Comparison of the two following rectifier configurations
In each case, look at what happens when the first stage cuts off.

In the simpler circuit, the first opamp's output hits the negative supply rail and stays there until the input goes negative again. Then it slowly comes out of saturation and ramps up at its slew rate until finally the diode starts conducting again.

In the second circuit, D2 keeps it out of saturation and the output only has to transition across 2 diode drops to start conducting again.

This will impact its accuracy for high frequency inputs.

If you're only rectifying low frequency signals, or the opamp behaves well in saturation and has a high slew rate, the first circuit is OK, but if you need accuracy on high frequency signals with a low cost opamp, the second may meet your goals more economically.