visibility of width $\delta_j$, the number of boxes that are within a circular region within the range $R_j = \sqrt{-\delta_R \log \epsilon}$ is approximately: $$N_B = \frac{\pi}{4}(2 \lceil R_j/L \rceil + 1)^2 .$$

The the order $q \le p$ is chosen to satisfy the error estimate:

$$\frac{r^q_{qj}}{1 - r_{qj}} \le \epsilon$$

where $r_qj = r\sqrt{e \delta/\delta_j(q + 1)}$.

To apply cheating (Section \[sec:cheating\]), we decrement the value of $q$, such that $q' = \rm{max}(q - p_d, 0)$.

Now we consider work required to update the Taylor coefficients in each box (Eq. \[eq:cbeta\]). The key is to compute and store the Hermite functions, and powers of $(\delta/\delta_j)^{|\beta|/2}$ separately, and perform the sum at the end, by taking advantage of the product form of the multi-index notation for multiple dimensions.

The Hermite function is a polynomial multiplied by a complex exponential. The argument for the complex exponential is the same for every order $q'$, so only one calculation is required per set of Hermite function evaluations. The way in which a complex exponential is computed by a CPU is highly implementation-dependant, so we simplify the analysis here and assume that it requires $N_{\rm cexp}$ floating point operations.

We observe that the Hermite 