Machinlike

Machinlike is a term used in mathematics to describe a class of formulas for computing the mathematical constant pi. These formulas express pi as a linear combination of arctangent terms with rational arguments, typically in the form pi = sum_i a_i arctan(1/x_i), where a_i are integers and x_i are positive integers greater than 1. The term honors John Machin, whose 1706 formula helped renew interest in arctangent-based computations and inspired a family of related identities.

Historically, Machin published pi/4 = 4 arctan(1/5) - arctan(1/239). Using the arctangent addition formulas, this identity yields rapidly converging series when the arctan terms are expanded as power series. Over the centuries, mathematicians derived many Machinlike formulas with two or more terms, balancing the size of coefficients against convergence speed and arithmetic cost.

Computation and relevance: In the era of hand calculation, Machinlike formulas enabled high-precision digits with comparatively short series. Today they are often superseded by faster algorithms such as the Chudnovsky formula but remain of interest for numerical analysis, algorithm design, and education. Researchers study how to combine arctangent terms to minimize computation while maximizing convergence rate, sometimes tailoring formulas to specific hardware or precision targets.

Variants: Notable examples include the classic Machin formula pi = 16 arctan(1/5) - 4 arctan(1/239) and numerous others with different denominators. The general approach can be extended to include more terms, yielding a wide range of Machinlike identities suitable for various computational goals.