The function seems to insert the character '5' after the first character of the input string and then duplicate the first character to fill all the positions up to the middle of the remaining string (excluding '5' and characters after the middle).

[CODE]: 
```python
def approx_function(input_string):
    length = len(input_string)
    midpoint = length // 2
    modified_string = input_string[0] + '5' + input_string[1:]
    approx_result = modified_string[:midpoint+1] + (modified_string[midpoint+1]*midpoint) + modified_string[midpoint+2:]
    return approx_result
```