Motivated by tremendous progress in neutrino oscillation experiments, we derive a new set of simple and compact formulas for three-flavor neutrino oscillation probabilities in matter of a constant density. A useful definition of the $eta$-gauge neutrino mass-squared difference $Delta^{}_* equiv eta Delta^{}_{31} + (1-eta) Delta^{}_{32}$ is introduced, where $Delta^{}_{ji} equiv m^2_j - m^2_i$ for $ji = 21, 31, 32$ are the ordinary neutrino mass-squared differences and $0 leq eta leq 1$ is a real and positive parameter. Expanding neutrino oscillation probabilities in terms of $alpha equiv Delta^{}_{21}/Delta^{}_*$, we demonstrate that the analytical formulas can be remarkably simplified for $eta = cos^2 theta^{}_{12}$, with $theta_{12}^{}$ being the solar mixing angle. As a by-product, the mapping from neutrino oscillation parameters in vacuum to their counterparts in matter is obtained at the order of ${cal O}(alpha^2)$. Finally, we show that our approximate formulas are not only valid for an arbitrary neutrino energy and any baseline length, but also still maintaining a high level of accuracy.
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