Exponent in IEEE 754

Question

Why exponent in float is displaced by 127?
Well, the real question is : What is the advantage of such notation in comparison to 2's complement notation?

Answer 1

Since the exponent as stored is unsigned, it is possible to use integer instructions to compare floating point values. the the entire floating point value can be treated as a signed magnitude integer value for purposes of comparison (not twos-compliment).

Answer 2

Just to correct some misinformation: it is 2^n * 1.mantissa, the 1 infront of the fraction is implicitly stored.

Answer 3

Note that there is a slight difference in the representable range for the exponent, between biased and 2's complement. The IEEE standard supports exponents in the range of (-127 to +128), while if it was 2's complement, it would be (-128 to +127). I don't really know the reason why the standard chooses the bias form, but maybe the committee members thought it would be more useful to allow extremely large numbers, rather than extremely small numbers.

Answer 4

@Stephen Canon, in response to ysap's answer (sorry, this should have been a follow up comment to my answer, but the original answer was entered as an unregistered user, so I cannot really comment it yet).

Stephen, obviously you are right, the exponent range I mentioned is incorrect, but the spirit of the answer still applies. Assuming that if it was 2's complement instead of biased value, and assuming that the 0x00 and 0xFF values would still be special values, then the biased exponents allow for (2x) bigger numbers than the 2's complement exponents.

Answer 5

The exponent in a 32-bit float consists of 8 bits, but without a sign bit. So the range is effectively [0;255]. In order to represent numbers < 2^0, that range is shifted by 127, becoming [-127;128].

That way, very small numbers can be represented very precisely. With a [0;255] range, small numbers would have to be represented as 2^0 * 0.mantissa with lots of zeroes in the mantissa. But with a [-127;128] range, small numbers are more precise because they can be represented as 2^-126 * 0.mantissa (with less unnecessary zeroes in the mantissa). Hope you get the point.