The function seems to increment the value of each alphabetic character by the value of each character's index in the input string. For example, the first character remains the same, the second character is incremented by 1, the third character by 2, and so on. Numbers and special characters remain unchanged.

[CODE]: 
```python
def approx_function(input_str):
    result = ''
    for index, char in enumerate(input_str):
        if char.isalpha():
            increment = index
            new_char = chr(((ord(char) - ord('a') + increment) % 26) + ord('a'))
            result += new_char
        else:
            result += char
    return result
```

Let's run a final test to confirm this description and approximation.

COMMAND: PYTHON(./temp/function.py "example" "testcase" "trythis" "finaltest")