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Merge branch 'benchmark' of https://github.com/heltonmc/Bessels.jl into benchmark
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‎README.md

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[![Build Status](https://github.com/heltonmc/Bessels.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/heltonmc/Bessels.jl/actions/workflows/CI.yml?query=branch%3Amaster)
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[![Coverage](https://codecov.io/gh/heltonmc/Bessels.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/heltonmc/Bessels.jl)
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A Julia implementation of Bessel's functions and modified Bessel's functions of the first and second kind.
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Implementations of Bessel's functions `besselj0`, `besselj1`, `bessely0`, `bessely1`, `besseli0`, `besseli1`, `besselk0`, `besselk1` in Julia.
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Most implementations are ported to Julia from the [Cephes](https://www.netlib.org/cephes/) math library originally written by Stephen L. Moshier in the C programming language
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which are (partly) used in [SciPy](https://docs.scipy.org/doc/scipy/reference/special.html#faster-versions-of-common-bessel-functions) and [GCC libquadmath](https://gcc.gnu.org/onlinedocs/libquadmath/).
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The library currently supports Bessel's function of the first and second kind for orders 0 and 1 and for any integer order for modified Bessel's functions for real arguments.
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# Quick start
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Only implemented for real arguments so far.
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```julia
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using Bessels
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# add the package
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pkg> add https://github.com/heltonmc/Bessels.jl.git
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julia> using Bessels
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# Bessel function of the first kind of order 0
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julia> besselj0(1.0)
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0.7651976865579665
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julia> besselj0(1.0f0)
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0.7651977f0
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# Bessel function of the first kind of order 1
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julia> besselj1(1.0)
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0.44005058574493355
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julia> besselj1(1.0f0)
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0.4400506f0
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# Bessel function of the second kind of order 0
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julia> bessely0(1.0)
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0.08825696421567697
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julia> bessely0(1.0f0)
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0.08825697f0
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# Bessel function of the second kind of order 1
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julia> bessely1(1.0)
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-0.7812128213002888
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julia> bessely1(1.0f0)
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-0.7812128f0
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# Modified Bessel function of the first kind of order 0
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julia> besseli0(1.0)
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1.2660658777520082
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julia> besseli0(1.0f0)
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1.266066f0
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# Scaled modified Bessel function of the first kind of order 0
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julia> besseli0x(1.0)
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0.46575960759364043
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julia> besseli0x(1.0f0)
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0.46575963f0
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# Scaled modified Bessel function of the first kind of order 1
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julia> besseli1x(1.0)
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0.2079104153497085
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julia> besseli1x(1.0f0)
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0.20791042f0
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# Modified Bessel function of the second kind of order 0
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julia> besselk0(1.0)
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0.42102443824070823
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julia> besselk0(1.0f0)
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0.42102447f0
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# Scaled modified Bessel function of the second kind of order 0
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julia> besselk0x(1.0)
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1.1444630798068947
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julia> besselk0x(1.0f0)
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1.1444632f0
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# Modified Bessel function of the second kind of order 1
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julia> besselk1(1.0)
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0.6019072301972346
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julia> besselk1(1.0f0)
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0.6019073f0
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# Scaled modified Bessel function of the second kind of order 1
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julia> besselk1x(1.0)
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1.636153486263258
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julia> besselk1x(1.0f0)
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1.6361537f0
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```
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# Benchmarks
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```julia
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# Bessels.jl
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julia> @benchmark besselj0(x) setup=(x=100.0*rand())
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BenchmarkTools.Trial: 10000 samples with 999 evaluations.
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Range (min … max): 7.465 ns … 28.946 ns ┊ GC (min … max): 0.00% … 0.00%
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Time (median): 25.860 ns ┊ GC (median): 0.00%
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Time (mean ± σ): 24.915 ns ± 4.090 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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Memory estimate: 0 bytes, allocs estimate: 0.
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# SpecialFunctions.jl
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julia> @benchmark besselj0(x) setup=(x=100.0*rand())
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BenchmarkTools.Trial: 10000 samples with 999 evaluations.
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Range (min … max): 9.050 ns … 79.412 ns ┊ GC (min … max): 0.00% … 0.00%
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Time (median): 51.677 ns ┊ GC (median): 0.00%
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Time (mean ± σ): 51.029 ns ± 5.653 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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Memory estimate: 0 bytes, allocs estimate: 0.
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```
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```julia
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# Bessels.jl
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julia> @benchmark besselk1(x) setup=(x=100.0*rand())
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BenchmarkTools.Trial: 10000 samples with 988 evaluations.
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Range (min … max): 47.781 ns … 97.672 ns ┊ GC (min … max): 0.00% … 0.00%
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Time (median): 47.950 ns ┊ GC (median): 0.00%
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Time (mean ± σ): 48.495 ns ± 2.656 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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Memory estimate: 0 bytes, allocs estimate: 0.
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# SpecialFunctions.jl
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julia> @benchmark besselk(1, x) setup=(x=10.0*rand())
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BenchmarkTools.Trial: 10000 samples with 681 evaluations.
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Range (min … max): 184.962 ns … 2.122 μs ┊ GC (min … max): 0.00% … 81.41%
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Time (median): 377.081 ns ┊ GC (median): 0.00%
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Time (mean ± σ): 388.243 ns ± 96.971 ns ┊ GC (mean ± σ): 0.04% ± 0.81%
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Memory estimate: 16 bytes, allocs estimate: 1.
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```
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Comparing the relative speed (`SpecialFunctions.jl / Bessels.jl`) for a vector of values between 0 and 100.
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## Float32
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# Supported functions
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| function | Relative speed |
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| ------------- | ------------- |
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| besselj0 | 1.7x |
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| besselj1 | 1.7x |
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| bessely0 | 1.6x |
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| bessely1 | 1.6x |
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| besseli0 | 24x |
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| besseli1 | 22x |
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| besselk0 | 13x |
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| besselk1 | 15x |
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- `besselj0(x)`
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- `besselj1(x)`
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- `bessely0(x)`
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- `bessely1(x)`
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- `besseli0(x)`
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- `besseli1(x)`
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- `besselk0(x)`
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- `besselk1(x)`
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- `besseli(nu, x)`
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- `besselk(nu, x)`
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* it looks like SpecialFunctions.jl doesn't always preserve the correct input type so some of the calculations may be done in Float64. This might skew the benchmarks for `Bessels.jl` as it should have all calculations done in the lower precision.
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Exponentially scaled versions of Modified Bessel's function can also be called with `besselix(nu, x)` and `besselkx(nu, x)`.
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```julia
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# SpecialFunctions.jl
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julia> @btime besselk(1, x) setup=(x=Float32(10.0*rand()))
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179.272 ns (1 allocation: 16 bytes)
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0.021948646168061196 # notice incorrect Float64 return type
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# Bessels.jl
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julia> @btime besselk1(x) setup=(x=Float32(10.0*rand()))
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22.967 ns (0 allocations: 0 bytes)
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0.0027777348f0 # notice correct Float32 return type
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```
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## Float64
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| function | Relative speed |
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| ------------- | ------------- |
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| besselj0 | 1.7x |
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| besselj1 | 1.9x |
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| bessely0 | 1.5x |
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| bessely1 | 1.5x |
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| besseli0 | 9x |
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| besseli1 | 7x |
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| besselk0 | 5x |
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| besselk1 | 5x |
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```julia
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# SpecialFunctions.jl
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julia> @btime besselk(1, x) setup=(x=10.0*rand())
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184.726 ns (1 allocation: 16 bytes) # notice the small difference in Float32 and Float64 implementations
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0.0007517428778913419
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# Benchmarks
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# Bessels.jl
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julia> @btime besselk1(x) setup=(x=10.0*rand())
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47.824 ns (0 allocations: 0 bytes)
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0.0057366790518002045
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```
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These results compare the median value from BenchmarkTools obtained on one machine for arguments between 0 and 100.
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We compare the relative sped to implementations provided by [SpecialFunctions.jl](https://github.com/JuliaMath/SpecialFunctions.jl).
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| function | `Float32` | `Float64`
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| ------------- | ------------- | ------------- |
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| besselj0 | 1.7x | 1.8x
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| besselj1 | 1.7x | 1.9x
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| bessely0 | 1.9x | 1.8x
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| bessely1 | 1.7x | 1.7x
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| besseli0 | 26x | 13.2x
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| besseli1 | 22x | 13.9x
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| besseli(20, x) | 5.4x | 2.1x |
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| besseli(120, x) | 6x | 4.5x |
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| besselk0 | 16x | 12.7x
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| besselk1 | 15x | 14.3x
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| besselk(20, x) | 5.4x | 5.7x |
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| besselk(120, x) | 2.9x | 3.4x |
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* SpecialFunctions.jl doesn't always preserve the correct input type so some of the calculations may be done in Float64. This might skew the benchmarks for `Bessels.jl`.

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