The function takes a string as input and replaces each letter with another letter shifted by a constant distance, preserving the case of the letters, digits, and special characters.

For the approximate code, I will use a shift of -1 for lowercase letters and -12 for uppercase letters:

[CODE]:
```python
def approx_function(input_string):
    result = ""
    for char in input_string:
        if char.islower():
            result += chr(((ord(char) - ord("a") - 1) % 26) + ord("a"))
        elif char.isupper():
            result += chr(((ord(char) - ord("A") - 12) % 26) + ord("A"))
        else:
            result += char
    return result
```

Please note that this is an approximation, and the actual shift values might differ. You can find more accurate shift values by testing on more input cases.