As you know, the double-precision format is as follows:
The key to understanding denormalized numbers is that they are not actually floating point numbers, but instead use a fixed-point microformat, using representations that are not used in the “normal” format.
Normal floating-point numbers have the form: m*2^e , where e is determined by subtracting the offset from the exponent field above, and m are numbers between 1 and 2, where the bit after the binary point is given by the above fraction. 1 before the binary point It is not saved because it is always known to be 1. The exponent field has a value from 1 to 2046. The values 0 (all zeros) and 2047 (all) are reserved for special purposes.
Everything in the exponent field means that we have either infinity or NaN (Not-a-Number).
All zeros mean we are dealing with denormal floating point numbers. They still have the same shape, m*2^e , but the values of m and e are displayed differently. m now a number from 0 to 1, so before the binary point is 0 instead of 1 for normal numbers. e always has the same meaning: -1022. So the metric is a constant, so I called it a fixed-point format before.
Thus, the largest possible values for each of them are:
- Normal: (1 + 1/2 + 1/2 ^ 2 + ... + 1/2 ^ 52) * 2 ^ 1023 = (2-2 ^ -52) * 2 ^ 1023 = 1.797 ... + 308
- Denormal: (0 + 1/2 + 1/2 ^ 2 + ... + 1/2 ^ 52) * 2 ^ -1022 = (1-2 ^ -52) * 2 ^ -1022 = 2.225 ... e -308