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### Gamma

##### Synopsis

    #include <boost/math/special_functions/gamma.hpp>

    namespace boost{ namespace math{

    template <class T>
    BOOST_MATH_GPU_ENABLED calculated-result-type tgamma(T z);

    template <class T, class Policy>
    BOOST_MATH_GPU_ENABLED calculated-result-type tgamma(T z, const Policy&);

    template <class T>
    BOOST_MATH_GPU_ENABLED calculated-result-type tgamma1pm1(T dz);

    template <class T, class Policy>
    BOOST_MATH_GPU_ENABLED calculated-result-type tgamma1pm1(T dz, const Policy&);

    }} // namespaces

##### Description

    template <class T>
    BOOST_MATH_GPU_ENABLED calculated-result-type tgamma(T z);

    template <class T, class Policy>
    BOOST_MATH_GPU_ENABLED calculated-result-type tgamma(T z, const Policy&);

Returns the "true gamma" (hence name tgamma) of value z:

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

The return type of this function is computed using the result type calculation rules: the result is `double` when T is an integer type, and T otherwise.

    template <class T>
    BOOST_MATH_GPU_ENABLED calculated-result-type tgamma1pm1(T dz);

    template <class T, class Policy>
    BOOST_MATH_GPU_ENABLED calculated-result-type tgamma1pm1(T dz, const Policy&);

Returns `tgamma(dz + 1) - 1`. Internally the implementation does not make use of the addition and subtraction implied by the definition, leading to accurate results even for very small `dz`.

The return type of this function is computed using the result type calculation rules: the result is `double` when T is an integer type, and T otherwise.

The final Policy argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the policy documentation for more details.

##### Accuracy

The following table shows the peak errors (in units of epsilon) found on various platforms with various floating point types, along with comparisons to other common libraries. Unless otherwise specified any floating point type that is narrower than the one shown will have effectively zero error.

Table 8.1. Error rates for tgamma

| GNU C++ version 7.1.0 linux double | GNU C++ version 7.1.0 linux long double | Sun compiler version 0x5150 Sun Solaris long double | Microsoft Visual C++ version 14.1 Win32 double
--- | --- | --- | --- | ---
factorials | Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 3.95ε (Mean = 0.783ε)) (Rmath 3.2.3: Max = 314ε (Mean = 93.4ε)) | Max = 2.67ε (Mean = 0.617ε) (<cmath>: Max = 1.66ε (Mean = 0.584ε)) (<math.h>: Max = 1.66ε (Mean = 0.584ε)) | Max = 172ε (Mean = 41ε) (<math.h>: Max = 0ε (Mean = 0ε)) | Max = 1.85ε (Mean = 0.566ε) (<math.h>: Max = 3.17ε (Mean = 0.928ε))
near 0 | Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 4.51ε (Mean = 1.92ε)) (Rmath 3.2.3: Max = 1ε (Mean = 0.335ε)) | Max = 2ε (Mean = 0.608ε) (<cmath>: Max = 1ε (Mean = 0.376ε)) (<math.h>: Max = 1ε (Mean = 0.376ε)) | Max = 2ε (Mean = 0.647ε) (<math.h>: Max = 0.5ε (Mean = 0.0791ε)) | Max = 1.5ε (Mean = 0.635ε) (<math.h>: Max = 1ε (Mean = 0.405ε))
near 1 | Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 4.41ε (Mean = 1.81ε)) (Rmath 3.2.3: Max = 1ε (Mean = 0.32ε)) | Max = 2.51ε (Mean = 1.02ε) (<cmath>: Max = 0.918ε (Mean = 0.203ε)) (<math.h>: Max = 0.918ε (Mean = 0.203ε)) | Max = 3.01ε (Mean = 1.06ε) (<math.h>: Max = 1ε (Mean = 0.175ε)) | Max = 1.1ε (Mean = 0.59ε) (<math.h>: Max = 1ε (Mean = 0.4ε))
near 2 | Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 7.95ε (Mean = 3.12ε)) (Rmath 3.2.3: Max = 1ε (Mean = 0.191ε)) | Max = 4.1ε (Mean = 1.55ε) (<cmath>: Max = 0.558ε (Mean = 0.298ε)) (<math.h>: Max = 0.558ε (Mean = 0.298ε)) | Max = 5.01ε (Mean = 1.89ε) (<math.h>: Max = 0ε (Mean = 0ε)) | Max = 2ε (Mean = 0.733ε) (<math.h>: Max = 0ε (Mean = 0ε))
near -10 | Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 2.6ε (Mean = 1.05ε)) (Rmath 3.2.3: Max = 34.9ε (Mean = 9.2ε)) | Max = 1.75ε (Mean = 0.895ε) (<cmath>: Max = 2.26ε (Mean = 1.08ε)) (<math.h>: Max = 2.26ε (Mean = 1.08ε)) | Max = 1.75ε (Mean = 0.819ε) (<math.h>: Max = 0ε (Mean = 0ε)) | Max = 1.86ε (Mean = 0.881ε) (<math.h>: Max = 0.866ε (Mean = 0.445ε))
near -55 | Max = 0ε (Mean = 0ε) (GSL 2.1: Max = 1.8ε (Mean = 0.782ε)) (Rmath 3.2.3: Max = 3.89e+04ε (Mean = 9.52e+03ε)) | Max = 2.69ε (Mean = 1.09ε) (<cmath>: Max = 1.79ε (Mean = 0.75ε)) (<math.h>: Max = 1.79ε (Mean = 0.75ε)) | Max = 98.5ε (Mean = 53.4ε) (<math.h>: Max = 0ε (Mean = 0ε)) | Max = 2.7ε (Mean = 1.35ε) (<math.h>: Max = 3.87e+04ε (Mean = 6.71e+03ε))

Table 8.2. Error rates for tgamma1pm1

| GNU C++ version 7.1.0 linux double | GNU C++ version 7.1.0 linux long double | Sun compiler version 0x5150 Sun Solaris long double | Microsoft Visual C++ version 14.1 Win32 double
--- | --- | --- | --- | ---
tgamma1pm1(dz) | Max = 0ε (Mean = 0ε) | Max = 1.12ε (Mean = 0.49ε) | Max = 6.61ε (Mean = 0.84ε) | Max = 3.31ε (Mean = 0.517ε)

The following error plot are based on an exhaustive search of the functions domain, MSVC-15.5 at `double` precision, and GCC-7.1/Ubuntu for `long double` and `__float128`.

##### Testing

The gamma is relatively easy to test: factorials and half-integer factorials can be calculated exactly by other means and compared with the gamma function. In addition, some accuracy tests in known tricky areas were computed at high precision using the generic version of this function.

The function `tgamma1pm1` is tested against values calculated very naively using the formula `tgamma(1+dz)-1` with a lanczos approximation accurate to around 100 decimal digits.

##### Implementation

The generic version of the `tgamma` function is implemented Sterling's approximation for `lgamma` for large z:

Following exponentiation, downward recursion is then used for small values of z.

For types of known precision the Lanczos approximation is used, a traits class `boost::math::lanczos::lanczos_traits` maps type T to an appropriate approximation.

For z in the range -20 < z < 1 then recursion is used to shift to z > 1 via:

For very small z, this helps to preserve the identity:

For z < -20 the reflection formula:

is used. Particular care has to be taken to evaluate the `z * sin(π * z)` part: a special routine is used to reduce z prior to multiplying by π to ensure that the result in is the range [0, π/2]. Without this an excessive amount of error occurs in this region (which is hard enough already, as the rate of change near a negative pole is exceptionally high).

Finally if the argument is a small integer then table lookup of the factorial is used.

The function `tgamma1pm1` is implemented using rational approximations devised by JM in the region `-0.5 < dz < 2`. These are the same approximations (and internal routines) that are used for lgamma, and so aren't detailed further here. The result of the approximation is `log(tgamma(dz+1))` which can fed into expm1 to give the desired result. Outside the range `-0.5 < dz < 2` then the naive formula `tgamma1pm1(dz) = tgamma(dz+1)-1` can be used directly.