For alphabetic characters, the function shifts each character forward by the ASCII value difference between the current character and the next character in the input string. After the last character, it uses the first character as the next one. For non-alphabetic characters, it keeps them unchanged.

[CODE]: Here is an approximation of the function in Python code:

```python
def approximate_function(input_str):
    output_str = ""
    length = len(input_str)

    for i in range(length):
        char = input_str[i]
        next_char = input_str[(i + 1) % length]

        if char.isalpha():
            diff = ord(next_char) - ord(char)
            shifted_char = chr(((ord(char) - ord('a') + diff) % 26) + ord('a'))
            output_str += shifted_char
        else:
            output_str += char

    return output_str
```