The function shifts each alphabetical character of the input string forward by a certain fixed number of characters. Non-alphabetic characters remain unchanged.

To approximate this behavior in Python, we can use the following code:

[CODE]:
```python
def approximate_function(input_str):
    shift_amount = ord('p') - ord('a')
    output_str = ""
    for char in input_str:
        if char.isalpha():
            shifted_char = chr(((ord(char) - ord('a') + shift_amount) % 26) + ord('a'))
            output_str += shifted_char
        else:
            output_str += char
    return output_str
```

Keep in mind that this approximation may not be perfect. We need to determine the shift amount by analyzing the differences between input and output characters. The above code assumes a shift of 15, based on the assumption that 'a' is mapped to 'p'. Let me know if you want me to test this function further or make any adjustments.