arb_hypgeom_gamma_taylor

The method reduces $x$ to $u = x - r \in [-1/2, 1/2]$ with $r$ an integer, evaluates the Taylor series of $1/\Gamma(1+u)$ with the coefficients stored in arb_hypgeom_gamma_coeffs, and multiplies or divides by a rising factorial. It applies for $|x| < 160$, for $r$ between about $-(40 + (\mathit{prec} - 40)/4)$ and $70 + (\mathit{prec}-40)/8$, and up to a working precision of ARB_HYPGEOM_GAMMA_TAB_PREC = 3456 bits. Wide inputs (below 40 bits of precision, or a radius above $2^{-16}$) get an enclosure from double-precision bounds for $\Gamma$ at the endpoints, taking the minimum of $\Gamma$ on $(0, \infty)$ into account. Otherwise the result has a radius of a few ulp, plus a bound for the error propagated from the radius of $x$ computed with an upper bound for the digamma function. See Taylor series of the reciprocal gamma function.