# std::tgamma, std::tgammaf, std::tgammal

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< cpp‎ | numeric‎ | math

Defined in header `<cmath>` |  |
| (1) |
float tgamma ( float num ); double tgamma ( double num ); long double tgamma ( long double num ); |  | (until C++23)
/*floating-point-type*/ tgamma ( /*floating-point-type*/ num ); |  | (since C++23) (constexpr since C++26)
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)
SIMD overload (since C++26) |  |
Defined in header `<simd>` |  |
template< /*math-floating-point*/ V > constexpr /*deduced-simd-t*/<V> tgamma ( const V& v_num ); | (S) | (since C++26)
Additional overloads (since C++11) |  |
Defined in header `<cmath>` |  |
template< class Integer > double tgamma ( Integer num ); | (A) | (constexpr since C++26)

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)

S) The SIMD overload performs an element-wise `std::tgamma` on v_num.

    (See `math-floating-point` and `deduced-simd-t` for their definitions.)

(since C++26)

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 gamma function of num, that is \(\Gamma(\mathtt{num}) = \displaystyle\int_0^\infty\!\! t^{\mathtt{num}-1} e^{-t}\, dt\)∫∞
0 t num-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

lgammalgammaflgammal (C++11)(C++11)(C++11) | natural logarithm of the gamma function (function)
betabetafbetal (C++17)(C++17)(C++17) | beta function (function)
C documentation for tgamma

### External links

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

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