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:

// 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; 
} 

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:

// 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; 
} 

Output:

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