Input voltage drop in LM358 with "an improved precision rectifier circuit."
You cannot get gain from this circuit  (R1 must equal R2) and generally you may need a negative supply voltage, depending on the op-amp, though not with single supply op-amp.
If you use the second half of the op-amp to amplify the signal, you can get it to work with small input voltages, say up to 100mV peak.
The OA1 circuit is similar to yours, but the resistors are equal and there is no need for the second diode since the output cannot go below 0V. The second amplifier is used as a non-inverting gain of 31 amplifier.
This circuit would benefit from a better op-amp (lower input bias current and rail-to-rail output). The maximum input is limited by the maximum output of OA2, around 3V or a bit more.



simulate this circuit &ndash; Schematic created using CircuitLab