Effect of employing approximate SHTs in the preconditioner. To simulate approximate SHTs, for every SHT computer, we have multiplied each coefficient of the result with a random variate drawn uniformly from the interval [1 − ϵ, 1 + ϵ], simulating numerical noise at the ϵ-level. Note that we carried out these simulations on a resolution of N_side = 128, using the downgrade procedure outlined in Appendix C. The experiment is otherwise identical to the one in Fig. 9. The approximate SHTs are employed both in the pseudo-inverse preconditioner and the mask-restricted multi-grid solver, while the (non-preconditioner) matrix-vector multiplication uses the full accuracy provided by Libsharp.