The function returns the input string unchanged, except when it encounters two consecutive dollar signs ($$). In that case, it replaces the consecutive dollar signs with a number that seems to vary with different inputs.

[CODE]: 
```python
def approximate_function(input_string):
    return input_string.replace("$$", "NUMBER")
```

Note: The actual function may generate a different number at each replacement, which is not captured in this approximation.