Machin-like formula

In mathematics, Machin-like formulas are a class of identities involving π = 3.14159... that generalize John Machin's formula from 1706:



which he used along with the Taylor series expansion of arctan to compute π to 100 decimal places.

Machin-like formulas have the form



with and integers.

The same method is still among the most efficient known for computing a large number of digits of π with digital computers.

Derivation

To understand where this formula comes from, start with following basic ideas:

(tangent double angle identity)
(tangent difference identity)
(approximately)
(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 can be found such that



Using elementary algebra, we can isolate :



Using the identities above, we substitute arctan(1) for π/4 and then expand the result.



Similarly, two applications of the double angle identity yields



and so

Two-term formulas

There are exactly three additional Machin-like formulas with two terms; these are Euler's

,


Hermann's,

,


and Hutton's

.

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:

Kikuo Takano (1982).


F. C. W. Störmer (1896).


The more efficient currently known Machin-like formulas for computing:

黃見利(Hwang Chien-Lih) (1997).


黃見利(Hwang Chien-Lih) (2003).

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John Machin, (bapt. 1686?—June 9, 1751),[1] a professor of astronomy at Gresham College, London, is best known for developing a quickly converging series for π in 1706 and using it to compute π to 100 decimal places.
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