  We present various improvements to the deformation method for computing the
zeta function of smooth projective hypersurfaces over finite fields using
$p$-adic cohomology. This includes new bounds for the $p$-adic and $t$-adic
precisions required to obtain provably correct results and gains in the
efficiency of the individual steps of the method. The algorithm that we thus
obtain has lower time and space complexities than existing methods. Moreover,
our implementation is more practical and can be applied more generally, which
we illustrate with examples of quintic curves and quartic surfaces.
