Binary arithmetic reimagined in Swift

What's New

Ultimathnum 0.3.0

  • Various API and non-API changes and improvements (0.y.z).
  • Trimmed lots of low-value [Mutable]DataInt.Body methods.
  • #5 Arithmetic in Karatsuba algorithm should be discard() not unchecked().


Everyone who hears these words of mine and does them is like a wise man who built his house on rock. The rain fell, the flood came, and the winds beat against that house, but it did not collapse because its foundation had been laid on rock. (Matthew 7:24-25)

Table of Contents

In programming, complexity is sand, and simplicity is a rock. The former is, by definition, a sum of parts. In this project, you will find unified abstractions and streamlined models that keep the parts few and similar to each other. So bring your favorite cup of tea, have a seat, then let's explore what it's all about.

Note that this project is mostly self-contained. It only extends the standard library when necessary. In practice, this means adding conversion to and from new types. All other functionality either belongs to the new types or is kept private through access control. This ensures interoperability without confusion or accidental dependencies.

The are many ways to abstract and there are pros and cons to all of them. Here, we keep it simple. We use a small set of primitives to derive most things. Our abstractions are purposeful and close to the machine. Likewise, you will not find many mutating methods. Instead, we opt for consuming methods. This cuts the project in half, compared to previous iterations of it.

If simplicity is a rock, then recovery is the capstone. Gone are they days of unavoidable traps because there now exists a recoverable alternative for every safe operation. Overflow, underflow, whatever—the framework will lazily prompt you to handle it. Furthermore, the abstractions and recovery mechanisms presented to you are ever-present in generic code as well.


Binary integers are dead.
Long live binary integers!

This project presents a novel binary integer abstraction that unifies the different sizes. It views all binary integers as infinite bit sequences with various modes of operation. It extends the maximum unsigned value to infinity and lets you recover from failure through intuitive, ergonomic, and generic error propagation mechanisms.

(~0 as IXL) // -1
(~1 as IXL) // -1 - 1
(~0 as UXL) //  ∞
(~1 as UXL) //  ∞ - 1

Some definitions have been updated to fit the new binary integer format. What was once called the sign bit is now the appendix bit. This means a so-called signed right shift treats an infinite unsigned integer like a negative signed integer, for example.

(~2 as UXL) >> 1 == (~1 as UXL)
(~1 as UXL) >> 1 == (~0 as UXL)
(~0 as UXL) >> 1 == (~0 as UXL)
( 0 as UXL) >> 1 == ( 0 as UXL)
( 1 as UXL) >> 1 == ( 0 as UXL)
( 2 as UXL) >> 1 == ( 1 as UXL)

In practice, an infinite integer is represented by its least significant body elements, followed by an infinite repetition of its appendix bit. All signed integers and all infinite integers store this bit in memory, but unsigned systems integers return zero through the type system. This makes all systems integers compatible with the un/signed two's complement format.

~(~(x)) == x for all x
│ some BinaryInteger │
│ body │ appendix... │

A binary integer must provide contiguous access to its endianness-sensitive body, which can be viewed through a data integer. Such view types keep track of memory alignments and may downsize their element type through reinterpretation.

Read Read & Write
DataInt<Element> MutableDataInt<Element>.Body
DataInt<Element>.Body MutableDataInt<Element>.Body

All binary integers agree on this format, which lets us perform arbitrary conversions with a fixed set of protocol witnesses. Compare this to the square matrix of doom that is formed by generic protocol requirements. In theory, a binary integer needs two initializers. One for aligned loads from compatible sources and one for unaligned loads from unknown sources. In practice, this depends on whether the optimizer can turn them into appropriate instructions.

init(load source: DataInt<U8>)                // unknown
init(load source: DataInt<Element.Magnitude>) // aligned
Here are the other conversion requirements...
@inlinable init(load source: consuming  UX.Signitude)
@inlinable init(load source: consuming  UX.Magnitude)
@inlinable borrowing func load(as type: UX.BitPattern.Type) -> UX.BitPattern

@inlinable init(load source: consuming  Element.Signitude)
@inlinable init(load source: consuming  Element.Magnitude)
@inlinable borrowing func load(as type: Element.BitPattern.Type) -> Element.BitPattern

Keep it simple.

You may realize that the infinite bit pattern definition implies a static size for all binary integers. Indeed, you can compare their sizes through metadata alone. Still, not all binary integers are trivial. A systems integer is represented in memory by its bit pattern. Additionally, a systems integer's size is a power of two in 8 through IX.max.

Size Signed Unsigned
pointer IX UX
8-bit I8 U8
16-bit I16 U16
32-bit I32 U32
64-bit I64 U64

Systems integers are intentionally simple so that the things you build with them may be simple. The only protocol requirements are multiplication and division algorithms for working with full precision in generic code, and byte swapping for efficient endianness conversions.

Trust me, I know what I'm doing...

Once you start using primitive types to form more complex types, you notice that some semantics compose better than others. A trusted input delegates some precondition validation to the programmer so that complex types can be built with less overhead. The type system will compel you to approve each trusted input with one of the following methods. Your choice will either make your code safer or easier to audit.

init(_:)         // error: traps
init(_:prune:)   // error: throws
init(exactly:)   // error: nil
init(unchecked:) // error: unchecked

We all know that dynamic validation comes at a price. The validation tax tends to be small in everyday code, but it can be noticeable when working with primitives. That is why some operations have unchecked alternatives. In this project, we validate unchecked operations in debug mode only. You may realize that it is sometimes correct to ignore errors completely. It is, therefore, best to view unchecked operations as semantic markers in performance-sensitive places.

U8.zero.decremented().error       // true, because the result is not representable
U8.zero.decremented().value       // 255,  this is the truncated bit pattern of -1
U8.zero.decremented().unchecked() // 255,  with precondition failure in debug mode

It doesn't matter how many times you fall.
It matters how many times you get back up.

Ergonomic error handling is one of the cornerstones of this project and a lot of effort has gone into ensuring that there's always a path to redemption. The Fallible<Value> wrapper plays an important part the recovery story. Use it to handle errors when you want and how you want. Here's a real example from the generic Fibonacci<Value> sequence:

/// Forms the sequence pair at `index + x.index`.
@inlinable public mutating func increment(by x: Self) throws {
    let ix = try i.plus (x.i).prune(Failure.overflow)
    let ax = try a.times(x.b).plus(b.minus(a).times(x.a)).prune(Failure.overflow)
    let bx = try b.times(x.b).plus(       (a).times(x.a)).prune(Failure.overflow)

    self.i = consume ix
    self.a = consume ax
    self.b = consume bx

Note that each operation propagates failure by setting Fallible<Value>'s error indicator, which is lazily checked in the throwing prune(_:) method. Alternatively, you may use optional(_:), result(_:) unwrap(_:), assert(_:) or plain value and error to validate your results. Also, note that each operation accepts both Value and Fallible<Value> inputs to make error handling as seamless as possible. Alternatively, you may use any of the common arithmetic operators for trapping or wrapping results:

static func  +(lhs: consuming Self, borrowing Self) -> Self // trapping
static func &+(lhs: consuming Self, borrowing Self) -> Self // wrapping

We don't know where it comes from, only that it exists.

You may create integers our of thin air with RootInt, which is a wrapper around Swift's StaticBigInt model. It comes with some additional bells and whistles. You may, for example, use it to query whether a type can represent an integer literal in generic code.

static func exactly(_ source: RootInt) -> Fallible<Self>

Each data integer operates on the contiguous in-memory representation of binary integers without taking ownership of them. The [Mutable]DataInt.Body type is fundamentally a buffer pointer. The [Mutable]DataInt type extends the bit pattern of its body with a repeating appendix bit. You may perform various buffer and arithmetic operations on these types, but remember that their operations are finite, unsigned, and unchecked by default.

overview MutableDataInt.Body:

/[de|in]crement(by: Bool) -> Fallible<Void>
/[de|in]crementSameSize(repeating: Bool, plus: Bool) -> Fallible<Void>
/[de|in]crement[SubSequence](by: Element) -> Fallible<[Void|Self]>
/[de|in]crement[SubSequence](by: Element, plus: Bool) -> Fallible<[Void|Self]>
/[de|in]crement[SubSequence](by: DataInt.Body, plus: Bool) -> Fallible<[Void|Self]>
/[de|in]crement[SubSequence](by: DataInt.Body, times: Element, plus: Element) -> Fallible<[Void|Self]>
/[de|in]crement[SubSequence](toggling: DataInt.Body, plus: Bool) -> Fallible<[Void|Self]>
overview MutableDataInt.Body:

/toggle(carrying: Bool) -> Fallible<Void>
overview DataInt:

.signum (of:  Self,    isSigned: Bool) -> Signum
.compare(lhs: Self, lhsIsSigned: Bool, rhs: Self, rhsIsSigned: Bool) -> Signum
overview [Mutable]DataInt.Body:

/signum() ->  Signum
/compared(to: [Mutable]DataInt.Body) -> Signum
overview [Mutable]DataInt:

/count(Bit.Entropy)     -> IX
/count(Bit.Nonappendix) -> IX
overview [Mutable]DataInt.Body:

/size()                   -> IX
/count(Bit)               -> IX
/count(Bit.Entropy)       -> IX
/count(Bit.Appendix)      -> IX
/count(Bit.Nonappendix)   -> IX
/count(Bit.Ascending)     -> IX
/count(Bit.Nonascending)  -> IX
/count(Bit.Descending)    -> IX
/count(Bit.Nondescending) -> IX
overview MutableDataInt.Body:

/remainder(Divisor<Element>) -> Element
/divisionSetQuotientGetRemainder(Divisor<Element>) -> Element
/divisionSetQuotientSetRemainderByLong2111MSB(dividing: Self, by: DataInt.Body)
/divisionGetQuotientSetRemainderByLong2111MSBIteration(Element) -> Element
overview [Mutable]DataInt:

/appendix -> Bit
/body     -> Body
/load()   -> Element
/next()   -> Element
/subscript(UX) -> Element
overview [Mutable]DataInt.Body:

/appendix -> Bit
/buffer() -> Unsafe[Mutable]BufferPointer
/count    -> IX
/indices  -> Range<IX>
/start    -> Unsafe[Mutable]Pointer
/subscript(unchecked: Void) -> Element
/subscript(unchecked: IX)   -> Element
overview MutableDataInt.Body:

/initialize(to:   DataInt.Body)
/initialize(load: DataInt.Body)
/initialize(repeating: Element)
overview MutableDataInt.Body:

/multiply(by: Element,  add: Element) -> Element
/initialize(to: DataInt.Body,  times: DataInt.Body)
/initializeByLongAlgorithm(to: DataInt.Body, times: DataInt.Body, plus: Element)
/initializeByKaratsubaAlgorithm(to: DataInt.Body, times: DataInt.Body)
/initialize(toSquareProductOf: DataInt.Body)
/initializeByLongAlgorithm(toSquareProductOf: DataInt.Body, plus: Element)
/initializeByKaratsubaAlgorithm(toSquareProductOf: DataInt.Body)
overview [Mutable]DataInt:

/normalized() -> Self
/subscript(PartialRangeFrom<UX>) -> Self
overview [Mutable]DataInt.Body:

/normalized() -> Self
/split(unchecked: IX) -> (low: Self, high: Self)
/subscript(unchecked: Range<IX>) ->  Self
/subscript(unchecked: PartialRangeFrom<IX>) -> Self
/subscript(unchecked: PartialRangeUpTo<IX>) -> Self
overview MutableDataInt.Body:

/[up|down]shift(environment: Element, major: IX, minor: IX)
/[up|down]shift(environment: Element, majorAtLeastOne: IX, minor: Void)
/[up|down]shift(environment: Element, major: IX, minorAtLeastOne: IX)

The data integer types let you downsize binary integer elements by reinterpretation. This versatile power stems from strict systems integer layout requirements. You may not upsize elements in this way, however, because the memory alignment of a smaller systems integer may not be compatible with a larger systems integer. Instead, you may use DataInt<U8> to load elements of any size. It performs an unaligned load when possible and handles the case where the load would read past the end. All binary integers can form a DataInt<U8> view since a byte is the smallest possible systems integer type.

At some point, you'll want to convert your binary integers to a human-readable format. When that happens, the description(as:) and init(_:as:) methods let you perform the common radix conversions via TextInt. The latter uses a fixed number of non-generic and non-inlinable algorithms, which are shared by all binary integers. This is an intentional size-over-performance optimization.

try! TextInt(radix:  12, letters: .uppercase)
try! IXL("123", as: .decimal).description(as: .hexadecimal) //  7b
try! IXL("123", as: .hexadecimal).description(as: .decimal) // 291

You may realize that the introduction of infinite values necessitates changes to the integer description format. The new format adds the # and & markers. The former is a spacer (cf. +) whereas the latter represents bitwise negation. In other words, +&123 translates to ∞ minus 123.

The case-insensitive base 36 regex.
let regex: Regex = #/^(\+|-)?(#|&)?([0-9A-Za-z]+)$/#

While this model prioritizes size, its operations are still fast enough for most purposes. The 210k-digit measurement illustrates this point. Keep in mind that hexadecimal radix conversions are linear operations, whereas decimal conversions are superlinear but instant for numbers intended to be read by humans.

MacBook Pro, 13-inch, M1, 2020, -O, code coverage disabled.
let fib1e6 = try! Fibonacci<UXL>(1_000_000)

let fib1e6r10 = fib1e6.element.description(as:     .decimal) // 0.924s (208988 digits)
let fib1e6r16 = fib1e6.element.description(as: .hexadecimal) // 0.002s (173561 digits)

try! UXL(fib1e6r10, as:     .decimal) // 0.040s (208988 digits)
try! UXL(fib1e6r16, as: .hexadecimal) // 0.002s (173561 digits)
TextInt optimizes base 2, 4, and 16 conversions (but not 8 or 32).

The BitCastable<BitPattern> protocol lets you perform type-safe bit casts in bulk. This is especially pertinent to binary integers since the abstraction is two representations bridged by a bit pattern transformation. You perform type-safe bit-casts by calling init(raw:).

@inlinable public func distance<Distance>(
    to other: Self,
    as type: Distance.Type = Distance.self
)   -> Fallible<Distance> where Distance: SignedInteger {
    if  Self.size < Distance.size {
        return Distance(load: other).minus(Distance(load: self))
    }   else if Self.isSigned {
        return other.minus(self).map(Distance.exactly)
    }   else {
        let distance = Fallible<Signitude>(raw: other.minus(self))
        let superoverflow = distance.value.isNegative != distance.error
        return Distance.exactly(distance.value).veto(superoverflow)

The above example shows a generic Strideable/distance(from:to:) esque method. In the narrowing unsigned case you find that the difference is reinterpreted as a same-size signed integer type via the init(raw:) bulk operation. Note that such type relationships are generically available to all binary integers. Also, note that this method is both fully generic and fully recoverable. The init(load:) method is similar, but it returns the bit pattern that fits.

Many types in this project let you perform bitwise logic through the common AND, OR, XOR, and NOT operations. This capability extends to generic code when a type conforms to BitOperable. Note that all binary integers conform to this protocol and that their transformations are always sound now that unsigned integers may end in infinitely repeating ones.

static prefix func ~(instance: consuming Self) -> Self
static func &(lhs: consuming Self, rhs: borrowing Self) -> Self
static func |(lhs: consuming Self, rhs: borrowing Self) -> Self
static func ^(lhs: consuming Self, rhs: borrowing Self) -> Self
Here are the derived operations that are available in the core module...
mutating  func toggle()
consuming func toggled() -> Self

static func &=(lhs: inout Self, rhs: Self)
static func |=(lhs: inout Self, rhs: Self)
static func ^=(lhs: inout Self, rhs: Self)

This project introduces various additional types. Some are more important than other, so here's a rundown of the three most prominent ones: Bit, Sign and Signum. Bit and Sign are a lot like Bool, but with bitwise operations and no short-circuits. Signum comes up a lot because it's the return type of the compared(to:) method. It also offers various conveniences for common transformations.

Type Values
Bit 0, 1
Sign +, -
Signum -1, 0, 1
Here's the some of the most common signum operations...
init(_ source: Bit) //....................  0 or 1
static func one(_ sign: Sign) -> Signum // -1 or 1
func negated() -> Self //................. toggles -1 and 1

Veni, vidi, vici. I came, I saw, I conquered.

 DoubleInt<I128>          DoubleInt<U128>
│ I256                  ││ U256                  │
│ U128      │ I128      ││ U128      │ U128      │
│ U64 │ U64 │ U64 │ I64 ││ U64 │ U64 │ U64 │ U64 │

The DoubleInt<Base> model lets you create software-defined systems integers larger than the ones supported by your hardware. Like all systems integers, it is stored without indirection. This makes it superior to arbitrary precision models in situations where the doubled precision is enough. It also has a niche on the stack when heap allocations are prohibitive.

Given that systems integers need some 2x width operations, it is both easy and tempting to accept and return DoubleInt<Base> instances. Swift's generic type system even allows it! The problem, however, is that it makes the model recursive and reliant on unspecialized runtime features. While runtime generics are awesome and overpowered in some cases, they're not appropriate for primitives. This is why the core module features Doublet<Base> instead. Here's a brief illustration of how that stops the recusion:

DoubleInt<Base> -> Doublet  <DoubleInt<Base>> // Doublet cannot upsize itself
DoubleInt<Base> -> DoubleInt<DoubleInt<Base>> -> ............................

Author: Slaps roof of car. This baby can fit infinity!

 InfiniInt<IX>            InfiniInt<UX>
│ IXL                   ││ UXL                   │
│ UX............. │ Bit ││ UX............. │ Bit │

It may or may not sound intuitive at first, but this infinite binary integer has a fixed size too. It is important to remember this when you consider the recovery modes of its arithmetic operations. It overflows and underflows just like the systems integers you know and love.

The main difference between a systems integer and an infinite integer is that the appendix bit is the only addressable infinity bit, which means you cannot cut infinity in half. Likewise, the log2(max+1) size must be promoted. To work with infinite numerical values, you must track where your values come from. Keep in mind that recovery from failure is the main purpose of infinity.

![NOTE] The introduction of infinity is what permits ~(~(x)) == x for all x.

Addition and subtraction at and around infinity just works.

UXL.min.decremented() // value: max, error: true
UXL.max.incremented() // value: min, error: true

Multiplication is also unchanged. All of the complicated stuff forms at one bit past the appendix bit. Imagine a really big integer and a product of twice that size with truncating behavior. It just works.

U32.max.times(U32.max) // value: 1, error: true
UXL.max.times(UXL.max) // value: 1, error: true

So, this is where things get tricky. Wait, no, it still just works in most cases. Since you know finite-by-finite division, I'm sure you intuit that finite-by-infinite division is trivial and that infinite-by-infinite division is at most one subtraction. The only weird case is infinite-by-finite division because the proper computation requires infinite memory. So, in this case, we just let the algorithm run and mark it as an error unless the divisor is one. This yields results similar to signed division.

dividend == divisor &* quotient &+ remainder // for all binary integers

Question: How do you test silly numbers?
Answer: With silly number generators, of course!

try! Fibonacci<UXL>(0) // (index: 0, element: 0, next: 1)
try! Fibonacci<UXL>(1) // (index: 1, element: 1, next: 1)
try! Fibonacci<UXL>(2) // (index: 2, element: 1, next: 2)
try! Fibonacci<UXL>(3) // (index: 3, element: 2, next: 3)
try! Fibonacci<UXL>(4) // (index: 4, element: 3, next: 5)
try! Fibonacci<UXL>(5) // (index: 5, element: 5, next: 8)

It uses a fast double-and-add algorithm:

MacBook Pro, 13-inch, M1, 2020, -O, code coverage disabled.
try! Fibonacci<UXL>( 1_000_000) // 0.04s
try! Fibonacci<UXL>(10_000_000) // 1.65s

But you can also step through it manually:

mutating func double()       throws // index * 2
mutating func increment()    throws // index + 1
mutating func decrement()    throws // index - 1
mutating func increment(by:) throws // index + x.index
mutating func decrement(by:) throws // index - x.index

You may have noticed that you can pick any two adjacent elements and express the sequence in terms of those elements. This observation allows us to climb up and down the index ladder. The idea is super simple:

f(x + 1 + 0) == f(x) * 0000 + f(x + 1) * 00000001
f(x + 1 + 1) == f(x) * 0001 + f(x + 1) * 00000001
f(x + 1 + 2) == f(x) * 0001 + f(x + 1) * 00000002
f(x + 1 + 3) == f(x) * 0002 + f(x + 1) * 00000003
f(x + 1 + 4) == f(x) * 0003 + f(x + 1) * 00000005
f(x + 1 + 5) == f(x) * 0005 + f(x + 1) * 00000008
f(x + 1 + y) == f(x) * f(y) + f(x + 1) * f(y + 1)

Going the other direction is a bit more complicated, but not much:

f(x - 0) == + f(x) * 00000001 - f(x + 1) * 0000
f(x - 1) == - f(x) * 00000001 + f(x + 1) * 0001
f(x - 2) == + f(x) * 00000002 - f(x + 1) * 0001
f(x - 3) == - f(x) * 00000003 + f(x + 1) * 0002
f(x - 4) == + f(x) * 00000005 - f(x + 1) * 0003
f(x - 5) == - f(x) * 00000008 + f(x + 1) * 0005
f(x - 6) == + f(x) * 00000013 - f(x + 1) * 0008
f(x - y) == ± f(x) * f(y + 1) ± f(x + 1) * f(y)

We have some cute algorithms and a generic sequence. Let's combine them and unit-test our models! We can rearrange the sequence addition formula in such a way that we call all basic arithmetic operations. Note that we don't need to know the inputs and outputs ahead of time because it's an equation. Neat!

f(x) * f(y) == (f(x+y+1) / f(x+1) - f(y+1)) * f(x+1) + f(x+y+1) % f(x+1)

Major version zero (0.y.z) is for initial development.
Anything MAY change at any time.
The public API SHOULD NOT be considered stable.

Add this package to your list of package dependencies.

.package(url: "https://github.com/oscbyspro/Ultimathnum.git", exact: "x.y.z"),

Choose target dependencies from this list of products.

.product(name: "Ultimathnum",  package: "Ultimathnum"), // umbrella
.product(name: "CoreKit",      package: "Ultimathnum"),
.product(name: "DoubleIntKit", package: "Ultimathnum"),
.product(name: "InfiniIntKit", package: "Ultimathnum"),
.product(name: "FibonacciKit", package: "Ultimathnum"),


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Last updated: Mon Jun 17 2024 07:23:37 GMT-0900 (Hawaii-Aleutian Daylight Time)