keendog, the cosine of the arcsine can be simplified.
When you do so, the formula d = R-(Rcos(asin(L/R)))
becomes d = R - sqrt(R^2-L^2) (using L instead of
your l to avoid confusion with the number 1).
Your number d is not the distance along a radial
drop (the way gravity pulls us). It is the distance
to the Earth along the line that is perpendicular to
the horizon.
In principle there is a value of the distance to Earth
whatever L may be. But the formula for d cannot be
evaluated for L > R because the arcsine makes no
sense for such values of L/R. The geometric explanation is
that the drop along the perpendicular line misses the Earth
when L > R.
For small values of L the difference between
your d and Fat Dog's exact radial drop distance,
sqrt(R^2+L^2)-R, is not too great. For example, if
L = 60 nm then the inaccuracy shows up only in the fourth
decimal place. However, at 600 nm the error is about 1.5%.