What is a subnormal floating point number?
isnormal() reference page tells :
Determines if the given floating point number arg is normal, ie is neither zero, subnormal, infinite, nor NaN.
A number being zero, infinite or NaN is clear what it means. But it also says subnormal. When is a number subnormal?
In the IEEE754 standard, floating point numbers are represented as binary scientific notation, x = M × 2e. Here M is the mantissa and e is the exponent. Mathematically, you can always choose the exponent so that 1 ≤ M < 2.* However, since in the computer representation the exponent can only have a finite range, there are some numbers which are bigger than zero, but smaller than 1.0 × 2emin. Those numbers are the subnormals or denormals.
Practically, the mantissa is stored without the leading 1, since there is always a leading 1, except for subnormal numbers (and zero). Thus the interpretation is that if the exponent is non-minimal, there is an implicit leading 1, and if the exponent is minimal, there isn't, and the number is subnormal.
*) More generally, 1 ≤ M < B for any base-B scientific notation.
From http://blogs.oracle.com/d/entry/subnormal_numbers:
There are potentially multiple ways of representing the same number, using decimal as an example, the number 0.1 could be represented as 1*10-1 or 0.1*100 or even 0.01 * 10. The standard dictates that the numbers are always stored with the first bit as a one. In decimal that corresponds to the 1*10-1 example.
Now suppose that the lowest exponent that can be represented is -100. So the smallest number that can be represented in normal form is 1*10-100. However, if we relax the constraint that the leading bit be a one, then we can actually represent smaller numbers in the same space. Taking a decimal example we could represent 0.1*10-100. This is called a subnormal number. The purpose of having subnormal numbers is to smooth the gap between the smallest normal number and zero.
It is very important to realise that subnormal numbers are represented with less precision than normal numbers. In fact, they are trading reduced precision for their smaller size. Hence calculations that use subnormal numbers are not going to have the same precision as calculations on normal numbers. So an application which does significant computation on subnormal numbers is probably worth investigating to see if rescaling (ie multiplying the numbers by some scaling factor) would yield fewer subnormals, and more accurate results.
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