Update documentation (#681)

* Update dependencies

* Update fenced code block examples

* Fix indents

* Add headers in docstrings

* Add backticks

* Fix admonition blocks

* Fix more doctests

docs/src/FAQ.md

@@ -63,7 +63,7 @@ For example, in `access(xs, n) = xs[n]`, the derivative of `access` with respect
 When no custom `frule` or `rrule` exists, if you try to call one of those, it will return `nothing` by default.
 As a result, you may encounter errors like
 
-```julia
+```plain
 MethodError: no method matching iterate(::Nothing)
 ```
 

docs/src/ad_author/opt_out.md

@@ -7,7 +7,7 @@ We provide two ways to know that a rule has been opted out of.
 `@opt_out` defines a `frule` or `rrule` matching the signature that returns `nothing`.
 
 If you are in a position to generate code, in response to values returned by function calls then you can do something like:
-```@julia
+```julia
 res = rrule(f, xs)
 if res === nothing
     y, pullback = perform_ad_via_decomposition(r, xs)  # do AD without hitting the rrule

docs/src/design/changing_the_primal.md

@@ -63,7 +63,7 @@ What about using `sincos`?
 ```@raw html
 <details open><summary>Example for `sin`</summary>
 ```
-```julia
+```julia-repl
 julia> using BenchmarkTools
 
 julia> @btime sin(x) setup=(x=rand());
@@ -76,7 +76,7 @@ julia> 3.838 + 4.795
 8.633
 ```
 vs computing both together:
-```julia
+```julia-repl
 julia> @btime sincos(x) setup=(x=rand());
   6.028 ns (0 allocations: 0 bytes)
 ```
@@ -96,7 +96,7 @@ So we can save time, if we can reuse that `exp(x)`.
 <details open><summary>Example for the logistic sigmoid</summary>
 ```
 If we have to computing separately:
-```julia
+```julia-repl
 julia> @btime 1/(1+exp(x)) setup=(x=rand());
   5.622 ns (0 allocations: 0 bytes)
 
@@ -108,7 +108,7 @@ julia> 5.622 + 6.036
 ```
 
 vs reusing `exp(x)`:
-```julia
+```julia-repl
 julia> @btime exp(x) setup=(x=rand());
   5.367 ns (0 allocations: 0 bytes)
 
@@ -148,8 +148,8 @@ x̄ = pullback_at(f, x, y, ȳ, intermediates)
 ```
 ```julia
 function augmented_primal(::typeof(sin), x)
-  y, cx = sincos(x)
-  return y, (; cx=cx)  # use a NamedTuple for the intermediates
+    y, cx = sincos(x)
+    return y, (; cx=cx)  # use a NamedTuple for the intermediates
 end
 
 pullback_at(::typeof(sin), x, y, ȳ, intermediates) = ȳ * intermediates.cx
@@ -163,9 +163,9 @@ pullback_at(::typeof(sin), x, y, ȳ, intermediates) = ȳ * intermediates.cx
 ```
 ```julia
 function augmented_primal(::typeof(σ), x)
-  ex = exp(x)
-  y = ex / (1 + ex)
-  return y, (; ex=ex)  # use a NamedTuple for the intermediates
+    ex = exp(x)
+    y = ex / (1 + ex)
+    return y, (; ex=ex)  # use a NamedTuple for the intermediates
 end
 
 pullback_at(::typeof(σ), x, y, ȳ, intermediates) = ȳ * y / (1 + intermediates.ex)
@@ -189,8 +189,8 @@ And storing all these things on the tape — inputs, outputs, sensitivities, int
 What if we generalized the idea of the `intermediate` named tuple, and had `augmented_primal` return a struct that just held anything we might want put on the tape.
 ```julia
 struct PullbackMemory{P, S}
-  primal_function::P
-  state::S
+    primal_function::P
+    state::S
 end
 # convenience constructor:
 PullbackMemory(primal_function; state...) = PullbackMemory(primal_function, state)
@@ -211,8 +211,8 @@ which is much cleaner.
 ```
 ```julia
 function augmented_primal(::typeof(sin), x)
-  y, cx = sincos(x)
-  return y, PullbackMemory(sin; cx=cx)
+    y, cx = sincos(x)
+    return y, PullbackMemory(sin; cx=cx)
 end
 
 pullback_at(pb::PullbackMemory{typeof(sin)}, ȳ) = ȳ * pb.cx
@@ -226,9 +226,9 @@ pullback_at(pb::PullbackMemory{typeof(sin)}, ȳ) = ȳ * pb.cx
 ```
 ```julia
 function augmented_primal(::typeof(σ), x)
-  ex = exp(x)
-  y = ex / (1 + ex)
-  return y, PullbackMemory(σ; y=y, ex=ex)
+    ex = exp(x)
+    y = ex / (1 + ex)
+    return y, PullbackMemory(σ; y=y, ex=ex)
 end
 
 pullback_at(pb::PullbackMemory{typeof(σ)}, ȳ) = ȳ * pb.y / (1 + pb.ex)
@@ -256,8 +256,8 @@ x̄ = pb(ȳ)
 ```
 ```julia
 function augmented_primal(::typeof(sin), x)
-  y, cx = sincos(x)
-  return y, PullbackMemory(sin; cx=cx)
+    y, cx = sincos(x)
+    return y, PullbackMemory(sin; cx=cx)
 end
 (pb::PullbackMemory{typeof(sin)})(ȳ) = ȳ * pb.cx
 ```
@@ -271,9 +271,9 @@ end
 ```
 ```julia
 function augmented_primal(::typeof(σ), x)
-  ex = exp(x)
-  y = ex / (1 + ex)
-  return y, PullbackMemory(σ; y=y, ex=ex)
+    ex = exp(x)
+    y = ex / (1 + ex)
+    return y, PullbackMemory(σ; y=y, ex=ex)
 end
 
 (pb::PullbackMemory{typeof(σ)})(ȳ) = ȳ * pb.y / (1 + pb.ex)
@@ -295,16 +295,16 @@ Let's go back and think about the changes we would have make to go from our orig
 To rewrite that original formulation in the new pullback form we have:
 ```julia
 function augmented_primal(::typeof(sin), x)
-  y = sin(x)
-  return y, PullbackMemory(sin; x=x)
+    y = sin(x)
+    return y, PullbackMemory(sin; x=x)
 end
 (pb::PullbackMemory)(ȳ) = ȳ * cos(pb.x)
 ```
 To go from that to:
 ```julia
 function augmented_primal(::typeof(sin), x)
-  y, cx = sincos(x)
-  return y, PullbackMemory(sin; cx=cx)
+    y, cx = sincos(x)
+    return y, PullbackMemory(sin; cx=cx)
 end
 (pb::PullbackMemory)(ȳ) = ȳ * pb.cx
 ```
@@ -317,17 +317,17 @@ end
 ```
 ```julia
 function augmented_primal(::typeof(σ), x)
-  y = σ(x)
-  return y, PullbackMemory(σ; y=y, x=x)
+    y = σ(x)
+    return y, PullbackMemory(σ; y=y, x=x)
 end
 (pb::PullbackMemory{typeof(σ)})(ȳ) = ȳ * pb.y * σ(-pb.x)
 ```
 to get to:
 ```julia
 function augmented_primal(::typeof(σ), x)
-  ex = exp(x)
-  y = ex/(1 + ex)
-  return y, PullbackMemory(σ; y=y, ex=ex)
+    ex = exp(x)
+    y = ex/(1 + ex)
+    return y, PullbackMemory(σ; y=y, ex=ex)
 end
 (pb::PullbackMemory{typeof(σ)})(ȳ) = ȳ * pb.y/(1 + pb.ex)
 ```
@@ -356,9 +356,9 @@ Replacing `PullbackMemory` with a closure that works the same way lets us avoid
 ```
 ```julia
 function augmented_primal(::typeof(sin), x)
-  y, cx = sincos(x)
-  pb = ȳ -> cx * ȳ  # pullback closure. closes over `cx`
-  return y, pb
+    y, cx = sincos(x)
+    pb = ȳ -> cx * ȳ  # pullback closure. closes over `cx`
+    return y, pb
 end
 ```
 ```@raw html
@@ -370,10 +370,10 @@ end
 ```
 ```julia
 function augmented_primal(::typeof(σ), x)
-  ex = exp(x)
-  y = ex / (1 + ex)
-  pb = ȳ -> ȳ * y / (1 + ex)  # pullback closure. closes over `y` and `ex`
-  return y, pb
+    ex = exp(x)
+    y = ex / (1 + ex)
+    pb = ȳ -> ȳ * y / (1 + ex)  # pullback closure. closes over `y` and `ex`
+    return y, pb
 end
 ```
 ```@raw html

docs/src/design/many_tangents.md

@@ -45,7 +45,7 @@ Structural tangents are derived from the structure of the input.
 Either automatically, as part of the AD, or manually, as part of a custom rule.
 
 Consider the structure of `DateTime`:
-```julia
+```julia-repl
 julia> dump(now())
 DateTime
   instant: UTInstant{Millisecond}
@@ -83,15 +83,15 @@ Where there is no natural tangent type for the outermost type but there is for s
 
 Consider if we had a representation of a country's GDP as output by some continuous time model like a Gaussian Process, where that representation is as a sequence of `TimeSample`s
 structured as follows:
-```julia
+```julia-repl
 julia> struct TimeSample
            time::DateTime
            value::Float64
        end
 ```
 
 We can look at its structure:
-```julia
+```julia-repl
 julia> dump(TimeSample(now(), 2.6e9))
 TimeSample
   time: DateTime

docs/src/index.md

@@ -191,6 +191,7 @@ end
 # output
 
 ```
+
 ```jldoctest index
 #### Find dfoo/dx via rrules
 #### First the forward pass, gathering up the pullbacks

docs/src/rule_author/converting_zygoterules.md

@@ -1,4 +1,4 @@
-# Converting ZygoteRules.@adjoint to `rrule`s
+# Converting `ZygoteRules.@adjoint` to `rrule`s
 
 [ZygoteRules.jl](https://github.com/FluxML/ZygoteRules.jl) is a legacy package similar to ChainRulesCore but supporting [Zygote.jl](https://github.com/FluxML/Zygote.jl) only.
 

docs/src/rule_author/example.md

@@ -38,8 +38,9 @@ end
 ```
 
 We can check this rule against a finite-differences approach using [`ChainRulesTestUtils`](https://github.com/JuliaDiff/ChainRulesTestUtils.jl):
-```julia
+```julia-repl
 julia> using ChainRulesTestUtils
+
 julia> test_rrule(foo_mul, Foo(rand(3, 3), 3.0), rand(3, 3))
 Test Summary:                                       | Pass  Total
 test_rrule: foo_mul on Foo{Float64},Matrix{Float64} |   10     10

docs/src/rule_author/which_functions_need_rules.md

@@ -34,8 +34,9 @@ function addone(a::AbstractArray)
 end
 ```
 complains that
-```julia
+```julia-repl
 julia> using Zygote
+
 julia> gradient(addone, a)
 ERROR: Mutating arrays is not supported
 ```
@@ -50,7 +51,7 @@ function ChainRules.rrule(::typeof(addone), a)
 end
 ```
 the gradient can be evaluated:
-```julia
+```julia-repl
 julia> gradient(addone, a)
 ([1.0, 1.0, 1.0],)
 ```
@@ -86,7 +87,7 @@ function exception(x)
 end
 ```
 does not work
-```julia
+```julia-repl
 julia> gradient(exception, 3.0)
 ERROR: Compiling Tuple{typeof(exception),Int64}: try/catch is not supported.
 ```
@@ -101,7 +102,7 @@ function ChainRulesCore.rrule(::typeof(exception), x)
 end
 ```
 
-```julia
+```julia-repl
 julia> gradient(exception, 3.0)
 (6.0,)
 ```
@@ -123,9 +124,11 @@ function mse(y, ŷ)
 end
 ```
 takes a lot longer to AD through
-```julia
-julia> y = rand(30)
-julia> ŷ = rand(30)
+```julia-repl
+julia> y = rand(30);
+
+julia> ŷ = rand(30);
+
 julia> @btime gradient(mse, $y, $ŷ)
   38.180 μs (993 allocations: 65.00 KiB)
 ```
@@ -142,7 +145,7 @@ function ChainRules.rrule(::typeof(mse), x, x̂)
 end
 ```
 which is much faster
-```julia
+```julia-repl
 julia> @btime gradient(mse, $y, $ŷ)
   143.697 ns (2 allocations: 672 bytes)
 ```
@@ -159,7 +162,7 @@ function sum3(array)
     return x+y+z
 end
 ```
-```julia
+```julia-repl
 julia> @btime gradient(sum3, rand(30))
   424.510 ns (9 allocations: 2.06 KiB)
 ```
@@ -176,7 +179,7 @@ function ChainRulesCore.rrule(::typeof(sum3), a)
 end
 ```
 turns out to be significantly faster 
-```julia
+```julia-repl
 julia> @btime gradient(sum3, rand(30))
   192.818 ns (3 allocations: 784 bytes)
 ```

docs/src/rule_author/writing_good_rules.md

@@ -110,7 +110,7 @@ Because `typeof(Bar)` is `DataType`, using this to define an `rrule`/`frule` wil
 
 You can check which to use with `Core.Typeof`:
 
-```julia
+```julia-repl
 julia> function foo end
 foo (generic function with 0 methods)
 
@@ -254,7 +254,7 @@ function ChainRulesCore.rrule(::typeof(double_it), x)
 end
 ```
 Ends up infering a return type of `Any`
-```julia
+```julia-repl
 julia> _, pullback = rrule(double_it, [2.0, 3.0])
 ([4.0, 6.0], var"#double_it_pullback#8"(Core.Box(var"#double_it_pullback#8"(#= circular reference @-2 =#))))
 
@@ -289,7 +289,7 @@ function ChainRulesCore.rrule(::typeof(double_it), x)
 end
 ```
 This infers just fine:
-```julia
+```julia-repl
 julia> _, pullback = rrule(double_it, [2.0, 3.0])
 ([4.0, 6.0], _double_it_pullback)