An explanation for this?

[sheldonison]

Sorry, I should've been more specific. I meant to ask for a formula independent of tetration; I'm assuming there's a power series of some kind defining 0<= q<=1\( \exp^{[q]}(x) \) Something that would reproduce this graph, since the linear model of tetration I've been using doesn't match up. Even a generalized power series for tetration would work, now that I think of it.

For base e exponent, as far as I know, the only way to generate the half iterate (or any other partial iterate) is indirectly, via \( \text{sexp}(\text{slog}(z)+0.5) \). I'm unaware of any other way to generate the power series.
- Shel