Code Golf: Leibniz formula for Pi

J, 14 chars

4*-/%>:+:i.1e6

Explanation

• 1e6 is number 1 followed by 6 zeroes (1000000).
• i.y generates the first y non negative numbers.
• +: is a function that doubles each element in the list argument.
• >: is a function that increments by one each element in the list argument.

So, the expression >:+:i.1e6 generates the first one million odd numbers:

1 3 5 7 …

• % is the reciprocal operator (numerator “1” can be omitted).
• -/ does an alternate sum of each element in the list argument.

So, the expression -/%>:+:i.1e6 generates the alternate sum of the reciprocals of the first one million odd numbers:

1 – 1/3 + 1/5 – 1/7 + …

• 4* is multiplication by four. If you multiply by four the previous sum, you have π.

That’s it! J is a powerful language for mathematics.

Edit: since generating 9! (362880) terms for the alternate sum is sufficient to have 5 decimal digit accuracy, and since the Leibniz formula can be written also this way:

4 – 4/3 + 4/5 – 4/7 + …

…you can write a shorter, 12 chars version of the program:

-/4%>:+:i.9!