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This is the formula by John Machin:

\begin{equation} \frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239} \end{equation}

This topic fascinates me, and I would like to know something more. For example, I do not know how this formula obtained, or do not know why it represent an algorithm that converges fast. I'd get you some reference (online or not) answer my curiosity, please.

Thank you very much

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You might begin with the Wikipedia article Machin-like Formulas and its references. There is also a separate article on the Machin formula. It is done by manipulation of the trigonometric identity $\tan(x+y)=\frac{\tan x +\tan y}{1-\tan x\tan y}$. This is an easy consequence of the usual addition laws for sine and cosine. Euler derived many Machin-like formulas. Some number theory is involved, since we end up looking for integer solutions of certain equations. The simplest such formula is $\pi/4=\arctan(1/2)+\arctan(1/3)$. – André Nicolas Dec 5 '12 at 7:23
Thank you! But could you recommend a good book about Machin formula? On MathWorld there are many references, but I would like a personal recommendation. – Mark Dec 5 '12 at 10:16
There was a nice expository paper on Machin formulas from the Gaussian integer point of view, by Calcut, American Math Monthly, 2009. "Nice" here is for my tastes. – André Nicolas Dec 5 '12 at 16:10
Thank you very much! – Mark Dec 5 '12 at 17:52
up vote 0 down vote accepted

The representation of pi with the arctan function at small values, is efficient for the calculation of pi, because many series such as the taylor series for the arctan function: $arctan(x)=x-x^3/3+x^5/5-x^7/7+x^9/9...$ converge faster, for smaller values of x, and if one has a unit fraction 1/k, they only have to calculate powers of k, sense 1 raised to any power is 1, as apposed to a non unit fraction a/b, where powers of both a, and b, have to be calculated. Sense it is also easy to multiply real numbers with integers haveing pi in the form $\pi=Karctan(1/p)+Barctan(1/q)...$ allows one to reduce the value of the argument in arctan(x), and therefore be able to use smaller values, which converge faster in many arctan series, for example although $\pi/4=arctan(1)=1-1/3+1/5-1/7+1/9..$, this series converges slowly, while $\pi/4=4(1/5-1/375+1/15625...)-(1/239-1/40955757..)$, converges much faster, thus making machin's formula more efficient.

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