  We solve the Hamiltonian and momentum constraints of general relativity for
two black holes with nearly extremal spins and relativistic boosts in the
puncture formalism. We use a non-conformally-flat ansatz with an attenuated
superposition of two Lorentz-boosted, conformally Kerr or conformally
Schwarzschild 3-metrics and their corresponding extrinsic curvatures. We
compare evolutions of these data with the standard Bowen-York conformally flat
ansatz (technically limited to intrinsic spins $\chi=S/M^2_{\text{ADM}}=0.928$
and boosts $P/M_{\text{ADM}}=0.897$), finding, typically, an order of magnitude
smaller burst of spurious radiation and agreement with inspiral and merger. As
a first case study, we evolve two equal-mass black holes from rest with an
initial separation of $d=12M$ and spins $\chi_i=S_i/m_i^2=0.99$, compute the
waveforms produced by the collision, the energy and angular momentum radiated,
and the recoil of the final remnant black hole. We find that the black-hole
trajectories curve at close separations, leading to the radiation of angular
momentum. We also study orbiting nonspinning and moderate-spin black-hole
binaries and compare these with standard Bowen-York data. We find a substantial
reduction in the nonphysical initial burst of radiation which leads to cleaner
waveforms. Finally, we study the case of orbiting binary black-hole systems
with spin magnitude $\chi_i=0.95$ in an aligned configuration and compare
waveform and final remnant results with those of the SXS Collaboration, finding
excellent agreement. This represents the first moving punctures evolution of
orbiting and spinning black holes exceeding the Bowen-York limit. Finally, we
study different choices of the initial lapse and lapse evolution equation in
the moving punctures approach to improve the accuracy and efficiency of the
simulations.
