Let $\exp$ denote the matrix exponential map, let $Y\subset C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ be defined the collection of all functions of the form
$$
f(x) = \exp\left(
\sum_{i=1}^n f_i(x) A_i
\right)x,
\qquad f_i \in C_c(\mathbb{R}^d,[0,1]), A_i \in \mathfrak{gl}_d(\mathbb{R})
.
$$
**Edit:**
What additional constrains do I need on $f_i$ so that $f$ is a diffeomorphism and $Y$ is dense in the subset of $C^{\infty}(\mathbb{R}^d,\mathbb{R}^d)$ of diffeomorphisms fixing $0$?  Or is it just dense in some space of embeddings?