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4_estimate_pi_(optional).jl
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4_estimate_pi_(optional).jl
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### A Pluto.jl notebook ###
# v0.19.40
using Markdown
using InteractiveUtils
# ╔═╡ 47789e98-5de0-413f-8677-29918719278a
begin
using Statistics, Random
include("utils.jl")
using .Utilities
end
# ╔═╡ c79e8873-789a-4db8-8af5-d915c465d72b
using .Utilities: still_missing
# ╔═╡ d685eff6-ab91-4bee-bf7a-957b2ae644f0
begin
using ForwardDiff # automatically finds the derivative of a function
using Plots
end
# ╔═╡ c6b3bebe-32da-11ef-09ae-ad02d9296cab
md"""### Estimate π using Monte Carlo
"""
# ╔═╡ 2f6e0b90-dfd5-4f70-bb25-4eecf9a30768
pi_err(x) = (value=x, error=abs(x-π))
# ╔═╡ 664e8ba5-6dd8-44af-a8e1-1df6f80510a2
begin
using KernelAbstractions
# using CUDA: CuArray
@kernel function estimate_pi_kernel(a, n)
i = @index(Global)
k = prod(@groupsize())
@inbounds a[i] = prod((i ÷ 2 * 2) / ((i-1) ÷ 2 * 2 + 1) for i in 1+i:k:n)
end
@time let N = 300_000_000, K = 64
A = ones(K)
# A = CuArray(ones(K))
device = @show get_backend(A) # CPU / GPU
run! = estimate_pi_kernel(device, K)
run!(A, N, ndrange=size(A))
synchronize(device)
2 * prod(A)
end |> pi_err
end
# ╔═╡ efe11271-fe46-4399-ad6b-089b5b5ceec3
function estimate_pi_mc(n=600_000_000)
rng = MersenneTwister(0)
mean(1:n) do _
rand(rng)^2 + rand(rng)^2 < 1
end * 4
end
# ╔═╡ 447bdcf9-260f-4cba-977b-6a97b2280d35
@time pi_err(estimate_pi_mc())
# ╔═╡ ec0cd66a-eed0-4ac5-a2ab-d021ba003ebd
let task = Threads.@spawn estimate_pi_mc()
@time fetch(@show task)
end
# ╔═╡ b4c16d52-45d6-421f-aefe-e9680977a176
@time let N = 600_000_000, k = @show Threads.nthreads()
mean(fetch.(Threads.@spawn estimate_pi_mc(N÷k) for _ in 1:k)) |> pi_err
end
# ╔═╡ 5ffdecbf-9a8b-49e5-adbb-90f932075076
md"""### Estimate π using the Wallis product
In 1655, John Wallis published what is now known as [Wallis product](https://en.wikipedia.org/wiki/Wallis_product), an infinite product:
$$π = \frac{2\cdot2\cdot4\cdot4\cdot6\cdot6\ldots}
{1\cdot3\cdot3\cdot5\cdot5\cdot7\ldots}$$
"""
# ╔═╡ 71f9f677-9721-4626-b8d6-eae70e25c2d5
function estimate_pi_wallis(N=300_000_000)
# generator (laze evaluation)
fracs = ((i ÷ 2 * 2) / ((i-1) ÷ 2 * 2 + 1) for i in 2:N)
# fracs = [(i ÷ 2 * 2) / ((i-1) ÷ 2 * 2 + 1) for i in 2:N]
2prod(fracs)
end
# ╔═╡ ed0e646e-f8c6-4cb3-95e3-e1d39fa0b4c9
@time estimate_pi_wallis() |> pi_err
# ╔═╡ 4b0fd4e7-fa60-4115-9548-02f88f3b7a82
md"""
!!! danger "Task"
Implement a multithreaded version of `estimate_pi_wallis`
"""
# ╔═╡ 60ef1f4b-89ab-42ec-973f-1025d0161b96
function estimate_pi_wallis_multithreaded(N=300_000_000)
missing # replace `missing` with your solution here
end
# ╔═╡ 658c4cca-a5c6-4c5d-80fb-f939728c1992
let
t0 = time()
result = @time estimate_pi_wallis_multithreaded()
dt = time() - t0
if ismissing(result)
Utilities.still_missing()
elseif dt > 0.6
Utilities.keep_working(md"Try reducing the time cost by distributing the computation on more threads.")
elseif abs(result - π) < 1e-8
Utilities.correct()
else
Utilities.keep_working()
end
end
# ╔═╡ 9aeec0a9-9742-41e8-8c64-61744a86f653
md"**Applying the `@Threads.threads` macro to a `for` loop:**"
# ╔═╡ 44d506a5-9b65-48c1-bbfb-5a1977abcde1
let N = 300_000_000, K = 240
times = [Float64[] for _ in 1:Threads.nthreads()]
pi = @time let
A = ones(K)
@Threads.threads for k in 1:K
t0 = time()
@inbounds A[k] = prod((i ÷ 2 * 2) / ((i-1) ÷ 2 * 2 + 1) for i in 1+k:K:N)
push!(times[Threads.threadid()], time() - t0)
end
2prod(A)
end
println(pi_err(pi))
[f(ts) for ts in times, f in [length, mean]]
end
# ╔═╡ 0c6a623d-ec6b-4234-8cf1-873e14158eb4
md"**Writing a CPU/GPU kernel for parallel computing:**"
# ╔═╡ c8bf3d51-570d-4668-a4b1-bfa6e8e475fe
md"""### Estimate π using Newton's method
Idea: ``\pi`` is a root of ``sin(x)``.
"""
# ╔═╡ 15603d34-3728-42ce-bac7-1608ef34ba3c
begin # Operator overloading
Base.adjoint(f::Function) = x -> ForwardDiff.derivative(f, x)
sin'(0), cos'(π/2)
end
# ╔═╡ 4b626ee4-44fb-4488-b630-888601587a54
function newton_method(f, x; max_iter=1000, tol=1e-12, buf=nothing)
for i in 1:max_iter
dx = f(x) / f'(x)
x -= dx
if buf != nothing
push!(buf, x)
end
if abs(dx) < tol
@info "converged after $i iterations"
return x
end
end
@warn "root not found after $max_iter iterations"
end
# ╔═╡ b6969c0f-c337-44a2-b458-10f916201484
@time newton_method(sin, 2.0) |> pi_err
# ╔═╡ fae4b543-03ac-4f27-bff6-07574fae5733
let f = sin, x = range(-1, 5, 1000)
y = f.(x)
x0 = 2
xs = []
newton_method(f, x0, max_iter=30, buf=xs)
@gif for x1 in xs
plot(x, y, label="$f(x)", title="Newton's method")
plot!([x0, x1], [f(x0), 0], label="tagent line", markershape=:circle)
plot!([x1, x1], [0, f(x1)], label=nothing, linestyle=:dash)
annotate!(x0, f(x0), ("x = $(Float16(x0))", 8, :top))
x0 = x1
end fps=2
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
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# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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