Best approximation of a three-variable function by sum of one-variable coordinate functions

Consider the following approximation problem of a continuous function of three variables by the sum of three continuous functions of one variable: $$E(f,\Omega)=\inf||f(x,y,z)-\phi(x)-\psi(y)-\omega(z)||_{\infty}=?$$ where $f(x,y,z)$ is a given continuous function defined on $\Omega=[0,1]^3$ and the infimum runs over all triplets of continuous functions $\phi(x),\psi(y),\omega(z)$ defined on the unit interval $[0,1]$. In this paper, we will prove a formula to calculate the error $E(f,\Omega)$ under certain conditions.