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Machin-Like Formulas


Machin-like formulas have the form

 mcot^(-1)u+ncot^(-1)v=1/4kpi,
(1)

where u, v, and k are positive integers and m and n are nonnegative integers. Some such formulas can be found by converting the inverse tangent decompositions for which c_n!=0 in the table of Todd (1949) to inverse cotangents. However, this gives only Machin-like formulas in which the smallest term is +/-1.

Generalized Machin-like formulas in which the argument of the cotangents are rational numbers, including Euler's

 1/4pi=5tan^(-1)(1/7)+2tan^(-1)(3/(79))
(2)

(Lehmer 1938, Wetherfield 1996), square roots such as

 1/2pi=2tan^(-1)(1/(sqrt(2)))+tan^(-1)(1/(sqrt(8))),
(3)

or even mixed quadratic surds (Lehmer 1938) have also been considered.

A trivial one-term Machin-like formula is given by the identity

 1/4pi=cot^(-1)1.
(4)

Two-term Machin-like formulas can be derived by writing

 cot^(-1)z=1/(2i)ln((z+i)/(z-i))
(5)

and looking for a_k and u_k such that

 sum_(k=1)^2a_kcot^(-1)u_k=1/4pi,
(6)

so

 product_(k=1)^2((u_k+i)/(u_k-i))^(a_k)=e^(2pii/4)=i.
(7)

Machin-like formulas exist iff (7) has a solution in integers. This is equivalent to finding positive integer values u, v, and k and integer values m and n such that

 mcot^(-1)u+ntan^(-1)v=1/4kpi,
(8)

which occur iff

 (1-i)^k(u+i)^m(v+i)^n
(9)

is real (Borwein and Borwein 1987, p. 345). Another equivalent formulation is to find all integer solutions to one of

1+x^2=2y^n
(10)
1+x^2=y^n
(11)

for n=3, 5, ....

There are only four such two-term formulas,

1/4pi=4tan^(-1)(1/5)-tan^(-1)(1/(239))
(12)
1/4pi=tan^(-1)(1/2)+tan^(-1)(1/3)
(13)
1/4pi=2tan^(-1)(1/2)-tan^(-1)(1/7)
(14)
1/4pi=2tan^(-1)(1/3)+tan^(-1)(1/7),
(15)

known as Machin's formula (Borwein and Bailey 2003, p. 105), Euler's Machin-like formula (Borwein and Bailey 2003, p. 105), Hermann's formula, and Hutton's formula. These follow from the identities

((5+i)/(5-i))^4((239+i)/(239-i))^(-1)=i
(16)
((2+i)/(2-i))((3+i)/(3-i))=i
(17)
((2+i)/(2-i))^2((7+i)/(7-i))^(-1)=i
(18)
((3+i)/(3-i))^2((7+i)/(7-i))=i.
(19)

Three-term Machin-like formulas include Gauss's Machin-like formula

 1/4pi=12cot^(-1)18+8cot^(-1)57-5cot^(-1)239,
(20)

Strassnitzky's formula

 1/4pi=cot^(-1)2+cot^(-1)5+cot^(-1)8,
(21)

which was used by Dase (Borwein and Bailey 2003, p. 106), and the following:

1/4pi=6cot^(-1)8+2cot^(-1)57+cot^(-1)239
(22)
1/4pi=4cot^(-1)5-cot^(-1)70+cot^(-1)99
(23)
1/4pi=8cot^(-1)10-cot^(-1)239-4cot^(-1)515
(24)
1/4pi=5cot^(-1)7+4cot^(-1)53+2cot^(-1)4443
(25)
1/4pi=3cot^(-1)4+cot^(-1)20+cot^(-1)1985
(26)

The first is due to Størmer, the second due to Rutherford, and the last appears in Borwein and Bailey (2003, p. 107). However, there are many other such formulas, a total of 105 of which are tabulated by Weisstein.

A total of 90 five-term Machin-like formulas are tabulated by Weisstein, including the two given by Borwein and Bailey (2003, pp. 62 and 111)

1/4pi=12cot^(-1)49+32cot^(-1)57-5cot^(-1)239+12cot^(-1)110443
(27)
1/4pi=44cot^(-1)57+7cot^(-1)239-12cot^(-1)682+24cot^(-1)12943,
(28)

the first of which was found by high school teacher K. Takano in 1982 and the second of which was known to Störmer in 1896.

Using trigonometric identities such as

 cot^(-1)x=2cot^(-1)(2x)-cot^(-1)(4x^3+3x),
(29)

it is possible to generate an infinite sequence of Machin-like formulas. Systematic searches therefore most often concentrate on formulas with particularly "nice" properties (such as "efficiency").

The efficiency of a generalized Machin-like formula (possibly with rational, quadratic surd, or other inverse cotangent arguments) is the time it takes to calculate pi with the power series for inverse cotangent given by

 pi=a_1cot^(-1)b_1+a_2cot^(-1)b_2+...,
(30)

and can be roughly characterized using Lehmer's "measure" formula

 e=sum1/(log_(10)b_i)
(31)

(Lehmer 1938). The number of terms required to achieve a given precision is roughly proportional to e, so lower e-values correspond to better sums. The best currently known efficiency is 1.51244, which is achieved by the 6-term series

 1/4pi=183cot^(-1)239+32cot^(-1)1023-68cot^(-1)5832 
 +12cot^(-1)110443-12cot^(-1)4841182-100cot^(-1)6826318
(32)

discovered by C.-L. Hwang (1997). Hwang (1997) also discovered the remarkable identities

 1/4pi=Pcot^(-1)2-Mcot^(-1)3+Lcot^(-1)5+Kcot^(-1)7+(N+K+L-2M+3P-5)cot^(-1)8+(2N+M-P+2-L)cot^(-1)18-(2P-3-M+L+K-N)cot^(-1)57-Ncot^(-1)239,
(33)

where K, L, M, N, and P are positive integers, and

 1/4pi=(N+2)cot^(-1)2-Ncot^(-1)3-(N+1)cot^(-1)N.
(34)

The following table gives the number N(n) of Machin-like formulas of n terms in the compilation by Weisstein. Except for previously known identities (which are included), the criteria for inclusion are the following:

1. first term <8 digits: measure <1.8.

2. first term = 8 digits: measure <1.9.

3. first term = 9 digits: measure <2.0.

4. first term =10 digits: measure <2.0.

nN(n)mine
110
241.85113
31061.78661
4391.58604
5901.63485
61201.51244
71131.54408
8181.65089
941.72801
10781.63086
11341.6305
121881.67458
13371.71934
1451.75161
15241.77957
16511.81522
1751.90938
185701.87698
1911.94899
20111.95716
2111.98938
Total15001.51244

See also

Euler's Machin-Like Formula, Gauss's Machin-Like Formula, Gregory Number, Hermann's Formula, Hutton's Formula, Inverse Cotangent, Machin's Formula, Pi, Størmer Number, Strassnitzky's Formula

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References

Arndt, J. "Arctan Formulas." http://www.jjj.de/hfloat/jjf.ps.Arndt, J. "Big ArcTan Formula Bucket." http://www.jjj.de/hfloat/fox.ps.Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 347-359, 1987.Berstel, J.; Pin, J.-E.; and Pocchiola, M. Mathématiques et informatique: Problémes résolus. New York: McGraw-Hill, 1991.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Castellanos, D. "The Ubiquitous Pi. Part I." Math. Mag. 61, 67-98, 1988.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 241-248, 1996.Gourdon, X. and Sebah, P. "Collection of Series for pi." http://numbers.computation.free.fr/Constants/Pi/piSeries.html.Hwang, C.-L. "More Machin-Type Identities." Math. Gaz. 81, 120-121, 1997.Lehmer, D. H. "On Arccotangent Relations for pi." Amer. Math. Monthly 45, 657-664, 1938.Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.Lewin, L. Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991.Nielsen, N. Der Euler'sche Dilogarithms. Leipzig, Germany: Halle, 1909.Séroul, R. "Machin Formulas." §9.3 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 240-252, 2000.Størmer, C. "Sur l'Application de la Théorie des Nombres Entiers Complexes à la Solution en Nombres Rationnels x_1, x_2, ..., c_1, c_2, ..., k de l'Equation...." Archiv for Mathematik og Naturvidenskab B 19, 75-85, 1896.Todd, J. "A Problem on Arc Tangent Relations." Amer. Math. Monthly 56, 517-528, 1949.Wetherfield, M. "The Enhancement of Machin's Formula by Todd's Process." Math. Gaz. 80, 333-344, 1996.Wetherfield, M. "Machin Revisited." Math. Gaz. 81, 121-123, 1997.Update a linkWilliams, R. "Arctangent Formulas for Pi." http://www.cacr.caltech.edu/~roy/upi/pi.formulas.html

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Machin-Like Formulas

Cite this as:

Weisstein, Eric W. "Machin-Like Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Machin-LikeFormulas.html

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