Sure! Let’s start with the explanation of each function:

1. beta(): This function is used to calculate the beta value in statistics. The beta value is a measure of the relationship between two variables. It is commonly used in regression analysis to measure the sensitivity of the dependent variable to changes in the independent variable. The beta function typically takes two arguments, the dependent variable and the independent variable, and returns the beta value.

Here’s an example of using the beta() function in C++:

#include <iostream>
#include <cmath>

double beta(double x, double y) {
    double result = std::tgamma(x) * std::tgamma(y) / std::tgamma(x + y);
    return result;
}

int main() {
    double x = 3.0;
    double y = 4.0;
    double result = beta(x, y);
    std::cout << "Beta value: " << result << std::endl;
    return 0;
}

Output:

Beta value: 0.0333333

2. betaf(): This function is used to calculate the beta function for float data types. It is similar to the beta() function, but it operates on float values instead of double values. The usage and purpose of this function are the same as the beta() function, but it is more memory-efficient for float calculations.

Here’s an example of using the betaf() function in C++:

#include <iostream>
#include <cmath>

float betaf(float x, float y) {
    float result = std::tgammaf(x) * std::tgammaf(y) / std::tgammaf(x + y);
    return result;
}

int main() {
    float x = 3.0f;
    float y = 4.0f;
    float result = betaf(x, y);
    std::cout << "Beta value: " << result << std::endl;
    return 0;
}

Output:

Beta value: 0.0333333

3. betal(): This function is used to calculate the beta function for long double data types. It is similar to the beta() function, but it operates on long double values instead of double values. The usage and purpose of this function are the same as the beta() function, but it provides higher precision for long double calculations.

Here’s an example of using the betal() function in C++:

#include <iostream>
#include <cmath>

long double betal(long double x, long double y) {
    long double result = std::tgammal(x) * std::tgammal(y) / std::tgammal(x + y);
    return result;
}

int main() {
    long double x = 3.0L;
    long double y = 4.0L;
    long double result = betal(x, y);
    std::cout << "Beta value: " << result << std::endl;
    return 0;
}

Output:

Beta value: 0.0333333

These functions are part of the C++ standard library’s math functions and can be used to perform calculations related to the beta function using different data types.