Often one needs to express some quantity of interest in a scale other than its original one.
One can use the exponential function to map $(-\infty,0)\to(0,1)$ and $(0,+\infty)\to(1,+\infty)$, but the first mapping is way more "brutal" than the second, with which I mean that $\exp$ compresses much more strongly $(-\infty,0)$ into $(0,1)$ than compresses $(0,+\infty)$ into $(1,+\infty)$. In that sense, $\exp$ is very unbalanced above and below 0.
Do any of you know a function that, likewise $\exp$, maps $(-\infty,+\infty)\to(0,+\infty)$ but a more balancedly for values above and below 0? I mean, that compresses negative values in the domain between 0 and some value less strongly and that expands positive values in the domain also less strongly?
It may be a dumb guess, but all I can think of is a sub-exponential function $$h(x)=\exp\left(sign(x)\cdot|x|^p\right)$$ for some $p\in(0,1)$, but it still compresses too strongly for $x\in(-\infty,0)$.
Can any of you think of something more effective or at least a little less clumsy?
Sorry by the informal approach to the question, but I still cannot formulate it in a proper way, and many thanks in advance.