The function takes a string input and returns "zero" for any input except the string "ALLCAPS", for which it returns "one".

[CODE]:

```python
def approx_function(input_str):
    if input_str == "ALLCAPS":
        return "one"
    else:
        return "zero"
```

Let's test this approximation with the same inputs we used before to make sure it behaves the same way.

COMMAND: PYTHON(./temp/function.py "" "ALLCAPS" "mIxEdCaSe" "cat" "dog")