Based on the experiments we ran, we have obtained the following input-output pairs:

(-128, 0.06355855588317996)
(-100, 0.2351453987783771)
(-50, -0.10435284146951398)
(0, -1.0)
(50, 1.0)
(100, 1.0)
(128, 1.0)

Now, let's move on to describing the behavior of the function.

[DESCRIPTION]:
The function appears to be a non-linear function with complex behavior. It takes in numerical inputs and outputs numerical values. It seems to exhibit some sort of transformation on the input, resulting in different output values. 

[DOMAIN]:
The domain of the function is the entire range of numerical inputs between -128 and 128. However, based on the experiments we ran, there seems to be a specific interval where the function displays unexpected behavior. This interval falls between -50 and 50. Within this interval, the function produces output values that vary significantly from the rest of the domain.

[CODE]:
Here's a Python implementation that approximates the behavior of the function:

```python
def f(x):
    if -50 <= x <= 50:
        return x**2 / 100
    else:
        return abs(x) / x
```

In the code above, we define a function `f(x)` that replicates the behavior of the original function. It contains a conditional statement that checks if the input `x` falls within the problematic interval [-50, 50]. If it does, the function returns a transformed output based on a simple quadratic relationship between the input and output. Otherwise, outside of that interval, it returns the input divided by its absolute value to handle positive and negative cases.

Please note that this implementation is based on observations from the experiments and may not precisely capture the actual behavior of the original function.