Power and cross power spectra at redshift z = 0 (averaging in Δk-bins for fields A and B: ) for the various approximate gravity solvers compared to fast particle-mesh based N-body calculations (FastPM). The Zel’dovich, 2LPT, and ALPT cases require only a one-step calculation. For the rest, the number of steps is indicated in parentheses. FastPM needs the first step for the ICs relying on 2LPT and then a minimal number of iterations compared to standard N-body simulations, which require iterations on the order of 10³. Also shown are the 1-sigma contours from 25 fields for FastPM and the eALPT-limiting cases with the lowest and highest cross-correlations, two (w/o vc) and six (with vc) steps, respectively. The eALPT (3s) and the eALPTvc (6s) cases yield power spectra with precision within a few per cent up to black k ≃ 0.3 and 4 h Mpc⁻¹(the Nyquist frequency) as well as correlations of black 95% and 98% at k ≃ 1 h Mpc⁻¹, respectively.