apl: mandelbrot real-axis batched z=z²+c (+9 tests, 296/296)

lib/apl/tests/programs/mandelbrot.apl

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+⍝ Mandelbrot — real-axis subset
+⍝
+⍝ For complex c, the Mandelbrot set is { c : |z_n| stays bounded } where
+⍝   z_0 = 0, z_{n+1} = z_n² + c.
+⍝ Restricting c (and z) to ℝ gives the segment c ∈ [-2, 1/4]
+⍝ where the iteration stays bounded.
+⍝
+⍝ Rank-polymorphic batched-iteration form:
+⍝   mandelbrot ← {⍵ ⍵⍵ ⍺⍺ +,(⍺⍺ × ⍺⍺) }
+⍝
+⍝ Pseudocode (as we don't have ⎕ system fns yet):
+⍝   z      ← 0×c                             ⍝ start at zero
+⍝   alive  ← 1+0×c                           ⍝ all "still in"
+⍝   for k iterations:
+⍝     alive ← alive ∧ 4 ≥ z×z                ⍝ still bounded?
+⍝     z     ← alive × c + z×z                ⍝ freeze escaped via mask
+⍝     count ← count + alive                  ⍝ tally surviving iters
+⍝
+⍝ Examples (count after 100 iterations):
+⍝   c=0      : 100  (z stays at 0)
+⍝   c=-1     : 100  (cycles 0,-1,0,-1,...)
+⍝   c=-2     : 100  (settles at 2 — boundary)
+⍝   c=0.25   : 100  (boundary — converges to 0.5)
+⍝   c=0.5    : 5    (escapes by iteration 6)
+⍝   c=1      : 3    (escapes quickly)
+⍝
+⍝ Real-axis Mandelbrot set: bounded for c ∈ [-2, 0.25].
+
+mandelbrot ← {z←alive←count←0×⍵ ⋄ {alive←alive∧4≥z×z ⋄ z←alive×⍵+z×z ⋄ count+←alive}⍣⍺⊢⍵}