Actually we know this for a long time. The universal approximation theorem states that any arbitrary function can be modelled through a nonlinear basis function so long as capacity is big enough. The practical bit here is knowing how many basis functions can be approximated with a two operators. That’s new!

eh, i didnt find that paragraph very helpful. it just restates what it means do decompose an expression into another one only relying on eml, and vaguely gestures at what this could mean, i was hoping for something more specific.

It's potentially useful for computer algebra with complex numbers - we might be able to simplify formulas using non-standard methods, but instead via pattern matching. We might use this to represent exact numbers internally, and only produce an inexact result when we later reduce the expression.

Consider it a bit like a "church encoding" for complex numbers. I'll try to demonstrate with an S-expression representation.

---

A small primer if you're not familiar. S-expressions are basically atoms (symbols/numbers etc), pairs, or null.

    S = <symbol>
      | <number>
      | (S . S)      ;; aka pair
      | ()           ;; aka null
    

There's some syntax sugar for right chains of pairs to form lists:

    (a b c)          == (a . (b . (c . ()))   ;; a proper list
    (a b . c)        == (a . (b . c))         ;; an improper list
    (#0=(a b c) #0#) == ((a b c) (a b c))     ;; a list with a repeated sublist using a reference

---

So, we have a function `eml(x, y) and a constant `1`. `x` and `y` are symbols.

Lets say we're going to replace `eml` with an infix operator `.`, and replace the unit 1 with `()`.

    C = <symbol>
      | <number>
      | (C . C)      ;; eml
      | ()           ;; 1

We have basically the same context-free structure - we can encode complex numbers as lists. Let's define ourselves a couple of symbols for use in the examples:

    ($define x (string->symbol "x"))
    ($define y (string->symbol "y"))

And now we can define the `eml` function as an alias for `cons`.

    ($define! eml cons)

    (eml x y)
    ;; Output: (x . y)

We can now write a bunch of functions which construct trees, representing the operations they perform. We use only `eml` or previously defined functions to construct each tree:

    ;; e^x

        ($define! exp     ($lambda (x) (eml x ())))
        
        (exp x)
        ;; Output: (x)
        ;; Note: (x) is syntax sugar for (x . ())

    ;; Euler's number `e`

        ($define! c:e     (exp ()))
        
        c:e          
        ;; Output: (())
        ;; Note: (()) is syntax sugar for (() . ())

    ;; exp(1) - ln(x)

        ($define! e1ml    ($lambda (x) (eml () x)))
        
        (e1ml x) 
        ;; Output: (() . x)

    ;; ln(x)

        ($define! ln      ($lambda (x) (e1ml (exp (e1ml x)))))
        
        (ln x)
        ;; Output: (() (() . x))

    ;; Zero

        ($define! c:0      (ln ()))
        
        c:0
        ;; Output: (() (()))

    ;; -infinity

        ($define! c:-inf   (ln 0))
        
        c:-inf
        ;; Output: (() (() () (())))

    ;; -x
        
        ($define! neg      ($lambda (x) (eml c:-inf (exp x))))

        (neg x)
        ;; Output: ((() (() () (()))) x)
        
    ;; +infinity
    
        ($define! c:+inf   (neg c:-inf))
        
        c:+inf
        ;; Output: (#0=(() (() () (()))) #0#)
        
    ;; 1/x
    
        ($define! recip    ($lambda (x) (exp (eml c:-inf x))))
        
        (recip x)
        ;; Output: (((() (() () (()))) . x))
  
    ;; x - y
    
        ($define! sub      ($lambda (x y) (eml (ln x) (exp y))))
    
        (sub x y)
        ;; Output: ((() (() . x)) y)
    
    ;; x + y
    
        ($define! add      ($lambda (x y) (sub x (neg y))))
    
        (add x y)
        ;; Output: ((() (() . x)) ((() (() () (()))) y))
    
    ;; x * y
    
        ($define! mul      ($lambda (x y) (exp (add (ln x) (exp (neg y))))))
        
        (mul x y)
        ;; Output: (((() (() () (() . x))) (#0=(() (() () (()))) ((#0# y)))))
        
    ;; x / y
    
        ($define! div      ($lambda (x y) (exp (sub (ln x) (ln y)))))
        
        (div x y)
        ;; Output: (((() (() () (() . x))) (() (() . y))))
        
    ;; x^y
    
        ($define! pow      ($lambda (x y) (exp (mul x (ln y)))))
  
        (pow x y)
        ;; Output: ((((() (() () (() . x))) (#0=(() (() () (()))) ((#0# (() (() . y))))))))
  

I'll stop there, but we continue for implementing all the trig, pi, etc using the same approach.

So basically, we have a way of constructing trees based on `eml`

Next, we pattern match. For example, to pattern match over addition, extract the `x` and `y` values, we can use:

    ($define! perform-addition
        ($lambda (add-expr)
            ($let ((((() (() . x)) ((() (() () (()))) y)) add-expr))
                (+ x y))))  

    ;; Note, + is provided by the language to perform addition of complex numbers

    (perform-addition (add 256 512))
    ;; Output: 768

So we didn't need to actually compute any `exp(x)` or `ln(y)` to perform this addition - we just needed to pattern match over the tree, which in this case the language does for us via deconstructing `$let`.

We can simplify the defintion of perform-addition by expanding the parameters of a call to `add` as the arguments to the function:

    ($define! $let-lambda
        ($vau (expr . body) env
            ($let ((params (eval expr env)))
                (wrap (eval (list* $vau (list params) #ignore body) env)))))
                
    ($define! perform-addition
        ($let-lambda (add x y)
            (+ x y)))

    ($define! perform-subtraction
        ($let-lambda (sub x y)
            (- x y)))


    ($define! sub-expr (sub 256 512))
    ;; Output: #inert
    sub-expr
    ;; Output: ((() (() . 256)) 512)

    (perform-subtraction sub-expr)
    ;; Output: -256

There's a bit more work involved for a full pattern matcher which will take some arbitrary `expr` and perform the relevant computation. I'm still working on that.

Examples are in the Kernel programming language, tested using klisp[1]

[1]:https://github.com/dbohdan/klisp