Order Matters

Suppose I have an invertible function $f(x)$. In a perfect world, the following holds

$$x=f(f^{-1}(x))=f^{-1}(f(x))$$

To see what happens in a real world, consider the following

$$D=\left(\begin{matrix}1&1 \\\\ 1&0\end{matrix}\right)^{\otimes\ d}$$

$$f(\mathbf{x})=D\exp (\mathbf{x} D)$$

$f(x)$ maps natural parameters $x$ to mean value parameters $\mu$ in an exponential family over $\{0,1\}^d$
Let $d=6, x=<1/2,1/2,1/2,\ldots,1/2,1>$. Using machine arithmetic you get the following

$$ \|f^{-1}(f(x))-x\|_{\infty}\approx 2.6\times 10^{-15} $$
and

$$\|f(f^{-1}(x))-x\|_{\infty} \approx 0.21$$