Let f(x) = fake(semi-exp(x))  to simplify notation.

This graph is f(f(z))/exp(z)-1, showing the very sharp switch from the region of good convergence, for positive reals, where f(f(z))/exp(z)-1 is very nearly zero, which is black. Here, I used the best approximation I have so far, which turns out to match the exponential function by a little better than 99% in the region of interest where the image is black. The boundary is pretty sharp. The conjecture is that there is a well defined boundary is exactly where f(z) takes on negative values, with imag(f(z))=0. At that boundary, the conjecture is that |f(f(z))/exp(z)-1|~=1. The 99.2% accuracy in this region may be as good as f(f(z))/exp(z) can be. With a 300 term Taylor series, accuracy peaks at 99.95% at z=2800, beyond which I would need to generate more Taylor series terms for convergence. 

graph from -20 real to +20 real, and -5 imag to +20 imag with grid lines every 5 units. 

The next graph is f(z), the asymptotic half iterate itself, using the same grid coordinates. You can see zeros for f on the real axis, as black dots, at -0.71, -4.26, and -15.21. The pattern goes on forever, as f grows at the negative real axis, oscillating between positive and negative. 

Notice the sharp blue line. That is the line of negative reals.  By taking f( blue line ) we more or less map that blue line to the negative real axis. But that is not consistant with exp(x) = f(f(x)) since exp(x) > 0 for all real x.

So at the blue line , the f(x) " stops making sense ". But the area sufficiently far away from the blue line and between the blue line and the positive real axis makes a very good half-iterate of exp(x) !  Well if we ignore the invariant of exp(z) ; exp(z) = exp(z + 2 pi i) that is.  But within that strip it works very well.

It is as if the function gets stretched to exp(x) by applying f(x) twice.

Take a look at the first picture again perhaps to get a feeling of what is happening.

The next graph is f(f(z)), same coordinates. You can clearly see the parts of the complex plane where f(f(z)) mimics the exponential, and the parts of the complex plane where it doesn't mimic the exponential function, which explains the first graph. And you can see the zeros, of f(f(z)), which are black dots. I'm pretty sure all the zeros of f(z) are on the negative real axis, and exp(z) has no zeros, so if you connect the black dots not on the real axis, this is where f(f(z))/exp(z) has to start to diverge from its asymptotic value of 1, and this is where f(z) is at the negative real axis. 

These pictures are not just pretty but give you some idea what fake function theory and tetration is all about.

It is also logical to see now why we started considering the zero's of these fake functions. There determine the overall structure and are thus cruxial ! 

related things ...