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std::cauchy_distribution(3)    C++ Standard Libary   std::cauchy_distribution(3)

NAME
     std::cauchy_distribution - std::cauchy_distribution

Synopsis
	Defined in header <random>
	template< class RealType = double >  (since C++11)
	class cauchy_distribution;

	Produces  random numbers according to a Cauchy distribution (also called
     Lorentz
	distribution):

	\({\small f(x;a,b)={(b\pi{[1+{(\frac{x-a}{b})}^{2}]} })}^{-1}\)f(x; a,b)
     = a
	a
	abI a!
	ac
	aL1 + a
	a
	a

	x - a
	b

	a
	a
	a 2
	ax
	aY
	a|a
	a
	a -1

	std::cauchy_distribution satisfies all requirements of	RandomNumberDis-
     tribution.

Template parameters
	RealType - The result type generated by the generator. The effect is un-
     defined if
		   this is not one of float, double, or long double.

Member types
	Member type	    Definition
	result_type (C++11) RealType
	param_type  (C++11)  the type of the parameter set, see RandomNumberDis-
     tribution.

Member functions
	constructor   constructs new distribution
	(C++11)       (public member function)
	reset	      resets the internal state of the distribution
	(C++11)       (public member function)

Generation
	operator()    generates the next random number in the distribution
	(C++11)       (public member function)

Characteristics
	a	      returns the distribution parameters
	b	      (public member function)
	(C++11)
	param	      gets or sets the distribution parameter object
	(C++11)       (public member function)
	min	      returns the minimum potentially generated value
	(C++11)       (public member function)
	max	      returns the maximum potentially generated value
	(C++11)       (public member function)

Non-member functions
	operator==
	operator!=		  compares two distribution objects
	(C++11) 		  (function)
	(C++11)(removed in C++20)
	operator<<		  performs stream input and  output  on  pseudo-
     random number
	operator>>		  distribution
	(C++11) 		  (function template)

Example
     // Run this code

      #include <algorithm>
      #include <cmath>
      #include <iomanip>
      #include <iostream>
      #include <map>
      #include <random>
      #include <vector>

      template<int  Height  = 5, int BarWidth = 1, int Padding = 1, int Offset =
     0, class Seq>
      void draw_vbars(Seq&& s, const bool DrawMinMax = true)
      {
	  static_assert(0 < Height and 0 < BarWidth and 0 <= Padding  and  0  <=
     Offset);

	  auto cout_n = [](auto&& v, int n = 1)
	  {
	      while (n-- > 0)
		  std::cout << v;
	  };

	  const    auto   [min,   max]	 =   std::minmax_element(std::cbegin(s),
     std::cend(s));

	  std::vector<std::div_t> qr;
	  for (typedef decltype(*std::cbegin(s)) V; V e : s)
	      qr.push_back(std::div(std::lerp(V(0), 8 * Height,
					      (e - *min) / (*max - *min)), 8));

	  for (auto h{Height}; h-- > 0; cout_n('\n'))
	  {
	      cout_n(' ', Offset);

	      for (auto dv : qr)
	      {
		  const auto q{dv.quot}, r{dv.rem};
		  unsigned char d[]{0xe2, 0x96, 0x88, 0}; // Full Block: 'a'
		  q < h ? d[0] = ' ', d[1] = 0 : q == h ? d[2] -= (7 - r) : 0;
		  cout_n(d, BarWidth), cout_n(' ', Padding);
	      }

	      if (DrawMinMax && Height > 1)
		  Height - 1 == h ? std::cout << "a~ " << *max:
				h ? std::cout << "a "
				  : std::cout << "a' " << *min;
	  }
      }

      int main()
      {
	  std::random_device rd{};
	  std::mt19937 gen{rd()};

	  auto cauchy = [&gen](const float x0, const float dh3/4)
	  {
	      std::cauchy_distribution<float> d{x0 /* a */, dh3/4 /* b */};

	      const int norm = 1'00'00;
	      const float cutoff = 0.005f;

	      std::map<int, int> hist{};
	      for (int n = 0; n != norm; ++n)
		  ++hist[std::round(d(gen))];

	      std::vector<float> bars;
	      std::vector<int> indices;
	      for (auto const& [n, p] : hist)
		  if (float x = p * (1.0 / norm); cutoff < x)
		  {
		      bars.push_back(x);
		      indices.push_back(n);
		  }

	      std::cout << "xa = " << x0 << ", dh3/4 = " << dh3/4 << ":\n";
	      draw_vbars<4,3>(bars);
	      for (int n : indices)
		  std::cout << std::setw(2) << n << "  ";
	      std::cout << "\n\n";
	  };

	  cauchy(/* xa = */ -2.0f, /* dh3/4 = */ 0.50f);
	  cauchy(/* xa = */ +0.0f, /* dh3/4 = */ 1.25f);
      }

Possible output:
      xa = -2, dh3/4 = 0.5:
			  aaa			  a~ 0.5006
			  aaa			  a
		      aaa aaa aaa		  a
      aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa a' 0.0076
      -7  -6  -5  -4  -3  -2  -1   0   1   2   3

      xa = 0, dh3/4 = 1.25:
				      aaa				      a~
     0.2539
				  aaa aaa aaa				  a
			      aaa aaa aaa aaa aaa			  a
      aaa  aaa	aaa  aaa  aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa a'
     0.0058
      -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   9

External links
	  Weisstein, Eric W. "Cauchy Distribution." From MathWorld a  A  Wolfram
     Web Resource.

http://cppreference.com 	   2024.06.10	     std::cauchy_distribution(3)
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