std::beta, std::betaf, std::betal
<cmath>
double beta ( double x, double y );
(until C++23)
/* floating-point-type */ y );
<cmath>
/* common-floating-point-type */ beta( Arithmetic1 x, Arithmetic2 y );
std::beta for all cv-unqualified floating-point types as the type of the parameters x and y.(since C++23)[edit] Parameters
[edit] Return value
If no errors occur, value of the beta function of x and y, that is \(\int_{0}^{1}{ {t}^{x-1}{(1-t)}^{y-1}\mathsf{d}t}\)∫10tx-1
(1-t)(y-1)
dt, or, equivalently, \(\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\)
[edit] 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.
[edit] 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 \(\frac{(x-1)!(y-1)!}{(x+y-1)!}\)⎜
⎝n
k⎞
⎟
⎠=
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)).
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)[edit] 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