UIntType, __a, __c, __m>::modulus;

  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
    constexpr _UIntType
    linear_congruential_engine<_UIntType, __a, __c, __m>::default_seed;
#endif

  /**
   * Seeds the LCR with integral value @p __s, adjusted so that the
   * ring identity is never a member of the convergence set.
   */
  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
    void
    linear_congruential_engine<_UIntType, __a, __c, __m>::
    seed(result_type __s)
    {
      if ((__detail::__mod<_UIntType, __m>(__c) == 0)
	  && (__detail::__mod<_UIntType, __m>(__s) == 0))
	_M_x = 1;
      else
	_M_x = __detail::__mod<_UIntType, __m>(__s);
    }

  /**
   * Seeds the LCR engine with a value generated by @p __q.
   */
  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m>
    template<typename _Sseq>
      auto
      linear_congruential_engine<_UIntType, __a, __c, __m>::
      seed(_Sseq& __q)
      -> _If_seed_seq<_Sseq>
      {
	const _UIntType __k0 = __m == 0 ? std::numeric_limits<_UIntType>::digits
	                                : std::__lg(__m);
	const _UIntType __k = (__k0 + 31) / 32;
	uint_least32_t __arr[__k + 3];
	__q.generate(__arr + 0, __arr + __k + 3);
	_UIntType __factor = 1u;
	_UIntType __sum = 0u;
	for (size_t __j = 0; __j < __k; ++__j)
	  {
	    __sum += __arr[__j + 3] * __factor;
	    __factor *= __detail::_Shift<_UIntType, 32>::__value;
	  }
	seed(__sum);
      }

  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m,
	   typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const linear_congruential_engine<_UIntType,
						__a, __c, __m>& __lcr)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      __os.flags(__ios_base::dec | __ios_base::fixed | __ios_base::left);
      __os.fill(__os.widen(' '));

      __os << __lcr._M_x;

      __os.flags(__flags);
      __os.fill(__fill);
      return __os;
    }

  template<typename _UIntType, _UIntType __a, _UIntType __c, _UIntType __m,
	   typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       linear_congruential_engine<_UIntType, __a, __c, __m>& __lcr)
    {
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec);

      __is >> __lcr._M_x;

      __is.flags(__flags);
      return __is;
    }

#if ! __cpp_inline_variables
  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>
    constexpr size_t
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::word_size;

  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>
    constexpr size_t
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::state_size;

  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>
    constexpr size_t
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::shift_size;

  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>
    constexpr size_t
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::mask_bits;

  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>
    constexpr _UIntType
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::xor_mask;

  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>
    constexpr size_t
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::tempering_u;
   
  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>
    constexpr _UIntType
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::tempering_d;

  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>
    constexpr size_t
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::tempering_s;

  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>
    constexpr _UIntType
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::tempering_b;

  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>
    constexpr size_t
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::tempering_t;

  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>
    constexpr _UIntType
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::tempering_c;

  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>
    constexpr size_t
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::tempering_l;

  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>
    constexpr _UIntType
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::
                                              initialization_multiplier;

  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>
    constexpr _UIntType
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::default_seed;
#endif

  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>
    void
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::
    seed(result_type __sd)
    {
      _M_x[0] = __detail::__mod<_UIntType,
	__detail::_Shift<_UIntType, __w>::__value>(__sd);

      for (size_t __i = 1; __i < state_size; ++__i)
	{
	  _UIntType __x = _M_x[__i - 1];
	  __x ^= __x >> (__w - 2);
	  __x *= __f;
	  __x += __detail::__mod<_UIntType, __n>(__i);
	  _M_x[__i] = __detail::__mod<_UIntType,
	    __detail::_Shift<_UIntType, __w>::__value>(__x);
	}
      _M_p = state_size;
    }

  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>
    template<typename _Sseq>
      auto
      mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			      __s, __b, __t, __c, __l, __f>::
      seed(_Sseq& __q)
      -> _If_seed_seq<_Sseq>
      {
	const _UIntType __upper_mask = (~_UIntType()) << __r;
	const size_t __k = (__w + 31) / 32;
	uint_least32_t __arr[__n * __k];
	__q.generate(__arr + 0, __arr + __n * __k);

	bool __zero = true;
	for (size_t __i = 0; __i < state_size; ++__i)
	  {
	    _UIntType __factor = 1u;
	    _UIntType __sum = 0u;
	    for (size_t __j = 0; __j < __k; ++__j)
	      {
		__sum += __arr[__k * __i + __j] * __factor;
		__factor *= __detail::_Shift<_UIntType, 32>::__value;
	      }
	    _M_x[__i] = __detail::__mod<_UIntType,
	      __detail::_Shift<_UIntType, __w>::__value>(__sum);

	    if (__zero)
	      {
		if (__i == 0)
		  {
		    if ((_M_x[0] & __upper_mask) != 0u)
		      __zero = false;
		  }
		else if (_M_x[__i] != 0u)
		  __zero = false;
	      }
	  }
        if (__zero)
          _M_x[0] = __detail::_Shift<_UIntType, __w - 1>::__value;
	_M_p = state_size;
      }

  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>
    void
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::
    _M_gen_rand(void)
    {
      const _UIntType __upper_mask = (~_UIntType()) << __r;
      const _UIntType __lower_mask = ~__upper_mask;

      for (size_t __k = 0; __k < (__n - __m); ++__k)
        {
	  _UIntType __y = ((_M_x[__k] & __upper_mask)
			   | (_M_x[__k + 1] & __lower_mask));
	  _M_x[__k] = (_M_x[__k + __m] ^ (__y >> 1)
		       ^ ((__y & 0x01) ? __a : 0));
        }

      for (size_t __k = (__n - __m); __k < (__n - 1); ++__k)
	{
	  _UIntType __y = ((_M_x[__k] & __upper_mask)
			   | (_M_x[__k + 1] & __lower_mask));
	  _M_x[__k] = (_M_x[__k + (__m - __n)] ^ (__y >> 1)
		       ^ ((__y & 0x01) ? __a : 0));
	}

      _UIntType __y = ((_M_x[__n - 1] & __upper_mask)
		       | (_M_x[0] & __lower_mask));
      _M_x[__n - 1] = (_M_x[__m - 1] ^ (__y >> 1)
		       ^ ((__y & 0x01) ? __a : 0));
      _M_p = 0;
    }

  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>
    void
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::
    discard(unsigned long long __z)
    {
      while (__z > state_size - _M_p)
	{
	  __z -= state_size - _M_p;
	  _M_gen_rand();
	}
      _M_p += __z;
    }

  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>
    typename
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::result_type
    mersenne_twister_engine<_UIntType, __w, __n, __m, __r, __a, __u, __d,
			    __s, __b, __t, __c, __l, __f>::
    operator()()
    {
      // Reload the vector - cost is O(n) amortized over n calls.
      if (_M_p >= state_size)
	_M_gen_rand();

      // Calculate o(x(i)).
      result_type __z = _M_x[_M_p++];
      __z ^= (__z >> __u) & __d;
      __z ^= (__z << __s) & __b;
      __z ^= (__z << __t) & __c;
      __z ^= (__z >> __l);

      return __z;
    }

  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, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const mersenne_twister_engine<_UIntType, __w, __n, __m,
	       __r, __a, __u, __d, __s, __b, __t, __c, __l, __f>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::dec | __ios_base::fixed | __ios_base::left);
      __os.fill(__space);

      for (size_t __i = 0; __i < __n; ++__i)
	__os << __x._M_x[__i] << __space;
      __os << __x._M_p;

      __os.flags(__flags);
      __os.fill(__fill);
      return __os;
    }

  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, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       mersenne_twister_engine<_UIntType, __w, __n, __m,
	       __r, __a, __u, __d, __s, __b, __t, __c, __l, __f>& __x)
    {
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      for (size_t __i = 0; __i < __n; ++__i)
	__is >> __x._M_x[__i];
      __is >> __x._M_p;

      __is.flags(__flags);
      return __is;
    }

#if ! __cpp_inline_variables
  template<typename _UIntType, size_t __w, size_t __s, size_t __r>
    constexpr size_t
    subtract_with_carry_engine<_UIntType, __w, __s, __r>::word_size;

  template<typename _UIntType, size_t __w, size_t __s, size_t __r>
    constexpr size_t
    subtract_with_carry_engine<_UIntType, __w, __s, __r>::short_lag;

  template<typename _UIntType, size_t __w, size_t __s, size_t __r>
    constexpr size_t
    subtract_with_carry_engine<_UIntType, __w, __s, __r>::long_lag;

  template<typename _UIntType, size_t __w, size_t __s, size_t __r>
    constexpr uint_least32_t
    subtract_with_carry_engine<_UIntType, __w, __s, __r>::default_seed;
#endif

  template<typename _UIntType, size_t __w, size_t __s, size_t __r>
    void
    subtract_with_carry_engine<_UIntType, __w, __s, __r>::
    seed(result_type __value)
    {
      // _GLIBCXX_RESOLVE_LIB_DEFECTS
      // 3809. Is std::subtract_with_carry_engine<uint16_t> supposed to work?
      // 4014. LWG 3809 changes behavior of some existing code
      std::linear_congruential_engine<uint_least32_t, 40014u, 0u, 2147483563u>
	__lcg(__value == 0u ? default_seed : __value % 2147483563u);

      const size_t __n = (__w + 31) / 32;

      for (size_t __i = 0; __i < long_lag; ++__i)
	{
	  _UIntType __sum = 0u;
	  _UIntType __factor = 1u;
	  for (size_t __j = 0; __j < __n; ++__j)
	    {
	      __sum += __detail::__mod<uint_least32_t,
		       __detail::_Shift<uint_least32_t, 32>::__value>
			 (__lcg()) * __factor;
	      __factor *= __detail::_Shift<_UIntType, 32>::__value;
	    }
	  _M_x[__i] = __detail::__mod<_UIntType,
	    __detail::_Shift<_UIntType, __w>::__value>(__sum);
	}
      _M_carry = (_M_x[long_lag - 1] == 0) ? 1 : 0;
      _M_p = 0;
    }

  template<typename _UIntType, size_t __w, size_t __s, size_t __r>
    template<typename _Sseq>
      auto
      subtract_with_carry_engine<_UIntType, __w, __s, __r>::
      seed(_Sseq& __q)
      -> _If_seed_seq<_Sseq>
      {
	const size_t __k = (__w + 31) / 32;
	uint_least32_t __arr[__r * __k];
	__q.generate(__arr + 0, __arr + __r * __k);

	for (size_t __i = 0; __i < long_lag; ++__i)
	  {
	    _UIntType __sum = 0u;
	    _UIntType __factor = 1u;
	    for (size_t __j = 0; __j < __k; ++__j)
	      {
		__sum += __arr[__k * __i + __j] * __factor;
		__factor *= __detail::_Shift<_UIntType, 32>::__value;
	      }
	    _M_x[__i] = __detail::__mod<_UIntType,
	      __detail::_Shift<_UIntType, __w>::__value>(__sum);
	  }
	_M_carry = (_M_x[long_lag - 1] == 0) ? 1 : 0;
	_M_p = 0;
      }

  template<typename _UIntType, size_t __w, size_t __s, size_t __r>
    typename subtract_with_carry_engine<_UIntType, __w, __s, __r>::
	     result_type
    subtract_with_carry_engine<_UIntType, __w, __s, __r>::
    operator()()
    {
      // Derive short lag index from current index.
      long __ps = _M_p - short_lag;
      if (__ps < 0)
	__ps += long_lag;

      // Calculate new x(i) without overflow or division.
      // NB: Thanks to the requirements for _UIntType, _M_x[_M_p] + _M_carry
      // cannot overflow.
      _UIntType __xi;
      if (_M_x[__ps] >= _M_x[_M_p] + _M_carry)
	{
	  __xi = _M_x[__ps] - _M_x[_M_p] - _M_carry;
	  _M_carry = 0;
	}
      else
	{
	  __xi = (__detail::_Shift<_UIntType, __w>::__value
		  - _M_x[_M_p] - _M_carry + _M_x[__ps]);
	  _M_carry = 1;
	}
      _M_x[_M_p] = __xi;

      // Adjust current index to loop around in ring buffer.
      if (++_M_p >= long_lag)
	_M_p = 0;

      return __xi;
    }

  template<typename _UIntType, size_t __w, size_t __s, size_t __r,
	   typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const subtract_with_carry_engine<_UIntType,
						__w, __s, __r>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::dec | __ios_base::fixed | __ios_base::left);
      __os.fill(__space);

      for (size_t __i = 0; __i < __r; ++__i)
	__os << __x._M_x[__i] << __space;
      __os << __x._M_carry << __space << __x._M_p;

      __os.flags(__flags);
      __os.fill(__fill);
      return __os;
    }

  template<typename _UIntType, size_t __w, size_t __s, size_t __r,
	   typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       subtract_with_carry_engine<_UIntType, __w, __s, __r>& __x)
    {
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      for (size_t __i = 0; __i < __r; ++__i)
	__is >> __x._M_x[__i];
      __is >> __x._M_carry;
      __is >> __x._M_p;

      __is.flags(__flags);
      return __is;
    }

#if ! __cpp_inline_variables
  template<typename _RandomNumberEngine, size_t __p, size_t __r>
    constexpr size_t
    discard_block_engine<_RandomNumberEngine, __p, __r>::block_size;

  template<typename _RandomNumberEngine, size_t __p, size_t __r>
    constexpr size_t
    discard_block_engine<_RandomNumberEngine, __p, __r>::used_block;
#endif

  template<typename _RandomNumberEngine, size_t __p, size_t __r>
    typename discard_block_engine<_RandomNumberEngine,
			   __p, __r>::result_type
    discard_block_engine<_RandomNumberEngine, __p, __r>::
    operator()()
    {
      if (_M_n >= used_block)
	{
	  _M_b.discard(block_size - _M_n);
	  _M_n = 0;
	}
      ++_M_n;
      return _M_b();
    }

  template<typename _RandomNumberEngine, size_t __p, size_t __r,
	   typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const discard_block_engine<_RandomNumberEngine,
	       __p, __r>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::dec | __ios_base::fixed | __ios_base::left);
      __os.fill(__space);

      __os << __x.base() << __space << __x._M_n;

      __os.flags(__flags);
      __os.fill(__fill);
      return __os;
    }

  template<typename _RandomNumberEngine, size_t __p, size_t __r,
	   typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       discard_block_engine<_RandomNumberEngine, __p, __r>& __x)
    {
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      __is >> __x._M_b >> __x._M_n;

      __is.flags(__flags);
      return __is;
    }


  template<typename _RandomNumberEngine, size_t __w, typename _UIntType>
    typename independent_bits_engine<_RandomNumberEngine, __w, _UIntType>::
      result_type
    independent_bits_engine<_RandomNumberEngine, __w, _UIntType>::
    operator()()
    {
      typedef typename _RandomNumberEngine::result_type _Eresult_type;
      const _Eresult_type __r
	= (_M_b.max() - _M_b.min() < std::numeric_limits<_Eresult_type>::max()
	   ? _M_b.max() - _M_b.min() + 1 : 0);
      const unsigned __edig = std::numeric_limits<_Eresult_type>::digits;
      const unsigned __m = __r ? std::__lg(__r) : __edig;

      typedef typename std::common_type<_Eresult_type, result_type>::type
	__ctype;
      const unsigned __cdig = std::numeric_limits<__ctype>::digits;

      unsigned __n, __n0;
      __ctype __s0, __s1, __y0, __y1;

      for (size_t __i = 0; __i < 2; ++__i)
	{
	  __n = (__w + __m - 1) / __m + __i;
	  __n0 = __n - __w % __n;
	  const unsigned __w0 = __w / __n;  // __w0 <= __m

	  __s0 = 0;
	  __s1 = 0;
	  if (__w0 < __cdig)
	    {
	      __s0 = __ctype(1) << __w0;
	      __s1 = __s0 << 1;
	    }

	  __y0 = 0;
	  __y1 = 0;
	  if (__r)
	    {
	      __y0 = __s0 * (__r / __s0);
	      if (__s1)
		__y1 = __s1 * (__r / __s1);

	      if (__r - __y0 <= __y0 / __n)
		break;
	    }
	  else
	    break;
	}

      result_type __sum = 0;
      for (size_t __k = 0; __k < __n0; ++__k)
	{
	  __ctype __u;
	  do
	    __u = _M_b() - _M_b.min();
	  while (__y0 && __u >= __y0);
	  __sum = __s0 * __sum + (__s0 ? __u % __s0 : __u);
	}
      for (size_t __k = __n0; __k < __n; ++__k)
	{
	  __ctype __u;
	  do
	    __u = _M_b() - _M_b.min();
	  while (__y1 && __u >= __y1);
	  __sum = __s1 * __sum + (__s1 ? __u % __s1 : __u);
	}
      return __sum;
    }

#if ! __cpp_inline_variables
  template<typename _RandomNumberEngine, size_t __k>
    constexpr size_t
    shuffle_order_engine<_RandomNumberEngine, __k>::table_size;
#endif

  namespace __detail
  {
    // Determine whether an integer is representable as double.
    template<typename _Tp>
      constexpr bool
      __representable_as_double(_Tp __x) noexcept
      {
	static_assert(numeric_limits<_Tp>::is_integer, "");
	static_assert(!numeric_limits<_Tp>::is_signed, "");
	// All integers <= 2^53 are representable.
	return (__x <= (1ull << __DBL_MANT_DIG__))
	  // Between 2^53 and 2^54 only even numbers are representable.
	  || (!(__x & 1) && __detail::__representable_as_double(__x >> 1));
      }

    // Determine whether x+1 is representable as double.
    template<typename _Tp>
      constexpr bool
      __p1_representable_as_double(_Tp __x) noexcept
      {
	static_assert(numeric_limits<_Tp>::is_integer, "");
	static_assert(!numeric_limits<_Tp>::is_signed, "");
	return numeric_limits<_Tp>::digits < __DBL_MANT_DIG__
	  || (bool(__x + 1u) // return false if x+1 wraps around to zero
	      && __detail::__representable_as_double(__x + 1u));
      }
  }

  template<typename _RandomNumberEngine, size_t __k>
    typename shuffle_order_engine<_RandomNumberEngine, __k>::result_type
    shuffle_order_engine<_RandomNumberEngine, __k>::
    operator()()
    {
      constexpr result_type __range = max() - min();
      size_t __j = __k;
      const result_type __y = _M_y - min();
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++17-extensions" // if constexpr
      // Avoid using slower long double arithmetic if possible.
      if constexpr (__detail::__p1_representable_as_double(__range))
	__j *= __y / (__range + 1.0);
      else
	__j *= __y / (__range + 1.0L);
#pragma GCC diagnostic pop
      _M_y = _M_v[__j];
      _M_v[__j] = _M_b();

      return _M_y;
    }

  template<typename _RandomNumberEngine, size_t __k,
	   typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const shuffle_order_engine<_RandomNumberEngine, __k>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::dec | __ios_base::fixed | __ios_base::left);
      __os.fill(__space);

      __os << __x.base();
      for (size_t __i = 0; __i < __k; ++__i)
	__os << __space << __x._M_v[__i];
      __os << __space << __x._M_y;

      __os.flags(__flags);
      __os.fill(__fill);
      return __os;
    }

  template<typename _RandomNumberEngine, size_t __k,
	   typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       shuffle_order_engine<_RandomNumberEngine, __k>& __x)
    {
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      __is >> __x._M_b;
      for (size_t __i = 0; __i < __k; ++__i)
	__is >> __x._M_v[__i];
      __is >> __x._M_y;

      __is.flags(__flags);
      return __is;
    }

#if __glibcxx_philox_engine // >= C++26

  template<typename _UIntType, size_t __w, size_t __n, size_t __r,
	   _UIntType... __consts>
    _UIntType
    philox_engine<_UIntType, __w, __n, __r, __consts...>::
    _S_mulhi(_UIntType __a, _UIntType __b)
    {
      using __type = typename __detail::_Select_uint_least_t<__w * 2>::type;
      const __type __num = static_cast<__type>(__a) * __b;
      return static_cast<_UIntType>(__num >> __w) & max();
    }

  template<typename _UIntType, size_t __w, size_t __n, size_t __r,
	   _UIntType... __consts>
    _UIntType
    philox_engine<_UIntType, __w, __n, __r, __consts...>::
    _S_mullo(_UIntType __a, _UIntType __b)
    {
      return static_cast<_UIntType>((__a * __b) & max());
    }

  template<typename _UIntType, size_t __w, size_t __n, size_t __r,
	   _UIntType... __consts>
    void
    philox_engine<_UIntType, __w, __n, __r, __consts...>::_M_transition()
    {
      ++_M_i;
      if (_M_i != __n)
	return;

      using __type = typename __detail::_Select_uint_least_t<__w * 2>::type;

      _M_philox();
      if constexpr (__n == 4)
	{
	  __type __uh
	    = (static_cast<__type>(_M_x[1]) << __w)
		| static_cast<__type>(_M_x[0]);
	  ++__uh;
	  __type __lh
	    = (static_cast<__type>(_M_x[3]) << __w)
		| static_cast<__type>(_M_x[2]);
	  __type __bigMask
	    = ~__type(0) >> ((sizeof(__type) * __CHAR_BIT__) - (__w * 2));
	  if ((__uh & __bigMask) == 0)
	    {
	      ++__lh;
	      __uh = 0;
	    }
	  _M_x[0] = static_cast<_UIntType>(__uh & max());
	  _M_x[1] = static_cast<_UIntType>((__uh >> (__w)) & max());
	  _M_x[2] = static_cast<_UIntType>(__lh & max());
	  _M_x[3] = static_cast<_UIntType>((__lh >> (__w)) & max());
	}
      else
	{
	  __type __num
	    = (static_cast<__type>(_M_x[1]) << __w)
		| static_cast<__type>(_M_x[0]);
	  ++__num;
	  _M_x[0] = static_cast<_UIntType>(__num & max());
	  _M_x[1] = static_cast<_UIntType>((__num >> __w) & max());
	}
      _M_i = 0;
    }

  template<typename _UIntType, size_t __w, size_t __n, size_t __r,
	   _UIntType... __consts>
    void
    philox_engine<_UIntType, __w, __n, __r, __consts...>::_M_philox()
    {
      array<_UIntType, __n> __outputSeq = _M_x;
      for (size_t __j = 0; __j < __r; ++__j)
	{
	  array<_UIntType, __n> __intermedSeq{};
	  if constexpr (__n == 4)
	    {
	      __intermedSeq[0] = __outputSeq[2];
	      __intermedSeq[1] = __outputSeq[1];
	      __intermedSeq[2] = __outputSeq[0];
	      __intermedSeq[3] = __outputSeq[3];
	    }
	  else
	    {
	      __intermedSeq[0] = __outputSeq[0];
	      __intermedSeq[1] = __outputSeq[1];
	    }
	  for (unsigned long __k = 0; __k < (__n/2); ++__k)
	    {
	      __outputSeq[2*__k]
		= _S_mulhi(__intermedSeq[2*__k], multipliers[__k])
		    ^ (((_M_k[__k] + (__j * round_consts[__k])) & max()))
		    ^ __intermedSeq[2*__k+1];

	      __outputSeq[(2*__k)+1]
		= _S_mullo(__intermedSeq[2*__k], multipliers[__k]);
	    }
	}
      _M_y = __outputSeq;
    }

  template<typename _UIntType, size_t __w, size_t __n, size_t __r,
	   _UIntType... __consts>
  template<typename _Sseq>
    void
    philox_engine<_UIntType, __w, __n, __r, __consts...>::seed(_Sseq& __q)
    requires __is_seed_seq<_Sseq>
    {
      seed(0);

      const unsigned __p = 1 + ((__w - 1) / 32);
      uint_least32_t __tmpArr[(__n/2) * __p];
      __q.generate(__tmpArr + 0, __tmpArr + ((__n/2) * __p));
      for (unsigned __k = 0; __k < (__n/2); ++__k)
	{
	  unsigned long long __precalc = 0;
	  for (unsigned __j = 0; __j < __p; ++__j)
	    {
	      unsigned long long __multiplicand = (1ull << (32 * __j));
	      __precalc += (__tmpArr[__k * __p + __j] * __multiplicand) & max();
	    }
	  _M_k[__k] = __precalc;
	}
    }
#endif // philox_engine

  template<typename _IntType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const uniform_int_distribution<_IntType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);

      __os << __x.a() << __space << __x.b();

      __os.flags(__flags);
      __os.fill(__fill);
      return __os;
    }

  template<typename _IntType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       uniform_int_distribution<_IntType>& __x)
    {
      using param_type
	= typename uniform_int_distribution<_IntType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _IntType __a, __b;
      if (__is >> __a >> __b)
	__x.param(param_type(__a, __b));

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      uniform_real_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);
	auto __range = __p.b() - __p.a();
	while (__f != __t)
	  *__f++ = __aurng() * __range + __p.a();
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const uniform_real_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.a() << __space << __x.b();

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       uniform_real_distribution<_RealType>& __x)
    {
      using param_type
	= typename uniform_real_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::skipws);

      _RealType __a, __b;
      if (__is >> __a >> __b)
	__x.param(param_type(__a, __b));

      __is.flags(__flags);
      return __is;
    }


  template<typename _ForwardIterator,
	   typename _UniformRandomNumberGenerator>
    void
    std::bernoulli_distribution::
    __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		    _UniformRandomNumberGenerator& __urng,
		    const param_type& __p)
    {
      __glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
      __detail::_Adaptor<_UniformRandomNumberGenerator, double>
	__aurng(__urng);
      auto __limit = __p.p() * (__aurng.max() - __aurng.min());

      while (__f != __t)
	*__f++ = (__aurng() - __aurng.min()) < __limit;
    }

  template<typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const bernoulli_distribution& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__os.widen(' '));
      __os.precision(std::numeric_limits<double>::max_digits10);

      __os << __x.p();

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }


  template<typename _IntType>
    template<typename _UniformRandomNumberGenerator>
      typename geometric_distribution<_IntType>::result_type
      geometric_distribution<_IntType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __param)
      {
	// About the epsilon thing see this thread:
	// http://gcc.gnu.org/ml/gcc-patches/2006-10/msg00971.html
	const double __naf =
	  (1 - std::numeric_limits<double>::epsilon()) / 2;
	// The largest _RealType convertible to _IntType.
	const double __thr =
	  std::numeric_limits<_IntType>::max() + __naf;
	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);

	double __cand;
	do
	  __cand = std::floor(std::log(1.0 - __aurng()) / __param._M_log_1_p);
	while (__cand >= __thr);

	return result_type(__cand + __naf);
      }

  template<typename _IntType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      geometric_distribution<_IntType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __param)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	// About the epsilon thing see this thread:
	// http://gcc.gnu.org/ml/gcc-patches/2006-10/msg00971.html
	const double __naf =
	  (1 - std::numeric_limits<double>::epsilon()) / 2;
	// The largest _RealType convertible to _IntType.
	const double __thr =
	  std::numeric_limits<_IntType>::max() + __naf;
	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);

	while (__f != __t)
	  {
	    double __cand;
	    do
	      __cand = std::floor(std::log(1.0 - __aurng())
				  / __param._M_log_1_p);
	    while (__cand >= __thr);

	    *__f++ = __cand + __naf;
	  }
      }

  template<typename _IntType,
	   typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const geometric_distribution<_IntType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__os.widen(' '));
      __os.precision(std::numeric_limits<double>::max_digits10);

      __os << __x.p();

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _IntType,
	   typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       geometric_distribution<_IntType>& __x)
    {
      using param_type = typename geometric_distribution<_IntType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::skipws);

      double __p;
      if (__is >> __p)
	__x.param(param_type(__p));

      __is.flags(__flags);
      return __is;
    }

  // This is Leger's algorithm, also in Devroye, Ch. X, Example 1.5.
  template<typename _IntType>
    template<typename _UniformRandomNumberGenerator>
      typename negative_binomial_distribution<_IntType>::result_type
      negative_binomial_distribution<_IntType>::
      operator()(_UniformRandomNumberGenerator& __urng)
      {
	const double __y = _M_gd(__urng);

	// XXX Is the constructor too slow?
	std::poisson_distribution<result_type> __poisson(__y);
	return __poisson(__urng);
      }

  template<typename _IntType>
    template<typename _UniformRandomNumberGenerator>
      typename negative_binomial_distribution<_IntType>::result_type
      negative_binomial_distribution<_IntType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __p)
      {
	typedef typename std::gamma_distribution<double>::param_type
	  param_type;
	
	const double __y =
	  _M_gd(__urng, param_type(__p.k(), (1.0 - __p.p()) / __p.p()));

	std::poisson_distribution<result_type> __poisson(__y);
	return __poisson(__urng);
      }

  template<typename _IntType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      negative_binomial_distribution<_IntType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	while (__f != __t)
	  {
	    const double __y = _M_gd(__urng);

	    // XXX Is the constructor too slow?
	    std::poisson_distribution<result_type> __poisson(__y);
	    *__f++ = __poisson(__urng);
	  }
      }

  template<typename _IntType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      negative_binomial_distribution<_IntType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	typename std::gamma_distribution<result_type>::param_type
	  __p2(__p.k(), (1.0 - __p.p()) / __p.p());

	while (__f != __t)
	  {
	    const double __y = _M_gd(__urng, __p2);

	    std::poisson_distribution<result_type> __poisson(__y);
	    *__f++ = __poisson(__urng);
	  }
      }

  template<typename _IntType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const negative_binomial_distribution<_IntType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__os.widen(' '));
      __os.precision(std::numeric_limits<double>::max_digits10);

      __os << __x.k() << __space << __x.p()
	   << __space << __x._M_gd;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _IntType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       negative_binomial_distribution<_IntType>& __x)
    {
      using param_type
	= typename negative_binomial_distribution<_IntType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::skipws);

      _IntType __k;
      double __p;
      if (__is >> __k >> __p >> __x._M_gd)
	__x.param(param_type(__k, __p));

      __is.flags(__flags);
      return __is;
    }


  template<typename _IntType>
    void
    poisson_distribution<_IntType>::param_type::
    _M_initialize()
    {
#if _GLIBCXX_USE_C99_MATH_FUNCS
      if (_M_mean >= 12)
	{
	  const double __m = std::floor(_M_mean);
	  _M_lm_thr = std::log(_M_mean);
	  _M_lfm = std::lgamma(__m + 1);
	  _M_sm = std::sqrt(__m);

	  const double __pi_4 = 0.7853981633974483096156608458198757L;
	  const double __dx = std::sqrt(2 * __m * std::log(32 * __m
							      / __pi_4));
	  _M_d = std::round(std::max<double>(6.0, std::min(__m, __dx)));
	  const double __cx = 2 * __m + _M_d;
	  _M_scx = std::sqrt(__cx / 2);
	  _M_1cx = 1 / __cx;

	  _M_c2b = std::sqrt(__pi_4 * __cx) * std::exp(_M_1cx);
	  _M_cb = 2 * __cx * std::exp(-_M_d * _M_1cx * (1 + _M_d / 2))
		/ _M_d;
	}
      else
#endif
	_M_lm_thr = std::exp(-_M_mean);
      }

  /**
   * A rejection algorithm when mean >= 12 and a simple method based
   * upon the multiplication of uniform random variates otherwise.
   * NB: The former is available only if _GLIBCXX_USE_C99_MATH_FUNCS
   * is defined.
   *
   * Reference:
   * Devroye, L. Non-Uniform Random Variates Generation. Springer-Verlag,
   * New York, 1986, Ch. X, Sects. 3.3 & 3.4 (+ Errata!).
   */
  template<typename _IntType>
    template<typename _UniformRandomNumberGenerator>
      typename poisson_distribution<_IntType>::result_type
      poisson_distribution<_IntType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __param)
      {
	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);
#if _GLIBCXX_USE_C99_MATH_FUNCS
	if (__param.mean() >= 12)
	  {
	    double __x;

	    // See comments above...
	    const double __naf =
	      (1 - std::numeric_limits<double>::epsilon()) / 2;
	    const double __thr =
	      std::numeric_limits<_IntType>::max() + __naf;

	    const double __m = std::floor(__param.mean());
	    // sqrt(pi / 2)
	    const double __spi_2 = 1.2533141373155002512078826424055226L;
	    const double __c1 = __param._M_sm * __spi_2;
	    const double __c2 = __param._M_c2b + __c1;
	    const double __c3 = __c2 + 1;
	    const double __c4 = __c3 + 1;
	    // 1 / 78
	    const double __178 = 0.0128205128205128205128205128205128L;
	    // e^(1 / 78)
	    const double __e178 = 1.0129030479320018583185514777512983L;
	    const double __c5 = __c4 + __e178;
	    const double __c = __param._M_cb + __c5;
	    const double __2cx = 2 * (2 * __m + __param._M_d);

	    bool __reject = true;
	    do
	      {
		const double __u = __c * __aurng();
		const double __e = -std::log(1.0 - __aurng());

		double __w = 0.0;

		if (__u <= __c1)
		  {
		    const double __n = _M_nd(__urng);
		    const double __y = -std::abs(__n) * __param._M_sm - 1;
		    __x = std::floor(__y);
		    __w = -__n * __n / 2;
		    if (__x < -__m)
		      continue;
		  }
		else if (__u <= __c2)
		  {
		    const double __n = _M_nd(__urng);
		    const double __y = 1 + std::abs(__n) * __param._M_scx;
		    __x = std::ceil(__y);
		    __w = __y * (2 - __y) * __param._M_1cx;
		    if (__x > __param._M_d)
		      continue;
		  }
		else if (__u <= __c3)
		  // NB: This case not in the book, nor in the Errata,
		  // but should be ok...
		  __x = -1;
		else if (__u <= __c4)
		  __x = 0;
		else if (__u <= __c5)
		  {
		    __x = 1;
		    // Only in the Errata, see libstdc++/83237.
		    __w = __178;
		  }
		else
		  {
		    const double __v = -std::log(1.0 - __aurng());
		    const double __y = __param._M_d
				     + __v * __2cx / __param._M_d;
		    __x = std::ceil(__y);
		    __w = -__param._M_d * __param._M_1cx * (1 + __y / 2);
		  }

		__reject = (__w - __e - __x * __param._M_lm_thr
			    > __param._M_lfm - std::lgamma(__x + __m + 1));

		__reject |= __x + __m >= __thr;

	      } while (__reject);

	    return result_type(__x + __m + __naf);
	  }
	else
#endif
	  {
	    _IntType     __x = 0;
	    double __prod = 1.0;

	    do
	      {
		__prod *= __aurng();
		__x += 1;
	      }
	    while (__prod > __param._M_lm_thr);

	    return __x - 1;
	  }
      }

  template<typename _IntType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      poisson_distribution<_IntType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __param)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	// We could duplicate everything from operator()...
	while (__f != __t)
	  *__f++ = this->operator()(__urng, __param);
      }

  template<typename _IntType,
	   typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const poisson_distribution<_IntType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<double>::max_digits10);

      __os << __x.mean() << __space << __x._M_nd;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _IntType,
	   typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       poisson_distribution<_IntType>& __x)
    {
      using param_type = typename poisson_distribution<_IntType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::skipws);

      double __mean;
      if (__is >> __mean >> __x._M_nd)
	__x.param(param_type(__mean));

      __is.flags(__flags);
      return __is;
    }


  template<typename _IntType>
    void
    binomial_distribution<_IntType>::param_type::
    _M_initialize()
    {
      const double __p12 = _M_p <= 0.5 ? _M_p : 1.0 - _M_p;

      _M_easy = true;

#if _GLIBCXX_USE_C99_MATH_FUNCS
      if (_M_t * __p12 >= 8)
	{
	  _M_easy = false;
	  const double __np = std::floor(_M_t * __p12);
	  const double __pa = __np / _M_t;
	  const double __1p = 1 - __pa;

	  const double __pi_4 = 0.7853981633974483096156608458198757L;
	  const double __d1x =
	    std::sqrt(__np * __1p * std::log(32 * __np
					     / (81 * __pi_4 * __1p)));
	  _M_d1 = std::round(std::max<double>(1.0, __d1x));
	  const double __d2x =
	    std::sqrt(__np * __1p * std::log(32 * _M_t * __1p
					     / (__pi_4 * __pa)));
	  _M_d2 = std::round(std::max<double>(1.0, __d2x));

	  // sqrt(pi / 2)
	  const double __spi_2 = 1.2533141373155002512078826424055226L;
	  _M_s1 = std::sqrt(__np * __1p) * (1 + _M_d1 / (4 * __np));
	  _M_s2 = std::sqrt(__np * __1p) * (1 + _M_d2 / (4 * (_M_t * __1p)));
	  _M_c = 2 * _M_d1 / __np;
	  _M_a1 = std::exp(_M_c) * _M_s1 * __spi_2;
	  const double __a12 = _M_a1 + _M_s2 * __spi_2;
	  const double __s1s = _M_s1 * _M_s1;
	  _M_a123 = __a12 + (std::exp(_M_d1 / (_M_t * __1p))
			     * 2 * __s1s / _M_d1
			     * std::exp(-_M_d1 * _M_d1 / (2 * __s1s)));
	  const double __s2s = _M_s2 * _M_s2;
	  _M_s = (_M_a123 + 2 * __s2s / _M_d2
		  * std::exp(-_M_d2 * _M_d2 / (2 * __s2s)));
	  _M_lf = (std::lgamma(__np + 1)
		   + std::lgamma(_M_t - __np + 1));
	  _M_lp1p = std::log(__pa / __1p);

	  _M_q = -std::log(1 - (__p12 - __pa) / __1p);
	}
      else
#endif
	_M_q = -std::log(1 - __p12);
    }

  template<typename _IntType>
    template<typename _UniformRandomNumberGenerator>
      typename binomial_distribution<_IntType>::result_type
      binomial_distribution<_IntType>::
      _M_waiting(_UniformRandomNumberGenerator& __urng,
		 _IntType __t, double __q)
      {
	_IntType __x = 0;
	double __sum = 0.0;
	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);

	do
	  {
	    if (__t == __x)
	      return __x;
	    const double __e = -std::log(1.0 - __aurng());
	    __sum += __e / (__t - __x);
	    __x += 1;
	  }
	while (__sum <= __q);

	return __x - 1;
      }

  /**
   * A rejection algorithm when t * p >= 8 and a simple waiting time
   * method - the second in the referenced book - otherwise.
   * NB: The former is available only if _GLIBCXX_USE_C99_MATH_FUNCS
   * is defined.
   *
   * Reference:
   * Devroye, L. Non-Uniform Random Variates Generation. Springer-Verlag,
   * New York, 1986, Ch. X, Sect. 4 (+ Errata!).
   */
  template<typename _IntType>
    template<typename _UniformRandomNumberGenerator>
      typename binomial_distribution<_IntType>::result_type
      binomial_distribution<_IntType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __param)
      {
	result_type __ret;
	const _IntType __t = __param.t();
	const double __p = __param.p();
	const double __p12 = __p <= 0.5 ? __p : 1.0 - __p;
	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);

#if _GLIBCXX_USE_C99_MATH_FUNCS
	if (!__param._M_easy)
	  {
	    double __x;

	    // See comments above...
	    const double __naf =
	      (1 - std::numeric_limits<double>::epsilon()) / 2;
	    const double __thr =
	      std::numeric_limits<_IntType>::max() + __naf;

	    const double __np = std::floor(__t * __p12);

	    // sqrt(pi / 2)
	    const double __spi_2 = 1.2533141373155002512078826424055226L;
	    const double __a1 = __param._M_a1;
	    const double __a12 = __a1 + __param._M_s2 * __spi_2;
	    const double __a123 = __param._M_a123;
	    const double __s1s = __param._M_s1 * __param._M_s1;
	    const double __s2s = __param._M_s2 * __param._M_s2;

	    bool __reject;
	    do
	      {
		const double __u = __param._M_s * __aurng();

		double __v;

		if (__u <= __a1)
		  {
		    const double __n = _M_nd(__urng);
		    const double __y = __param._M_s1 * std::abs(__n);
		    __reject = __y >= __param._M_d1;
		    if (!__reject)
		      {
			const double __e = -std::log(1.0 - __aurng());
			__x = std::floor(__y);
			__v = -__e - __n * __n / 2 + __param._M_c;
		      }
		  }
		else if (__u <= __a12)
		  {
		    const double __n = _M_nd(__urng);
		    const double __y = __param._M_s2 * std::abs(__n);
		    __reject = __y >= __param._M_d2;
		    if (!__reject)
		      {
			const double __e = -std::log(1.0 - __aurng());
			__x = std::floor(-__y);
			__v = -__e - __n * __n / 2;
		      }
		  }
		else if (__u <= __a123)
		  {
		    const double __e1 = -std::log(1.0 - __aurng());
		    const double __e2 = -std::log(1.0 - __aurng());

		    const double __y = __param._M_d1
				     + 2 * __s1s * __e1 / __param._M_d1;
		    __x = std::floor(__y);
		    __v = (-__e2 + __param._M_d1 * (1 / (__t - __np)
						    -__y / (2 * __s1s)));
		    __reject = false;
		  }
		else
		  {
		    const double __e1 = -std::log(1.0 - __aurng());
		    const double __e2 = -std::log(1.0 - __aurng());

		    const double __y = __param._M_d2
				     + 2 * __s2s * __e1 / __param._M_d2;
		    __x = std::floor(-__y);
		    __v = -__e2 - __param._M_d2 * __y / (2 * __s2s);
		    __reject = false;
		  }

		__reject = __reject || __x < -__np || __x > __t - __np;
		if (!__reject)
		  {
		    const double __lfx =
		      std::lgamma(__np + __x + 1)
		      + std::lgamma(__t - (__np + __x) + 1);
		    __reject = __v > __param._M_lf - __lfx
			     + __x * __param._M_lp1p;
		  }

		__reject |= __x + __np >= __thr;
	      }
	    while (__reject);

	    __x += __np + __naf;

	    const _IntType __z = _M_waiting(__urng, __t - _IntType(__x),
					    __param._M_q);
	    __ret = _IntType(__x) + __z;
	  }
	else
#endif
	  __ret = _M_waiting(__urng, __t, __param._M_q);

	if (__p12 != __p)
	  __ret = __t - __ret;
	return __ret;
      }

  template<typename _IntType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      binomial_distribution<_IntType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __param)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	// We could duplicate everything from operator()...
	while (__f != __t)
	  *__f++ = this->operator()(__urng, __param);
      }

  template<typename _IntType,
	   typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const binomial_distribution<_IntType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<double>::max_digits10);

      __os << __x.t() << __space << __x.p()
	   << __space << __x._M_nd;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _IntType,
	   typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       binomial_distribution<_IntType>& __x)
    {
      using param_type = typename binomial_distribution<_IntType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _IntType __t;
      double __p;
      if (__is >> __t >> __p >> __x._M_nd)
	__x.param(param_type(__t, __p));

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      std::exponential_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);
	while (__f != __t)
	  *__f++ = -std::log(result_type(1) - __aurng()) / __p.lambda();
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const exponential_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__os.widen(' '));
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.lambda();

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       exponential_distribution<_RealType>& __x)
    {
      using param_type
	= typename exponential_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _RealType __lambda;
      if (__is >> __lambda)
	__x.param(param_type(__lambda));

      __is.flags(__flags);
      return __is;
    }


  /**
   * Polar method due to Marsaglia.
   *
   * Devroye, L. Non-Uniform Random Variates Generation. Springer-Verlag,
   * New York, 1986, Ch. V, Sect. 4.4.
   */
  template<typename _RealType>
    template<typename _UniformRandomNumberGenerator>
      typename normal_distribution<_RealType>::result_type
      normal_distribution<_RealType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __param)
      {
	result_type __ret;
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);

	if (_M_saved_available)
	  {
	    _M_saved_available = false;
	    __ret = _M_saved;
	  }
	else
	  {
	    result_type __x, __y, __r2;
	    do
	      {
		__x = result_type(2.0) * __aurng() - 1.0;
		__y = result_type(2.0) * __aurng() - 1.0;
		__r2 = __x * __x + __y * __y;
	      }
	    while (__r2 > 1.0 || __r2 == 0.0);

	    const result_type __mult = std::sqrt(-2 * std::log(__r2) / __r2);
	    _M_saved = __x * __mult;
	    _M_saved_available = true;
	    __ret = __y * __mult;
	  }

	__ret = __ret * __param.stddev() + __param.mean();
	return __ret;
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      normal_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __param)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)

	if (__f == __t)
	  return;

	if (_M_saved_available)
	  {
	    _M_saved_available = false;
	    *__f++ = _M_saved * __param.stddev() + __param.mean();

	    if (__f == __t)
	      return;
	  }

	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);

	while (__f + 1 < __t)
	  {
	    result_type __x, __y, __r2;
	    do
	      {
		__x = result_type(2.0) * __aurng() - 1.0;
		__y = result_type(2.0) * __aurng() - 1.0;
		__r2 = __x * __x + __y * __y;
	      }
	    while (__r2 > 1.0 || __r2 == 0.0);

	    const result_type __mult = std::sqrt(-2 * std::log(__r2) / __r2);
	    *__f++ = __y * __mult * __param.stddev() + __param.mean();
	    *__f++ = __x * __mult * __param.stddev() + __param.mean();
	  }

	if (__f != __t)
	  {
	    result_type __x, __y, __r2;
	    do
	      {
		__x = result_type(2.0) * __aurng() - 1.0;
		__y = result_type(2.0) * __aurng() - 1.0;
		__r2 = __x * __x + __y * __y;
	      }
	    while (__r2 > 1.0 || __r2 == 0.0);

	    const result_type __mult = std::sqrt(-2 * std::log(__r2) / __r2);
	    _M_saved = __x * __mult;
	    _M_saved_available = true;
	    *__f = __y * __mult * __param.stddev() + __param.mean();
	  }
      }

  template<typename _RealType>
    bool
    operator==(const std::normal_distribution<_RealType>& __d1,
	       const std::normal_distribution<_RealType>& __d2)
    {
      if (__d1._M_param == __d2._M_param
	  && __d1._M_saved_available == __d2._M_saved_available)
	return __d1._M_saved_available ? __d1._M_saved == __d2._M_saved : true;
      else
	return false;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const normal_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.mean() << __space << __x.stddev()
	   << __space << __x._M_saved_available;
      if (__x._M_saved_available)
	__os << __space << __x._M_saved;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       normal_distribution<_RealType>& __x)
    {
      using param_type = typename normal_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      double __mean, __stddev;
      bool __saved_avail;
      if (__is >> __mean >> __stddev >> __saved_avail)
	{
	  if (!__saved_avail || (__is >> __x._M_saved))
	    {
	      __x._M_saved_available = __saved_avail;
	      __x.param(param_type(__mean, __stddev));
	    }
	}

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      lognormal_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	  while (__f != __t)
	    *__f++ = std::exp(__p.s() * _M_nd(__urng) + __p.m());
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const lognormal_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.m() << __space << __x.s()
	   << __space << __x._M_nd;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       lognormal_distribution<_RealType>& __x)
    {
      using param_type
	= typename lognormal_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _RealType __m, __s;
      if (__is >> __m >> __s >> __x._M_nd)
	__x.param(param_type(__m, __s));

      __is.flags(__flags);
      return __is;
    }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      std::chi_squared_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	while (__f != __t)
	  *__f++ = 2 * _M_gd(__urng);
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      std::chi_squared_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const typename
		      std::gamma_distribution<result_type>::param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	while (__f != __t)
	  *__f++ = 2 * _M_gd(__urng, __p);
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const chi_squared_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.n() << __space << __x._M_gd;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       chi_squared_distribution<_RealType>& __x)
    {
      using param_type
	= typename chi_squared_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _RealType __n;
      if (__is >> __n >> __x._M_gd)
	__x.param(param_type(__n));

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    template<typename _UniformRandomNumberGenerator>
      typename cauchy_distribution<_RealType>::result_type
      cauchy_distribution<_RealType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __p)
      {
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);
	_RealType __u;
	do
	  __u = __aurng();
	while (__u == 0.5);

	const _RealType __pi = 3.1415926535897932384626433832795029L;
	return __p.a() + __p.b() * std::tan(__pi * __u);
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      cauchy_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	const _RealType __pi = 3.1415926535897932384626433832795029L;
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);
	while (__f != __t)
	  {
	    _RealType __u;
	    do
	      __u = __aurng();
	    while (__u == 0.5);

	    *__f++ = __p.a() + __p.b() * std::tan(__pi * __u);
	  }
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const cauchy_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.a() << __space << __x.b();

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       cauchy_distribution<_RealType>& __x)
    {
      using param_type = typename cauchy_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _RealType __a, __b;
      if (__is >> __a >> __b)
	__x.param(param_type(__a, __b));

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      std::fisher_f_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	while (__f != __t)
	  *__f++ = ((_M_gd_x(__urng) * n()) / (_M_gd_y(__urng) * m()));
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      std::fisher_f_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	typedef typename std::gamma_distribution<result_type>::param_type
	  param_type;
	param_type __p1(__p.m() / 2);
	param_type __p2(__p.n() / 2);
	while (__f != __t)
	  *__f++ = ((_M_gd_x(__urng, __p1) * n())
		    / (_M_gd_y(__urng, __p2) * m()));
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const fisher_f_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.m() << __space << __x.n()
	   << __space << __x._M_gd_x << __space << __x._M_gd_y;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       fisher_f_distribution<_RealType>& __x)
    {
      using param_type
	= typename fisher_f_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _RealType __m, __n;
      if (__is >> __m >> __n >> __x._M_gd_x >> __x._M_gd_y)
	__x.param(param_type(__m, __n));

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      std::student_t_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	while (__f != __t)
	  *__f++ = _M_nd(__urng) * std::sqrt(n() / _M_gd(__urng));
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      std::student_t_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	typename std::gamma_distribution<result_type>::param_type
	  __p2(__p.n() / 2, 2);
	while (__f != __t)
	  *__f++ =  _M_nd(__urng) * std::sqrt(__p.n() / _M_gd(__urng, __p2));
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const student_t_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.n() << __space << __x._M_nd << __space << __x._M_gd;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       student_t_distribution<_RealType>& __x)
    {
      using param_type
	= typename student_t_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _RealType __n;
      if (__is >> __n >> __x._M_nd >> __x._M_gd)
	__x.param(param_type(__n));

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    void
    gamma_distribution<_RealType>::param_type::
    _M_initialize()
    {
      _M_malpha = _M_alpha < 1.0 ? _M_alpha + _RealType(1.0) : _M_alpha;

      const _RealType __a1 = _M_malpha - _RealType(1.0) / _RealType(3.0);
      _M_a2 = _RealType(1.0) / std::sqrt(_RealType(9.0) * __a1);
    }

  /**
   * Marsaglia, G. and Tsang, W. W.
   * "A Simple Method for Generating Gamma Variables"
   * ACM Transactions on Mathematical Software, 26, 3, 363-372, 2000.
   */
  template<typename _RealType>
    template<typename _UniformRandomNumberGenerator>
      typename gamma_distribution<_RealType>::result_type
      gamma_distribution<_RealType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __param)
      {
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);

	result_type __u, __v, __n;
	const result_type __a1 = (__param._M_malpha
				  - _RealType(1.0) / _RealType(3.0));

	do
	  {
	    do
	      {
		__n = _M_nd(__urng);
		__v = result_type(1.0) + __param._M_a2 * __n; 
	      }
	    while (__v <= 0.0);

	    __v = __v * __v * __v;
	    __u = __aurng();
	  }
	while (__u > result_type(1.0) - 0.0331 * __n * __n * __n * __n
	       && (std::log(__u) > (0.5 * __n * __n + __a1
				    * (1.0 - __v + std::log(__v)))));

	if (__param.alpha() == __param._M_malpha)
	  return __a1 * __v * __param.beta();
	else
	  {
	    do
	      __u = __aurng();
	    while (__u == 0.0);
	    
	    return (std::pow(__u, result_type(1.0) / __param.alpha())
		    * __a1 * __v * __param.beta());
	  }
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      gamma_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __param)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);

	result_type __u, __v, __n;
	const result_type __a1 = (__param._M_malpha
				  - _RealType(1.0) / _RealType(3.0));

	if (__param.alpha() == __param._M_malpha)
	  while (__f != __t)
	    {
	      do
		{
		  do
		    {
		      __n = _M_nd(__urng);
		      __v = result_type(1.0) + __param._M_a2 * __n;
		    }
		  while (__v <= 0.0);

		  __v = __v * __v * __v;
		  __u = __aurng();
		}
	      while (__u > result_type(1.0) - 0.0331 * __n * __n * __n * __n
		     && (std::log(__u) > (0.5 * __n * __n + __a1
					  * (1.0 - __v + std::log(__v)))));

	      *__f++ = __a1 * __v * __param.beta();
	    }
	else
	  while (__f != __t)
	    {
	      do
		{
		  do
		    {
		      __n = _M_nd(__urng);
		      __v = result_type(1.0) + __param._M_a2 * __n;
		    }
		  while (__v <= 0.0);

		  __v = __v * __v * __v;
		  __u = __aurng();
		}
	      while (__u > result_type(1.0) - 0.0331 * __n * __n * __n * __n
		     && (std::log(__u) > (0.5 * __n * __n + __a1
					  * (1.0 - __v + std::log(__v)))));

	      do
		__u = __aurng();
	      while (__u == 0.0);

	      *__f++ = (std::pow(__u, result_type(1.0) / __param.alpha())
			* __a1 * __v * __param.beta());
	    }
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const gamma_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.alpha() << __space << __x.beta()
	   << __space << __x._M_nd;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       gamma_distribution<_RealType>& __x)
    {
      using param_type = typename gamma_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _RealType __alpha_val, __beta_val;
      if (__is >> __alpha_val >> __beta_val >> __x._M_nd)
	__x.param(param_type(__alpha_val, __beta_val));

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    template<typename _UniformRandomNumberGenerator>
      typename weibull_distribution<_RealType>::result_type
      weibull_distribution<_RealType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __p)
      {
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);
	return __p.b() * std::pow(-std::log(result_type(1) - __aurng()),
				  result_type(1) / __p.a());
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      weibull_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);
	auto __inv_a = result_type(1) / __p.a();

	while (__f != __t)
	  *__f++ = __p.b() * std::pow(-std::log(result_type(1) - __aurng()),
				      __inv_a);
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const weibull_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.a() << __space << __x.b();

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       weibull_distribution<_RealType>& __x)
    {
      using param_type = typename weibull_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _RealType __a, __b;
      if (__is >> __a >> __b)
	__x.param(param_type(__a, __b));

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    template<typename _UniformRandomNumberGenerator>
      typename extreme_value_distribution<_RealType>::result_type
      extreme_value_distribution<_RealType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __p)
      {
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);
	return __p.a() - __p.b() * std::log(-std::log(result_type(1)
						      - __aurng()));
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      extreme_value_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __p)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	__detail::_Adaptor<_UniformRandomNumberGenerator, result_type>
	  __aurng(__urng);

	while (__f != __t)
	  *__f++ = __p.a() - __p.b() * std::log(-std::log(result_type(1)
							  - __aurng()));
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const extreme_value_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      __os << __x.a() << __space << __x.b();

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       extreme_value_distribution<_RealType>& __x)
    {
      using param_type
	= typename extreme_value_distribution<_RealType>::param_type;
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      _RealType __a, __b;
      if (__is >> __a >> __b)
	__x.param(param_type(__a, __b));

      __is.flags(__flags);
      return __is;
    }


  template<typename _IntType>
    void
    discrete_distribution<_IntType>::param_type::
    _M_initialize()
    {
      if (_M_prob.size() < 2)
	{
	  _M_prob.clear();
	  return;
	}

      const double __sum = std::accumulate(_M_prob.begin(),
					   _M_prob.end(), 0.0);
      __glibcxx_assert(__sum > 0);
      // Now normalize the probabilites.
      __detail::__normalize(_M_prob.begin(), _M_prob.end(), _M_prob.begin(),
			    __sum);
      // Accumulate partial sums.
      _M_cp.reserve(_M_prob.size());
      std::partial_sum(_M_prob.begin(), _M_prob.end(),
		       std::back_inserter(_M_cp));
      // Make sure the last cumulative probability is one.
      _M_cp[_M_cp.size() - 1] = 1.0;
    }

  template<typename _IntType>
    template<typename _Func>
      discrete_distribution<_IntType>::param_type::
      param_type(size_t __nw, double __xmin, double __xmax, _Func __fw)
      : _M_prob(), _M_cp()
      {
	const size_t __n = __nw == 0 ? 1 : __nw;
	const double __delta = (__xmax - __xmin) / __n;

	_M_prob.reserve(__n);
	for (size_t __k = 0; __k < __nw; ++__k)
	  _M_prob.push_back(__fw(__xmin + __k * __delta + 0.5 * __delta));

	_M_initialize();
      }

  template<typename _IntType>
    template<typename _UniformRandomNumberGenerator>
      typename discrete_distribution<_IntType>::result_type
      discrete_distribution<_IntType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __param)
      {
	if (__param._M_cp.empty())
	  return result_type(0);

	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);

	const double __p = __aurng();
	auto __pos = std::lower_bound(__param._M_cp.begin(),
				      __param._M_cp.end(), __p);

	return __pos - __param._M_cp.begin();
      }

  template<typename _IntType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      discrete_distribution<_IntType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __param)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)

	if (__param._M_cp.empty())
	  {
	    while (__f != __t)
	      *__f++ = result_type(0);
	    return;
	  }

	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);

	while (__f != __t)
	  {
	    const double __p = __aurng();
	    auto __pos = std::lower_bound(__param._M_cp.begin(),
					  __param._M_cp.end(), __p);

	    *__f++ = __pos - __param._M_cp.begin();
	  }
      }

  template<typename _IntType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const discrete_distribution<_IntType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<double>::max_digits10);

      std::vector<double> __prob = __x.probabilities();
      __os << __prob.size();
      for (auto __dit = __prob.begin(); __dit != __prob.end(); ++__dit)
	__os << __space << *__dit;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

namespace __detail
{
  template<typename _ValT, typename _CharT, typename _Traits>
    basic_istream<_CharT, _Traits>&
    __extract_params(basic_istream<_CharT, _Traits>& __is,
		     vector<_ValT>& __vals, size_t __n)
    {
      __vals.reserve(__n);
      while (__n--)
	{
	  _ValT __val;
	  if (__is >> __val)
	    __vals.push_back(__val);
	  else
	    break;
	}
      return __is;
    }
} // namespace __detail

  template<typename _IntType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       discrete_distribution<_IntType>& __x)
    {
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      size_t __n;
      if (__is >> __n)
	{
	  std::vector<double> __prob_vec;
	  if (__detail::__extract_params(__is, __prob_vec, __n))
	    __x.param({__prob_vec.begin(), __prob_vec.end()});
	}

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    void
    piecewise_constant_distribution<_RealType>::param_type::
    _M_initialize()
    {
      if (_M_int.size() < 2
	  || (_M_int.size() == 2
	      && _M_int[0] == _RealType(0)
	      && _M_int[1] == _RealType(1)))
	{
	  _M_int.clear();
	  _M_den.clear();
	  return;
	}

      const double __sum = std::accumulate(_M_den.begin(),
					   _M_den.end(), 0.0);
      __glibcxx_assert(__sum > 0);

      __detail::__normalize(_M_den.begin(), _M_den.end(), _M_den.begin(),
			    __sum);

      _M_cp.reserve(_M_den.size());
      std::partial_sum(_M_den.begin(), _M_den.end(),
		       std::back_inserter(_M_cp));

      // Make sure the last cumulative probability is one.
      _M_cp[_M_cp.size() - 1] = 1.0;

      for (size_t __k = 0; __k < _M_den.size(); ++__k)
	_M_den[__k] /= _M_int[__k + 1] - _M_int[__k];
    }

  template<typename _RealType>
    template<typename _InputIteratorB, typename _InputIteratorW>
      piecewise_constant_distribution<_RealType>::param_type::
      param_type(_InputIteratorB __bbegin,
		 _InputIteratorB __bend,
		 _InputIteratorW __wbegin)
      : _M_int(), _M_den(), _M_cp()
      {
	if (__bbegin != __bend)
	  {
	    for (;;)
	      {
		_M_int.push_back(*__bbegin);
		++__bbegin;
		if (__bbegin == __bend)
		  break;

		_M_den.push_back(*__wbegin);
		++__wbegin;
	      }
	  }

	_M_initialize();
      }

  template<typename _RealType>
    template<typename _Func>
      piecewise_constant_distribution<_RealType>::param_type::
      param_type(initializer_list<_RealType> __bl, _Func __fw)
      : _M_int(), _M_den(), _M_cp()
      {
	_M_int.reserve(__bl.size());
	for (auto __biter = __bl.begin(); __biter != __bl.end(); ++__biter)
	  _M_int.push_back(*__biter);

	_M_den.reserve(_M_int.size() - 1);
	for (size_t __k = 0; __k < _M_int.size() - 1; ++__k)
	  _M_den.push_back(__fw(0.5 * (_M_int[__k + 1] + _M_int[__k])));

	_M_initialize();
      }

  template<typename _RealType>
    template<typename _Func>
      piecewise_constant_distribution<_RealType>::param_type::
      param_type(size_t __nw, _RealType __xmin, _RealType __xmax, _Func __fw)
      : _M_int(), _M_den(), _M_cp()
      {
	const size_t __n = __nw == 0 ? 1 : __nw;
	const _RealType __delta = (__xmax - __xmin) / __n;

	_M_int.reserve(__n + 1);
	for (size_t __k = 0; __k <= __nw; ++__k)
	  _M_int.push_back(__xmin + __k * __delta);

	_M_den.reserve(__n);
	for (size_t __k = 0; __k < __nw; ++__k)
	  _M_den.push_back(__fw(_M_int[__k] + 0.5 * __delta));

	_M_initialize();
      }

  template<typename _RealType>
    template<typename _UniformRandomNumberGenerator>
      typename piecewise_constant_distribution<_RealType>::result_type
      piecewise_constant_distribution<_RealType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __param)
      {
	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);

	const double __p = __aurng();
	if (__param._M_cp.empty())
	  return __p;

	auto __pos = std::lower_bound(__param._M_cp.begin(),
				      __param._M_cp.end(), __p);
	const size_t __i = __pos - __param._M_cp.begin();

	const double __pref = __i > 0 ? __param._M_cp[__i - 1] : 0.0;

	return __param._M_int[__i] + (__p - __pref) / __param._M_den[__i];
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      piecewise_constant_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __param)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);

	if (__param._M_cp.empty())
	  {
	    while (__f != __t)
	      *__f++ = __aurng();
	    return;
	  }

	while (__f != __t)
	  {
	    const double __p = __aurng();

	    auto __pos = std::lower_bound(__param._M_cp.begin(),
					  __param._M_cp.end(), __p);
	    const size_t __i = __pos - __param._M_cp.begin();

	    const double __pref = __i > 0 ? __param._M_cp[__i - 1] : 0.0;

	    *__f++ = (__param._M_int[__i]
		      + (__p - __pref) / __param._M_den[__i]);
	  }
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const piecewise_constant_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      std::vector<_RealType> __int = __x.intervals();
      __os << __int.size() - 1;

      for (auto __xit = __int.begin(); __xit != __int.end(); ++__xit)
	__os << __space << *__xit;

      std::vector<double> __den = __x.densities();
      for (auto __dit = __den.begin(); __dit != __den.end(); ++__dit)
	__os << __space << *__dit;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       piecewise_constant_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      size_t __n;
      if (__is >> __n)
	{
	  std::vector<_RealType> __int_vec;
	  if (__detail::__extract_params(__is, __int_vec, __n + 1))
	    {
	      std::vector<double> __den_vec;
	      if (__detail::__extract_params(__is, __den_vec, __n))
		{
		  __x.param({ __int_vec.begin(), __int_vec.end(),
			      __den_vec.begin() });
		}
	    }
	}

      __is.flags(__flags);
      return __is;
    }


  template<typename _RealType>
    void
    piecewise_linear_distribution<_RealType>::param_type::
    _M_initialize()
    {
      if (_M_int.size() < 2
	  || (_M_int.size() == 2
	      && _M_int[0] == _RealType(0)
	      && _M_int[1] == _RealType(1)
	      && _M_den[0] == _M_den[1]))
	{
	  _M_int.clear();
	  _M_den.clear();
	  return;
	}

      double __sum = 0.0;
      _M_cp.reserve(_M_int.size() - 1);
      _M_m.reserve(_M_int.size() - 1);
      for (size_t __k = 0; __k < _M_int.size() - 1; ++__k)
	{
	  const _RealType __delta = _M_int[__k + 1] - _M_int[__k];
	  __sum += 0.5 * (_M_den[__k + 1] + _M_den[__k]) * __delta;
	  _M_cp.push_back(__sum);
	  _M_m.push_back((_M_den[__k + 1] - _M_den[__k]) / __delta);
	}
      __glibcxx_assert(__sum > 0);

      //  Now normalize the densities...
      __detail::__normalize(_M_den.begin(), _M_den.end(), _M_den.begin(),
			    __sum);
      //  ... and partial sums... 
      __detail::__normalize(_M_cp.begin(), _M_cp.end(), _M_cp.begin(), __sum);
      //  ... and slopes.
      __detail::__normalize(_M_m.begin(), _M_m.end(), _M_m.begin(), __sum);

      //  Make sure the last cumulative probablility is one.
      _M_cp[_M_cp.size() - 1] = 1.0;
     }

  template<typename _RealType>
    template<typename _InputIteratorB, typename _InputIteratorW>
      piecewise_linear_distribution<_RealType>::param_type::
      param_type(_InputIteratorB __bbegin,
		 _InputIteratorB __bend,
		 _InputIteratorW __wbegin)
      : _M_int(), _M_den(), _M_cp(), _M_m()
      {
	for (; __bbegin != __bend; ++__bbegin, (void) ++__wbegin)
	  {
	    _M_int.push_back(*__bbegin);
	    _M_den.push_back(*__wbegin);
	  }

	_M_initialize();
      }

  template<typename _RealType>
    template<typename _Func>
      piecewise_linear_distribution<_RealType>::param_type::
      param_type(initializer_list<_RealType> __bl, _Func __fw)
      : _M_int(), _M_den(), _M_cp(), _M_m()
      {
	_M_int.reserve(__bl.size());
	_M_den.reserve(__bl.size());
	for (auto __biter = __bl.begin(); __biter != __bl.end(); ++__biter)
	  {
	    _M_int.push_back(*__biter);
	    _M_den.push_back(__fw(*__biter));
	  }

	_M_initialize();
      }

  template<typename _RealType>
    template<typename _Func>
      piecewise_linear_distribution<_RealType>::param_type::
      param_type(size_t __nw, _RealType __xmin, _RealType __xmax, _Func __fw)
      : _M_int(), _M_den(), _M_cp(), _M_m()
      {
	const size_t __n = __nw == 0 ? 1 : __nw;
	const _RealType __delta = (__xmax - __xmin) / __n;

	_M_int.reserve(__n + 1);
	_M_den.reserve(__n + 1);
	for (size_t __k = 0; __k <= __nw; ++__k)
	  {
	    _M_int.push_back(__xmin + __k * __delta);
	    _M_den.push_back(__fw(_M_int[__k] + __delta));
	  }

	_M_initialize();
      }

  template<typename _RealType>
    template<typename _UniformRandomNumberGenerator>
      typename piecewise_linear_distribution<_RealType>::result_type
      piecewise_linear_distribution<_RealType>::
      operator()(_UniformRandomNumberGenerator& __urng,
		 const param_type& __param)
      {
	__detail::_Adaptor<_UniformRandomNumberGenerator, double>
	  __aurng(__urng);

	const double __p = __aurng();
	if (__param._M_cp.empty())
	  return __p;

	auto __pos = std::lower_bound(__param._M_cp.begin(),
				      __param._M_cp.end(), __p);
	const size_t __i = __pos - __param._M_cp.begin();

	const double __pref = __i > 0 ? __param._M_cp[__i - 1] : 0.0;

	const double __a = 0.5 * __param._M_m[__i];
	const double __b = __param._M_den[__i];
	const double __cm = __p - __pref;

	_RealType __x = __param._M_int[__i];
	if (__a == 0)
	  __x += __cm / __b;
	else
	  {
	    const double __d = __b * __b + 4.0 * __a * __cm;
	    __x += 0.5 * (std::sqrt(__d) - __b) / __a;
          }

        return __x;
      }

  template<typename _RealType>
    template<typename _ForwardIterator,
	     typename _UniformRandomNumberGenerator>
      void
      piecewise_linear_distribution<_RealType>::
      __generate_impl(_ForwardIterator __f, _ForwardIterator __t,
		      _UniformRandomNumberGenerator& __urng,
		      const param_type& __param)
      {
	__glibcxx_function_requires(_ForwardIteratorConcept<_ForwardIterator>)
	// We could duplicate everything from operator()...
	while (__f != __t)
	  *__f++ = this->operator()(__urng, __param);
      }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_ostream<_CharT, _Traits>&
    operator<<(std::basic_ostream<_CharT, _Traits>& __os,
	       const piecewise_linear_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_ostream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __os.flags();
      const _CharT __fill = __os.fill();
      const std::streamsize __precision = __os.precision();
      const _CharT __space = __os.widen(' ');
      __os.flags(__ios_base::scientific | __ios_base::left);
      __os.fill(__space);
      __os.precision(std::numeric_limits<_RealType>::max_digits10);

      std::vector<_RealType> __int = __x.intervals();
      __os << __int.size() - 1;

      for (auto __xit = __int.begin(); __xit != __int.end(); ++__xit)
	__os << __space << *__xit;

      std::vector<double> __den = __x.densities();
      for (auto __dit = __den.begin(); __dit != __den.end(); ++__dit)
	__os << __space << *__dit;

      __os.flags(__flags);
      __os.fill(__fill);
      __os.precision(__precision);
      return __os;
    }

  template<typename _RealType, typename _CharT, typename _Traits>
    std::basic_istream<_CharT, _Traits>&
    operator>>(std::basic_istream<_CharT, _Traits>& __is,
	       piecewise_linear_distribution<_RealType>& __x)
    {
      using __ios_base = typename basic_istream<_CharT, _Traits>::ios_base;

      const typename __ios_base::fmtflags __flags = __is.flags();
      __is.flags(__ios_base::dec | __ios_base::skipws);

      size_t __n;
      if (__is >> __n)
	{
	  vector<_RealType> __int_vec;
	  if (__detail::__extract_params(__is, __int_vec, __n + 1))
	    {
	      vector<double> __den_vec;
	      if (__detail::__extract_params(__is, __den_vec, __n + 1))
		{
		  __x.param({ __int_vec.begin(), __int_vec.end(),
			      __den_vec.begin() });
		}
	    }
	}
      __is.flags(__flags);
      return __is;
    }


  template<typename _IntType, typename>
    seed_seq::seed_seq(std::initializer_list<_IntType> __il)
    {
      _M_v.reserve(__il.size());
      for (auto __iter = __il.begin(); __iter != __il.end(); ++__iter)
	_M_v.push_back(__detail::__mod<result_type,
		       __detail::_Shift<result_type, 32>::__value>(*__iter));
    }

  template<typename _InputIterator>
    seed_seq::seed_seq(_InputIterator __begin, _InputIterator __end)
    {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++17-extensions" // if constexpr
      if constexpr (__is_random_access_iter<_InputIterator>::value)
	_M_v.reserve(std::distance(__begin, __end));
#pragma GCC diagnostic pop

      for (_InputIterator __iter = __begin; __iter != __end; ++__iter)
	_M_v.push_back(__detail::__mod<result_type,
		       __detail::_Shift<result_type, 32>::__value>(*__iter));
    }

  template<typename _RandomAccessIterator>
    void
    seed_seq::generate(_RandomAccessIterator __begin,
		       _RandomAccessIterator __end)
    {
      typedef typename iterator_traits<_RandomAccessIterator>::value_type
        _Type;

      if (__begin == __end)
	return;

      std::fill(__begin, __end, _Type(0x8b8b8b8bu));

      const size_t __n = __end - __begin;
      const size_t __s = _M_v.size();
      const size_t __t = (__n >= 623) ? 11
		       : (__n >=  68) ? 7
		       : (__n >=  39) ? 5
		       : (__n >=   7) ? 3
		       : (__n - 1) / 2;
      const size_t __p = (__n - __t) / 2;
      const size_t __q = __p + __t;
      const size_t __m = std::max(size_t(__s + 1), __n);

#ifndef __UINT32_TYPE__
      struct _Up
      {
	_Up(uint_least32_t v) : _M_v(v & 0xffffffffu) { }

	operator uint_least32_t() const { return _M_v; }

	uint_least32_t _M_v;
      };
      using uint32_t = _Up;
#endif

      // k == 0, every element in [begin,end) equals 0x8b8b8b8bu
	{
	  uint32_t __r1 = 1371501266u;
	  uint32_t __r2 = __r1 + __s;
	  __begin[__p] += __r1;
	  __begin[__q] = (uint32_t)__begin[__q] + __r2;
	  __begin[0] = __r2;
	}

      for (size_t __k = 1; __k <= __s; ++__k)
	{
	  const size_t __kn = __k % __n;
	  const size_t __kpn = (__k + __p) % __n;
	  const size_t __kqn = (__k + __q) % __n;
	  uint32_t __arg = (__begin[__kn]
			    ^ __begin[__kpn]
			    ^ __begin[(__k - 1) % __n]);
	  uint32_t __r1 = 1664525u * (__arg ^ (__arg >> 27));
	  uint32_t __r2 = __r1 + (uint32_t)__kn + _M_v[__k - 1];
	  __begin[__kpn] = (uint32_t)__begin[__kpn] + __r1;
	  __begin[__kqn] = (uint32_t)__begin[__kqn] + __r2;
	  __begin[__kn] = __r2;
	}

      for (size_t __k = __s + 1; __k < __m; ++__k)
	{
	  const size_t __kn = __k % __n;
	  const size_t __kpn = (__k + __p) % __n;
	  const size_t __kqn = (__k + __q) % __n;
	  uint32_t __arg = (__begin[__kn]
				 ^ __begin[__kpn]
				 ^ __begin[(__k - 1) % __n]);
	  uint32_t __r1 = 1664525u * (__arg ^ (__arg >> 27));
	  uint32_t __r2 = __r1 + (uint32_t)__kn;
	  __begin[__kpn] = (uint32_t)__begin[__kpn] + __r1;
	  __begin[__kqn] = (uint32_t)__begin[__kqn] + __r2;
	  __begin[__kn] = __r2;
	}

      for (size_t __k = __m; __k < __m + __n; ++__k)
	{
	  const size_t __kn = __k % __n;
	  const size_t __kpn = (__k + __p) % __n;
	  const size_t __kqn = (__k + __q) % __n;
	  uint32_t __arg = (__begin[__kn]
			    + __begin[__kpn]
			    + __begin[(__k - 1) % __n]);
	  uint32_t __r3 = 1566083941u * (__arg ^ (__arg >> 27));
	  uint32_t __r4 = __r3 - __kn;
	  __begin[__kpn] ^= __r3;
	  __begin[__kqn] ^= __r4;
	  __begin[__kn] = __r4;
	}
    }

// [rand.util.canonical]
// generate_canonical(RNG&)

#ifndef _GLIBCXX_USE_OLD_GENERATE_CANONICAL

#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wc++14-extensions" // for variable templates
#pragma GCC diagnostic ignored "-Wc++17-extensions" // if constexpr

  // __generate_canonical_pow2 is used when Urbg::max()-Urbg::min() is
  // a power of two less 1. It works by calling urng() as many times as
  // needed to fill the target mantissa, accumulating entropy into an
  // integer value, converting that to the float type, and then dividing
  // by the range of the integer value (a constexpr power of 2,
  // so only adjusts the exponent) to produce a result in [0..1].
  // In case of an exact 1.0 result, we re-try.
  //
  // It needs to work even when the integer type used is only as big
  // as the float mantissa, such as uint64_t for long double. So,
  // commented-out assignments represent computations the Standard
  // prescribes but cannot be performed, or are not used. Names are
  // chosen to match the description in the Standard.
  //
  // When the result is close to zero, the strict Standard-prescribed
  // calculation may leave more low-order zeros in the mantissa than
  // is usually necessary. When spare entropy has been extracted, as
  // is usual for float and double, some or all of the spare entropy
  // can commonly be pulled into the result for better randomness.
  // Defining _GLIBCXX_GENERATE_CANONICAL_STRICT discards it instead.
  //
  // When k calls to urng() yield more bits of entropy, log2_Rk_max,
  // than fit into UInt, we discard some of it by overflowing, which
  // is OK.  On converting the integer representation of the sample
  // to the float value, we must divide out the (possibly-truncated)
  // size log2_Rk.
  //
  // This implementation works with std::bfloat16, which can exactly
  // represent 2^32, but not with std::float16_t, limited to 2^15.

  template<typename _RealT, size_t __d, typename _Urbg>
    _RealT
    __generate_canonical_pow2(_Urbg& __urng)
    {
      using _UInt = typename __detail::_Select_uint_least_t<__d>::type;

      // Parameter __d is the actual target number of bits.
      // Commented-out assignments below are of values specified in
      //  the Standard, but not used here for reasons noted.
      // r = 2;  // Redundant, we only support radix 2.
      using _Rng = decltype(_Urbg::max());
      const _Rng __rng_range_less_1 = _Urbg::max() - _Urbg::min();
      // R = _UInt(__rng_range_less_1) + 1;  // May wrap to 0.
      const auto __log2_R = __builtin_popcountg(__rng_range_less_1);
      const auto __log2_uint_max = sizeof(_UInt) * __CHAR_BIT__;
      // rd = _UInt(1) << __d;  // Could overflow, UB.
      const unsigned __k = (__d + __log2_R - 1) / __log2_R;
      const unsigned __log2_Rk_max = __k * __log2_R;
      const unsigned __log2_Rk =  // Bits of entropy actually obtained:
	__log2_uint_max < __log2_Rk_max ? __log2_uint_max : __log2_Rk_max;
      // Rk = _UInt(1) << __log2_Rk;  // Likely overflows, UB.
      _GLIBCXX14_CONSTEXPR const _RealT __Rk
	 = _RealT(_UInt(1) << (__log2_Rk - 1)) * _RealT(2.0);
#if defined(_GLIBCXX_GENERATE_CANONICAL_STRICT)
      const unsigned __log2_x = __log2_Rk - __d; // # of spare entropy bits.
#else
      const unsigned __log2_x = 0;
#endif
      _GLIBCXX14_CONSTEXPR const _UInt __x = _UInt(1) << __log2_x;
      _GLIBCXX14_CONSTEXPR const _RealT __rd = __Rk / _RealT(__x);
      // xrd = __x << __d;  // Could overflow.

      while (true)
	{
	  _UInt __sum = _UInt(__urng() - _Urbg::min());
	  for (unsigned __i = __k - 1, __shift = 0; __i > 0; --__i)
	    {
	      __shift += __log2_R;
	      __sum |= _UInt(__urng() - _Urbg::min()) << __shift;
	    }
	  const _RealT __ret = _RealT(__sum >> __log2_x) / _RealT(__rd);
	  if (__ret < _RealT(1.0))
	    return __ret;
	}
    }


  template<typename _UInt>
    struct __gen_canon_log_res
    {
       unsigned __floor_log;
       _UInt __floor_pow;

       constexpr __gen_canon_log_res
       update(_UInt __base) const
       { return {__floor_log + 1, __floor_pow * __base}; }
    };


  template <typename _UInt1, typename _UInt2,
            typename _UComm = __conditional_t<(sizeof(_UInt2) > sizeof(_UInt1)),
					      _UInt2, _UInt1>>
    constexpr __gen_canon_log_res<_UInt1>
    __gen_canon_log(_UInt1 __val, _UInt2 __base)
    {
#if __cplusplus >= 201402L
      __gen_canon_log_res<_UInt1> __res{0, _UInt1(1)};
      if (_UComm(__base) > _UComm(__val))
	return __res;

      const _UInt1 __base1(__base);
      do
        { 
	  __val /= __base1;
          __res = __res.update(__base1);
        }
      while (__val >= __base1);
      return __res;
#else
      return (_UComm(__val) >= _UComm(__base))
	     ? __gen_canon_log(__val / _UInt1(__base), _UInt1(__base))
	         .update(_UInt1(__base))
	     : __gen_canon_log_res<_UInt1>{0, _UInt1(1)};
#endif
    }

  // This version must be used when the range of possible RNG results,
  // Urbg::max()-Urbg::min(), is not a power of two less one. The UInt
  // type passed must be big enough to represent Rk, R^k, a power of R
  // (the range of values produced by the rng) up to twice the length
  // of the mantissa.

  template<typename _RealT, size_t __d, typename _Urbg>
    _RealT
    __generate_canonical_any(_Urbg& __urng)
    {
      // Names below are chosen to match the description in the Standard.
      // Parameter d is the actual target number of bits.
#if (__cplusplus >= 201402L) || defined(__SIZEOF_INT128__)
#  define _GLIBCXX_GEN_CANON_CONST constexpr
#else
#  define _GLIBCXX_GEN_CANON_CONST const
#endif

      using _UIntR = typename make_unsigned<decltype(_Urbg::max())>::type;
      // Cannot overflow, as _Urbg::max() - _Urbg::min() is not power of
      // two minus one
      constexpr _UIntR __R = _UIntR(_Urbg::max() - _Urbg::min()) + 1;
      constexpr unsigned __log2R
	= sizeof(_UIntR) * __CHAR_BIT__ - __builtin_clzg(__R) - 1;
      // We overstimate number of required bits, by computing
      // r such that l * log2(R) >= d, so:
      // R^l >= (2 ^ log2(R)) ^ l == 2 ^ (log2(r) * l) >= 2^d
      // And then requiring l * bit_width(R) bits.
      constexpr unsigned __l = (__d + __log2R - 1) / __log2R;
      constexpr unsigned __bits = (__log2R + 1) * __l;
      using _UInt = typename __detail::_Select_uint_least_t<__bits>::type;

      _GLIBCXX_GEN_CANON_CONST _UInt __rd = _UInt(1) << __d;
      _GLIBCXX_GEN_CANON_CONST auto __logRrd = __gen_canon_log(__rd, __R);
      _GLIBCXX_GEN_CANON_CONST unsigned __k
	 = __logRrd.__floor_log + (__rd > __logRrd.__floor_pow);

      _GLIBCXX_GEN_CANON_CONST _UInt __Rk
	 = (__k > __logRrd.__floor_log)
	   ? _UInt(__logRrd.__floor_pow) * _UInt(__R)
	   : _UInt(__logRrd.__floor_pow);
      _GLIBCXX_GEN_CANON_CONST _UInt __x =  __Rk / __rd;

      while (true)
	{
	  _UInt __Ri{1};
	  _UInt __sum(__urng() - _Urbg::min());
	  for (int __i = __k - 1; __i > 0; --__i)
	    {
	      __Ri *= _UInt(__R);
	      __sum += _UInt(__urng() - _Urbg::min()) * __Ri;
	    }
	  const _RealT __ret = _RealT(__sum / __x) / _RealT(__rd);
	  if (__ret < _RealT(1.0))
	    return __ret;
	}
#undef _GLIBCXX_GEN_CANON_CONST 
    }

#if !defined(_GLIBCXX_GENERATE_CANONICAL_STRICT)
  template <typename _Tp>
    const bool __is_rand_dist_float_v = is_floating_point<_Tp>::value;
#else
  template <typename _Tp> const bool __is_rand_dist_float_v = false;
  template <> const bool __is_rand_dist_float_v<float> = true;
  template <> const bool __is_rand_dist_float_v<double> = true;
  template <> const bool __is_rand_dist_float_v<long double> = true;
#endif

  // Note, this works even when (__range + 1) overflows:
  template <typename _Rng>
    constexpr bool __is_power_of_2_less_1(_Rng __range)
      { return ((__range + 1) & __range) == 0; };

_GLIBCXX_BEGIN_INLINE_ABI_NAMESPACE(_V2)
  /** Produce a random floating-point value in the range [0..1)
   *
   * The result of `std::generate_canonical<RealT,digits>(urng)` is a
   * random floating-point value of type `RealT` in the range [0..1),
   * using entropy provided by the uniform random bit generator `urng`.
   * A value for `digits` may be passed to limit the precision of the
   * result to so many bits, but normally `-1u` is passed to get the
   * native precision of `RealT`. As many `urng()` calls are made as
   * needed to obtain the required entropy. On rare occasions, more
   * `urng()` calls are used. It is fastest when the value of
   * `Urbg::max()` is a power of two less one, such as from a
   * `std::philox4x32` (for `float`) or `philox4x64` (for `double`).
   *
   *  @since C++11
   */
  template<typename _RealT, size_t __digits,
	   typename _Urbg>
    _RealT
    generate_canonical(_Urbg& __urng)
    {
#ifdef __glibcxx_concepts
      static_assert(uniform_random_bit_generator<_Urbg>);
#endif      
      static_assert(__is_rand_dist_float_v<_RealT>,
	"template argument must be a floating point type");
      static_assert(__digits != 0 && _Urbg::max() > _Urbg::min(),
	"random samples with 0 bits are not meaningful");
      static_assert(std::numeric_limits<_RealT>::radix == 2,
	"only base-2 float types are supported");
#if defined(__STDCPP_FLOAT16_T__)
      static_assert(! is_same_v<_RealT, _Float16>,
	"float16_t type is not supported, consider using bfloat16_t");
#endif

      const unsigned __d_max = std::numeric_limits<_RealT>::digits;
      const unsigned __d = __digits > __d_max ? __d_max : __digits;

      // If the RNG range is a power of 2 less 1, the float type mantissa
      // is enough bits. If not, we need more.
      if constexpr (__is_power_of_2_less_1(_Urbg::max() - _Urbg::min()))
	return __generate_canonical_pow2<_RealT, __d>(__urng);
      else // Need up to 2x bits.
	return __generate_canonical_any<_RealT, __d>(__urng);
    }
_GLIBCXX_END_INLINE_ABI_NAMESPACE(_V2)

#pragma GCC diagnostic pop

#else // _GLIBCXX_USE_OLD_GENERATE_CANONICAL

  // This is the pre-P0952 definition, to reproduce old results.

  template<typename _RealType, size_t __bits,
	  typename _UniformRandomNumberGenerator>
    _RealType
    generate_canonical(_UniformRandomNumberGenerator& __urng)
    {
      static_assert(std::is_floating_point<_RealType>::value,
		   "template argument must be a floating point type");

      const size_t __b
       = std::min(static_cast<size_t>(std::numeric_limits<_RealType>::digits),
		   __bits);
      const long double __r = static_cast<long double>(__urng.max())
			   - static_cast<long double>(__urng.min()) + 1.0L;
      const size_t __log2r = std::log(__r) / std::log(2.0L);
      const size_t __m = std::max<size_t>(1UL,
					 (__b + __log2r - 1UL) / __log2r);
      _RealType __ret;
      _RealType __sum = _RealType(0);
      _RealType __tmp = _RealType(1);
      for (size_t __k = __m; __k != 0; --__k)
	{
	  __sum += _RealType(__urng() - __urng.min()) * __tmp;
	  __tmp *= __r;
	}
      __ret = __sum / __tmp;
      if (__builtin_expect(__ret >= _RealType(1), 0))
	{
# if _GLIBCXX_USE_C99_MATH_FUNCS
	  __ret = std::nextafter(_RealType(1), _RealType(0));
# else
	  __ret = _RealType(1)
	    - std::numeric_limits<_RealType>::epsilon() / _RealType(2);
# endif
	}
      return __ret;
    }

#endif // _GLIBCXX_USE_OLD_GENERATE_CANONICAL

_GLIBCXX_END_NAMESPACE_VERSION
} // namespace

#endif
