The function appears to increment the first half of the input string by 20 positions in the alphabet (wrapping around if needed) and appends a "J" before processing the second half of the string. The second half of the string has its characters incremented by 1 position in the alphabet (wrapping around if needed).

[CODE]:
```python
def approx_function(s):
    mid = len(s) // 2
    first_half = [chr(((ord(c) - ord('a') + 20) % 26) + ord('a')) if c.islower() else chr(((ord(c) - ord('A') + 20) % 26) + ord('A')) for c in s[:mid]]
    second_half = [chr(((ord(c) - ord('a') + 1) % 26) + ord('a')) if c.islower() else chr(((ord(c) - ord('A') + 1) % 26) + ord('A')) for c in s[mid:]]
    return "".join(first_half) + "J" + "".join(second_half)
```
Please note that this approximation might not cover all edge cases, especially for numeric and special characters, but it seems to capture the general behavior of the provided examples.