✨ An arithmagick overhaul in Swift.

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

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

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

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>

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.

Each type of 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. As such, it leverages A. Karatsuba's multiplication algorithm, as well as C. Burnikel's and J. Ziegler's fast recursive division.

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.

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)

Note: The Digit type is Int when Self is signed, and UInt otherwise.

This project is inspired by Int128 and DoubleWidth by Apple.