A previously unexplored method, combining logical and mathematical elements, is shown to yield substantial numerical improvements in the area of Diophantine approximations. Kreisel illustrated the method abstractly by noting that effective bounds on the number of elements are ensured if Herbrand terms from ineffective proofs of $\Sigma_2$-finiteness theorems satisfy certain simple growth conditions. Here several efficient growth conditions for the same purpose are presented that are actually satisfied in practice, in particular, by the proofs of Roth's theorem due to Roth himself and to Esnault and Viehweg. The analysis of the former yields an exponential bound of order $\exp(70\varepsilon^{-2}d^2)$ in place of $\exp(285\varepsilon^{-2}d^2$) given by Davenport and Roth in 1955, where $\alpha$ is (real) algebraic of degree $d \geq 2$ and $|\alpha - pq^{-1}| < q^{-2 - \varepsilon}$. (Thus the new bound is less than the fourth root of the old one.) The new bounds extracted from the other proof are polynomial of low degree (in $\varepsilon^{-1}$ and $\log d$). Corollaries: Apart from a new bound for the number of solutions of the corresponding Diophantine equations and inequalities (among them Thue's inequality), $\log \log q_\nu < C_{\alpha, \varepsilon} \nu^{5/6 + \varepsilon}$, where $q_\nu$ are the denominators of the convergents to the continued fraction of $\alpha$.