acb_hypgeom_log_rising_ui

The function computes $(x)_n$, takes its principal logarithm and determines the branch correction $2\pi i m$. The correction is found with Hare's algorithm [Hare1997] in double-precision arithmetic, which counts how often the partial products cross the negative real axis, when this is safe ($n$ and $|x|$ at most $10^6$, $|\operatorname{Im} x| > 10^{-6}$ and an accurate input); otherwise the imaginary part is computed directly as $\sum_{k} \arg(x+k)$, first at low precision to determine $m$ and if necessary at full precision. Real $x$ needs no correction for $x > 0$ and gets the imaginary part $\pi \min(n, \lceil -x \rceil)$ for negative non-integer $x$; a real $x$ containing a nonpositive integer gives an indeterminate result.