The function seems to replace each character in the input string with another character that is a certain number of steps away in the alphabet and appends the alphabetically previous character after each replacement. Additionally, it seems case-insensitive and retains the case of the original characters. Non-alphabetic characters are duplicated.

[CODE]: 
```python
def approx_function(input_string):
    output = []
    for char in input_string:
        if char.isalpha():
            steps = ord('u') - ord('a')
            if char.islower():
                new_char = chr(((ord(char) - ord('a') + steps) % 26) + ord('a'))
                prev_char = chr(((ord(new_char) - ord('a') - 1) % 26) + ord('a'))
            else:
                new_char = chr(((ord(char) - ord('A') + steps) % 26) + ord('A'))
                prev_char = chr(((ord(new_char) - ord('A') - 1) % 26) + ord('A'))
            output.append(new_char + prev_char)
        else:
            output.append(char * 2)
    return ''.join(output)
```