a_{\rm direct},$$ where $a$ denotes the magnitude of the acceleration. [Figure \[fig:error\]]{} plots the mean relative error, $\overline\varepsilon$, and that at the 99th percentile, $\varepsilon_{99\%}$, versus the CPU time needed by an ordinary PC (Pentium III/933MHz/Linux/compiler: gcc version 2.95.2) for both constant and mass-dependent tolerance parameter. This figure allows several interesting observations.

1.  For the same $\theta=\,$const and stellar system, the errors decrease with increasing $N$. This is because, at constant [*relative*]{} force error per individual interaction (as is the case for $\theta=\,$const), the total error of some body’s force scales with the inverse square root of the number of individual interactions contributing, which increases with $N$.

2.  At the same operational effort, as measured by the CPU time, the mass-dependent tolerance parameter employing [equation [(\[open:mass\])]{}]{} results in smaller errors than $\theta=\,$const, for $N>10^4$. This advantage becomes more pronounced for larger $N$, because, with $\theta=\,$const, the [ *absolute*]{} force errors of individual interactions contributing to some body’s force vary stronger with increasing $N$ (due to the larger range of cell masses), such that balancing them becomes more beneficial.

3.  Finally, a relative error of $\varepsilon_{99\%}$ of a few per cent or $\overline\varepsilon$ of a few $10^{-3}$ at $N=10^5$, which is 