NoiseLang: Where N = 5 is a Dirac delta

During my telecommunications degree I took a course on signals and noise, I spent a lot of evenings writing probability by hand: expectations, variances, the odds of two random variables landing in some region. It always sucked, when I tried to run it on a computer, so much boilerplate.

That wish became NoiseLang. I started it about nine years ago, however, I never finished it. Only recently, I brought it back thanks to AI tools and something far more ambitious than what I could have built alone the first time.

Everything is a distribution

The whole language hangs on one idea, that every value is a probability distribution. A plain number is a Dirac spike, a distribution with all its weight on a single value. Since constants and random variables are the same kind of object, every operator in the language maps distributions to distributions.

A name always refers to one fixed node, the same way X is the same X across a whole page of math. So X + X is 2X and X - X is exactly 0,. If you want variable independence you write separate draws, ie, using ~ multiple times, or ~[N] to draw N independent variables into a vector.

X ~ unif_int(1, 6)
Y ~ unif_int(1, 6)
X + Y                 # two independent dice, a real 2d6 distribution

Nothing runs until you ask for some results, for example, P(X + Y < 10), at that moment it forces the runtime to run millions of simulations (across all cores, if available) and return an estimate with a standard error attached.

Bday = unif_int(1, 365)
days ~[23] Bday          # 23 people in a room
P(has_duplicates(days))  # the birthday paradox, about 0.507

This is much easier to watch than to describe, so I built some cool demos!

Every value is a distribution

A = 5D1 ~ unif_int(1, 6)D2 ~ unif_int(1, 6)S  = A + D1 + D2E(S)

draws 0 A ≈ —

Figure 1. The same histogram through four steps: a constant is a single spike, the tilde spreads it into a die, adding two dice curves it toward a bell, and a query reads the mean back.

A number is a distribution

A = 5 looks like a plain constant, and it is, but Noise sees it as a probability distribution where every draw lands on 5. Statisticians call this a Dirac delta, a single infinitely-thin spike.

The tilde spreads it out

D1 ~ unif_int(1, 6) binds a fair die, a discrete uniform where all six faces are equally likely. The spike fans out into six flat bars, so D1 is now uncertain, but you still write it like any other variable.

Join them and a bell appears

Add the constant and two independent dice with S = A + D1 + D2. The flat shapes convolve into a triangle peaked at 12, already curving toward a bell, and if you join a few more they sharpen into a true Gaussian. That's the central limit theorem showing up for free!

A query collapses millions of draws

Nothing runs until you ask. E(S), the expected value, fires a Monte Carlo pass that averages millions of rolls into one number, and by the law of large numbers the running mean settles on 12. You wrote a few lines of math, and an expert-level kernel ran underneath.

Why it sat for nine years

The design was never the hard part, because a parser and a tree-walking interpreter for this language is a weekend of work. The problem was everything else, writing a efficient Monte Carlo runtime, instead of a naive interpreter, conditional bayesian inference, and more.

Current version is a compiler, a JIT (using the amazing Cranelift), a WASM backend, and a pile of careful numerical code, so for a cute-weekend project it stayed permanently out of reach.

Building the ambitious version with an agent

At my day job and side projects, I am experimenting with the boundaries of what today’s AI agents can do. For example, I am also porting a game I built 15 years ago for iOs, in archaic Objective-C to a modern game engine (with relative success).

With NoiseLand, I realized, AI is great at building the JIT parts, the runtime parts, the numerical parts, but it sucks at coming up with good language design ideas, many times overriding existing language features for different purposes, or coming with with different syntax for non-orthogonal features.

One IR, three backends

Under the hood, ~ and the distribution constructors build an append-only DAG called the RvGraph. This graph is the single source of truth, which later are converted into three different code paths:

a columnar batch interpreter that works everywhere and acts as the correctness oracle;
a Cranelift JIT that fuses a whole expression into one native kernel;
a WASM emitter that does the same for the browser.

One shared module defines what the graph means, so the two code generators stay thin and cannot drift apart. Anything a backend can’t successfully compile falls back to the interpreter, and the results stay identical across backends and core counts. All tests run in all three code paths and compared to be bit-identical.

Making the Monte Carlo loop cheap

All the performance work is about one loop: draw a few million samples, evaluate the expression on each, and reduce the results. A handful of techniques carry most of it, while keeping the results deterministic (that was the hard part).

Kernel fusion keeps every intermediate value in registers, so an arithmetic-heavy expression stay on registers. The PRNG (xoshiro256++) compiles into the kernel, and the ln, sin, and cos become inline polynomial approximations, speeding up the kernel by a factor of 2.

My favorite trick is in the RNG. Generating random numbers is a serial dependency chain, so instead of fighting that, the kernel runs four independent streams at once and lets the out-of-order core overlap them. This trick ended up beating a hand-written SIMD kernel!

On my 14-core M4 Pro, a one-line P(...) sustains around 5.8 billion samples per second and scales about 9.6× from one core to all of them. Per core, the generated kernel runs within about 1.15× of hand-written Rust compiled by LLVM. The same fused loop, emitted as WASM, runs at roughly half to three-quarters of native speed inside V8.

X ~ rand::unif(-1, 1)Y ~ rand::unif(-1, 1) C = X^2 + Y^2 < 1 4 * P(C)

darts 0 inside 0 π ≈ —

Figure 4. Monte Carlo π. The circle covers π/4 of the square, so four times the teal share converges on π as the darts accumulate.

Two random draws

This is the same ~ as before, used twice. X ~ unif(-1, 1) and Y ~ unif(-1, 1) each draw a number anywhere in [-1, 1], and together they pick a random point in the square.

Ask one yes/no question

C = X² + Y² < 1 is true exactly when the point falls inside the unit circle. In Noise that comparison is itself a random variable, a Bernoulli that lands true or false on each draw.

Now throw the darts

Each dart is one draw of (X, Y), teal when it lands inside the circle and grey when it doesn't. They scatter evenly, because the uniform draws don't favor any point over another.

Four times the fraction is π

The circle covers π/4 of the square's area, so the teal share hovers around π/4, and that makes 4 * P(C) an estimate of π. The more darts you throw, the sharper the estimate gets.

Where Noise sits

NoiseLang is a toy language, you probably should not use it for anything serious, however I wish this language existed during my university days.

For a language nerd, it’s a small, static random-variable algebra with forward Monte Carlo, expression-based, rejection-based conditioning, language.

You might ask, how does it compare to NumPy or Stan? NumPy makes you write the simulation yourself, and Stan makes you declare a model and wait for a sampler. Noise lets you write the probability as math while running Monte Carlo under the hood to get the answer.

Table 1. Where Noise sits. Stan and PyMC beat it at fitting a posterior to lots of data, and NumPy beats it at raw array crunching, but Noise gets you from a probability question to a visible answer faster than any of them.

Stan and PyMC beat Noise at the thing they’re built for, fitting a posterior to lots of continuous data with their HMC/NUTS samplers, and NumPy beats it at raw array crunching. Conditioning in Noise is rejection-based, so it works great for a handful of discrete observations but becomes useless for ten thousand continuous measurements, and there is no stateful simulation yet (no Markov chains yet). Where Noise wins when you have a probability question and you wanna know the answer without much hassle.

So use Noise for the whiteboard stage of a problem, when you want to run the math you just wrote, and move to Stan or PyMC when you need a real posterior, or to NumPy and JAX when you need to go to production.

Back to signals and noise

Going back to my university days, there was this subject called “Señales Aleatorias Y Ruido”, which is a spanish translation of “Random Signals and Noise”.

Cover of the textbook Señales Aleatorias y Ruido — The textbook, in all its glory.

This subject was the nightmare of many students, including myself, in fact, I failed it. Truth be told, I didn’t put enough effort into it during the first year, but it changed when I had the take the same subject again in the second year. The professor was great, and made the subject interesting. The things that blew my mind was how he could model why FM survives a noisy channel when AM doesn’t.

So, here is my tribute to the subject, a one-screen Noise program that models why FM survives a noisy channel when AM doesn’t.

msg = 0.3 * signal::sine(3); am_modulate(m)        = 1 + m;fm_modulate(m, swing) = exp(i * swing * m); static ~ signal::noise_white_complex(0.4);rx_am = am_modulate(msg) + static;rx_fm = fm_modulate(msg, 3) + static; am_demodulate(c)        = abs(c) - 1;fm_demodulate(c, swing) = arg(c) / swing;rec_am = am_demodulate(rx_am);rec_fm = fm_demodulate(rx_fm, 3); Print("AM error", E(mse(rec_am, msg)));Print("FM error", E(mse(rec_fm, msg)));

scroll through the steps →

Figure 3. AM vs FM end to end. The carrier is a phasor, AM writes the message into its length and FM into its angle, so the same static corrupts what AM reads while barely touching what FM reads.

The message

In Noise a value can carry a whole signal, and here it's a gentle tone, a slow three-cycle sine. This is what we want to send through a noisy channel and get back intact.

AM puts the message in the length

The carrier is a complex number, a spinning arrow. AM's modulator puts the message in the arrow's length, so the tip slides in and out, crossing the unit circle as the message rises and falls.

FM puts it in the angle

FM puts the same message in the angle instead, so the tip rides along the unit circle while its length stays fixed. The information is identical, just encoded in rotation.

Add the same static

Now identical white noise hits both tips. It smears AM along exactly the radius it reads from, but it only nudges FM around the circle, barely changing the angle. You can see it in the shape of the two clouds.

Recover

Demodulate: read the length back for AM and the angle back for FM, laid over the original message. You can already see it, the AM trace comes back ragged while the FM one hugs the original.

Same storm, very different damage

mse averages the error over the waveform, and E averages that over many draws of the static. Same signal energy, same noise energy, yet FM comes back about 5× cleaner, and in the small-noise limit the advantage approaches swing² = 9. That is the bandwidth it spent on the angle paying off.

Run NoiseLang in the browser

Everything above runs on @noiselang/core, thanks to the Rust engine compiled to WebAssembly.

npm install @noiselang/core

import { run } from "@noiselang/core";

const result = await run(`
  X ~ rand::unif(-1, 1);
  Y ~ rand::unif(-1, 1);
  4 * P(X^2 + Y^2 < 1)
`);

console.log(result.value); // "3.1415…" — the last statement's value
console.log(result.output); // everything Print(...) emitted

run never throws, failures come back on result.error with a source span. There is also runWithIntrospection, the API behind the variable inspector at noiselang.com.

NoiseLang is playable in the browser at noiselang.com. Open it, type X ~ unif(-1, 1); Y ~ unif(-1, 1); 4 * P(X^2 + Y^2 < 1), and watch a few million draws estimate π from your browser tab.