  We give an interim report on some improvements and generalizations of the
Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample
hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in
linear time in $p$. These are illustrated with a number of examples including
K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is
a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor
on moduli space; this verifies that the coplanar divisor is not a
Noether-Lefschetz divisor in the sense of Hassett.
