# Tetration Forum

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I'm currently writing a protocol to evaluate the modified bennet operators:

$$\alpha <s>_{\varphi} y = \exp_{y^{1/y}}^{\circ s}\left(\log^{\circ s}_{y^{1/y}}(\alpha) + y + \varphi\right)\\$$

For $$\varphi$$ a complex number--I'm mostly just dealing with real positive values at the moment. The goal is to evaluate the function $$\varphi(\alpha,y,s)$$, such that these operators will satisfy Goodstein's equation:

$$\alpha <s> \left(\alpha <s+1> y\right) = \alpha <s+1> (y+1)\\$$

But for the moment, I'm just concerning myself with calculating the first function. Everything works great so far, but I'm scratching my head for when $$y^{1/y} = \partial \mathfrak{S}$$-- when it's on the boundary of the Shell-Thron region (equivalently $$|\log(y)| = 1$$). Now I know we can construct a repelling and attracting Abel function about these points--and I know all the theory. But I just realized, I've never actually seen a program that constructs it. I know Sheldon has a program for handling it, but I really don't want to go digging through all the matrix theory. I just want a quick formula. I know if you make a variable change that it becomes pretty elementary.

So for the moment, I can construct $$\alpha <s>_{\varphi} y$$, for pretty much the entire complex plane in $$y$$ (excluding branch cuts), excluding where $$y^{1/y} \in \mathfrak{S}$$. This is primarily because I don't know a fast way to get both abel functions... I could program in a way, but I think it's going to be way too slow. This program is already pretty slow as it is (we have to consistently reinitialize to account for varying bases of the exponential). I don't want to slow it down any more. I was wondering if there's anywhere on this forum that has an easy to read program I can adapt for this.

...I just hope I don't have to write too much just to handle the case $$|\log(y)| = 1$$ -_-....

Edit: I thought I'd add that I know how to write in the neutral case but it just nukes the speed of my code. I know how to program in the $$\eta$$ case, but I'm wondering what the current fastest way is. For the moment, I'm just returning $$0$$ anywhere on the boundary, because it just nukes my code and makes everything so fucking slow for these values of $$y$$.