beta(), betaf() and betal() functions in C++ STL

The beta(), betaf() and betal() are built-in functions in C++ STL that are used to compute the beta functionof two positive real values. The function takes two variables x and y as input and returns the beta function of x and y. Beta function (Also known as Euler integral of first kind) of x and y can be defined as:

     $$B(x, y)=\int_0^1 t^{x-1} (1-t)^{y-1} dt $$

Syntax

double beta(double x, double y)
or 
long double betal(long double x, long double y)
or 
float betaf(float x, float y)

Parameters: The function accepts two mandatory parameters x and y which specifies the values of a floating-point or integral type. The parameters can be of double, double or float, float or long double, long double data-type.

Return Value: The function returns the value of beta function of x and y. The return type depends on the parameters passed. It is same as that of the parameter.

Note: The function runs in and above C++ 17(7.1).

Below program illustrates the beta(), betaf() and betal() functions:

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// C++ program to illustrate the three functions
// Being a special function, beta is only guaranteed
// to be in cmath if the user defines
//  __STDCPP_WANT_MATH_SPEC_FUNCS__ before including
// any standard library headers.
#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
#include <bits/stdc++.h>
using namespace std;
  
int main()
{
    // Computes the beta function of 5 and 4 and print it
    // If provided arguments are not of type double
    // they are implicitly type-casted to the higher type.
  
    // first example of beta()
    cout << beta(5, 4) << "\n";
  
    // second example of betaf()
    cout << betaf(10.0, 4.0) << "\n";
  
    // third example of betal()
    cout << betal(10.0, 6.7) << "\n";
    return 0;
}

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Output:

0.00357143
0.00034965
1.65804e-005

Application of Beta function: It is used to compute Binomial Coefficients.The binomial coefficient in terms of beta function can be expressed as:

      $$\binom{n}{k}= \frac{1}{(n+1)B(n-k+1, k+1)}$$

The above relation can be used to compute the binomial coeeficient. An illustration has been shown below:

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// C++ program to print the pascal triangle
// Being a special function, beta is only guaranteed
// to be in cmath if the user defines
//  __STDCPP_WANT_MATH_SPEC_FUNCS__ before including
// any standard library headers.
#define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
#include <bits/stdc++.h>
#include <cmath>
using namespace std;
  
// Function to return the value of binomial Coefficient
double binomialCoefficient(int n, int k)
{
    // Calculate the value of nCr using above formula.
    double ans = 1 / ((n + 1) * beta(n - k + 1, k + 1));
    return ans;
}
// Driver Code
int main()
{
    // Print the binomial Coefficient nCr where n=5 and r=2
    cout << binomialCoefficient(5, 2) << "\n";
  
    return 0;
}

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Output:

10


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