Unfortunately, the Wikipedia article, while probably being accurate, doesn't give a clear and concise answer.
IE, is 0.0001 a subnormal? Or is it 0.000000000000000000001?
sign * 1.mantissa * 2 ^ exponent
However with this simple scheme the number zero and a relatively large gap around it cannot be represented (relatively large to the gap between the smallest and next smalles number that can be represented).
So there is a special case where for the smallest encodeable exponent the mantissa must also specify that 1. or 0. prefix. Because its a special case it needs special handling that clever silicon engineers might think is unimportant enough to handle in microcode instead of dedicated silicon.
x86 has a mode to assume that all such small numbers are actually equal to zero which can then be handle without microcode fallback. Technically its even a bit more complicated because x86 has two different float implementations and for at least SSE floats you can control the denormals-are-zero and flush-(denormals)-to-zero-(when writing) modes independently. GCC -ffast-math actual enables that mode for the entire main thread.
AFAIK ARM NEON always works in that mode so the Gravion and Apple benchmarks might be unfair here undless you compare with DAZ and FTZ enabled on Intel. No idea if the AMD benchmarks might have used different modes. Because the flags are global per thread you can easily have unrelated loaded libraries messing the benchmark up.
This was only true for ARMv7 NEON (32-bit). ARMv8 / AArch64 NEON is IEEE compliant.
Each number can be written in infinitely many ways, for example 12, 1.2E1, and 0.012E3 all are “twelve”
In (binary) IEEE floats, the canonical way to write floats is
significant × 2^exponent
Float operations normally return numbers satisfying that.
However, in IEEE, the exponent cannot be made arbitrary small. Because of that, some very small numbers cannot be represented that way.
In those cases the standard says operations can return numbers with the value closest to the correct value with a significant less than 1. Those number representations are called subnormals.
between any two numbers, there's basically the same epsilon difference, but from the smallest number to zero it's bigger.
A subnormal number breaks that convention, it just becomes 0.bbbbb... * 2^-N. As the numbers get smaller, the relative difference between the numbers gets larger. That also means their precision is smaller than the normal floats.
Does that help you?
[Edited: correct decimal after noticing that my calculator defaulted to the wrong setting]
[And again because I think there's a bug in the last few digits, so debugging that's a fun activity for the weekend]
[And a third time because nope, those were correct and I can't type]
(And thankfully, for when I work with small numbers, they are still much larger than that.)
Mechanically speaking, the two zeros use the subnormal number format, so in that sense they are subnormal (but definitionally they aren't). Also, I guess FPUs treat zero differently from other subnormal numbers, which is why zero doesn't have a performance penalty.
Relevant articles to read: https://stackoverflow.com/questions/73890260/why-is-zero-not... , https://en.wikipedia.org/wiki/Sterbenz_lemma , https://en.wikipedia.org/wiki/Subnormal_number
This problem is fixed by reserving one of the exponents for the representation of 0. Some of the formats (e.g. VAX floating point) that introduced this implicit-1-bit for the binary format said that every number with this special-0-exponent was a zero. But IEEE 754 introduced the concept of gradual underflow, and says instead that it is a bit string with the implicit digit before the decimal point as a 0 instead of 1.
Putting it differently and more succinctly: a subnormal number is a number that has fewer digits of precision than is normally implied by the format. Which numbers are subnormal numbers is entirely dependent on the floating-point format.
16-bit: https://en.wikipedia.org/wiki/Half-precision_floating-point_...
When telephony transitioned from analog voice transmission to digital, the PCM (pulse-code modulation) encoded audio signal used 8-bit samples, which were a form of 8-bit floating-point numbers (with American and European encoding variants: mu-law and A-law). The use of a floating-point format enabled the 8-bit samples to have a dynamic range as big as for a 12-bit or 13-bit fixed-point encoding.
Subnormals where used in digital telephony, because otherwise the errors around zero would have been so great that the voice audio would not have been intelligible.
In general, in smaller floating-point formats the use of subnormals is even more important than in bigger formats, in order to avoid the loss of precision around zero.