Computing π
The video covering Machin-like formula can be found at https://youtu.be/M_fTdDx8IlY. Example code is given in Machin.lsp, written in CLISP. The code can be run online via CodingGround. To run locally have CLISP installed then type clisp Machin.lsp in the terminal in the directory where Machin.lsp is saved. If you would rather compile it first, use clisp -c Machin.lsp then clisp Machin.fas.
Use 333 binary digits of pi for 100 decimal digits requiring about 75 iterations per arctangent. For 10,000 decimal digits of precision for pi use 33220 binary digits and around 7160 iteration terms. If interested in how CLISP computes pi see CLISP's source code.
Reference links:
- "James Gregory Tercentenary memorial volume"
- "A History of Pi" by Petr Beckmann
- Leibniz's π paper
- "On the Leibnizian quadrature of the circle"
- The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha
- How Euler Did Even More by C. Edward Sandifer
- Euler's π paper E705
- John Machin and Robert Simson on Inverse-tangent Series for π
- Synopsis palmariorum matheseos by William Jones
- CLISP source code for π
The video covering the sublinear convergence order of the Gregory-Leibniz series can be found here with code here with the ability to run online via CodingGround.