# DoubleFloats.jl

### Math with 85+ accurate bits.

#### Extended precision float and complex types

- N.B.
`Double64`

is the most performant type^{β}

## Installation

`pkg> add DoubleFloats`

or

```
julia> using Pkg
julia> Pkg.add("DoubleFloats")
```

## More Performant Than Float128, BigFloat

*these results are from BenchmarkTools, on one machine*

There is another package, Quadmath.jl, which exports Float128 from GNU’s libquadmath. Float128s have 6 more significant bits than Double64s, and a much wider exponent range (Double64s exponents have the same range as Float64s). Big128 is BigFloat after setprecision(BigFloat, 128).

Benchmarking: vectors (`v`

) of 1000 values and 50x50 matrices (`m`

).

Double64 | Float128 | Big128 | Double64 | Float128 | Big128 | ||
---|---|---|---|---|---|---|---|

`dot(v,v)` |
1 | 3 | 7 | `exp.(m)` |
1 | 2 | 6 |

`v .+ v` |
1 | 7 | 16 | `m * m` |
1 | 3 | 9 |

`v .* v` |
1 | 12 | 25 | `det(m)` |
1 | 5 | 11 |

relative performance: smaller is faster, the larger number takes proportionately longer.

## Examples

### Double64, Double32, Double16

```
julia> using DoubleFloats
julia> dbl64 = sqrt(Double64(2)); 1 - dbl64 * inv(dbl64)
0.0
julia> dbl32 = sqrt(Double32(2)); 1 - dbl32 * inv(dbl32)
0.0
julia> dbl16 = sqrt(Double16(2)); 1 - dbl16 * inv(dbl16)
0.0
julia> typeof(ans) === Double16
true
```

note: floating-point constants must be used with care, they are evaluated as Float64 values before additional processing

```
julia> Double64(0.2)
0.2
julia> showall(ans)
2.0000000000000001110223024625156540e-01
julia> Double64(2)/10
0.2
julia> showall(ans)
1.9999999999999999999999999999999937e-01
julia> df64"0.2"
0.2
julia> showall(ans)
1.9999999999999999999999999999999937e-01
```

### Complex functions

```
julia> x = ComplexDF64(sqrt(df64"2"), cbrt(df64"3"))
1.4142135623730951 + 1.4422495703074083im
julia> showall(x)
1.4142135623730950488016887242096816 + 1.4422495703074083823216383107800998im
julia> y = acosh(x)
1.402873733241199 + 0.8555178360714634im
julia> x - cosh(y)
7.395570986446986e-32 + 0.0im
```

### show, string, parse

```
julia> using DoubleFloats
julia> x = sqrt(Double64(2)) / sqrt(Double64(6))
0.5773502691896257
julia> string(x)
"5.7735026918962576450914878050194151e-01"
julia> show(IOContext(Base.stdout,:compact=>false),x)
5.7735026918962576450914878050194151e-01
julia> showall(x)
0.5773502691896257645091487805019415
julia> showtyped(x)
Double64(0.5773502691896257, 3.3450280739356326e-17)
julia> showtyped(parse(Double64, stringtyped(x)))
Double64(0.5773502691896257, 3.3450280739356326e-17)
julia> Meta.parse(stringtyped(x))
:(Double64(0.5773502691896257, 3.3450280739356326e-17))
julia> x = ComplexDF32(sqrt(df32"2"), cbrt(df32"3"))
1.4142135 + 1.4422495im
julia> string(x)
"1.414213562373094 + 1.442249570307406im"
julia> stringtyped(x)
"ComplexD32(Double32(1.4142135, 2.4203233e-8), Double32(1.4422495, 3.3793125e-8))"
```

see https://juliamath.github.io/DoubleFloats.jl/stable/ for more information

## Accuracy

results for f(x), x in 0..1

function | abserr | relerr |
---|---|---|

exp | 1.0e-31 | 1.0e-31 |

log | 1.0e-31 | 1.0e-31 |

sin | 1.0e-31 | 1.0e-31 |

cos | 1.0e-31 | 1.0e-31 |

tan | 1.0e-31 | 1.0e-31 |

asin | 1.0e-31 | 1.0e-31 |

acos | 1.0e-31 | 1.0e-31 |

atan | 1.0e-31 | 1.0e-31 |

sinh | 1.0e-31 | 1.0e-29 |

cosh | 1.0e-31 | 1.0e-31 |

tanh | 1.0e-31 | 1.0e-29 |

asinh | 1.0e-31 | 1.0e-29 |

atanh | 1.0e-31 | 1.0e-30 |

results for f(x), x in 1..2

function | abserr | relerr |
---|---|---|

exp | 1.0e-30 | 1.0e-31 |

log | 1.0e-31 | 1.0e-31 |

sin | 1.0e-31 | 1.0e-31 |

cos | 1.0e-31 | 1.0e-28 |

tan | 1.0e-30 | 1.0e-30 |

atan | 1.0e-31 | 1.0e-31 |

sinh | 1.0e-30 | 1.0e-31 |

cosh | 1.0e-30 | 1.0e-31 |

tanh | 1.0e-31 | 1.0e-31 |

asinh | 1.0e-31 | 1.0e-31 |

### isapprox

`isapprox`

uses this default`rtol=eps(1.0)^(37/64)`

.

## Good Ways To Use This

In addition to simply `using DoubleFloats`

and going from there, these two suggestions are easily managed
and will go a long way in increasing the robustness of the work and reliability in the computational results.

If your input values are Float64s, map them to Double64s and proceed with your computation. Then unmap your output values as Float64s, do additional work using those Float64s. With Float32 inputs, used Double32s similarly. Where throughput is important, and your algorithms are well-understood, this approach be used with the numerically sensitive parts of your computation only. If you are doing that, be careful to map the inputs to those parts and unmap the outputs from those parts just as described above.

## Questions

Usage questions can be posted on the Julia Discourse forum. Use the topic `Numerics`

(a "Discipline") and a put the package name, DoubleFloats, in your question ("topic").

## Contributions

Contributions are very welcome, as are feature requests and suggestions. Please open an issue if you encounter any problems. The contributing page has a few guidelines that should be followed when opening pull requests.

β: If you want to get involved with moving `Double32`

performance forward, great. I would provide guidance. Otherwise, for most purposes you are better off using `Float64`

than `Double32`

(`Float64`

has more significant bits, wider exponent range, and is much faster).