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This comes from http://programmers.blogoverflow.com/2012/08/20-controversial-programming-opinions/

"Given that Pi can be estimated using the function 4 * (1 – 1/3 + 1/5 – 1/7 + …) with more terms giving greater accuracy, write a function that calculates Pi to an accuracy of 5 decimal places."

  • Note, the estimation must be done by calculating the sequence given above.
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6  
You should probably add some more rules, otherwise you will get answers like (python) p=lambda:3.14159 –  Matt Aug 29 '12 at 15:44
1  
Have you seen codegolf.stackexchange.com/questions/506/… , which is very similar? At the very least, trig functions should be banned for this problem because they allow for trivial solutions such as this QBASIC program: ?INT(4E5*ATN(1))/1E5 –  PleaseStand Aug 29 '12 at 17:30
    
I think you should require that the algorithm be one of successive approximation: the longer you compute, the closer you get to pi. –  David Carraher Aug 29 '12 at 22:59
    
@DavidCarraher, although that's mathematically inevitable using this series, from a numerical analytical point of view it's highly dubious. A slowly converging alternating series is a poster child for loss of significance. –  Peter Taylor Aug 30 '12 at 7:03
2  
Dupe, but it's so old it's not here: stackoverflow.com/q/407518/12274 –  J B Sep 1 '12 at 21:42
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20 Answers

JavaScript, 46 58 56 chars

function(){for(a=i=0;i<1e6;a+=8/++i/~-(i+=3));return a}

This version is a function expression; add two characters (e.g. " f") if you want it named. This version clobbers the globals a and i; this could be prevented if we added "a,i" to the parameter list.

Makes use of a reformulated version of the algorithm in order to circumvent the need for subtraction.

 1/1 - 1/3  +   1/5 - 1/7   +    1/9 - 1/11  + ...
(3/3 - 1/3) + (7/35 - 5/35) + (11/99 - 9/99) + ...
    2/3     +      2/35     +       2/99     + ...
  2/(1*3)   +    2/(5*7)    +     2/(9*11)   + ...

Here's a "plain" version without this adjustment:

function(){for(a=0,i=1;i<1e6;i+=2)a+=[,4,,-4][i%4]/i;return a}

which clocks in at 64 62 characters.

Thanks to @ardnew for the suggestion to get rid of the 4* before the return.


History

function(){for(a=i=0;i<1e6;a+=8/++i/~-(i+=3));return a}     // got rid of `i+=4`; restructured
// Old versions below.
function(){for(a=0,i=1;i<1e6;i+=4)a+=8/i/-~-~i;return a}    // got rid of `4*`
function(){for(a=0,i=1;i<1e6;i+=4)a+=2/i/-~-~i;return 4*a}
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o.O very nice job, factoring out the subtraction. –  acolyte Aug 30 '12 at 17:01
1  
great work, but needs to be written as a proper function –  ardnew Aug 30 '12 at 17:55
    
@ardnew: Thanks, I must've missed that detail when I read the problem description. I've updated it, and it's now a callable function expression (lambda); not sure if this is allowed or if it has to be given a name. If that's the case, it's just an additional two characters anyway. –  FireFly Aug 30 '12 at 18:11
1  
@FireFly you can also shave off 2 chars by changing a+=2/i/-~-~i;return 4*a to a+=8/i/-~-~i;return a –  ardnew Aug 30 '12 at 18:25
    
@ardnew: oh, awesome; didn't think of that. :D –  FireFly Aug 30 '12 at 18:50
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Mathematica 42 39 34 33 31 26 21

Archimedes' Approach 26 chars

N@#*Sin[180 Degree/#]&

This reaches the criterion when input is 822.

Question: Does anyone know how he computed the Sin of 180 degrees? I don't.


Leibniz' Approach (Gregory's series) 32 chars

This is the same function the problem poser gave as an example. It reaches the criterion in approximately one half million iterations.

N@4 Sum[(-1)^k/(2 k+1),{k,0,10^6}]

Madhava-Leibniz Approach 37 chars

This variation uses a few more characters but converges to criterion in only 9 iterations!

N@Sqrt@12 Sum[(-1/3)^k/(2 k + 1), {k, 0, 9}]
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those all compute it by the algorithm given in the problem definition? –  acolyte Aug 30 '12 at 17:00
    
@acolyte Leibniz' approach (now the first one listed) is indeed the one mentioned in the description of the problem. It's very slow to converge. A slight variation on it (Madhava-Leibniz) converges very quickly. –  David Carraher Aug 30 '12 at 20:07
    
Sine of 180° is pretty easy. It's 180°/N that can get tricky outside of the usual suspects for N. –  J B Sep 5 '12 at 21:35
    
Please explain, @J.B. Tricky to measure? –  David Carraher Sep 5 '12 at 23:02
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Java (67 chars)

float r(){float p=0,s=4,i=1E6f;while(--i>0)p+=(s=-s)/i--;return p;}

Note that this avoids loss of significance by adding the numbers up in the correct order.

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this is fully compliant C code too. if posted as C, you could change while(--i>0) to while(i--) and save 2 chars –  ardnew Aug 30 '12 at 18:13
1  
@ardnew, true, but with C there are much more interesting tricks to play... –  Peter Taylor Aug 30 '12 at 19:03
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Python 59 bytes

print reduce(lambda x,p:p/2*x/p+2*10**999,range(6637,1,-2))

This prints out 1000 digits; slightly more than the required 5. Instead of using the prescribed iteration, it uses this:

pi = 2 + 1/3*(2 + 2/5*(2 + 3/7*(2 + 4/9*(2 + 5/11*(2 + ...)))))

The 6637 (the innermost denominator) can be formulated as:

digits * 2*log2(10)

This implies a linear convergence. Each deeper iteration will produce one more binary bit of pi.

If, however, you insist on using the tan-1 identity, a similar convergence can be achieved, if you don't mind going about the problem slightly differently. Taking a look at the partial sums:

4.0, 2.66667, 3.46667, 2.89524, 3.33968, 2.97605, 3.28374, ...

it is apparent that each term jumps back and forth to either side of the convergence point; the series has alternating convergence. Additionally, each term is closer to the convergence point than the previous term was; it is absolutely monotonic with respect to its convergence point. The combination of these two properties implies that the arithmetic mean of any two neighboring terms is closer to the convergence point than either of the terms themselves. To give you a better idea of what I mean, consider the following image:

Partial Sums

The outer series is the original, and the inner series is found by taking the average of each of the neighboring terms. A remarkable difference. But what's truly remarkable, is that this new series also has alternating convergence, and is absolutely monotonic with respect to its convergence point. That means that this process can be applied over and over again, ad nauseum.

Ok. But how?

Some formal definitions. Let P1(n) be the nth term of the first sequence, P2(n) be the nth term of the second sequence, and similarly Pk(n) the nth term of the kth sequence as defined above.

P1 = [P1(1), P1(2), P1(3), P1(4), P1(5), ...]

P2 = [(P1(1) +P1(2))/2, (P1(2) +P1(3))/2, (P1(3) +P1(4))/2, (P1(4) +P1(5))/2, ...]

P3 = [(P1(1) +2P1(2) +P1(3))/4, (P1(2) +2P1(3) +P1(4))/4, (P1(3) +2P1(4) +P1(5))/4, ...]

P4 = [(P1(1) +3P1(2) +3P1(3) +P1(4))/8, (P1(2) +3P1(3) +3P1(4) +P1(5))/8, ...]

Not surprisingly, these coefficients follow exactly the binomial coefficients, and can expressed as a single row of Pascal's Triangle. Since an arbitrary row of Pascal's Triangle is trivial to calculate, an arbitrarily 'deep' series can be found, simply by taking the first n partial sums, multiply each by the corresponding term in the nth row of Pascal's Triangle, and dividing by 2n-1.

In this way, full 32-bit floating point precision (~14 decimal places) can be achieved with just 36 iterations, at which point the partial sums haven't even converged on the second decimal place. This is obviously not golfed:

# used for pascal's triangle
t = 36; v = 1.0/(1<<t-1); e = 1
# used for the partial sums of pi
p = 4; d = 3; s = -4.0

x = 0
while t:
  t -= 1
  p += s/d; d += 2; s *= -1
  x += p*v
  v = v*t/e; e += 1

print "%.14f"%x

If you wanted arbitrary precision, this can be achieved with a little modification. Here once again calculating 1000 digits:

# used for pascal's triangle
f = t = 3318; v = 1; e = 1
# used for the partial sums of pi
p = 4096*10**999; d = 3; s = -p

x = 0
while t:
  t -= 1
  p += s/d; d += 2; s *= -1
  x += p*v
  v = v*t/e; e += 1

print x>>f+9

The initial value of p begins 210 larger, to counteract the integer division effects of s/d as d becomes larger, causing the last few digits to not converge. Notice here again that 3318 is also:

digits * log2(10)

The same number of iteratations as the first algorithm (halved because t decreases by 1 instead of 2 each iteration). Once again, this indicates a linear convergence: one binary bit of pi per iteration. In both cases, 3318 iterations are required to calculate 1000 digits of pi, as slightly better quota than 1 million iterations to calculate 5.

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J, 26 chars

+/+/_2((4 _4)&%)>:+:i.100

Moved from 100 items of sequence to 1e6 items. Also now it's a code tagged and could be copypasted from browser to the console without errors.

+/+/_2((4 _4)&%)\>:+:i.1e6
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2  
-/4%>:2*i.1e6 -- 13 characters. (Thanks to b_jonas in #jsoftware for making me realise that -/ works to compute a sum with alternating sign. [This is since all operators in J are of equal precedence and right-associative, so -/ 1 2 3 4 <=> 1 - (2 - (3 - 4)) <=> 1 - 2 + 3 - 4.]) –  FireFly Aug 30 '12 at 20:46
    
that's neat and twice as awesome. Or even 2^10 more awesome! –  fftw Aug 31 '12 at 1:16
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C (GCC) (44 chars)

float p(i){return i<1E6?4./++i-p(++i):0;}

That's 41 chars, but it also has to be compiled with -O2 to get the optimiser to eliminate the tail recursion. This also relies on undefined behaviour with respect to the order in which the ++ are executed; thanks to ugoren for pointing this out. I've tested with gcc 4.4.3 under 64-bit Linux .

Note that unless the optimiser also reorders the sum, it will add from the smallest number, so it avoids loss of significance.

Call as p().

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Your recursive call is q(), not p(). And I don't think -O2 should be counted (but if you do count it, it's 4 chars because of the required space). –  ugoren Sep 5 '12 at 14:57
    
Also: 1. gcc 4.1.1 doesn't optimize the recursion (and I don't see how it could), so the stack overflows. 2. it should be called as p(0). 3. Save a char by return++i.... 4. Two ++i makes undefined behavior. –  ugoren Sep 5 '12 at 15:03
    
@ugoren, thanks for your comments. In order: q - that'll teach me to double-check after renaming. I think I'm following normal practice in counting -O2 as 3 chars, but we can open it up on meta if you want; meta.codegolf.stackexchange.com/questions/19 is the only relevant discussion I can find. I've added the version of gcc which I'm using, and which allows me to call it as p(). Saving the char stops the optimiser and gives segfault. I will clarify that I'm using undefined behaviour, as per meta.codegolf.stackexchange.com/questions/21 –  Peter Taylor Sep 5 '12 at 21:51
    
I added an answer to the meta question about flags. About p() - are you sure calling p() from any context would work? Or is it just what happened to be on the stack in your test? –  ugoren Sep 6 '12 at 7:20
    
@ugoren, maybe I got lucky consistently. Even if I call it twice in a row, the second one still returns the correct value. gcc does seem to produce slightly different code for p() vs p(0), but I don't know what behaviour it documents and I'm not really a C programmer. –  Peter Taylor Sep 6 '12 at 22:28
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APL (14)

   

4×-/÷1-⍨2×⍳1e6
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13, --/4÷1-2×⍳1e6 –  Timtech Feb 24 at 22:29
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Haskell, 32

foldr(\k->(4/(2*k+1)-))0[0..8^7]

GHCi> foldr(\k->(4/(2*k+1)-))0[0..8^7]
3.141593130426724

Counting a function name it's

34

π=foldr(\k->(4/(2*k+1)-))0[0..8^7]
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R - 25 chars

sum(c(4,-4)/seq(1,1e6,2))
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C, 69 chars

float p,b;void main(a){b++<9e6?p+=a/b++,main(-a):printf("%f\n",4*p);}
  • Run with no command line parameters (so a is initialized to 1).
  • Must be compiled with optimization.
  • void main is strange and non-standard, but makes things work. Without it, the recursion is implemented as a real call, leading to a stack overflow. An alternative is adding return.
  • Two characters 4* can be saved, if running with three command line parameters.
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Perl - 43 39 chars

not sure the rules on anonymous subroutines, but here's another implementation using @FireFly's series construction

sub{$s+=8/((4*$_+2)**2-1)for 0..1e6;$s}

sub p{$s+=(-1)**$_*4/(2*$_+1)for 0..1e6;$s}

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Clojure - 79 chars

(fn [](* 4(apply +(map #(*(Math/pow -1 %1)(/ 1.0(+ 1 %1 %1)))(range 377000)))))

This creates a function of no arguments which will calculate a float which approximates pi correctly to five decimal places. Note that this does not bind the function to a name such as pi, so this code must either be evaluated in place with eval as (<code>) or bound to a name in which case the solution is

(defn p[](* 4(apply +(map #(*(Math/pow -1 %1)(/ 1.0(+ 1 %1 %1)))(range 377000)))))

for 82 chars

About

(defn nth-term-of-pi [n] (* (Math/pow -1 n) (/ 1.0 (+ 1 n n))))
(defn pi [c] (* 4 (apply + (map nth-term-of-pi (range c)))))
(def  pi-accuracy-constant (loop [c 1000] (if (< (pi c) 3.14159) (recur (inc c)) c)))
; (pi pi-accuracy-constant) is then the value of pi to the accuracy of five decimal places
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PHP - 56 55 chars

<?for($j=$i=-1;1e6>$j;){$p+=($i=-$i)/($j+=2);}echo$p*4;  

I don't know that I can get it much smaller without breaking the algorithm rule.

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1  
How about this for 45? <?for(;1e6>$j;)$p+=($i=-$i|4)/~-$j+=2;echo$p; –  primo Sep 1 '12 at 23:26
    
I was trying to come up with that, but couldn't get the bitwise ops to work. Thanks for the suggestion! –  TwoScoopsofPig Sep 18 '12 at 20:21
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Java - 92 84 chars

I cannot beat by far Peter Taylor's result, but here is mine:

double d(){float n=0,k=0,x;while(n<9E5){x=1/(1+2*n++);k+=(n%2==0)?-x:x;}return 4*k;}

Ungolfed version:

double d() {
    float n = 0, k = 0, x;
    while (n < 9E5) {
        x = 1 / (1 + 2 * n++);
        k += (n % 2 == 0) ? -x : x;
    }
    return 4 * k;
}

Edit: Saved a few characters using ternary operator.

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Python - 56 chars

Meh, My python-fu is not strong enough. I couldn't see any more shortcuts but maybe a more experienced golfer could find something to trim here?

t=s=0
k=i=1
while t<1e6:t,s,i,k=t+1,k*4./i+s,i+2,-k
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Ruby - 54 chars

def a()p=0;1000000.times{|i|p+=8/(4*i*(4*i+2))};p;end;

My first try on console

def a()i=1;p=0;while i<2**100 do p+=8/(i*(i+2));i+=4;end;p;end;

63 chars.

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Perl (76 chars)

$y=1e4;for$x(0..1e4-1){$y--while sqrt($x**2+$y**2)>1e4;$a+=$y}print 4*$a/1e8

(Result: 3.14159052)

Not the shortest possible solution, but maybe interesting. It's a geometrical one. I calculate the area under a circle.

I got another funny approach, but it's really slow. It counts the number of discrete points in a square that are below a quarter circle and calculates pi from it:

$i=shift;for$x(0..$i){for$y(0..$i){$h++if sqrt($x**2+$y**2)<$i}}print$h*4/$i**2

It expects the number of iterations as command line argument. Here you can see how run time relates to accuracy. ;)

$ time perl -e '$i=shift;for$x(0..$i){for$y(0..$i){$h++if sqrt($x**2+$y**2)<$i}}print$h*4/$i**2' 100
3.1796
real    0m0.011s
user    0m0.005s
sys 0m0.003s

$ time perl -e '$i=shift;for$x(0..$i){for$y(0..$i){$h++if sqrt($x**2+$y**2)<$i}}print$h*4/$i**2' 1000
3.14552
real    0m0.354s
user    0m0.340s
sys 0m0.004s

$ time perl -e '$i=shift;for$x(0..$i){for$y(0..$i){$h++if sqrt($x**2+$y**2)<$i}}print$h*4/$i**2' 10000
3.14199016
real    0m34.941s
user    0m33.757s
sys 0m0.097s
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k (25 chars)

4*+/%(i#1 -1)'1+2!i:1000000

Slightly shorter:

+/(i#4 -4)%1+2*!i:1000000
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Python (49)

print 4*sum((-1)**i/(2*i+1.)for i in range(9**6))
3.14159453527
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Javascript - 33 Characters

p=x=>4*(1-(x&2))/x+(x>1?p(x-2):0)

Call p passing a positive odd number x and it will calculate Pi with (x-1)/2 terms.

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