Calculating Pi with dc(1)

This is something I couldn't finish in time for March 14th = Pi Day. It's a little trick using dc, which should be pre-installed in almost any Unix-like environment.

If you have some free time, try running the following command. After a while, the value of Pi (or something close to it) will be displayed.

$ echo '6k10K^1-[d2-d1<A]dsAx0dsasF[[lar-sa0sF]sZ]sC[[la+sa1sF]sZlF1=C1r/lZxz0<m]dsmx4la*p' | dc

If the processing time is too long and you can't wait, change the 6 at the beginning of the string passed to echo to a 5.

Explanation

This is a formula that those of you who spent your middle school days killing time with BASIC might find familiar. It's the one with rather slow convergence.

$$\pi = 4 \arctan{1} = 4 \sum_{n=0}^{\infty}{\frac{(-1)^n}{2n+1}}$$

6k			# Precision is 6 digits
10K^1-[d2-d1<A]dsAx	# Enumerate all odd numbers
0dsasF			# Initialize various registers
[[lar-sa0sF]sZ]sC	# Macro to override for odd-numbered terms
[
	[la+sa1sF]sZ	# Assume it's an even-numbered term
	lF1=C		# Override if it's an odd-numbered term
	1r/		# Convert to reciprocal
	lZx		# Addition process
	z0<m		# Loop until stack height is 1
]dsmx			# Up to here is arctan(1)
4la*p			# Display Pi