Preparing the next gravitational million-body simulations: Evolution of
  single and binary stars in Nbody6++GPU, MOCCA and McLuster
  We present the implementation of updated stellar evolution recipes in the
codes \texttt{Nbody6++GPU, MOCCA} and \texttt{McLuster}. We test them through
numerical simulations of star clusters containing $1.1\times 10^5$ stars (with
$2.0\times 10^4$ in primordial hard binaries) performing high-resolution direct
$N$-body (\texttt{Nbody6++GPU}) and Monte-Carlo (\texttt{MOCCA}) simulations to
an age of 10~Gyr. We compare models implementing either delayed or
core-collapse supernovae mechanisms, a different mass ratio distribution for
binaries, and white dwarf natal kicks enabled/disabled. Compared to
\texttt{Nbody6++GPU}, the \texttt{MOCCA} models appear to be denser, with a
larger scatter in the remnant masses, and a lower binary fraction on average.
The \texttt{MOCCA} models produce more black holes (BHs) and helium white
dwarfs (WDs), whilst \texttt{Nbody6++GPU} models are characterised by a much
larger amount of WD-WD binaries. The remnant kick velocity and escape speed
distributions are similar for the BHs and neutron stars (NSs), and some NSs
formed via electron-capture supernovae, accretion-induced collapse or
merger-induced collapse escape the cluster in all simulations. The escape speed
distributions for the WDs, on the other hand, are very dissimilar. We
categorise the stellar evolution recipes available in \texttt{Nbody6++GPU},
\texttt{MOCCA} and \texttt{Mcluster} into four levels: the one implemented in
previous \texttt{Nbody6++GPU} and \texttt{MOCCA} versions (\texttt{level A}),
state-of-the-art prescriptions (\texttt{level B}), some in a testing phase
(\texttt{level C}), and those that will be added in future versions of our
codes.
