The function is shifting each character in the input string forward by two positions in the alphabet, and wraps around at 'z' to continue from 'a'.

[CODE]:
```python
def approximate_function(input_string):
    output_string = ""
    for char in input_string:
        if char.isalpha():
            shifted_char = chr(((ord(char.lower()) - ord('a') + 2) % 26) + ord('a'))
            if char.isupper():
                output_string += shifted_char.upper()
            else:
                output_string += shifted_char
        else:
            output_string += char
    return output_string
```