# Approximate the Dottie number

The Dottie number is the fixed point of the cosine function, or the solution to the equation cos(x)=x.1

Your task will be to make code that approximates this constant. Your code should represent a function that takes an integer as input and outputs a real number. The limit of your function as the input grows should be the Dottie number.

You may output as a fraction, a decimal, or an algebraic representation of a number. Your output should be capable of being arbitrarily precise, floats and doubles are not sufficient for this challenge. If your language is not capable of arbitrary precision numbers, then you must either implement them or choose a new language.

This is a question so answers will be scored in bytes, with fewer bytes being better.

### Tips

One way of calculating the constant is to take any number and repeatedly apply the cosine to it. As the number of applications tends towards infinity the result tends towards the fixed point of cosine.

Here is a fairly accurate approximation of the number.

0.739085133215161


1: Here we will take cosine in radians

• So, if we are using Python, we must implement our own type or import Decimal? Aug 4 '17 at 15:28
• How accurate must our submissions be? Aug 4 '17 at 15:30
• Goes to Jelly tutorial to steal ÆẠȷ¡ realizes it's invalid. Tries Brachylog; oh no Brachylog doesn't even do floats. Aug 4 '17 at 15:30
• I feel like the "arbitrarily precise" requirement is a bit too stringent. Why not consider an answer valid once x=cos(x)? Aug 4 '17 at 16:58
• I would like to see this in Haskell, APL, and some Lisp flavor. Oct 25 '17 at 8:50

# MATL, 3430 19 bytes

11 bytes off thanks to Sanchises!

48i:"'cos('wh41hGY$ The last decimal figures in the output may be off. However, the number of correct figures starting from the left increases with the input, and the result converges to the actual constant. Try it online! ### Explanation For input n, and starting at x=1, this applies the function x ↦ cos(x) with n-digit variable-precision arithmetic n times. 48 % Push 48, which is ASCII for '1': initial value for x as a string i:" % Do n times, where n is the input 'cos(' % Push this string w % Swap. Moves current string x onto the top of the stack h % Concatenate 41 % Push 41, which is ASCII for ')' h % Concatenate. This gives the string 'cos(x)', where x is the % current number GY$      %   Evaluate with variable-prevision arithmetic using n digits
%   The result is a string, which represents the new x
% End (implicit). Display (implicit). The stack contains the last x

• Why not just apply it n times at n digits precision? This seems overly complicated. Aug 4 '17 at 21:13
• This is incredible. I want to see it in APL. Oct 25 '17 at 8:48

# Python 3, 58 bytes

lambda n:S('cos('*n+'0'+')'*n).evalf(n)
from sympy import*


Try it online!

# GNU bc -l, 30

Score includes +1 for -l flag to bc.

for(a=1;a/A-b/A;b=c(a))a=b
a


The final newline is significant and necessary.

-l does 2 things:

• enable the "math" library, including c() for cos(x)
• sets precision (scale) to 20 decimal places (bc has arbitrary precision calculation)

I'm not really clear on the precision requirement. As it is, this program calculates to 20 decimal places. If a different precision is required, then scale=n; needs to be inserted at the start of the program, where n is the number of decimal places. I don't know if I should add this to my score or not.

Note also that for some numbers of decimal places (e.g. 21, but not 20), the calculation oscillates either side of the solution in the last digit. Thus in the comparison of current and previous iterations, I divide both sides by 10 (A) to erase the last digit.

• According to current consensus on scoring flags, the score of this answer is now 29. Jun 28 '20 at 19:13

# dzaima/APL, 55 bytes

⎕←⊃{⍵,⍨-/P,((P÷⍨×)/¨(2×⍳N)⍴¨⊃⍵)÷!2L×⍳N}⍣{⍵≢∪⍵}P←10L*N←⎕


Using big integer (no big decimals!) arithmetic (where $$\10^N\$$ is the equivalent of a 1), iterate the first $$\N\$$ terms of the Taylor series (an overestimate, but that's fine) until a duplicate has been encountered. May be off by a bit due to lost precision in the end, but, as with other answers, those differences will disappear with higher $$\N\$$.

No TIO link as TIO's dzaima/APL hasn't been updated to support bigintegers.

Example I/O:

1
9L

10
7390851332L

100
7390851332151606416553120876738734040134117589007574649656806357732846548835475945993761069317665318L

200
73908513321516064165531208767387340401341175890075746496568063577328465488354759459937610693176653184980124664398716302771490369130842031578044057462077868852490389153928943884509523480133563127677224L


# Wolfram Language (Mathematica), 22 bytes

Nest[Cos@#&,0,9#]~N~#&


Try it online!

# R (+Rmpfr), 55 bytes

function(n,b=Rmpfr::mpfr(1,n)){for(i in 1:n)b=cos(b);b}


Dennis has now added Rmpfr to TIO so this will work; added some test cases.

### Explanation:

Takes the code I wrote from this challenge to evaluate cos n times starting at 1, but first I specify the precision I want the values to be in by creating an object b of class mpfr with value 1 and precision n, n>=2, so we get more precision as we go along.

Try it online!

• Try again. :) In the future, if anything is missing from TIO, don't hesitate to drop a message in talk.tryitonline.net. Aug 4 '17 at 16:05
• @Dennis Thank you! I'll keep that in mind in the future! Aug 4 '17 at 16:08

# Octave, 42 bytes

@(n)digits(n)*0+vpasolve(sym('cos(x)-x'));


Try it online!

Pretty much a duplicate of my answer to Approximate the Plastic Number, but somewhat shorter due to more relaxed requirements.

{$MaxPrecision=#}~Block~Cos~FixedPoint~N[1,#]&  Try it online! # PHP, 50 bytes $a=$argv[1];$i=$j=0;while($i<$a){$j=cos($j);$i++;}


Try it online!

• Welcome to the site! :) Aug 4 '17 at 18:13
• I believe that for($a=$argv[1];$a--;)$j=cos($j);echo$j; (40 bytes) is enough. Aug 5 '17 at 11:44

# Pyth, 57 54 bytes

u_.tG1lX$globals()$"neg"$__import__("decimal").Decimal  This would be much shorter if we didn't need the Decimal to be up to spec, but it is what it is. Edit 1: -3 bytes because we need a number anyways, so we can use Xs returned copy of globals() length as our starting value, moving it to the end and removing a $ and some whitespace.

Try it online!

# JavaScript (Node.js), 84 bytes

n=>"0."+(F=(J,Z=c=0n)=>J?F(J*-I*I/++c/++c/B/B,Z+J):I-Z>>2n?(I=Z,F(B)):I)(B=I=10n**n)


Try it online!

Has a precision of roughly n-1 digits. BigInt is used and cos(x) is calculated using its Taylor expansion. The I-Z>>2n part is used only to prevent looping forever (with a cost of 4 bytes and some precision). Although theoretical applicable for arbitrary precision, practical range is n<63 because of stack overflow.

### Shorter (82 bytes), no worries about stack overflow, but far fewer precision

n=>"0."+eval("for(I=B=10n**n;n--;I=Z)for(Z=J=B,c=0n;J;)Z+=(J=J*-I*I/++c/++c/B/B)")


### Much shorter (80 bytes), larger range until stack overflow (n<172), but same precision as the 82-byte.

n=>"0."+(F=(J,Z=c=0n)=>J?F(J*-I*I/++c/++c/B/B,Z+J):n--?(I=Z,F(B)):I)(B=I=10n**n)


### If arbitrary precision is not the main point, then 25 bytes:

F=n=>n?Math.cos(F(n-1)):1


# Factor + math.polynomials math.factorials, 66 bytes

[| x | 1 x [ sq neg x [0,b) [ 2 * n! recip ] map polyval ] times ]


Try it online!

Factor has rational numbers, but no arbitrary-precision decimals. This answer tries to exploit it as much as possible: iterate the evaluation of the fist n terms of Taylor series (more precisely, Maclaurin series) n times, with the starting value of 1. The output is given as a rational number; the floating-point representation is also shown on TIO to check the value in human-readable form.

The function is very slow to calculate and very slow to converge, but in theory it must converge to the Dottie number as n increases, since the iterated function approaches the cosine function, the iteration count increases to infinity, and the whole computation is done in rationals (and therefore exact).

### How it works

Given a positive integer $$\n\$$, the code approximates the Dottie number as follows:

$$\cos{x} = \sum_{i=0}^{\infty}{\frac{(-1)^i}{(2i)!}x^{2i}} \approx \sum_{i=0}^{n-1}{\frac{(-x^2)^i}{(2i)!}} \\ \text{Dottie number } = \cos{x} \text{ iterated } \infty \text{ times on } 1 \\ \approx \sum_{i=0}^{n-1}{\frac{(-x^2)^i}{(2i)!}} \text{ iterated } n \text{ times on } 1$$

[| x |  ! an anonymous function that takes one arg x as a local variable
1 x [ ... ] times  ! repeat the inner function x times on the value 1...
sq neg           !   val -> -val^2
x [0,b)          !   0..x-1
[ 2 * n! recip ] map  ! convert each number i to 1/(2i)!
polyval          !   evaluate the array as polynomial at the value of -val^2
]


# Perl 5, 41 Bytes

use bignum;sub f{$_[0]?cos(f($_[0]-1)):0}


Bignum is required for the arbitrary precision. Defines a function f that recursively applies cosine to 0 N times.

TIO doesn't seem to have bignum so no link :(

## Mathematica 44 Bytes

FindRoot[Cos@x-x,{x,0},WorkingPrecision->#]&


FindRoot uses Newton's method by default.

# Axiom, 174 bytes

f(n:PI):Complex Float==(n>10^4=>%i;m:=digits(n+10);e:=10^(-n-7);a:=0;repeat(b:=a+(cos(a)-a)/(sin(a)+1.);if a~=0 and a-b<e then break;a:=b);a:=floor(b*10^n)/10.^n;digits(m);a)


ungolfed and commented

-- Input: n:PI numero di cifre
-- Output la soluzione x a cos(x)=x con n cifre significative dopo la virgola
-- Usa il metodo di Newton a_0:=a  a_(n+1)=a_n-f(a_n)/f'(a_n)
fo(n:PI):Complex Float==
n>10^4=>%i
m:=digits(n+10)
e:=10^(-n-7)
a:=0     -- Punto iniziale
repeat
b:=a+(cos(a)-a)/(sin(a)+1.)
if a~=0 and a-b<e then break
a:=b
a:=floor(b*10^n)/10.^n
digits(m)
a


results:

(3) -> for i in 1..10 repeat output[i,f(i)]
[1.0,0.7]
[2.0,0.73]
[3.0,0.739]
[4.0,0.739]
[5.0,0.73908]
[6.0,0.739085]
[7.0,0.7390851]
[8.0,0.73908513]
[9.0,0.739085133]
[10.0,0.7390851332]
Type: Void
Time: 0.12 (IN) + 0.10 (EV) + 0.12 (OT) + 0.02 (GC) = 0.35 sec
(4) -> f 300
(4)
0.7390851332 1516064165 5312087673 8734040134 1175890075 7464965680 635773284
6 5488354759 4599376106 9317665318 4980124664 3987163027 7149036913 084203157
8 0440574620 7786885249 0389153928 9438845095 2348013356 3127677223 158095635
3 7765724512 0437341993 6433512538 4097800343 4064670047 9402143478 080271801
8 8377113613 8204206631
Type: Complex Float
Time: 0.03 (IN) + 0.07 (OT) = 0.10 sec


I would use the Newton method because it would be faster than 'repeated cos(x) method'

 800   92x
1000  153x
2000  379x


where in the first column there is the number of digit and in the second column there is how much Newton method is faster than use repeated cos(x) method, here. Good Morning

# 05AB1E, 3 bytes

FÅ¾


Try it online!

FÅ¾  # full program
F    # for N in [0, 1, ...,
# ..., implicit input...
F    # minus 1]...
Å¾  # push cosine of...
# implicit input...
# (implicit) or top of stack if not first iteration...