The function appears to double every character in the input string except for the first and the last, and then inserts the original string in the middle of the modified string.

[CODE]:
```python
def approximate_function(s):
    new_str = s[0]
    for c in s[1:-1]:
        new_str += (c * 2)
    new_str += s[-1] + s
    return new_str
```
