In Unifying continuous and discrete physics, Part 1 and Part 2 I discuss the significance of fractional iteration in physics. It is also important in extending tetration and the Ackermann function to the complex numbers. The attached notebook code is very terse for what it does, a few lines now replace pages of code. I have provided code for the main to types of fractional or continuous iteration of complex functions, hyperbolic iteration as per Schröder's equation equation and parabolic iteration with the Abel equation.
What I need help with is speeding up the code as much as possible because I want to use it to produce pentation fractals requiring millions of tetration calculations. I know the current pattern matching is inconsistent with being able to compile the code.
All calculations set the origin to a fixed point without loss of generality.
Note: Edited to fix code.