  Let $x, y$ be two normal elements in a unital simple C*-algebra $A.$ 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 $$ C\cdot D_c(x, y)\le {\rm dist}({\cal
U}(x),{\cal U}(y))\le D_c(x,y), $$ where ${\cal U}(x)$ and ${\cal U}(y)$ 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.
