HAKMEM is a collection of tricks and trivia from the MIT AI lab written in 1972. I’ve mentioned HAKMEM here before. The image below came from HAKMEM item 123, explained here.

I ran across HAKMEM item 143 while writing my previous two blog posts about the arithmetic-geometric mean (AGM). This entry, due to Gene Salamin, says that

for large *n*. In fact, *n* doesn’t have to be very large. The expression on the right converges exponentially as *n* grows.

Here’s the Mathematica code that produced the plot above.

Plot[{2 n ArithmeticGeometricMean[1, 4 Exp[-n]], Pi}, {n, 1, 5}, PlotRange -> All]

If you have *e* to a large number of decimals, you could compute π efficiently by letting *n* be a power of 2, computing *e*^{–n} by repeatedly squaring 1/*e*, and using the AGM. Computing the AGM is simple. In Python notation, iterate

a, b = (a+b)/2, sqrt(a*b)

until *a* and *b* are equal modulo your tolerance.

If you want to search / copypaste HAKMEM:

reformatted as html (Henry Baker)

https://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html

and an ocr pdf

https://w3.pppl.gov/~hammett/work/2009/AIM-239-ocr.pdf