Machin-like formula

Machin-like formulas are expressions that compute the circle constant pi as a sum of arctangents of simple fractions. They are named for John Machin, whose formula was published by William Jones in 1706 together with the first 100 decimal digits of pi, a record at the time. Machin's own formula states that pi over four equals four times the arctangent of one fifth minus the arctangent of one over 239. The plain arctangent series converges extremely slowly, needing about five billion terms to give ten correct digits, whereas Machin's formula needs only six terms. Other mathematicians soon developed variants. All of them can be derived by repeated use of the angle addition formula for arctangent, which can be pictured as multiplying complex numbers and adding their angles. A formula is judged mainly by its Lehmer's measure, and there are exactly four two-term solutions, all found by Machin in 1705 and 1706. Longer formulas, such as an eleven-term one found in 1993, were used for record computations of pi, including a 2002 calculation of 1,241,100,000,000 digits that used two formulas to cross-check each other.