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ole Remove broken link in Real module readme
The link target for `[Real]` wasn't defined. I decided toremove the link altogether because the readme would essentially link to itself.
Latest commit 0ed7e78 Nov 12, 2019


SE-0246 proposed an API for "basic math functions" that would make operations like sine and logarithm available in generic contexts. It was accepted, but because of limitations in the compiler, the API could not be added to the standard library in a source-stable manner. The Real module provides that API as a separate module so that you can use it right away to get access to the improved API for these operations in your projects.

Protocols and Methods

The module defines three protocols. The most general is ElementaryFunctions, which makes the following functions available:

  • Exponential functions: exp, expMinusOne
  • Logarithmic functions: log, log(onePlus:)
  • Trigonometric functions: cos, sin, tan
  • Inverse trigonometric functions: acos, asin, atan
  • Hyperbolic functions: cosh, sinh, tanh
  • Inverse hyperbolic functions: acosh, asinh, atanh
  • Power and root functions: pow, sqrt, root

ElementaryFunctions refines AdditiveArithmetic, and so also provides addition, subtraction, and the .zero property.

The RealFunctions protocol refines ElementaryFunctions, and adds operations that are difficult to define or implement over fields more general than the real numbers:

  • atan2(y:x:), which computes atan(y/x) with sign chosen by the quadrant of the point (x,y) in the Cartesian plane.
  • hypot, which computes sqrt(x*x + y*y) without intermediate overflow or underflow.
  • erf and erfc, the error function and its complement.
  • Exponential functions: exp2 and exp10
  • Logarithmetic functions: log2 and log10
  • Gamma functions: gamma, logGamma, and signGamma, which evaluate the gamma function, its logarithm, and its sign.

The protocol that you will use most often is Real, which describes a floating-point type equipped with the full set of basic math functions. This is a great protocol to use in writing generic code, because it has all the basics that you need to implement most numeric functions.

Using Real

First, either import Real directly or import the Numerics umbrella module.

Suppose we were experimenting with some basic machine learning, and needed a generic [sigmoid function][Sigmoid] activation function:

import Numerics

func sigmoid<T: Real>(_ x: T) -> T {
  1 / (1 + .exp(-x))

Or suppose we were implementing a DFT, and wanted to precompute weights for the transform; DFT weights are roots of unity:

import Numerics

extension Real {
  // The real and imaginary parts of e^{-2πik/n}
  static func dftWeight(k: Int, n: Int) -> (r: Self, i: Self) {
    precondition(0 <= k && k < n, "k is out of range")
    guard let N = Self(exactly: n) else {
      preconditionFailure("n cannot be represented exactly.")
    let theta = -2 * .pi * (Self(k) / N)
    return (r: .cos(theta), i: .sin(theta))

This gives us an implementation that works for Float, Double, and Float80 if the target supports it. When new basic floating-point types are added to Swift, like Float16 or Float128, it will work for them as well. Not having this protocol is a significant missing feature for numerical computing in Swift, and I'm really looking forward to seeing what people do with it.


  • The C standard math library (libm) via the NumericShims target.
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