std::tgamma,std::tgammaf,std::tgammal(3)

NAME
std::tgamma,std::tgammaf,std::tgammal - std::tgamma,std::tgammaf,std::tgammal

Synopsis
   Defined in header <cmath>
   float tgamma ( float num );
   double tgamma ( double num ); (until C++23)
   long double tgamma ( long double num );
   /* floating-point-type */ (since C++23)
               tgamma ( /* floating-point-type */ num (constexpr since C++26)
   ); (1)
   float tgammaf( float num ); (2) (since C++11)
                        (constexpr since C++26)
   long double tgammal( long double num ); (3) (since C++11)
                        (constexpr since C++26)
   Additional overloads (since C++11)
   Defined in header <cmath>
   template< class Integer > (A) (constexpr since C++26)
   double tgamma ( Integer num );
   1-3) Computes the gamma function of num.
   The library provides overloads of std::tgamma 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 (since C++11)
   treated as double.

Parameters
   num - floating-point or integer value

Return value
   If no errors occur, the value of the gamma function of num, that is
   \(\Gamma(\mathtt{num}) = \displaystyle\int_0^\infty\!\! t^{\mathtt{num}-1} e^{-t}\,
   dt\)∫∞
   0tnum-1
   e^-t dt, is returned.
   If a domain error occurs, an implementation-defined value (NaN where supported) 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.
   If a range error due to underflow occurs, the correct value (after rounding) 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 or a domain error may
   occur.
   If the implementation supports IEEE floating-point arithmetic (IEC 60559),
     * If the argument is ±0, ±∞ is returned and FE_DIVBYZERO is raised.
     * If the argument is a negative integer, NaN is returned and FE_INVALID is raised.
     * If the argument is -∞, NaN is returned and FE_INVALID is raised.
     * If the argument is +∞, +∞ is returned.
     * If the argument is NaN, NaN is returned.

Notes
   If num is a natural number, std::tgamma(num) is the factorial of num - 1. Many
   implementations calculate the exact integer-domain factorial if the argument is a
   sufficiently small integer.
   For IEEE-compatible type double, overflow happens if 0 < num && num < 1 / DBL_MAX or
   if num > 171.7.
   POSIX requires that a pole error occurs if the argument is zero, but a domain error
   occurs when the argument is a negative integer. It also specifies that in future,
   domain errors may be replaced by pole errors for negative integer arguments (in
   which case the return value in those cases would change from NaN to ±∞).
   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::tgamma(num) has the same effect as std::tgamma(static_cast<double>(num)).

Example
// Run this code
 #include <cerrno>
 #include <cfenv>
 #include <cmath>
 #include <cstring>
 #include <iostream>
 // #pragma STDC FENV_ACCESS ON
 int main()
 {
     std::cout << "tgamma(10) = " << std::tgamma(10)
               << ", 9! = " << 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 << '\n'
               << "tgamma(0.5) = " << std::tgamma(0.5)
               << ", sqrt(pi) = " << std::sqrt(std::acos(-1)) << '\n';
     // special values
     std::cout << "tgamma(1) = " << std::tgamma(1) << '\n'
               << "tgamma(+Inf) = " << std::tgamma(INFINITY) << '\n';
     // error handling
     errno = 0;
     std::feclearexcept(FE_ALL_EXCEPT);
     std::cout << "tgamma(-1) = " << std::tgamma(-1) << '\n';
     if (errno == EDOM)
         std::cout << " errno == EDOM: " << std::strerror(errno) << '\n';
     if (std::fetestexcept(FE_INVALID))
         std::cout << " FE_INVALID raised\n";
 }

Possible output:
 tgamma(10) = 362880, 9! = 362880
 tgamma(0.5) = 1.77245, sqrt(pi) = 1.77245
 tgamma(1) = 1
 tgamma(+Inf) = inf
 tgamma(-1) = nan
     errno == EDOM: Numerical argument out of domain
     FE_INVALID raised

See also
   lgamma
   lgammaf
   lgammal natural logarithm of the gamma function
   (C++11) (function)
   (C++11)
   (C++11)
   beta
   betaf
   betal beta function
   (C++17) (function)
   (C++17)
   (C++17)
   C documentation for
   tgamma

External links
   Weisstein, Eric W. "Gamma Function." From MathWorld — A Wolfram Web Resource.