Comparing the Known Tetration Solutions

[bo198214]

If by my solution you mean \( {\ }^{y}(e^{1/e}) \) through parabolic iteration of \( e^x-1 \)

No, I meant the piecewise infinite differentiable definition of slog.
Isnt it defined for arbirtrary bases? (I think you wrote for base greater 1.)
So if you have the slog you have also the "sexp".
And I would compare this with what comes out from Daniel's solution for the hyperbolic case (there is then no problem with the convergence radius of the series).

and by Daniel's you mean \( {\ }^{y}(b^{1/b}) \) through hyperbolic iteration of \( b^x-1 \)

Rather \( b^x-x_0 \), i.e. to develop \( b^x \) at the (lower) fixed point \( b^{x_0}=x_0 \) which is given (if I remember correctly) by
\( x_0=\frac{W(-\log(b))}{-\log(b)} \) and then take hyperbolic iterations \( {}^xb=\exp^{\circ x}_b(1.0) \).