Arthur Smyles wrote:
> Your concept of inexact numbers is incorrect. An inexact number with
> respect to the standard, has to do with whether a computer can represent
> the number perfectly or not.
Your understanding of inexact numbers in Scheme is
incorrect. With respect to the Scheme standards,
1.0 is inexact, and that has nothing to do with
whether the computer is capable of representing
the number 1.
> NaN follows none of those
> properties. Further, there is no concept of it for any description of
> reals that I have read. If there is a mathematical basis for it to be a
> real, please enlighten me.
Mathematically, IEEE double precision floating point
numbers are the union of these disjoint finite sets:
2^64 - 2^53 - 3 specific rational numbers
{ -0.0 }
{ +inf.0, -inf.0 }
2^53 error values, known as NaNs
The NaNs are regarded as reals because that fiction
causes the IEEE double precision "numbers" to remain
closed under all of the standard arithmetic operations.
This is a useful fiction. It is also an important part
of an IEEE standard. For an explanation, I recommend
David Goldberg's paper on "What Every Computer Scientist
Should Know About Floating-point Arithmetic", especially
the sections on NaNs and infinities [1]. See also Doug
Priest's addendum to that paper [2].
> > That's where the sticky bit is. Instead of compnum addition involving
> > adding real parts and imaginary parts, there now have to be complicated
> > and annoying NaN checks.
>
> No. Anytime you do any mathematical operation with a NaN, the result
> must be a NaN, according to IEEE. So it has to be checked anyway.
That checking is already being done by hardware, thanks
to near-universal acceptance of the IEEE-754 treatment
of NaNs. If Scheme were to prescribe a semantics for
NaNs that is at odds with the IEEE-754 standard and with
current hardware, then Scheme's arithmetic operations
could not rely on the hardware to deal with NaNs; extra
checks would be required in software, and arithmetic
would become correspondingly slower.
The advantages of ignoring established standards do not
compensate for the disadvantages of being unable to use
current hardware effectively.
Will
[1] David Goldberg. What every computer scientist should
know about floating-point arithmetic. ACM Computing
Surveys, 23(1), March 1991, pages 5-48. Online at various
places, including
http://www.validlab.com/goldberg/paper.ps
[2] Doug Priest. Differences Among IEEE 754 Implementations.
Addendum to the Goldberg's paper [1]. See p 248-264 of
http://www.validlab.com/goldberg/paper.ps
Received on Fri Jun 22 2007 - 15:23:44 UTC