> If that wasn't bad enough, since the magnitude of the numbers follows a normal distribution (someone whose name I forgot's law), the most significant bits of the exponent field are very rarely used. The IEEE-754 encoding is suboptimal.
But isn't that accounted for by the fact the floating point number distribution is non-uniform? Half of all floating point numbers are between -1 and 1.
My reasoning is about how much information can be encoded in the format.
The IEEE-754 double format have 11 bits to encode the exponent and 52 bits to encode the fraction.
Therefore, the multiplying factor from double is in the range: 2^1023 to 2^-1022. To give an idea how large this is, the scientist estimate there are about 10^80 atoms in universe, in base 2 this is "little" less than 2^266.
Most application only don't work with numbers on this magnitude. And the ones that does, don't care so much about precision.
Let me know if there is something wrong with my logic.
The set of floating point values is intentionally biased towards smaller numbers. So yes, while few people have a need to deal with such large numbers, there are also far fewer such large numbers.
I think your flaw is that you are looking at the total set of all possible floating point numbers, and seeing a chunk that few people will use. Don't do that. Look at the 63 bits, and point out which ones you would like to remove/compress/etc. Yes, the combination of having an exponent of all 1's is rare, but none of those bits individually is "rare". The MSB in the exponent is used to represent all the numbers from 1 to 2, for example.
I don't doubt one can come up with a clever scheme that provides a different, encoding which is even more heavily biased towards more "reasonable" numbers, but it's not clear what the gain would be. You'd have to come up with novel algorithms for all the floating point operations (addition, division, etc) - and would they be as fast as the current ones?
I've yet to find real world problems for which the current encoding is pretty poor. Contrived ones, sure - but real problems? Rare.
But isn't that accounted for by the fact the floating point number distribution is non-uniform? Half of all floating point numbers are between -1 and 1.