Tetration Forum
Tetration online calculator - Printable Version

+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Computation (https://tetrationforum.org/forumdisplay.php?fid=8)
+--- Thread: Tetration online calculator (/showthread.php?tid=1830)

Tetration online calculator - MorgothV8 - 11/14/2026

I've created a web page for tetration, super-log, super-root: https://hyp4.duckdns.org/ (it is self-hosted on my own VPS server from OVH Cloud).

There is also and API that anybody can use to get values calculated by this tool: curl -s 'https://hyp4.duckdns.org?prec=20&func=ate&base=2&value=65536'

All details are on documentation links on that page.

RE: Tetration online calculator - MorgothV8 - 11/14/2026

What the page does

https://hyp4.duckdns.org/ puts my program tet (source, thread Arbitrary Tetration in rust) on the web, so that nobody has to build it to get a number:
  • tet: the value b^^h for a complex base b and a complex height h;
  • ate (slog): the height h with b^^h = z;
  • ssrt (sroot): the base b with b^^h = z.
Each number can be typed as one complex number or as its real and imaginary part, and it may be an expression: 1.5+0.5i, pi/e, e^(1/e), sqrt(2), 2^^0.5, ate(2, 16). The precision is the number of decimal digits that you ask for; the server has no largest one of its own.

The page is a plain form, so it also works without JavaScript and in a text browser such as links. The same thing is an API that anybody may call. It answers in JSON, or with format=text in the two lines (real part, imaginary part) that the program prints:

Code:
$ curl 'https://hyp4.duckdns.org/?prec=20&func=ate&base=2&value=65536'
{"func":"ate","prec":20,"base":"2","value":"65536","result":"height","re":"4.0000000000000000000","im":"0","seconds":1.721,"cached":false,"command":"tet --ate 20 2 65536"}
$ curl 'https://hyp4.duckdns.org/?func=tet&base=2&height=0.5&format=text'
1.4587818160364217007
0
$ curl 'https://hyp4.duckdns.org/?prec=30&func=ssrt&value=16&height=3&format=text'
2.00000000000000000000000000000
0
$ curl 'https://hyp4.duckdns.org/?func=tet&base=1.5+0.5i&height=1+0.7i&format=text'
1.1766384980936338172
6.0310294707303037947e-1

It runs on a small server with two processor cores, so it wants some patience. A request waits two minutes. A calculation that takes longer goes on by itself for up to three hours, and the same request later gets its result (2^^0.51 to 300 digits took four and a half minutes). Four calculations run at a time; while all four are busy the answer is "try again later". Results are kept in memory, so what somebody asked for before is answered at once.

Whose work this is

The program implements what was constructed, argued about and tested in the open on this forum, and it is checked against that:
  • Bases inside the Shell-Thron region: regular iteration (Schröder, Koenigs) at the attracting fixed point.
  • Real bases above e^(1/e): Kneser's tetration, by a theta-mapping iteration that is built on the idea of sheldonison's (Sheldon Levenstein) programs, the fast accurate Kneser sexp algorithm and fatou.gp. fatou.gp is also my independent reference: 2^^0.5 agrees with it in all 100 digits that I compared.
  • Complex and negative bases outside the Shell-Thron region, and the real bases 0 < b < e^(-e): the two-sided form of that iteration, in the two-fixed-point formulation of the papers of William Paulsen and Samuel Cowgill, and after sheldonison's complex-base program.
  • Dmitrii Kouznetsov's Cauchy-integral construction, which was the heart of the program until this month and is still its fallback and its independent cross-check.
  • The uniqueness theorem of bo198214 (Henryk Trappmann) and Kouznetsov for holomorphic Abel functions at a pair of fixed points, without which there would be no "the" tetration to aim at.
And around those: mike3 (Cauchy Integral Experiment, tetration for ALL bases, real and complex), andydude (Andrew Robbins: the natural slog and Designing a Tetration Library), Gottfried Helms (the matrix operator approach and the fixpoint comparisons), jaydfox (Jay D. Fox: accelerated slog), JmsNxn (James Nixon: the beta method and infinite compositions), tommy1729, Ember Edison, MphLee, and everybody whose questions and counterexamples here decided what "getting it right" means. The list with all links is at the top of the README.

Thank you all, and thank you bo198214 for keeping this forum going for so many years. As I wrote in the other thread, the code was written with a lot of help from AI models; the ideas are yours.

What it is not

It is not a proof and not a certified bound. The digits are checked by convergence, by the functional equation, on independent contours and, where there is one, against an independent reference. That is evidence, not a guarantee. Not everything is covered either: bases on the Shell-Thron boundary itself (a neutral fixed point, e^(1/e) included) are not, and close to that boundary a calculation can take very long.

If a value looks wrong to you, or differs from what your own program gives, please tell me here. That is the most useful thing that can happen to it.

Documentation: the page and the API, the program, its methods and their limits.