  We compute the one-loop corrections to \tth up to order
$\mathcal{O}(\epsilon^2)$ in the dimensional regularization parameter. We apply
the projector method to compute polarized amplitudes, which generalize massless
helicity amplitudes to the massive case. We employ a semi-numerical strategy to
evaluate the scattering amplitudes. We express the form factors through scalar
integrals analytically, and obtain separately integration by parts reduction
identities in compact form. We integrate numerically the corresponding master
integrals with an enhanced implementation of the Auxiliary Mass Flow algorithm.
Using a numerical fit method, we concatenate the analytic and the numeric
results, to obtain fast and reliable evaluation of the scattering amplitude.
This approach improves numerical stability and evaluation time. Our results are
implemented in the \texttt{Mathematica} package \texttt{TTH}.
