  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.
