Before revision:  When applying this algorithm to u \cdot F, where u is uniformly random in the product of unitary groups, the algorithm performs }  (n, \delta) \cdot L (F) \cdot \Gamma(F) \log \Gamma(F) + \log \log \epsilon^{-1 } operations on average. Here n is the number of variables, \delta the maximum degree, L(F) denotes the evaluation cost of F, and \Gamma(F) reflects an aspect of the numerical condition of F. Moreover, we prove that for inputs given by random Gaussian algebraic branching programs of size (n,\delta), the algorithm runs on average in time polynomial in n and \delta.
After revision:  When applying this algorithm to u \cdot F, where u is uniformly random in the product of unitary groups, the algorithm performs } (n, \delta) \cdot L (F) \cdot \Gamma(F) \log \Gamma(F) + \log \log \epsilon^{-1 } (n,\delta), the algorithm runs on average in time polynomial in n and \delta.