$$\sin(x)\cos(x) = \frac{\sin(2x)}{2}$$
Plotting the L.H.S. and the R.H.S. of this identity shows that these functions “seem” to be identical, but it is not a proof.
Or is it?
Is there a mathematical theory that can be used to reason, given two one-argument elementary function formulas of size (number of primitive operations) at most s and involving only integer constants of magnitude at most m, that if these functions are always sufficiently close (according to some criteria) then they must be equivalent on the intersection of their domains? There are only finitely many elementary expressions of size at most s and involving constants of magnitude at most m, therefore a function $\Delta(s,m)$ exists with the property that if two elementary formulas of size at most s and involving integer constants of magnitude at most m never differ by more than $\Delta(s,m)$ then they must be identical on the shared domain.
I will readily accept that $\Delta(s,m)$ might be incomputable. But is any nonzero lower bound within reach of computability?
EDIT: for the original example, $m=2$ and $s=4$ (dividing by $2$ is multiplying by $2^{-1}$, hence an extra operation). Is $\Delta(4,2) < 2^{-32}$?
