Ask HN: How do you calculate Pi to N digits? | Hacker News Reader

Ask HN: How do you calculate Pi to N digits?

6 pointseddtries10 comments

How do you actually calculate pi to say 1000 digits? or 10,000? Most info I found on google take you to pre-solved lists of digits OR code that is optimised beyond recognition so I'm unsure how you actually do the calculation to begin with.

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https://en.wikipedia.org/wiki/Approximations_of_π

The method of using inscribed and circumscribed n-gons conceptually is the simplest (1), but converges very slowly. See https://arxiv.org/pdf/2008.07995.pdf.

(1) at least to non-mathematicians. Pure mathematicians will want to prove the claims that the circumference of a circle is larger than/smaller than that of its inscribed/circumscribed polygon and that both can get arbitrarily close to it if the number of sides gets large enough.

The Digit extraction methods (https://en.wikipedia.org/wiki/Approximations_of_%CF%80#Digit...) section still amazes me, that we can calculate the Nth digit without needing to calculate the digits before it.

Matt Parker makes a new video on calculating Pi (by hand) every year: https://www.youtube.com/playlist?list=PLhtC92GarkjyYbxI3-4qz...

The more practical methods:

- Chudnovsky algorithm: https://youtu.be/LhlqCJjbEa0

- Newton's method: https://youtu.be/CKl1B8y4qXw?t=160

Search "infinite series for pi". Several sources contain historical surveys of series: the earliest-discovered converge slowly, the best recent ones I think quadruple the number of digits on each iteration.

There was a paper

https://maa.org/sites/default/files/pdf/pubs/amm_supplements... A Spigot Algorithm for the Digits of π Author(s): Stanley Rabinowitz and Stan Wagon Source: The American Mathematical Monthly, Vol. 102, No. 3 (Mar., 1995), pp. 195-203

Fortran code for the algorithm is at https://fortran-lang.discourse.group/t/fortran-code-snippets...

To approximate an arbitrary number of digits of pi with sympy:

  #! pip install sympy gmpy
  import sympy
  sympy.evalf(sympy.pi)
  sympy.N(sympy.pi)

https://docs.sympy.org/latest/modules/evalf.html

RosettaCode > [Python] lists quite a few more: https://rosettacode.org/wiki/Pi#Python

I want to try write the underlying code myself, but thanks!

gmpy2.const_pi(precision=0: =53bits) probably isn't too helpful then either: https://gmpy2.readthedocs.io/en/latest/mpfr.html#gmpy2.const... https://github.com/aleaxit/gmpy/issues/253#issue-499509692

RosettaCode has only one solution to the "Approximate Pi with Rationals (not float64s)" in Python problem.

I remember having seen a few statistical manifestation of Pi demos on YouTube

"Calculating Pi with Darts by Physics Girl [and Veritasium Derek] (PBS LearningMedia)" https://youtube.com/watch?v=M34TO71SKGk

Monte Carlo method: https://en.wikipedia.org/wiki/Monte_Carlo_method

"Inferred value of Pi is never 3.1415" (with pymc) https://discourse.pymc.io/t/inferred-value-of-pi-is-never-3-... https://github.com/pymc-devs/pymc

https://www.bellard.org/pi/

dig into the links a bit

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