std::lgamma, std::lgammaf, std::lgammal

Defined in header `<cmath>`
float lgamma ( float arg ); float lgammaf( float arg );	(1)	(since C++11)
double lgamma ( double arg );	(2)	(since C++11)
long double lgamma ( long double arg ); long double lgammal( long double arg );	(3)	(since C++11)
double lgamma ( IntegralType arg );	(4)	(since C++11)

1-3) Computes the natural logarithm of the absolute value of the gamma function of arg.

4) A set of overloads or a function template accepting an argument of any integral type. Equivalent to (2) (the argument is cast to double).

Parameters

arg

-

value of a floating-point or Integral type

Return value

If no errors occur, the value of the logarithm of the gamma function of arg, that is $log e | \int \infty 0 t arg-1 e -t d t |$ , is returned.

If a pole error occurs, +HUGE_VAL, +HUGE_VALF, or +HUGE_VALL is returned.

If a range error due to overflow occurs, ±HUGE_VAL, ±HUGE_VALF, or ±HUGE_VALL is returned.

Error handling

Errors are reported as specified in math_errhandling.

If arg is zero or is an integer less than zero, a pole error may occur.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

If the argument is 1, +0 is returned
If the argument is 2, +0 is returned
If the argument is ±0, +∞ is returned and FE_DIVBYZERO is raised
If the argument is a negative integer, +∞ is returned and FE_DIVBYZERO is raised
If the argument is ±∞, +∞ is returned.
If the argument is NaN, NaN is returned

Notes

If arg is a natural number, std::lgamma(arg) is the logarithm of the factorial of arg-1.

The POSIX version of lgamma is not thread-safe: each execution of the function stores the sign of the gamma function of arg in the static external variable signgam. Some implementations provide lgamma_r, which takes a pointer to user-provided storage for singgam as the second parameter, and is thread-safe.

There is a non-standard function named gamma in various implementations, but its definition is inconsistent. For example, glibc and 4.2BSD version of gamma executes lgamma, but 4.4BSD version of gamma executes tgamma.

Example

Run this code

#include <iostream>
#include <cmath>
#include <cerrno>
#include <cstring>
#include <cfenv>
// #pragma STDC FENV_ACCESS ON
const double pi = std::acos(-1); // or std::numbers::pi since C++20
int main()
{
    std::cout << "lgamma(10) = " << std::lgamma(10)
              << ",  log(9!) = " << std::log(2*3*4*5*6*7*8*9) << '\n'
              << "lgamma(0.5) = " << std::lgamma(0.5)
              << " , log(sqrt(pi)) = " << std::log(std::sqrt(pi)) << '\n';
    // special values
    std::cout << "lgamma(1) = " << std::lgamma(1) << '\n'
              << "lgamma(+Inf) = " << std::lgamma(INFINITY) << '\n';
    // error handling
    errno = 0;
    std::feclearexcept(FE_ALL_EXCEPT);
    std::cout << "lgamma(0) = " << std::lgamma(0) << '\n';
    if (errno == ERANGE)
        std::cout << "    errno == ERANGE: " << std::strerror(errno) << '\n';
    if (std::fetestexcept(FE_DIVBYZERO))
        std::cout << "    FE_DIVBYZERO raised\n";
}

Output:

lgamma(10) = 12.8018,  log(9!) = 12.8018
lgamma(0.5) = 0.572365 , log(sqrt(pi)) = 0.572365
lgamma(1) = 0
lgamma(+Inf) = inf
lgamma(0) = inf
    errno == ERANGE: Numerical result out of range
    FE_DIVBYZERO raised

External links

Weisstein, Eric W. "Log Gamma Function." From MathWorld--A Wolfram Web Resource.

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