Efficient Symmetric level-index arithmetic in D?

[H. S. Teoh]

On Fri, Mar 01, 2019 at 03:12:15PM +0000, Simen Kjærås via Digitalmars-d wrote:
> On Friday, 1 March 2019 at 15:02:04 UTC, Simen Kjærås wrote:
> > It is very unlikely that a software implementation would be useful
> > for normal numbers, but may possibly be useful for some esoteric
> > calculations, maybe.
> 
> For an example of when such a representation may be useful, Numberphile had
> a video on the number 10^10^10^10^10^1.1:
> 
>   https://youtu.be/1GCf29FPM4k

That's a pretty "small" number in the grand scheme of things. Look up
Graham's number, for example, and be boggled at how many times it
explodes your mental visualization of "infinity" and yet it's still
*finite*. :-D (And if it doesn't cause your brain to explode many times
over, you haven't truly appreciated how huge it is -- read it again.
:-P)

And BTW, Graham's number G is beyond even the reach of SIL -- many
orders of representation beyond. (Yes I said "orders of representation"
rather than "orders of magnitude": orders of magnitude are
insignificant(!) when talking about such huge numbers!) It's so big that
the difference between G, factorial(G), and e^G is basically
indiscernible -- i.e., these operations don't significantly change its
general magnitude such that we can meaningfully tell them apart. They're
mathematically different, certainly, but that difference is miniscule
(indeed, indiscernible) compared to the general, unimaginably humongous
magnitude that they all lie in.


> To give an idea of the accuracy of the calculations for this kind of
> number, the units for that number, in a scientific paper, is given as
> 'Planck times, millennia, or whatever'. A factor of 10^55 is
> inconsequential here. That would never be the case for floating points
> numbers, where the accuracy is always a certain percentage of the
> absolute value.
[...]

Yes. When you're dealing with numbers of the magnitude of Ackermann's
function of Graham's number, a difference of 10^55 is not even on the
radar, it's basically non-existent. "Roundoff error" would be the
understatement of the millenium (or several millenia!). :-D


T

-- 
"No, John.  I want formats that are actually useful, rather than over-featured megaliths that address all questions by piling on ridiculous internal links in forms which are hideously over-complex." -- Simon St. Laurent on xml-dev