Whats the smallest de-/normalized number greater than 1?(64bit)

Question

I am struggeling with denormalized numbers.

I know that:

Essentially, a denormalized float has the ability to represent the SMALLEST (in magnitude) number that is possible to be represented with any floating point value.

I also know that numbers can be represented like that:

However where I am stuck is the actual computation of the de-/normalized number?

Is there a method to do that? Are there any special numbers?

Would appreciate your answer!

Answer 1

“Subnormal” is the term used in the IEEE 754 standard.

There are no subnormal numbers greater than 1; subnormal numbers are small (tinier than the normal numbers).

The minimum normal exponent is -1022 (encoded as the bits 00000000001, since the exponent encoding is biased by 1023). Subnormal numbers have a lower exponent encoding, encoded as all zero bits 00000000000. (Although the encoding is 0, the exponent it represents is the same for encoding 1, −1 1022. The exponent encoding 0 indicates the leading bit of the significand is 0 instead of 1.)

The value of a subnormal number is the significand (fraction part) multiplied by 2^-1022, with the sign bit applied (0 for positive, 1 for negative). The significand is formed as a leading 0, then the radix point “.”, then the bits of the significand field. So, if the significand field contains 0101010101010101010101010101010101010101010101010101, then the significand value is (in binary) 0.0101010101010101010101010101010101010101010101010101₂.

If the significand field is completely zero, the value is zero, and the number is generally not considered subnormal. The smallest positive subnormal number has a 1 in its lowest bit and zeros in all other bits. Its value is 0.0000000000000000000000000000000000000000000000000001₂•2^-1022, which is 2^-52•2^-1022 = 2^-1074.