We present a two-dimensional distributional-finite-difference algorithm for modeling the propagation of seismic waves in heterogeneous media in the time domain. We revisit the classic staggered finite-difference algorithm by substituting the standard finite-difference operators with the recently introduced distributional-finite-difference operators with methodological differences we underline. We show that the distributional-finite-difference operators improve simulation accuracy while maintaining the simple structure of the finite-difference algorithm. Thanks to its weak formalism, the newly proposed algorithm accurately and naturally accounts for the free surface, which is a substantial improvement for finite-difference approaches. Moreover, we propose an efficient factorization for the distributional-finite-difference operators. It limits the computational cost of the proposed algorithm to twice that of the finite-difference algorithm. Numerical examples demonstrate that the extra computational burden is well compensated by the superior accuracy of the distributional-finite-difference operators.