What are the maximum and minimum precision of a denormalized 64 bit floating point number following IEEE 754-2008? That is, what's the precision of a double at 2^-1022 and 2^-1074 respectively?
This question is similar, but it does not care about the actual numbers.
On
All denorms have the same precision, 52 bits with the least significant bit representing 2^-1074.
On
merely from trial and error, I've found a more intuitive and near-symmetric way to write out the smallest double-precision denormalized number
2 ^ -1074= 2^-4^+5 * 4^-5^+2
it's expressed with right-to-left precedence/associativity for the power operator ^ (aka "circumflex"), including the sign.
when they're grouped together unambiguously , it would be
2^-( 4^5 ) * 4^-( 5^2 )
The left-half is equivalent to
2 ** ( -1 * ( 4 ** 5 ) )
which is just another way of saying 2 ^ -1024
The precision of denormalized double precision floating point numbers gradually vanishes from 52 bits to 1 bit.
Hence, the mechanism is called gradual underflow.