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A hyperbolic singularity in the wave function of s-wave interacting atoms is the root problem for any accurate numerical simulation. Here, we apply the transcorrelated method, whereby the wave-function singularity is explicitly described by a two-body Jastrow factor, and then folded into the Hamiltonian via a similarity transformation. The resulting nonsingular eigenfunctions are approximated by stochastic Fock-space diagonalization with energy errors scaling with 1/M in the number M of single-particle basis functions. The performance of the transcorrelated method is demonstrated on the example of strongly correlated fermions with unitary interactions. The current method provides the most accurate ground-state energies so far for three and four fermions in a rectangular box with periodic boundary conditions.