Define a sigmoid as any bounded, odd, increasing function from $\mathbb{R} \rightarrow \mathbb{R}$, and a pretty sigmoid as a sigmoid which is convex over $\mathbb{R^-}$ and concave over $\mathbb{R^+}.$

Let $P$ be the smallest set of functions from $\mathbb{R} \rightarrow \mathbb{R}$ containing polynomials and closed under exponentiation and product.

If there a function $f$ in $P$ such that $\lim_{x\rightarrow -\infty} f(x) = -\infty$, $\lim_{x\rightarrow\infty} f(x) = 0$ and $e^f - \frac{1}{2}$ is a (pretty) sigmoid.