[SideQuark]

The exp and ln are infinite series. Exp is roughly the infinite series for cos AND the infinite series for sin. Hiding that every op is an infinite series behind a name doesn’t make things free. It just makes even trivial ops like 1+2 vastly more work.

[freehorse]

They are not infinite series per se. They can be represented by infinite series in several ways but there are standard ways to define them that do not involve infinite series. The logarithm in particular is not even represented by an infinite series (in form of Taylor expansion) defined in the whole complex plane. And knowledge/use of trigonometric functions greatly precedes such infinite series representations.

Moreover, the point is not always numerical computation. I don’t think anybody argues that eml sounds like an efficient way to compute elementary functions numerically. It may or may not still be useful for symbolic computations.

The article is about producing all elementary functions, which 1/(x-y) clearly doesn’t, as it doesn’t produce any transcendental function. Like many of such universality-style results it may not have practical applications, but may still be interesting on its own right.

[SideQuark]

Any transcendental function can be produced by arithmetic, since its complete for R.

Go ahead and show how to compute exp or ln without an infinite series without circular reasoning. You can’t, since they’re transcendental.

There are infinitely many ways to make these binary operators. Picking extremely high compute cost ones really doesn’t make a good basis for computation.

[freehorse]

> Any transcendental function can be produced by arithmetic, since its complete for R.

Not without some form of limit process or construction. You can approximate e with the basic arithmetic operations but not actually get an exact form in finite steps. And you definitely cannot transverse an infinite binary tree, so the main point of the result in the article is missed by your arguments.

Again, you are mixing separate things. Nobody said that eml is some way to approximate elementary functions more efficiently. It is a way to express elementary functions in a finite amount of operations. Meaning, computing symbolically, not numerically. Eg I may care that exp(3)*exp(2)=exp(5) without caring to approximate exp(5) numerically. The paper is literally under "Computer Science > Symbolic Computation", not "numerical analysis" or "engineering" after all.

And to be precise:

> Go ahead and show how to compute exp or ln without an infinite series without circular reasoning. You can’t, since they’re transcendental.

You don't necessarily need "infinite series", you need some limit process. A basic example is that exp(x) can be approximated by (1 + x/n)^n for large n. For the logarithm you can use a formula involving the arithmetic–geometric mean which you can approximate using an iterative process/recursion without infinite series. You can also approximate the exponential by using Newton's method together with that, see [0].

[0] Fast Computations of the Exponential Function https://link.springer.com/chapter/10.1007/3-540-49116-3_28

[SideQuark]

Yep, I’ve written numerical methods papers, and am very well aware of the field.

A limit process is a definition. Try computing with it. You’ll end up with an infinite sequence, or an approximation.

An iterative process is an infinite series. They’re equivalent.

Newtons method is the same. Completely equivalent to an infinite series as you increase precision.

And both require constants, infinitely precise. So you’re still not doing anything the 1/(x-y) operation cannot do, and to do those series you’ll compute using things amenable to being done via ops easy to do by hand or machine via the 1/(x-y) op.