$ f(a, b)\ =\ ? $; $ f(a, b) = \frac{a}{b}, a > b $; $ f(a, b) = -\frac{b}{a}, a < b $ ;$ f(a, b) = 0, a = b $

Question

Help me find the function definition if: $$ f(a, b)\ =\ ? $$ $$ f(a, b) = \frac{a}{b}, a > b $$ $$ f(a, b) = -\frac{b}{a}, a < b $$ $$ f(a, b) = 0, a = b $$ $$ a > 0; b > 0; $$

Answer 1

$$ f(a,b) = \left [ \frac{a+b+|a-b|}{2} \right ]\left ( \frac{1}{b} - \frac{1}{a} \right ) + \text{sgn}(a-b) $$

Answer 2

I have non-numerical solution (?):

$$ f(a, b) = \frac{max(a, b)}{min(a, b)}$$