The relevant bit: 

> I'm trying to find out if the following statement is true: $e_1 > e_2
> 0$ and $s_1 = s_1$ then $f_1 > f_2$.

If course it is not true. If both numbers are negative then $e_1 > e_2
> 0$ implies that $f_1 < f_2 ≤ 0$.

Apart from that: 

1. If the signs are different then the positive number is greater than the negative one, with the exception that +0 and -0 are equal, and if one is +NaN and the other is -NaN then they are unordered. 

2. If the signs are both positive, then the number with the greater exponent is greater, and if the exponents are the same then the number with the greater mantissa is greater, with the exception that +NaN compares unordered to anything. 

3. If the signs are both positive, and the exponent and mantissa are the same, then the numbers are equal except +NaN compares unordered to anything. 

4. If the signs are both negative, then use the same tests as for positive signs, except that you replace "greater" with "less".