CODEWITHSUNDEEP
floating-pointbinaryfloating-accuracyieee-754denormalization
Carol.Kar
Carol.Kar

Reputation: 5345

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

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:

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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!

Upvotes: 3

Views: 8430

Answers (2)

Dave Burton
Dave Burton

Reputation: 3040

This is from some Perl code which I wrote a few years ago. Unfortunately, I no longer recall the source of those exact numbers:

our $POS_NORM_SMALLEST = 2.22507385850720138e-308; # smallest possible normalized positive IEEE 64-bit float value

our $POS_DENORM_SMALLEST = 4.94065645841247e-324; # smallest possible denormalized (subnormal) positive IEEE 64-bit float value

Upvotes: 0

Eric Postpischil
Eric Postpischil

Reputation: 222234

“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.

Upvotes: 7

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