std::tgamma...std::tgammal(3)

💖 std::tgamma...std::tgammal(3) 💖

okay so bestie, welcome to the std::tgamma...std::tgammal(3) page straight outta the C++ Standard Libary manual — and yes, we're about to get into the gamma function. it's giving math but make it cute. 🎀✨

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🌸 NAME 🌸

std::tgamma,std::tgammaf,std::tgammal — std::tgamma,std::tgammaf,std::tgammal. literally the gamma-function trio, no cap. 💅

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✨ 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 >
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)

A) Additional overloads are provided for all integer types, which are (since C++11) treated as double. ✨

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🦋 Parameters 🦋

• num — floating-point or integer value. that's the whole assignment, bestie. 🫶

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🌸 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. 🌟

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💅 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.

⚠️ bestie, this is the part where the math gets spicy: pole errors and domain errors are real things here, so don't sleep on the return values.

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🎀 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. iconic behavior, honestly. 👑

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 way that is lowkey chaotic — iykyk. 💀

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)). fr fr. ✨

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🍓 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

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🧚 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

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🌈 External links 🌈

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

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🎀 source/version: http://cppreference.com · date: 2024.06.10 · page: std::tgamma...std::tgammal(3) — byeee bestie, go compute some gamma and slay! 💅💖