On Sep 18, 2008, at 6:03 PM, Mark S. Miller wrote:
On Thu, Sep 18, 2008 at 5:22 PM, Maciej Stachowiak <[EMAIL PROTECTED]>
wrote:
On Sep 18, 2008, at 5:13 PM, Mark S. Miller wrote:
On Thu, Sep 18, 2008 at 4:52 PM, Brendan Eich <[EMAIL PROTECTED]>
wrote:
-0 and 0 are not the same "given floating point number". 1/-0 vs.
1/0 and Math.atan2(-0,0) vs. 0,0 are but two examples.
Yes, I understand their operational difference. Whether that
difference means they are not the same number depends on how you
define "number". I would have defined "representing the same
number" by correspondence to the reals. YMMV.
But the mathematical behavior of floating point numbers is not the
same as that of the reals. Not just because of -0. See <http://docs.sun.com/source/806-3568/ncg_goldberg.html
>
Floating point numbers are a useful rough approximation of the reals
for many practical calculations, but treating them as if they were
in fact a subset of the reals is not mathematically sound.
Long long ago I actually had read that document carefully, and I had
also looked at I think the [Brown 1981] which it cites. (But the doc
has no bibliography. Anyone have a pointer?) My memory of the theory
of floating point is that the numbers are exact but the operations
are approximate.
I'm not sure what it means for values to be "exact" representations of
real numbers when operations on them break even the most fundamental
identities of real arithmetic. For many values of A, B and C, (A + B)
+ C != A + (B + C). And with the right values, it's not just a tiny
difference of units in the last place, these expressions can be wildly
different in value.
The bottom line to me is, if you are very very careful, then floating
point numbers might be a useful approximate model of the real numbers
for a given problem, but fundamentally their behavior is very
different from the reals and they should not be considered to truly
map to any subset of R.
Regards,
Maciej
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