find operators that are stronger than pow but weaker than pentation (then implement them)

[Patashu]

pow and tetration are 'boring' for reaching higher layers in the sense that they just count up layers one at a time past a certain point - everything devolves into min, max, cmp, succ and add operations. but pentation is way too strong, lacks research into non-integer arguments and explodes way too rapidly for even tiny integers.

what would be some kind of in-between operator that e.g. a crazy incremental game could use to surge up the layers to 1^^1.8e308?

real numbered hyper operators > 3 and < 5? (0 is successor, 1 is add, 2 is mul, 3 is pow, 4 is tetrate, 5 is pentate.)

functions arbitrarily crafted to have approximately hyper operator >= 0 <= 3 strength behaviour on the layers of their arguments, with magnitudes interpreted as partial layers?

[Patashu]

Maybe we could find inspiration in big O notation and ordinal collapsing functions/fast growing hierarchies/slow hierarchies, which are basically the language for describing how fast functions grow.

https://www.reddit.com/r/incremental_games/comments/2ztcfk/linear_polynomial_exponential_and_more_growth/

https://oeis.org/wiki/Growth_of_sequences

https://en.wikipedia.org/wiki/Big_O_notation

http://googology.wikia.com/wiki/Fast-growing_hierarchy / http://googology.wikia.com/wiki/Slow-growing_hierarchy / http://googology.wikia.com/wiki/Introduction_to_ordinal_collapsing_functions

[marath007]

I had considered a sort of loglift but it behaved weird. essentially i was doing 10^(log10(num1)*log10(num2)) for a multiplication and increase the amount of logs as needed.

other solutions I had is operations on the top exponent directly.

but both solutions lacks of smooth transitions.

[aarextiaokhiao]

Ideas: Basic layer X logarithm operators.

For logAdd(value, log, layer) function, 'log' variable is added to 'layer'th layer logarithm of 'value' variable. Same as logSub, logMul, logDiv, logPow, and logRoot.

Examples:
Decimal.logAdd(1e10, 5, 1) = 1e(10+5) = 1e15
Decimal.logSub(1e10, 4, 1) = 1e(10-4) = 1e6
Decimal.logMul(1e9e15, 10, 2) = 1e1e(log10(9e15)*10) ~= 1e3e159
Decimal.logDiv(1e1e1e10, 5, 1) = 1e(1e1e10/5) ~= 1e1e1e10

[Patashu]

https://sites.google.com/site/pointlesslargenumberstuff/home/2/weakoperators is an interesting article that describes 'Weak' Hyperoperators. So called 'Strong' Tetration and Pentation evaluates right to left so as to get the most massive numbers, but the 'Weak' hyper operators go left to right and get results that grow slower. It takes about two 'Weak' hyperoperators to match the boost from one 'Strong' hyperoperator, so:

• weak tetration is between exponentiation and tetration (and is very easy to define for continuous heights!)
• weak pentation is on par with tetration
• weak hexation is between tetration and pentation
• weak heptation is on par with pentation

http://mrob.com/pub/math/hyper4.html#lower4 MROB also explores weak tetration, and even explains how to compute weak sroot and weak slog (the two inverse operators). Cool!

[Patashu]

https://math.blogoverflow.com/tag/super-square-root/ Since I don't have a better place to put this, identities and growth rate for tetration. https://andydude.github.io/tetration/archives/tetration2/ident.html Some more.

[Patashu]

random idea would be: you could slog both numbers, then do some math on the slogs, then base 10 tetrate them back into hueg numbers. but then aren't you just doling linear operations and making it look superexponential, kind of like how true exponential eventually just has a level that's 'your number, as an exponent'? maybe, but if you mix math-on-slogs and normal operations maybe something interesting could come out of it.

[Patashu]

articles on fractional hyperoperators, since I think for the most part it does come down to 'can we define operators with non-integer rank, since anything else is just combinations of existing operators rather than a fundamentally new one':

https://www.hindawi.com/journals/mpe/2016/4356371/ Fractional Mathematical Operators and Their Computational Approximation
https://math.stackexchange.com/questions/1269643/continuum-between-addition-multiplication-and-exponentiation Continuum between addition, multiplication and exponentiation?
https://math.stackexchange.com/questions/107245/continuum-between-linear-and-logarithmic continuum between linear and logarithmic
http://tetration.org/Ackermann/index.html http://tetration.org/Dynamics/index.html tetration.org - Continuous Ackermann Function and Continuous Iteration
https://math.stackexchange.com/questions/1227761/example-x-y-and-z-values-for-x-uparrow-alpha-y-z-where-alpha-in-bbb/1241979#1241979 On non-integer rank Hyperoperations
https://arxiv.org/abs/2106.03935 A Family of Bounded and Analytic Hyper-Operators