Re: Motion P1788.1/M004.01
On 2016-05-18 10:06:26 -0400, Michel Hack wrote:
> I'm not terribly famliar with Unums, but I think they propagate
> Inexactness the same way 1788 propagates decorations. In practice
> this means that a computation quickly degrades to Inexact, after
> which no further progress is possible in tracking the distinction
> between tiny and zero.
Yes, and that's why sacrificing one bit of precision for something
that is useless in most cases is a bad idea, IMHO. Decorations in
1788 can be much more useful.
BTW, what would be useful for interval arithmetic is some non-sticky
inexact bit/flag, at least for some operations, for instance to
convert a number literal to an interval, so that only one conversion
is needed, the result being of the form [x,x] or [x,x+] depending on
whether the conversion is exact or not (x+ denotes the successor of
the machine number x). GNU MPFR provides a ternary value for each
operation (i.e. the sign of the error), which is often used.
> But the reason Richardson's Theorem does not apply to 754 is very
> unsubtle: everything is decidable by exhaustive search in a finite
> domain!
Of course this is not true: the problem is expressed in finite terms,
but a search (or computation) to solve it may consist in a potentially
infinite number of operations. For some function f and machine number x,
the fact that f(x) is exactly representable or not may be undecidable;
in theory, on such a case, one would iterate indefinitely in the Ziv
loop. For elementary functions, this is always decidable, but this is
still unknown for some special functions, such as erf.
In practice, this is much worse: even things that are decidable or
computable in theory (with unlimited space) are not necessarily in
practice to due the limited space. For instance, sin(x) on a large
value of x (possible if one has a 64-bit exponent) needs a huge
precision, which can yield a crash due to the lack of memory. And
solving the TMD on a particular f(x) might also involve a huge
precision (though it is not known whether this is possible).
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <https://www.vinc17.net/>
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Work: CR INRIA - computer arithmetic / AriC project (LIP, ENS-Lyon)