/C++

# std::cauchy_distribution

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

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) =

⎝bπ

⎣1 +

x - a/b

2

-1

std::cauchy_distribution satisfies all requirements of RandomNumberDistribution.

### Template parameters

 RealType - The result type generated by the generator. The effect is undefined 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 RandomNumberDistribution.

### Member functions

(C++11)
constructs new distribution
(public member function)
(C++11)
resets the internal state of the distribution
(public member function)
##### Generation
(C++11)
generates the next random number in the distribution
(public member function)
##### Characteristics
returns the distribution parameters
(public member function)
(C++11)
gets or sets the distribution parameter object
(public member function)
(C++11)
returns the minimum potentially generated value
(public member function)
(C++11)
returns the maximum potentially generated value
(public member function)

### Non-member functions

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

### Example

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

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((Height > 0) && (BarWidth > 0) && (Padding >= 0) && (Offset >= 0));
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(*cbegin(s)) V; V e : s)
qr.push_back(std::div(std::lerp(V(0), Height*8, (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: '█'
q < h ? d[0] = ' ', d[1] = 0 : q == h ? d[2] -= (7 - r) : 0;
}
if (DrawMinMax && Height > 1)
Height - 1 == h ? std::cout << "┬ " << *max:
h ? std::cout << "│ "
: std::cout << "┴ " << *min;
}
}

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

auto cauchy = [&gen](const float x0, const float 𝛾) {
std::cauchy_distribution<float> d{ x0 /* a */, 𝛾 /* 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 << "x₀ = " << x0 << ", 𝛾 = " << 𝛾 << ":\n";
draw_vbars<4,3>(bars);
for (int n : indices) { std::cout << "" << std::setw(2) << n << "  "; }
std::cout << "\n\n";
};

cauchy(/* x₀ = */ -2.0f, /* 𝛾 = */ 0.50f);
cauchy(/* x₀ = */ +0.0f, /* 𝛾 = */ 1.25f);
}

Possible output:

x₀ = -2, 𝛾 = 0.5:
███                     ┬ 0.5006
███                     │
▂▂▂ ███ ▁▁▁                 │
▁▁▁ ▁▁▁ ▁▁▁ ▃▃▃ ███ ███ ███ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0076
-7  -6  -5  -4  -3  -2  -1   0   1   2   3

x₀ = 0, 𝛾 = 1.25:
███                                 ┬ 0.2539
▅▅▅ ███ ▃▃▃                             │
▁▁▁ ███ ███ ███ ▁▁▁                         │
▁▁▁ ▁▁▁ ▁▁▁ ▁▁▁ ▃▃▃ ▅▅▅ ███ ███ ███ ███ ███ ▅▅▅ ▃▃▃ ▂▂▂ ▁▁▁ ▁▁▁ ▁▁▁ ┴ 0.0058
-8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4   5   6   7   9

Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource.