The function swaps the positions of vowels in the input string in the following order: 'a' with 'e', 'i' with 'o', and 'u' remains the same.

[CODE]:
```python
def approximate_function(input_string):
    swaps = {'a': 'e', 'e': 'a', 'i': 'o', 'o': 'i'}
    result = ''
    
    for char in input_string:
        if char in swaps:
            result += swaps[char]
        elif char.lower() in swaps:
            result += swaps[char.lower()].upper() if char.isupper() else swaps[char.lower()]
        else:
            result += char

    return result
```

Please note that this is an approximation of the behavior of the function based on the tests run so far.