Uniqueness of half-iterate of exp(x) ?

[hixidom]

Or maybe it's just plain wrong. A test using Matlab suggests that the f(x) I provided is not the solution. That's so weird; It seemed airtight to me, but there must be some pathology in my derivation. Hopefully someone with a better understanding of the W function can spot it for me.

Code:

x=(.5:.01:3);
f=@(x)exp(lambertw(x.^2));
for i=1:numel(x);
    ffx(i)=f(f(x(i)));
end
figure(1);
clf();
plot(x,exp(x),x,ffx)