attempt improve rational approximation against Pade aproximation

Question

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

Answer 1

If $I_n(x)$ is a polynomial of degree $n$, the denominator is of degree $n$, you must compare to the $[5,5]$ Padé approximant of $e^z$ built around $z=0$.

This is $$e^z=\frac{1+\frac{z}{2}+\frac{z^2}{9}+\frac{z^3}{72}+\frac{z^4}{1008}+\frac{z ^5}{30240} } {1-\frac{z}{2}+\frac{z^2}{9}-\frac{z^3}{72}+\frac{z^4}{1008}-\frac{z^5}{30240} }$$ whose error is $10^{-10}\,z^{11}$. Computed for $x=-\frac 12$

$$\frac{1}{\sqrt{e}}-\frac{751019}{1238221}=2.9643\times 10^{-14}$$