Computing horizon height at far-plane for current camera?

Anyone know a smart snippet to do this?

On the CPU/scripting side (not GPU/shading side), we know a camera's far-plane, near-plane, FoV and transform and matrices. We also know it only ever rotates around the Y (look left or right) or X axes (look up or down), but never around Z axis (tilting).

With this info, I want to compute whether the "horizon" (world ground level y=0) is currently inside the cam's far-plane and if so, at which camera-space coordinate Y[-1..1] that horizon will show. Since the camera never tilts, the horizon will be (roughly) a straight horizontal line, at least that approximation is good enough for this here purpose.