Cody and Hillstrom (1967) gives minimax rational approximations for $\ln \mathrm{\Gamma }\left(x\right)$ for the ranges $0.5\le x\le 1.5$, $1.5\le x\le 4$, $4\le x\le 12$; precision is variable. Hart et al. (1968) gives minimax polynomial and rational approximations to $\mathrm{\Gamma }\left(x\right)$ and $\ln \mathrm{\Gamma }\left(x\right)$ in the intervals $0\le x\le 1$, $8\le x\le 1000$, $12\le x\le 1000$; precision is variable. Cody et al. (1973) gives minimax rational approximations for $\psi \left(x\right)$ for the ranges $0.5\le x\le 3$ and ; precision is variable.

For additional approximations see Hart et al. (1968, Appendix B), Luke (1975, pp. 22–23), and Weniger (2003).

Luke (1969b) gives the coefficients to 20D for the Chebyshev-series expansions of $\mathrm{\Gamma }\left(1+x\right)$, $1/\mathrm{\Gamma }\left(1+x\right)$, $\mathrm{\Gamma }\left(x+3\right)$, $\ln \mathrm{\Gamma }\left(x+3\right)$, $\psi \left(x+3\right)$, and the first six derivatives of $\psi \left(x+3\right)$ for $0\le x\le 1$. These coefficients are reproduced in Luke (1975). Clenshaw (1962) also gives 20D Chebyshev-series coefficients for $\mathrm{\Gamma }\left(1+x\right)$ and its reciprocal for $0\le x\le 1$. See Luke (1975, pp. 22–23) for additional expansions.

See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of $\mathrm{\Gamma }\left(z\right)$.

For rational approximations to $\psi \left(z\right)+\gamma$ see Luke (1975, pp. 13–16).