Comment by testaccount28
12 hours ago
derivation of -x seems wrong. we can look at the execution trace on a stack machine, but it's actually not hard to see. starting from the last node before the output, we see that the tree has the form
eml(z, eml(x, 1))
= e^z - ln(eml(x, 1))
= e^z - ln(e^x)
= e^z - x
and the claim is that, after it's expanded, z will be such that this whole thing is equal to -x. but with some algebra, this is happening only if
e^z = 0,
and there is no complex number z that satisfies this equation. indeed if we laboriously expand the given formula for z (the left branch of the tree), we see that it goes through ln(0), and compound expressions.
x^-1 has the same problem.
both formulae work ...sort of... if we allow ln(0) = Infinity and some other moxie, such as x / Infinity = 0 for all finite x.
yeah, it's annoying that author talks about RPN notation, but only gives found formulas in form of images
looks like it computes ln(1)=0, then computes e-ln(0)=+inf, then computes e-ln(+inf)=-inf
ah, the paper acknowledges this. my bad for jumping to the diagrams!
On page 11, the paper explicitly states:
> EML-compiled formulas work flawlessly in symbolic Mathematica and IEEE754 floating-point… This is because some formulas internally might rely on the following properties of extended reals: ln 0 = −∞, e^(−∞) = 0.
And then follows with:
> But EML expressions in general do not work ‘out of the box’ in pure Python/Julia or numerical Mathematica.
Thus, the paper’s completeness claim depends on a non-standard arithmetic convention (ln(0) = -∞), not just complex numbers as it primarily advertises. While the paper is transparent about this, it is however, buried on page 11 rather than foregrounded as a core caveat. Your comment deserves credit for flagging it.
I would not call a "non-standard arithmetic convention" that ln(0) = -∞.
This is the standard convention when doing operations in the extended real number line, i.e. in the set of the real numbers completed with positive and negative infinities.
When the overflow exception is disabled, any modern CPU implements the operations with floating-point numbers as operations in the extended real number line.
So in computing this convention has been standard for more than 40 years, while in mathematics it has been standard for a couple of centuries or so.
As always in mathematics, when computing expressions, i.e. when computing any kind of function, you must be aware very well which are the sets within which you operate.
If you work with real numbers (i.e. in a computer you enable the FP overflow exception), then ln(0) is undefined. However, if you work with the extended real number line, which is actually the default setting in most current programming languages, then ln(0) is well defined and it is -∞.
The author does address this further on page 14 of SI and provides an alternative of:
−z = 1 − (e − ((e − 1) − z))