All elementary functions from a single binary operator • Hacker News Board

Re: All elementary functions from a single binary operator

#171

Post by AlotOfReading » Mon, Apr 13, 2026, 2:19 PM UTC

xpe wrote: post #160

AlotOfReading wrote: post #24

Earlier quoted context omitted.

You already have an FPU that approximates exp() and ln() really fast, because float integer conversions approximate the power 2 functions respectively. Doing it accurately runs face-first into the tablemaker's dilemma, but you could do this with just 2 conversions, 2 FMAs (for power adjustments), and a subtraction per. A lot of cases would be even faster. Whether that's worth it will be situational.

What are some of your favorite sources to dig into this field as a whole?

Anything discussing fast inverse sqrt will go over the logarithmic side [0], as it's the key insight behind that code. Exp is just the other direction. It's not widely documented in text otherwise, to my knowledge.

[0] https://github.com/francisrstokes/githublog/blob/main/2024/5...

Re: All elementary functions from a single binary operator

#172

Post by lugao » Mon, Apr 13, 2026, 2:22 PM UTC

lmf4lol wrote: post #154

Stupid question maybe (I am no mathematician), but aren't exp and ln really primitives? Aren't they implemented in terms of +,-,/,* etc? Or do we assume that we have an infinite lookup table for all possible inputs?

I think the point here is to explore the reduction of these functions to finite binary trees using a single binary operator and a single stopping constant. The operator used could be arbitrarily complex; the objective is to prove that other expressions in a certain family — in this case, the elementary functions — can be expanded as a finite (often incomplete) binary tree of that same operation.

In other words, this result does not aim to improve computability or bound the complexity of calculating the numerical value. Rather, it aims to exhibit this uniform, finite tree structure for the entire family of elementary expressions.

Re: All elementary functions from a single binary operator

#173

Post by entaloneralie » Mon, Apr 13, 2026, 2:23 PM UTC

StilesCrisis wrote: post #130

entaloneralie wrote: post #22

This is amazing! I love seeing FRACTRAN-shaped things on the homepage :) This reminds me of how 1-bit stacks are encoded in binary: A stack of zeros and ones can be encoded in a single number by keeping with bit-shifting and incrementing. Pushing a 0 onto the stack is equivalent to doubling the number. Pushing a 1 is equivalent to doubling and adding 1. Popping is equivalent to dividing by 2, where the remainder is t…

Did you just explain base-2 numbers on the HN forums as if it were novel?

I never claimed it was novel, chill.

Re: All elementary functions from a single binary operator

#174

Post by stared » Mon, Apr 13, 2026, 2:26 PM UTC

eugene3306 wrote: post #61

This makes a good benchmark LLMs: ``` look at this paper: https://arxiv.org/pdf/2603.21852 now please produce 2x+y as a composition on EMLs ``` Opus(paid) - claimed that "2" is circular. Once I told it that ChatGPT have already done this, finished successfully. ChatGPT(free) - did it from the first try. Grok - produced estimation of the depth of the formula. Gemini - success Deepseek - Assumed some pre-existing knowl…

If you like creating such things, consider contributing to Terminal Bench Science, https://www.tbench.ai/news/tb-science-announcement.

Re: All elementary functions from a single binary operator

#175

Post by tgtweak » Mon, Apr 13, 2026, 2:27 PM UTC

This could have some interesting hardware implications as well - it suggests that a large dedicated silicon instruction set could accelerate any mathematical algorithm provided it can be mapped to this primitive. It also suggests a compiler/translation layer should be possible as well as some novel visualization methods for functions and methods.

Re: All elementary functions from a single binary operator

#176

Post by SeanSullivan86 » Mon, Apr 13, 2026, 2:29 PM UTC

entaloneralie wrote: post #22

This is amazing! I love seeing FRACTRAN-shaped things on the homepage :) This reminds me of how 1-bit stacks are encoded in binary: A stack of zeros and ones can be encoded in a single number by keeping with bit-shifting and incrementing. Pushing a 0 onto the stack is equivalent to doubling the number. Pushing a 1 is equivalent to doubling and adding 1. Popping is equivalent to dividing by 2, where the remainder is t…

Wouldn't you also need to keep track of the stack's size, to know if there are leading zeros?

Re: All elementary functions from a single binary operator

#177

Post by [deleted] » Mon, Apr 13, 2026, 2:32 PM UTC

tgtweak wrote: post #175

This could have some interesting hardware implications as well - it suggests that a large dedicated silicon instruction set could accelerate any mathematical algorithm provided it can be mapped to this primitive. It also suggests a compiler/translation layer should be possible as well as some novel visualization methods for functions and methods.

[deleted]

Re: All elementary functions from a single binary operator

#178

Post by benleejamin » Mon, Apr 13, 2026, 2:32 PM UTC

tgtweak wrote: post #175

This could have some interesting hardware implications as well - it suggests that a large dedicated silicon instruction set could accelerate any mathematical algorithm provided it can be mapped to this primitive. It also suggests a compiler/translation layer should be possible as well as some novel visualization methods for functions and methods.

I'm not too familiar with the hardware world, but does EML look like the kind of computation that's hardware-friendly? Would love for someone with more expertise to chime in here.

Re: All elementary functions from a single binary operator

#179

Post by mmastrac » Mon, Apr 13, 2026, 2:37 PM UTC

I couldn't find any information on this, but is it possible that given how nicely exponentiation and logarithms differentiate and integrate, is it possible that this operator may be useful to simplify the process of finding symbolic solutions to integrals and derivatives?

Re: All elementary functions from a single binary operator

#180

Post by freehorse » Mon, Apr 13, 2026, 2:44 PM UTC

lmf4lol wrote: post #154

Stupid question maybe (I am no mathematician), but aren't exp and ln really primitives? Aren't they implemented in terms of +,-,/,* etc? Or do we assume that we have an infinite lookup table for all possible inputs?

You have a gate (called here "eml") that takes x and y and gives `exp(x) - log(y)`. Then you implement all other operations and elementary functions, including addition, multiplication etc, using only compositions of this gate/function (and the constant 1). You don't have addition as you start, you only have eml and 1. You define addition in terms of those.