Dear swift-evolution,

I less want to make concrete proposals than start a conversation about

expanded numerical support in Swift. In that spirit, here are some thoughts:

Arbitrary-precision integers (bigints) are a logical first step. Bigints

then enable practical support for rationals. (With bigints, rational

operations need not fail because of integer overflow.) Adding at least

these two numerical types would bring Swift far closer wrt numerical

support wrt languages like Clojure or Ruby.

Many languages make a distinction between fixed precision integers and

bigints (i.e. Haskell's Int vs Integer). I propose the definition of two

related types: ArbitraryPrecisionInteger and Integer.

ArbitraryPrecisionInteger is a struct wrapping a pointer to a byte buffer,

much like how Swift's Array, Set, and Dictionary collection types work

(c.f.

https://www.mikeash.com/pyblog/friday-qa-2015-04-17-lets-build-swiftarray.html).

It wraps an arbitrarily large integer within a container adhering to value

semantics. Perhaps this can be exposed as part of the stdlib, or a

'private' implementation detail.

Integer is intended to be used by application developers, and most of the

arithmetic/comparison functionality exposed by the stdlib deals with

arguments and return values of type Integer. Integers are enums:

enum Integer {

case Small(Int)

case Big(ArbitraryPrecisionInteger)

}

Like ArbitraryPrecisionInteger, Integer represents an arbitrary-precision

integer. It wraps either a fixed-size (64-bit) signed integer, or a bigint.

Integer's invariant is that integers within the range of an Int are always

stored as an Int. This is enforced by the mathematical operations defined

on arguments of type Integer. This allows for a slow path (consisting of

operations that are conducted on ArbitraryPrecisionIntegers), and a fast

path (consisting of checked operations on Ints, with automatic promotion to

ArbitraryPrecisionInteger as necessary). Fast path operations, in the best

case, need not allocate heap memory or invoke retain/release operations.

Integers can be unconditionally constructed (promoted) from Ints, and can

be conditionally demoted to Ints ( () -> Int? ). (All Ints are Integers,

but not all Integers are Ints.)

A useful compiler support feature might be a "BigIntegerLiteralConvertible"

protocol, which allows literal integers out of the range of Ints to be

assigned to Integer values. Swift can already tell if an integer literal is

out of range (e.g. "let x : Int8 = 123456"), so there is some precedent.

This would be a more elegant solution than requiring bigints to be

initialized via strings (as some other arbitrary precision arithmetic

libraries do).

Rational numbers are represented by the Rational struct which encapsulates

two Integers, the numerator and denominator. Rationals can be conditionally

constructed explicitly from two Integers (as long as the denominator isn't

0), or from a single Integer (all Integers are Rationals, but the opposite

is not true). A good invariant for Rationals might be having them always

represented in the most simplified form - for example, Rational(2, 6)

should be represented internally as 1/3 when constructed.

Future topics to explore include complex numbers, arbitrary-precision

floating-point numbers, and fixed-precision and/or decimal number types.

Given that I'm little more than a dabbler in these topics (most of my

expertise came from trying to reverse-assemble Clojure's numerical

support), feedback from someone with experience and/or expertise wrt

numerical/scientific computing, bignum libraries, numerical towers, etc.

would be hugely appreciated.

Best regards,

Austin

## ···

On Thu, Dec 3, 2015 at 6:52 PM, Austin Zheng <austinzheng@gmail.com> wrote:

Thanks, Chris and Dmitri! I will do some research and write something up

over the weekend.

Austin

On Thu, Dec 3, 2015 at 2:24 PM, Dmitri Gribenko <gribozavr@gmail.com> > wrote:

On Thu, Dec 3, 2015 at 1:14 PM, Austin Zheng <austinzheng@gmail.com> >> wrote:

>

> Hi all,

>

> There are quite a few programming languages that provide support for

numeric types apart from the customary floating-point and fixed-width

integer types. Prominent examples of additional numeric types include

rational numbers, arbitrary-width integer types, and fixed-point numbers.

Many of these numeric types are applicable to a wide variety of problem

domains.

>

> Swift seems like it would be a good fit for stdlib implementation of

some of these numeric types. Structs and value semantics, literal

initialization, and operator overloading would allow such types to be

treated as first-class citizens. Is the community amenable to such a

proposal, which would entail the data structures themselves, arithmetic

operations, and interoperation between different numeric types to form a

numerical tower of sorts?

Hi Austin,

We are interested in improving our numerics support, and we are

definitely interested in hearing your ideas in this space. You don't

have to write a full proposal though. Just an extended email to

swift-evolution would be a good start.

You can find the current prototype for library support for integers

here:

https://github.com/apple/swift/blob/master/test/Prototypes/Integers.swift.gyb

Dmitri

--

main(i,j){for(i=2;;i++){for(j=2;j<i;j++){if(!(i%j)){j=0;break;}}if

(j){printf("%d\n",i);}}} /*Dmitri Gribenko <gribozavr@gmail.com>*/