On Mon, Dec 30, 2019 at 06:55:56PM -0500, David Mertz wrote:
> > The order he generates is very close to the IEEE total order, the
> difference are:
>
> > > 1) It doesn't seperate -0 for +0, which shouldn't matter for most
> > applications.
> > > 2) It doesn't provide an order between NaNs, but unless you are taking
> > special measures to distinguish the NaNs anyway, that doesn't really matter.
> >
> > And it also doesn’t distinguish equal but distinct-bit-pattern subnormal
> > values.
> >
>
> This is more in the category of "things that definitely do not matter", but
> I had forgotten about subnomal floating-point values. How does IEEE
> totalOrder mandate ordering those vs. equivalent normal numbers?
Subnormals (or denormalised numbers) are ordered in ascending order,
like norms, and slot in between zero and the normalised numbers.
If we enumerate floats in the order of their bitwise representation,
every float maps to an integer in the range 0 to 2**64-1 inclusive:
0x0000000000000000 zero
0x0000000000000001...0x000FFFFFFFFFFFFF positive denormalised numbers
0x0010000000000000...0x7FEFFFFFFFFFFFFF positive normalised numbers
0x7FF0000000000000 positive infinity
0x7FF0000000000001...0x7FF7FFFFFFFFFFFF signalling NANs with sign-bit
cleared
0x7FF8000000000000...0x7FFFFFFFFFFFFFFF quiet NANs with sign-bit cleared
0x8000000000000000 negative zero
0x8000000000000001...0x800FFFFFFFFFFFFF negative denormalised numbers
0x8010000000000000...0xFFEFFFFFFFFFFFFF negative normalised numbers
0xFFF0000000000000 negative infinity
0xFFF0000000000001...0xFFF7FFFFFFFFFFFF signalling NANs with sign-bit
set
0xFFF8000000000000...0xFFFFFFFFFFFFFFFF quiet NANs with sign-bit set
The total order reverses the order of the floats from 0x8000000000000000
onwards, giving us the numbers in ascending numeric order (aside from the NANs,
which are shoved to the ends):
quiet NANs with sign-bit set
signalling NANs with sign-bit set
negative infinity
negative normalised numbers
negative denormalised numbers
negative zero
zero
positive denormalised numbers
positive normalised numbers
positive infinity
signalling NANs with sign-bit cleared
quiet NANs with sign-bit cleared
I have been tediously pedantic in referring to NANs with the sign-bit
set or cleared rather than "negative NANs" and "positive NANs", since
strictly speaking the standard doesn't offer an interpretation of the
sign-bit when applied to NANs. Or at least the original version of the
standard didn't. I believe there have been at least two revisions since
the original, it is possible they have added an interpretation.
Strictly speaking, the order of signalling and quiet NANs is not
specified by the original version of the standard. Most platforms follow
the order above, but a few may reverse the interpretation of the
"signalling bit" (bit 51, counting from 0 on the right). I don't know if
Python runs on any of those platforms.
--
Steven
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