Going back to basics: accelerating exoplanet transit modelling using Taylor-series expansion of the orbital motion

A significant fraction of an exoplanet transit model evaluation time is spent calculating projected distances between the planet and its host star. This is a relatively fast operation for a circular orbit, but slower for an eccentric one. However, because the planet's position and its time derivatives are constant for any specific point in orbital phase, the projected distance can be calculated rapidly and accurately in the vicinity of the transit by expanding the planet's $x$ and $y$ positions in the sky plane into a Taylor series at mid-transit. Calculating the projected distance for an elliptical orbit using the four first time derivatives of the position vector (velocity, acceleration, jerk, and snap) is $\sim100$ times faster than calculating it using the Newton's method, and also significantly faster than calculating $z$ for a circular orbit because the approach does not use numerically expensive trigonometric functions. The speed gain in the projected distance calculation leads to 2-25 times faster transit model evaluation speed, depending on the transit model complexity and orbital eccentricity. Calculation of the four position derivatives using numerical differentiation takes $\sim1\,\mu$s with a modern laptop and needs to be done only once for a given orbit, and the maximum error the approximation introduces to a transit light curve is below 1~ppm for the major part of the physically plausible orbital parameter space.