PiEngine

PiEngine is a high-performance Pi calculator capable of computing billions of digits with extreme precision. It features dual implementations in OCaml and C, serving as a benchmark for high-precision arithmetic and functional vs. systems programming paradigms.

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Recent Achievement: 1,000,000,000 Digits

We have successfully reached the 1 Billion Digits milestone! By leveraging the Chudnovsky algorithm and binary splitting, PiEngine efficiently handles massive numbers. Note that computing 1B digits and beyond is primarily dependent on available RAM, as the intermediate values in the binary splitting process grow significantly.

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Features

• Dual Implementations (Branch-separated):
  • OCaml (main branch): Utilizing the powerful Zarith library for clean, high-performance functional code.
  • C (v3-C branch): Leveraging GMP and MPFR for maximum low-level efficiency.
• Binary Splitting: Optimized series summation that reduces the complexity of large multiplications.
• High Performance: Designed to scale from thousands to billions of digits.
• Cross-Platform: Tested on Linux and macOS.
• Robust CLI: Complete control over precision, output, and verbose timing metrics.

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How the Algorithm Works

Chudnovsky Series

PiEngine uses the world-record Chudnovsky algorithm, which converges at approximately 14 digits per term:

$$\frac{1}{\pi} = 12 \sum_{k=0}^{\infty} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k+3/2}}$$

Binary Splitting

To compute this series efficiently, we use Binary Splitting. Instead of computing each term and adding it to a running total, we recursively split the series into two halves. This allows us to use large, efficient multiplications (Karatsuba or FFT-based) provided by GMP and Zarith, significantly speeding up the process for millions of digits. For very large digit counts, the memory usage becomes the primary bottleneck.

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Roadmap and Next Goal: 10,000,000,000 Digits

Our next target is the 10 Billion Digits milestone. To reach this level of performance and handle the increasing computational load, future updates will focus on:

• Parallelization: Implementing multi-threaded binary splitting to utilize all available CPU cores.
• Memory Optimization: Reducing the memory footprint during the division and square root phases.
• Disk-based Computation: Exploring swapping strategies for digit counts that exceed available RAM.

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Getting Started

Prerequisites

For OCaml (on main branch)

• OPAM
• Dune
• zarith library

opam install zarith dune

For C (on v3-C branch)

• GMP
• MPFR

# Ubuntu/Debian
sudo apt install libgmp-dev libmpfr-dev
# Arch Linux
sudo pacman -S gmp mpfr

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Build and Run

OCaml Version (Default - main branch)

The OCaml version offers a balance of safety and performance.

dune build
dune exec bin/main.exe -- --digits 1000000 --verbose --output pi.txt

C Version (Switch to v3-C branch)

The C version provides low-level control and direct access to GMP primitives.

git checkout v3-C
make
./piengine --digits 1000000 --verbose --output pi.txt

CLI Options

Both versions support identical CLI flags:

• -d, --digits NUM: Number of digits to compute (default: 1000).
• -o, --output FILE: Write the result to a file instead of stdout.
• -v, --verbose: Display detailed timing and performance information.
• -h, --help: Show help message.

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Project Structure

The project is organized into two primary branches:

• main: Contains the OCaml implementation.
  • bin/: CLI application.
  • lib/: Core Chudnovsky logic.
  • test/: Regression and comparison tests.
• v3-C: Contains the C implementation.
  • src/: Core logic and CLI.
  • tests/: Unit tests.

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Milestones

Milestone	Digits	Status
Milestone 1	1,000	Complete
Milestone 2	1,000,000	Complete
Milestone 3	100,000,000	Complete
Milestone 4	1,000,000,000	Achieved
Milestone 5	10,000,000,000	In Progress / Planned

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Testing

We ensure correctness by comparing results between the OCaml and C implementations, as well as against known reference constants.

# Run OCaml tests (on main branch)
./test_pi.sh

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“Mathematics is the music of reason.”