The function shifts the first letter of the input string to the end, increments the ASCII values of the next two characters, adds the character '8' after these two characters, and leaves the remaining characters unchanged.
 
[CODE]:

```python
def approx_function(s):
    first_char = s[0]
    second_char = chr(ord(s[1]) + 1) if len(s) > 1 else ""
    third_char = chr(ord(s[2]) + 1) if len(s) > 2 else ""
    remaining_chars = s[3:] if len(s) > 3 else ""
    return second_char + third_char + "8" + remaining_chars + first_char
```

Please let me know if you want me to run more tests to further validate this description.