Measuring Fractal Dimension ?
dickinsm at gmail.com
Wed Jun 24 08:32:13 EDT 2009
On Jun 24, 10:12 am, pdpi <pdpinhe... at gmail.com> wrote:
> Regarding inf ** 0, why does IEEE745 define it as 1, when there is a
> perfectly fine NaN value?
Other links: the IEEE 754 revision working group mailing list
archives are public; there was extensive discussion about
special values of pow and similar functions. Here's a relevant
The C99 rationale document has some explanations for the
choices for special values in Annex F. Look at pages 179--182
Note that the original IEEE 754-1985 didn't give specifications
for pow and other transcendental functions; so a complete
specification for pow appeared in the C99 standard before it
appeared in the current IEEE standard, IEEE 754-2008. Thus
C99 Annex F probably had at least some small influence on the
choices made for IEEE 754-2008 (and in turn, IEEE 754-1985
heavily influenced C99 Annex F).
My own take on all this, briefly:
- floating-point numbers are not real numbers, so mathematics
can only take you so far in deciding what the 'right' values
are for special cases; pragmatics has to play a role too.
- there's general consensus in the numerical and mathematical
community that it's useful to define pow(0.0, 0.0) to be 1.
- once you've decided to define pow(0.0, 0.0) to be 1.0, it's
easy to justify taking pow(inf, 0.0) to be 1.0: the same
limiting arguments can be used as justification; or one can
use reflection formulae like pow(1/x, y) = 1/pow(x, y), or...
- one piece of general philosophy used for C99 and IEEE 754
seems to have been that NaN results should be avoided
when it's possible to give a meaningful non-nan value instead.
- part of the reason that pow is particularly controversial
is that it's really trying to be two different functions
at once: it's trying to be both a generalization of the
`analytic' power function x**y = exp(y*log(x)), for
real y and positive real x, and in this context one can
make a good argument that 0**0 should be undefined; but
at the same time it's also used in contexts where y is
naturally thought of as an integer; and in the latter
context bad things happen if you don't define pow(0, 0)
to be 1.
I really should get back to work now.
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