that these interactions will dominate the [*total*]{} error of any body’s acceleration. It is, therfore, expedient to balance the absolute errors of the individual interactions, which can be approximately achieved by using a mass-dependent tolerance parameter as follows $$\label{open:mass}
        {\theta^{p+2}\over(1-\theta)^2} =
        {\theta_{\rm min}^{p+2}\over(1-\theta_{\rm min})^2}
        \left({{{\sf M}^{}}\over{{\sf M}^{}}_{\rm tot}}\right)^{(2-d)/d}$$ where ${{\sf M}^{}}_{\rm tot}$ is the mass of the root cell, while $\theta_{\rm min}=
\theta({{\sf M}^{}}_{\rm tot})$ is the new tolerance parameter. If $r_{\rm A,max}\sim
r_{\rm B,max}$ and ${{\sf M}^{}}_{{\rm A}}\sim{{\sf M}^{}}_{{\rm B}}$, this method results in approximately constant absolute acceleration errors. Note that [equation [(\[open:mass\])]{}]{} results in a very weak decrease of $\theta$ with increasing mass: $\theta\propto{{\sf M}^{-1/15}}$ for $d=3$, $p=3$ and $\theta\ll1$.

Instead of [equation [(\[error:abs\])]{}]{}, one can obtain a stricter error limit incorporating the first $p+1$ multipole moments (see Appendix A), which may be turned into an MAC [@WS]. However, our choice [(\[open:mass\])]{} is (i) much simpler and (ii) overcomes already the main disadvantage of $\theta=\,$const, the variations of the absolute individual errors.

Direct Summation {#sec:sum}
----------------

For small $N$ the exact force computation via direct summation is not only more accurate than approximate methods but also more efficient. Therefore, we replace the approximate technique by direct summation whenever 