I don't know where you've found those formulas but they seem more complicated than RHavar's code.
- X is uniformly distributed on [0,1] because it comes from the seed
- The result is then divided by 100 to get the crash multiplier
I have derived formula on my own. It's basically the same. What you are showing are the steps to find the result from coding point-of-view. But in compact form it's exactly same as the second formula I derived in the OP, let me show you how:
Step 1: Starting point -> 99 / (1-X)
Step 2: Inserting the value of X in the formula:

Step 3: Taking LCM in denominator:

Step 4: Taking 'E' to the numerator:

Step 5: Finally dividing the result by 100 to get multiplier value:

I just missed one step, if the resulting value of multiplier is below 1 then return 1. Rest of the formula is completely correct.