Errors of the M2L kernel. Each point in the left panels represents a single cell → cell interaction with expansion order p as indicated. The maximum of the relative (bottom) and absolute force error (top; normalised to the mean force ) within the sink cell is plotted against the opening angle θ. The curves in the top panel correspond to the relation (10), while the dashed lines are the power laws . The right panels show the distributions of the ratio of the maximum absolute (top) and relative (bottom) force error to either the simple estimates (11) (dashed, σ on the left) or the improved estimates (14) (solid, σ on the right). The reported probabilities for error underestimation refer to the improved estimates. Vertical scale is linear in the number of interactions.