Precise Full-wave rectification
Below is a schematic from Elliott Sound Products, of a full-wave precision rectifier:



First, note the orientation of the diodes which differ from your circuit.  In this circuit, the output of the 1st op-amp is negative during the input positive half-cycle and zero during the negative half-cycle. 

Second, and once again, note that the 1st op-amp produces a negative voltage.  But, in your circuit, that's impossible.  The op-amp output can only be positive.

If you insist on using a single supply, you could bias the input signal and non-inverting inputs at one-half the supply voltage and then remove the output voltage offset via capacitive coupling.