_acb_dirichlet_refine_hardy_z_zero

Below $4 \log_2 t + 40$ bits, the Illinois method is used, with iterates at $\text{prec} + \log_2 t + 8$ bits but $Z$ evaluated at only $\text{prec} + 12$ bits (its value near the zero is needed to about $\text{prec} - \log_2 t$ bits after the binary point), so that the Riemann–Siegel main sum stays within reach of dfloat_powsum_sieved; when an iterate is so close to the zero that the sign of $Z$ is undetermined, the zero is bracketed by the points at a quarter of the tolerance on either side instead of raising the precision. From $\log_2 t + 96$ bits, the zero is first refined to $p_1 = (\text{prec} + \log_2 t)/2 + 16$ bits, where $Z$ is much cheaper, and finished by one secant step through the endpoints of that bracket and the signs of $Z$ at a quarter of the tolerance on either side of the result: four evaluations at the target precision rather than a dozen for the Illinois method alone. Above $4 \log_2 t + 40$ bits, Newton's method is used.