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Comment by throwaway81523 | Hacker News Reader

It's news to me that "elementary functions" include roots of arbitrary polynomials, but the wiki article in fact says that they're included at least some of the time. I remember reading about the Risch algorithm (for finding closed form antiderivatives) a long time ago and elementary functions were just the ordinary ones found on calculators.

Interestingly, the abs (absolute value) function is non-elementary. I wonder if exp-minus-log can represent it.

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EML can represent the real absolute value, so long as we agree with the original author's proviso that we define log(0) and exp(-∞), by way of sqrt(x^2) as f(x) = exp((1/2)log x). Traditionally, log(0) isn't defined, but the original author stipulated it to be -∞, and that all arithmetic works over the "extended reals", which makes

    abs(0)
    = f(0)            ; by defn
    = exp(1/2 log 0)  ; by defn
    = exp(-∞/2)       ; log 0 rule
    = exp(-∞)         ; extended real arith
    = 0               ; exp(-∞) rule

If we don't agree with this, then abs() could be defined with a hole punched out of the real line. The logarithm function isn't exactly elegant in this regard with its domain restrictions. :)

It's ok for elementary functions to have singularities, like 1/x at x=0. But I'm not sure what happens with your version of abs, since the log function has branches. log(1) is any of 0, 2*pi*i, 4*pi*i, etc.

abs(x) = sqrt(x*x), no?

I think the issue might be the branch cut in the sqrt function. Per the wiki article, elementary functions have to be differentiable in the complex plane at all but a finite number of points.

The origianl article use complex numbers, in particular to get sin and cos from eml:

> e^{iφ} = cosφ + i sin φ

So x may be a complex number and sqrt(x*x) is a complex number that sometimes is equal to x and sometimes to -x depending on how lucky you were selecting the branches of sqrt.

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