# MultiFloats.jl

MultiFloats.jl is a Julia package for extended-precision floating-point arithmetic using 100–400 accurate bits (≈30–120 accurate digits). In this range, it is by far the fastest extended-precision floating-point library that the author is aware of. At 100-bit precision, MultiFloats.jl is roughly 40x faster than Base.BigFloat and 2x faster than DoubleFloats.jl.

MultiFloats.jl achieves this speed by performing arithmetic with native Float64 operations using immutable data structures that do not dynamically allocate memory. It stores extended-precision numbers in the double-double representation, generalized to an arbitrary number of components. This idea takes inspiration from Jonathan Shewchuk's work on adaptive-precision floating-point arithmetic and Yozo Hida/Xiaoye Li/David Bailey's algorithms for quad-double arithmetic, combined in a unique fashion with Julia's unique JIT architecture and metaprogramming capabilities.

MultiFloats.jl currently provides basic arithmetic operations (+, -, *, /, sqrt), comparison operators (==, !=, <, >, <=, >=), and floating-point introspection methods (isfinite, eps, minfloat, etc.). Work on trigonometric functions, exponentials, and logarithms is currently in progress.

## Usage

MultiFloats.jl provides the types Float64x2, Float64x3, ..., Float64x8 representing extended-precision numbers with 2x, 3x, ..., 8x the precision of Float64. These are all instances of the parametric type MultiFloat{T,N}, where T = Float64 and N = 2, 3, ..., 8.

Instances of Float64x2, Float64x3, ..., Float64x8 are convertible to and from Float64 and BigFloat, as shown in the following example.

julia> using MultiFloats

julia> x = Float64x4(2.0)

julia> y = sqrt(x)
1.41421356237309504880168872420969807856967187537694807317667973799

julia> y * y - x
-1.1566582006914837e-66


A comparison with sqrt(BigFloat(2)) reveals that all displayed digits are correct in this example.

Note: MultiFloats.jl also provides a Float64x1 type that has the same precision as Float64, but behaves like Float64x2Float64x8 in terms of supported operations. This is occasionally useful for testing, since any code that works for Float64x1 should also work for Float64x2Float64x8 and vice versa.

## Benchmarks

Two basic linear algebra tasks are used below to compare the performance of extended-precision floating-point libraries:

• QR factorization of a random 400×400 matrix
• Computing the pseudoinverse of a random 400×250 matrix (using GenericSVD.jl)

See benchmark code here. The timings reported below are averages of 10 runs performed under identical conditions on an Intel Core i7-8650U (Surface Book 2 13.5").

MultiFloats Float64x2 Julia Base BigFloat ArbNumerics ArbFloat Decimals Decimal DecFP Dec128 DoubleFloats Double64 Quadmath Float128
400×400 qr time 0.257 sec 10.303 sec (40x slower) 17.871 sec (69x slower) ❌ Error 9.448 sec (36x slower) 0.535 sec (2x slower) 2.403 sec (9x slower)
accurate digits 26.0 25.9 25.9 ❌ Error 27.6 26.1 28.1
400×250 pinv time 1.709 sec 96.655 sec (56x slower) 133.085 sec (77x slower) ❌ Error ❌ Error 3.668 sec (2x slower) 15.576 sec (9x slower)
accurate digits 25.6 25.8 25.8 ❌ Error ❌ Error 25.4 27.9

## Feature Comparison

MultiFloats BigFloat ArbNumerics Decimals DecFP DoubleFloats Quadmath
user-selectable precision ✔️ ✔️ ✔️
avoids dynamic memory allocation ✔️ ✔️ ⚠️ ✔️
basic arithmetic +, -, *, /, sqrt ✔️ ✔️ ✔️ ✔️ ✔️ ✔️
transcendental functions sin, cos, exp, log ❌ (WIP) ✔️ ✔️ ✔️ ✔️ ✔️
compatible with GenericSVD.jl ✔️ ✔️ ✔️ ✔️ ✔️
floating-point introspection minfloat, eps ✔️ ✔️ ✔️ ✔️ ✔️ ✔️