Minimality of the exponential-recovery identity

Determine whether the displayed identity recovering $e^z$ from the activation function $F(z)=e^z+\ln z$ and rational operations is the simplest construction of its kind.

Background

The proposed Möbius-layer architecture uses the complex activation $F(z)=e^z+\ln z=\eml(z,z^{-1})$. A rational identity involving $F(z)$, $F(2z)$, and $F(3z)$ recovers the exponential, after which the logarithm and the original EML operator can also be recovered.

The identity has relatively high reverse-Polish complexity, and the paper explicitly leaves unresolved whether a simpler identity exists. This is a concrete symbolic-complexity problem independent of the broader question of whether the resulting network can be optimized efficiently.