Relative differences of binned PolSpice covariance matrices calculated using approximations of the pseudo-spectrum covariance with respect to the exact computation: , for TTTT (left-hand side) and EEEE (right-hand side), with binning Δℓ = 50. On the first row we plot the relative differences for the diagonal, i.e. b = b′, while on the second row we plot the differences for the first off-diagonal, i.e. b′ = b + 1. Similarly to the case of pseudo-covariances in Fig. 6, we find acceptable accuracy for the NKA (dash light blue), INKA (dash-dot dark blue) and FRI (dash-double-dot purple) approximations, while our ACC approximation (solid red) improves overall the others. The PolSpice covariance matrices have been calculated using Eqs. (55) and (56). The last row displays the corresponding binned underlying renormalized spectrum TT or EE, to highlight the fact that the differences in the covariances are on the peaks and in the troughs of the spectra, i.e. where the curvature of the spectrum is maximal.