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

Defined in header `<cmath>`
	(1)
float lgamma ( float num ); double lgamma ( double num ); long double lgamma ( long double num );			(until C++23)
/* floating-point-type / lgamma ( / floating-point-type */ num );			(since C++23) (constexpr since C++26)
float lgammaf( float num );		(2)	(since C++11) (constexpr since C++26)
long double lgammal( long double num );		(3)	(since C++11) (constexpr since C++26)
Additional overloads (since C++11)
Defined in header `<cmath>`
template< class Integer > double lgamma ( Integer num );		(A)	(constexpr since C++26)

1-3) Computes the natural logarithm of the absolute value of the gamma function of num. The library provides overloads of std::lgamma for all cv-unqualified floating-point types as the type of the parameter. (since C++23)

A) Additional overloads are provided for all integer types, which are treated as double.

(since C++11)

Parameters

num

-

floating-point or integer value

Return value

If no errors occur, the value of the logarithm of the gamma function of num, that is $log e | \int \infty 0 t num-1 e -t d t |$ , 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 num 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 num is a natural number, std::lgamma(num) is the logarithm of the factorial of num - 1.

The POSIX version of lgamma is not thread-safe: each execution of the function stores the sign of the gamma function of num 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.

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their argument num of integer type, std::lgamma(num) has the same effect as std::lgamma(static_cast<double>(num)).

Example

Run this code

#include <cerrno>
#include <cfenv>
#include <cmath>
#include <cstring>
#include <iostream>
 
// #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(std::tgamma(10))
              << ", exp(lgamma(10)) = " << std::exp(std::lgamma(10)) << '\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, exp(lgamma(10)) = 362880
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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