Are there any numerically stable methods for computing the condition number of a near-singular matrix that are fast?

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I know of a number of techniques for computing the condition number that are either slow (doing a whole singular value decomposition) or fast (ratio of norm of matrix and its inverse obtained via LU decomposition) but maybe unstable. I just want to determine whether a matrix is non-singular or not (i.e. whether its condition number is below, say, 1e-8) in an efficient way. Does anyone have anything faster than SVD? Thanks. I'm looking to deploy said method with LAPACK and cuSOLVER.