### A Pluto.jl notebook ###
# v0.20.17

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    #! format: off
    return quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
    #! format: on
end

# ╔═╡ 6d3acd2f-0854-49bf-807a-2b10d00b804d
using LogarithmicNumbers

# ╔═╡ 3c700749-ea91-41cb-badc-d42e7399eb0b
using LogExpFunctions

# ╔═╡ 143d4954-6710-4702-9a8b-fb83154476b2
using DoubleFloats

# ╔═╡ 1c640715-9bef-4935-9dce-f94ff2a3740b
begin
	using PlutoUI, PlutoTeachingTools
	using Plots, LaTeXStrings
	using BenchmarkTools
end

# ╔═╡ 0b431bf7-1f57-40c4-ad0c-012cbdbf9528
md"> Astro 528: High-Performance Scientific Computing for Astrophysics (Fall 2025)"

# ╔═╡ 0b4ae83b-290e-46f5-854d-01db4c754187
md"""
## Setup Accounts
- Thanks if you already logged into Lynx cluster and got a GitHub account.
- If you haven't yet, pelase try before class on Friday.
- If you have trouble, please ask for help before Friday class.
"""

# ╔═╡ 959f2c12-287c-4648-a585-0c11d0db812d
md"""
# Week 1 Discussion Topics
- Introductions
- Scientific software
   - Floating point arithmetic
   - Testing scientific codes
- Challenges of testing codes
- ICDS clusters: Lynx, Roar Collab, LIGO,...
- Programming languages (depending on on time)
- Start on Labs
---
"""

# ╔═╡ 17afed3c-4f28-424d-9758-a483fa1de279
md"""
# What makes scientific software special?
"""

# ╔═╡ 354621a9-9ab1-4bbf-84a1-4d86c2696933
hint(md"""
- Emphasis on floating-point arithmetic
- Numerically intensive (large data sets and/or many itterations)
- Sophisticated tool sets
- Rapidly changing specifications.
""")

# ╔═╡ ade401f7-fa34-42b4-a8e1-e0949561f9cb
md"""
## Comparing to Computer Software Industry

- How is scientific sofware development different?
- ...similar?

"""

# ╔═╡ e2c01b6f-f92b-4fea-bf5e-ecfdddc8717a
md"""
## Floating Point Arithmetic

- When should we be paranoid?
- What can we do about it?
"""

# ╔═╡ d97ea77a-85fb-456c-b9a3-fea23179d412
md"## Example 1: Catastrophic Cancelation"

# ╔═╡ 61cba1a3-77bd-4dbe-af6a-ed3db6b0a5f8
L"\frac{1}{1-x^2}"

# ╔═╡ 9c6f88b4-21a2-480e-9e2f-95053043386b
begin
	ϵ = 1e-16.*2 .^(1:31)
	x = 1 .- ϵ
	y1 = 1 ./(1 .- x.^2)
	y2 = @. 1/(ϵ*(1+x))
	plt = scatter(ϵ,log10.(abs.(y2.-y1)),label=:none,xscale=:log10)
	xlabel!(plt, "ϵ = 1-x")
	ylabel!(plt, L"\log_{10} \left| \mathrm{Error} \right|")
end

# ╔═╡ b39708ea-16f6-4536-b58b-a3e5a95db067
md"""
### Avoiding Catstrophic Cancelation
- Look for where subtract two number of similar size
```math
\frac{1}{1-x^2}
```
- What if x ≃ 1?
"""

# ╔═╡ 6c2d3a07-3a2a-4c6d-a047-e6d87a5de7f8
hint(md"""Rewrite expression to avoid near cancelation:  
E.g., if $x\simeq 1$, then $\epsilon \equiv 1-x$
```math
\frac{1}{1-x^2} = \frac{1}{\left(1-x\right)\left(1+x\right)} = \frac{1}{\epsilon \left(1+x\right)} = \frac{1}{\epsilon(2-\epsilon)}
```

""")

# ╔═╡ 10a632fc-5047-4f00-879c-54b98bc173ca
md"""
## Example 2: Underflow
"""

# ╔═╡ c7eeb1c7-028a-479f-8492-c0c151f1a5eb
exp(-746)

# ╔═╡ df9a17af-36a6-4134-a44d-f0e42bf7e736
md"Why would you ever calculate exp(-746)?"

# ╔═╡ 61d3b850-5398-424e-9aef-1c4db92b0587
hint(md"Probability of many observations")

# ╔═╡ a41741e6-039d-4a80-a9f0-10f447589880
md"""
## Avoiding underflow in statistical computations
"""

# ╔═╡ 379aa32c-7914-4da5-a2d4-201a0acf2115
md"""
- Instead of working with very small probabilities (e.g., $p$) perform computations in terms of the log probability (e.g., $\log(p)$).

```math
p = \exp\left[-\frac{1}{2}\sum_i \frac{Δ_i^2}{σ_i^2} \right] 
```
"""

# ╔═╡ 98d31fcb-232f-4313-a17f-270dfc56d50e
hint(md"""

Work with log of numbers
```math
\log p = -\frac{1}{2}\sum_i \frac{Δ_i^2}{σ_i^2} 
```
- Often one's final results involve taking ratios of probabilities that largely cancel anyway.  E.g., conditional probabilities, Bayes theorem
```math
p(A | B) = \frac{p(A, B)}{p(B)} = \frac{p(A) p(B | A)}{p(B)}
```

""")

# ╔═╡ fed5c970-9ba4-4acb-b2ae-223ee40792b3
md"""
Packages like [LogarithmicNumbers.jl](https://github.com/cjdoris/LogarithmicNumbers.jl) or Package [LogExpFunction.jl](https://juliastats.org/Package [LogExpFunction.jl](https://juliastats.org/LogExpFunctions.jl/stable/) .jl/stable/) can help.
"""

# ╔═╡ f81e772f-00b6-4c61-b7a3-de6f96ca7c83
let
	a = exp(Logarithmic,-746)
	b = exp(Logarithmic,-747)
	c = a/b
end

# ╔═╡ c72e1905-bf29-49a7-a8ad-d5b86424032b
let
	a = exp(-746)
	b = exp(-747)
	logsumexp([a,b])
end

# ╔═╡ a1458625-5545-4433-97e1-d41a1790ad7f
md"## Example 3: Round-off Error"

# ╔═╡ 3d576e8d-2100-403d-a295-62a1ca92728a
md"""
```math
\mathrm{Error} = \frac{n}{d} - \sum_{i=1}^n \frac{1}{d}
```
"""

# ╔═╡ 6e1ddb6a-9403-41db-8b7f-c53a1d744548
function roundoff_error_test(n::Integer; unit::Real = 1/17)
	n*unit - reduce(+,unit for i in 1:n )
end

# ╔═╡ 6f912a3d-c621-4873-b2e4-f34d6986640d
md"d = $(@bind denom NumberField(1:1000; default=17))"

# ╔═╡ 2e11d1c6-4c49-42df-b285-34d3936d2489
roundoff_error_test(100, unit = 1/Float64(denom))

# ╔═╡ 235cbf58-1a75-412f-a80f-b8d0fde1c317
roundoff_error_test(100, unit = 1/Float32(denom))

# ╔═╡ 82d7aed4-ee1d-4b8f-ba79-3eed3fe8a255
roundoff_error_test(100,unit = 1/Double64(denom) )

# ╔═╡ edd6cf50-990c-4d10-a595-1b3c69c93c28
md"### Round-off error versus number of terms"

# ╔═╡ 4a2908b0-be48-4968-a5ee-2cf0229c0764
begin
	log2nmax = 24
	n_list = 2 .^(4:log2nmax)
	errs_float32 = roundoff_error_test.(n_list, unit=1/Float32(denom))
	errs_float64 = roundoff_error_test.(n_list, unit=1/Float64(denom))
	(;errs_float32, errs_float64)
end;

# ╔═╡ 575fd9df-91ab-4152-bdc8-52ade89eeef1
begin
	local plt = plot(legend=:topleft,xscale=:log10)
	scatter!(n_list,log10.(abs.(errs_float32)), markercolor=:blue, label="Float32")
	plot!(n_list,log10.(abs.(errs_float32)), linecolor=:blue, label=:none)
	scatter!(n_list,log10.(abs.(errs_float64)), markercolor=:green, label="Float64")
	plot!(n_list,log10.(abs.(errs_float64)), linecolor=:green, label=:none)
	xlabel!(L"$n$")
	ylabel!(L"$log_{10} \left|\mathrm{Error}\right|$")
	title!(LaTeXString("\$d = " * string(denom) * "\$"))
	plt
end

# ╔═╡ 14ec952c-696c-44f7-957a-57f540c4a56f
md"#### Why would you add so many numbers?"

# ╔═╡ e3888187-4170-443f-92a5-4a9d87863bc8
BigFloat

# ╔═╡ 992c6ab6-9bb3-4fa4-ba56-9525fb8c6710
md"""## Extended Precission

- Would using extended precision number help?
  + Julia Base: `BigFloat`  (included in Julia)
  * [QuadMath.jl](https://github.com/JuliaMath/Quadmath.jl): `Float128` (IEEE compliant, faster than BigFloat)
  + [DoubleFloats.jl](https://github.com/JuliaMath/DoubleFloats.jl): `Double64` (faster)
  + [MultiFloats.jl](https://docs.juliahub.com/MultiFloats/k11Ei/2.3.0/): `Float64x2`  (even faster)
  + [ArbNumerics.jl](https://github.com/JeffreySarnoff/ArbNumerics.jl): `ArbFloat{200}` (slow)
  + [AccurateArithmetic.jl](https://github.com/JuliaMath/AccurateArithmetic.jl): `sum_kbn`, `dot_oro`, ...
  + ...
- When would it not help?

"""

# ╔═╡ dd3a260e-e949-44d1-8fa9-5f2b2ce74400
errs_double64 = roundoff_error_test.(n_list, unit=1/Double64(denom))

# ╔═╡ 35bf1b18-e65b-425a-9d59-c169355d5c29
begin
	local plt = plot(legend=:topleft,xscale=:log10)
	scatter!(n_list,log10.(abs.(errs_float32)), markercolor=:blue, label="Float32")
	plot!(n_list,log10.(abs.(errs_float32)), linecolor=:blue, label="Float32")
	scatter!(n_list,log10.(abs.(errs_float64)), markercolor=:green, label="Float64")
	plot!(n_list,log10.(abs.(errs_float64)), linecolor=:green, label="Float64")
	scatter!(n_list,log10.(abs.(errs_double64)), markercolor=:red, label="Double64")
	plot!(n_list,log10.(abs.(errs_double64)), linecolor=:red, label="Double64")
	xlabel!("n")
	ylabel!(L"$log_{10} \left|\mathrm{Error}\right|$")
	title!(LaTeXString("\$d = " * string(denom) * "\$"))
	plt
end

# ╔═╡ 84241913-7c53-47c1-b5f0-1f0c9a30a76f
md"## Performance vs Precision"

# ╔═╡ 77a00d38-e61a-4825-9bc3-ef0ce0cdff2a
@benchmark $roundoff_error_test(100,unit=Float32(1/17))

# ╔═╡ 9d3c906e-5153-4ead-b405-01db7c039471
@benchmark $roundoff_error_test(100,unit=Float64(1/17))

# ╔═╡ d7170965-0171-4662-8eab-7d856895d4b5
@benchmark $(roundoff_error_test)(100,unit=Double64(1/17))

# ╔═╡ 5362f60b-1b2c-400f-9205-c750e6fa8d47
md"""
## Challenges of Testing Scientific Codes

!!! hint
    - How can you test a code solving an unsolved problem?
    - End-to-end tests may take a long time
    - Does it make sense for scientific codes?

"""

# ╔═╡ 4e53021f-19b6-4289-b79f-6770debc5e3c
md"""
## Is anyone going to reuse my code?

!!! hint
	- You
	- Your research group
	- Collaborators / future team members
    - Other research groups
	- Unexpected users

"""

# ╔═╡ 8759b216-cc38-42ed-b85c-04d508579c54
md"# Helper Code"

# ╔═╡ a21b553b-eecb-4105-a0ed-d936e500788b
WidthOverDocs()

# ╔═╡ afe9b7c1-d031-4e1f-bd5b-5aeed30d7048
md"ToC on side $(@bind toc_aside CheckBox(;default=true))"

# ╔═╡ 080d3a94-161e-4482-9cf4-b82ffb98d0ed
TableOfContents(aside=toc_aside)

# ╔═╡ 00000000-0000-0000-0000-000000000001
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "7ed9347888fac59a618302ee38216dd0379c480d"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.12+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXau_jll", "Xorg_libXdmcp_jll"]
git-tree-sha1 = "bfcaf7ec088eaba362093393fe11aa141fa15422"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.1+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "e3150c7400c41e207012b41659591f083f3ef795"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.3+0"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "c5bf2dad6a03dfef57ea0a170a1fe493601603f2"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.5+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f4fc02e384b74418679983a97385644b67e1263b"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll"]
git-tree-sha1 = "68da27247e7d8d8dafd1fcf0c3654ad6506f5f97"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "44ec54b0e2acd408b0fb361e1e9244c60c9c3dd4"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "5b0263b6d080716a02544c55fdff2c8d7f9a16a0"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.10+0"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f233c83cad1fa0e70b7771e0e21b061a116f2763"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.2+0"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "801a858fc9fb90c11ffddee1801bb06a738bda9b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.7+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "00af7ebdc563c9217ecc67776d1bbf037dbcebf4"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.44.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a63799ff68005991f9d9491b6e95bd3478d783cb"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.6.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "446b23e73536f84e8037f5dce465e92275f6a308"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.7+1"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "c3b0e6196d50eab0c5ed34021aaa0bb463489510"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.14+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b6a34e0e0960190ac2a4363a1bd003504772d631"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.61.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "4bba74fa59ab0755167ad24f98800fe5d727175b"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.12.1+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "125eedcb0a4a0bba65b657251ce1d27c8714e9d6"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.17.4+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "56d643b57b188d30cccc25e331d416d3d358e557"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.13.4+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "646634dd19587a56ee2f1199563ec056c5f228df"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.4+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "91d05d7f4a9f67205bd6cf395e488009fe85b499"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.28.1+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "07b6a107d926093898e82b3b1db657ebe33134ec"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.50+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll"]
git-tree-sha1 = "11e1772e7f3cc987e9d3de991dd4f6b2602663a5"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.8+0"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b4d631fd51f2e9cdd93724ae25b2efc198b059b1"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.7+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "14cc7083fc6dff3cc44f2bc435ee96d06ed79aa7"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "10164.0.1+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e7b67590c14d487e734dcb925924c5dc43ec85f3"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "4.1.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "fbf139bce07a534df0e699dbb5f5cc9346f95cc1"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.9.2+0"
"""

# ╔═╡ Cell order:
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# ╠═77a00d38-e61a-4825-9bc3-ef0ce0cdff2a
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