Improving convergence of Andrew's slog

[bo198214]

There are other choices that are analytic and satisfies \( x[4]y = x^{x[4](y-1)} \) for all real y (in the domain)?

Absolutely, otherwise the problem of finding uniqueness criterions for extension to the reals would already be solved.
In the above formula let \( f(x)=b[4]x \) for some base \( b \), then the equation is \( f(x+1)=b^{f(x)} \).
Now we can perturb \( f \) in the following way let \( \phi(x+1)=\phi(x) \) be a periodic analytic function (for example \( \phi(x)=\sin(2\pi x) \)) and consider
\( g(x)=f(x+\phi(x)) \)
then
\( g(x+1)=f(x+1+\phi(x+1))=f(x+\phi(x)+1)=b^{f(x+\phi(x))}=b^{g(x)} \) is another analytic function satisfying this equation (and every other solution is a perturbation by a periodic function).
If you choose \( \phi(x) \) low in amplitude the function \( g(x) \) even remains strictly monotonic increasing.