# Good rational approximations of pi

Write a program that prints out all the good rational approximations of pi with denominator < 1000000, in increasing denominator order. a/b is a "good rational approximation" of pi if it is closer to pi than any other rational with denominator no bigger than b.

The output should have 167 lines total, and start and end like this:

3/1
13/4
16/5
19/6
22/7
179/57
...
833719/265381
1146408/364913
3126535/995207


Shortest program wins.

• Does it have to be the string a/b or is a b also allowed? Commented Jul 21, 2023 at 17:24

## Golfscript, 71 70 69 chars

2\!:^2^..292^15.2/3]{(.)2/.9>+{\+.((}*;.}do;;]-1%{^0@{2$*+\}/"/"\n}/;  (Assumes that you don't pass it anything on stdin) I don't want to hear any more whinging by people who don't have built-in constants for pi. I don't even have floating point numbers! See http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations for the background. # No input, so the stack contains "" 2\!:^2^..292^15.2/3] # ^ is used to store 1 because that saves a char by allowing the elimination of whitespace # Otherwise straightforward: stack now contains [2 1 2 1 1 1 292 1 15 7 3] # Pi as a continued fraction is 3+1/(7+1/(15+1/(...))) # If you reverse the array now on the stack you get the first 10 continuants followed by 2 # (rather than 3) # That's a little hack to avoid passing the denominator 1000000 { # Stack holds: ... [c_n c_{n-1} ... c_0] (.)2/.9>+ # Stack holds ... [c_{n-1} ... c_0] c_n (1+c_n)/2+((1+c_n)/2 > 9 ? 1 : 0) # (1+c_n)/2 > 9 is an ad-hoc approximation of the "half rule" # which works in this case but not in general # Let k = (1+c_n)/2+((1+c_n)/2 > 9 ? 1 : 0) # We execute the next block k times { # ... [c_{n-1} ... c_0] z \+.(( # ... [z c_{n-1} ... c_0] [c_{n-1} ... c_0] z-1 }* # So we now have ... [c_n c_{n-1} ... c_0] [(c_n)-1 c_{n-1} ... c_0] ... # [(c_n)-k+1 c_{n-1} ... c_0] [c_{n-1} ... c_0] c_n-k ; # Go round the loop until the array runs out . }do # Stack now contains all the solutions as CFs in reverse order, plus two surplus: # [2 1 2 1 1 1 292 1 15 7 3] [1 2 1 1 1 292 1 15 7 3] ... [6 3] [5 3] [4 3] [3] [2] [] # Ditch the two surplus ones, bundle everything up in an array, and reverse it ;;]-1% # For each CF... { # Stack holds ... [(c_n)-j c_{n-1} ... c_0] # We now need to convert the CF into a rational in canonical form # We unwind from the inside out starting with (c_n)-j + 1/infinity, # representing infinity as 1/0 ^0@ # ... 1 0 [c_n-j c_{n-1} ... c_0] # Loop over the terms of the CF { # ... numerator denominator term-of-CF 2$*+\
# ... (term-of-CF * numerator + denominator) numerator
}/

# Presentation
"/"\n
# ... numerator "/" denominator newline
}/

# Pop that final newline to avoid a trailing blank line which isn't in the spec
;

• Well, technically GolfScript has both floating point numbers and constant for PI. It's called "#{Math.PI}".
– null
Commented May 28, 2012 at 17:52
• @GlitchMr, in what way is a string a floating point number? Commented Aug 17, 2012 at 12:17
• I'd really like to see this unrolled with comments. Commented Nov 29, 2012 at 11:23
• Amazing. The very first line 2\!:^2^..292^15.2/3] already blew my mind. Commented Dec 10, 2012 at 15:27
• @PeterTaylor Tied. Can we do better? Commented Feb 20, 2014 at 6:09

## Mathematica, 67 63

This isn't going to be fast, but I believe it is technically correct.

Round[π,1/Range@1*^6]//.x_:>First/@Split[x,#2≥#&@@Abs[π-{##}]&]


Round[π, x] gives the closest fraction to π in steps of x. This is "listable" so Round[π,1/Range@1*^6] does this for all fractions down to 1/10^6 in order. The resulting list with many "bad" rational approximations is then repeatedly (//.) processed by removing any elements which are farther from π than the preceding one.

• Pretty cool, but I can't test it because I don't have Mathematica. Commented Dec 20, 2011 at 17:35
• @Keith, here's the logic. Round[Pi, x] gives the closest fraction to Pi in steps of x. This is "listable" so Round[Pi,1/Range@1*^6] does this for all fractions down to 1/10^6 in order. The resulting list with many "bad" rational approximations is then repeatedly (//.) processed by removing any elements which are farther from pi than the preceding one. Commented Dec 20, 2011 at 18:41
• Mathematica beating GolfScript. Neat. Commented Nov 29, 2012 at 17:47
• In 61: Select[Round[f=Pi,1/Range@1*^6],If[#<f,f=#;True]&@Abs[#-Pi]&] ... but useless given the dominant bias Commented Feb 21, 2014 at 3:28
• Yarr, Matie. Thar be magic in this code. Commented Feb 21, 2014 at 3:31

## Perl, 77 chars

$e=$p=atan2 0,-1;($f=abs$p-($==$p*$_+.5)/$_)<$e&&($e=$f,say"$=/$_")for 1..1e6  A minor challenge is that Perl doesn't have a built-in π constant available, so I first had to calculate it as atan2(0,-1). I'm sure this will be beaten by languages more suited for the job, but it's not bad for a language mainly designed for text processing. • You could change 999999 to 1e6 and save 3 chars. – Toto Commented Dec 10, 2011 at 10:28 • @M42: Thanks! Down to 82 chars now. Commented Dec 10, 2011 at 16:05 • Really nice,$= to get integer. Sorry, I can't upvote twice.
– Toto
Commented Dec 11, 2011 at 13:36
• I can't get this to run: String found where operator expected at prog.pl line 1, near "say"$=/$_"" Commented Dec 20, 2011 at 17:25
• @KeithRandall: You need the -M5.01 switch (and Perl 5.10.0 or later) for the say command. Sorry for not mentioning that. Commented Dec 20, 2011 at 22:34

## JS (95 characters)

for(i=k=1,m=Math;i<1e6;i++)if((j=m.abs((x=m.round(m.PI*i))/i-m.PI))<k)k=j,console.log(x+'/'+i)


It does print 167 lines.

## Python, 9693 89 characters

a=b=d=1.
while b<=1e6:
e=3.14159265359-a/b;x=abs(e)
if x<d:print a,b;d=x
a+=e>0;b+=e<0


### Python, 95 93 characters, different algorithm

p=3.14159265359;d=1
for a in range(3,p*1e6):
b=round(a/p);e=abs(p-a/b)
if e<d:print a,b;d=e


note: It was less characters to write p=3.14159265359; than from math import*. Darn those wordy imports!

• Some shortenings: 1.0 -> 1., 10**6 -> 1e6 Commented Dec 10, 2011 at 6:34
• I have updated with your improvements. Thanks much. Commented Dec 10, 2011 at 6:40
• @KeithRandall, but the second of those makes the output violate the spec. Commented Dec 10, 2011 at 7:32
• In second approach there is no need for variable p. That is 4 chars.
– Ante
Commented Dec 10, 2011 at 14:23
• @PeterTaylor: I don't understand. How does it violate the spec? Commented Dec 11, 2011 at 0:48

### Ruby 1.9, 84 characters

m=1;(1..1e6).map{|d|n=(d*q=Math::PI).round;k=(n-q*d).abs/d;k<m&&(m=k;puts [n,d]*?/)}

• @Peter Taylor You're right. You have to use Ruby 1.9. Commented Dec 10, 2011 at 11:46

## C99, 113 characters

main(d,n){double e=9,p=2*asin(1),c,a=1;for(;n=d*p+.5,c=fabsl(p-a*n/d),d<1e6;++d)c<e&&printf("%d/%d\n",n,d,e=c);}


Need to compile with -lm, and probably full of undefined behaviour, but it works for me.

### Scala - 180 chars

import math._
def p(z:Int,n:Int,s:Double):Unit=
if(n==1e6)0 else{val q=1.0*z/n
val x=if(abs(Pi-q)<s){println(z+"/"+n)
abs(Pi-q)}else s
if(Pi-q<0)p(z,n+1,x)else p(z+1,n,x)}
p(3,1,1)


// ungolfed: 457

val pi=math.Pi
@annotation.tailrec
def toPi (zaehler: Int = 3, nenner: Int = 1, sofar: Double=1): Unit = {
if (nenner == 1000000) ()
else {
val quotient = 1.0*zaehler/nenner
val diff = (pi - quotient)
val next = if (adiff < sofar) {
println (zaehler + "/" + nenner)
}
else sofar
if (diff < 0) toPi (zaehler, nenner + 1, next)
else toPi (zaehler + 1, nenner, next)
}
}


The tailrec annotation is just a check, to verify, that it is tail-recursive, which is often a performance improvement.

• I can't get this to work: pi.scala:1 error: not found: value math Commented Dec 20, 2011 at 17:34
• Are you using Scala 2.8? Commented Dec 21, 2011 at 18:49
• My scala says "unknown version", weird. On ideone.com, they use 2.8.0 and I still get errors. Commented Dec 21, 2011 at 20:19
• Try it at simplyscala.com - works for me. For scala-2.8, replacing math with Math might be sufficient. I mentioned simplyscala on this metathread, if you happen to search for it again: meta.codegolf.stackexchange.com/a/401/373 Commented Dec 21, 2011 at 20:48
• OK, that works. Commented Dec 22, 2011 at 0:52

# Mathematica 18 17 chars

I chose to use, as a measure of "best", the number of terms in a continued fraction representation of π. By this criterion, the best rational approximations of π are its convergents.

There are 10 convergents of π with a denominator less than one million. This is fewer than the requested 167 terms, but I am including it here because it may be of interest to others.

Convergents[π, 10]

(* out *)
{3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317,
312689/99532, 833719/265381, 1146408/364913}


If you really want to see the denominator for the first convergent, it will cost an additional 11 characters:

Convergents[π, 10] /. {3 -> "3/1"}
(* out *)
{"3/1", 22/7, 333/106, 355/113, 103993/33102, 104348/33215,
208341/66317, 312689/99532, 833719/265381, 1146408/364913}


For those who are interested, the following shows the relations among the convergents, partial quotients, and continued fraction expression of convergents of π:

Table[ContinuedFraction[π, k], {k, 10}]
w[frac_] := Row[{Fold[(#1^-1 + #2) &, Last[#], Rest[Reverse[#]]] &[Text@Style[#, Blue, Bold, 14] & /@ ToString /@ ContinuedFraction[frac]]}];
w /@ FromContinuedFraction /@ ContinuedFraction /@ Convergents[π, 10]


Please excuse the inconsistent formatting of the continued fractions.

• That's about half-way to a solution, but it's the easiest half. My GolfScript solution hard-codes a suitable representation of the the continued fraction in only 2 characters more. Commented Aug 17, 2012 at 12:23
• But you didn't use continued fractions for your solution to this question, did you? Commented Aug 17, 2012 at 17:21
• Yes. It was the obvious way to do it. Commented Aug 17, 2012 at 19:42
• In addition to being concise, this is much faster than most or all of the other solutions that have been posted. Commented Feb 21, 2014 at 3:36

# C# 140 129 chars

double n=3,d=1,e=d;while(n<4e5){double w=n/d-Math.PI,a=Math.Abs(w);if(a<e){e=a;Console.WriteLine(n+"/"+d);}if(w>0)d++;else n++;}


Uncompressed code

var numerator = 3d;
var denominator = 1d;
var delta = 4d;
while (numerator < 4e5)
{
var newDelta = (numerator / denominator) - Math.PI;
var absNewDelta = Math.Abs(newDelta);
if (absNewDelta < delta)
{
delta = absNewDelta;
Console.WriteLine(string.Format("{0}/{1}", numerator, denominator));
}

if (newDelta > 0)
{
denominator++;
}
else
{
numerator++;
}
}

• var is not always your friend. By removing it in favour of double you gain the ability to merge declarations, lose the requirement to use double literals, and can save 16 chars. OTOH the question asks for a program, so you'll lose a few to adding a class declaration and a Main method. Commented Dec 11, 2012 at 9:36

# J, 69 65

### New

],@.(<&j{.)/({~(i.<./)@j=.|@-l)@(%~(i:3x)+<.@*l=.1p1&)"0>:_i.1e3


Still a brute force approach but much faster and a tad shorter.

### Old

A simple "brute force":

(#~({:<<./@}:)\@j)({~(i.<./)@j=.|@-l)@(%~(i:6x)+<.@*l=.1p1&)"0>:i.1e3


make a list of a/bs and then discard those that are farther from π for some b'<b.

Note: Change 1e3 to 1e6` for the full list. Go do something else and return later.