Approximating the number of integers free of large prime factors

Define psi(x, y) to be the number of positive integers n less than or equal to x such that n has no prime divisor larger than y. We present a simple algorithm that approximates psi(x,y) in O(y{log log x/log y + 1/log log y}) floating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than other known approximations, including the well-known approximation psi(x, y) approximate to xp(log x/log y), where p(u) is Dickman's function.