Why this exists. Every likelihood in sfa evaluates the composed error's density and none needs its distribution function, which is why there was not one. Copula models need it: a Gaussian copula on the composed error contains \(\Phi^{-1}(F(\varepsilon))\), so an \(F\) that saturates at 0 or 1 does not merely lose precision, it returns an infinity and takes the copula density with it.
How it is computed. Conditioning on \(u\) leaves a normal distribution function in closed form, so
$$F(q) = E_u\left[\Phi\left((q + su)/\sigma_v\right)\right],$$
with \(s = +1\) for a production frontier and \(-1\) for a cost frontier. The noise is therefore never drawn or integrated, and the tails inherit the accuracy of pnorm, which is reliable twenty standard deviations out. The expectation over \(u\) is taken by Gauss-Legendre quadrature on \(u = \sigma_u t/(1-t)\), and accumulated by log-sum-exp so that terms which would underflow individually still contribute.
The upper tail. Requesting lower.tail = FALSE negates the argument of \(\Phi\) rather than forming \(1 - F\). At \(q = 30\) with \(\sigma_u = \sigma_v = 1\) the complement is exactly 1 in double precision while the direct computation returns a finite log-probability below \(-10^{2}\).
Accuracy. There is no closed form except at \(\lambda = \sigma_u/\sigma_v\) equal to 0 (the normal) or 1, where Amsler, Schmidt and Tsay show \(P(Q) = \Phi(Q/(\sqrt{2}\sigma_u))^2\) for the cost case. Against that standard the default quadrature is accurate to about \(10^{-13}\) relative at probabilities as small as \(10^{-115}\). It is deliberately used everywhere rather than special-cased at \(\lambda = 1\): the production form of that identity is a complement, which would reintroduce the saturation the function exists to avoid.