  Excursion set theory is a powerful and widely used tool for describing the
distribution of dark matter haloes, but it is normally applied with simplifying
approximations. We use numerical sampling methods to study the mass functions
predicted by the theory without approximations. With a spherical top-hat window
and a constant $\delta=1.5$ threshold, the theory accurately predicts mass
functions with the $M_{200}$ mass definition, both unconditional and
conditional, in simulations of a range of matter-dominated cosmologies. For
$\Lambda$CDM at the present epoch, predictions lie between the
$M_\mathrm{200m}$ and $M_\mathrm{200c}$ mass functions. In contrast, with the
same window function, a nonconstant threshold based on ellipsoidal collapse
predicts uniformly too few haloes. This work indicates a new way to simply and
accurately evaluate halo mass functions, clustering bias, and assembly
histories for a range of cosmologies. We provide a simple fitting function that
accurately represents the predictions of the theory for a wide range of
parameters.
