A quick Julia library for calculating Fresnel Integrals using the error function from SpecialFunctions.jl.
using FresnelIntegrals
using Plots
z = -30:0.001:30
plot(x->real(fresnelc(x)), x->real(fresnels(x)),z,legend=false)
title!("Euler Spiral")
xlabel!("C(z)")
ylabel!("S(z)")
As expected, this is much faster than the numeric integral solution
using QuadGK
using FresnelIntegrals
julia> @benchmark fresnelc(1.8)
BenchmarkTools.Trial: 10000 samples with 772 evaluations.
Range (min … max): 161.448 ns … 1.744 μs ┊ GC (min … max): 0.00% … 0.00%
Time (median): 163.927 ns ┊ GC (median): 0.00%
Time (mean ± σ): 164.460 ns ± 16.519 ns ┊ GC (mean ± σ): 0.00% ± 0.00%
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161 ns Histogram: log(frequency) by time 168 ns <
Memory estimate: 0 bytes, allocs estimate: 0.
julia> @benchmark quadgk(t->cos(π*t^2/2),0,1.8)
BenchmarkTools.Trial: 10000 samples with 187 evaluations.
Range (min … max): 548.321 ns … 48.949 μs ┊ GC (min … max): 0.00% … 98.43%
Time (median): 554.893 ns ┊ GC (median): 0.00%
Time (mean ± σ): 599.010 ns ± 558.147 ns ┊ GC (mean ± σ): 2.25% ± 3.29%
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548 ns Histogram: log(frequency) by time 754 ns <
Memory estimate: 368 bytes, allocs estimate: 2.