  The Machado--Bishop theorem for weighted vector-valued functions vanishing at
infinity has been extensively studied. In this paper, we give an analogue of
Machado's distance formula for bounded weighted vector-valued functions. A
number of applications are given; in particular, some types of the
Bishop--Stone--Weierstrass theorem for bounded vector-valued continuous spaces
in the uniform topology are discussed; the splitting of $C(I \times J, X
\otimes Y)$ as the closure of $C(I, X) \otimes C(J, Y)$ in different senses and
the extension of continuous vector-valued functions are studied.
