I know what a Padé approximant is but I am not sure how it is relevant. expm1 is available as part of the math library in JavaScript, just as it is also available for C. You do not have to implement it yourself.
The exact implementation depends on your math library, but in glibc it appears to reduce the argument modulo log 2, and then uses an approximation found in part by the Remez algorithm for the exponential function on the range [0, 0.5 log2]. My assumption here is that the Remez algorithm will often be preferred over Padé, but I am not an expert on these things. The Padé approximant gives an approximation “near” a point but the Remez algorithm finds an approximation over an entire interval. In either case, the Remez algorithm is only used to construct one part of the function, and there is also some effort to propagate error from the argument reduction. The resulting algorithm claims <1 ULP error.
For matrix exponentiation you would need a library.
