# 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:

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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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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In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. The principal inverses are listed in the following table.

Name Usual notation Definition Domain of x for real result Range of usual principal value
The integers (from the Latin integer, which means with untouched integrity, whole, entire) are the set of numbers including the whole numbers (0, 1, 2, 3, …) and their negatives (0, −1, −2, −3, …).
computer is a machine which manipulates data according to a list of instructions.

Computers take numerous physical forms. The first devices that resemble modern computers date to the mid-20th century (around 1940 - 1941), although the computer concept and various machines
Leonhard Euler

Portrait by Johann Georg Brucker
Born March 15 1707
Basel, Switzerland
Died September 18 [O.S.

University of Tokyo (東京大学 Tōkyō daigaku
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A supercomputer is a computer that led the world (or was close to doing so) in terms of processing capacity, particularly speed of calculation, at the time of its introduction.
Kikuo Takano was a Japanese poet and mathematician. He was born on Sado Island in 1927. He graduated from Utsunomiya Agricultural College in 1948.

He began to write poems from the next day Japan had ended role in World War II.
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