Discussion
Computer Science > Symbolic Computation
peterlk: Reminds me a bit of the coolest talk I ever got to see in person: https://youtu.be/FITJMJjASUs?si=Fx4hmo77A62zHqzyIt’s a derivation of the Y combinator from ruby lambdas
nonfamous: How would an architecture with a highly-optimized hardware implementation of EML compare with a traditional math coprocessor?
wildzzz: Dreadfully slow for integer math but probably some similar performance to something like a CORDIC for specific operations. If you can build an FPU that does exp() and ln() really fast, it's simple binary tree traversal to find the solution.
jekude: What would physical EML gates be implemented in reality?Posts like these are the reason i check HN every day
selcuka: So, like brainf*ck (the esoteric programming language), but for maths?
Lerc: But even tighter. With eml and 1 you could encode a funtion in rpn as bits.Although you also need to encode where to put the input.The real question is what emoji to use for eml when written out.
zephen: Judging by the title, I thought I would have a good laugh, like when the doctor discovered numerical integration and published a paper.But no...This is about continuous math, not ones and zeroes. Assuming peer review proves it out, this is outstanding.
hyperhello: > eml(x,y)=exp(x)-ln(y)Exp and ln, isn't the operation its own inverse depending on the parameter? What a neat find.
Analemma_: If you've never worked through a derivation/explanation of the Y combinator, definitely find one (there are many across the internet) and work through it until the light bulb goes off. It's pretty incredible, it almost seems like "matter ex nihilo" which shouldn't work, and yet does.It's one of those facts that tends to blow minds when it's first encountered, I can see why one would name a company after it.
tripdout: Interesting, but is the required combination of EML gates less complex than using other primitives?
simplesighman: > For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operationsI read the paper. Is there a table covering all other math operations translated to eml(x,y) form?
zephen: > The real question is what emoji to use for eml when written out.Some Emil or another, I suppose. Maybe the one from Ratatouille, or maybe this one: https://en.wikipedia.org/wiki/Emil_i_L%C3%B6nneberga
Charon77: Not brainf*ck. This is the SUBLEQ equivalent of math https://en.wikipedia.org/wiki/One-instruction_set_computer#S...
selcuka: So brainf*ck in binary?
jmyeet: I was curious about that too. Gemini actually gave a decent list. Trig functions come from Euler's identity: e^ix = cos x + i sin x which means: e^-ix = cos -x + i sin -x = cos x - i sin x so adding them together: e^ix + e^-ix = 2 cos x cos x = (e*ix - e^-ix) / 2 So I guess the real part of that.Multiplication, division, addition and subtraction are all straightforward. So are hyperbolic trig functions. All other trig functions can be derived as per above.
krick: > using EML trees as trainable circuits ..., I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4That's awesome. I always wondered if there is some way to do this.
thaumasiotes: > isn't the operation its own inverse depending on the parameter?This is a function from ℝ² to ℝ. It can't be its own inverse; what would that mean?
AlotOfReading: 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.
saratogacx: last page of the PDF has several tree's that represent a few common math functions.
entaloneralie: I love seeing FRACTRAN-shaped things on the homepage :) I use daily a programming language based on this idea:Rejoice is a concatenative programming language in which data is encoded as multisets that compose by multiplication. Think Fractran, without the rule-searching, or Forth without a stack.https://wiki.xxiivv.com/site/rejoice
qiller: For completeness, there is also Peirce’s arrow aka NOR operation which is functionally complete. Fun applications iirc VMProtect copy protection system has an internal VM based on NOR.Quick google seach brings up https://github.com/pr701/nor_vm_core, which has a basic idea
DoctorOetker: I'm still reading this, but if this checks out, this is one of the most significant discoveries in years.Why use splines or polynomials or haphazardly chosen basis functions if you can just fit (gradient descent) your data or wave functions to the proper computational EML tree?Got a multidimensional and multivariate function to model (with random samples or a full map)? Just do gradient descent and convert it to approximant EML trees.Perform gradient descent on EML function tree "phi" so that the derivatives in the Schroedinger equation match.But as I said, still reading, this sounds too good to be true, but I have witnessed such things before :)
gilgoomesh: > Why use splines or polynomials or haphazardly chosen basis functions if you can just fit (gradient descent) your data or wave functions to the proper computational EML tree?Same reason all boolean logic isn't performed with combinations of NAND – it's computationally inefficient. Polynomials are (for their expressivity) very quick to compute.
hyperhello: eml(1,eml(x,1)) = eml(eml(1,x),1) = exp(ln(x)) = ln(exp(x)) = x
BobbyTables2: How does one actually add with this?
nick238: Don't know adding, but multiplication has diagram on the last page of the PDF.xy = eml(eml(1, eml(eml(eml(eml(1, eml(eml(1, eml(1, x)), 1)), eml(1, eml(eml(1, eml(y, 1)), 1))), 1), 1)), 1)From Table 4, I think addition is slightly more complicated?
simplesighman: Thanks for posting that. You had a transcribing typo which was corrected in the ECMAScript below. Here's the calculation for 5 x 7: const eml = (x,y) => Math.exp(x) - Math.log(y); const mul = (x,y) => eml(eml(1,eml(eml(eml(1,eml(eml(1,eml(1,x)),1)),eml(1,eml(eml(1,eml(y,1)),1))),1)),1); console.log(mul(5,7)); > 35.00000000000001
paulpauper: I don't think this is ever making it past the editor of any journal, let alone peer review.Elementary functions such as exponentiation, logarithms and trigonometric functions are the standard vocabulary of STEM education. Each comes with its own rules and a dedicated button on a scientific calculator;What?and No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, √ , and log has always required multiple distinct operations. Here we show that a single binary operatorYeah, this is done by using tables and series. His method does not actually facilitate the computation of these functions.There is no such things as "continuous mathematics". Maybe he meant to say continuous function?The whole thing comes off a gibberish or rediscovering something that already exists.
