On the other hand, I’m not sure if modern compilers even bother converting division/multiplications by powers of two to shifts these days, so maybe it isn’t worth it…
This comes up all of the time with stuff like lat/lng values. People are convinced you have to use doubles because of the inaccuracy of floats. Completely skipping over the fact that you could have fixed point accuracy to 6 decimal digits with the same number of bits as a float. You just reduce your max/min value that you can represent. Which, for lat/lng, you are already heavily bounded.
I think it is fair to argue that basic libraries don't support trig on common fixed point sizes. But that is ultimately my lament. Floats are amazing for what they need to do. For what many people need, though, it feels overkill because it is overkill.
Using any floating point representation for lat/long makes no sense at all to me. Floating point is designed for representing values where expected error scales with magnitude, not for values where error is constant with magnitude; the latter requires fixed point. I truly don't understand why someone would optimize their position representation for maximum accuracy off the coast of Africa, instead of having it uniform across the world.
Again, I do not mean this as a criticism of floats. For simulations and for numbers where you do have to support completely arbitrary values, there is a reason floats are a thing.
So fixed-point addition and subtraction are definitely faster, multiplication is a wash if you're doing binary-based fixed point (but slower if you're doing decimal-based fixed point), and fixed-point division is definitely slower than floating-point division.
My gut would still be that it is typically a wash for most everyone as far as speeds go? If default libraries supported it more directly, I would think it would largely be a win for a lot of reasoning. In particular, silly stuff like 1e32 + 1e1 would not be nearly as surprising to most people. And the entire class of bugs around stuff like doing something until it reaches 0.9 would almost certainly go away if we guaranteed precision to a set number decimal places.
Alas, default libraries do not support this, though. So the above is admittedly wishful thinking on my end. And I could as easily describe a world where people insist on arbitrary numbers of fixed point values and how that would be its own set of landmines.