REFERENCES.md

References

"The Computation of Classical Constants" by the Chudnovsky Bros

This is a classic paper introducing Chudnovsky's algorithm for computing pi and its efficient computation. I consider it landmark.

• Link

Some more helpful materials:

• Tito Piezas articles

• Ramanujan's Series for 1/pi

• A Detailed Proof of Chudnovsky Formula: Excellent first-principles presentation of the formula.

"Matters Computational" (aka fxtbook)

Joerg Arndt's tour de force on computer arithmetic algorithms. He's been in the business for a while.

• Link

Binary Splitting

There are a few references on binary splitting. The canonical one for me is the Haible paper.

• Haible & Papanikolaou

• Gourdon & Sebah

Number-Theoretic Transforms

• Link: "Faster arithmetic for number-theoretic transforms" by David Harvey. These slides are generally useful for the implementation of NTTs. Includes Shoup's modular arithmetic trick.

• Link: A simple NTT tutorial going through the basics.

• Link: Mikko Tommila's notes on the NTT.

• Speeding up the number theoretic transform by Longa and Naehrig

• The FFT - an algorithm the whole family can use: Just a fun paper title. Not useful for this project.

Primes and Modular Arithmetic

• Generalized Mersenne Primes: Solinas gives a method for reducing modular reduction of "nice" moduli to adds, subtracts, and shifts. Finnicky to get right though. Understanding this paper requires you to know what a LFSR is.

• Montgomery Multiplication

Arbitrary-precision arithmetic software

Useful for rooting your nose around.

• Link: A general list.

• Link: The bigz library, very easy to read, but not very sophisticated.

Fabrice Bellard's libbf

This is an impressive and small library, in typical Bellard style. It's quite difficult to read though.

• Link

• Technical notes

• GitHub Mirror

Assembly and SBCL

Paul Khuong has a wonderful series on writing low-level code (read: assembly) in SBCL.

• SBCL: The Ultimate Assembly Code Breadboard

• How to Define New Intrinsics

• SSE Intrinsics

Quick x864-64 assembly references:

• Link

• godbolt Useful for checking C code and getting assembler inspiration.

Closed Source Mathematical Software :(

The authors of these programs have done very impressive things. So impressive that they feel their computational secrets must not be shared. Too bad; the world is better when math software is open.

• y-cruncher: Purportedly the best pi calculating program on the planet. Broke many a record.

• PiFast: Was the fastest program for a long time.

• SuperPi: Port of a program that broke a record in the 90s. Became popular with benchmarkers.

Out-of-Core FFTs

• Fast Fourier Transforms---For Fun and Profit: The OG paper about it.

• Determining an Out-of-Core FFT Decomposition Strategy by Thomas Cormen: Has info about Swarztrauber's method and out-of-core FFTs.

• FFTs in External or Hierarchical Memory by David Bailey.

• See also Arndt.

Miscellaneous links

• Numbers, Constants, and Computation: A classic website (all the way back to 1999!) by Gourdon and Sebah. I remember reading this when I was just starting to get into math & programming.

• Jason Papadopoulos's pages: Pi programs, FFTs, etc. Another early inspiration for my study.

• Modern Computer Arithmetic by Brent and Zimmerman