followed by the usage of a grid of \(N\times N\) equidistant points on \(u\) and \(w\). A
two‑dimensional spline (either bicubic or univariate) is then used to
interpolate between points on the considered interval. The implementation of Piantadosi’s encoding procedure is then identical
to the previous case, with the only difference that the series of points
are now constructed as \(x \in \left[(0,0),\text{...},(0,N),(1,0),\text{...},(1,N)\right]\). The implementation of the encoding procedure for all
meta-GGA exchange functionals in the LibXC library is presented in a
Jupyter notebook in XXX. Once again, the accuracy of the procedure
depends only on two variables, the number of points used to interpolate the enhancement factor, \(N^{2}\), and the accuracy of the encoder parameter, \(\beta\). The major hurdle in the procedure is that the number of
digits required to represent the parameter is now much higher than for
the previous case. Interpolations with \(N>20\) become computationally expensive since they require > 400 points, and result in
parameters with more than 1500 digits, regardless of the value of \(\beta\). For well-behaved functionals, however, \(N=20\) and \(\beta=8\) result in parameters with ~1000 digits,
and overall errors < 1 %, similarly to the GGA case.