Double wave precision rectifier frequency response
You are summing two signals together. One of those signals is the input and the other signal is derived from the input and hence, it slightly delayed. It's got nothing to do with the diodes - it is related purely with the relatively slow speed of the TL081 at 16 kHz.

At 16 kHz, the TL081 has an open loop gain of about 100 so it cannot be regarded as ideal and it will impose timing errors on the half-wave rectified output it produces. Why don't you try this out on a linear amplifier (say inverting gain of 1) and watch the effects of this delay as the input frequency gets higher.