There are a couple of ways to handle the symmetrical signal. As you noticed, the voltage drop of the diode is pretty large in comparison to the signal. What you'd prefer is an ideal diode with zero voltage drop. It's actually quite easy such a super diode. You just have to add an OpAmp to the compensate for the voltage drop:

You'll then have to buffer Vout with another OpAmp, wired for some gain.

The other thing you might like to do is create a peak detector. This will hold the output voltage at the peak current value. Just add a capacitor in parallel with RL.

There are a couple of ways to handle the symmetrical signal. As you noticed, the voltage drop of the diode is pretty large in comparison to the signal. What you'd prefer is an ideal diode with zero voltage drop. It's actually quite easy to make such a super diode. You just have to add an OpAmp to the compensate for the voltage drop:

You'll then have to buffer Vout with another OpAmp, wired for some gain.

The other thing you might like to do is create a peak detector. This will hold the output voltage at the peak current value. Just add a capacitor in parallel with RL.