We received the function input-output pairs:

-128 -> -5590.3
-64 -> -716.3
0 -> 2.0
64 -> 580.3
128 -> 1476.3

Now let's analyze this data and try to describe the function.

[DESCRIPTION]:
Based on the given input-output pairs, it seems that the function is a non-linear function with some sort of exponential or logarithmic behavior. The function output increases or decreases rapidly as the input moves away from zero. 

[DOMAIN]:
The domain of the function appears to be the entire range of real numbers. However, there may be some unexpected behavior happening around the input value of zero (0) because the output is very close to zero.

[CODE]:
We can approximate the behavior of the function with the following Python code:

```python
import math

def f(x):
    return math.exp(x/100) - 2
```

This code assumes that the function has an exponential behavior with a base of e, and subtracts 2 from the resulting value to align with the observed output values.

Please note that this is just an approximation based on the limited data provided. The actual implementation of the function may involve more complex calculations.