std::beta, std::betaf, std::betal

From cppreference.com
 
 
 
 
Defined in header <cmath>
(1)
float       beta ( float x, float y );

double      beta ( double x, double y );

long double beta ( long double x, long double y );
(since C++17)
(until C++23)
/* floating-point-type */ beta( /* floating-point-type */ x,
                                /* floating-point-type */ y );
(since C++23)
float       betaf( float x, float y );
(2) (since C++17)
long double betal( long double x, long double y );
(3) (since C++17)
Defined in header <cmath>
template< class Arithmetic1, class Arithmetic2 >
/* common-floating-point-type */ beta( Arithmetic1 x, Arithmetic2 y );
(A) (since C++17)
1-3) Computes the Beta function of x and y. The library provides overloads of std::beta for all cv-unqualified floating-point types as the type of the parameters x and y. (since C++23)
A) Additional overloads are provided for all other combinations of arithmetic types.

Parameters

x, y - floating-point or integer values

Return value

If no errors occur, value of the beta function of x and y, that is 1
0
tx-1
(1-t)(y-1)
dt
, or, equivalently,
Γ(x)Γ(y)
Γ(x+y)
is returned.

Error handling

Errors may be reported as specified in math_errhandling

  • If any argument is NaN, NaN is returned and domain error is not reported
  • The function is only required to be defined where both x and y are greater than zero, and is allowed to report a domain error otherwise.

Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header tr1/cmath and namespace std::tr1.

An implementation of this function is also available in boost.math.

std::beta(x, y) equals std::beta(y, x).

When x and y are positive integers, std::beta(x, y) equals
(x-1)!(y-1)!
(x+y-1)!
. Binomial coefficients can be expressed in terms of the beta function:

n
k


=
1
(n+1)Β(n-k+1,k+1)
.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their first argument num1 and second argument num2:

  • If num1 or num2 has type long double, then std::beta(num1, num2) has the same effect as std::beta(static_cast<long double>(num1),
              static_cast<long double>(num2))
    .
  • Otherwise, if num1 and/or num2 has type double or an integer type, then std::beta(num1, num2) has the same effect as std::beta(static_cast<double>(num1),
              static_cast<double>(num2))
    .
  • Otherwise, if num1 or num2 has type float, then std::beta(num1, num2) has the same effect as std::beta(static_cast<float>(num1),
              static_cast<float>(num2))
    .
(until C++23)

If num1 and num2 have arithmetic types, then std::beta(num1, num2) has the same effect as std::beta(static_cast</* common-floating-point-type */>(num1),
          static_cast</* common-floating-point-type */>(num2))
, where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2, arguments of integer type are considered to have the same floating-point conversion rank as double.

If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided.

(since C++23)

Example

#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <numbers>
#include <string>
 
long binom_via_beta(int n, int k)
{
    return std::lround(1 / ((n + 1) * std::beta(n - k + 1, k + 1)));
}
 
long binom_via_gamma(int n, int k)
{
    return std::lround(std::tgamma(n + 1) /
                      (std::tgamma(n - k + 1) * 
                       std::tgamma(k + 1)));
}
 
int main()
{
    std::cout << "Pascal's triangle:\n";
    for (int n = 1; n < 10; ++n)
    {
        std::cout << std::string(20 - n * 2, ' ');
        for (int k = 1; k < n; ++k)
        {
            std::cout << std::setw(3) << binom_via_beta(n, k) << ' ';
            assert(binom_via_beta(n, k) == binom_via_gamma(n, k));
        }
        std::cout << '\n';
    }
 
    // A spot-check
    const long double p = 0.123; // a random value in [0, 1]
    const long double q = 1 - p;
    const long double π = std::numbers::pi_v<long double>;
    std::cout << "\n\n" << std::setprecision(19)
              << "β(p,1-p)   = " << std::beta(p, q) << '\n'
              << "π/sin(π*p) = " << π / std::sin(π * p) << '\n';
}

Output:

Pascal's triangle:
 
                  2
                3   3
              4   6   4
            5  10  10   5
          6  15  20  15   6
        7  21  35  35  21   7
      8  28  56  70  56  28   8
    9  36  84 126 126  84  36   9
 
β(p,1-p)   = 8.335989149587307836
π/sin(π*p) = 8.335989149587307834

See also

(C++11)(C++11)(C++11)
gamma function
(function)

External links

Weisstein, Eric W. "Beta Function." From MathWorld — A Wolfram Web Resource.