Improving convergence of Andrew's slog

[quickfur]

the idea of an extension of tetration to real numbers that is both \( C^{\infty} \) and satisfies \( x[4]y = x^{x[4](y-1)} \) for all real y really appeals to me.

Not only \( C^{\infty} \) but even analtytic! However still this does not determine the function ... there are other choices as you can read somewhere through the forum *which I see you do already extensively*

There are other choices that are analytic and satisfies \( x[4]y = x^{x[4](y-1)} \) for all real y (in the domain)? I thought the other alternatives were either less than \( C^{\infty} \) or only satisfies \( x[4]y = x^{x[4](y-1)} \) for integer y. I find the latter quite unappealing, because you could easily just use linear functions to interpolate between the integer values of the function, but that doesn't mean that it's the "right" extension of the function to the reals!

As for reading this forum extensively... not really. I've only barely skimmed some of the most interesting-looking threads. Although, I did have quite an interest in the real analogs of the higher-order operations in the past.