pkg> add DoubleFloatsor
julia> using Pkg
julia> Pkg.add("DoubleFloats")Comparing Double64 and BigFloat after setting BigFloat precision to 106 bits.
| op | speedup |
|---|---|
| + | 11x |
| * | 18x |
| \ | 7x |
| trig | 3x-6x |
| these results are from BenchmarkTools, on one machine |
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
truenote: 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-01julia> x = ComplexD64(sqrt(d64"2"), cbrt(d64"3"))
1.4142135623730951 + 1.4422495703074083im
julia> y = acosh(x)
1.402873733241199 + 0.8555178360714634im
julia> x - cosh(y)
7.395570986446986e-32 + 0.0imjulia> 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))
julia> x = ComplexD32(sqrt(d32"2"), cbrt(d32"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))"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 |
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.
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.
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