limited only by currently used precision. Search algorithm could then terminate and switch to some high-precision or symbolic algebra verification method. Unfortunately, equivalence problem is undecidable in general [@richardson]. One cannot exclude, that both numbers deviate at some far more distant decimal place. In the case of (ii) i.e. truly transcendental number, however, sequence will converge indefinitely, by generation of approximations, with progressively more complicated formulas. Realistic convergence examples, computed using extended precision (`long double` in C) to prevent round-off errors, are presented in Figure \[SmokingGun\]. The example target $z=\sqrt{2}^{\sqrt{3}}$ from introduction was selected. Number is obviously in $\exp-\log$ class. But is not *explicite* listed in any of , so it must be entered using appropriate sequence of buttons. For RPN calculator it is obvioulsy 2, SQRT, 3, SQRT, POW with $K=5$. In Fig. \[SmokingGun\] is is visible at $K=5.28$, where decimal part show inner loop progress.

![\[SmokingGun\] Typical behavior of absolute error $\epsilon$ for subsequence of progressively better approximations in terms of $\exp-\log$ formulas generated with ”calculators” , as a function of Kolmogorov complexity $K$. For we have used $x=2$. For (blue) and (black) sudden error drop-off to machine epsilon (zero) is observed for small $K$, marking possible ”exact” 