Theorem

Given the pdf $f$, the compressor $G:\mathbb{R}\to(0,1)$ which minimizes $$D(Q_{G,N})\approx \frac{B^{2}}{12N^{2}}\int\limits _{-\infty}^{\infty} \frac{f(x)}{G'(x)^{2}} \, dx$$is determined by $$G'(x)=\frac{f(x)^{1/3}}{\int\limits _{-\infty}^{\infty}f(y)^{1/3} \, dy }$$For the (asymptotically) optimal companding scheme using this $G$ $$D(Q_{G,N})\approx \frac{1}{12N^{2}}\left( \int\limits _{-\infty}^{\infty} f(x)^{1/3} \, dx \right)^{3}$$

Intuition

This can essentially be thought of as the optimal N-level quantizer.