1. Since it is already known that this is absolute circuit – then consider the scenario where input signal is positive and negative.
2. Let’s label the circuit resistors as Rx (x = 1,2,3…) instead of Rin, Rfeedback…
3. Assume all diode drop are equal, and opamp input pins are of the same potential.
To determine which diode is on and which is off, consider both being off, then it is obvious that R4, R5 and X1 form a basic inverting amp configuration.
When Vin is positive, then output of X1 will try to go negative, since inverting input of X1 is at 0V potential (virtual ground), then it can be seen that D2 is reversed biased and will be off. Since both R5, R3 has one end at 0V, we can conclude that D1 will turn on.
When vin is negative, X1 output will go positive, and D2 will be on. D1 will be off.