Tetration/iteratedexp has incorrect behavior for large heights on small bases with large payloads

[MathCookie17]

Tetration/iteratedexp has incorrect behavior for large heights on small bases with large payloads

[MathCookie17]

OK, this is turning out to be harder than I thought. I've coded up a solution that I think works for integer-height (and infinite-height) tetration, but in order to do fractional height I need layeradd to work for these small bases, which means I need an idea of what slog means on these smaller bases. slog_1.3(20) sure seems meaningless since no power tower of only 1.3s will ever get up to 20, but whatever it is, slog_1.3(1.3^20) (1.3^20 is around 190.0496377) should be 1 more than it.

The solution I'm thinking of is to write a function called "excess_slog", so named because it handles numbers that are above ("in excess of") the limit of usual slog for these smaller bases. excess_slog returns a pair (an array of two entries since this is TypeScript rather than a language with built-in tuples) whose first entry is a Decimal and whose second entry is a number - that number is either 0, 1, or 2. If it's 0, that means it's the usual slog, which operates in the lower range (the value is below the lower b^x = x solution) - for example, excess_slog(1.4, 1.3), i.e. the base 1.3 slog of 1.4, would return [1.9393492860381027, 0]. If the number is 1, that means we're in the middle range (the value is between the two b^x = x solutions - in this range, slog of the upper solution is -Infinity and slog of the lower solution is +Infinity, so slog is actually decreasing), and if the number is 2, that means we're in the upper range (slog of the upper solution is -Infinity here too). What I'm unsure of is what should be set as the 0 value for each range. For range 0, obviously excess_slog(1) is [0, 0], but what about the other two? I'm thinking I use the geometric mean of the two b^x = x solutions as the 0 value for range 1 (since those are the endpoints of range 1), but since range 2 only has a left endpoint (it shoots off to infinity on the right), I can't think of any choice for excess_slog(x) = [0, 2] that isn't arbitrary - the ideas that have come to my mind all involve the upper b^x = x solution (I'll call it "u" here), such as u + 1, u * 2, u^2, or perhaps u * l (where l is the lower b^x = x solution), but all of those are pretty arbitrary.

Any thoughts on this idea? Is the two-element array approach to return both a value and the range it's in a good idea, or do you have a better one? What should I use as the 0 values in the middle and high ranges of this "excess_slog" function? And if I do make the excess_slog function, should it be public like the rest of the Decimal functions, or do we say "every other Decimal function returns something like a number, a boolean, or another Decimal, an array of a Decimal and a number would be too confusing, so just make it private"?

[Patashu]

Hmm, this sounds a bit above my paygrade (unless I spent time of my own researching it). It'd be great if we had arbitrary analytical tetration at this point but it remains hard to port to javascript :p I would suggest:

• This sounds extremely like an internal implementation detail, so keep it private.
• However you implement it, come up with some unit tests that make reasonable assertions on how it works. That way, it doesn't matter how it's implemented as long as it satisfies the unit tests.

[MathCookie17]

Fixed.