  As neutrino oscillation data becomes ever more precise, the use of
approximate formulae for the oscillation probabilities ${\mathcal
P}_{\alpha\beta}$ must be examined to ensure that the approximation is
adequate. Here, the oscillation probability ${\mathcal P}_{ee}$ is investigated
in the context of the Daya Bay experiment; the oscillation probability
${\mathcal P}_{\mu\mu}$ is investigated in terms of the T2K disappearance
experiment; and the probability ${\mathcal P}_{\mu e}$ is investigated in terms
of the T2K appearance experiment. Daya Bay requires ${\mathcal P}_{ee}$ in
vacuum and thus the simple analytic formula negates the need for an approximate
formula. However, improved data from T2K will soon become sensitive to the
hierarchy, and thus require a more careful treatment of that aspect. For the
other cases, we choose an expansion by Akhmedov et al. which systematically
includes all terms through second order in $\sin\theta_{13}$ and in $\alpha =:
\Delta_{21}/\Delta_{31}$ ($\Delta_{jk} =: m^2_j - m^2_k$). For the T2K
disappearance experiment the approximation is quite accurate. However, for the
T2K appearance experiment the approximate formula is not precise enough for
addressing such questions as hierarchy or the existence of CP violation in the
lepton sector. We suggest the use of numerical calculations of the oscillation
probabilities, which are stable, accurate, and efficient and eliminate the
possibility that differences in analyses are emanating from different
approximations.
