Efficient Symmetric level-index arithmetic in D?

[Simen Kjærås]

On Friday, 1 March 2019 at 14:00:46 UTC, Olivier FAURE wrote:
> On Wednesday, 27 February 2019 at 20:47:34 UTC, Michelle Long 
> wrote:
>> https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4927227/
>> https://en.wikipedia.org/wiki/Symmetric_level-index_arithmetic
>
> tl;dr ?

It's a different way of storing and operating on numbers, 
essentially.

For redonkulously humongous numbers, IEEE-754 is not good enough. 
You know scientific notation? Imagine repeated scientific 
notation: 10^10^...^10^10^A. By having A in [0, 1), we don't need 
a number before the first 10 as we're used to, and we'll just add 
more steps in the exponentiation tower to get A small enough. For 
instance, 2 is 10^0.301. Also, since we're mathy, we like using e 
instead of 10. It's more... natural. :p

So, the number 1234567 can be written as approximately 
e^e^e^0.97113083, for 3 e's and a final exponent of 0.97113083. 
Since the number of e's will always be a whole number, and the 
exponent always positive and < 1, we can add them together as 
φ(3.97113083).

We can also introduce the sign bit back in, and have 
-φ(3.97113083) represent the number -1234567.

One problem now is we can only represent big numbers. We can 
introduce another bit, called the reciprocal bit, to indicate 
that a number is actually a reciprocal. Thus, -1/1234567 would be 
-φ(3.97113083)^-1. And bingo, we can represent the whole real 
number line.

That sums up what SLI is, and sorta explains when it might be 
useful. It might in fact be faster in silicon than regular 
floating point, and so may be interesting for hardware 
manufacturers. It is very unlikely that a software implementation 
would be useful for normal numbers, but may possibly be useful 
for some esoteric calculations, maybe.

I'd never heard of it before, but found it an interesting idea, 
like the unum format that has been posted here once or twice. 
Interesting on a theoretical level, but probably not useful for D.

--
   Simen