  A new computational framework for evaluation of the gamma function
$\Gamma(z)$ over the complex plane is developed. The algorithm is based on
interpolation by rational functions, and generalizes the classical methods of
Lanczos \cite{Lanczos} and Spouge \cite{Spouge} (which we show are also
interpolatory). This framework utilizes the exact poles of the gamma function.
By relaxing this condition and allowing the poles to vary, a near-optimal
rational approximation is possible, which is demonstrated using the adaptive
Antoulous Anderson (AAA) algorithm, developed in \cite{AAA,AAA_2020}. The
resulting approximations are competitive with Stirling's formula in terms of
overall efficiency.
