Print cos(2π/17) exactly

Question

One way to construct a regular heptadecagon starts with drawing a horizontal line of length 1 from the center to a vertex. Then the distance along that line from the center to the second vertex is cos(2π/17). All other vertices are easy to get.

The goal of this challenge is to print (or produce in any other conventional way) an ASCII math expression involving the operations +, -, *, /, sqrt() and the natural numbers, which gives cos(2π/17) when evaluated. The power operator is not allowed: e.g. sqrt(17) cannot be written as 17^(1/2).

For example (formula taken from Wikipedia):

1/16*(
    -1+
    sqrt(17)+
    sqrt(34-2*sqrt(17))+
    2*sqrt(
        17+
        3*sqrt(17)-
        sqrt(34-2*sqrt(17))-
        2*sqrt(34+2*sqrt(17))
    )
)

Whitespace doesn't matter. The length and complexity of the expression doesn't matter, as long as it's syntactically correct (as defined above) and numerically equivalent to cos(2π/17).

Built-in functions are allowed.

Answer 1

CJam (60 bytes)

1-4{"(- +sqrt( * -4*
))/2"S/@a*N/\a*}:F~'-\+WF'-\+W-4FWFF'/2

Online demo

Output (with added newlines to avoid wrapping):

(--(--(-1+sqrt(1*1-4*-4))/2+sqrt(-(-1+sqrt(1*1-4*-4))/2*-(-1+sqrt(1*1-4*-4))/2-4*-1)
)/2+sqrt(-(--(-1+sqrt(1*1-4*-4))/2+sqrt(-(-1+sqrt(1*1-4*-4))/2*-(-1+sqrt(1*1-4*-4))/
2-4*-1))/2*-(--(-1+sqrt(1*1-4*-4))/2+sqrt(-(-1+sqrt(1*1-4*-4))/2*-(-1+sqrt(1*1-4*-4)
)/2-4*-1))/2-4*(-(--1+sqrt(-1*-1-4*-4))/2+sqrt((--1+sqrt(-1*-1-4*-4))/2*(--1+sqrt(-1
*-1-4*-4))/2-4*-1))/2))/2/2

I suspect this is the kind of answer OP was secretly hoping for. Rather than apply general-purpose compression to a string, it uses the field structure which Gauss first exploited to present this value in surds, in his Disquisitiones Arithmeticae.

I worked from this presentation (pages 18 and 19) but adapted it slightly in order to only ever require the positive root of a quadratic. Unrolled and in pseudo-code, the expression breaks down as

# Positive root of x^2 + bx + c = 0
F(b, c) = (-b + sqrt(b*b - 4*c))/2
# (8,1)
p = F(1, -4)
# (4,1)
q = F(-p, -1)
# -(8,3)
r = F(-1, -4)
# (4,3)
s = F(r, -1)
# Result
F(-q, s)/2

Answer 2

JavaScript (ES6), 93

Edit 1 byte saved thx @ETHproductions

An anonymous function.

x=>"(19)+8-29))+29+39)-8-29))-2*8+29)))-1)/16".replace(/8|9/g,n=>["sqrt(34","*sqrt(17"][n-8])

Output:

(1*sqrt(17)+sqrt(34-2*sqrt(17))+2*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt(34+2*sqrt(17)))-1)/16

TEST

F=x=>"(19)+8-29))+29+39)-8-29))-2*8+29)))-1)/16".replace(/8|9/g,n=>["sqrt(34","*sqrt(17"][n-8])

// Math check

k=Math.cos(Math.PI*2/17)
x=F()

v=eval(x.replace(/sqrt/g,"Math.sqrt"))
       
O.textContent=k+' (Math.cos(Math.PI*2/17))\n'+v+' ('+x+')\n'+(v==k)

<pre id=O></pre>

Answer 3

Mathematica, 26 bytes

FunctionExpand@Cos[2Pi/17]

Output

1/(4 Sqrt[2/(15+Sqrt[17]-Sqrt[2 (17-Sqrt[17])]+Sqrt[2 (34+6 Sqrt[17]+Sqrt[2 (17-Sqrt[17])]-Sqrt[34 (17-Sqrt[17])]+8 Sqrt[2 (17+Sqrt[17])])])])

---

Alternative form

CForm@FunctionExpand@Cos[2Pi/17]

gives

1/(4.*Sqrt(2/(15 + Sqrt(17) - Sqrt(2*(17 - Sqrt(17))) + Sqrt(2*(34 + 6*Sqrt(17) + Sqrt(2*(17 - Sqrt(17))) - Sqrt(34*(17 - Sqrt(17))) + 8*Sqrt(2*(17 + Sqrt(17))))))))

Answer 4

Bubblegum, 50 bytes

Hex dump:

78 DA D3 28 2E 2C 2A D1 30 34
D7 D4 06 33 8C 4D 74 61 02 5A
46 50 31 43 73 6D B8 98 B1 2E
16 65 30 31 6D 24 31 30 D2 35
D4 D4 37 34 03 00 AC BC 1B 67

Contains a pesky null byte, so you can’t try it online, but it prints this string:

(sqrt(17)+sqrt(34-sqrt(17)*2)+sqrt(17+sqrt(17)*3-sqrt(34-sqrt(17)*2)-sqrt(34+sqrt(17)*2)*2)*2-1)/16

Answer 5

golflua, 122 chars

Adapted from this Math.se post.

f='sqrt('y='(1-'..f..'17))'x='('..y..'/2+'..f..y..'*'..y..'/4+4))'w('(2/'..x..'+'..f..x..'/2*'..f..'17+4*'..f..'17))))/4')

which is pretty damn long :( (mostly because of the wasteful .. for concatenating strings).

Output

(2/((1-sqrt(17))/2+sqrt((1-sqrt(17))*(1-sqrt(17))/4+4))+sqrt(((1-sqrt(17))/2+sqrt((1-sqrt(17))*(1-sqrt(17))/4+4))/2*sqrt(17+4*sqrt(17))))/4

Which, when evaluated, returns 0.93247222940436

Answer 6

Ruby, 71

I think this could be shorter in another language.

'(s/2)+s-s*2))+s*2+s*72)-4*s-s*2))-8*s+s*2)))-1)/16'.gsub(?s,'sqrt(34')

The above string expression evaluates to the below string. The idea was to rearrange the expression so that a single substitution of s -> sqrt(34 would be the only operation needed to decompress it.

(sqrt(34/2)+sqrt(34-sqrt(34*2))+sqrt(34*2+sqrt(34*72)-4*sqrt(34-sqrt(34*2))-8*sqrt(34+sqrt(34*2)))-1)/16

I assume "produce in any other conventional way" means an expression is enough. If this is not the case: Printing the string in quotation marks would need 2 extra bytes at the beginning of the source code: p. For 5 bytes it can be printed without quotation marks by adding puts. Enclosing it in ->{} yields an anonymous function for 4 extra bytes.

Answer 7

Bubblegum, 39 bytes

0000000: d3 35 d4 37 34 d3 2e 2e 2c 2a d1 30 34 d7 84 b3  .5.74...,*.04...
0000010: 75 8d b4 60 62 da c6 26 98 e2 a8 d2 ba 58 f4 00  u..`b..&.....X..
0000020: 31 82 0b 32 da 02 00                             1..2...

Try it online!

Verification

$ base64 -d > exact-cos.bg <<< 0zXUNzTTLi4sKtEwNNeEs3WNtGBi2sYmmOKo0rpY9AAxggsy2gIA
$ wc -c exact-cos.bg
39 exact-cos.bg
$ bubblegum exact-cos.bg; echo
-1/16+sqrt(17)/16+sqrt(-2*sqrt(17)+34)/16+sqrt(-2*sqrt(2*sqrt(17)+34)-sqrt(-2*sqrt(17)+34)+3*sqrt(17)+17)/8
$ cat verify-cos
#!/usr/bin/python3

from sympy import *

print(Eq(S(input()), cos(2*pi/17)))
$ bubblegum exact-cos.bg | ./verify-cos
True

Answer 8

CJam, 68

"2* 17 sqrt( 23) 243-41) (1+0+423+1+41-0-4243+41))"S/~5{s/\*}/W")/"G

Try it online

Answer 9

R 68 bytes

s=sqrt;x=s(17);y=s(34-2*x);cat((x+y+2*s(17+3*x-y-2*s(34+2*x))-1)/16)

Answer 10

Vim, 70 keystrokes

:im a *sqrt(<CR>
:im b a17)<CR>
i(1b-1+1a34-2b)+2b<bsp>+3b-1a34-2b)-2a34+2b)))/16<esc>

Answer 11

Perl 5, 78 bytes

A subroutine:

{'(1S)+T-2S))+2S+3S)-T-2S))-2*T+2S)))-1)/16'=~s/S/*sqrt(17/gr=~s/T/sqrt(34/gr}

View the returned value by means of print sub{...}->() or the like.

Hat tip.