Approximating the number of integers without large prime factors

denotes the number of positive integers and free of prime factors . Hildebrand and Tenenbaum gave a smooth approximation formula for in the range , where is a fixed positive number . In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate . The computational complexity of this algorithm is . We give numerical results which show that this algorithm provides accurate estimates for and is faster than conventional methods such as algorithms exploiting Dickman's function.