DoubleFloats.jl
Math with 85+ accurate bits.
Documentation
- Current — most recently tagged version of the documentation.
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-30 | 1.0e-30 |
acos | 1.0e-30 | 1.0e-29 |
atan | 1.0e-31 | 1.0e-30 |
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 | notes |
---|---|---|---|
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-24 | 1.0e-28 | near asymptote |
asin | 1.0e-30 | 1.0e-30 | |
acos | 1.0e-30 | 1.0e-29 | |
atan | 1.0e-31 | 1.0e-30 | |
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 |
Installation
pkg> add DoubleFloats
or
julia> using Pkg
julia> Pkg.add("DoubleFloats")
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)
2.0000000000000001110223024625156540e-01
julia> Double64(2)/10
1.9999999999999999999999999999999937e-01
julia> d64"0.2"
1.9999999999999999999999999999999937e-01
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> 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))
golden ratio
julia> using DoubleFloats
julia> ϕ = Double32(MathConstants.golden)
1.61803398874989490
julia> phi = "1.61803398874989484820+"
julia> ϕ⁻¹ = inv(ϕ)
6.18033988749894902e-01
julia> ϕ == 1 + ϕ⁻¹
true
julia> ϕ === ϕ * ϕ⁻¹ + ϕ⁻¹
true
typed value | computed value | ~abs(golden - computed) |
---|---|---|
MathConstants.golden |
1.61803_39887_49894_84820_45868+ | 0.0 |
Float64(MathConstants.golden) |
1.61803_39887_49895 | 1.5e-16 |
Double32(MathConstants.golden) |
1.61803_39887_49894_90 | 5.2e-17 |
Performance
Double64 relative to BigFloat
op | speedup |
---|---|
+ | 11x |
* | 18x |
\ | 7x |
trig | 3x-6x |
- results from testing with BenchmarkTools on one machine
- BigFloat precision was set to 106 bits, for fair comparison
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 and Contributions
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 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.