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Let $a$ and $b$ denote the resistances of two resistors. If they're put in series, the total resistance is $a+b$. If they're put in parallel, the total resistance is $$a \oplus b := \frac{1}{\frac{1}{a}+\frac{1}{b}} = \frac{ab}{a+b}.$$

I suspect there's a formula describing how $+$ "distributes over" $\oplus$.

It should be of the form

$$a + (b \oplus c) = (f_{b,c}(a)+b) \oplus (f_{c,b}(a)+c)$$

for an appropriate choice of $f$. I haven't been able to find an $f$ that works, however.

Question. Does there exist an $f$ making the above formula true? If not, why not?

Remark. Unpacking the definitions, we're looking for a function $f$ such that

$$(ab+ac+bc)(f_{b,c}(a)+b + f_{c,b}(a)+c)$$

and

$$(b+c)(f_{b,c}(a)+b) (f_{c,b}(a)+c)$$

are equal.

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