  Direct $N$-body simulations of star clusters are accurate but expensive,
largely due to the numerous $\mathcal{O} (N^2)$ pairwise force calculations. To
solve the post-million-body problem, it will be necessary to use approximate
force solvers, such as tree codes. In this work, we adapt a tree-based,
optimized Fast Multipole Method (FMM) to the collisional $N$-body problem. The
use of a rotation-accelerated translation operator and an error-controlled cell
opening criterion leads to a code that can be tuned to arbitrary accuracy. We
demonstrate that our code, Taichi, can be as accurate as direct summation when
$N> 10^4$. This opens up the possibility of performing large-$N$, star-by-star
simulations of massive stellar clusters, and would permit large parameter space
studies that would require years with the current generation of direct
summation codes. Using a series of tests and idealized models, we show that
Taichi can accurately model collisional effects, such as dynamical friction and
the core-collapse time of idealized clusters, producing results in strong
agreement with benchmarks from other collisional codes such as NBODY6++GPU or
PeTar. Parallelized using OpenMP and AVX, Taichi is demonstrated to be more
efficient than other CPU-based direct $N$-body codes for simulating large
systems. With future improvements to the handling of close encounters and
binary evolution, we clearly demonstrate the potential of an optimized FMM for
the modeling of collisional stellar systems, opening the door to accurate
simulations of massive globular clusters, super star clusters, and even
galactic nuclei.
