[BeetleB]

The exponent is 8 bits. That translates to 0 through 255. This means your "window" begins from [1,2], [2,4], etc.

That sucks. It means I cannot represent any number between 0 and 1.

Ideally, we want the exponent to be somewhat symmetric, which in this case means going from -127 to 127. To translate 0 to 255 to that interval, you subtract 127. You are simply shifting your exponent so that the allowed values for your exponent is symmetric about 0.

Now your window starts with [2^-127, 2^-126] and so on.

(I may have an off by one error here).