prompter: how is the precise value of pi calculated?

assistant: The precise value of pi (π) is calculated using various mathematical algorithms and techniques. 

The most common method is to use an iterative algorithm, such as the Archimedes method or the Monte Carlo method, to approximate the value of π to a large number of decimal places. 

The decimal representation of π can also be calculated using infinite series, such as the Leibniz formula for π, or using numerical analysis techniques such as Fourier analysis. 

In addition, the value of π can also be calculated using symbolic computation methods, such as the use of elliptic functions or modular forms.

assistant: There are several different ways to calculate pi, but since pi has an infinite number of decimal places it is impossible to obtain a definitive answer. A simple way to calculate pi is to use the Gregory-Leibniz series. To do this you need to calculate this formula: π=(4/1)-(4/3)+(4/5)-(4/7)+(4/9)-(4/11)+(4/13)-(4/15) ⋯
The longer the formula you use, the more precise the answer will be.

prompter: This series converges incredibly slowly. Can you give me something that converges faster?

assistant: Sure! Another way to calculate π is using the Bailey–Borwein–Plouffe (BBP) formula. The BBP formula states that π = ∑ n=0 ∞ (1 / (16^n) * (4 / (8n + 1) - 2 / (8n + 4) - 1 / (8n + 5) - 1 / (8n + 6))).

The sigma (∑) symbol means summarization, that is, π equals to the sum of 1 / (16^n) * (4 / (8n + 1) - 2 / (8n + 4) - 1 / (8n + 5) - 1 / (8n + 6)) for any n, from 0 to infinity.

prompter: I know that there exist supercomputers that compute new digits to Pi. They must have some algorithm to compute Pi. Can you tell me which algorithm these supercomputers use?

assistant: The algorithm for calculating the digits of pi that is most often used by supercomputers is the Chudnovsky algorithm, which is based on Ramanujan’s pi formula. It has been used in most of the recent world record calculations, such as calculating 100 trillion digits of pi, which was achieved on March 21, 2022. The algorithm has a time complexity of O(n(log n)^3).

assistant: Yes, there are several algorithms that supercomputers use to compute new digits of Pi. One of the most widely used algorithms is the Bailey-Borwein-Plouffe (BBP) formula. The BBP formula is a fast and efficient method for calculating the nth hexadecimal digit of Pi without having to calculate the preceding digits. This makes it ideal for use on supercomputers.

Another commonly used algorithm is the Gauss-Legendre algorithm. This is an iterative algorithm that uses the Gauss-Legendre formula to calculate Pi with increasing precision. The Gauss-Legendre algorithm is especially well-suited for parallel computation, making it a popular choice for use on supercomputing clusters.

Both the BBP and Gauss-Legendre algorithms are based on mathematical formulas and theorems related to Pi and they are designed to take advantage of the power and speed of modern supercomputers. With these algorithms, supercomputers are capable of computing trillions of digits of Pi with high precision.

assistant: As of 2022, Pi is usually calculated with the program y-cruncher, witch is based on the Chudnovsky algorithm.

assistant: Chudnovsky algorithm may be used to calculate digits of pi. It was used to find up to 100 trillion digits on March 21, 2022.

prompter: Explain to me Chudnovsky Algorithm, and why we do not use any other algorithm

prompter: Tell me more about the background of the different algorithms, and their history, to evaluate the places of Pi. What were the most significant breakthroughs of the last 50 years?

assistant: Google calculated 100 trillion digits of pi and they said this in a blog post:
"Program: y-cruncher v0.7.8, by Alexander J. Yee
Algorithm: Chudnovsky algorithm"
Source : https://cloud.google.com/blog/products/compute/calculating-100-trillion-digits-of-pi-on-google-cloud

assistant: Current record for pi precision is at 100,000,000,000,000 digits, calculated June 8, 2022 using software called y-cruncher. Previous 8 precision records for pi were also calculated using versions of this software.
The mathematical formula behind the record-breaking software is known as Chudnovsky algorithm, a fast method for calculating the digits of π, based on Ramanujan’s π formulae.
A detailed proof of the Chudnovsky formula with means of basic complex analysis is available at https://doi.org/10.48550/arXiv.1809.00533

assistant: The way that Pi is calculated is the ratio of the circumference of a circle to its diameter.