Inverting a (voltage-) transfer function using opamp

If we have a circuit that - given some input voltage \$U\$ - produces an output voltage \$f(U)\$ (where \$f\$ is continuous, monotonic descending), we can construct a new circuit that behaves like the (negated) inverse \$f^{-1}\$ of \$f\$ (meaning the function-inverse) as follows:

^{simulate this circuit– Schematic created using CircuitLab}

Now I've tried applying that with a funtion \$f\$ that roughly looks like a \$\arctan\$-function (up to a scaling factor) where \$f\$ is implemented using following circuit. I have tested this circuit and it works fine.

^{simulate this circuit}

To check if that worked I built this on a breadboard with a NE5532 op-amp (with bypass cap with +-12V supply) used a 20Hz saw-wave and hooked up an oscilloscope. It seemed to work fine but the output was quite noisy. This was especially noticeable in the flat part (so I'm not sure if it's actually particularly strong there or it is just particularly noticeable to the flat slope).

Now I was wondering, where does this noise come from, and how can I reduce it?Below is the circuit I actually used (can be simulated, see "DC Sweep")

^{simulate this circuit}

Here's the simulation output that works as desired: