How is pi calculated to trillions of digits?

You know 3.14159265... but how do we calculate these digits in the first place? Measure it? Well, definitely not up to trillions of digits of accuracy, and Archimedes' method is insanely slow for trillions of digits to be calculated. Here is a simple explanation of Machin-like algorithms, which only requires high school knowledge of complex numbers.

*REFERENCES AND FURTHER READINGS*:

1. A comprehensive history of pi and its computation: http://www.cecm.sfu.ca/~jborwein/pi-slides.pdf

2a. A brief history of pi (up to 1996): https://www.davidhbailey.com//dhbpapers/pi-quest.pdf
2b. A more layman version of the above: https://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80

3. Machin-like formula: https://en.wikipedia.org/wiki/Machin-like_formula

4a. BBP algorithm easy derivation paper: https://www.jstor.org/stable/3620961?seq=1
4b. The above adapted into an easy calculus exercise: https://drive.google.com/open?id=1yuHtAAYI54_A1r1Q43-WtZd-S3SjZEST

5a. Official Google cloud article on breaking record of pi calculation: https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud
5b. A personal blog for the same event (with a bit more technical details): http://www.numberworld.org/blogs/2019_3_14_pi_record/

6. Proof of Chudnovsky formula: https://arxiv.org/pdf/1809.00533.pdf (if you are really into this monstrous formula)

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Видео How is pi calculated to trillions of digits? канала Mathemaniac