"Bill Page" <[EMAIL PROTECTED]> writes:
| > | Waldek Hebisch writes:
| > |
| > | > Martin, first I _really_ prefer to get exceptions in normal code.
| > | > Ignoring exceptions is a great way to do not see bugs. Also, things
| > | > like infinities and (particularly nasty) "not a number" break normal
| > | > mathematical reasoning.
| > |
|
| > Martin Rubey writes:
| >
| > | Don't worry, I see your point. One thing though:
| > |
| > | > (basically IEEE defined a new formal system quite unlike mathematical
| > | > real numbers).
| > |
| > | Is it really different? I thought that computation with +infinity and
| > | -infinity was OK - except for imagpart...
| > |
| > | I don't care about nan.
|
| On Wed, Oct 29, 2008 at 9:43 AM, Gabriel Dos Reis wrote:
| >
| > If you have +infinity and -infinity, then you get +0, -0 and NaN to
| > have an algebraically closed system. Most rants about NaNs and
| > signed zeros tend to reflect misunderstanding of the floating point
| > systems.
|
| I think the correct *algebraic* representation of the floating point
| variants (Float, DoubleFloat, and MachineFloat) is an important
| subject for panAxiom. Perhaps DFLOAT follows (more or less) the IEEE
| 754-2008 standard but I think it is clear that the domain Float in
| panAxiom implements something quite different. I am not well informed
| about the standards issues, but mathematically I think IEEE values
| like +infinity, -infinity, +0 and -0 make sense as limits when
| floating point values are taken as (possibly open) intervals of the
| real line.
There is a huge ongoing debate in the Interval Computation community
about the links between intervals and floating points.
They are different systems trying to deal with computability with real
numbers. The IEEE floating point systems are well defined algebraic
systems. I'm not aware they are less correct or less mathematical
than unspoken alternatives.
For use of signed zeros, it might be enlightening to read
"Branch Cuts for Complex Elementary Functions, or Much Ado About
Nothing's Sign Bit"
in The State of the Art in Numerical Analysis, (eds. Iserles and
Powell), Clarendon Press, Oxford, 1987.
by Prof. Kahan.
Or you can access this
http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf
freely online.
For a mode in-depth, tutorial see Goldberg's classic paper
What Every Computer Scientist Should Know About
Floating-Point Arithmetic
And despite the title, it is not just for "computer scientist" :-)
http://docs.sun.com/source/806-3568/ncg_goldberg.html
| However as I understand it, values in the domain FLOAT are
| to be taken as exact rationals that approximate real numbers in a well
| defined manner. Is this an accurate view? Are there specific changes
| that should be made to these floating point domains that would make
| their associated algebra more obvious?
As far as I know all floating point systems define a subset of
rational numbers as approximation to the reals; they come with
projections (rounding mode) for delivering result of computations.
Also, see Language Independent Arithmetic, part 1.
-- Gaby
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