# Golf the xᵗʰ root of x

While bored in high-school (when I was half my current age...), I found that $$\f(x) = x^{x^{-1}}\$$ had some interesting properties, including e.g. that the maximum $$\f\$$ for $$\0 ≤ x\$$ is $$\f(e)\$$, and that the binding energy per nucleon of an isotope can be approximated as $$\6 × f(x ÷ 21)\$$...

Anyway, write the shortest function or program that calculates the xth root of x for any number in your language's domain.

## Examples cases

### For all languages

     -1   >       -1
¯0.2   >    -3125
¯0.5   >        4
0.5   >     0.25
1   >        1
2   >    1.414
e   >    1.444
3   >    1.442
100   >    1.047
10000   >    1.001


### For languages that handle complex numbers

   -2   >        -0.7071i
i   >            4.81
2i   >    2.063-0.745i
1+2i   >   1.820-0.1834i
2+2i   >   1.575-0.1003i


### For languages that handle infinities

-1/∞   >   0    (or ∞ or ̃∞)
0   >   0    (or 1 or ∞)
1/∞   >   0
∞   >   1
-∞   >   1


### For languages that handle both infinities and complex numbers

 -∞-2i   >   1      (or ̃∞)


 ̃∞ denotes directed infinity.

• Here is a Wolfram Alpha plot for positive real x. If you omit the x limits in the query, Wolfram Alpha will include negative values of x where the function value depends on a choice of "branch" for the complex logarithm (or for a similar complex function). Feb 24, 2016 at 14:36
• What about for languages that do not handle power of decimals? Mar 31, 2016 at 3:24
• @KennyLau Feel free to post with a note that says so, especially if the algorithm would work, had the language supported it.
Mar 31, 2016 at 5:37

# Awk, 8 bytes

$1^$1^-1


A shell command to check it

awk '{print $1^$1^-1}'


Longer version (+1) with

$1^(1/$1)


# MATLAB, 11bytes

@(x)x^(1/x)


Creates an anonymous function. This will work with all test cases including complex numbers and infinities.

This also works with Octave. You can try online here. Simply enter something like:

f=@(x)x^(1/x)


At the command line of the online interpreter, then do:

f(-1)


To test.

# C#, 25 bytes, type: double

Bare form of a lambda expression:

x=>System.Math.Pow(x,1/x)


The expression can be assigned to a function variable, e.g. (Func<double, double>), ...

System. can be omitted, if using System; is used (but more bytes).

Complete test program (143 bytes) with the input of the question:

using System;class P{static void Main(){Array.ForEach(new double[]{-.2,-.5,.5,1,2,Math.E,3,100,10000},x=>Console.WriteLine(Math.Pow(x,1/x)));}}


Result:

-3125
4
0,25
1
1,4142135623731
1,44466786100977
1,44224957030741
1,0471285480509
1,0009214583193


# C#, 37 bytes, type: Complex

x=>System.Numerics.Complex.Pow(x,1/x)


Also the assembly for System.Numerics needs to be referenced. The division operator is overloaded for complex arguments.

Test program (189 bytes) with the input of the question for complex numbers:

using System;using C=System.Numerics.Complex;class P{static void Main(){Array.ForEach(new C[]{new C(-2,0),new C(0,1),new C(0,2),new C(1,2),new C(2,2)},x=>Console.WriteLine(C.Pow(x,1/x)));}}


Result:

(4,32963728535968E-17, -0,707106781186548)
(4,81047738096535, 0)
(2,0628722350809, -0,745007062179724)
(1,81984053615444, -0,183434723804562)
(1,57500291115344, -0,100274868564155)


Test program (264 bytes) with infinity:

using System;using C=System.Numerics.Complex;class P{static void Main(){var p=double.PositiveInfinity;var n=double.NegativeInfinity;Array.ForEach(new C[]{new C(-1/p,0),new C(0,0),new C(1/p,0),new C(p,0),new C(n,0),new C(n,2) },x=>Console.WriteLine(C.Pow(x,1/x)));}}


Result:

(0, 0)
(0, 0)
(0, 0)
(1, 0)
(1, 0)
(1, 0)


# Vitsy, 4 + 1 (function declaration) = 5 bytes

Hey, look, my language actually does this. Go figure.

Di^^
D    Duplicate the top item (input)
i   Push -1.
^  Pop top (-1), put the second to top to the power of the popped item (x^-1)
^ Same as above, except we now have the desired number.

This is a function that leaves the value requested on the stack. For view-ability, I have also made the program output on the Try it Online! link.

Try it Online!

• -1: Anonymous functions are allowed. Jun 14, 2016 at 8:51
• @EʀɪᴋᴛʜᴇGᴏʟғᴇʀ Unfortunately, that's what I thought. However, I was informed that I would have to include the return character before the function in order for it to be valid. Jun 15, 2016 at 16:21

# PARI/GP, 9 bytes

x->x^x^-1


This is shorter than the alternatives x->x^(1/x) and x->sqrtn(x,x). It uses the right-associativity of ^ and the strong binding of the unary -. GP isn't quite functional enough to curry (a,b)->sqrtn(a,b) into (x)->sqrtn(x,x); the best it can manage is x->call(sqrtn,[x,x]) which is far too long.

# Bean, 11 bytes

Hexdump:

00000000 26 4c a0 43 95 4c a5 3a 8e 20 43  &L C.L¥:. C
0000000b


Equivalent JavaScript:

A**(1/A)


Implicitly converts first line of input to number as A and implicitly outputs the result of the expression.

# Codename Dragon, 16 bytes

a=<>;dispa^(1/a)


I'm abusing the lack of a syntax error.

Ungolfed:

disp((a=<>)^(1/a))


This is a simple program that takes input through <>.

You can find an interpreter here, no github site yet.

# Jolf, 5 bytes

Try it here!

^j/1j
j    input
^ /1j to the 1/j pow.


## Gnuplot, 13 bytes

f(x)=x**(1/x)


or

f(x)=x**x**-1


# Brachylog, 8 7 bytes

Thanks to Fatalize for saving a byte.

^?^-1=.


## Explanation

^?^-1=.
=.   unify output with
^?^-1     input raised to the 1/input power

• You can save one byte: you don't need the first ?, it is implicitely the current variable at the beginning of the predicate. Mar 16, 2016 at 20:28

# Lua, 20

n=(...)print(n^n^-1)


How it works:

n=(...)              Take input and store at n
print(
n^n^-1  n^(n^(-1))
)


# dc, 8 6 chars

Outputs to the top of the stack:

?d1r/^


Or if I can expect the input at the top of the stack, this 5 char solution works too:

d1r/^


Explanation:

?        take input from STDIN and execute it (numbers are executed by pushing to the stack)
d1r     duplicate top of stack, push 1 and swap them
/^   divide and exponentiate


Alas, dc does not allow for fractional exponents so this answer does not actually work for any given inputs above 1, and gives the wrong answer for 0 < input < 1. It does the same thing as many of the other programs though.

# Factor, 14 10 bytes

1 over / ^


Expects input on top of the stack.

## Convex, 3 bytes

Convex is a new language that I am developing that is heavily based on CJam and Golfscript. The interpreter and IDE can be found here. Input is an integer into the command line arguments. Indexes are one-based. Uses the CP-1252 encoding. This answer in non-competing as it was created after this challenge.

_¹#


Explanation:

    Implied input
_   Duplicate top of stack
¹  Find reciprocal
# Power
Implied output

• Can be found... where?
– cat
Apr 5, 2016 at 21:47
• @cat oops. Fixed. Note that I literally just updated the interpreter though, but this still works. Apr 6, 2016 at 2:04

# Racket, 28 22 bytes

(λ(n)(expt n(/ 1 n)))


@cat gets credit for the 6 byte save. Thanks!

Seriously, Racket is impossible to golf with :). Nothing to see here, move along!

• You can do (λ(n)(expt n(/ 1 n))) as an anonymous function for 22 bytes
– cat
Apr 5, 2016 at 21:43
• We need a golfing dialect of Racket, like we need a golfing dialect of Factor except people have already made LISPs for golf and the code becomes mostly parentheses.
– cat
Apr 6, 2016 at 2:08
• I forgot that anonymous functions are generally okay. We definitely could use a lovely LISP for golfing. But yeah, I can only imagine the parens are still going to reduce competitiveness. Apr 6, 2016 at 6:44

Python 3: 34

n=complex(input());print(n**(1/n))


Unless I'm missing something this would do nicely. I like that Python normally uses spaces, but this solution doesn't need them.

• Why the int(...)?
Jun 14, 2016 at 7:19
• Python 3 won't implicitly take an input as an integer. So it needs to be converted for the arithmetic in the second statement. Unless it does work for this scenarios in which case my solution will be shorter Jun 14, 2016 at 7:29
• OP: for any number in your language's domain
Jun 14, 2016 at 7:37
• @Adám I'm not sure that I understand that part then. I just thought it meant it would take any number in the normal base. For Python that is base 10. Jun 14, 2016 at 8:05
• If Python natively handles real numbers, then input may be a real (i.e. non-integer).
Jun 14, 2016 at 8:11

# 05AB1E, 3 bytes

Dzm


Explanation:

D    # Duplicate the input
z   # Inverse, 1 ÷ input
m  # Power, calculating input ** (1 ÷ input)

• Wait a minute! If it couldn't parse floats, then floats were not in the domain, and so an integer solution was acceptable.
Jun 14, 2016 at 11:01
• @Adám Oh, I didn't know that! The code would still be the same though :p. Jun 14, 2016 at 11:02
• Weird how -.2 still returns -3124.999999999999 Mar 21, 2017 at 14:36
• And, also, I think he meant you can remove the non-competing tag. Mar 21, 2017 at 14:36
• @carusocomputing Hmm, how is that weird then? Mar 21, 2017 at 14:42

# Quetzalcoatl, 4 bytes

c.I?


Explanation:

c    # Input
.   # Duplicate
I  # Inverse
? # Power

• So... "Quetzalcoatl" can have an operator which finds x^x^-1... Jun 14, 2016 at 8:57
• @EʀɪᴋᴛʜᴇGᴏʟғᴇʀ What operator are you referring to? Jun 14, 2016 at 14:39
• Like Python's a**a**-1. Jun 14, 2016 at 14:43

# TI-Basic, 4 bytes

Ans^Ansֿ ¹


Since TI-Basic is tokenized, this code is a mere four bytes.

• We already have a TI-BASIC answer at 3 bytes. Feb 23, 2016 at 23:37
• Is ans accepted as input method? Feb 24, 2016 at 0:06

# Mouse-2002, 13 bytes

x:x.b/x.&ROOT


Other programs assume the number's on the top of the stack and so shall I.

x:    ~ put TOS in x
x.    ~ push x
b /   ~ divide by 1
x.    ~ push x again
&ROOT ~ y to the x root


# Scala, 33 bytes

def f=(x:Double)=>Math.pow(x,1/x)


## Perl, 11 bytes

Added 2 bytes for -pn.

$_**=1/$_


Try it here !

# Mumps, 14 Bytes

R I W I**(1/I)


R reads from input into variable I, and the W (write) statement does the mathematical calculations.

I will warn you: the output will be ugly as there are no implied newlines in Mumps, and at least the way I read the challenge, the OP didn't require them. If you did want cleaner output, add 2 bytes and add a newline:

R I W !,I**(1/I)


## SQL (MySQL), 60 bytes

CREATE FUNCTION G(X REAL) RETURNS REAL RETURN POWER(X,1./X);


Caps lock is cruise control for cool golf.

• Can you do )RETURNS instead of ) RETURNS for -1 byte? Jun 14, 2016 at 8:59
• @EʀɪᴋᴛʜᴇGᴏʟғᴇʀ Possibly; I don't have a MySQL server handy right now to test it, though.
– user45941
Jun 14, 2016 at 9:00

# Desmos, 14 bytes

a=1
\sqrt[a]{a}


Demonstration

• That looks interesting. So \sqrt is a root-of function, and not square-root?
Jun 6, 2016 at 20:04
• @Adám It is TeX. \sqrt is the Square Root symbol, and the brackets change the root. Jun 6, 2016 at 20:50

# tcl, 23

I post two approaches which occupy the exact same number of bytes.

puts [expr $n**(1./$n)]


puts [expr $n**$n**-1.]


Available to test on http://rextester.com/EMW19680

## QBIC, 9 bytes

:?a^(1/a)


Gets a numeric input from the command line and does the math...

# SmileBASIC, 18 bytes

INPUT X?POW(X,1/X)


# Common Lisp, 24 bytes

(lambda(x)(expt x(/ x)))


Try it online!

# Excel, 10 bytes

=A1^(1/A1)


Self explanatory.