  We present a new package ZpL for the mathematical software system SM. It
implements a sharp tracking of precision on p-adic numbers, following the
theory of ultrametric precision introduced in [4]. The underlying algorithms
are mostly based on automatic dierentiation techniques. We introduce them,
study their complexity and discuss our design choices. We illustrate the
bene-ts of our package (in comparison with previous implementations) with a
large sample of examples coming from linear algebra, com-mutative algebra and
dierential equations.
