Machinlike formulas have the form
(1) 
Maclaurinlike formulas can be derived by writing
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
There are only four such Formulas,
(8)  
(9)  
(10)  
(11) 
(12)  
(13)  
(14)  
(15) 
Machinlike formulas with two terms can also be generated which do not have integral arc cotangent arguments such as
Euler's
(16) 
(17) 
Threeterm Machinlike formulas include Gauss's MachinLike Formula
(18) 
(19) 
(20)  
(21)  
(22)  
(23)  
(24) 
Using trigonometric identities such as
(25) 
The efficiency of a Formula is the time it takes to calculate with the Power series for arctangent
(26) 
(27) 
(28) 

(29) 
(30) 
The following table gives the number of Machinlike formulas of terms in the compilation by Wetherfield and Hwang. Except for previously known identities (which are included), the criteria for inclusion are the following:
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 MachinLike Formula, Gauss's MachinLike 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. 347359, 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, 6798, 1988.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: SpringerVerlag, pp. 241248, 1996.
Hwang, C.L. ``More MachinType Identities.'' Math. Gaz., 120121, March 1997.
Lehmer, D. H. ``On Arccotangent Relations for .'' Amer. Math. Monthly 45, 657664, 1938.
Lewin, L. Polylogarithms and Associated Functions. New York: NorthHolland, 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 , , ..., , , ..., de l'Equation....'' Archiv for Mathematik og Naturvidenskab B 19, 7585, 1896.
Todd, J. ``A Problem on Arc Tangent Relations.'' Amer. Math. Monthly 56, 517528, 1949.
Weisstein, E. W. ``MachinLike Formulas.'' Mathematica notebook MachinFormulas.m.
Wetherfield, M. ``The Enhancement of Machin's Formula by Todd's Process.'' Math. Gaz. 80, 333344, 1996.
Wetherfield, M. ``Machin Revisited.'' Math. Gaz., 121123, March 1997.
© 19969 Eric W. Weisstein