`DoubleDouble.jl`

is a Julia package for performing extended-precision arithmetic using pairs of floating-point numbers. This is commonly known as "double-double" arithmetic, as the most common format is a pair of C-doubles (`Float64`

in Julia), although `DoubleDouble.jl`

will actually work for any floating-point type. Its aim is to provide accurate results without the overhead of `BigFloat`

types.

The core routines are based on the ideas and algorithms of Dekker (1971).

The main type is `Double`

, with two floating-point fields: `hi`

, storing the leading bits, and `lo`

storing the remainder. `hi`

is stored to full precision and rounded to nearest; hence, for any `Double`

`x`

, we have `abs(x.lo) <= 0.5 * eps(x.hi)`

. Although these types can be created directly, the usual interface is the `Double`

function:

```
julia> using DoubleDouble
julia> x = Double(pi)
Double{Float64}(3.141592653589793,1.2246467991473532e-16)
julia> eps(x.hi)
4.440892098500626e-16
```

The other type defined is `Single`

, which is simply a wrapper for a
floating-point type, but whose results will be promoted to `Double`

.

By exploiting this property, we can compute exact products of floating point numbers.

```
julia> u, v = 64 * rand(), 64 * rand()
(15.59263373822506,39.07676672446341)
julia> w = Single(u) * Single(v)
Double{Float64}(609.3097112086186, -5.3107663829696295e-14)
```

Note that the product of two `Single`

s is a `Double`

: the `hi`

element of this
double is equal to the usual rounded product, and the `lo`

element contains the exact
difference between the exact value and the rounded.

This can be used to get an accurate remainder

```
julia> r = rem(w, 1.0)
Double{Float64}(0.309711208618584, 1.6507898617445858e-17)
julia> Float64(r)
0.309711208618584
```

This is much more accurate than taking ordinary products, and gives the same answer as using `BigFloat`

s:

```
julia> rem(u*v, 1.0)
0.3097112086186371
julia> Float64(rem(big(u) * big(v), 1.0))
0.309711208618584
```

However, since the `DoubleDouble`

version is carried out using ordinary floating-point operations, it is of the order of 1000x faster than the `BigFloat`

version.

If a number cannot be exactly represented by a floating-point number, it may be rounded incorrectly when used later, e.g.

```
julia> pi * 0.1
0.3141592653589793
julia> Float64(big(pi) * 0.1)
0.31415926535897937
```

We can also do this computation using `Double`

s (note that the promotion rules mean that only one needs to be specified):

```
julia> Float64(Double(pi) * 0.1)
0.31415926535897937
julia> Float64(pi * Single(0.1))
0.31415926535897937
```

The fused multiply-add (FMA) operation is an intrinsic floating-point
operation that allows the evaluation of `a * b + c`

, with rounding occurring only
at the last step. This operation is unavailable on 32-bit x86 architecture, and available
only on the most recent x86_64 chips, but can be emulated via double-double arithmetic:

```
julia> 0.1 * 0.1 + 0.1
0.11000000000000001
julia> Float64(big(0.1) * 0.1 + 0.1)
0.11
julia> Base.fma(a::Float64,b::Float64,c::Float64) = Float64(Single(a) * Single(b) + Single(c))
fma (generic function with 1 method)
julia> fma(0.1, 0.1, 0.1)
0.11
```

09/27/2013

2 months ago

40 commits