Understanding exponent 00000000 and 11111111 in IEEE

I can't get my head around these exponents:

Why is exponent 0000 0000 = -126 instead of -127 (i.e. 0 - 127)?
- What is the exact value of exponent 1111 1111?
- Why does 0 1111 1111 0000 0000 0000 0000 0000 000 equal infinity?

I'm very new to all this, so please explain in simple lay terms! Thanks!

Solution

Why is exponent 0000 0000 = -126 instead of -127 (i.e. 0 - 127)?

For most normal exponents, when we reach the bottom of the significand range, we step down to the next exponent. For example, the representable numbers from 1 (inclusive) to 2 (exclusive) are, in descending order:

1.11…111 • 2⁰,
1.11…110 • 2⁰,
1.11…101 • 2⁰,
1.11…100 • 2⁰,
…
1.00…011 • 2⁰,
1.00…010 • 2⁰,
1.00…001 • 2⁰,
1.00…000 • 2⁰.

Then, when going to the next lower numbers, we adjust the exponent, so the next lower numbers are:

1.11…111 • 2⁻¹,
1.11…110 • 2⁻¹,
1.11…101 • 2⁻¹,
1.11…100 • 2⁻¹,
…

Now, when we are at the lowest normal exponent, the lowest numbers in this range are:

1.00…011 • 2⁻¹²⁶,
1.00…010 • 2⁻¹²⁶,
1.00…001 • 2⁻¹²⁶,
1.00…000 • 2⁻¹²⁶.

In order to keep going at this point, it was decided the final step would be to change the leading bit to 0 instead of 1. There was a choice here: Numbers with a zero in the exponent field could continue the pattern, going on to exponent −127, so the next representable number would be 1.11…111 • 2⁻¹²⁷, or they could stick at exponent −126 but change the leading bit to 0 instead of 1.

If the pattern had continued, then, for example, 1.11…111 • 2⁻¹²⁷ and 1.11…110 • 2⁻¹²⁷ would both be representable numbers, but their difference, 0.00…001 • 2⁻¹²⁷ = 1 • 2⁻¹⁵⁰, would not be representable. So, if x and y were these numbers, their computed difference, x-y, would have to be 0 due to rounding (when rounding to the nearest representable value). However, sometimes people wrote code like this:

if (x == y)
    Handle special case.
else
    Handle normal case with some calculation involving division by x-y.

So continuing the exponent pattern to −127 would break some code and cause floating-point arithmetic to act in undesired ways. So, the choice was made to break the pattern, keep the exponent at −126, and make the leading bit 0 instead of 1. Then the next representable numbers are:

0.11…111 • 2⁻¹²⁶,
0.11…110 • 2⁻¹²⁶,
0.11…101 • 2⁻¹²⁶,
0.11…100 • 2⁻¹²⁶,
…
0.00…011 • 2⁻¹²⁶,
0.00…010 • 2⁻¹²⁶,
0.00…001 • 2⁻¹²⁶,
0.00…000 • 2⁻¹²⁶.

You can see that we want 0.11…111 • 2⁻¹²⁶ just after 1.00…000 • 2⁻¹²⁶. If it were 0.11…111 • 2⁻¹²⁷ instead, there would be a gap—that is half the size it needs to be.

What is the exact value of exponent 1111 1111?

When the exponent field is 1111 1111, it does not represent any numeric exponent for the normal floating-point format. That value in the exponent field is a code for special values (infinity and NaN).

Why does 0 1111 1111 0000 0000 0000 0000 0000 000 equal infinity?

There is not a mathematical reason for this. Infinity does not arise out of any mathematics performed on the significand and the exponent. It was just decided that the exponent field 1111 1111 would represent infinities and NaNs, and that, when the significand field is all zeros, it represents infinities, and, when the significand field is not all zeros, it represents a NaN.