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.