LetXandYbe two real-valued random variables. Let (X1,Y1), (X2,Y2),... be independent identically distributed copies of (X,Y). Suppose there are two players A and B. Player A has access toX1,X2,... and player B has access toY1,Y2,.... Without communication, what joint probability distributions can players A and B jointly simulate? That is, ifk,mare fixed positive integers, what probability distributions on {1,...,m}2are equal to the distribution of (f(X1,...,Xk),g(Y1,...,Yk)) for somef,g: Rk→ {1,...,m}? WhenXandYare standard Gaussians with fixed correlation ρ ∈ (-1, 1), we show that the set of probability distributions that can be noninteractively simulated fromkGaussian samples is the same for anyk≥m2. Previously, it was not even known if this number of samplesm2would be finite or not, except whenm≤ 2. Consequently, a straightforward brute-force search deciding whether or not a probability distribution on {1,...,m}2is within distance 0kcorrelated Gaussian samples has run time bounded by (5/ε)m(log(ε/2)/ log |ρ|)m2, improving a bound of Ghazi, Kamath and Raghavendra. A nonlinear central limit theorem (i.e. invariance principle) of Mossel then generalizes this result to decide whether or not a probability distribution on {1,...,m}2is within distance 0ksamples of a given finite discrete distribution (X,Y) in run time that does not depend onk, with constants that again improve a bound of Ghazi, Kamath and Raghavendra.