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We present analytical and numerical progress on black-hole binary spinprecession at second post-Newtonian order using multi-timescale methods. Inaddition to the commonly used effective spin which acts as a constant ofmotion, we exploit the weighted spin difference and show that suchreparametrization cures the coordinate singularity that affected the previousformulation for the case of equal-mass binaries. The dynamics on the precessiontimescale is written down in closed form in both coprecessing and inertialframes. Radiation reaction can then be introduced in a quasi-adiabatic fashionsuch that, at least for binaries on quasi-circular orbits, gravitationalinspirals reduce to solving a single ordinary differential equation. We providea broad review of the resulting phenomenology and rewrite the relevant physicsin terms of the newly adopted parametrization. This includes the spin-orbitresonances, the up-down instability, spin propagation at past time infinity,and new precession estimators to be used in gravitational-wave astronomy. Ourfindings are implemented in version 2 of the public Python module PRECESSION.Performing a precession-averaged post-Newtonian evolution from/to arbitrarilylarge separation takes $\lesssim 0.1$ s on a single off-the-shelf processor - a50x speedup compared to our previous implementation. This allows for a widevariety of applications including propagating gravitational-wave posteriorsamples as well as population-synthesis predictions of astrophysical nature.

Efficient multi-timescale dynamics of precessing black-hole binaries