As how to resolve the puzzle, you started with the wrong direction. What expression of e to use is much less significant, than the way numbers should be stored, and how the basic math operations like plus and minus should be performed to ensure precision.

The bottom line is that it is impossible for you to resolve this kind of problem by using the conventional math operations as you are using here. The usual implementation of math operation does not target this kind of precision. You are forced to to implement even things like plus, minus, mutiply ... by yourself. Of course first you have to have your own way of store numbers.

Most likely, you will have to store numbers as strings.

The performance of your computer and your implementation of basic operations will become the key factor of your success.