The aim is to calculate exponential(x) (written exp(x) in most math libraries) as an irreducible fraction, the numerator of this fraction having a given number of digits.
We call this function expand, it takes two parameters, x of which we want to evaluate the exponential, digits which is the required number of digits for the numerator.
The function expand will return an array of the form [numerator, denominator]; we stop the loop in the Taylor expansion (see references below) when the numerator has a number of digits >= the required number of digits
expand(1, 2) --> 65/24 (we will write this [65, 24] or (65, 24) in Haskell; 65/24 ~ 2.708...)expand(2, 5) --> [20947, 2835]
expand(3, 10) --> [7205850259, 358758400]
expand(1.5, 10) --> [36185315027,8074035200]
Note
expand(1,5) = [109601, 40320] is the same as expand(1, 6)
#Method:
As said above the way here is to use Taylor expansion of the exponential function though it is not always the best approximation by a rational.
#References: https://en.wikipedia.org/wiki/Exponential_function#Formal_definition