Bug introduced in 9.0 and persisting through 12.1
I get the following output with a fresh Mathematica (ver 10.0.2.0 on Mac) session
FullSimplify[Exp[-100*(i-0.5)^2]] (* 0. *) Simplify[Exp[-100*(i-0.5)^2]] (* E^(-100. (-0.5+i)^2) *)
FullSimplify seems to be a bit overambitious and kills the expression completely. Is there anything that explains this behavior or is this simply a bug?
As suggested in the comments, I did an additional test:
FullSimplify[Exp[-100*(i-1/2)^2]] (* E^(-25 (1-2 i)^2) *)
Apparently, the float point math causes the problem.