In mathematics, Machin-like formulas are a class of identities involving π = 3.14159... that generalize John Machin's formula from 1706:
- <math>\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239},</math>
which he used along with the Taylor series expansion of arctan to compute π to 100 decimal places. Machin-like formulas have the form
- <math>\frac{\pi}{4} = \sum_{n}^N a_n \arctan\frac{1}{b_n}</math>
with <math>a_n</math> and <math>b_n</math> integers. The same method is still among the most efficient known for computing a large number of digits of π with digital computers.
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Derivation
To understand where this formula comes from, start with following basic ideas:
- <math>\frac{\pi}{4} = \arctan(1)</math>
- <math>\tan(2\arctan(a)) = \frac{2 a} { 1 - a ^ 2}</math> (tangent double angle identity)
- <math>\tan(a-\arctan(b)) = \frac{\tan(a)-b} { 1 + \tan(a) b}</math> (tangent difference identity)
- <math>\frac{\pi}{16} = 0.196349\dots</math> (approximately)
- <math>\arctan\left(\frac{1}{5}\right) = \arctan(0.2) = 0.197395\dots </math> (approximately)
In other words, for small numbers, arctangent is to a good approximation just the identity function. This leads to the possibility that a number <math>q</math> can be found such that
- <math>\frac{\pi}{16} = \arctan(\frac{1}{5}) - \frac{1}{4} \arctan(q). </math>
Using elementary algebra, we can isolate <math>q</math>:
- <math>q = \tan\left(4 \arctan\left(\frac{1}{5}\right) - \frac{\pi}{4}\right) </math>
Using the identities above, we substitute arctan(1) for π/4 and then expand the result.
- <math>q = \frac{\tan\left(4 \arctan\left(\frac{1}{5}\right)\right) - 1} { 1 + \tan\left(4 \arctan\left(\frac{1}{5}\right)\right)} </math>
Similarly, two applications of the double angle identity yields
- <math>\tan\left(4 \arctan\left(\frac{1}{5}\right)\right) = \frac{120}{119}</math>
and so
- <math>q = \frac{\frac{120}{119} - 1}{1 +\frac{120}{119}} = \frac{1}{239}.</math>
Two-term formulas
There are exactly three additional Machin-like formulas with two terms; these are Euler's
- <math>\frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3}</math>,
Hermann's,
- <math>\frac{\pi}{4} = 2 \arctan\frac{1}{2} - \arctan\frac{1}{7}</math>,
and Hutton's
- <math>\frac{\pi}{4} = 2 \arctan\frac{1}{3} + \arctan\frac{1}{7}</math>.
More terms
The current record for digits of π, 1,241,100,000,000, by Yasumasa Kanada of Tokyo University, was obtained in 2002. A 64-node Hitachi supercomputer with 1 terabyte of main memory, performing 2 trillion operations per second, was used to evaluate the following Machin-like formulas:
- <math> \frac{\pi}{4} = 12 \arctan\frac{1}{49} + 32 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} + 12 \arctan\frac{1}{110443}</math>
- Kikuo Takano (1982).
- <math> \frac{\pi}{4} = 44 \arctan\frac{1}{57} + 7 \arctan\frac{1}{239} - 12 \arctan\frac{1}{682} + 24 \arctan\frac{1}{12943}</math>
- F. C. W. Störmer (1896).
The more efficient currently known Machin-like formulas for computing:
- <math>
\begin{align} \frac{\pi}{4} =& 183\arctan\frac{1}{239} + 32\arctan\frac{1}{1023} - 68\arctan\frac{1}{5832} + 12\arctan\frac{1}{110443}\\ & - 12\arctan\frac{1}{4841182} - 100\arctan\frac{1}{6826318}\\ \end{align} </math>
- 黃見利(Hwang Chien-Lih) (1997).
- <math>
\begin{align} \frac{\pi}{4} =& 183\arctan\frac{1}{239} + 32\arctan\frac{1}{1023} - 68\arctan\frac{1}{5832} + 12\arctan\frac{1}{113021}\\ & - 100\arctan\frac{1}{6826318} - 12\arctan\frac{1}{33366019650} + 12\arctan\frac{1}{43599522992503626068}\\ \end{align} </math>
- 黃見利(Hwang Chien-Lih) (2003).
