I'm trying to figure out a formula for tetration that will work for non-integer heights.
I know the usual recurrence relation for tetration ($x \in \mathbb{R}, \text{ }n \in \mathbb{N})$:
$${^{n}x} = \begin{cases} 1 &\text{if }n=0 \\ \\ x^{\left(^{(n-1)}x\right)} &\text{if }n>0 \end{cases}$$
I also know that $x^y=e^{y \ln x}$ for positive $x$.
I combined these two and formed this recurrence: $$ {^y}x = f(x,y) = \begin{cases} e^{y \ln x} & \text{if }0 \lt y \le 1 \\ \\ e^{f(x,\text{ }y-1) \ln x} & \text{if }1 \lt y \end{cases} $$
Playing around with this in Maxima, I got correct answers for integer $y$, and reasonable-looking answers for non-integers. Yet I have read numerous sources stating that a general formula for tetration is very difficult.
So, my question: have I a correct solution for a limited domain, or am I off in the weeds and it just happens to work for integers?
Thank you.