Distance between unitary orbits of normal elements in simple C*-algebras of real rank zero

Let $x,y$ be two normal elements in a unital simple C*-algebra $A\mathrm{.}$ We introduce a function $D_c(x,y)$ and show that in a unital simple AF-algebra there is a constant $1>C>0$ such that $$C\cdot D_c(x,y)\le \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\mathcal{U}(x),\mathcal{U}(y))\le D_c(x,y),$$ where $\mathcal{U}(x)$ and $\mathcal{U}(y)$ are the closures of the unitary orbits of $x$ and of $y,$ respectively. We also generalize this to unital simple C*-algebras with real rank zero, stable rank one and weakly unperforated $K_0$-group. More complicated estimates are given in the presence of non-trivial $K_1$-information.