How do I make this equation more round?

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This is my equation: $e^{-0.5*{2^y} - 0.5 *{ 2^x}}$

My goal is to create a 3D semi-circle that starts off at a corner with a z value of 1, then as it goes to the opposing corners it decays to a z of zero. My equation as it stands now makes a semi box with a rounded edge.

I'm not the most eloquent person, so if this doesn't make sense, I can clarify.

Edit Going off of the great help from Théophile. Looking at the bottom picture of the figure, how do I increase the size of the innermost circle, without changing any of the decay properties?Wolframalpha Figure