  Many relevant applications in gravitational wave physics share a significant
common problem: the seven-dimensional parameter space of gravitational
waveforms from precessing compact binary inspirals and coalescences is large
enough to prohibit covering the space of waveforms with sufficient density. We
find that by using the reduced basis method together with a parametrization of
waveforms based on their phase and precession, we can construct ultra-compact
yet high-accuracy representations of this large space. As a demonstration, we
show that less than $100$ judiciously chosen precessing inspiral waveforms are
needed for $200$ cycles, mass ratios from $1$ to $10$ and spin magnitudes $\le
0.9$. In fact, using only the first $10$ reduced basis waveforms yields a
maximum mismatch of $0.016$ over the whole range of considered parameters. We
test whether the parameters selected from the inspiral regime result in an
accurate reduced basis when including merger and ringdown; we find that this is
indeed the case in the context of a non-precessing effective-one-body model.
This evidence suggests that as few as $\sim 100$ numerical simulations of
binary black hole coalescences may accurately represent the seven-dimensional
parameter space of precession waveforms for the considered ranges.
