bondscell_results_$1b768306-0e0e-414d-a501-5c62a2db720fqueued¤logsrunning¦outputbodyV

Computations as Trees

mimetext/htmlrootassigneelast_run_timestampAڎҚapersist_js_state·has_pluto_hook_features§cell_id$1b768306-0e0e-414d-a501-5c62a2db720fdepends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$aa99b123-7164-4ed7-833f-b99b19118a0dqueued¤logsrunning¦outputbody[+] → 14 mimetext/plainrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$aa99b123-7164-4ed7-833f-b99b19118a0ddepends_on_disabled_cells§runtimer1published_object_keysdepends_on_skipped_cells§errored$be693f92-de60-4b43-82b3-ee6cceea310aqueued¤logsrunning¦outputbody^

$$\frac{\partial}{\partial x} (2xy + (x-1)^2) = 2y + 2(x-1)$$

$$\frac{\partial}{\partial y} (2xy + (x-1)^2) = 2x$$

If x = 3 and y = 5, then these evaluate to 14 and 6.

mimetext/htmlrootassigneelast_run_timestampAڎҞ persist_js_state·has_pluto_hook_features§cell_id$be693f92-de60-4b43-82b3-ee6cceea310adepends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$1f1b384a-6588-45a5-9dd3-6de3face8bfbqueued¤logsrunning¦outputbody)ad_steps (generic function with 1 method)mimetext/plainrootassigneelast_run_timestampAڎVpersist_js_state·has_pluto_hook_features§cell_id$1f1b384a-6588-45a5-9dd3-6de3face8bfbdepends_on_disabled_cells§runtimecµpublished_object_keysdepends_on_skipped_cells§errored$e55dfad2-db50-459f-ab54-fa7637fc3638queued¤logsrunning¦outputbodyq

How can we visualize both the forward and the reverse pass?

We can further visualize each steps we just took. First we do the forwards calculation, where we also build up our tree, then we go down the tree in the opposite direction to accumulate our gradient.

mimetext/htmlrootassigneelast_run_timestampAڎҴpersist_js_state·has_pluto_hook_features§cell_id$e55dfad2-db50-459f-ab54-fa7637fc3638depends_on_disabled_cells§runtimeOpublished_object_keysdepends_on_skipped_cells§errored$a0c0f48d-e0b0-4f45-8e0e-d9ab5743958bqueued¤logsrunning¦outputbodyrTracked{Float64} val: Float64 3.0 name: Symbol x df: Nothing nothing deps: Array{Tracked}((0,)) Tracked[] mimetext/plainrootassigneelast_run_timestampAڎJRapersist_js_state·has_pluto_hook_features§cell_id$a0c0f48d-e0b0-4f45-8e0e-d9ab5743958bdepends_on_disabled_cells§runtime;hpublished_object_keysdepends_on_skipped_cells§errored$7dca2c78-6faf-450a-8096-1c5b9f7f1295queued¤logsrunning¦outputbody+leaky_relu (generic function with 1 method)mimetext/plainrootassigneelast_run_timestampAڎzfpersist_js_state·has_pluto_hook_features§cell_id$7dca2c78-6faf-450a-8096-1c5b9f7f1295depends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$13487e65-5e48-4a37-9bea-f262dd7b6d56queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎK'persist_js_state·has_pluto_hook_features§cell_id$13487e65-5e48-4a37-9bea-f262dd7b6d56depends_on_disabled_cells§runtimeѵpublished_object_keysdepends_on_skipped_cells§errored$96286b65-1a22-4458-a399-46579248cce4queued¤logslinemsg)Saved animation to /tmp/jl_9s36bOHY54.giftext/plaincell_id$96286b65-1a22-4458-a399-46579248cce4kwargsidPlots_7e84f40bfile9/home/runner/.julia/packages/Plots/GIume/src/animation.jlgroupanimationlevelInforunning¦outputbodypmimetext/htmlrootassigneelast_run_timestampAڎ^Fpersist_js_state·has_pluto_hook_features§cell_id$96286b65-1a22-4458-a399-46579248cce4depends_on_disabled_cells§runtimeǗpublished_object_keysdepends_on_skipped_cells§errored$1a154bb7-93a3-4973-8908-788db77ac294queued¤logsrunning¦outputbody. mimetext/htmlrootassignees2last_run_timestampAڎdApersist_js_state·has_pluto_hook_features§cell_id$1a154bb7-93a3-4973-8908-788db77ac294depends_on_disabled_cells§runtimeʵpublished_object_keysdepends_on_skipped_cells§errored$8f1ec8d7-89ab-4bc5-9eef-2b0a69019854queued¤logsrunning¦outputbodyN

We'll try to approximate a sine curve:

mimetext/htmlrootassigneelast_run_timestampAڎҵ{persist_js_state·has_pluto_hook_features§cell_id$8f1ec8d7-89ab-4bc5-9eef-2b0a69019854depends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$795c510e-6280-42d9-8820-290ca07be88dqueued¤logsrunning¦outputbodyprefixMain.var"workspace#3".TrackedelementstypeArrayprefix_shortobjectidc657c0653d02cfb4mime!application/vnd.pluto.tree+objectrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$795c510e-6280-42d9-8820-290ca07be88ddepends_on_disabled_cells§runtime͔published_object_keysdepends_on_skipped_cells§errored$27b39d7d-fc08-4ccc-aea4-b64f8a4f5726queued¤logsrunning¦outputbodyPB
Move me!
1 mimetext/htmlrootassigneelast_run_timestampAڎeSmpersist_js_state·has_pluto_hook_features§cell_id$27b39d7d-fc08-4ccc-aea4-b64f8a4f5726depends_on_disabled_cells§runtimehcfpublished_object_keysdepends_on_skipped_cells§errored$85098c31-04a1-421c-b3f6-25fec9360588queued¤logsrunning¦outputbody

What if we use a (leaky) rectified linear unit (ReLU) as activation?

mimetext/htmlrootassigneelast_run_timestampAڎҵ԰persist_js_state·has_pluto_hook_features§cell_id$85098c31-04a1-421c-b3f6-25fec9360588depends_on_disabled_cells§runtimeypublished_object_keysdepends_on_skipped_cells§errored$8ab0f55d-a393-4a8a-a48c-9ced26033f57queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎgpersist_js_state·has_pluto_hook_features§cell_id$8ab0f55d-a393-4a8a-a48c-9ced26033f57depends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$e01114fa-a847-424a-931f-8f42e623109fqueued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎmpersist_js_state·has_pluto_hook_features§cell_id$e01114fa-a847-424a-931f-8f42e623109fdepends_on_disabled_cells§runtimeFpublished_object_keysdepends_on_skipped_cells§errored$2ba83769-89b4-4ba0-8726-9210f394b4e7queued¤logsrunning¦outputbodyV

Operations on Tracked

mimetext/htmlrootassigneelast_run_timestampAڎҨpersist_js_state·has_pluto_hook_features§cell_id$2ba83769-89b4-4ba0-8726-9210f394b4e7depends_on_disabled_cells§runtimeصpublished_object_keysdepends_on_skipped_cells§errored$d48b9f23-b1b8-43f8-a05d-2234b69bf1f0queued¤logsrunning¦outputbody[+] → 34 mimetext/plainrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$d48b9f23-b1b8-43f8-a05d-2234b69bf1f0depends_on_disabled_cells§runtime͚published_object_keysdepends_on_skipped_cells§errored$a5671f2f-48ca-4896-9428-147dc671d2b9queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎ^̰persist_js_state·has_pluto_hook_features§cell_id$a5671f2f-48ca-4896-9428-147dc671d2b9depends_on_disabled_cells§runtimeTpublished_object_keysdepends_on_skipped_cells§errored$d5565717-239b-41ab-b311-d59b383130edqueued¤logsrunning¦outputbodyٛ

Ok, let's create our function grad which will accumulate all intermediate gradients into a dictionary:

mimetext/htmlrootassigneelast_run_timestampAڎҴFpersist_js_state·has_pluto_hook_features§cell_id$d5565717-239b-41ab-b311-d59b383130eddepends_on_disabled_cells§runtimeɵpublished_object_keysdepends_on_skipped_cells§errored$81eb8a2d-a3a9-45af-a5a5-b96aefd48712queued¤logsrunning¦outputbody34mimetext/plainrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$81eb8a2d-a3a9-45af-a5a5-b96aefd48712depends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$2188a663-5a85-4ce4-bc8d-20383481e59bqueued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎ_persist_js_state·has_pluto_hook_features§cell_id$2188a663-5a85-4ce4-bc8d-20383481e59bdepends_on_disabled_cells§runtime published_object_keysdepends_on_skipped_cells§errored$db4f871a-3ca1-4bd3-a2c5-da7d6f599281queued¤logsrunning¦outputbody15mimetext/plainrootassigneelast_run_timestampAڎV'persist_js_state·has_pluto_hook_features§cell_id$db4f871a-3ca1-4bd3-a2c5-da7d6f599281depends_on_disabled_cells§runtime=published_object_keysdepends_on_skipped_cells§errored$53ae4070-8818-4d21-8648-19df9319918aqueued¤logsrunning¦outputbodyق

Let's write our own neural network!

mimetext/htmlrootassigneelast_run_timestampAڎҵ+ǰpersist_js_state·has_pluto_hook_features§cell_id$53ae4070-8818-4d21-8648-19df9319918adepends_on_disabled_cells§runtime published_object_keysdepends_on_skipped_cells§errored$79f71f9d-b491-4a2c-85a4-29ae8da4f312queued¤logsrunning¦outputbody 3y + 2 * (x - 1) 35=y + 2 * (x - 1) [*] → 15 + 2 * (x - 1) [*] → 15 + 2 * (3=x - 1) [*] → 15 + 2[-] → 2 [*] → 15 + [*] → 4 [+] → 19 mimetext/htmlrootassigneelast_run_timestampAڎ۰persist_js_state·has_pluto_hook_features§cell_id$79f71f9d-b491-4a2c-85a4-29ae8da4f312depends_on_disabled_cells§runtimeCpublished_object_keysdepends_on_skipped_cells§errored$e450329b-46ee-4f53-b6f0-891d6596938cqueued¤logsrunning¦outputbody[+] → 14 mimetext/plainrootassigneelast_run_timestampAڎ|[persist_js_state·has_pluto_hook_features§cell_id$e450329b-46ee-4f53-b6f0-891d6596938cdepends_on_disabled_cells§runtimewpublished_object_keysdepends_on_skipped_cells§errored$a7c8cb6a-6e17-4d8f-8958-fe3527c5b8e7queued¤logsrunning¦outputbodyelements6text/plain3text/plaintypeTupleobjectidfa3317f9f54f06c9mime!application/vnd.pluto.tree+objectrootassigneelast_run_timestampAڎvpersist_js_state·has_pluto_hook_features§cell_id$a7c8cb6a-6e17-4d8f-8958-fe3527c5b8e7depends_on_disabled_cells§runtimeY|published_object_keysdepends_on_skipped_cells§errored$d1312080-272c-4bd5-bcfe-bf47f8320961queued¤logsrunning¦outputbody`

The Tracked Data Structure

mimetext/htmlrootassigneelast_run_timestampAڎҨpersist_js_state·has_pluto_hook_features§cell_id$d1312080-272c-4bd5-bcfe-bf47f8320961depends_on_disabled_cells§runtime:published_object_keysdepends_on_skipped_cells§errored$885cd51d-895f-4996-a23b-780498b5b810queued¤logsrunning¦outputbody

All overloads will do the operation (e.g. sum x and y), but also remember the pullback map and input variables for the reverse pass.

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Move me!
1 mimetext/htmlrootassigneelast_run_timestampAڎjTqpersist_js_state·has_pluto_hook_features§cell_id$ec723b90-1c8b-4a87-8a86-fe17d6e85f22depends_on_disabled_cells§runtimedpublished_object_keysdepends_on_skipped_cells§errored$c9d70049-4793-41db-9075-e9e73e926c1aqueued¤logsrunning¦outputbody[+] → 8 mimetext/plainrootassigneelast_run_timestampAڎL)persist_js_state·has_pluto_hook_features§cell_id$c9d70049-4793-41db-9075-e9e73e926c1adepends_on_disabled_cells§runtime+published_object_keysdepends_on_skipped_cells§errored$17f9e2c0-aa70-4d4e-ba42-51782df43785queued¤logsrunning¦outputbody~

Computational Tree (Forward Pass)

mimetext/htmlrootassigneelast_run_timestampAڎқpersist_js_state·has_pluto_hook_features§cell_id$17f9e2c0-aa70-4d4e-ba42-51782df43785depends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$0d186a1b-eb89-4be8-a237-e5a9608f4a30queued¤logsrunning¦outputbody'
Move me!
1 mimetext/htmlrootassigneelast_run_timestampAڎp@!persist_js_state·has_pluto_hook_features§cell_id$0d186a1b-eb89-4be8-a237-e5a9608f4a30depends_on_disabled_cells§runtimedpublished_object_keysdepends_on_skipped_cells§errored$d2d21d41-9a4e-428b-b696-6878901849baqueued¤logsrunning¦outputbodyo1×100 Matrix{Float64}: 0.0 0.0634239 0.126592 0.189251 0.251148 … -0.126592 -0.0634239 -2.44929e-16mimetext/plainrootassigneeŷlast_run_timestampAڎoQpersist_js_state·has_pluto_hook_features§cell_id$d2d21d41-9a4e-428b-b696-6878901849badepends_on_disabled_cells§runtime.published_object_keysdepends_on_skipped_cells§errored$0b094198-cf44-41d7-a8dc-fd8fd0716bb4queued¤logsrunning¦outputbodyS

Helper Functions

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Move me!
1 mimetext/htmlrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$2eb6af9c-9ac7-4b64-93fb-c5a2f037d303depends_on_disabled_cells§runtime`Rpublished_object_keysdepends_on_skipped_cells§errored$0b5e6560-81fd-4182-bba5-aca702fb3048queued¤logsrunning¦outputbody2=u mimetext/plainrootassigneelast_run_timestampAڎapersist_js_state·has_pluto_hook_features§cell_id$0b5e6560-81fd-4182-bba5-aca702fb3048depends_on_disabled_cells§runtimeZֵpublished_object_keysdepends_on_skipped_cells§errored$8a70e835-dae6-4727-8204-95c87d5c23daqueued¤logsrunning¦outputbody?

We need some more overloads

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Besides the mathematics of reverse mode differentiation, there are many interesting computer science issues regarding the implementation of reverse diff. Probably the jury is still out which is best.

The implementation below is based on Julia's dispatch system. This means we create a type Tracked for keeping track of our input variables and everything we'll need to calculate later.

The other main approach (not illustrated here) is source to source translation of code. In Julia the zygote package translates at the high level, while the increasingly popular Enzyme package translates at the lower level.

mimetext/htmlrootassigneelast_run_timestampAڎҨi۰persist_js_state·has_pluto_hook_features§cell_id$5e02b2c0-10cf-4746-840d-7017648f89f8depends_on_disabled_cells§runtime|published_object_keysdepends_on_skipped_cells§errored$a6c2d3a2-326a-41dd-864d-aa3662466222queued¤logsrunning¦outputbodyelements3=x text/plain5=y text/plaintypeTupleobjectid4d5843ff827a456bmime!application/vnd.pluto.tree+objectrootassigneelast_run_timestampAڎ)persist_js_state·has_pluto_hook_features§cell_id$a6c2d3a2-326a-41dd-864d-aa3662466222depends_on_disabled_cells§runtime*published_object_keysdepends_on_skipped_cells§errored$2c3f3103-9ccb-446e-a289-87bbd0a1b97bqueued¤logsrunning¦outputbody5=y mimetext/plainrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$2c3f3103-9ccb-446e-a289-87bbd0a1b97bdepends_on_disabled_cells§runtime͎upublished_object_keysdepends_on_skipped_cells§errored$e52aa672-69a9-419b-a992-e7a3d1364fb6queued¤logsrunning¦outputbodyprefix*Text{Main.var"workspace#3".Tracked{Int64}}elements[+] → 34text/plain[*] → 30text/plain[*] → 6text/plain2text/plain3=xtext/plain5=ytext/plain[^2] → 4text/plain[-] → 2text/plain 3=xtext/plain 1text/plaintypeArrayprefix_shortobjectidd36836b3dea4f895mime!application/vnd.pluto.tree+objectrootassigneelast_run_timestampAڎƋpersist_js_state·has_pluto_hook_features§cell_id$e52aa672-69a9-419b-a992-e7a3d1364fb6depends_on_disabled_cells§runtime갋published_object_keysdepends_on_skipped_cells§errored$7299bfea-e23e-4993-bebd-9d8762427294queued¤logsrunning¦outputbodyٌ

Pretty Viz: Forward Pass + Reverse Mode Diff

mimetext/htmlrootassigneelast_run_timestampAڎҜްpersist_js_state·has_pluto_hook_features§cell_id$7299bfea-e23e-4993-bebd-9d8762427294depends_on_disabled_cells§runtimeާpublished_object_keysdepends_on_skipped_cells§errored$4da7a594-aec5-4c8d-ab20-6913ed6ff8acqueued¤logsrunning¦outputbodyP mimetext/htmlrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$4da7a594-aec5-4c8d-ab20-6913ed6ff8acdepends_on_disabled_cells§runtimeDpublished_object_keysdepends_on_skipped_cells§errored$4bfc2f7d-a5b0-44c7-8bb6-f1b834c1cc51queued¤logsrunning¦outputbody

Tracked is a tree – We just need to tell AbstractTrees.jl how to get the children for each node and we get tree printing and iteration over all nodes for free.

mimetext/htmlrootassigneelast_run_timestampAڎҴYpersist_js_state·has_pluto_hook_features§cell_id$4bfc2f7d-a5b0-44c7-8bb6-f1b834c1cc51depends_on_disabled_cells§runtime׵published_object_keysdepends_on_skipped_cells§errored$0d2a3187-46f1-4705-bb3e-ecf74e34840dqueued¤logsrunning¦outputbody[+] → 34 mimetext/plainrootassigneezlast_run_timestampAڎqpersist_js_state·has_pluto_hook_features§cell_id$0d2a3187-46f1-4705-bb3e-ecf74e34840ddepends_on_disabled_cells§runtime͠ϵpublished_object_keysdepends_on_skipped_cells§errored$437285d4-ec53-4bb7-9966-fcfb5352e205queued¤logsrunning¦outputbody+show_tree (generic function with 2 methods)mimetext/plainrootassigneelast_run_timestampAڎ:persist_js_state·has_pluto_hook_features§cell_id$437285d4-ec53-4bb7-9966-fcfb5352e205depends_on_disabled_cells§runtimexnpublished_object_keysdepends_on_skipped_cells§errored$61513f88-c126-43ab-9515-85a2c8a472bfqueued¤logsrunning¦outputbodyprefixMain.var"workspace#3".Trackedelements[^2] → 9 text/plain5=y text/plaintypeArrayprefix_shortobjectidfb72d1e218d626d3mime!application/vnd.pluto.tree+objectrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$61513f88-c126-43ab-9515-85a2c8a472bfdepends_on_disabled_cells§runtime=+published_object_keysdepends_on_skipped_cells§errored$73d638bf-30c1-4694-b3a8-4b29c5e3fa65queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎLOpersist_js_state·has_pluto_hook_features§cell_id$73d638bf-30c1-4694-b3a8-4b29c5e3fa65depends_on_disabled_cells§runtimezrpublished_object_keysdepends_on_skipped_cells§errored$ccafd0b9-95aa-4e58-8afc-26cd3ee61cc9queued¤logsrunning¦outputbody^

Straight away we get the primal result of our calculation:

mimetext/htmlrootassigneelast_run_timestampAڎҴyݰpersist_js_state·has_pluto_hook_features§cell_id$ccafd0b9-95aa-4e58-8afc-26cd3ee61cc9depends_on_disabled_cells§runtimeMpublished_object_keysdepends_on_skipped_cells§errored$95ab6c83-0757-40b3-a116-6d872a074f69queued¤logsrunning¦outputbody[*] → 15 mimetext/plainrootassigneelast_run_timestampAڎMpersist_js_state·has_pluto_hook_features§cell_id$95ab6c83-0757-40b3-a116-6d872a074f69depends_on_disabled_cells§runtime*published_object_keysdepends_on_skipped_cells§errored$86aa821b-a373-4814-953e-535f3a33c002queued¤logsrunning¦outputbody
mimetext/htmlrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$86aa821b-a373-4814-953e-535f3a33c002depends_on_disabled_cells§runtime USܵpublished_object_keysdepends_on_skipped_cells§errored$29360c2a-c47b-48a3-b225-248eb3ec8a42queued¤logsrunning¦outputbody:*mimetext/plainrootassigneelast_run_timestampAڎ`!2persist_js_state·has_pluto_hook_features§cell_id$29360c2a-c47b-48a3-b225-248eb3ec8a42depends_on_disabled_cells§runtime@published_object_keysdepends_on_skipped_cells§errored$814786be-a760-4023-a614-83e727681075queued¤logsrunning¦outputbodyvTracked{Float64} val: Float64 2.5 name: Symbol ##303 df: Nothing nothing deps: Array{Tracked}((0,)) Tracked[] mimetext/plainrootassigneelast_run_timestampAڎJpersist_js_state·has_pluto_hook_features§cell_id$814786be-a760-4023-a614-83e727681075depends_on_disabled_cells§runtime0published_object_keysdepends_on_skipped_cells§errored$ac097299-0a31-474c-ab26-a4fb24bb9046queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎ mimeimage/svg+xmlrootassigneelast_run_timestampAڎCpersist_js_state·has_pluto_hook_features§cell_id$0911a08a-4290-455b-9b26-0bf2862296dadepends_on_disabled_cells§runtimeA#;published_object_keysdepends_on_skipped_cells§errored$20571e1a-9687-4d42-98b3-b3bc0b207b3bqueued¤logsrunning¦outputbodyX

Implementation in Code

mimetext/htmlrootassigneelast_run_timestampAڎҞ'persist_js_state·has_pluto_hook_features§cell_id$20571e1a-9687-4d42-98b3-b3bc0b207b3bdepends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$01eacbd4-ef37-4524-aecc-2ef9a1044cf8queued¤logsrunning¦outputbodyٜ

To also get the gradient, we'll use PreOrderDFS to traverse the tree we just created from the top down.

mimetext/htmlrootassigneelast_run_timestampAڎҴgpersist_js_state·has_pluto_hook_features§cell_id$01eacbd4-ef37-4524-aecc-2ef9a1044cf8depends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$99a3507b-ca03-429f-acde-e2d1ebb32054queued¤logsrunning¦outputbody%grad (generic function with 1 method)mimetext/plainrootassigneelast_run_timestampAڎepersist_js_state·has_pluto_hook_features§cell_id$99a3507b-ca03-429f-acde-e2d1ebb32054depends_on_disabled_cells§runtimes|published_object_keysdepends_on_skipped_cells§errored$7fcccb65-4a7b-4527-97be-d25f481f6eafqueued¤logsrunning¦outputbodyO

We can verify that it does the right thing:

mimetext/htmlrootassigneelast_run_timestampAڎҴRpersist_js_state·has_pluto_hook_features§cell_id$7fcccb65-4a7b-4527-97be-d25f481f6eafdepends_on_disabled_cells§runtimeیpublished_object_keysdepends_on_skipped_cells§errored$6733267e-1251-49a7-86a0-ff45ba408205queued¤logsrunning¦outputbodyP

Tracking variables

mimetext/htmlrootassigneelast_run_timestampAڎқɰpersist_js_state·has_pluto_hook_features§cell_id$6733267e-1251-49a7-86a0-ff45ba408205depends_on_disabled_cells§runtimevpublished_object_keysdepends_on_skipped_cells§errored$266bf2ed-d625-47be-8a87-21e12052b27fqueued¤logsrunning¦outputbody٘

Let's use three dense layers, all of the form

$$x \mapsto \tanh.(W \cdot x + b)$$

mimetext/htmlrootassigneelast_run_timestampAڎҵagpersist_js_state·has_pluto_hook_features§cell_id$266bf2ed-d625-47be-8a87-21e12052b27fdepends_on_disabled_cells§runtime;published_object_keysdepends_on_skipped_cells§errored$ea94bdf9-b0b1-42ad-ba44-aaba73ed1192queued¤logsrunning¦outputbody|

Let's look at our example from above, $z = x^2+y$. We'll introduce an intermediate variable $u = x^2$ so we can write $z$ as $u + y$.

We will first start with $1 \cdot dz$. The one here represents the fact that $\frac{\partial z}{\partial z} = 1$.

We then want to pull that back until we get something in terms of $dx$ and $dy$, so we then get the derivatives wrt $x$ and $y$

The pullback map for $+$ will first map $dz$ to $d(u + y) = du + dy$ and then we can use the pullback map for ^ to expand $du$ to $2x \cdot dx$.

Using that recursive method we end up with $dz = 2x \cdot dx + dy$, which is exactly what we'd get if we did this by hand. This tells us that $\frac{\partial z}{\partial x} = 2x$ and $\frac{\partial z}{\partial y} = y$.

mimetext/htmlrootassigneelast_run_timestampAڎҴ2'persist_js_state·has_pluto_hook_features§cell_id$ea94bdf9-b0b1-42ad-ba44-aaba73ed1192depends_on_disabled_cells§runtime صpublished_object_keysdepends_on_skipped_cells§errored$98807f0d-f8a5-4fe0-9983-2b95290347d9queued¤logsrunning¦outputbodyه

Tada! 🎉

(It's even reasonably fast for neural nets)

mimetext/htmlrootassigneelast_run_timestampAڎҵIpersist_js_state·has_pluto_hook_features§cell_id$98807f0d-f8a5-4fe0-9983-2b95290347d9depends_on_disabled_cells§runtime/صpublished_object_keysdepends_on_skipped_cells§errored$8b486bd8-25a1-4895-99c8-0541c9e3c8b0queued¤logsrunning¦outputbody3
Move me!
1 mimetext/htmlrootassigneelast_run_timestampAڎ&xpersist_js_state·has_pluto_hook_features§cell_id$8b486bd8-25a1-4895-99c8-0541c9e3c8b0depends_on_disabled_cells§runtime 7[ypublished_object_keysdepends_on_skipped_cells§errored$bcff2aa8-2387-44c7-a28f-39cd505a7adfqueued¤logsrunning¦outputbodyq

We can also visualize what Julia does in the forward pass on the code itself:

mimetext/htmlrootassigneelast_run_timestampAڎҵpersist_js_state·has_pluto_hook_features§cell_id$bcff2aa8-2387-44c7-a28f-39cd505a7adfdepends_on_disabled_cells§runtime1published_object_keysdepends_on_skipped_cells§errored$b47754ac-cb17-4ea3-a97d-20371fd341a5queued¤logsrunning¦outputbodyelements5text/plain3text/plaintypeTupleobjectidf8402269077b4228mime!application/vnd.pluto.tree+objectrootassigneelast_run_timestampAڎiupersist_js_state·has_pluto_hook_features§cell_id$b47754ac-cb17-4ea3-a97d-20371fd341a5depends_on_disabled_cells§runtimeCnpublished_object_keysdepends_on_skipped_cells§errored$520f280d-c78e-433b-a0a2-8ef05b04f7ccqueued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎspersist_js_state·has_pluto_hook_features§cell_id$520f280d-c78e-433b-a0a2-8ef05b04f7ccdepends_on_disabled_cells§runtimeӵpublished_object_keysdepends_on_skipped_cells§errored$faf69c6f-b179-4ca2-aa22-641e9560e17bqueued¤logsrunning¦outputbody@ad_steps (macro with 1 method)mimetext/plainrootassigneelast_run_timestampAڎ°persist_js_state·has_pluto_hook_features§cell_id$faf69c6f-b179-4ca2-aa22-641e9560e17bdepends_on_disabled_cells§runtime Ipublished_object_keysdepends_on_skipped_cells§errored$d82adc20-4c8c-4f2c-9839-d03ad7e7f581queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎDbpersist_js_state·has_pluto_hook_features§cell_id$d82adc20-4c8c-4f2c-9839-d03ad7e7f581depends_on_disabled_cells§runtimePcpublished_object_keysdepends_on_skipped_cells§errored$18b1c55d-a6b5-44f6-b0b3-50bdb0aa9d96queued¤logsrunning¦outputbodyH[+] → 18 ├─ [*] → 15 │ ├─ 3=x │ └─ 5=y └─ 3=x mimetext/plainrootassigneewlast_run_timestampAڎҞpersist_js_state·has_pluto_hook_features§cell_id$18b1c55d-a6b5-44f6-b0b3-50bdb0aa9d96depends_on_disabled_cells§runtimecpublished_object_keysdepends_on_skipped_cells§errored$8919faec-753d-41d0-b0a8-c84c025b2919queued¤logsrunning¦outputbody3.0=x mimetext/plainrootassigneexfloatlast_run_timestampAڎtpersist_js_state·has_pluto_hook_features§cell_id$8919faec-753d-41d0-b0a8-c84c025b2919depends_on_disabled_cells§runtimeGpublished_object_keysdepends_on_skipped_cells§errored$0d006e4d-72e3-42ab-beb3-e5304a4d9ba9queued¤logsrunning¦outputbody3=x mimetext/plainrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$0d006e4d-72e3-42ab-beb3-e5304a4d9ba9depends_on_disabled_cells§runtime0Apublished_object_keysdepends_on_skipped_cells§errored$4813e57f-557a-4179-afd2-7925687b3d35queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎҲOpersist_js_state·has_pluto_hook_features§cell_id$4813e57f-557a-4179-afd2-7925687b3d35depends_on_disabled_cells§runtime#published_object_keysdepends_on_skipped_cells§errored$86fa378b-815d-4c3d-9121-1338ee54f30fqueued¤logslinemsg)Saved animation to /tmp/jl_98t05Ct494.giftext/plaincell_id$86fa378b-815d-4c3d-9121-1338ee54f30fkwargsidPlots_7e84f40bfile9/home/runner/.julia/packages/Plots/GIume/src/animation.jlgroupanimationlevelInforunning¦outputbody|mimetext/htmlrootassigneelast_run_timestampAڎRưpersist_js_state·has_pluto_hook_features§cell_id$86fa378b-815d-4c3d-9121-1338ee54f30fdepends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$6b1fb808-e993-4c2b-b81b-6710f8206de7queued¤logsrunning¦outputbody(to_json (generic function with 1 method)mimetext/plainrootassigneelast_run_timestampAڎfǰpersist_js_state·has_pluto_hook_features§cell_id$6b1fb808-e993-4c2b-b81b-6710f8206de7depends_on_disabled_cells§runtimezpublished_object_keysdepends_on_skipped_cells§errored$8110f306-a7bb-43a2-bb36-6182c59b4b2equeued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎpersist_js_state·has_pluto_hook_features§cell_id$8110f306-a7bb-43a2-bb36-6182c59b4b2edepends_on_disabled_cells§runtimeDapublished_object_keysdepends_on_skipped_cells§errored$99b6ab91-a022-449c-988c-0e5c5719c910queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎ>persist_js_state·has_pluto_hook_features§cell_id$99b6ab91-a022-449c-988c-0e5c5719c910depends_on_disabled_cells§runtime̲published_object_keysdepends_on_skipped_cells§errored$db3389c0-cb00-404c-81a8-8c4ccdab22f4queued¤logsrunning¦outputbody
Move me!
1 mimetext/htmlrootassigneelast_run_timestampAڎt'persist_js_state·has_pluto_hook_features§cell_id$db3389c0-cb00-404c-81a8-8c4ccdab22f4depends_on_disabled_cells§runtimenipublished_object_keysdepends_on_skipped_cells§errored$b0cc4665-eb45-48ea-9a33-5acf56d2a283queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAڎLgpersist_js_state·has_pluto_hook_features§cell_id$b0cc4665-eb45-48ea-9a33-5acf56d2a283depends_on_disabled_cells§runtimeUpublished_object_keysdepends_on_skipped_cells§errored$d4e9b202-242e-4420-986b-12d2ab57af93queued¤logsrunning¦outputbody&grad (generic function with 2 methods)mimetext/plainrootassigneelast_run_timestampAڎ؛persist_js_state·has_pluto_hook_features§cell_id$d4e9b202-242e-4420-986b-12d2ab57af93depends_on_disabled_cells§runtimeWgpublished_object_keysdepends_on_skipped_cells§errored±cell_dependencies_$1b768306-0e0e-414d-a501-5c62a2db720fprecedence_heuristic 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v1.0.17+0  Installed libaom_jll ─────────────────── v3.13.3+0  Installed StatsBase ──────────────────── v0.34.11  Installed JLFzf ──────────────────────── v0.1.11  Installed Xorg_libxcb_jll ────────────── v1.17.1+0  Installed Qt6Svg_jll ─────────────────── v6.10.2+0  Installed eudev_jll ──────────────────── v3.2.14+0  Installed libpng_jll ─────────────────── v1.6.58+0  Installed FreeType2_jll ──────────────── v2.14.3+1  Installed TensorCore ─────────────────── v0.1.1  Installed Parsers ────────────────────── v2.8.5  Installed Dbus_jll ───────────────────── v1.16.2+0  Installed Format ─────────────────────── v1.3.7  Installed ColorVectorSpace ───────────── v0.11.0  Installed ColorTypes ─────────────────── v0.12.1  Installed libevdev_jll ───────────────── v1.13.4+0  Installed Plots ──────────────────────── v1.41.6  Installed Xorg_libXext_jll ───────────── v1.3.8+0  Installed Zstd_jll ───────────────────── v1.5.7+1  Installed libinput_jll ───────────────── v1.28.1+0  Installed Ogg_jll ────────────────────── v1.3.6+0  Installed Vulkan_Loader_jll ──────────── v1.3.243+0  Installed Libtiff_jll ────────────────── v4.7.2+0  Installed Reexport ───────────────────── v1.2.2  Installed LogExpFunctions ────────────── v1.0.1  Installed Libffi_jll ─────────────────── v3.4.7+0  Installed AliasTables ────────────────── v1.1.3  Installed Xorg_libXrender_jll ────────── v0.9.12+0  Installed Xorg_libXi_jll ─────────────── v1.8.3+0  Installed Qt6ShaderTools_jll ─────────── v6.10.2+1  Installed Ghostscript_jll ────────────── v9.55.1+0  Installed DocStringExtensions ────────── v0.9.5  Installed Xorg_xcb_util_renderutil_jll ─ v0.3.10+0  Installed Qt6Declarative_jll ─────────── v6.10.2+1  Installed Xorg_xcb_util_cursor_jll ───── v0.1.6+0  Installed Graphite2_jll ──────────────── v1.3.16+0  Installed libass_jll ─────────────────── v0.17.4+0  Installed MacroTools ─────────────────── v0.5.16  Installed Xorg_libXcursor_jll ────────── v1.2.4+0  Installed Wayland_jll ────────────────── v1.24.0+0  Installed Libuuid_jll ────────────────── v2.42.0+0  Installed Xorg_libICE_jll ────────────── v1.1.2+0  Installed Pixman_jll ─────────────────── v0.46.4+0  Installed StableRNGs ─────────────────── v1.0.4  Installed Latexify ───────────────────── v0.16.10  Installed Xorg_xkeyboard_config_jll ──── v2.47.0+1  Installed FixedPointNumbers ──────────── v0.8.6  Installed Xorg_xtrans_jll ────────────── v1.6.0+0  Installed libdrm_jll ─────────────────── v2.4.125+1  Installed RecipesBase ────────────────── v1.3.4  Installed LAME_jll ───────────────────── v3.100.3+0  Installed GettextRuntime_jll ─────────── v0.22.4+0  Installed FFMPEG_jll ─────────────────── v8.1.0+0  Installed LaTeXStrings ───────────────── v1.4.0  Installed IOCapture ──────────────────── v1.0.0  Installed libvorbis_jll ──────────────── v1.3.8+0  Installed Glib_jll ───────────────────── v2.86.3+0  Installed LLVMOpenMP_jll ─────────────── v18.1.8+0  Installed Xorg_libXrandr_jll ─────────── v1.5.6+0  Installed Unzip ──────────────────────── v0.2.0  Installed Xorg_libXfixes_jll ─────────── v6.0.2+0  Installed SortingAlgorithms ──────────── v1.2.2  Installed Qt6Wayland_jll ─────────────── v6.10.2+1  Installed Libglvnd_jll ───────────────── v1.7.1+1  Installed Qt6Base_jll ────────────────── v6.10.2+2  Installed PlutoTest ──────────────────── v0.2.4  Installed Xorg_libX11_jll ────────────── v1.8.13+0  Installed Xorg_libXdmcp_jll ──────────── v1.1.6+0  Installing 76 artifacts  Installed artifact Xorg_xcb_util_renderutil 32.1 KiB  Installed artifact libass 427.8 KiB  Installed artifact Bzip2 503.5 KiB  Installed artifact LERC 267.4 KiB  Installed artifact Pixman 390.8 KiB  Installed artifact Xorg_libpciaccess 26.2 KiB  Installed artifact Xorg_xtrans 48.2 KiB  Installed artifact Xorg_xcb_util_wm 147.9 KiB  Installed artifact mtdev 79.6 KiB  Installed artifact Qt6ShaderTools 2.1 MiB  Installed artifact Xorg_xkeyboard_config 545.0 KiB  Installed artifact Qt6Wayland 1.4 MiB  Installed artifact Xorg_libXi 1.6 MiB  Installed artifact Libmount 6.9 MiB  Installed artifact libinput 1.0 MiB  Installed artifact GLFW 187.9 KiB  Installed artifact Libglvnd 833.5 KiB  Installed artifact eudev 3.9 MiB  Installed artifact Libtiff 2.3 MiB  Installed artifact Xorg_libSM 164.1 KiB  Installed artifact libpng 329.4 KiB  Installed artifact Xorg_xcb_util 39.5 KiB  Installed artifact Vulkan_Loader 177.3 KiB  Installed artifact libdecor 44.1 KiB  Installed artifact Dbus 489.8 KiB  Installed artifact Xorg_libXau 36.6 KiB  Installed artifact Pango 2.1 MiB  Installed artifact libfdk_aac 2.8 MiB  Installed artifact Libuuid 3.9 MiB  Installed artifact Xorg_xcb_util_cursor 27.6 KiB  Installed artifact Xorg_libXext 286.6 KiB  Installed artifact Xorg_xcb_util_image 54.3 KiB  Installed artifact xkbcommon 298.7 KiB  Installed artifact x264 2.1 MiB  Installed artifact JpegTurbo 1.4 MiB  Installed artifact GettextRuntime 543.0 KiB  Installed artifact XZ 1.6 MiB  Installed artifact Xorg_xcb_util_keysyms 17.8 KiB  Installed artifact fzf 3.3 MiB  Installed artifact FriBidi 78.9 KiB  Installed artifact Graphite2 120.3 KiB  Installed artifact x265 1.4 MiB  Installed artifact Ghostscript 31.4 MiB  Installed artifact Xorg_libXinerama 25.5 KiB  Installed artifact GR 19.3 MiB  Installed artifact Xorg_libX11 4.8 MiB  Installed artifact Zstd 646.0 KiB  Installed artifact Xorg_libXdmcp 67.7 KiB  Installed artifact HarfBuzz 1.7 MiB  Installed artifact Xorg_libXrandr 97.3 KiB  Installed artifact Xorg_libxkbfile 91.1 KiB  Installed artifact Xorg_xkbcomp 274.8 KiB  Installed artifact Xorg_libXrender 435.9 KiB  Installed artifact libvorbis 271.0 KiB  Installed artifact Qt6Svg 386.4 KiB  Installed artifact Wayland 390.6 KiB  Installed artifact libevdev 117.1 KiB  Installed artifact Libffi 44.2 KiB  Installed artifact Opus 960.9 KiB  Installed artifact Xorg_libxcb 2.1 MiB  Installed artifact FreeType2 1.5 MiB  Installed artifact LAME 292.5 KiB  Installed artifact libdrm 368.0 KiB  Installed artifact libva 235.7 KiB  Installed artifact Fontconfig 984.8 KiB  Installed artifact Ogg 250.3 KiB  Installed artifact LLVMOpenMP 661.6 KiB  Installed artifact Xorg_libICE 384.9 KiB  Installed artifact Cairo 2.2 MiB  Installed artifact libaom 6.4 MiB  Installed artifact FFMPEG 11.9 MiB  Installed artifact Glib 7.7 MiB  Installed artifact Xorg_libXfixes 59.0 KiB  Installed artifact Xorg_libXcursor 227.1 KiB  Installed artifact Qt6Declarative 23.8 MiB  Installed artifact Qt6Base 23.1 MiB  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Manifest.toml` Instantiating... === Precompiling... === LinearAlgebraJH Resolving... ===  Installed libfdk_aac_jll ─────────────── v2.0.4+0  Installed x265_jll ───────────────────── v4.1.0+0  Installed GR_jll ─────────────────────── v0.73.26+0  Installed Opus_jll ───────────────────── v1.6.1+0  Installed libdecor_jll ───────────────── v0.2.2+0  Installed Hyperscript ────────────────── v0.0.5  Installed GR ─────────────────────────── v0.73.26  Installed Contour ────────────────────── v0.6.3  Installed Libmount_jll ───────────────── v2.42.0+0  Installed Xorg_xcb_util_wm_jll ───────── v0.4.2+0  Installed Grisu ──────────────────────── v1.0.2  Installed JpegTurbo_jll ──────────────── v3.1.5+0  Installed LERC_jll ───────────────────── v4.1.0+0  Installed Xorg_xcb_util_image_jll ────── v0.4.1+0  Installed RecipesPipeline ────────────── v0.6.12  Installed DelimitedFiles ─────────────── v1.9.1  Installed Measures ───────────────────── v0.3.3  Installed Xorg_xkbcomp_jll ───────────── v1.4.7+0  Installed PlutoUI ────────────────────── v0.7.83  Installed Xorg_xcb_util_jll ──────────── v0.4.1+0  Installed PlotUtils ──────────────────── v1.4.4  Installed libva_jll ──────────────────── v2.23.0+0  Installed Cairo_jll ──────────────────── v1.18.7+0  Installed EpollShim_jll ──────────────── v0.0.20230411+1  Installed Statistics ─────────────────── v1.11.1  Installed Xorg_libSM_jll ─────────────── v1.2.6+0  Installed Missings ───────────────────── v1.2.0  Installed Showoff ────────────────────── v1.0.3  Installed ColorSchemes ───────────────── v3.31.0  Installed Xorg_libXinerama_jll ───────── v1.1.7+0  Installed Fontconfig_jll ─────────────── v2.17.1+0  Installed Xorg_libXau_jll ────────────── v1.0.13+0  Installed FFMPEG ─────────────────────── v0.4.5  Installed Xorg_libxkbfile_jll ────────── v1.2.0+0  Installed IrrationalConstants ────────── v0.2.6  Installed Xorg_xcb_util_keysyms_jll ──── v0.4.1+0  Installed xkbcommon_jll ──────────────── v1.13.0+0  Installed Bzip2_jll ──────────────────── v1.0.9+0  Installed HarfBuzz_jll ───────────────── v8.5.1+0  Installed PlotThemes ─────────────────── v3.3.0  Installed NaNMath ────────────────────── v1.1.4  Installed Xorg_libpciaccess_jll ──────── v0.19.0+0  Installed x264_jll ───────────────────── v10164.0.1+0  Installed UnicodeFun ─────────────────── v0.4.1  Installed Pango_jll ──────────────────── v1.57.1+0  Installed Colors ─────────────────────── v0.13.1  Installed PtrArrays ──────────────────── v1.4.0  Installed DataStructures ─────────────── v0.19.5  Installed XZ_jll ─────────────────────── v5.8.3+0  Installed StatsAPI ───────────────────── v1.8.0  Installed fzf_jll ────────────────────── v0.61.1+0  Installed GLFW_jll ───────────────────── v3.4.1+1  Installed mtdev_jll ──────────────────── v1.1.7+0  Installed FriBidi_jll ────────────────── v1.0.17+0  Installed libaom_jll ─────────────────── v3.13.3+0  Installed StatsBase ──────────────────── v0.34.11  Installed JLFzf ──────────────────────── v0.1.11  Installed Xorg_libxcb_jll ────────────── v1.17.1+0  Installed Qt6Svg_jll ─────────────────── v6.10.2+0  Installed eudev_jll ──────────────────── v3.2.14+0  Installed libpng_jll ─────────────────── v1.6.58+0  Installed FreeType2_jll ──────────────── v2.14.3+1  Installed TensorCore ─────────────────── v0.1.1  Installed Parsers ────────────────────── v2.8.5  Installed Dbus_jll ───────────────────── v1.16.2+0  Installed Format ─────────────────────── v1.3.7  Installed ColorVectorSpace ───────────── v0.11.0  Installed ColorTypes ─────────────────── v0.12.1  Installed libevdev_jll ───────────────── v1.13.4+0  Installed Plots ──────────────────────── v1.41.6  Installed Xorg_libXext_jll ───────────── v1.3.8+0  Installed Zstd_jll ───────────────────── v1.5.7+1  Installed libinput_jll ───────────────── v1.28.1+0  Installed Ogg_jll ────────────────────── v1.3.6+0  Installed Vulkan_Loader_jll ──────────── v1.3.243+0  Installed Libtiff_jll ────────────────── v4.7.2+0  Installed Reexport ───────────────────── v1.2.2  Installed LogExpFunctions ────────────── v1.0.1  Installed Libffi_jll ─────────────────── v3.4.7+0  Installed AliasTables ────────────────── v1.1.3  Installed Xorg_libXrender_jll ────────── v0.9.12+0  Installed Xorg_libXi_jll ─────────────── v1.8.3+0  Installed Qt6ShaderTools_jll ─────────── v6.10.2+1  Installed Ghostscript_jll ────────────── v9.55.1+0  Installed DocStringExtensions ────────── v0.9.5  Installed Xorg_xcb_util_renderutil_jll ─ v0.3.10+0  Installed Qt6Declarative_jll ─────────── v6.10.2+1  Installed Xorg_xcb_util_cursor_jll ───── v0.1.6+0  Installed Graphite2_jll ──────────────── v1.3.16+0  Installed libass_jll ─────────────────── v0.17.4+0  Installed MacroTools ─────────────────── v0.5.16  Installed Xorg_libXcursor_jll ────────── v1.2.4+0  Installed Wayland_jll ────────────────── v1.24.0+0  Installed Libuuid_jll ────────────────── v2.42.0+0  Installed Xorg_libICE_jll ────────────── v1.1.2+0  Installed Pixman_jll ─────────────────── v0.46.4+0  Installed StableRNGs ─────────────────── v1.0.4  Installed Latexify ───────────────────── v0.16.10  Installed Xorg_xkeyboard_config_jll ──── v2.47.0+1  Installed FixedPointNumbers ──────────── v0.8.6  Installed Xorg_xtrans_jll ────────────── v1.6.0+0  Installed libdrm_jll ─────────────────── v2.4.125+1  Installed RecipesBase ────────────────── v1.3.4  Installed LAME_jll ───────────────────── v3.100.3+0  Installed GettextRuntime_jll ─────────── v0.22.4+0  Installed FFMPEG_jll ─────────────────── v8.1.0+0  Installed LaTeXStrings ───────────────── v1.4.0  Installed IOCapture ──────────────────── v1.0.0  Installed libvorbis_jll ──────────────── v1.3.8+0  Installed Glib_jll ───────────────────── v2.86.3+0  Installed LLVMOpenMP_jll ─────────────── v18.1.8+0  Installed Xorg_libXrandr_jll ─────────── v1.5.6+0  Installed Unzip ──────────────────────── v0.2.0  Installed Xorg_libXfixes_jll ─────────── v6.0.2+0  Installed SortingAlgorithms ──────────── v1.2.2  Installed Qt6Wayland_jll ─────────────── v6.10.2+1  Installed Libglvnd_jll ───────────────── v1.7.1+1  Installed Qt6Base_jll ────────────────── v6.10.2+2  Installed PlutoTest ──────────────────── v0.2.4  Installed Xorg_libX11_jll ────────────── v1.8.13+0  Installed Xorg_libXdmcp_jll ──────────── v1.1.6+0  Installing 76 artifacts  Installed artifact Xorg_xcb_util_renderutil 32.1 KiB  Installed artifact libass 427.8 KiB  Installed artifact Bzip2 503.5 KiB  Installed artifact LERC 267.4 KiB  Installed artifact Pixman 390.8 KiB  Installed artifact Xorg_libpciaccess 26.2 KiB  Installed artifact Xorg_xtrans 48.2 KiB  Installed artifact Xorg_xcb_util_wm 147.9 KiB  Installed artifact mtdev 79.6 KiB  Installed artifact Qt6ShaderTools 2.1 MiB  Installed artifact Xorg_xkeyboard_config 545.0 KiB  Installed artifact Qt6Wayland 1.4 MiB  Installed artifact Xorg_libXi 1.6 MiB  Installed artifact Libmount 6.9 MiB  Installed artifact libinput 1.0 MiB  Installed artifact GLFW 187.9 KiB  Installed artifact Libglvnd 833.5 KiB  Installed artifact eudev 3.9 MiB  Installed artifact Libtiff 2.3 MiB  Installed artifact Xorg_libSM 164.1 KiB  Installed artifact libpng 329.4 KiB  Installed artifact Xorg_xcb_util 39.5 KiB  Installed artifact Vulkan_Loader 177.3 KiB  Installed artifact libdecor 44.1 KiB  Installed artifact Dbus 489.8 KiB  Installed artifact Xorg_libXau 36.6 KiB  Installed artifact Pango 2.1 MiB  Installed artifact libfdk_aac 2.8 MiB  Installed artifact Libuuid 3.9 MiB  Installed artifact Xorg_xcb_util_cursor 27.6 KiB  Installed artifact Xorg_libXext 286.6 KiB  Installed artifact Xorg_xcb_util_image 54.3 KiB  Installed artifact xkbcommon 298.7 KiB  Installed artifact x264 2.1 MiB  Installed artifact JpegTurbo 1.4 MiB  Installed artifact GettextRuntime 543.0 KiB  Installed artifact XZ 1.6 MiB  Installed artifact Xorg_xcb_util_keysyms 17.8 KiB  Installed artifact fzf 3.3 MiB  Installed artifact FriBidi 78.9 KiB  Installed artifact Graphite2 120.3 KiB  Installed artifact x265 1.4 MiB  Installed artifact Ghostscript 31.4 MiB  Installed artifact Xorg_libXinerama 25.5 KiB  Installed artifact GR 19.3 MiB  Installed artifact Xorg_libX11 4.8 MiB  Installed artifact Zstd 646.0 KiB  Installed artifact Xorg_libXdmcp 67.7 KiB  Installed artifact HarfBuzz 1.7 MiB  Installed artifact Xorg_libXrandr 97.3 KiB  Installed artifact Xorg_libxkbfile 91.1 KiB  Installed artifact Xorg_xkbcomp 274.8 KiB  Installed artifact Xorg_libXrender 435.9 KiB  Installed artifact libvorbis 271.0 KiB  Installed artifact Qt6Svg 386.4 KiB  Installed artifact Wayland 390.6 KiB  Installed artifact libevdev 117.1 KiB  Installed artifact Libffi 44.2 KiB  Installed artifact Opus 960.9 KiB  Installed artifact Xorg_libxcb 2.1 MiB  Installed artifact FreeType2 1.5 MiB  Installed artifact LAME 292.5 KiB  Installed artifact libdrm 368.0 KiB  Installed artifact libva 235.7 KiB  Installed artifact Fontconfig 984.8 KiB  Installed artifact Ogg 250.3 KiB  Installed artifact LLVMOpenMP 661.6 KiB  Installed artifact Xorg_libICE 384.9 KiB  Installed artifact Cairo 2.2 MiB  Installed artifact libaom 6.4 MiB  Installed artifact FFMPEG 11.9 MiB  Installed artifact Glib 7.7 MiB  Installed artifact Xorg_libXfixes 59.0 KiB  Installed artifact Xorg_libXcursor 227.1 KiB  Installed artifact Qt6Declarative 23.8 MiB  Installed artifact Qt6Base 23.1 MiB  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Manifest.toml` Instantiating... === Precompiling... === PlutoUIJH Resolving... ===  Installed libfdk_aac_jll ─────────────── v2.0.4+0  Installed x265_jll ───────────────────── v4.1.0+0  Installed GR_jll ─────────────────────── v0.73.26+0  Installed Opus_jll ───────────────────── v1.6.1+0  Installed libdecor_jll ───────────────── v0.2.2+0  Installed Hyperscript ────────────────── v0.0.5  Installed GR ─────────────────────────── v0.73.26  Installed Contour ────────────────────── v0.6.3  Installed Libmount_jll ───────────────── v2.42.0+0  Installed Xorg_xcb_util_wm_jll ───────── v0.4.2+0  Installed Grisu ──────────────────────── v1.0.2  Installed JpegTurbo_jll ──────────────── v3.1.5+0  Installed LERC_jll ───────────────────── v4.1.0+0  Installed Xorg_xcb_util_image_jll ────── v0.4.1+0  Installed RecipesPipeline ────────────── v0.6.12  Installed DelimitedFiles ─────────────── v1.9.1  Installed Measures ───────────────────── v0.3.3  Installed Xorg_xkbcomp_jll ───────────── v1.4.7+0  Installed PlutoUI ────────────────────── v0.7.83  Installed Xorg_xcb_util_jll ──────────── v0.4.1+0  Installed PlotUtils ──────────────────── v1.4.4  Installed libva_jll ──────────────────── v2.23.0+0  Installed Cairo_jll ──────────────────── v1.18.7+0  Installed EpollShim_jll ──────────────── v0.0.20230411+1  Installed Statistics ─────────────────── v1.11.1  Installed Xorg_libSM_jll ─────────────── v1.2.6+0  Installed Missings ───────────────────── v1.2.0  Installed Showoff ────────────────────── v1.0.3  Installed ColorSchemes ───────────────── v3.31.0  Installed Xorg_libXinerama_jll ───────── v1.1.7+0  Installed Fontconfig_jll ─────────────── v2.17.1+0  Installed Xorg_libXau_jll ────────────── v1.0.13+0  Installed FFMPEG ─────────────────────── v0.4.5  Installed Xorg_libxkbfile_jll ────────── v1.2.0+0  Installed IrrationalConstants ────────── v0.2.6  Installed Xorg_xcb_util_keysyms_jll ──── v0.4.1+0  Installed xkbcommon_jll ──────────────── v1.13.0+0  Installed Bzip2_jll ──────────────────── v1.0.9+0  Installed HarfBuzz_jll ───────────────── v8.5.1+0  Installed PlotThemes ─────────────────── v3.3.0  Installed NaNMath ────────────────────── v1.1.4  Installed Xorg_libpciaccess_jll ──────── v0.19.0+0  Installed x264_jll ───────────────────── v10164.0.1+0  Installed UnicodeFun ─────────────────── v0.4.1  Installed Pango_jll ──────────────────── v1.57.1+0  Installed Colors ─────────────────────── v0.13.1  Installed PtrArrays ──────────────────── v1.4.0  Installed DataStructures ─────────────── v0.19.5  Installed XZ_jll ─────────────────────── v5.8.3+0  Installed StatsAPI ───────────────────── v1.8.0  Installed fzf_jll ────────────────────── v0.61.1+0  Installed GLFW_jll ───────────────────── v3.4.1+1  Installed mtdev_jll ──────────────────── v1.1.7+0  Installed FriBidi_jll ────────────────── v1.0.17+0  Installed libaom_jll ─────────────────── v3.13.3+0  Installed StatsBase ──────────────────── v0.34.11  Installed JLFzf ──────────────────────── v0.1.11  Installed Xorg_libxcb_jll ────────────── v1.17.1+0  Installed Qt6Svg_jll ─────────────────── v6.10.2+0  Installed eudev_jll ──────────────────── v3.2.14+0  Installed libpng_jll ─────────────────── v1.6.58+0  Installed FreeType2_jll ──────────────── v2.14.3+1  Installed TensorCore ─────────────────── v0.1.1  Installed Parsers ────────────────────── v2.8.5  Installed Dbus_jll ───────────────────── v1.16.2+0  Installed Format ─────────────────────── v1.3.7  Installed ColorVectorSpace ───────────── v0.11.0  Installed ColorTypes ─────────────────── v0.12.1  Installed libevdev_jll ───────────────── v1.13.4+0  Installed Plots ──────────────────────── v1.41.6  Installed Xorg_libXext_jll ───────────── v1.3.8+0  Installed Zstd_jll ───────────────────── v1.5.7+1  Installed libinput_jll ───────────────── v1.28.1+0  Installed Ogg_jll ────────────────────── v1.3.6+0  Installed Vulkan_Loader_jll ──────────── v1.3.243+0  Installed Libtiff_jll ────────────────── v4.7.2+0  Installed Reexport ───────────────────── v1.2.2  Installed LogExpFunctions ────────────── v1.0.1  Installed Libffi_jll ─────────────────── v3.4.7+0  Installed AliasTables ────────────────── v1.1.3  Installed Xorg_libXrender_jll ────────── v0.9.12+0  Installed Xorg_libXi_jll ─────────────── v1.8.3+0  Installed Qt6ShaderTools_jll ─────────── v6.10.2+1  Installed Ghostscript_jll ────────────── v9.55.1+0  Installed DocStringExtensions ────────── v0.9.5  Installed Xorg_xcb_util_renderutil_jll ─ v0.3.10+0  Installed Qt6Declarative_jll ─────────── v6.10.2+1  Installed Xorg_xcb_util_cursor_jll ───── v0.1.6+0  Installed Graphite2_jll ──────────────── v1.3.16+0  Installed libass_jll ─────────────────── v0.17.4+0  Installed MacroTools ─────────────────── v0.5.16  Installed Xorg_libXcursor_jll ────────── v1.2.4+0  Installed Wayland_jll ────────────────── v1.24.0+0  Installed Libuuid_jll ────────────────── v2.42.0+0  Installed Xorg_libICE_jll ────────────── v1.1.2+0  Installed Pixman_jll ─────────────────── v0.46.4+0  Installed StableRNGs ─────────────────── v1.0.4  Installed Latexify ───────────────────── v0.16.10  Installed Xorg_xkeyboard_config_jll ──── v2.47.0+1  Installed FixedPointNumbers ──────────── v0.8.6  Installed Xorg_xtrans_jll ────────────── v1.6.0+0  Installed libdrm_jll ─────────────────── v2.4.125+1  Installed RecipesBase ────────────────── v1.3.4  Installed LAME_jll ───────────────────── v3.100.3+0  Installed GettextRuntime_jll ─────────── v0.22.4+0  Installed FFMPEG_jll ─────────────────── v8.1.0+0  Installed LaTeXStrings ───────────────── v1.4.0  Installed IOCapture ──────────────────── v1.0.0  Installed libvorbis_jll ──────────────── v1.3.8+0  Installed Glib_jll ───────────────────── v2.86.3+0  Installed LLVMOpenMP_jll ─────────────── v18.1.8+0  Installed Xorg_libXrandr_jll ─────────── v1.5.6+0  Installed Unzip ──────────────────────── v0.2.0  Installed Xorg_libXfixes_jll ─────────── v6.0.2+0  Installed SortingAlgorithms ──────────── v1.2.2  Installed Qt6Wayland_jll ─────────────── v6.10.2+1  Installed Libglvnd_jll ───────────────── v1.7.1+1  Installed Qt6Base_jll ────────────────── v6.10.2+2  Installed PlutoTest ──────────────────── v0.2.4  Installed Xorg_libX11_jll ────────────── v1.8.13+0  Installed Xorg_libXdmcp_jll ──────────── v1.1.6+0  Installing 76 artifacts  Installed artifact Xorg_xcb_util_renderutil 32.1 KiB  Installed artifact libass 427.8 KiB  Installed artifact Bzip2 503.5 KiB  Installed artifact LERC 267.4 KiB  Installed artifact Pixman 390.8 KiB  Installed artifact Xorg_libpciaccess 26.2 KiB  Installed artifact Xorg_xtrans 48.2 KiB  Installed artifact Xorg_xcb_util_wm 147.9 KiB  Installed artifact mtdev 79.6 KiB  Installed artifact Qt6ShaderTools 2.1 MiB  Installed artifact Xorg_xkeyboard_config 545.0 KiB  Installed artifact Qt6Wayland 1.4 MiB  Installed artifact Xorg_libXi 1.6 MiB  Installed artifact Libmount 6.9 MiB  Installed artifact libinput 1.0 MiB  Installed artifact GLFW 187.9 KiB  Installed artifact Libglvnd 833.5 KiB  Installed artifact eudev 3.9 MiB  Installed artifact Libtiff 2.3 MiB  Installed artifact Xorg_libSM 164.1 KiB  Installed artifact libpng 329.4 KiB  Installed artifact Xorg_xcb_util 39.5 KiB  Installed artifact Vulkan_Loader 177.3 KiB  Installed artifact libdecor 44.1 KiB  Installed artifact Dbus 489.8 KiB  Installed artifact Xorg_libXau 36.6 KiB  Installed artifact Pango 2.1 MiB  Installed artifact libfdk_aac 2.8 MiB  Installed artifact Libuuid 3.9 MiB  Installed artifact Xorg_xcb_util_cursor 27.6 KiB  Installed artifact Xorg_libXext 286.6 KiB  Installed artifact Xorg_xcb_util_image 54.3 KiB  Installed artifact xkbcommon 298.7 KiB  Installed artifact x264 2.1 MiB  Installed artifact JpegTurbo 1.4 MiB  Installed artifact GettextRuntime 543.0 KiB  Installed artifact XZ 1.6 MiB  Installed artifact Xorg_xcb_util_keysyms 17.8 KiB  Installed artifact fzf 3.3 MiB  Installed artifact FriBidi 78.9 KiB  Installed artifact Graphite2 120.3 KiB  Installed artifact x265 1.4 MiB  Installed artifact Ghostscript 31.4 MiB  Installed artifact Xorg_libXinerama 25.5 KiB  Installed artifact GR 19.3 MiB  Installed artifact Xorg_libX11 4.8 MiB  Installed artifact Zstd 646.0 KiB  Installed artifact Xorg_libXdmcp 67.7 KiB  Installed artifact HarfBuzz 1.7 MiB  Installed artifact Xorg_libXrandr 97.3 KiB  Installed artifact Xorg_libxkbfile 91.1 KiB  Installed artifact Xorg_xkbcomp 274.8 KiB  Installed artifact Xorg_libXrender 435.9 KiB  Installed artifact libvorbis 271.0 KiB  Installed artifact Qt6Svg 386.4 KiB  Installed artifact Wayland 390.6 KiB  Installed artifact libevdev 117.1 KiB  Installed artifact Libffi 44.2 KiB  Installed artifact Opus 960.9 KiB  Installed artifact Xorg_libxcb 2.1 MiB  Installed artifact FreeType2 1.5 MiB  Installed artifact LAME 292.5 KiB  Installed artifact libdrm 368.0 KiB  Installed artifact libva 235.7 KiB  Installed artifact Fontconfig 984.8 KiB  Installed artifact Ogg 250.3 KiB  Installed artifact LLVMOpenMP 661.6 KiB  Installed artifact Xorg_libICE 384.9 KiB  Installed artifact Cairo 2.2 MiB  Installed artifact libaom 6.4 MiB  Installed artifact FFMPEG 11.9 MiB  Installed artifact Glib 7.7 MiB  Installed artifact Xorg_libXfixes 59.0 KiB  Installed artifact Xorg_libXcursor 227.1 KiB  Installed artifact Qt6Declarative 23.8 MiB  Installed artifact Qt6Base 23.1 MiB  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Manifest.toml` Instantiating... === Precompiling... === PlotsJH Resolving... ===  Installed libfdk_aac_jll ─────────────── v2.0.4+0  Installed x265_jll ───────────────────── v4.1.0+0  Installed GR_jll ─────────────────────── v0.73.26+0  Installed Opus_jll ───────────────────── v1.6.1+0  Installed libdecor_jll ───────────────── v0.2.2+0  Installed Hyperscript ────────────────── v0.0.5  Installed GR ─────────────────────────── v0.73.26  Installed Contour ────────────────────── v0.6.3  Installed Libmount_jll ───────────────── v2.42.0+0  Installed Xorg_xcb_util_wm_jll ───────── v0.4.2+0  Installed Grisu ──────────────────────── v1.0.2  Installed JpegTurbo_jll ──────────────── v3.1.5+0  Installed LERC_jll ───────────────────── v4.1.0+0  Installed Xorg_xcb_util_image_jll ────── v0.4.1+0  Installed RecipesPipeline ────────────── v0.6.12  Installed DelimitedFiles ─────────────── v1.9.1  Installed Measures ───────────────────── v0.3.3  Installed Xorg_xkbcomp_jll ───────────── v1.4.7+0  Installed PlutoUI ────────────────────── v0.7.83  Installed Xorg_xcb_util_jll ──────────── v0.4.1+0  Installed PlotUtils ──────────────────── v1.4.4  Installed libva_jll ──────────────────── v2.23.0+0  Installed Cairo_jll ──────────────────── v1.18.7+0  Installed EpollShim_jll ──────────────── v0.0.20230411+1  Installed Statistics ─────────────────── v1.11.1  Installed Xorg_libSM_jll ─────────────── v1.2.6+0  Installed Missings ───────────────────── v1.2.0  Installed Showoff ────────────────────── v1.0.3  Installed ColorSchemes ───────────────── v3.31.0  Installed Xorg_libXinerama_jll ───────── v1.1.7+0  Installed Fontconfig_jll ─────────────── v2.17.1+0  Installed Xorg_libXau_jll ────────────── v1.0.13+0  Installed FFMPEG ─────────────────────── v0.4.5  Installed Xorg_libxkbfile_jll ────────── v1.2.0+0  Installed IrrationalConstants ────────── v0.2.6  Installed Xorg_xcb_util_keysyms_jll ──── v0.4.1+0  Installed xkbcommon_jll ──────────────── v1.13.0+0  Installed Bzip2_jll ──────────────────── v1.0.9+0  Installed HarfBuzz_jll ───────────────── v8.5.1+0  Installed PlotThemes ─────────────────── v3.3.0  Installed NaNMath ────────────────────── v1.1.4  Installed Xorg_libpciaccess_jll ──────── v0.19.0+0  Installed x264_jll ───────────────────── v10164.0.1+0  Installed UnicodeFun ─────────────────── v0.4.1  Installed Pango_jll ──────────────────── v1.57.1+0  Installed Colors ─────────────────────── v0.13.1  Installed PtrArrays ──────────────────── v1.4.0  Installed DataStructures ─────────────── v0.19.5  Installed XZ_jll ─────────────────────── v5.8.3+0  Installed StatsAPI ───────────────────── v1.8.0  Installed fzf_jll ────────────────────── v0.61.1+0  Installed GLFW_jll ───────────────────── v3.4.1+1  Installed mtdev_jll ──────────────────── v1.1.7+0  Installed FriBidi_jll ────────────────── v1.0.17+0  Installed libaom_jll ─────────────────── v3.13.3+0  Installed StatsBase ──────────────────── v0.34.11  Installed JLFzf ──────────────────────── v0.1.11  Installed Xorg_libxcb_jll ────────────── v1.17.1+0  Installed Qt6Svg_jll ─────────────────── v6.10.2+0  Installed eudev_jll ──────────────────── v3.2.14+0  Installed libpng_jll ─────────────────── v1.6.58+0  Installed FreeType2_jll ──────────────── v2.14.3+1  Installed TensorCore ─────────────────── v0.1.1  Installed Parsers ────────────────────── v2.8.5  Installed Dbus_jll ───────────────────── v1.16.2+0  Installed Format ─────────────────────── v1.3.7  Installed ColorVectorSpace ───────────── v0.11.0  Installed ColorTypes ─────────────────── v0.12.1  Installed libevdev_jll ───────────────── v1.13.4+0  Installed Plots ──────────────────────── v1.41.6  Installed Xorg_libXext_jll ───────────── v1.3.8+0  Installed Zstd_jll ───────────────────── v1.5.7+1  Installed libinput_jll ───────────────── v1.28.1+0  Installed Ogg_jll ────────────────────── v1.3.6+0  Installed Vulkan_Loader_jll ──────────── v1.3.243+0  Installed Libtiff_jll ────────────────── v4.7.2+0  Installed Reexport ───────────────────── v1.2.2  Installed LogExpFunctions ────────────── v1.0.1  Installed Libffi_jll ─────────────────── v3.4.7+0  Installed AliasTables ────────────────── v1.1.3  Installed Xorg_libXrender_jll ────────── v0.9.12+0  Installed Xorg_libXi_jll ─────────────── v1.8.3+0  Installed Qt6ShaderTools_jll ─────────── v6.10.2+1  Installed Ghostscript_jll ────────────── v9.55.1+0  Installed DocStringExtensions ────────── v0.9.5  Installed Xorg_xcb_util_renderutil_jll ─ v0.3.10+0  Installed Qt6Declarative_jll ─────────── v6.10.2+1  Installed Xorg_xcb_util_cursor_jll ───── v0.1.6+0  Installed Graphite2_jll ──────────────── v1.3.16+0  Installed libass_jll ─────────────────── v0.17.4+0  Installed MacroTools ─────────────────── v0.5.16  Installed Xorg_libXcursor_jll ────────── v1.2.4+0  Installed Wayland_jll ────────────────── v1.24.0+0  Installed Libuuid_jll ────────────────── v2.42.0+0  Installed Xorg_libICE_jll ────────────── v1.1.2+0  Installed Pixman_jll ─────────────────── v0.46.4+0  Installed StableRNGs ─────────────────── v1.0.4  Installed Latexify ───────────────────── v0.16.10  Installed Xorg_xkeyboard_config_jll ──── v2.47.0+1  Installed FixedPointNumbers ──────────── v0.8.6  Installed Xorg_xtrans_jll ────────────── v1.6.0+0  Installed libdrm_jll ─────────────────── v2.4.125+1  Installed RecipesBase ────────────────── v1.3.4  Installed LAME_jll ───────────────────── v3.100.3+0  Installed GettextRuntime_jll ─────────── v0.22.4+0  Installed FFMPEG_jll ─────────────────── v8.1.0+0  Installed LaTeXStrings ───────────────── v1.4.0  Installed IOCapture ──────────────────── v1.0.0  Installed libvorbis_jll ──────────────── v1.3.8+0  Installed Glib_jll ───────────────────── v2.86.3+0  Installed LLVMOpenMP_jll ─────────────── v18.1.8+0  Installed Xorg_libXrandr_jll ─────────── v1.5.6+0  Installed Unzip ──────────────────────── v0.2.0  Installed Xorg_libXfixes_jll ─────────── v6.0.2+0  Installed SortingAlgorithms ──────────── v1.2.2  Installed Qt6Wayland_jll ─────────────── v6.10.2+1  Installed Libglvnd_jll ───────────────── v1.7.1+1  Installed Qt6Base_jll ────────────────── v6.10.2+2  Installed PlutoTest ──────────────────── v0.2.4  Installed Xorg_libX11_jll ────────────── v1.8.13+0  Installed Xorg_libXdmcp_jll ──────────── v1.1.6+0  Installing 76 artifacts  Installed artifact Xorg_xcb_util_renderutil 32.1 KiB  Installed artifact libass 427.8 KiB  Installed artifact Bzip2 503.5 KiB  Installed artifact LERC 267.4 KiB  Installed artifact Pixman 390.8 KiB  Installed artifact Xorg_libpciaccess 26.2 KiB  Installed artifact Xorg_xtrans 48.2 KiB  Installed artifact Xorg_xcb_util_wm 147.9 KiB  Installed artifact mtdev 79.6 KiB  Installed artifact Qt6ShaderTools 2.1 MiB  Installed artifact Xorg_xkeyboard_config 545.0 KiB  Installed artifact Qt6Wayland 1.4 MiB  Installed artifact Xorg_libXi 1.6 MiB  Installed artifact Libmount 6.9 MiB  Installed artifact libinput 1.0 MiB  Installed artifact GLFW 187.9 KiB  Installed artifact Libglvnd 833.5 KiB  Installed artifact eudev 3.9 MiB  Installed artifact Libtiff 2.3 MiB  Installed artifact Xorg_libSM 164.1 KiB  Installed artifact libpng 329.4 KiB  Installed artifact Xorg_xcb_util 39.5 KiB  Installed artifact Vulkan_Loader 177.3 KiB  Installed artifact libdecor 44.1 KiB  Installed artifact Dbus 489.8 KiB  Installed artifact Xorg_libXau 36.6 KiB  Installed artifact Pango 2.1 MiB  Installed artifact libfdk_aac 2.8 MiB  Installed artifact Libuuid 3.9 MiB  Installed artifact Xorg_xcb_util_cursor 27.6 KiB  Installed artifact Xorg_libXext 286.6 KiB  Installed artifact Xorg_xcb_util_image 54.3 KiB  Installed artifact xkbcommon 298.7 KiB  Installed artifact x264 2.1 MiB  Installed artifact JpegTurbo 1.4 MiB  Installed artifact GettextRuntime 543.0 KiB  Installed artifact XZ 1.6 MiB  Installed artifact Xorg_xcb_util_keysyms 17.8 KiB  Installed artifact fzf 3.3 MiB  Installed artifact FriBidi 78.9 KiB  Installed artifact Graphite2 120.3 KiB  Installed artifact x265 1.4 MiB  Installed artifact Ghostscript 31.4 MiB  Installed artifact Xorg_libXinerama 25.5 KiB  Installed artifact GR 19.3 MiB  Installed artifact Xorg_libX11 4.8 MiB  Installed artifact Zstd 646.0 KiB  Installed artifact Xorg_libXdmcp 67.7 KiB  Installed artifact HarfBuzz 1.7 MiB  Installed artifact Xorg_libXrandr 97.3 KiB  Installed artifact Xorg_libxkbfile 91.1 KiB  Installed artifact Xorg_xkbcomp 274.8 KiB  Installed artifact Xorg_libXrender 435.9 KiB  Installed artifact libvorbis 271.0 KiB  Installed artifact Qt6Svg 386.4 KiB  Installed artifact Wayland 390.6 KiB  Installed artifact libevdev 117.1 KiB  Installed artifact Libffi 44.2 KiB  Installed artifact Opus 960.9 KiB  Installed artifact Xorg_libxcb 2.1 MiB  Installed artifact FreeType2 1.5 MiB  Installed artifact LAME 292.5 KiB  Installed artifact libdrm 368.0 KiB  Installed artifact libva 235.7 KiB  Installed artifact Fontconfig 984.8 KiB  Installed artifact Ogg 250.3 KiB  Installed artifact LLVMOpenMP 661.6 KiB  Installed artifact Xorg_libICE 384.9 KiB  Installed artifact Cairo 2.2 MiB  Installed artifact libaom 6.4 MiB  Installed artifact FFMPEG 11.9 MiB  Installed artifact Glib 7.7 MiB  Installed artifact Xorg_libXfixes 59.0 KiB  Installed artifact Xorg_libXcursor 227.1 KiB  Installed artifact Qt6Declarative 23.8 MiB  Installed artifact Qt6Base 23.1 MiB  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Manifest.toml` Instantiating... === Precompiling... === HypertextLiteralJH Resolving... ===  Installed libfdk_aac_jll ─────────────── v2.0.4+0  Installed x265_jll ───────────────────── v4.1.0+0  Installed GR_jll ─────────────────────── v0.73.26+0  Installed Opus_jll ───────────────────── v1.6.1+0  Installed libdecor_jll ───────────────── v0.2.2+0  Installed Hyperscript ────────────────── v0.0.5  Installed GR ─────────────────────────── v0.73.26  Installed Contour ────────────────────── v0.6.3  Installed Libmount_jll ───────────────── v2.42.0+0  Installed Xorg_xcb_util_wm_jll ───────── v0.4.2+0  Installed Grisu ──────────────────────── v1.0.2  Installed JpegTurbo_jll ──────────────── v3.1.5+0  Installed LERC_jll ───────────────────── v4.1.0+0  Installed Xorg_xcb_util_image_jll ────── v0.4.1+0  Installed RecipesPipeline ────────────── v0.6.12  Installed DelimitedFiles ─────────────── v1.9.1  Installed Measures ───────────────────── v0.3.3  Installed Xorg_xkbcomp_jll ───────────── v1.4.7+0  Installed PlutoUI ────────────────────── v0.7.83  Installed Xorg_xcb_util_jll ──────────── v0.4.1+0  Installed PlotUtils ──────────────────── v1.4.4  Installed libva_jll ──────────────────── v2.23.0+0  Installed Cairo_jll ──────────────────── v1.18.7+0  Installed EpollShim_jll ──────────────── v0.0.20230411+1  Installed Statistics ─────────────────── v1.11.1  Installed Xorg_libSM_jll ─────────────── v1.2.6+0  Installed Missings ───────────────────── v1.2.0  Installed Showoff ────────────────────── v1.0.3  Installed ColorSchemes ───────────────── v3.31.0  Installed Xorg_libXinerama_jll ───────── v1.1.7+0  Installed Fontconfig_jll ─────────────── v2.17.1+0  Installed Xorg_libXau_jll ────────────── v1.0.13+0  Installed FFMPEG ─────────────────────── v0.4.5  Installed Xorg_libxkbfile_jll ────────── v1.2.0+0  Installed IrrationalConstants ────────── v0.2.6  Installed Xorg_xcb_util_keysyms_jll ──── v0.4.1+0  Installed xkbcommon_jll ──────────────── v1.13.0+0  Installed Bzip2_jll ──────────────────── v1.0.9+0  Installed HarfBuzz_jll ───────────────── v8.5.1+0  Installed PlotThemes ─────────────────── v3.3.0  Installed NaNMath ────────────────────── v1.1.4  Installed Xorg_libpciaccess_jll ──────── v0.19.0+0  Installed x264_jll ───────────────────── v10164.0.1+0  Installed UnicodeFun ─────────────────── v0.4.1  Installed Pango_jll ──────────────────── v1.57.1+0  Installed Colors ─────────────────────── v0.13.1  Installed PtrArrays ──────────────────── v1.4.0  Installed DataStructures ─────────────── v0.19.5  Installed XZ_jll ─────────────────────── v5.8.3+0  Installed StatsAPI ───────────────────── v1.8.0  Installed fzf_jll ────────────────────── v0.61.1+0  Installed GLFW_jll ───────────────────── v3.4.1+1  Installed mtdev_jll ──────────────────── v1.1.7+0  Installed FriBidi_jll ────────────────── v1.0.17+0  Installed libaom_jll ─────────────────── v3.13.3+0  Installed StatsBase ──────────────────── v0.34.11  Installed JLFzf ──────────────────────── v0.1.11  Installed Xorg_libxcb_jll ────────────── v1.17.1+0  Installed Qt6Svg_jll ─────────────────── v6.10.2+0  Installed eudev_jll ──────────────────── v3.2.14+0  Installed libpng_jll ─────────────────── v1.6.58+0  Installed FreeType2_jll ──────────────── v2.14.3+1  Installed TensorCore ─────────────────── v0.1.1  Installed Parsers ────────────────────── v2.8.5  Installed Dbus_jll ───────────────────── v1.16.2+0  Installed Format ─────────────────────── v1.3.7  Installed ColorVectorSpace ───────────── v0.11.0  Installed ColorTypes ─────────────────── v0.12.1  Installed libevdev_jll ───────────────── v1.13.4+0  Installed Plots ──────────────────────── v1.41.6  Installed Xorg_libXext_jll ───────────── v1.3.8+0  Installed Zstd_jll ───────────────────── v1.5.7+1  Installed libinput_jll ───────────────── v1.28.1+0  Installed Ogg_jll ────────────────────── v1.3.6+0  Installed Vulkan_Loader_jll ──────────── v1.3.243+0  Installed Libtiff_jll ────────────────── v4.7.2+0  Installed Reexport ───────────────────── v1.2.2  Installed LogExpFunctions ────────────── v1.0.1  Installed Libffi_jll ─────────────────── v3.4.7+0  Installed AliasTables ────────────────── v1.1.3  Installed Xorg_libXrender_jll ────────── v0.9.12+0  Installed Xorg_libXi_jll ─────────────── v1.8.3+0  Installed Qt6ShaderTools_jll ─────────── v6.10.2+1  Installed Ghostscript_jll ────────────── v9.55.1+0  Installed DocStringExtensions ────────── v0.9.5  Installed Xorg_xcb_util_renderutil_jll ─ v0.3.10+0  Installed Qt6Declarative_jll ─────────── v6.10.2+1  Installed Xorg_xcb_util_cursor_jll ───── v0.1.6+0  Installed Graphite2_jll ──────────────── v1.3.16+0  Installed libass_jll ─────────────────── v0.17.4+0  Installed MacroTools ─────────────────── v0.5.16  Installed Xorg_libXcursor_jll ────────── v1.2.4+0  Installed Wayland_jll ────────────────── v1.24.0+0  Installed Libuuid_jll ────────────────── v2.42.0+0  Installed Xorg_libICE_jll ────────────── v1.1.2+0  Installed Pixman_jll ─────────────────── v0.46.4+0  Installed StableRNGs ─────────────────── v1.0.4  Installed Latexify ───────────────────── v0.16.10  Installed Xorg_xkeyboard_config_jll ──── v2.47.0+1  Installed FixedPointNumbers ──────────── v0.8.6  Installed Xorg_xtrans_jll ────────────── v1.6.0+0  Installed libdrm_jll ─────────────────── v2.4.125+1  Installed RecipesBase ────────────────── v1.3.4  Installed LAME_jll ───────────────────── v3.100.3+0  Installed GettextRuntime_jll ─────────── v0.22.4+0  Installed FFMPEG_jll ─────────────────── v8.1.0+0  Installed LaTeXStrings ───────────────── v1.4.0  Installed IOCapture ──────────────────── v1.0.0  Installed libvorbis_jll ──────────────── v1.3.8+0  Installed Glib_jll ───────────────────── v2.86.3+0  Installed LLVMOpenMP_jll ─────────────── v18.1.8+0  Installed Xorg_libXrandr_jll ─────────── v1.5.6+0  Installed Unzip ──────────────────────── v0.2.0  Installed Xorg_libXfixes_jll ─────────── v6.0.2+0  Installed SortingAlgorithms ──────────── v1.2.2  Installed Qt6Wayland_jll ─────────────── v6.10.2+1  Installed Libglvnd_jll ───────────────── v1.7.1+1  Installed Qt6Base_jll ────────────────── v6.10.2+2  Installed PlutoTest ──────────────────── v0.2.4  Installed Xorg_libX11_jll ────────────── v1.8.13+0  Installed Xorg_libXdmcp_jll ──────────── v1.1.6+0  Installing 76 artifacts  Installed artifact Xorg_xcb_util_renderutil 32.1 KiB  Installed artifact libass 427.8 KiB  Installed artifact Bzip2 503.5 KiB  Installed artifact LERC 267.4 KiB  Installed artifact Pixman 390.8 KiB  Installed artifact Xorg_libpciaccess 26.2 KiB  Installed artifact Xorg_xtrans 48.2 KiB  Installed artifact Xorg_xcb_util_wm 147.9 KiB  Installed artifact mtdev 79.6 KiB  Installed artifact Qt6ShaderTools 2.1 MiB  Installed artifact Xorg_xkeyboard_config 545.0 KiB  Installed artifact Qt6Wayland 1.4 MiB  Installed artifact Xorg_libXi 1.6 MiB  Installed artifact Libmount 6.9 MiB  Installed artifact libinput 1.0 MiB  Installed artifact GLFW 187.9 KiB  Installed artifact Libglvnd 833.5 KiB  Installed artifact eudev 3.9 MiB  Installed artifact Libtiff 2.3 MiB  Installed artifact Xorg_libSM 164.1 KiB  Installed artifact libpng 329.4 KiB  Installed artifact Xorg_xcb_util 39.5 KiB  Installed artifact Vulkan_Loader 177.3 KiB  Installed artifact libdecor 44.1 KiB  Installed artifact Dbus 489.8 KiB  Installed artifact Xorg_libXau 36.6 KiB  Installed artifact Pango 2.1 MiB  Installed artifact libfdk_aac 2.8 MiB  Installed artifact Libuuid 3.9 MiB  Installed artifact Xorg_xcb_util_cursor 27.6 KiB  Installed artifact Xorg_libXext 286.6 KiB  Installed artifact Xorg_xcb_util_image 54.3 KiB  Installed artifact xkbcommon 298.7 KiB  Installed artifact x264 2.1 MiB  Installed artifact JpegTurbo 1.4 MiB  Installed artifact GettextRuntime 543.0 KiB  Installed artifact XZ 1.6 MiB  Installed artifact Xorg_xcb_util_keysyms 17.8 KiB  Installed artifact fzf 3.3 MiB  Installed artifact FriBidi 78.9 KiB  Installed artifact Graphite2 120.3 KiB  Installed artifact x265 1.4 MiB  Installed artifact Ghostscript 31.4 MiB  Installed artifact Xorg_libXinerama 25.5 KiB  Installed artifact GR 19.3 MiB  Installed artifact Xorg_libX11 4.8 MiB  Installed artifact Zstd 646.0 KiB  Installed artifact Xorg_libXdmcp 67.7 KiB  Installed artifact HarfBuzz 1.7 MiB  Installed artifact Xorg_libXrandr 97.3 KiB  Installed artifact Xorg_libxkbfile 91.1 KiB  Installed artifact Xorg_xkbcomp 274.8 KiB  Installed artifact Xorg_libXrender 435.9 KiB  Installed artifact libvorbis 271.0 KiB  Installed artifact Qt6Svg 386.4 KiB  Installed artifact Wayland 390.6 KiB  Installed artifact libevdev 117.1 KiB  Installed artifact Libffi 44.2 KiB  Installed artifact Opus 960.9 KiB  Installed artifact Xorg_libxcb 2.1 MiB  Installed artifact FreeType2 1.5 MiB  Installed artifact LAME 292.5 KiB  Installed artifact libdrm 368.0 KiB  Installed artifact libva 235.7 KiB  Installed artifact Fontconfig 984.8 KiB  Installed artifact Ogg 250.3 KiB  Installed artifact LLVMOpenMP 661.6 KiB  Installed artifact Xorg_libICE 384.9 KiB  Installed artifact Cairo 2.2 MiB  Installed artifact libaom 6.4 MiB  Installed artifact FFMPEG 11.9 MiB  Installed artifact Glib 7.7 MiB  Installed artifact Xorg_libXfixes 59.0 KiB  Installed artifact Xorg_libXcursor 227.1 KiB  Installed artifact Qt6Declarative 23.8 MiB  Installed artifact Qt6Base 23.1 MiB  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Manifest.toml` Instantiating... === Precompiling... === AbstractTreesJH Resolving... ===  Installed libfdk_aac_jll ─────────────── v2.0.4+0  Installed x265_jll ───────────────────── v4.1.0+0  Installed GR_jll ─────────────────────── v0.73.26+0  Installed Opus_jll ───────────────────── v1.6.1+0  Installed libdecor_jll ───────────────── v0.2.2+0  Installed Hyperscript ────────────────── v0.0.5  Installed GR ─────────────────────────── v0.73.26  Installed Contour ────────────────────── v0.6.3  Installed Libmount_jll ───────────────── v2.42.0+0  Installed Xorg_xcb_util_wm_jll ───────── v0.4.2+0  Installed Grisu ──────────────────────── v1.0.2  Installed JpegTurbo_jll ──────────────── v3.1.5+0  Installed LERC_jll ───────────────────── v4.1.0+0  Installed Xorg_xcb_util_image_jll ────── v0.4.1+0  Installed RecipesPipeline ────────────── v0.6.12  Installed DelimitedFiles ─────────────── v1.9.1  Installed Measures ───────────────────── v0.3.3  Installed Xorg_xkbcomp_jll ───────────── v1.4.7+0  Installed PlutoUI ────────────────────── v0.7.83  Installed Xorg_xcb_util_jll ──────────── v0.4.1+0  Installed PlotUtils ──────────────────── v1.4.4  Installed libva_jll ──────────────────── v2.23.0+0  Installed Cairo_jll ──────────────────── v1.18.7+0  Installed EpollShim_jll ──────────────── v0.0.20230411+1  Installed Statistics ─────────────────── v1.11.1  Installed Xorg_libSM_jll ─────────────── v1.2.6+0  Installed Missings ───────────────────── v1.2.0  Installed Showoff ────────────────────── v1.0.3  Installed ColorSchemes ───────────────── v3.31.0  Installed Xorg_libXinerama_jll ───────── v1.1.7+0  Installed Fontconfig_jll ─────────────── v2.17.1+0  Installed Xorg_libXau_jll ────────────── v1.0.13+0  Installed FFMPEG ─────────────────────── v0.4.5  Installed Xorg_libxkbfile_jll ────────── v1.2.0+0  Installed IrrationalConstants ────────── v0.2.6  Installed Xorg_xcb_util_keysyms_jll ──── v0.4.1+0  Installed xkbcommon_jll ──────────────── v1.13.0+0  Installed Bzip2_jll ──────────────────── v1.0.9+0  Installed HarfBuzz_jll ───────────────── v8.5.1+0  Installed PlotThemes ─────────────────── v3.3.0  Installed NaNMath ────────────────────── v1.1.4  Installed Xorg_libpciaccess_jll ──────── v0.19.0+0  Installed x264_jll ───────────────────── v10164.0.1+0  Installed UnicodeFun ─────────────────── v0.4.1  Installed Pango_jll ──────────────────── v1.57.1+0  Installed Colors ─────────────────────── v0.13.1  Installed PtrArrays ──────────────────── v1.4.0  Installed DataStructures ─────────────── v0.19.5  Installed XZ_jll ─────────────────────── v5.8.3+0  Installed StatsAPI ───────────────────── v1.8.0  Installed fzf_jll ────────────────────── v0.61.1+0  Installed GLFW_jll ───────────────────── v3.4.1+1  Installed mtdev_jll ──────────────────── v1.1.7+0  Installed FriBidi_jll ────────────────── v1.0.17+0  Installed libaom_jll ─────────────────── v3.13.3+0  Installed StatsBase ──────────────────── v0.34.11  Installed JLFzf ──────────────────────── v0.1.11  Installed Xorg_libxcb_jll ────────────── v1.17.1+0  Installed Qt6Svg_jll ─────────────────── v6.10.2+0  Installed eudev_jll ──────────────────── v3.2.14+0  Installed libpng_jll ─────────────────── v1.6.58+0  Installed FreeType2_jll ──────────────── v2.14.3+1  Installed TensorCore ─────────────────── v0.1.1  Installed Parsers ────────────────────── v2.8.5  Installed Dbus_jll ───────────────────── v1.16.2+0  Installed Format ─────────────────────── v1.3.7  Installed ColorVectorSpace ───────────── v0.11.0  Installed ColorTypes ─────────────────── v0.12.1  Installed libevdev_jll ───────────────── v1.13.4+0  Installed Plots ──────────────────────── v1.41.6  Installed Xorg_libXext_jll ───────────── v1.3.8+0  Installed Zstd_jll ───────────────────── v1.5.7+1  Installed libinput_jll ───────────────── v1.28.1+0  Installed Ogg_jll ────────────────────── v1.3.6+0  Installed Vulkan_Loader_jll ──────────── v1.3.243+0  Installed Libtiff_jll ────────────────── v4.7.2+0  Installed Reexport ───────────────────── v1.2.2  Installed LogExpFunctions ────────────── v1.0.1  Installed Libffi_jll ─────────────────── v3.4.7+0  Installed AliasTables ────────────────── v1.1.3  Installed Xorg_libXrender_jll ────────── v0.9.12+0  Installed Xorg_libXi_jll ─────────────── v1.8.3+0  Installed Qt6ShaderTools_jll ─────────── v6.10.2+1  Installed Ghostscript_jll ────────────── v9.55.1+0  Installed DocStringExtensions ────────── v0.9.5  Installed Xorg_xcb_util_renderutil_jll ─ v0.3.10+0  Installed Qt6Declarative_jll ─────────── v6.10.2+1  Installed Xorg_xcb_util_cursor_jll ───── v0.1.6+0  Installed Graphite2_jll ──────────────── v1.3.16+0  Installed libass_jll ─────────────────── v0.17.4+0  Installed MacroTools ─────────────────── v0.5.16  Installed Xorg_libXcursor_jll ────────── v1.2.4+0  Installed Wayland_jll ────────────────── v1.24.0+0  Installed Libuuid_jll ────────────────── v2.42.0+0  Installed Xorg_libICE_jll ────────────── v1.1.2+0  Installed Pixman_jll ─────────────────── v0.46.4+0  Installed StableRNGs ─────────────────── v1.0.4  Installed Latexify ───────────────────── v0.16.10  Installed Xorg_xkeyboard_config_jll ──── v2.47.0+1  Installed FixedPointNumbers ──────────── v0.8.6  Installed Xorg_xtrans_jll ────────────── v1.6.0+0  Installed libdrm_jll ─────────────────── v2.4.125+1  Installed RecipesBase ────────────────── v1.3.4  Installed LAME_jll ───────────────────── v3.100.3+0  Installed GettextRuntime_jll ─────────── v0.22.4+0  Installed FFMPEG_jll ─────────────────── v8.1.0+0  Installed LaTeXStrings ───────────────── v1.4.0  Installed IOCapture ──────────────────── v1.0.0  Installed libvorbis_jll ──────────────── v1.3.8+0  Installed Glib_jll ───────────────────── v2.86.3+0  Installed LLVMOpenMP_jll ─────────────── v18.1.8+0  Installed Xorg_libXrandr_jll ─────────── v1.5.6+0  Installed Unzip ──────────────────────── v0.2.0  Installed Xorg_libXfixes_jll ─────────── v6.0.2+0  Installed SortingAlgorithms ──────────── v1.2.2  Installed Qt6Wayland_jll ─────────────── v6.10.2+1  Installed Libglvnd_jll ───────────────── v1.7.1+1  Installed Qt6Base_jll ────────────────── v6.10.2+2  Installed PlutoTest ──────────────────── v0.2.4  Installed Xorg_libX11_jll ────────────── v1.8.13+0  Installed Xorg_libXdmcp_jll ──────────── v1.1.6+0  Installing 76 artifacts  Installed artifact Xorg_xcb_util_renderutil 32.1 KiB  Installed artifact libass 427.8 KiB  Installed artifact Bzip2 503.5 KiB  Installed artifact LERC 267.4 KiB  Installed artifact Pixman 390.8 KiB  Installed artifact Xorg_libpciaccess 26.2 KiB  Installed artifact Xorg_xtrans 48.2 KiB  Installed artifact Xorg_xcb_util_wm 147.9 KiB  Installed artifact mtdev 79.6 KiB  Installed artifact Qt6ShaderTools 2.1 MiB  Installed artifact Xorg_xkeyboard_config 545.0 KiB  Installed artifact Qt6Wayland 1.4 MiB  Installed artifact Xorg_libXi 1.6 MiB  Installed artifact Libmount 6.9 MiB  Installed artifact libinput 1.0 MiB  Installed artifact GLFW 187.9 KiB  Installed artifact Libglvnd 833.5 KiB  Installed artifact eudev 3.9 MiB  Installed artifact Libtiff 2.3 MiB  Installed artifact Xorg_libSM 164.1 KiB  Installed artifact libpng 329.4 KiB  Installed artifact Xorg_xcb_util 39.5 KiB  Installed artifact Vulkan_Loader 177.3 KiB  Installed artifact libdecor 44.1 KiB  Installed artifact Dbus 489.8 KiB  Installed artifact Xorg_libXau 36.6 KiB  Installed artifact Pango 2.1 MiB  Installed artifact libfdk_aac 2.8 MiB  Installed artifact Libuuid 3.9 MiB  Installed artifact Xorg_xcb_util_cursor 27.6 KiB  Installed artifact Xorg_libXext 286.6 KiB  Installed artifact Xorg_xcb_util_image 54.3 KiB  Installed artifact xkbcommon 298.7 KiB  Installed artifact x264 2.1 MiB  Installed artifact JpegTurbo 1.4 MiB  Installed artifact GettextRuntime 543.0 KiB  Installed artifact XZ 1.6 MiB  Installed artifact Xorg_xcb_util_keysyms 17.8 KiB  Installed artifact fzf 3.3 MiB  Installed artifact FriBidi 78.9 KiB  Installed artifact Graphite2 120.3 KiB  Installed artifact x265 1.4 MiB  Installed artifact Ghostscript 31.4 MiB  Installed artifact Xorg_libXinerama 25.5 KiB  Installed artifact GR 19.3 MiB  Installed artifact Xorg_libX11 4.8 MiB  Installed artifact Zstd 646.0 KiB  Installed artifact Xorg_libXdmcp 67.7 KiB  Installed artifact HarfBuzz 1.7 MiB  Installed artifact Xorg_libXrandr 97.3 KiB  Installed artifact Xorg_libxkbfile 91.1 KiB  Installed artifact Xorg_xkbcomp 274.8 KiB  Installed artifact Xorg_libXrender 435.9 KiB  Installed artifact libvorbis 271.0 KiB  Installed artifact Qt6Svg 386.4 KiB  Installed artifact Wayland 390.6 KiB  Installed artifact libevdev 117.1 KiB  Installed artifact Libffi 44.2 KiB  Installed artifact Opus 960.9 KiB  Installed artifact Xorg_libxcb 2.1 MiB  Installed artifact FreeType2 1.5 MiB  Installed artifact LAME 292.5 KiB  Installed artifact libdrm 368.0 KiB  Installed artifact libva 235.7 KiB  Installed artifact Fontconfig 984.8 KiB  Installed artifact Ogg 250.3 KiB  Installed artifact LLVMOpenMP 661.6 KiB  Installed artifact Xorg_libICE 384.9 KiB  Installed artifact Cairo 2.2 MiB  Installed artifact libaom 6.4 MiB  Installed artifact FFMPEG 11.9 MiB  Installed artifact Glib 7.7 MiB  Installed artifact Xorg_libXfixes 59.0 KiB  Installed artifact Xorg_libXcursor 227.1 KiB  Installed artifact Qt6Declarative 23.8 MiB  Installed artifact Qt6Base 23.1 MiB  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Manifest.toml` Instantiating... === Precompiling... === PlutoTestJH Resolving... ===  Installed libfdk_aac_jll ─────────────── v2.0.4+0  Installed x265_jll ───────────────────── v4.1.0+0  Installed GR_jll ─────────────────────── v0.73.26+0  Installed Opus_jll ───────────────────── v1.6.1+0  Installed libdecor_jll ───────────────── v0.2.2+0  Installed Hyperscript ────────────────── v0.0.5  Installed GR ─────────────────────────── v0.73.26  Installed Contour ────────────────────── v0.6.3  Installed Libmount_jll ───────────────── v2.42.0+0  Installed Xorg_xcb_util_wm_jll ───────── v0.4.2+0  Installed Grisu ──────────────────────── v1.0.2  Installed JpegTurbo_jll ──────────────── v3.1.5+0  Installed LERC_jll ───────────────────── v4.1.0+0  Installed Xorg_xcb_util_image_jll ────── v0.4.1+0  Installed RecipesPipeline ────────────── v0.6.12  Installed DelimitedFiles ─────────────── v1.9.1  Installed Measures ───────────────────── v0.3.3  Installed Xorg_xkbcomp_jll ───────────── v1.4.7+0  Installed PlutoUI ────────────────────── v0.7.83  Installed Xorg_xcb_util_jll ──────────── v0.4.1+0  Installed PlotUtils ──────────────────── v1.4.4  Installed libva_jll ──────────────────── v2.23.0+0  Installed Cairo_jll ──────────────────── v1.18.7+0  Installed EpollShim_jll ──────────────── v0.0.20230411+1  Installed Statistics ─────────────────── v1.11.1  Installed Xorg_libSM_jll ─────────────── v1.2.6+0  Installed Missings ───────────────────── v1.2.0  Installed Showoff ────────────────────── v1.0.3  Installed ColorSchemes ───────────────── v3.31.0  Installed Xorg_libXinerama_jll ───────── v1.1.7+0  Installed Fontconfig_jll ─────────────── v2.17.1+0  Installed Xorg_libXau_jll ────────────── v1.0.13+0  Installed FFMPEG ─────────────────────── v0.4.5  Installed Xorg_libxkbfile_jll ────────── v1.2.0+0  Installed IrrationalConstants ────────── v0.2.6  Installed Xorg_xcb_util_keysyms_jll ──── v0.4.1+0  Installed xkbcommon_jll ──────────────── v1.13.0+0  Installed Bzip2_jll ──────────────────── v1.0.9+0  Installed HarfBuzz_jll ───────────────── v8.5.1+0  Installed PlotThemes ─────────────────── v3.3.0  Installed NaNMath ────────────────────── v1.1.4  Installed Xorg_libpciaccess_jll ──────── v0.19.0+0  Installed x264_jll ───────────────────── v10164.0.1+0  Installed UnicodeFun ─────────────────── v0.4.1  Installed Pango_jll ──────────────────── v1.57.1+0  Installed Colors ─────────────────────── v0.13.1  Installed PtrArrays ──────────────────── v1.4.0  Installed DataStructures ─────────────── v0.19.5  Installed XZ_jll ─────────────────────── v5.8.3+0  Installed StatsAPI ───────────────────── v1.8.0  Installed fzf_jll ────────────────────── v0.61.1+0  Installed GLFW_jll ───────────────────── v3.4.1+1  Installed mtdev_jll ──────────────────── v1.1.7+0  Installed FriBidi_jll ────────────────── v1.0.17+0  Installed libaom_jll ─────────────────── v3.13.3+0  Installed StatsBase ──────────────────── v0.34.11  Installed JLFzf ──────────────────────── v0.1.11  Installed Xorg_libxcb_jll ────────────── v1.17.1+0  Installed Qt6Svg_jll ─────────────────── v6.10.2+0  Installed eudev_jll ──────────────────── v3.2.14+0  Installed libpng_jll ─────────────────── v1.6.58+0  Installed FreeType2_jll ──────────────── v2.14.3+1  Installed TensorCore ─────────────────── v0.1.1  Installed Parsers ────────────────────── v2.8.5  Installed Dbus_jll ───────────────────── v1.16.2+0  Installed Format ─────────────────────── v1.3.7  Installed ColorVectorSpace ───────────── v0.11.0  Installed ColorTypes ─────────────────── v0.12.1  Installed libevdev_jll ───────────────── v1.13.4+0  Installed Plots ──────────────────────── v1.41.6  Installed Xorg_libXext_jll ───────────── v1.3.8+0  Installed Zstd_jll ───────────────────── v1.5.7+1  Installed libinput_jll ───────────────── v1.28.1+0  Installed Ogg_jll ────────────────────── v1.3.6+0  Installed Vulkan_Loader_jll ──────────── v1.3.243+0  Installed Libtiff_jll ────────────────── v4.7.2+0  Installed Reexport ───────────────────── v1.2.2  Installed LogExpFunctions ────────────── v1.0.1  Installed Libffi_jll ─────────────────── v3.4.7+0  Installed AliasTables ────────────────── v1.1.3  Installed Xorg_libXrender_jll ────────── v0.9.12+0  Installed Xorg_libXi_jll ─────────────── v1.8.3+0  Installed Qt6ShaderTools_jll ─────────── v6.10.2+1  Installed Ghostscript_jll ────────────── v9.55.1+0  Installed DocStringExtensions ────────── v0.9.5  Installed Xorg_xcb_util_renderutil_jll ─ v0.3.10+0  Installed Qt6Declarative_jll ─────────── v6.10.2+1  Installed Xorg_xcb_util_cursor_jll ───── v0.1.6+0  Installed Graphite2_jll ──────────────── v1.3.16+0  Installed libass_jll ─────────────────── v0.17.4+0  Installed MacroTools ─────────────────── v0.5.16  Installed Xorg_libXcursor_jll ────────── v1.2.4+0  Installed Wayland_jll ────────────────── v1.24.0+0  Installed Libuuid_jll ────────────────── v2.42.0+0  Installed Xorg_libICE_jll ────────────── v1.1.2+0  Installed Pixman_jll ─────────────────── v0.46.4+0  Installed StableRNGs ─────────────────── v1.0.4  Installed Latexify ───────────────────── v0.16.10  Installed Xorg_xkeyboard_config_jll ──── v2.47.0+1  Installed FixedPointNumbers ──────────── v0.8.6  Installed Xorg_xtrans_jll ────────────── v1.6.0+0  Installed libdrm_jll ─────────────────── v2.4.125+1  Installed RecipesBase ────────────────── v1.3.4  Installed LAME_jll ───────────────────── v3.100.3+0  Installed GettextRuntime_jll ─────────── v0.22.4+0  Installed FFMPEG_jll ─────────────────── v8.1.0+0  Installed LaTeXStrings ───────────────── v1.4.0  Installed IOCapture ──────────────────── v1.0.0  Installed libvorbis_jll ──────────────── v1.3.8+0  Installed Glib_jll ───────────────────── v2.86.3+0  Installed LLVMOpenMP_jll ─────────────── v18.1.8+0  Installed Xorg_libXrandr_jll ─────────── v1.5.6+0  Installed Unzip ──────────────────────── v0.2.0  Installed Xorg_libXfixes_jll ─────────── v6.0.2+0  Installed SortingAlgorithms ──────────── v1.2.2  Installed Qt6Wayland_jll ─────────────── v6.10.2+1  Installed Libglvnd_jll ───────────────── v1.7.1+1  Installed Qt6Base_jll ────────────────── v6.10.2+2  Installed PlutoTest ──────────────────── v0.2.4  Installed Xorg_libX11_jll ────────────── v1.8.13+0  Installed Xorg_libXdmcp_jll ──────────── v1.1.6+0  Installing 76 artifacts  Installed artifact Xorg_xcb_util_renderutil 32.1 KiB  Installed artifact libass 427.8 KiB  Installed artifact Bzip2 503.5 KiB  Installed artifact LERC 267.4 KiB  Installed artifact Pixman 390.8 KiB  Installed artifact Xorg_libpciaccess 26.2 KiB  Installed artifact Xorg_xtrans 48.2 KiB  Installed artifact Xorg_xcb_util_wm 147.9 KiB  Installed artifact mtdev 79.6 KiB  Installed artifact Qt6ShaderTools 2.1 MiB  Installed artifact Xorg_xkeyboard_config 545.0 KiB  Installed artifact Qt6Wayland 1.4 MiB  Installed artifact Xorg_libXi 1.6 MiB  Installed artifact Libmount 6.9 MiB  Installed artifact libinput 1.0 MiB  Installed artifact GLFW 187.9 KiB  Installed artifact Libglvnd 833.5 KiB  Installed artifact eudev 3.9 MiB  Installed artifact Libtiff 2.3 MiB  Installed artifact Xorg_libSM 164.1 KiB  Installed artifact libpng 329.4 KiB  Installed artifact Xorg_xcb_util 39.5 KiB  Installed artifact Vulkan_Loader 177.3 KiB  Installed artifact libdecor 44.1 KiB  Installed artifact Dbus 489.8 KiB  Installed artifact Xorg_libXau 36.6 KiB  Installed artifact Pango 2.1 MiB  Installed artifact libfdk_aac 2.8 MiB  Installed artifact Libuuid 3.9 MiB  Installed artifact Xorg_xcb_util_cursor 27.6 KiB  Installed artifact Xorg_libXext 286.6 KiB  Installed artifact Xorg_xcb_util_image 54.3 KiB  Installed artifact xkbcommon 298.7 KiB  Installed artifact x264 2.1 MiB  Installed artifact JpegTurbo 1.4 MiB  Installed artifact GettextRuntime 543.0 KiB  Installed artifact XZ 1.6 MiB  Installed artifact Xorg_xcb_util_keysyms 17.8 KiB  Installed artifact fzf 3.3 MiB  Installed artifact FriBidi 78.9 KiB  Installed artifact Graphite2 120.3 KiB  Installed artifact x265 1.4 MiB  Installed artifact Ghostscript 31.4 MiB  Installed artifact Xorg_libXinerama 25.5 KiB  Installed artifact GR 19.3 MiB  Installed artifact Xorg_libX11 4.8 MiB  Installed artifact Zstd 646.0 KiB  Installed artifact Xorg_libXdmcp 67.7 KiB  Installed artifact HarfBuzz 1.7 MiB  Installed artifact Xorg_libXrandr 97.3 KiB  Installed artifact Xorg_libxkbfile 91.1 KiB  Installed artifact Xorg_xkbcomp 274.8 KiB  Installed artifact Xorg_libXrender 435.9 KiB  Installed artifact libvorbis 271.0 KiB  Installed artifact Qt6Svg 386.4 KiB  Installed artifact Wayland 390.6 KiB  Installed artifact libevdev 117.1 KiB  Installed artifact Libffi 44.2 KiB  Installed artifact Opus 960.9 KiB  Installed artifact Xorg_libxcb 2.1 MiB  Installed artifact FreeType2 1.5 MiB  Installed artifact LAME 292.5 KiB  Installed artifact libdrm 368.0 KiB  Installed artifact libva 235.7 KiB  Installed artifact Fontconfig 984.8 KiB  Installed artifact Ogg 250.3 KiB  Installed artifact LLVMOpenMP 661.6 KiB  Installed artifact Xorg_libICE 384.9 KiB  Installed artifact Cairo 2.2 MiB  Installed artifact libaom 6.4 MiB  Installed artifact FFMPEG 11.9 MiB  Installed artifact Glib 7.7 MiB  Installed artifact Xorg_libXfixes 59.0 KiB  Installed artifact Xorg_libXcursor 227.1 KiB  Installed artifact Qt6Declarative 23.8 MiB  Installed artifact Qt6Base 23.1 MiB  Project No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Project.toml`  Manifest No packages added to or removed from `~/.julia/scratchspaces/c3e4b0f8-55cb-11ea-2926-15256bba5781/pkg_envs/env_nveuvwnhlj/Manifest.toml` Instantiating... === Precompiling... === enabled÷restart_recommended_msgrestart_required_msgbusy_packageswaiting_for_permission,waiting_for_permission_but_probably_disabled«cell_inputs_$1b768306-0e0e-414d-a501-5c62a2db720fcell_id$1b768306-0e0e-414d-a501-5c62a2db720fcodemd"## Computations as Trees"metadatashow_logsèdisabled®skip_as_script«code_folded$aa99b123-7164-4ed7-833f-b99b19118a0dcell_id$aa99b123-7164-4ed7-833f-b99b19118a0dcodex^2+ymetadatashow_logsèdisabled®skip_as_script«code_folded$be693f92-de60-4b43-82b3-ee6cceea310acell_id$be693f92-de60-4b43-82b3-ee6cceea310acodeٰmd""" $\frac{\partial}{\partial x} (2xy + (x-1)^2) = 2y + 2(x-1)$ $\frac{\partial}{\partial y} (2xy + (x-1)^2) = 2x$ If x = 3 and y = 5, then these evaluate to 14 and 6. """metadatashow_logsèdisabled®skip_as_script«code_folded$1f1b384a-6588-45a5-9dd3-6de3face8bfbcell_id$1f1b384a-6588-45a5-9dd3-6de3face8bfbcodefunction ad_steps(x::Expr; color_fwd="red", color_bwd="green", font_size=".8em") x = EX(x) repr(x) = sprint(show, x; context=:compact=>true) span_fwd = @htl "" span_bwd = @htl "" d1 = Dict( let e = eval(i.x) i => @htl " $(span_fwd)$(repr(e isa Tracked ? e.val : e))" end for i in PostOrderDFS(x) if isempty(children(i)) ) res = accumulate(Iterators.filter(x -> !isempty(children(x)), PostOrderDFS(x)); init=d1) do d,i d = copy(d) e = eval(i.x) d[i] = @htl " $(span_fwd)$(repr(e isa Tracked ? e.val : e))" d end pushfirst!(res, d1) f = eval(x.x) d = Dict{Any, Any}(f => 1) let d1 = copy(res[end]) d1[x] = @htl "$(d1[x]) $(span_bwd)1" push!(res, d1) end for (x, e) in zip(PreOrderDFS.((f, x))...) x.df === nothing && continue dy = x.df(d[x]) for (yᵢ, dyᵢ, e) in zip(x.deps, dy, children(e)) d1 = copy(res[end]) if haskey(d, yᵢ) d1[e] = @htl "$(get(d1, e, ""))$(span_bwd) + $(repr(dyᵢ))" else d1[e] = @htl "$(get(d1, e, "")) $(span_bwd)$(repr(dyᵢ))" end push!(res, d1) d[yᵢ] = get(d, yᵢ, 0) + dyᵢ end end res endmetadatashow_logsèdisabled®skip_as_script«code_folded$e55dfad2-db50-459f-ab54-fa7637fc3638cell_id$e55dfad2-db50-459f-ab54-fa7637fc3638codemd""" ## How can we visualize both the forward and the reverse pass? We can further visualize each steps we just took. First we do the forwards calculation, where we also build up our tree, then we go down the tree in the opposite direction to accumulate our gradient. """metadatashow_logsèdisabled®skip_as_script«code_folded$a0c0f48d-e0b0-4f45-8e0e-d9ab5743958bcell_id$a0c0f48d-e0b0-4f45-8e0e-d9ab5743958bcodeDump(aa)metadatashow_logsèdisabled®skip_as_script«code_folded$7dca2c78-6faf-450a-8096-1c5b9f7f1295cell_id$7dca2c78-6faf-450a-8096-1c5b9f7f1295codeleaky_relu(x) = x > 0 ? x : .1xmetadatashow_logsèdisabled®skip_as_script«code_folded$13487e65-5e48-4a37-9bea-f262dd7b6d56cell_id$13487e65-5e48-4a37-9bea-f262dd7b6d56codegfunction Base.:+(x::Tracked, y::Tracked) Tracked(x.val + y.val, :+, Δ -> (Δ, Δ), Tracked[x, y]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$96286b65-1a22-4458-a399-46579248cce4cell_id$96286b65-1a22-4458-a399-46579248cce4codebegin _rand(s...) = randn(s...) / √prod(s) W1, W2, W3 = _rand(32, 1), _rand(32, 32), _rand(1, 32) b1, b2, b3 = _rand(32), _rand(32), _rand(1) params = [@t(W1), @t(W2), @t(W3), @t(b1), @t(b2), @t(b3)] p = @animate for i in 1:10000 loss = norm(NN(input, params) - @t(ŷ)) ∇ = grad(loss) for p in params p.val .-= 1e-3 .* ∇[p] end plot(input', [NN(input, params).val; ŷ]'; label=["prediction" "training data"]) end every 200 gif(p; fps=5) endmetadatashow_logsèdisabled®skip_as_script«code_folded$1a154bb7-93a3-4973-8908-788db77ac294cell_id$1a154bb7-93a3-4973-8908-788db77ac294code?s2 = @htl """ """metadatashow_logsèdisabled®skip_as_script«code_folded$8f1ec8d7-89ab-4bc5-9eef-2b0a69019854cell_id$8f1ec8d7-89ab-4bc5-9eef-2b0a69019854code0md""" We'll try to approximate a sine curve: """metadatashow_logsèdisabled®skip_as_script«code_folded$795c510e-6280-42d9-8820-290ca07be88dcell_id$795c510e-6280-42d9-8820-290ca07be88dcode((x^2+y).deps[2]).depsmetadatashow_logsèdisabled®skip_as_script«code_folded$27b39d7d-fc08-4ccc-aea4-b64f8a4f5726cell_id$27b39d7d-fc08-4ccc-aea4-b64f8a4f5726codeU#@ad_steps 3y*x + 2(x-1)*x @ad_steps (2x*y + (x-1)^2) # type the steps for the macrometadatashow_logsèdisabled®skip_as_script«code_folded$85098c31-04a1-421c-b3f6-25fec9360588cell_id$85098c31-04a1-421c-b3f6-25fec9360588codeLmd"### What if we use a (leaky) rectified linear unit (ReLU) as activation?"metadatashow_logsèdisabled®skip_as_script«code_folded$8ab0f55d-a393-4a8a-a48c-9ced26033f57cell_id$8ab0f55d-a393-4a8a-a48c-9ced26033f57codecfunction Base.sin(x::Tracked) Tracked(sin(x.val), :sin, Δ -> (Δ * cos(x.val),), Tracked[x,]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$e01114fa-a847-424a-931f-8f42e623109fcell_id$e01114fa-a847-424a-931f-8f42e623109fcodezfunction LinearAlgebra.norm(x::Tracked) res = norm(x.val) Tracked(res, :norm, Δ -> (Δ/res .* x.val,), Tracked[x,]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$2ba83769-89b4-4ba0-8726-9210f394b4e7cell_id$2ba83769-89b4-4ba0-8726-9210f394b4e7codemd"### Operations on Tracked"metadatashow_logsèdisabled®skip_as_script«code_folded$d48b9f23-b1b8-43f8-a05d-2234b69bf1f0cell_id$d48b9f23-b1b8-43f8-a05d-2234b69bf1f0code(2x*y + (x-1)^2)metadatashow_logsèdisabled®skip_as_script«code_folded$a5671f2f-48ca-4896-9428-147dc671d2b9cell_id$a5671f2f-48ca-4896-9428-147dc671d2b9code٢function Base.broadcasted(::typeof(+), x::Tracked{<:VecOrMat}, y::Tracked{<:Vector}) Tracked(x.val .+ y.val, :.+, Δ -> (Δ, sum(Δ; dims=2)), Tracked[x, y]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$d5565717-239b-41ab-b311-d59b383130edcell_id$d5565717-239b-41ab-b311-d59b383130edcodermd""" Ok, let's create our function `grad` which will accumulate all intermediate gradients into a dictionary: """metadatashow_logsèdisabled®skip_as_script«code_folded$81eb8a2d-a3a9-45af-a5a5-b96aefd48712cell_id$81eb8a2d-a3a9-45af-a5a5-b96aefd48712code5(2x*y + (x-1)^2).val # The result of `2x*y + (x-1)^2`metadatashow_logsèdisabled®skip_as_script«code_folded$2188a663-5a85-4ce4-bc8d-20383481e59bcell_id$2188a663-5a85-4ce4-bc8d-20383481e59bcode+AbstractTrees.children(x::Tracked) = x.depsmetadatashow_logsèdisabled®skip_as_script«code_folded$db4f871a-3ca1-4bd3-a2c5-da7d6f599281cell_id$db4f871a-3ca1-4bd3-a2c5-da7d6f599281code(x*y).valmetadatashow_logsèdisabled®skip_as_script«code_folded$53ae4070-8818-4d21-8648-19df9319918acell_id$53ae4070-8818-4d21-8648-19df9319918acode0md""" ## Let's write our own neural network! """metadatashow_logsèdisabled®skip_as_script«code_folded$79f71f9d-b491-4a2c-85a4-29ae8da4f312cell_id$79f71f9d-b491-4a2c-85a4-29ae8da4f312code@visual_debug 3y + 2(x-1)metadatashow_logsèdisabled®skip_as_script«code_folded$e450329b-46ee-4f53-b6f0-891d6596938ccell_id$e450329b-46ee-4f53-b6f0-891d6596938ccodex^2+ymetadatashow_logsèdisabled®skip_as_script«code_folded$a7c8cb6a-6e17-4d8f-8958-fe3527c5b8e7cell_id$a7c8cb6a-6e17-4d8f-8958-fe3527c5b8e7codegrad(w, x), grad(w, y)metadatashow_logsèdisabled®skip_as_script«code_folded$d1312080-272c-4bd5-bcfe-bf47f8320961cell_id$d1312080-272c-4bd5-bcfe-bf47f8320961code"md"### The Tracked Data Structure"metadatashow_logsèdisabled®skip_as_script«code_folded$885cd51d-895f-4996-a23b-780498b5b810cell_id$885cd51d-895f-4996-a23b-780498b5b810codeْmd""" All overloads will do the operation (e.g. sum `x` and `y`), but also remember the pullback map and input variables for the reverse pass. """metadatashow_logsèdisabled®skip_as_script«code_folded$d5559dba-9fe4-11ec-3744-ebd1408e7dc4cell_id$d5559dba-9fe4-11ec-3744-ebd1408e7dc4codeOusing AbstractTrees, PlutoUI, HypertextLiteral, PlutoTest, LinearAlgebra, Plotsmetadatashow_logsèdisabled®skip_as_script«code_folded$49333dc7-b809-4bb1-b285-20a12d8b9209cell_id$49333dc7-b809-4bb1-b285-20a12d8b9209codelet ŷ_t = @t(ŷ) loss = norm(NN(input, params) - ŷ_t) ∇ = grad(loss) params′ = deepcopy(params) for (p, p′) in zip(params, params′) p′.val .-= 1e-3 .* ∇[p] end plot(input', [NN(input, params).val; ŷ]'; label=["prediction" "training data"]) quiver!(input[1:4:end], NN(input, params).val[1:4:end]; quiver=(zero(input[1:4:end]), 3(NN(input, params′).val .- NN(input, params).val)[1:4:end])) endmetadatashow_logsèdisabled®skip_as_script«code_folded$f6ce8448-d9ce-4453-9e47-dc6443d50f55cell_id$f6ce8448-d9ce-4453-9e47-dc6443d50f55codes1 = html""" """metadatashow_logsèdisabled®skip_as_script«code_folded$ec723b90-1c8b-4a87-8a86-fe17d6e85f22cell_id$ec723b90-1c8b-4a87-8a86-fe17d6e85f22code@ad_steps (x+y)metadatashow_logsèdisabled®skip_as_script«code_folded$c9d70049-4793-41db-9075-e9e73e926c1acell_id$c9d70049-4793-41db-9075-e9e73e926c1acodex+ymetadatashow_logsèdisabled®skip_as_script«code_folded$17f9e2c0-aa70-4d4e-ba42-51782df43785cell_id$17f9e2c0-aa70-4d4e-ba42-51782df43785code)md"### Computational Tree (Forward Pass)"metadatashow_logsèdisabled®skip_as_script«code_folded$0d186a1b-eb89-4be8-a237-e5a9608f4a30cell_id$0d186a1b-eb89-4be8-a237-e5a9608f4a30code@ad_steps (x*y)metadatashow_logsèdisabled®skip_as_script«code_folded$d2d21d41-9a4e-428b-b696-6878901849bacell_id$d2d21d41-9a4e-428b-b696-6878901849bacode2ŷ = sin.(input) #.+ .2input .^ 2 .- cos.(5input)metadatashow_logsèdisabled®skip_as_script«code_folded$0b094198-cf44-41d7-a8dc-fd8fd0716bb4cell_id$0b094198-cf44-41d7-a8dc-fd8fd0716bb4code"md""" --- ### Helper Functions """metadatashow_logsèdisabled®skip_as_script«code_folded$b2622dc4-2fd3-4d80-8efd-a6a985a6b2e9cell_id$b2622dc4-2fd3-4d80-8efd-a6a985a6b2e9code٧function Base.broadcasted(::typeof(leaky_relu), x::Tracked) Tracked(leaky_relu.(x.val), :tanh, Δ -> (map((x, Δ) -> x > 0 ? Δ : .1Δ, x.val, Δ),), Tracked[x,]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$506d408e-dc2b-4e12-b917-286e3f4079a2cell_id$506d408e-dc2b-4e12-b917-286e3f4079a2codegrad(w)metadatashow_logsèdisabled®skip_as_script«code_folded$2eb6af9c-9ac7-4b64-93fb-c5a2f037d303cell_id$2eb6af9c-9ac7-4b64-93fb-c5a2f037d303code@ad_steps y^2metadatashow_logsèdisabled®skip_as_script«code_folded$0b5e6560-81fd-4182-bba5-aca702fb3048cell_id$0b5e6560-81fd-4182-bba5-aca702fb3048code١begin # The second argument of Tracked is a symbol only used for printing x = Tracked{Int}(3, :x) y = Tracked{Int}(5, :y) u = Tracked{Int}(2, :u) endmetadatashow_logsèdisabled®skip_as_script«code_folded$8a70e835-dae6-4727-8204-95c87d5c23dacell_id$8a70e835-dae6-4727-8204-95c87d5c23dacode%md""" We need some more overloads """metadatashow_logsèdisabled®skip_as_script«code_folded$337bc837-174f-4ed6-bee1-c9e80dc03b52cell_id$337bc837-174f-4ed6-bee1-c9e80dc03b52code+input = collect(range(0, 2π; length=100))'metadatashow_logsèdisabled®skip_as_script«code_folded$9a141034-17cb-4d85-a5a2-4724a38dd269cell_id$9a141034-17cb-4d85-a5a2-4724a38dd269code٬macro visual_debug(expr) s1 quote $(esc(:(PlutoTest.@eval_step_by_step($expr)))) .|> PlutoTest.SlottedDisplay |> PlutoTest.frames |> PlutoTest.with_slotted_css end endmetadatashow_logsèdisabled®skip_as_script«code_folded$be28f764-a415-46ed-a503-68e8d831f8a8cell_id$be28f764-a415-46ed-a503-68e8d831f8a8codeP(aa,bb) = (promote(x,2.5)) # promotes x to be a float and also y to be a Trackedmetadatashow_logsèdisabled®skip_as_script«code_folded$5e02b2c0-10cf-4746-840d-7017648f89f8cell_id$5e02b2c0-10cf-4746-840d-7017648f89f8codenmd""" Besides the mathematics of reverse mode differentiation, there are many interesting computer science issues regarding the implementation of reverse diff. Probably the jury is still out which is best. The implementation below is based on Julia's dispatch system. This means we create a type `Tracked` for keeping track of our input variables and everything we'll need to calculate later. The other main approach (not illustrated here) is source to source translation of code. In Julia the zygote package translates at the high level, while the increasingly popular Enzyme package translates at the lower level. """metadatashow_logsèdisabled®skip_as_script«code_folded$a6c2d3a2-326a-41dd-864d-aa3662466222cell_id$a6c2d3a2-326a-41dd-864d-aa3662466222codex,ymetadatashow_logsèdisabled®skip_as_script«code_folded$2c3f3103-9ccb-446e-a289-87bbd0a1b97bcell_id$2c3f3103-9ccb-446e-a289-87bbd0a1b97bcode(x^2+y).deps[2]metadatashow_logsèdisabled®skip_as_script«code_folded$e52aa672-69a9-419b-a992-e7a3d1364fb6cell_id$e52aa672-69a9-419b-a992-e7a3d1364fb6codeT# `PreOrderDFS` traverses this tree from the top down Text.(collect(PreOrderDFS(z)))metadatashow_logsèdisabled®skip_as_script«code_folded$7299bfea-e23e-4993-bebd-9d8762427294cell_id$7299bfea-e23e-4993-bebd-9d8762427294code4md"### Pretty Viz: Forward Pass + Reverse Mode Diff"metadatashow_logsèdisabled®skip_as_script«code_folded$4da7a594-aec5-4c8d-ab20-6913ed6ff8accell_id$4da7a594-aec5-4c8d-ab20-6913ed6ff8accodeQTableOfContents(title="📚 Table of Contents", indent=true, depth=4, aside=true)metadatashow_logsèdisabled®skip_as_script«code_folded$4bfc2f7d-a5b0-44c7-8bb6-f1b834c1cc51cell_id$4bfc2f7d-a5b0-44c7-8bb6-f1b834c1cc51codeٮmd""" `Tracked` is a tree -- We just need to tell *AbstractTrees.jl* how to get the children for each node and we get tree printing and iteration over all nodes for free. """metadatashow_logsèdisabled®skip_as_script«code_folded$0d2a3187-46f1-4705-bb3e-ecf74e34840dcell_id$0d2a3187-46f1-4705-bb3e-ecf74e34840dcodez = (2x*y + (x-1)^2)metadatashow_logsèdisabled®skip_as_script«code_folded$437285d4-ec53-4bb7-9966-fcfb5352e205cell_id$437285d4-ec53-4bb7-9966-fcfb5352e205codefunction show_tree(x; height=400) id = gensym() @htl """ $s2
""" endmetadatashow_logsèdisabled®skip_as_script«code_folded$61513f88-c126-43ab-9515-85a2c8a472bfcell_id$61513f88-c126-43ab-9515-85a2c8a472bfcode(x^2+y).depsmetadatashow_logsèdisabled®skip_as_script«code_folded$73d638bf-30c1-4694-b3a8-4b29c5e3fa65cell_id$73d638bf-30c1-4694-b3a8-4b29c5e3fa65codeyfunction Base.:*(x::Tracked, y::Tracked) Tracked(x.val * y.val, :*, Δ -> (Δ * y.val', x.val' * Δ), Tracked[x, y]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$ccafd0b9-95aa-4e58-8afc-26cd3ee61cc9cell_id$ccafd0b9-95aa-4e58-8afc-26cd3ee61cc9codeDmd""" Straight away we get the primal result of our calculation: """metadatashow_logsèdisabled®skip_as_script«code_folded$95ab6c83-0757-40b3-a116-6d872a074f69cell_id$95ab6c83-0757-40b3-a116-6d872a074f69codex*ymetadatashow_logsèdisabled®skip_as_script«code_folded$86aa821b-a373-4814-953e-535f3a33c002cell_id$86aa821b-a373-4814-953e-535f3a33c002code;show_tree(norm(NN(input, params) - @t(ŷ)); height = 1000)metadatashow_logsèdisabled®skip_as_script«code_folded$29360c2a-c47b-48a3-b225-248eb3ec8a42cell_id$29360c2a-c47b-48a3-b225-248eb3ec8a42code(x*y).namemetadatashow_logsèdisabled®skip_as_script«code_folded$814786be-a760-4023-a614-83e727681075cell_id$814786be-a760-4023-a614-83e727681075codeDump(bb)metadatashow_logsèdisabled®skip_as_script«code_folded$ac097299-0a31-474c-ab26-a4fb24bb9046cell_id$ac097299-0a31-474c-ab26-a4fb24bb9046codeufunction Base.:^(x::Tracked, n::Int) Tracked(x.val^n, Symbol("^$n"), Δ -> (Δ * n * x.val^(n-1),), Tracked[x,]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$d9304d8d-9a34-46f2-908d-42d5cc5f5c5fcell_id$d9304d8d-9a34-46f2-908d-42d5cc5f5c5fcode👽 = Tracked{Int}(12, :👽)metadatashow_logsèdisabled®skip_as_script«code_folded$788903f0-a372-4265-bbf3-5b507a705100cell_id$788903f0-a372-4265-bbf3-5b507a705100codeVNN(x, (W1, W2, W3, b1, b2, b3)) = W3 * tanh.(W2 * tanh.(W1 * @t(x) .+ b1) .+ b2) .+ b3metadatashow_logsèdisabled®skip_as_script«code_folded$bb75e790-a628-4821-b7b1-11cd34a73d38cell_id$bb75e790-a628-4821-b7b1-11cd34a73d38code(x*y).depsmetadatashow_logsèdisabled®skip_as_script«code_folded$094bf7ee-7c9b-458c-98dc-521768831654cell_id$094bf7ee-7c9b-458c-98dc-521768831654code7function show_steps(steps; height=400) id1 = gensym() @htl """ $s2
Move me!
1 """ endmetadatashow_logsèdisabled®skip_as_script«code_folded$e6ee1921-57aa-4194-b2eb-8e32fa4a6c44cell_id$e6ee1921-57aa-4194-b2eb-8e32fa4a6c44codeImacro t(s) :(Tracked{typeof($(esc(s)))}($(esc(s)), $(QuoteNode(s)))) endmetadatashow_logsèdisabled®skip_as_script«code_folded$2141849b-675e-406c-8df4-34b2706507afcell_id$2141849b-675e-406c-8df4-34b2706507afcodeقfunction Base.:/(x::Tracked, y::Tracked) Tracked(x.val / y.val, :/, Δ -> (Δ / y.val, -Δ * x.val / y.val^2), Tracked[x, y]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$0911a08a-4290-455b-9b26-0bf2862296dacell_id$0911a08a-4290-455b-9b26-0bf2862296dacode$plot(leaky_relu; label="leaky relu")metadatashow_logsèdisabled®skip_as_script«code_folded$20571e1a-9687-4d42-98b3-b3bc0b207b3bcell_id$20571e1a-9687-4d42-98b3-b3bc0b207b3bcodemd"## Implementation in Code"metadatashow_logsèdisabled®skip_as_script«code_folded$01eacbd4-ef37-4524-aecc-2ef9a1044cf8cell_id$01eacbd4-ef37-4524-aecc-2ef9a1044cf8codesmd""" To also get the gradient, we'll use `PreOrderDFS` to traverse the tree we just created from the top down. """metadatashow_logsèdisabled®skip_as_script«code_folded$99a3507b-ca03-429f-acde-e2d1ebb32054cell_id$99a3507b-ca03-429f-acde-e2d1ebb32054codefunction grad(f::Tracked) d = Dict{Any, Any}(f => 1) for x in PreOrderDFS(f) # recursively traverse all dependents x.df === nothing && continue # ignore untracked variables like constants dy = x.df(d[x]) # evaluate pullback for (yᵢ, dyᵢ) in zip(x.deps, dy) # store the gradient in d # if we have already stored a gradient for this variable, we need to add them d[yᵢ] = get(d, yᵢ, 0) .+ dyᵢ end end return d endmetadatashow_logsèdisabled®skip_as_script«code_folded$7fcccb65-4a7b-4527-97be-d25f481f6eafcell_id$7fcccb65-4a7b-4527-97be-d25f481f6eafcode5md""" We can verify that it does the right thing: """metadatashow_logsèdisabled®skip_as_script«code_folded$6733267e-1251-49a7-86a0-ff45ba408205cell_id$6733267e-1251-49a7-86a0-ff45ba408205codemd"### Tracking variables"metadatashow_logsèdisabled®skip_as_script«code_folded$266bf2ed-d625-47be-8a87-21e12052b27fcell_id$266bf2ed-d625-47be-8a87-21e12052b27fcodeZmd""" Let's use three dense layers, all of the form $x \mapsto \tanh.(W \cdot x + b)$ """metadatashow_logsèdisabled®skip_as_script«code_folded$ea94bdf9-b0b1-42ad-ba44-aaba73ed1192cell_id$ea94bdf9-b0b1-42ad-ba44-aaba73ed1192codemd""" Let's look at our example from above, $z = x^2+y$. We'll introduce an intermediate variable $u = x^2$ so we can write $z$ as $u + y$. We will first start with $1 \cdot dz$. The one here represents the fact that $\frac{\partial z}{\partial z} = 1$. We then want to *pull* that *back* until we get something in terms of $dx$ and $dy$, so we then get the derivatives wrt $x$ and $y$ The pullback map for $+$ will first map $dz$ to $d(u + y) = du + dy$ and then we can use the pullback map for ^ to expand $du$ to $2x \cdot dx$. Using that recursive method we end up with $dz = 2x \cdot dx + dy$, which is exactly what we'd get if we did this by hand. This tells us that $\frac{\partial z}{\partial x} = 2x$ and $\frac{\partial z}{\partial y} = y$. """metadatashow_logsèdisabled®skip_as_script«code_folded$98807f0d-f8a5-4fe0-9983-2b95290347d9cell_id$98807f0d-f8a5-4fe0-9983-2b95290347d9codeEmd""" ### Tada! 🎉 (It's even reasonably fast for neural nets) """metadatashow_logsèdisabled®skip_as_script«code_folded$8b486bd8-25a1-4895-99c8-0541c9e3c8b0cell_id$8b486bd8-25a1-4895-99c8-0541c9e3c8b0code@ad_steps (x/y)metadatashow_logsèdisabled®skip_as_script«code_folded$bcff2aa8-2387-44c7-a28f-39cd505a7adfcell_id$bcff2aa8-2387-44c7-a28f-39cd505a7adfcodeWmd""" We can also visualize what Julia does in the forward pass on the code itself: """metadatashow_logsèdisabled®skip_as_script«code_folded$b47754ac-cb17-4ea3-a97d-20371fd341a5cell_id$b47754ac-cb17-4ea3-a97d-20371fd341a5code((x*y).df)(1) # For referencemetadatashow_logsèdisabled®skip_as_script«code_folded$520f280d-c78e-433b-a0a2-8ef05b04f7cccell_id$520f280d-c78e-433b-a0a2-8ef05b04f7cccodeٍfunction Base.broadcasted(::typeof(tanh), x::Tracked) res = tanh.(x.val) Tracked(res, :tanh, Δ -> (Δ .* (1 .- res.^2),), Tracked[x,]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$faf69c6f-b179-4ca2-aa22-641e9560e17bcell_id$faf69c6f-b179-4ca2-aa22-641e9560e17bcodeهmacro ad_steps(ex) quote ex = $(QuoteNode(ex)) steps = ad_steps(ex) show_steps([TTREE(EX(ex), step) for step in steps]) end endmetadatashow_logsèdisabled®skip_as_script«code_folded$d82adc20-4c8c-4f2c-9839-d03ad7e7f581cell_id$d82adc20-4c8c-4f2c-9839-d03ad7e7f581codebegin struct EX x::Any function EX(ex) if Meta.isexpr(ex, :call) && ex.args[1] === :+ && length(ex.args) > 3 new(Expr(:call, :+, Expr(:call, :+, ex.args[2:end-1]...), ex.args[end])) else new(ex) end end end show_tree(ex::Expr) = show_tree(EX(ex)) function Base.show(io::IO, ex::EX) Base.show_unquoted(io, Meta.isexpr(ex.x, :call) ? ex.x.args[1] : ex.x) if Meta.isexpr(ex.x, :call) && ex.x.args[1] === :^ print(io, ex.x.args[3]) end end function AbstractTrees.children(ex::EX) if Meta.isexpr(ex.x, :call) ex.x.args[1] === :^ ? [EX(ex.x.args[2])] : EX.(ex.x.args[2:end]) else EX[] end end Base.:(==)(ex1::EX, ex2::EX) = ex1.x == ex2.x Base.hash(ex::EX, i::UInt) = hash(ex.x, i) endmetadatashow_logsèdisabled®skip_as_script«code_folded$18b1c55d-a6b5-44f6-b0b3-50bdb0aa9d96cell_id$18b1c55d-a6b5-44f6-b0b3-50bdb0aa9d96codew = x*y + xmetadatashow_logsèdisabled®skip_as_script«code_folded$8919faec-753d-41d0-b0a8-c84c025b2919cell_id$8919faec-753d-41d0-b0a8-c84c025b2919code&xfloat = convert( Tracked{Float64}, x)metadatashow_logsèdisabled®skip_as_script«code_folded$0d006e4d-72e3-42ab-beb3-e5304a4d9ba9cell_id$0d006e4d-72e3-42ab-beb3-e5304a4d9ba9codexmetadatashow_logsèdisabled®skip_as_script«code_folded$4813e57f-557a-4179-afd2-7925687b3d35cell_id$4813e57f-557a-4179-afd2-7925687b3d35codeف# These operations are # d(x/y)= (y dx - x dy)/y^2 # ∇(x/y) = [1/y; -x/y^2] # Notice that the Δ -> (?,?) is just the gradientmetadatashow_logsèdisabled®skip_as_script«code_folded$86fa378b-815d-4c3d-9121-1338ee54f30fcell_id$86fa378b-815d-4c3d-9121-1338ee54f30fcode7let NN(x, (W1, W2, W3, b1, b2, b3)) = W3 * leaky_relu.(W2 * leaky_relu.(W1 * @t(x) .+ b1) .+ b2) .+ b3 _rand(s...) = randn(s...) / √prod(s) W1, W2, W3 = _rand(32, 1), _rand(32, 32), _rand(1, 32) b1, b2, b3 = _rand(32), _rand(32), _rand(1) params = [@t(W1), @t(W2), @t(W3), @t(b1), @t(b2), @t(b3)] p = @animate for i in 1:10000 loss = norm(NN(input, params) - @t(ŷ)) ∇ = grad(loss) for p in params p.val .-= 1e-3 .* ∇[p] end plot(input', [NN(input, params).val; ŷ]'; label=["prediction" "training data"]) end every 200 gif(p; fps=5) endmetadatashow_logsèdisabled®skip_as_script«code_folded$6b1fb808-e993-4c2b-b81b-6710f8206de7cell_id$6b1fb808-e993-4c2b-b81b-6710f8206de7codefunction to_json(x) d = Dict{Symbol, Any}( :innerHTML => sprint(AbstractTrees.printnode, x), :children => Any[to_json(c) for c in children(x)], #:collapsed => !isempty(children(x)), ) endmetadatashow_logsèdisabled®skip_as_script«code_folded$8110f306-a7bb-43a2-bb36-6182c59b4b2ecell_id$8110f306-a7bb-43a2-bb36-6182c59b4b2ecodebegin struct TTREE x d::Dict{Any, Any} end function Base.show(io::IO, x::TTREE) show(io, x.x) print(io, " ") show(io, MIME("text/html"), get(x.d, x.x, @htl(""))) end AbstractTrees.children(x::TTREE) = (TTREE(i, x.d) for i in children(x.x)) endmetadatashow_logsèdisabled®skip_as_script«code_folded$99b6ab91-a022-449c-988c-0e5c5719c910cell_id$99b6ab91-a022-449c-988c-0e5c5719c910codebegin struct Tracked{T} <: Number val::T name::Symbol df deps::Vector{Tracked} end # the two argument version as in Tracked{Int}(3,:x) Tracked{T}(x, name=gensym()) where {T} = Tracked{T}(x, name, nothing, Tracked[]) ## The above is typical Julia, the below shows up often when you ## define a new number type and you want ordinary numbers to work together ## with your newly defined numbers. ## This is similar to when in mathematics, strictly speaking integers ## are separate from the reals, but we all treat integers as embedded ## in the reals. # Converting a Tracked variable, converts like the number Base.convert(T::Type{Tracked{S}}, x::Tracked) where {S} = T(convert(S, x.val), x.name, x.df, x.deps) # This tells Julia to convert any number added to a `Tracked` to a `Tracked` first Base.promote_rule(::Type{Tracked{S}}, ::Type{T}) where {S<:Number, T<:Number} = Tracked{promote_type(S, T)} # All this is just for nicer printing function Base.show(io::IO, x::Tracked) if x.df === nothing print(io, Base.isgensym(x.name) ? x.val : "$(x.val)=$(x.name)") else #print(io, "⋅Tracked(") print(io,"[") print(io, x.name) print(io,"]") #print(io, ", ") print(io," → ") show(io, x.val) #print(io, ")") end end function Base.show(io::IO, x::Tracked{<:AbstractArray}) io = IOContext(io, :compact=>true) print(io, Base.isgensym(x.name) ? x.val : string(x.name)) end Base.show(io::IO, ::MIME"text/plain", x::Tracked) = print_tree(io, x) Base.:(==)(x::Tracked, y::Tracked) = x === y endmetadatashow_logsèdisabled®skip_as_script«code_folded$db3389c0-cb00-404c-81a8-8c4ccdab22f4cell_id$db3389c0-cb00-404c-81a8-8c4ccdab22f4code@ad_steps ((x*y)*u)metadatashow_logsèdisabled®skip_as_script«code_folded$b0cc4665-eb45-48ea-9a33-5acf56d2a283cell_id$b0cc4665-eb45-48ea-9a33-5acf56d2a283codehfunction Base.:-(x::Tracked, y::Tracked) Tracked(x.val - y.val, :-, Δ -> (Δ, -Δ), Tracked[x, y]) endmetadatashow_logsèdisabled®skip_as_script«code_folded$d4e9b202-242e-4420-986b-12d2ab57af93cell_id$d4e9b202-242e-4420-986b-12d2ab57af93code)grad(f::Tracked, x::Tracked) = grad(f)[x]metadatashow_logsèdisabled®skip_as_script«code_folded«notebook_id$40b713b8-6f1b-11f1-8fa9-4f7bea6be0d0in_temp_dir¨metadata