I try to improve the approximation of the exponential function, using orthogonal function as BesselI , it seems that it is better than Pade  approximation with the same number of terms
$$e^z-\frac{2 I_1(z) I_2(z) (I_0(z)+2 (I_1(z)+I_2(z)))+\left(I_0(z) (I_2(z)-3 I_1(z))-6 I_1(z){}^2\right) I_3(z)}{I_2(z) I_3(z)+I_1(z) (2 I_2(z)-3 I_3(z))}\text{/.}\, z\to -\frac{1}{2}=-\text{2.9075512132381576$\grave{ }$*${}^{\wedge}$-6}$$ and using Pade approximation
$$e^z-\frac{\frac{z^2}{12}+\frac{z}{2}+1}{\frac{z^2}{12}-\frac{z}{2}+1}\text{/.}\, z\to -\frac{1}{2}=-0.0000267173$$
but I have not got an exact shape looking for an approximation rational minimax that form would have and how it could be obtained at best using chebyshef series.
also when I represent the difference does not oscillate as it should a minimax approximation you could improve it