alt Hacker NewsRelated is the paper [What is a closed-form number?], which explores the field E, defined as the smallest subfield of ℂ closed under exp and log. I believe the set of numbers that can be generated using exp-minus-log is a strict subset of this.
In a similar vein to this post, the paper points out that general polynomials do not have solutions in E, so of course exp-minus-log is similarly incomplete.
What is intriguing is that we don’t even know whether many simple equations like exp(-x) = x (i.e. the [omega constant]) have solutions in E. We of course suspect they don’t, but this conjecture is not proven: https://en.wikipedia.org/wiki/Schanuel%27s_conjecture
What is a closed-form number?: http://timothychow.net/closedform.pdf omega constant: https://en.wikipedia.org/wiki/Omega_constant
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> Related is the paper [What is a closed-form number?], which explores the field E, defined as the smallest subfield of ℂ closed under exp and log. I believe the set of numbers that can be generated using exp-minus-log is a strict subset of this.
is that a typo / accidental mis-phrasing?
exp-minus-log construction is closed for the operations it supports, and spans both exp and log, so E must be either identical to or a subset of exp-minus-log; not the other way around.
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EML is spanned by a single binary operator, while the article you reference describing ("what is a closed-form number") just tacitly assumes +, -, x, / are available for free, so even in just this sense the EML construction is superior. Since EML can construct the larger presumed basic operations of E, E must be contained in it, but since the E implicitly has +, - besides exp(x) and ln(x) the reverse can also be said, so the sets and functions spanned by E and EML should be equivalent. So what is novel? precisely what the recent article describes: all the tacitly (+,-,x,/) and explicitly assumed (exp and ln) operations can be spanned with just 1 (non-unique) binary operation; and on top of that:
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the recent article describes freely available code to conduct such searches and find alternative binary operations, search for functions or constants.
The EML paper provides code and machinery to conduct a search for the value x in exp(-x)=x : use a multiprecision library to get an arbitrarily precise representation, and search for some EML expression to find candidates.