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Template:Tetration

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The {{tetration}} mathematical function template returns the tetration
b ↑↑ h
, a power tower with nonnegative real as base
b
, and with nonnegative integer in [0..32] as height
h
, i.e.

otherwise returns an error message.

Beware: Calculated with the MediaWiki #expr parser function extension, which has a significand precision of 53 bits (52 explicitly stored) (approximately 16 decimal digits, 53 log10 2 ≈ 15.955).[1]

Usage

{{tetration|base|height}}

or

{{^^|base|height}}

Valid arguments

If neither the base nor the height is given, the empty product 1 is returned.

  • First argument: base of power tower, as a nonnegative real number;
  • Second argument: height of power tower, as a nonnegative integer in [0..32] (defaults to 1).

Examples

Examples with valid input

Returns
b ↑↑ h
(or INF if overflow occurs, i.e. above ≈ 1.7976931348623157 x 10 308).[1]
Code Result Code Result
{{^^}} 1   {{^^|2}} 2
{{^^|0|0}} 1   {{^^|-1|0}} 1
{{^^|2|0}} 1   {{^^|2|1}} 2
{{^^|2|2}} 4   {{^^|2|3}} 16
{{^^|2|4}} 65536   {{^^|2|5}} INF
{{^^|3|3}} 7625597484987   {{^^|3|4}} INF

NOTE: {{^^|b|h}} for negative
b
gives complex numbers (except for
b =  −  1
) and we get NAN (Not A Number) since only real numbers are handled by the #expr: parser function.

{{#expr: (-1)^(-1) }} = -1
{{#expr: (-1.1)^(-1.1) }} = NAN
{{#expr: 1 / ( (-1.1)^(1.1) ) }} = NAN


Table of values of the tetration function (using {{^^|b|h}} template, except for manually entered last column entries)
-3 -0.037037037037037 NAN NAN NAN NAN NAN  
-2.5 NAN NAN NAN NAN NAN NAN  
-2 0.25 NAN NAN NAN NAN NAN  
-1.5 NAN NAN NAN NAN NAN NAN  
-1 -1 -1 -1 -1 -1 -1  
-0.5 NAN NAN NAN NAN NAN NAN  
0 1 0 1 0 1 1 undefined
0.1 0.79432823472428 0.16057272043212 0.69091922875998 0.20374209674284 0.62554405820608 0.4229202149354 0.39901297826025
0.2 0.7247796636777 0.31145890709838 0.60575856902197 0.37721845362499 0.5449235984186 0.46970963528893 0.46962192293561
0.3 0.69684530193595 0.43214988609837 0.59434417379411 0.48891128772632 0.55508396180265 0.52895696102551 0.52895644844295
0.4 0.69314484315515 0.5298707364733 0.6153797974711 0.56900457469329 0.59370446435584 0.58504317471451 0.58504317197440
0.5 0.70710678118655 0.61254732653607 0.65404086004207 0.63549784581337 0.64371864172287 0.64118574451391 0.64118574450499
0.6 0.73602192281783 0.68661684945978 0.70416577097103 0.69788152946118 0.700125434673 0.69953450919474 0.69953450919473
0.7 0.77905591267045 0.75739548555921 0.76326958144204 0.76167209672267 0.76210620802468 0.76201343081072 0.76201343081072
0.8 0.83651164207302 0.82972398762427 0.83098165747098 0.83074848273819 0.83079170890087 0.83078494866324 0.83078494866324
0.9 0.90953257608296 0.90861953913804 0.90870695076419 0.90869858185342 0.90869938310125 0.90869931309174 0.90869931309174
1 1 1 1 1 1 1 1
1.1 1.1105342410546 1.1116498000258 1.111768001503 1.1117805265341 1.1117818537373 1.1117820110418 1.11178201104184
1.2 1.244564747204 1.2547181707692 1.2570430412015 1.2575759820011 1.2576981823156 1.2577345413765 1.25773454137653
1.3 1.4064566732379 1.446293346286 1.4614888698918 1.4673271088311 1.469576402415 1.47098896009 1.47098896009015
1.4 1.6016928982022 1.7141634748254 1.780276017527 1.8203220817394 1.8450158019786 1.886662556747 1.88666330624633
1.5 1.8371173070874 2.1062033521489 2.3490053186119 2.5920257049075 2.8604414974606 INF
1.6 2.1212505710976 2.7101269178649 3.5743004801538 5.3652029167449 12.449305553378 INF
1.7 2.4646948994849 3.698164974975 7.1160358861268 43.639804309351 11395826597.324 INF

Table of values of the tetration function (using #expr: case by case, except for manually entered last column entries)
0 1 0 1 0 1   undefined
0.1 0.79432823472428 0.16057272043212 0.69091922875998 0.20374209674284 0.62554405820608   0.39901297826025
0.2 0.7247796636777 0.31145890709838 0.60575856902197 0.37721845362499 0.5449235984186   0.46962192293561
0.3 0.69684530193595 0.43214988609837 0.59434417379411 0.48891128772632 0.55508396180265   0.52895644844295
0.4 0.69314484315515 0.5298707364733 0.6153797974711 0.56900457469329 0.59370446435584   0.58504317197440
0.5 0.70710678118655 0.61254732653607 0.65404086004207 0.63549784581337 0.64371864172287   0.64118574450499
0.6 0.73602192281783 0.68661684945978 0.70416577097103 0.69788152946118 0.700125434673   0.69953450919473
0.7 0.77905591267045 0.75739548555921 0.76326958144204 0.76167209672267 0.76210620802468   0.76201343081072
0.8 0.83651164207302 0.82972398762427 0.83098165747098 0.83074848273819 0.83079170890087   0.83078494866324
0.9 0.90953257608296 0.90861953913804 0.90870695076419 0.90869858185342 0.90869938310125   0.90869931309174
1 1 1 1 1 1   1
1.1 1.1105342410546 1.1116498000258 1.111768001503 1.1117805265341 1.1117818537373   1.11178201104184
1.2 1.244564747204 1.2547181707692 1.2570430412015 1.2575759820011 1.2576981823156   1.25773454137653
1.3 1.4064566732379 1.446293346286 1.4614888698918 1.4673271088311 1.469576402415   1.47098896009015
1.4 1.6016928982022 1.7141634748254 1.780276017527 1.8203220817394 1.8450158019786   1.88666330624633
1.5 1.8371173070874 2.1062033521489 2.3490053186119 2.5920257049075 2.8604414974606  
1.6 2.1212505710976 2.7101269178649 3.5743004801538 5.3652029167449 12.449305553378  
1.7 2.4646948994849 3.698164974975 7.1160358861268 43.639804309351 11395826597.324  

Examples with invalid input

Returns an error message.

Code Result
{{^^|a|0}} Expression error: Unrecognized word "a".
{{^^|a|1}} Expression error: Unrecognized word "a".
{{^^|2i|3}} Expression error: Unrecognized word "i".
{{^^|2i+1|3}} Expression error: Unrecognized word "i".
{{^^|2|33}} Tetration error: Second argument (height) must be a nonnegative integer up to 32
{{^^|2|-1}} Tetration error: Second argument (height) must be a nonnegative integer up to 32
{{^^|2|0.5}} Tetration error: Second argument (height) must be a nonnegative integer

Test

The following few results for the tetration function (using {{^^|b|h}} template) fully agree (all 14 decimal digits) with Mathematica's Power @@ Table[b, {h}].

Table of values of the tetration function (using {{^^|b|h}} template) compared with Mathematica's Power @@ Table[b, {h}]
h {{^^|(1/e)^e|h}} Power @@ Table[(1/e)^e, {h}]
0 1 1
1 0.065988035845313 0.0659880358453125
2 0.83579318845359 0.835793188453588
3 0.10311386287497 0.103113862874971
4 0.75556267703185  
5 0.12824249755517  
6 0.70567587269525  
7 0.14686728817137  
8 0.6708386179955 0.670838617995497
9 0.16145506261086  
10 0.644757912042  
11 0.17331687438423  
12 0.62430009159136 0.624300091591359
13 0.18322807430165  
14 0.6077051016648  
15 0.19168272746053 0.191682727460533
16 0.59389800988891 0.593898009888908
17 0.19901358938169  
18 0.58218033596699  
19 0.20545459299035 0.205454592990346
20 0.57207596716079  
21 0.21117592903209
22 0.56324775217168  
23 0.21630493523809  
24 0.5554493934993 0.555449393499302
25 0.22093914155261  
26 0.54849624036848  
27 0.22515474374947 0.225154743749465
28 0.54224679029599  
29 0.2290122927811  
30 0.53659054896353  
31 0.53659054896353  
32 0.53659054896353 0.531439817592522

See also

Notes