Transformation better suited for very small numbers

Question

displacement = transform(output)

Aim is to get the transformation function and suitable output.

So we have displacement range from $-0.05$ to $+0.05$. displacement can be as small as $10^{-12}$.

Properties of output

1. Small changes in displacement of the order of 10^-12 should have decent change in output

expected

displacement output
0.05          100.000
.
.
10^-12
0              0.000
-10^-12
.
.
.
-0.05         -100.000