arb_hypgeom_gamma_stirling

For $x < -5$ the reflection formula $\Gamma(x) = \pi / (\sin(\pi x) \Gamma(1-x))$ is applied first. The argument is then shifted by a rising factorial to $x + r \ge \beta \cdot \mathit{prec}$, where $\beta$ is between 0.17 and 0.24 depending on the precision, and the number of terms is chosen so that the bound for the remainder of the Stirling series is below $2^{-\mathit{prec}}$. The working precision is increased by the number of exponent bits of large $x$ and limited by the accuracy of the input. Arguments with more than about $10\,\mathit{prec} + 4096$ exponent bits give an indeterminate result. See The Stirling series.