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We discuss the best methods available for computing the gamma function Γ(z) in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also cover the log-gamma function logΓ(z), the digamma function ψ(z), and derivatives Γ^(n)(z) and ψ^(n)(z). Besides attempting to summarize the existing state of the art, we present some newarXiv:2109.08392v1 fatcat:jq7vdq6xijf5de5gaqdehvy5uu