  The equational complexity function
$\beta_\mathscr{V}:\mathbb{N}\to\mathbb{N}$ of an equational class of algebras
$\mathscr{V}$ bounds the size of equation required to determine membership of
$n$-element algebras in $\mathscr{V}$. Known examples of finitely generated
varieties $\mathscr{V}$ with unbounded equational complexity have growth in
$\Omega(n^c)$, usually for $c\geq \frac{1}{2}$. We show that much slower growth
is possible, exhibiting $O(\log_2^3(n))$ growth amongst varieties of
semilattice ordered inverse semigroups and additive idempotent semirings. We
also examine a quasivariety analogue of equational complexity, and show that a
finite group has polylogarithmic quasi-equational complexity function, bounded
if and only if all Sylow subgroups are abelian.
