I did a bit of research earlier in my life[0] to study the handling of loops and without using Monte Carlo simulation. The result was actually workable if incredibly resource intensive to the point of being impractical. If I had chosen to do it again, I might’ve accepted using Monte Carlo simulations while still supporting loops.

[0]: https://github.com/kccqzy/probabilistic-program-inference/bl... Shameless self promotion I know! I put quite a bit of effort into that README and the code.

The engine is Rust, the JIT is built on Cranelift, there is also a WASM backend so everything runs in the browser too.

Full disclosure, I could only finish it now because of AI agents. In my experience they are amazing at the runtime and the numerical code, but pretty bad at language design, so I kept that part for myself.

It's a toy language. Ask me anything!

I remember encountering this idea written in a book written by Ed Catmull of Pixar fame (can't find the title sorry, but it was written in the 80s), but generally comes from signal processing as a way of avoiding aliasing artifacts..

The core idea is to make programming, which is a discrete and discontinuous domain, into a well-behaved band limited signal. Otherwise you get aliasing (or jaggies), which can happen even INSIDE a surface, if the shader's like that.

The code idea for this is the step function which is the integral of the dirac delta. step(x) returns 1 for all x >0 and 0 otherwise. Step is not a well-behaved function in the sense, that it changes infinitely quickly at x=0. But once we know what we want, we can replace it with something like that, that's well behaved.

Consider the example pseudocode

     color = x> 5? green:blue;

Now if we put the code into a shader, we get jaggies. So to combat the value changing infinitely fast, we go for a function that's like step, but changes smoothly* from 0 to 1 around x=0. Enter smoothstep: color = blue + smoothstep(x-(5+EPSILION),(x-EPSILON), x)*(green-blue)

And so we defined a 'transition zone' of +-EPSILON(an arbitrary number). While any smooth function can work, smoothstep is chosen because it has a smooth first and second derivative (meaning even if you want to get the rate of change, something that often pops up in computer graphics, the result will be still well behaved).

Pixar's Renderman shading language (which is remarkably similar to GLSL/HLSL/C), used to do this automatically for you. Essentially it could take arbitrary code peppered with if statements, and turn it into a continuous function.

Which is kinda cool imo.

It's also a cool trick in the age of AI. Since you have a function that's well-behaved, you can do things like gradient descent to train an AI to synthetize a function for you. You can even say, that you don't need exact results, you can accept some error.

In this case your program optimization problem can be reframed from doing idempotent transformations on the list of instructions, to getting a program that generates a target function whose error is no greater than some (mathematical) reference function.

Seems worth an investigation and maybe mention on the article.