Complexity of Approximate Realizations of Lipschitz Functions by Schemes in Continuous Bases

We show that any function satisfying the Lipschitz condition on a given closed interval can be approximately computed by a scheme (nonbranching program) in the basis composed of functions
$$ x-y,\quad |x|,\quad x*y=\min(\max(x,0),1)\min(\max(y,0),1), $$
and all constants from the closed interval $[0,1]$; here the complexity of the scheme is $O(1/\sqrt{\varepsilon})$, where $\varepsilon$ is the accuracy of the approximation. This estimate of complexity, is in general, order-sharp.