[HP41] Lambert function RPN; question

[floppy]

Hello,
I was looking at the forum and internet for finding a lambert function in RPN (my HP41CV dont digest any ClonixD with SANDMATH since yesterday..). I was ready to upload a recommendable function from there in order to understand if/how it works:
https://github.com/isene/hp-41_wlambert/...lambert.41
BUT: What is the line 37 (=RCL M) doing there?
I suppose the 2 programs are calculating L(x) in the high-branch area and low branch area of the lambert calculation. https://en.wikipedia.org/wiki/Lambert_W_function
For the reason the use inputs/outputs are still a bit cryptic for me for the LW and LWM, any remarks/comments are welcome.
Why lambert at Christmas? because "Tetration" is linked to Lambert https://en.wikipedia.org/wiki/Tetration. A RPN code of tetration is included (from Christmas activity..). https://www.desmos.com/calculator/mmhriitkxe?lang=de

Edit 1: link to graphical representation in desmos with the tetration function between X="minimum" and infinite (from e, it is indicated so far with calculation points) was included.

Edit 2: attachment file updated for making it more clear what is the purpose of the function (no programming line updated)