A new approach to fast evaluation of the electron intracule and extracule densities on large grids of points is described. Substantial (50- to 100-fold) speed ups over the conventional algorithms are attained through the use of precomputed intermediates in the grid-dependent phase of calculations. These intermediates are evaluated only once in a grid-invariant procedure that employs efficient two-stage integral screening to reduce computational effort. In addition to delivering high performance, the new algorithm facilitates calculations of analytical gradients and Hessians of the intracule and extracule electron densities. For regular grids with shared components of Cartesian coordinates, the present method allows the factorization of the primitive quartet contributions that makes the cost of calculations proportional to the cubic root of the number of grid points.