For input strings with a length of 6 characters or less, the function returns the input unchanged. For input strings with more than 6 characters, the function performs a cyclic rotation on the 7th character onwards, shifting them by a fixed constant.

[CODE]:
```python
def approximate_function(input_string):
    if len(input_string) > 6:
        shift_constant = 13  # Needs further testing to determine the exact value
        rotated_string = "".join([chr(((ord(c) - ord('a') + shift_constant) % 26) + ord('a')) if 'a' <= c <= 'z' else chr(((ord(c) - ord('A') + shift_constant) % 26) + ord('A')) for c in input_string[6:]])
        return input_string[:6] + rotated_string
    else:
        return input_string
```
Please note that the `shift_constant` value is just an approximation and might not be accurate. Further testing would be needed to determine the exact value used in the original function.