Iterated Log Coding

It reminds me of schemes like <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Elias_delta_coding" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Elias_delta_coding</a> , <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Elias_omega_coding" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Elias_omega_coding</a>

It&#x27;s a neat encoding but the writeup is needlessly confusing. The introduction would <i>really</i> benefit from a graph or two. Or even just a textual sequence showing how the value evolves as you iterate on it.<p>Similarly, while the end result is quite elegant the consequences of the choices made to get there aren&#x27;t really explained. It&#x27;s quite a bit of extra effort to reason out what would have happened if things had been done differently.

How feasible is it to do arithmetic directly in this representation?

Given an integer n, what is the number of bits required to represent every integer in the range -n..n ?

This feels like you can use e (or I guess any number) for the logarithm base rather than 2. I wonder if that’d make it a bit more uniform in some sense.

Isn&#x27;t this just a variant of Dirac&#x27;s solution for representing any number using sqrt, log, and the number 2?

Neat, looks like it would work well for values clustered around 0.5, and which hang more closely around 0 and 1.

Nice and elegant! Surprising it was not discovered earlier (unless it was and we&#x27;re unaware of it).

&gt; Any value representable in n bits is representable in n+1 bits<p>&gt; [1 1 1 1 1 1 1] 2.004e+19728<p>Does that mean that the 8 bits version has numbers larger than this? Doesn&#x27;t seem very useful for 10^100 is already infinity for all practical purposes.

I thought it was some log file hackery, but it&#x27;s actually about logarithms.

holy Father! this is amazing, the things a human mind can do, ask an LLM for that and it&#x27;ll give you 10 more complicated and unmaintainable ways to do it.

Cute! I wonder if it would be amenable for use as a variable-width encoding for, say, DCT coefficients in a JPEG-like codec..?

Could be good for AI representation theory, cool idea

Feels a bit like binary encoding with + and - instead of 0 and 1