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.

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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DoctorOetker 2 days ago | parent [-]

> 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.

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:

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.

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xigoi 2 days ago | parent [-]

> 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.

Since E is by definition closed under exp, log and subtraction, it is clearly also closed under EML.

	▲	DoctorOetker 2 days ago \| parent [-]
		SabrinaJewson claims it is a STRICT subset: EML ⊂ E I remind the trivial results that both E ⊆ EML and EML ⊆ E and hence EML = E apart from construction: which is minimal for EML but highly redundant for E. the EML paper shows that this minimal construction for EML is not unique so other binary operations may be found with perhaps more interesting properties, or admitting shorter binary trees for commonly used functions and values (which may reflect subjective "simplification" of expressions in mathematics.