For large numbers $|p|\gg1$ we want to compute $x=1-e^p$.
Is the problem for $|p|\gg 1$ well conditioned.
We have $$\kappa = \frac{\|x'\|}{\|x\|}\cdot \|p\|=\frac{\|-e^p\|}{\|1-e^p\|}\cdot \|p\|$$
If $p>0$ we have $$\kappa=\frac{e^p}{-1+e^p} \cdot p=\frac{pe^p}{e^p-1}\approx p$$ so the problem is ill conditioned.
If $p<0$ $$\kappa=\frac{e^p}{1-e^p}\cdot -p=\frac{-p e^p}{1-e^p}\approx -p e^p \to 0 ~(p\to-\infty),$$ so the problem is well conditioned. I am not sure if my work is correct.