AbstractClassicalapproximation results show that anycircuit of sizeand depthhas an‐error probabilistic polynomial over the reals of degree. We improve this upper bound to, which is much better for small values of. We then use this result to show that‐wise independence foolscircuits of sizeand depthup to error at most, improving on Tal's strengthening of Braverman's result that‐wise independence suffices. To our knowledge, this is the first PRG construction forthat achieves optimal dependence on the error. We also prove lower bounds on the best polynomial approximations to. We show that any polynomial approximating thefunction onbits to a small constant error must have degree at least. This result improves exponentially on a result of Meka, Nguyen, and Vu (Theory Comput. 2016).