Approximating the approximant: A numerical code for polynomial compression of discrete integral operators. The action of various one-dimensional integral operators, discretized by a suitable quadrature method, can be compressed and accelerated by means of Chebyshev series approximation. Our approach has a different conception with respect to other well-known fast methods: its effectiveness rests on the “smoothing effect” of integration, and it works in linear as well as nonlinear instances, with both smooth and nonsmooth kernels. We describe a Matlab toolbox which implements Chebyshev-like compression of discrete integral operators, and we present several numerical tests, where the basic (O(n^2)) complexity is shown to be reduced to (O(mn)), with (mll n).