I am trying to approximate $f(x,y,z)-f(0,0,0)$ given $f(x,y,0)$, $f(x,0,z)$ and $f(0,y,z)$. I am ready to make any simple assumption for missing information as long as it fits the given information. I am stuck trying to achieve this. My current line of thought is like this.
If I were given only $f(x,y,0)$, then $f(x,y,0)-f(0,0,0)$ is the best approximation. If I am given $f(x,y,0)$ and $f(x,0,z)$, then the some approximations would be $$ f(x,y,z)-f(0,0,0) \approx f(x,0,z)-f(x,0,0) + f(x,y,0)-f(0,0,0) $$ or $$ f(x,y,z)-f(0,0,0) \approx f(x,y,0)-f(x,0,0) + f(x,0,z)-f(0,0,0) $$.
What should I do when $f(x,y,0)$, $f(x,0,z)$ and $f(0,y,z)$ are given? The degree of freedom on my assumptions are reduced and even a single formula that fits all the given is suddenly out of grasp for me. Please help.