Machin-Like Formulas

Machin-like formulas have the form

$$m\cot^{-1}u+n\cot^{-1}v={\textstyle{1\over 4}}k\pi,$$ (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 $c_n\not=0$ in the table of Todd (1949) to Inverse Cotangents. However, this gives only Machin-like formulas in which the smallest term is $\pm 1$.

Maclaurin-like formulas can be derived by writing

$$\cot^{-1} z = {1\over 2i}\ln\left({z+i\over z-i}\right)$$ (2)

and looking for $a_k$ and $u_k$ such that

$$\sum_k a_k \cot^{-1} u_k={\textstyle{1\over 4}}\pi,$$ (3)

so

$$\prod_k \left({u_k+i\over u_k-i}\right)^{a_k} = e^{2\pi i/4}=i.$$ (4)

Machin-like formulas exist Iff (4) has a solution in Integers. This is equivalent to finding Integer values such that

$$(1-i)^k(u+i)^m(v+i)^n$$ (5)

is Real (Borwein and Borwein 1987, p. 345). An equivalent formulation is to find all integral solutions to one of

$$1+x^2=2y^n$$ (6)

$$1+x^2=y^n$$ (7)

for $n=3$, 5, ....

There are only four such Formulas,

$${\textstyle{1\over 4}}\pi = 4\tan^{-1}({\textstyle{1\over 5}})-\tan^{-1}({\textstyle{1\over 239}})$$ (8)

$${\textstyle{1\over 4}}\pi = \tan^{-1}({\textstyle{1\over 2}})+\tan^{-1}({\textstyle{1\over 3}})$$ (9)

$${\textstyle{1\over 4}}\pi = 2\tan^{-1}({\textstyle{1\over 2}})-\tan^{-1}({\textstyle{1\over 7}})$$ (10)

$${\textstyle{1\over 4}}\pi = 2\tan^{-1}({\textstyle{1\over 3}})+\tan^{-1}({\textstyle{1\over 7}}),$$ (11)

known as Machin's Formula, Euler's Machin-Like Formula, Hermann's Formula, and Hutton's Formula. These follow from the identities

$$\left({5+i\over 5-i}\right)^4\left({239+i\over 239-i}\right)^{-1} = i$$ (12)

$$\left({2+i\over 2-i}\right)\left({3+i\over 3-i}\right) = i$$ (13)

$$\left({2+i\over 2-i}\right)^2\left({7+i\over 7-i}\right)^{-1} = i$$ (14)

$$\left({3+i\over 3-i}\right)^2\left({7+i\over 7-i}\right) = i.$$ (15)

Machin-like formulas with two terms can also be generated which do not have integral arc cotangent arguments such as Euler's

$${\textstyle{1\over 4}}\pi = 5\tan^{-1}({\textstyle{1\over 7}})+2\tan^{-1}({\textstyle{3\over 79}})$$ (16)

(Wetherfield 1996), and which involve inverse Square Roots, such as

$${\pi\over 2}=2\tan^{-1}\left({1\over\sqrt{2}}\right)+\tan^{-1}\left({1\over\sqrt{8}}\right).$$ (17)

Three-term Machin-like formulas include Gauss's Machin-Like Formula

$${\textstyle{1\over 4}}\pi = 12\cot^{-1}18+8\cot^{-1}57-5\cot^{-1}239,$$ (18)

Strassnitzky's Formula

$${\textstyle{1\over 4}}\pi=\cot^{-1}2+\cot^{-1}5+\cot^{-1}8,$$ (19)

and the following,

$${\textstyle{1\over 4}}\pi = 6\cot^{-1}8+2\cot^{-1}57+\cot^{-1}239$$ (20)

$${\textstyle{1\over 4}}\pi = 4\cot^{-1}5-1\cot^{-1}70+\cot^{-1}99$$ (21)

$${\textstyle{1\over 4}}\pi = 1\cot^{-1}2+1\cot^{-1}5+\cot^{-1}8$$ (22)

$${\textstyle{1\over 4}}\pi = 8\cot^{-1}10-1\cot^{-1}239-4\cot^{-1}515$$ (23)

$${\textstyle{1\over 4}}\pi = 5\cot^{-1}7+4\cot^{-1}53+2\cot^{-1}4443.$$ (24)

The first is due to Størmer, the second due to Rutherford, and the third due to Dase.

Using trigonometric identities such as

$$\cot^{-1}x = 2\cot^{-1}(2x)-\cot^{-1}(4x^3+3x),$$ (25)

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 Formula is the time it takes to calculate $\pi$ with the Power series for arctangent

$$\pi=a_1\cot(b_1)+a_2\cot(b_2)+\ldots,$$ (26)

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

$$e\equiv \sum {1\over\log_{10} b_i}.$$ (27)

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

${\textstyle{1\over 4}}\pi=183\cot^{-1}239+32\cot^{-1}1023-68\cot^{-1}5832$
$+12\cot^{-1}110443-12\cot^{-1}4841182-100\cot^{-1}6826318\quad$	(28)

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

${\textstyle{1\over 4}}\pi=P\cot^{-1} 2-M\cot^{-1}3+L\cot^{-1}5+K\cot^{-1}7$
$\quad +(N+K+L-2M+3P-5)\cot^{-1}8$
$\quad +(2N+M-P+2-L)\cot^{-1}18$
$\quad -(2P-3-M+L+K-N)\cot^{-1}57-N\cot^{-1}{239},$
	(29)

where $K$, $L$, $M$, $N$, and $P$ are Positive Integers, and

$${\textstyle{1\over 4}}\pi=(N+2)\cot^{-1}2-N\cot^{-1}3-(N+1)\cot^{-1}N.$$ (30)

The following table gives the number $N(n)$ of Machin-like formulas of $n$ terms in the compilation by Wetherfield and Hwang. 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 $<2.0$.

$n$	$N(n)$	$\min e$
1	1	0
2	4	1.85113
3	106	1.78661
4	39	1.58604
5	90	1.63485
6	120	1.51244
7	113	1.54408
8	18	1.65089
9	4	1.72801
10	78	1.63086
11	34	1.6305
12	188	1.67458
13	37	1.71934
14	5	1.75161
15	24	1.77957
16	51	1.81522
17	5	1.90938
18	570	1.87698
19	1	1.94899
20	11	1.95716
21	1	1.98938
Total	1500	1.51244

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

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 347-359, 1987.

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.

Hwang, C.-L. ``More Machin-Type Identities.'' Math. Gaz., 120-121, March 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.

Størmer, C. ``Sur l'Application de la Théorie des Nombres Entiers Complexes à la Solution en Nombres Rationels $x_1$, $x_2$, ..., $c_1$, $c_2$, ..., $k$ 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.

Weisstein, E. W. ``Machin-Like Formulas.'' Mathematica notebook MachinFormulas.m.

Wetherfield, M. ``The Enhancement of Machin's Formula by Todd's Process.'' Math. Gaz. 80, 333-344, 1996.

Wetherfield, M. ``Machin Revisited.'' Math. Gaz., 121-123, March 1997.