We present a new, fast method for computing the inspiral trajectory and
gravitational waves from extreme mass-ratio inspirals that can incorporate all
known (and future) self-force results. Using near-identity (averaging)
transformations we formulate equations of motion that do not explicitly depend
upon the orbital phases of the inspiral, making them fast to evaluate, and
whose solutions track the evolving constants of motion, orbital phases and
waveform phase of a full self-force inspiral to $O(\eta)$, where $\eta$ is the
(small) mass ratio. As a concrete example, we implement these equations for
inspirals of non-spinning (Schwarzschild) binaries. Our code computes inspiral
trajectories in milliseconds which is a speed up of 2-5 orders of magnitude
(depending on the mass-ratio) over previous self-force inspiral models which
take minutes to hours to evaluate. Computing two-year duration waveforms using
our new model we find a mismatch better than $\sim 10^{-4}$ with respect to
waveforms computed using the (slower) full self-force models. The speed of our
new approach is comparable with kludge models but has the added benefit of
easily incorporating self-force results which will, once known, allow the
waveform phase to be tracked to sub-radian accuracy over an inspiral.