I'm working on a mobile game and I'd like to avoid using floating point math. I'm having trouble coming up with an algorithm for moving a point a specific distance toward another point. I have no issues doing this with floating point using trig functions or vector math.
Does anyone recall how this was done in those long-gone days before we had FPUs to lean on?
Before there were FPUs there was software that performed floating point operations.
Hardware FPUs are pretty efficient these days. They are pipelined and should perform close to fixed point math. The only downside, that maybe you're alluding to, is that these circuits generally consume a good chunk of power.
For your case, the easiest thing that comes to mind is to divide the distance into integer chunks between the two points. For example, if the distance is 100 you could scale it to 100, or 1000, or 10000 depending on your desired resolution. Then have, for example, 10,000 steps to indicate where along the path your object is. But this sounds like it's way too complicated.
For reference, how the Commodore 64 did it with software routines: