Numberick

✨ An arithmagick overhaul in Swift.

[!IMPORTANT] The development of this project has moved over to Ultimathnum.

It turns out that I need \~(\~(x)) to equal x for all x.\ This is not possible with arbitrary unsigned integers at the moment.\ So I'm working on a novel abstraction that permits this.\ It unifies all sizes and brings recovery mechanisms to generic code.\ Also, I'm open to work.

Table of Contents

• NBKCoreKit
• NBKDoubleWidthKit
• NBKFlexibleWidthKit
• Installation
• Acknowledgements

NBKCoreKit (Sources, Tests, Benchmarks)

A new protocol hierarchy that refines Swift's standard library.

Protocols

• NBKBinaryInteger
• NBKBitPatternConvertible
• NBKCoreInteger
• NBKFixedWidthInteger
• NBKSignedInteger
• NBKUnsignedInteger

Models

• NBKChunkedInt
• NBKEndianness
• NBKPrimeSieve
• NBKStaticBigInt

NBKDoubleWidthKit (Sources, Tests, Benchmarks)

A composable, large, fixed-width, two's complement, binary integer.

🧩 Composable

NBKDoubleWidth is a generic software model for working with fixed-width integers larger than one machine word. Its bit width is double the bit width of its High component. In this way, you may construct new integer types:

typealias  Int256 = NBKDoubleWidth< Int128>
typealias UInt256 = NBKDoubleWidth<UInt128>

💕 Two's Complement

Like other binary integers, NBKDoubleWidth has two's complement semantics.

The two's complement representation of  0 is an infinite sequence of 0s.
The two's complement representation of -1 is an infinite sequence of 1s.

🏰 Fixed-Width Integer

Each NBKDoubleWidth has a fixed bit width, and so do its halves. This design comes with a suite of overflow and bit-casting operations. The even split also lends itself to divide-and-conquer strategies.

📖 Trivial UInt Collection

NBKDoubleWidth models a trivial UInt collection, where UInt is an unsigned machine word. It contains at least two words, but the exact count depends on the platform's architecture. You should, therefore, use properties like startIndex and endIndex instead of hard-coded indices.

// Int256 and UInt256, as constructed on a 64-bit platform:
┌───────────────────────────┐ ┌───────────────────────────┐
│           Int256          │ │          UInt256          │
├─────────────┬─────────────┤ ├─────────────┬─────────────┤
│    Int128   │   UInt128   │ │   UInt128   │   UInt128   │
├──────┬──────┼──────┬──────┤ ├──────┬──────┼──────┬──────┤
│  Int │ UInt │ UInt │ UInt │ │ UInt │ UInt │ UInt │ UInt │
└──────┴──────┴──────┴──────┘ └──────┴──────┴──────┴──────┘

Swift's type system enforces proper layout insofar as Int and UInt are the only types in the standard library that meet its type requirements. Specifically, only Int and UInt have NBKCoreInteger<UInt> Digit types.

🚀 Single Digit Arithmagick

Alongside its ordinary arithmagick operations, NBKDoubleWidth provides single-digit operations, where a digit is an un/signed machine word. These operations are more efficient for small calculations. Here are some examples:

Int256(1) + Int(1), UInt256(1) + UInt(1)
Int256(2) - Int(2), UInt256(2) - UInt(2)
Int256(3) * Int(3), UInt256(3) * UInt(3)
Int256(4) / Int(4), UInt256(4) / UInt(4)
Int256(5) % Int(5), UInt256(5) % UInt(5)

NBKFlexibleWidthKit (Sources, Tests, Benchmarks)

[!IMPORTANT] It's a work in progress. I may rework it at any time.

Models

• NBKFibonacciXL
• UIntXL

Fibonacci

NBKFibonacciXL(0) // (index: 0, element: 0, next: 1)
NBKFibonacciXL(1) // (index: 1, element: 1, next: 1)
NBKFibonacciXL(2) // (index: 2, element: 1, next: 2)
NBKFibonacciXL(3) // (index: 3, element: 2, next: 3)
NBKFibonacciXL(4) // (index: 4, element: 3, next: 5)
NBKFibonacciXL(5) // (index: 5, element: 5, next: 8)

It uses a fast double-and-add algorithm:

NBKFibonacciXL(10_000_000) // 2.3s on MacBook Pro, 13-inch, M1, 2020

But you can also step through it manually:

mutating func increment() { ... } // index + 1
mutating func decrement() { ... } // index - 1
mutating func    double() { ... } // index * 2

SemVer 2.0.0

Major version zero (0.y.z) is for initial development.

Anything MAY change at any time.

The public API SHOULD NOT be considered stable.

Using SwiftPM

Add this package to your list of package dependencies.

.package(url: "https://github.com/oscbyspro/Numberick.git", .upToNextMinor(from: "0.17.0")),

Choose target dependencies from the products in Package.swift.

.product(name: "Numberick",           package: "Numberick"),
.product(name: "NBKCoreKit",          package: "Numberick"),
.product(name: "NBKDoubleWidthKit",   package: "Numberick"),
.product(name: "NBKFlexibleWidthKit", package: "Numberick"),

Using CocoaPods

Choose target dependencies from the pods listed in the root directory.

pod "Numberick",                   "~> 0.17.0"
pod "Numberick-NBKCoreKit",        "~> 0.17.0"
pod "Numberick-NBKDoubleWidthKit", "~> 0.17.0"

This project is inspired by Int128 and DoubleWidth by Apple.