feat: add encoder tuning harness for 4K60 fractal videos

make_clips.sh renders lossless FFV1 4K60 test clips through the Makefile's
colour conversion; sweep.sh encodes them with a list of ffmpeg configs and
scores each with VMAF 4K (mean/p1/min), VMAF-NEG, CAMBI banding and PSNR-Y.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

.gitignore

@@ -3,4 +3,4 @@ Cargo.lock
 dist
 frames*
 out.mp4
-
+__pycache__

CLAUDE.md

@@ -6,9 +6,15 @@ This file provides guidance to Claude Code (claude.ai/code) when working with co
 
 A deep-zoom fractal explorer (Rust + wgpu + egui + WGSL). It zooms past the
 ~10¹³× limit of plain `f64` using **perturbation theory**: one high-precision
-reference orbit is computed on the CPU (arbitrary precision via `dashu-float`),
+reference orbit is computed on the CPU (arbitrary precision via `bignum::Big`:
+`rug` natively, `malachite-float` on the web),
 and every pixel is rendered on the GPU as a cheap `f32` delta from it, with
-rebasing to avoid glitches. The `f32` GPU tier reaches roughly 10³⁰×. Runs
+rebasing to avoid glitches. Plain `f32` deltas run out of exponent range
+once a pixel is ~2^-124 wide (~10³⁴× at 1080p), so from 2^-122 per pixel
+(`view::DEEP_PIXEL_SIZE`) a `DEEP` shader variant starts each pixel with
+rescaled deltas (f32 mantissa × 2^i32). There's no practical depth limit:
+`half_height` is a `view::Scale` (f64 mantissa × 2^i32), floored only at
+`Scale::MIN` = 2^-(2^20) to keep shader exponent sums in i32. Runs
 natively (Vulkan/Metal/DX12) and in the browser (WebGPU only — WebGL2 can't do
 storage buffers, which the fragment shader needs for the reference orbit).
 
@@ -17,9 +23,12 @@ storage buffers, which the fragment shader needs for the reference orbit).
 ```sh
 cargo run --release          # native, run (release matters: fractal math is hot)
 cargo test                   # reference-orbit math, share-link round-trip, WGSL validation
+cargo test --features wasm   # same, on the web build's malachite big-float backend
 cargo test --test shader_valid   # just the WGSL parse/validate tests (naga, no GPU needed)
 cargo clippy
 cargo fmt                    # rustfmt.toml just pins edition = "2024"
+cargo build --release --no-default-features   # headless-only binary: no eframe/egui (the default `gui` feature)
+tools/bench/bench.sh --rev HEAD   # perf of uncommitted edits vs HEAD (hyperfine; see tools/bench/README.md)
 ```
 
 Web build (WebGPU):
@@ -27,14 +36,16 @@ Web build (WebGPU):
 ```sh
 rustup target add wasm32-unknown-unknown
 cargo install wasm-bindgen-cli --version 0.2.128   # must match the wasm-bindgen crate version
-./build-web.sh                                     # -> ./dist
+./build-web.sh                                     # -> ./dist (builds with --features wasm)
 python3 -m http.server -d dist 8080
 ```
 
 Native CLI flags (`src/cli.rs`, applied in `FractalApp::apply_cli`): `--kind`,
 `--power`, `--julia re,im`, `--phoenix-p re,im`, `--lambda-l re,im`,
 `--palette`, `--share <fragment>`,
-`--view re,im,half_height[,iterations]`, `--de`, `--buddhabrot`,
+`--view re,im,half_height[,iterations]`, `--rendering-kind`,
+`--yaw`/`--pitch` (3D camera, degrees), `--de`, `--antialias` (2×2),
+`--buddhabrot`,
 `--buddha-palette`. `--headless` (`src/headless.rs`) skips the window
 entirely: it builds the same view from the other flags, creates its own
 offscreen wgpu device, and renders straight to a PNG (`--width`/`--height`,
@@ -42,20 +53,43 @@ default 1920×1080, `--export-path out.png`) without needing a GPU-backed
 window/event loop. Not yet supported with `--buddhabrot`. Run
 `mandelbrot --help` for the full list.
 
-`--headless` also has an animation mode, for feeding into `ffmpeg`: add
+`--headless` also has an animation mode, for feeding into `ffmpeg`: give any
-`--to-view re,im,half_height[,iterations]` (or `--to-share <fragment>`, which
+end-state flag alongside the start flags (`--view`/`--share`/`--kind`/
-only pulls position/zoom/iterations out of the link) alongside a start view
+`--julia`/...), plus `--frames N` or `--fps`/`--duration`. End-state flags:
-(`--view`/`--share`/`--kind`/`--julia`), plus `--frames N` or
+`--to-view re,im,half_height[,iterations]` or `--to-share <fragment>` (only
-`--fps`/`--duration`. `--export-path` then names an output *directory* of
+position/zoom/iterations are pulled out of the link), `--to-iterations`,
-`frame-00001.png`, `frame-00002.png`, ... instead of a single file. Only the
+`--to-julia`, `--to-phoenix-p`, `--to-lambda-l`, `--to-complex-power`
-camera (center + half-height) is animated — kind, colors, and per-kind
+(or `--to-complex-power-re`/`--to-complex-power-im` to move one component),
-constants stay fixed at whatever the start flags set. `view::interpolate_view`
+and `--to-kind` (per-step formula blend via `KindMorph`, camera untouched).
-does the interpolation: half-height geometrically (log-linear, since zoom
+Anything without a target stays at its start value; colors stay fixed.
-spans many decades), center linearly through the complex plane at full
+`--export-path` then names an output *directory* of `frame-00001.png`,
-`Big` precision; `--linear` swaps the default smoothstep easing for constant
+`frame-00002.png`, ... instead of a single file. `--export-path -` writes
-pacing. Iteration count auto-scales with zoom depth per frame (same
+to stdout instead (refused on a terminal): the PNG for a still, or for an
-`auto_iteration_count` the interactive app uses while zooming), overriding
+animation raw RGBA8 frames in order (`unpad_rgba`, no PNG encode) for
-any iteration count from `--view`/`--share`/`--to-view`/`--to-share`.
+`ffmpeg -f rawvideo -pix_fmt rgba -s WxH -r FPS -i -`. A single writer
+thread reorders the frames. Its buffer is bounded by the orbit workers not
+starting a frame more than `window` past the last one written, not by
+blocking the writer, which could deadlock. `headless.rs::AnimTargets`
+collects the targets; the export pipeline is rebuilt only when the
+`PipelineKey` changes between frames (kind morph). `view::interpolate_view`
+does the camera: half-height geometrically (log-linear, since zoom spans many
+decades), center linearly through the complex plane at full `Big` precision;
+constants interpolate linearly. `--linear` swaps the default smoothstep
+easing for constant pacing. `--shards N --shard K` (1-based) renders only
+the K-th of N contiguous parts (`shard_range`), still timed against the whole
+animation (global `t`, global `frame-NNNNN.png` numbers; stdout streams just
+that part), so separately rendered clips join seamlessly. Without `--to-iterations` (or a share link's),
+iteration count auto-scales with zoom depth per frame (same
+`auto_iteration_count` the interactive app uses while zooming).
+`--to-yaw`/`--to-pitch` (degrees, from `--yaw`/`--pitch`, yaw unwrapped so
+`--to-yaw 720` is two turns) orbit the 3D camera with `--rendering-kind 3d`.
+Frames are pipelined across every core (`run_animation`): each frame's
+state is a pure function of `t` (`apply_frame`), so all frames'
+`FractalApp::reference_job`s are snapshotted up front and `RefJob::compute`d
+by a worker pool. The main thread renders them on the GPU as they arrive
+(out of order), and another pool PNG-encodes and writes them
+(`encode_png`, `Compression::Fast`). Channels are bounded. Once orbits and
+encoding are off the main thread, the GPU is usually the bottleneck.
 
 There's no GPU in most sandboxes: `cargo check`/`cargo test --test shader_valid`
 are the fast, headless way to validate a change. `cargo test` also runs but
@@ -74,9 +108,32 @@ runs out), rebase: `e ← y_n − X_0`, restart the reference index at 0. This i
 what makes deep zoom cheap — one expensive high-precision orbit, then every
 pixel is a handful of `f32` complex multiplies.
 
-- `src/view.rs` — `ViewState`; center is arbitrary-precision `FBig` (`Big`
+- `src/bignum/` — `Big`, the arbitrary-precision binary float, with one
-  type alias), pixel scale stays `f64` (still in-range at 10³⁰×). Precision
+  backend per library behind the same inherent API + operators: `rug`
-  (bits) scales with zoom depth (`precision_for`).
+  (GMP/MPFR, default, fastest, can't target wasm32) and `malachite-float`
+  (pure Rust, the `wasm` feature, required for the web build; a
+  `compile_error!` enforces it). Cargo features are additive, so rug is a
+  non-wasm32 target dependency and the backend is picked by
+  `cfg(feature = "wasm")`. Both follow the precision rule: a result has the
+  larger operand precision, rounded to nearest; shifts are exact. Malachite's
+  zero has no precision, so its wrapper stores `prec` alongside. Any new
+  `Big` operation must be added to both backends (`cargo test` and
+  `cargo test --features wasm` run the same tests on each).
+- `src/view.rs` — `ViewState`; center is arbitrary-precision `Big`
+  (re-exported as `view::Big`). The pixel scale (`half_height`) is a `Scale`, an f64
+  mantissa with its own i32 exponent, so it goes past f64's ~1e-308. Never
+  collapse it (or a center difference) to a plain `f64` on a path used at
+  depth. Rescale first: `Scale::scaled_f64(k)`, or shift the `Big` by
+  `-scale_exp` before `to_f64()`, as `dc_offset`/`drift_from` do.
+  `Display`/`FromStr` use scientific notation of any exponent (share links,
+  `--view`, the zoom field). Precision (bits) scales with zoom depth
+  (`precision_for`). `needs_deep` switches rendering to the deep pipeline
+  once a pixel of the full-resolution render is below `DEEP_PIXEL_SIZE`
+  (2^-122; the f32 path is exact down to 2^-124 with AA's quarter-pixel
+  offsets, measured, and the deep path is ~40% slower, so the switch is as
+  late as that allows). `deep_scale_exp` gives the scale exponent.
+  `make_uniforms(aspect, height_px)` takes that full-resolution height, the
+  same during the interaction-downscaled pass so the pipeline doesn't flip.
 - `src/fractal/kind.rs` — the `FractalKind` enum (Mandelbrot, Burning Ship,
   Tricorn, Multibrot, Celtic, Perpendicular, Buffalo, Phoenix, Lambda,
   Complex Multibrot) plus everything that only needs to switch on it:
@@ -88,10 +145,21 @@ pixel is a handful of `f32` complex multiplies.
   `f32` pairs — that's the reference orbit the GPU perturbs from. At
   precision ≤ `F64_MAX_PRECISION` (80 bits, i.e. shallow views) it takes a
   plain-`f64` fast path (`compute_reference_f64`), so each kind's formula
-  exists twice in this file (f64 + `FBig`) and both must stay in sync;
+  exists twice in this file (f64 + `Big`) and both must stay in sync;
-  `f64_fast_path_matches_big` checks they agree. Requests are made with 1.5×
+  `f64_fast_path_matches_big` checks they agree. The result is a `RefOrbit`:
+  `points` plus a parallel `exps`. A point below 2^-100 (only possible on the
+  `Big` path) is stored as a normalized mantissa with its exponent in `exps`
+  (the true value is `points[n]·2^exps[n]`). That happens when the orbit
+  passes near 0 at a deep minibrot. `has_scaled()` then forces the deep
+  pipeline, the only one that reads `exps`. Requests are made with 1.5×
   iteration headroom (`reference_iterations` in `app.rs`), so auto-iterations
-  creeping up during a zoom doesn't recompute the orbit every frame.
+  creeping up during a zoom doesn't recompute the orbit every frame. The
+  interactive reference buffers start at 2^17 points and grow on demand
+  (`FractalRenderer::ensure_ref_capacity`) up to `MAX_REF_POINTS` (2^24, the
+  128 MiB WebGPU default binding size). That is also the hard iteration
+  ceiling (`app.rs::MAX_ITERATIONS`), because the shader treats an exhausted
+  reference as escaped. The UI slider only goes to 100k when dragged; typed
+  values can go higher.
 - `src/shaders/*.wgsl` — none of these are standalone WGSL modules; WGSL has
   no `#include`, so each is compiled by concatenating plain-text fragments
   with `concat!`/`include_str!` at the `create_shader_module` call site (see
@@ -110,16 +178,19 @@ pixel is a handful of `f32` complex multiplies.
   pipeline creation. Read those constants in the shader, never `u.kind` /
   `u.is_julia` / `u.de_coloring` (they're still uploaded for layout reasons).
   `renderer.rs` builds one pipeline set per `PipelineKey` lazily on first
-  use, and `tests/shader_valid.rs` compiles every kind × Julia × DE variant to
+  use, and `tests/shader_valid.rs` compiles every kind × Julia × DE × morph ×
-  SPIR-V. So a new kind needs no pipeline-list change, only its `KIND_*`
+  deep variant to SPIR-V. So a new kind needs no pipeline-list change, only its `KIND_*`
   constant. `buddhabrot.wgsl` does the same with its own `override KIND`.
   Interior pixels exit early through **periodicity detection**. It uses
   Brent-style checkpoints plus two guards: the cycle's multiplier must be
   clearly attracting (`PERIOD_MAX_MULT2`), and the contracting return must
   repeat in `PERIOD_CONFIRMATIONS` consecutive windows. Both guards are
   needed: without them, exterior pixels at cusps and minibrot edges turned
-  black. Retune them only against f64 ground truth on such views. Phoenix is
+  black. Retune them only against f64 ground truth on such views. Each
-  excluded (two-term map).
+  window saves its iterate closest to the critical point, not the one at the
+  checkpoint. At an arbitrary phase the relative tolerance is far coarser
+  than a deep minibrot's scale, and a black disk surrounded the minibrot
+  (seen at ~1e-13 zoom). Phoenix is excluded (two-term map).
   `advance_delta(z, e)` is the per-kind delta step (`z` = reference point,
   `e` = current delta); the caller adds `step_add` (= `dc`) afterward — this
   relies on `c` being additive in every current kind's formula (a kind where
@@ -133,18 +204,47 @@ pixel is a handful of `f32` complex multiplies.
   are `advance_delta_kind`/`fprime_kind`; `advance_delta`/`fprime` wrap them
   to blend two kinds during the kind-switch morph (`u.morph_from`,
   `u.morph_w`: each step is `(1-w)·f_kind + w·f_from`, mirrored on the CPU by
-  the `morph` argument of `compute_reference`, in both its f64 and `FBig`
+  the `morph` argument of `compute_reference`, in both its f64 and `Big`
   paths). The blend only exists in pipelines built with the `MORPH` override
   (part of `PipelineKey`, on while `morph_w > 0`); those also skip periodicity
   detection and the cardioid bypass. App side: `KindMorph` in
   `app.rs`; the uniforms use the morph the *current reference* was built with
   (`ref_morph`), not the live one, so orbit and delta formula never disagree.
+  **Deep views** (`DEEP` override, `u.scale_exp != 0`) handle zooms where
+  f32 deltas underflow. `make_uniforms` sets `scale_exp = E` (≈ log2 of the
+  half-height) and uploads `span`/`dc_offset` × 2^-E. The per-pixel `offset`
+  and `px` are therefore in units of 2^E. `iterate_sample` first runs a
+  **deep prologue**:
+  - The delta is carried as `w·2^sx` and the DE derivative as `v·2^sv`
+    (separate exponents, since they drift apart near the critical point).
+  - Each step goes through `advance_delta_scaled` →
+    `deep_step_kind` → `advance_delta_scaled_kind`. These return the step at
+    its own output scale `t`. Next to the critical point (X tiny or 0), the
+    linear term vanishes and the step's value is ~e^p, far below 2^sx.
+    `deep_step_kind` measures X and e in a common unit (the kinds are
+    p-homogeneous) and the loop moves `sx` there. Assuming the e² terms merely
+    flush when negligible was wrong exactly there: pixels near deep minibrots
+    lost their delta and followed the reference forever.
+  - Rebasing uses X at full range (`ref_fe`).
+  - Once `|e| > 2^DEEP_EXIT_LOG2` (and dzs is normal), the state converts to
+    f32 and the ordinary loop continues from the same `n`/`m`.
+  - Periodicity detection restarts after the prologue with a sentinel save,
+    because saving the hand-off `z` (an arbitrary phase) made exterior pixels
+    shadowing a periodic nucleus reference read as interior.
+
+  The deep path is exact at any depth: forcing it everywhere (raise
+  `DEEP_PIXEL_SIZE`, raise `DEEP_EXIT_LOG2` to about -8) must reproduce the
+  plain f32 renders on non-chaotic views. That's the check to rerun after
+  changing it. Known gaps: Lambda's critical point is 1/2, so its step keeps
+  the input scale. Lambda set mode's reference sits at the origin, so it
+  never reaches deep zooms anyway.
 - `src/fractal/renderer.rs` — `FractalRenderer` (wgpu pipelines, uniform +
   storage buffers, bind groups), `Uniforms` (repr(C) layout that must match
   the WGSL `Uniforms` struct field-for-field, including padding; it includes
   CPU-precomputed data: `cm_coef`, the Complex Multibrot binomial
-  coefficients from `app.rs::complex_binomials`, and `light_count` for the
+  coefficients from `app.rs::complex_binomials`, `light_count` for the
-  packed `GpuLight` buffer from `lights.rs::gpu_lights`), and
+  packed `GpuLight` buffer from `lights.rs::gpu_lights`, and `scale_exp`,
+  the deep view scale), `ref_exp_buffer` (binding 3, `RefOrbit::exps`), and
   `FractalCallback` (the `egui_wgpu::CallbackTrait` impl: `prepare()` uploads
   changed buffers and decides whether to re-run the iterate pass, the cheap
   colourise pass, or just blit the cached texture). Also `ExportRender`, a
@@ -158,7 +258,10 @@ pixel is a handful of `f32` complex multiplies.
 - `src/app.rs` — `FractalApp` (the egui app + all UI). Key methods:
   `should_request`/`ensure_reference` (decide when the reference is stale and
   dispatch/collect it), `make_uniforms` (assemble the per-frame `Uniforms`),
-  `tick_animations` (drives the "morph c/p/λ" and auto-zoom animations),
+  `tick_animations` (drives the interactive animations: colour cycle,
+auto-zoom, c/p/λ circle drift via `ConstOrbit`, per-component complex-power
+oscillation via `AxisOsc`, 3D camera orbit, kind cycling through
+`switch_kind`),
   `default_view_for` (wraps `FractalKind::default_set_view`, adding the
   kind-independent Julia case). `JULIA_PRESETS` and `SET_PRESETS` are sized as
   `[T; FractalKind::<last variant> as usize + 1]` — adding a new `FractalKind`
@@ -174,7 +277,9 @@ Touches, in order: `kind.rs` (enum variant + `ALL` slot + `label`/
 `description`/`formula`/`share_tag`/`from_share_tag`/`default_set_view`
 arms), `reference.rs` (CPU iteration formula arm, and a test comparing
 against a naive `f64` iteration), `common.wgsl` (matching `KIND_*` const),
-`mandelbrot.wgsl` (matching `advance_delta`/`fprime` arms), `buddhabrot.wgsl`
+`mandelbrot.wgsl` (matching `advance_delta`/`fprime` arms, plus the deep
+path's `advance_delta_scaled_kind` arm, its degree in `deep_step_kind` and,
+if not z²-like, a `deep_fprime` arm), `buddhabrot.wgsl`
 (matching arm in `advance()`, if the kind makes sense as a Buddhabrot),
 `renderer.rs` `Uniforms` (only if the kind needs a new per-kind constant,
 e.g. Phoenix's `phoenix_p`), `app.rs` (`JULIA_PRESETS`/`SET_PRESETS` slot,
@@ -211,7 +316,10 @@ sample per pixel. When AA is on, `fs_refine` reads that texture and runs the
 2×2 grid only on pixels whose 4-neighbours differ (interior/exterior edge, or
 `ci`/DE beyond `AA_CI_EPS`/`AA_DE_EPS`), copying the rest. Colourise then
 reads the refined texture. PNG export (`fs_color`) still supersamples every
-pixel.
+pixel, except in 3D: the raymarcher needs the whole height field, so a 3D
+`ExportRender` (`RaymarchExport`) runs the interactive chain instead, with
+its tiles iterating `fs_data` into its own data texture and the last tile adding
+refine + colourise into the target.
 
 The 3D view (`colorize.wgsl::ray_marching`) sphere-traces the DE height
 field straight from the data texture. It's cheap: rays start on the z = 0

Cargo.toml

@@ -3,19 +3,31 @@ name = "mandelbrot"
 version = "0.1.0"
 edition = "2024"
 
+[features]
+default = ["gui"]
+# Windowed egui app. Without it only `--headless` rendering is built.
+gui = ["dep:eframe", "dep:egui"]
+# Pure-Rust big floats for the web build (`rug` needs GMP/MPFR, which can't
+# target wasm32). Required for wasm32, optional natively (to test that backend).
+wasm = ["dep:malachite-float", "dep:malachite-base"]
+
 [dependencies]
 bytemuck = { version = "1.25.2", features = ["derive"] }
-dashu-float = "0.6.0"
+eframe = { version = "0.36.2", default-features = false, features = ["wgpu", "default_fonts", "x11", "wayland", "accesskit"], optional = true }
-eframe = { version = "0.36.2", default-features = false, features = ["wgpu", "default_fonts", "x11", "wayland", "accesskit"] }
+egui = { version = "0.36.2", optional = true }
-egui = "0.36.2"
+ecolor = { version = "0.36.2", features = ["bytemuck"] }
+wgpu = "30.0.1"
 glam = "0.33.8"
 log = "0.4.34"
 png = "0.18.1"
+malachite-float = { version = "0.12", optional = true }
+malachite-base = { version = "0.12", optional = true }
 
 [target.'cfg(not(target_arch = "wasm32"))'.dependencies]
 env_logger = "0.11.11"
 clap = { version = "4.5.51", features = ["derive"] }
 pollster = "1.0.1"
+rug = { version = "1.30.0", default-features = false, features = ["float", "std"] }
 
 [target.'cfg(target_arch = "wasm32")'.dependencies]
 futures-channel = { version = "0.3.34", default-features = false, features = ["alloc", "std"] }
@@ -32,7 +44,7 @@ opt-level = 3
 # codegen-units = 1
 debug = true
 
-# Dev: keep our own crate debuggable, but optimize dependencies (dashu, wgpu,
+# Dev: keep our own crate debuggable, but optimize dependencies (big floats, wgpu,
 # egui) so the explorer is actually interactive during development.
 [profile.dev]
 opt-level = 1

Makefile

@@ -0,0 +1,35 @@
+.PHONY: hevc-nvenc x265 av1
+
+FFMPEG=ffmpeg
+IN_FMT=rgba
+
+RESOLUTION=1920x1080
+FPS=60
+OUT=out.mp4
+
+FFMPEG_CMD=$(FFMPEG) -f rawvideo -pix_fmt $(IN_FMT) -s $(RESOLUTION) -framerate $(FPS) -i - -vf "scale=out_color_matrix=bt709:out_range=tv:flags=accurate_rnd+full_chroma_int+bitexact,format=p010le"
+
+hevc-nvenc:
+	$(FFMPEG_CMD) \
+		-c:v hevc_nvenc -preset p7 -tune hq -rc vbr -cq 14 -b:v 0 -maxrate 200M -bufsize 400M \
+		-profile:v main10 -pix_fmt p010le \
+		-spatial-aq 1 -aq-strength 6 -temporal-aq 1 \
+		-rc-lookahead 32 -bf 4 -b_ref_mode middle -multipass fullres \
+		-colorspace bt709 -color_primaries bt709 -color_trc bt709 -color_range tv \
+		-tag:v hvc1 -movflags +faststart \
+		$(OUT)
+
+x265:
+	$(FFMPEG_CMD) \
+		-c:v libx265 -crf 18 -preset medium -pix_fmt yuv420p10le \
+		-x265-params "aq-mode=3:no-sao=1" \
+		-colorspace bt709 -color_primaries bt709 -color_trc bt709 -color_range tv \
+		-tag:v hvc1 -movflags +faststart \
+		$(OUT)
+
+av1:
+	$(FFMPEG_CMD) \
+		-c:v libsvtav1 -crf 24 -preset 6 -pix_fmt yuv420p10le \
+		-svtav1-params "film-grain=0:enable-overlays=0" \
+		-colorspace bt709 -color_primaries bt709 -color_trc bt709 -color_range tv \
+		$(OUT)

README.md

@@ -3,7 +3,8 @@
 A fast, interactive deep-zoom fractal explorer — Mandelbrot and Julia sets —
 built with **Rust + wgpu + egui + WGSL**. It zooms far past the ~10¹³× limit of
 plain `f64` using **perturbation theory**: one high-precision reference orbit is
-computed on the CPU (arbitrary precision via `dashu-float`), and every pixel is
+computed on the CPU (arbitrary precision via `rug` natively, `malachite-float`
+on the web), and every pixel is
 rendered on the GPU as a cheap `f32` delta from it, with **rebasing** to avoid
 glitches. Runs natively (Vulkan/Metal/DX12) and in the browser (WebGPU).
 
@@ -27,6 +28,9 @@ The `f32` GPU tier reaches roughly **10³⁰× magnification** with sharp detail
 cargo run --release
 ```
 
+Native builds use `rug` (GMP/MPFR) for the reference orbit, which needs a C
+toolchain and `m4` (on Windows, MSYS2).
+
 ## Build & run — web (WebGPU)
 
 Requires the `wasm32-unknown-unknown` target and `wasm-bindgen-cli` (matching the
@@ -37,6 +41,7 @@ rustup target add wasm32-unknown-unknown
 cargo install wasm-bindgen-cli --version 0.2.128   # once
 
 ./build-web.sh                       # outputs ./dist (index.html, .js, .wasm)
+                                     # (builds with --features wasm: pure-Rust big floats)
 python3 -m http.server -d dist 8080  # serve over http
 ```
 
@@ -60,7 +65,8 @@ qualifies. Deploy by serving the `dist/` directory as static files.
 
 ## How it works
 
-- `src/view.rs` — view state. Center is arbitrary precision (`FBig`); the pixel
+- `src/view.rs` — view state. Center is arbitrary precision (`Big`, from
+  `src/bignum/`); the pixel
   scale stays `f64` (even at 10³⁰× it is ~10⁻³³, within `f64` range).
 - `src/fractal/reference.rs` — high-precision reference orbit `Z_{n+1}=Z_n²+C`.
 - `src/shaders/mandelbrot.wgsl` — per-pixel perturbation `e_{n+1}=2·Z_n·e_n+e_n²+δc`
@@ -84,6 +90,7 @@ reference computation to a Web Worker.
 
 ```sh
 cargo test
+cargo test --features wasm   # same suite on the web build's big-float backend
 ```
 
 Covers the reference orbit (vs. a naive `f64` iteration, Mandelbrot and Julia)

build-web.sh

@@ -8,7 +8,7 @@ export PATH="$HOME/.cargo/bin:$PATH"
 OUT="${1:-dist}"
 
 echo "==> cargo build (wasm32, release)"
-cargo build --release --target wasm32-unknown-unknown
+cargo build --release --target wasm32-unknown-unknown --features wasm
 
 echo "==> wasm-bindgen -> $OUT"
 mkdir -p "$OUT"

src/bignum/mod.rs

@@ -0,0 +1,225 @@
+//! Arbitrary-precision binary float [`Big`] (one coordinate of a deep-zoom
+//! center, or of a reference orbit point), over one of two backends:
+//!
+//! - native (default): `rug` (GMP/MPFR), the fastest, but C code that can't
+//!   target `wasm32-unknown-unknown`;
+//! - `wasm` feature: a pure-Rust library, required for the web build.
+//!
+//! Both expose the same inherent API plus the operators below, and follow
+//! the same precision rule: an operation's result has the larger of its
+//! operands' precisions (in bits), rounded to nearest. Shifts (`<<`/`>>` by
+//! an `isize`) are exact multiplications by powers of two.
+
+#[cfg(not(feature = "wasm"))]
+mod rug_backend;
+#[cfg(not(feature = "wasm"))]
+pub use rug_backend::Big;
+
+#[cfg(feature = "wasm")]
+mod web_backend;
+#[cfg(feature = "wasm")]
+pub use web_backend::Big;
+// Native `--features wasm` builds (to test the web backend) still link rug.
+#[cfg(all(feature = "wasm", not(target_arch = "wasm32")))]
+use rug as _;
+
+#[cfg(all(target_arch = "wasm32", not(feature = "wasm")))]
+compile_error!("the web build needs `--features wasm` (rug can't target wasm32)");
+
+use core::ops::{Add, Mul, Neg, Shl, Shr, Sub};
+
+/// Implements `$Trait` for every owned/borrowed combination of `Big`
+/// operands, through the backend's by-reference `$imp`.
+macro_rules! forward_binop {
+    ($Trait:ident, $method:ident, $imp:ident) => {
+        impl $Trait<&Big> for &Big {
+            type Output = Big;
+            fn $method(self, rhs: &Big) -> Big {
+                self.$imp(rhs)
+            }
+        }
+        impl $Trait<Big> for &Big {
+            type Output = Big;
+            fn $method(self, rhs: Big) -> Big {
+                self.$imp(&rhs)
+            }
+        }
+        impl $Trait<&Big> for Big {
+            type Output = Big;
+            fn $method(self, rhs: &Big) -> Big {
+                (&self).$imp(rhs)
+            }
+        }
+        impl $Trait<Big> for Big {
+            type Output = Big;
+            fn $method(self, rhs: Big) -> Big {
+                (&self).$imp(&rhs)
+            }
+        }
+    };
+}
+
+forward_binop!(Add, add, add_ref);
+forward_binop!(Sub, sub, sub_ref);
+forward_binop!(Mul, mul, mul_ref);
+
+impl Neg for Big {
+    type Output = Big;
+    fn neg(self) -> Big {
+        self.negated()
+    }
+}
+
+impl Neg for &Big {
+    type Output = Big;
+    fn neg(self) -> Big {
+        self.clone().negated()
+    }
+}
+
+impl Shl<isize> for Big {
+    type Output = Big;
+    fn shl(self, k: isize) -> Big {
+        self.mul_pow2(k)
+    }
+}
+
+impl Shl<isize> for &Big {
+    type Output = Big;
+    fn shl(self, k: isize) -> Big {
+        self.clone().mul_pow2(k)
+    }
+}
+
+impl Shr<isize> for Big {
+    type Output = Big;
+    fn shr(self, k: isize) -> Big {
+        self.mul_pow2(-k)
+    }
+}
+
+impl Shr<isize> for &Big {
+    type Output = Big;
+    fn shr(self, k: isize) -> Big {
+        self.clone().mul_pow2(-k)
+    }
+}
+
+/// Significant decimal digits of a value, as a backend's
+/// `to_decimal_parts` returns them: the value is `±0.digits × 10^exp10`,
+/// `digits` has no trailing zeros, and it is empty for zero.
+pub struct DecimalParts {
+    pub negative: bool,
+    pub digits: String,
+    pub exp10: isize,
+}
+
+impl DecimalParts {
+    /// Build from a significand digit string (maybe with trailing zeros)
+    /// whose value is `0.digits × 10^exp10`.
+    fn new(negative: bool, digits: &str, exp10: isize) -> Self {
+        let digits = digits.trim_end_matches('0').to_string();
+        Self {
+            negative: negative && !digits.is_empty(),
+            digits,
+            exp10,
+        }
+    }
+}
+
+impl Big {
+    /// Decimal string with `sig_digits` significant digits, in plain
+    /// positional notation (`-0.000123`, `4500`), trailing zeros trimmed.
+    pub fn to_decimal_string(&self, sig_digits: usize) -> String {
+        let DecimalParts {
+            negative,
+            digits,
+            exp10,
+        } = self.to_decimal_parts(sig_digits.max(1));
+        if digits.is_empty() {
+            return "0".to_string();
+        }
+        let sign = if negative { "-" } else { "" };
+        let len = digits.len() as isize;
+        if exp10 <= 0 {
+            let zeros = "0".repeat((-exp10) as usize);
+            format!("{sign}0.{zeros}{digits}")
+        } else if exp10 >= len {
+            let zeros = "0".repeat((exp10 - len) as usize);
+            format!("{sign}{digits}{zeros}")
+        } else {
+            let (int, frac) = digits.split_at(exp10 as usize);
+            format!("{sign}{int}.{frac}")
+        }
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    fn big(x: f64) -> Big {
+        Big::from_f64(x, 200)
+    }
+
+    #[test]
+    fn decimal_string_is_positional() {
+        assert_eq!(
+            big(-0.7515).to_decimal_string(20),
+            "-0.75149999999999994582"
+        );
+        assert_eq!(big(-0.7515).to_decimal_string(5), "-0.7515");
+        assert_eq!(big(0.0).to_decimal_string(5), "0");
+        assert_eq!(big(1.0).to_decimal_string(5), "1");
+        assert_eq!(big(123.456).to_decimal_string(5), "123.46");
+        assert_eq!(big(1e20).to_decimal_string(5), "100000000000000000000");
+        assert_eq!(big(0.1).to_decimal_string(5), "0.1");
+        let tiny = big(1.0) >> 200;
+        let s = tiny.to_decimal_string(5);
+        assert_eq!(s, format!("0.{}6223", "0".repeat(60)));
+    }
+
+    #[test]
+    fn decimal_round_trip() {
+        let s = "-0.743643887037158704752191506114774";
+        let x = Big::from_decimal_str(s, 200).unwrap();
+        assert_eq!(x.to_decimal_string(33), s);
+        assert_eq!(
+            Big::from_decimal_str("1.5e-20", 64).unwrap().to_f64(),
+            1.5e-20
+        );
+        assert!(Big::from_decimal_str("abc", 64).is_none());
+    }
+
+    #[test]
+    fn precision_is_max_of_operands() {
+        let a = Big::from_f64(1.0, 100);
+        let b = Big::from_f64(3.0, 200);
+        assert_eq!((&a + &b).precision(), 200);
+        assert_eq!((&a * &b).precision(), 200);
+        assert_eq!((&b - &a).precision(), 200);
+        assert_eq!(a.clone().with_precision(300).precision(), 300);
+    }
+
+    #[test]
+    fn exact_shifts_and_log2() {
+        let x = Big::from_f64(0.75, 64) >> 5000;
+        assert_eq!(x.log2_floor(), Some(-5001));
+        assert_eq!((x << 5000).to_f64(), 0.75);
+        assert_eq!(Big::zero(64).log2_floor(), None);
+        assert!(Big::from_f64(-1.0, 64).is_negative());
+        assert!(!Big::zero(64).is_negative());
+    }
+
+    #[test]
+    fn transcendentals() {
+        let x = Big::from_f64(0.5, 128);
+        assert!((x.ln().to_f64() - 0.5f64.ln()).abs() < 1e-15);
+        assert!((x.exp().to_f64() - 0.5f64.exp()).abs() < 1e-15);
+        let (s, c) = x.sin_cos();
+        assert!((s.to_f64() - 0.5f64.sin()).abs() < 1e-15);
+        assert!((c.to_f64() - 0.5f64.cos()).abs() < 1e-15);
+        let y = Big::from_f64(-0.3, 128);
+        assert!((y.atan2(&x).to_f64() - (-0.3f64).atan2(0.5)).abs() < 1e-15);
+    }
+}

src/bignum/rug_backend.rs

@@ -0,0 +1,128 @@
+//! [`Big`] over `rug::Float` (MPFR), for native builds.
+
+use core::cmp::Ordering;
+
+use rug::{Assign, Float};
+
+use super::DecimalParts;
+
+/// Arbitrary-precision binary float. See the module docs.
+#[derive(Clone, Debug, PartialEq)]
+pub struct Big(Float);
+
+/// MPFR precision for `bits`, clamped to what it accepts.
+fn prec(bits: usize) -> u32 {
+    (bits as u32).clamp(rug::float::prec_min(), rug::float::prec_max())
+}
+
+impl Big {
+    /// `x` at `bits` of precision (exact when `bits >= 53`). Non-finite `x`
+    /// reads as 0.
+    pub fn from_f64(x: f64, bits: usize) -> Self {
+        let x = if x.is_finite() { x } else { 0.0 };
+        Self(Float::with_val(prec(bits), x))
+    }
+
+    pub fn zero(bits: usize) -> Self {
+        Self(Float::new(prec(bits)))
+    }
+
+    /// Parse a decimal (`-0.75`, `1.5e-20`, any number of digits) at `bits`
+    /// (at least 53) of precision.
+    pub fn from_decimal_str(s: &str, bits: usize) -> Option<Self> {
+        let parsed = Float::parse(s.trim()).ok()?;
+        let x = Float::with_val(prec(bits.max(53)), parsed);
+        x.is_finite().then_some(Self(x))
+    }
+
+    pub fn precision(&self) -> usize {
+        self.0.prec() as usize
+    }
+
+    /// Change the precision, rounding to nearest if it shrinks.
+    pub fn with_precision(mut self, bits: usize) -> Self {
+        self.0.set_prec(prec(bits));
+        self
+    }
+
+    pub fn to_f64(&self) -> f64 {
+        self.0.to_f64()
+    }
+
+    pub fn is_zero(&self) -> bool {
+        self.0.is_zero()
+    }
+
+    pub fn is_negative(&self) -> bool {
+        self.0.cmp0() == Some(Ordering::Less)
+    }
+
+    pub fn abs(self) -> Self {
+        Self(self.0.abs())
+    }
+
+    pub fn sqr(&self) -> Self {
+        Self(Float::with_val(self.0.prec(), self.0.square_ref()))
+    }
+
+    /// `floor(log2|x|)`, or `None` for zero. Exact, at any exponent.
+    pub fn log2_floor(&self) -> Option<isize> {
+        // MPFR normalizes the significand to [0.5, 1).
+        self.0.get_exp().map(|e| e as isize - 1)
+    }
+
+    pub fn ln(&self) -> Self {
+        Self(Float::with_val(self.0.prec(), self.0.ln_ref()))
+    }
+
+    pub fn exp(&self) -> Self {
+        Self(Float::with_val(self.0.prec(), self.0.exp_ref()))
+    }
+
+    /// `atan2(self, x)`: the angle of `(x, self)`.
+    pub fn atan2(&self, x: &Big) -> Self {
+        let p = self.0.prec().max(x.0.prec());
+        Self(Float::with_val(p, self.0.atan2_ref(&x.0)))
+    }
+
+    pub fn sin_cos(&self) -> (Self, Self) {
+        let p = self.0.prec();
+        let (mut s, mut c) = (Float::new(p), Float::new(p));
+        (&mut s, &mut c).assign(self.0.sin_cos_ref());
+        (Self(s), Self(c))
+    }
+
+    /// `sig` significant decimal digits (rounded to nearest).
+    pub fn to_decimal_parts(&self, sig: usize) -> DecimalParts {
+        if self.0.is_zero() {
+            return DecimalParts::new(false, "", 0);
+        }
+        let (negative, digits, exp) = self.0.to_sign_string_exp(10, Some(sig));
+        DecimalParts::new(negative, &digits, exp.unwrap_or(0) as isize)
+    }
+
+    pub(super) fn add_ref(&self, rhs: &Big) -> Big {
+        let p = self.0.prec().max(rhs.0.prec());
+        Big(Float::with_val(p, &self.0 + &rhs.0))
+    }
+
+    pub(super) fn sub_ref(&self, rhs: &Big) -> Big {
+        let p = self.0.prec().max(rhs.0.prec());
+        Big(Float::with_val(p, &self.0 - &rhs.0))
+    }
+
+    pub(super) fn mul_ref(&self, rhs: &Big) -> Big {
+        let p = self.0.prec().max(rhs.0.prec());
+        Big(Float::with_val(p, &self.0 * &rhs.0))
+    }
+
+    pub(super) fn negated(self) -> Big {
+        Big(-self.0)
+    }
+
+    /// `self · 2^k`, exact. `|k|` stays far below `i32::MAX` (precision and
+    /// zoom depth are capped around 2^20 bits).
+    pub(super) fn mul_pow2(self, k: isize) -> Big {
+        Big(self.0 << k as i32)
+    }
+}

src/bignum/web_backend.rs

@@ -0,0 +1,173 @@
+//! [`Big`] over `malachite-float`, pure Rust, for the web build.
+
+use malachite_base::num::basic::traits::Zero;
+use malachite_base::num::conversion::string::options::ToSciOptions;
+use malachite_base::num::conversion::traits::{RoundingFrom, ToSci};
+use malachite_base::rounding_modes::RoundingMode::Nearest;
+use malachite_float::Float;
+
+use super::DecimalParts;
+
+/// Arbitrary-precision binary float. See the module docs.
+///
+/// The precision is stored next to the value: malachite's zero has none,
+/// but "zero at `p` bits" must still lift a sum with it to `p` bits.
+#[derive(Clone, Debug)]
+pub struct Big {
+    f: Float,
+    prec: u64,
+}
+
+impl PartialEq for Big {
+    fn eq(&self, other: &Self) -> bool {
+        self.f == other.f
+    }
+}
+
+fn prec(bits: usize) -> u64 {
+    (bits as u64).max(1)
+}
+
+impl Big {
+    fn new(f: Float, prec: u64) -> Self {
+        // Only finite values reach here; anything else (overflow) is a bug.
+        debug_assert!(f.is_finite(), "non-finite Big: {f}");
+        Self { f, prec }
+    }
+
+    /// `x` at `bits` of precision (exact when `bits >= 53`). Non-finite `x`
+    /// reads as 0.
+    pub fn from_f64(x: f64, bits: usize) -> Self {
+        let p = prec(bits);
+        if !x.is_finite() {
+            return Self::zero(bits);
+        }
+        Self::new(Float::from_primitive_float_prec(x, p).0, p)
+    }
+
+    pub fn zero(bits: usize) -> Self {
+        Self::new(Float::ZERO, prec(bits))
+    }
+
+    /// Parse a decimal (`-0.75`, `1.5e-20`, any number of digits) at `bits`
+    /// (at least 53) of precision.
+    pub fn from_decimal_str(s: &str, bits: usize) -> Option<Self> {
+        let p = prec(bits.max(53));
+        let (f, _) = Float::from_sci_string_prec(s.trim(), p)?;
+        f.is_finite().then(|| Self::new(f, p))
+    }
+
+    pub fn precision(&self) -> usize {
+        self.prec as usize
+    }
+
+    /// Change the precision, rounding to nearest if it shrinks.
+    pub fn with_precision(mut self, bits: usize) -> Self {
+        let p = prec(bits);
+        if self.f.get_prec().is_some() {
+            self.f.set_prec(p);
+        }
+        self.prec = p;
+        self
+    }
+
+    pub fn to_f64(&self) -> f64 {
+        f64::rounding_from(&self.f, Nearest).0
+    }
+
+    pub fn is_zero(&self) -> bool {
+        self.f.is_zero()
+    }
+
+    pub fn is_negative(&self) -> bool {
+        self.f.is_sign_negative() && !self.f.is_zero()
+    }
+
+    pub fn abs(self) -> Self {
+        if self.f.is_sign_negative() {
+            self.negated()
+        } else {
+            self
+        }
+    }
+
+    pub fn sqr(&self) -> Self {
+        Self::new(self.f.square_prec_ref(self.prec).0, self.prec)
+    }
+
+    /// `floor(log2|x|)`, or `None` for zero. Exact, at any exponent.
+    pub fn log2_floor(&self) -> Option<isize> {
+        // The significand is normalized to [0.5, 1).
+        self.f.get_exponent().map(|e| e as isize - 1)
+    }
+
+    pub fn ln(&self) -> Self {
+        Self::new(self.f.ln_prec_ref(self.prec).0, self.prec)
+    }
+
+    pub fn exp(&self) -> Self {
+        Self::new(self.f.exp_prec_ref(self.prec).0, self.prec)
+    }
+
+    /// `atan2(self, x)`: the angle of `(x, self)`.
+    pub fn atan2(&self, x: &Big) -> Self {
+        let p = self.prec.max(x.prec);
+        Self::new(self.f.atan2_prec_ref_ref(&x.f, p).0, p)
+    }
+
+    pub fn sin_cos(&self) -> (Self, Self) {
+        let (s, c, _, _) = self.f.sin_cos_prec_ref(self.prec);
+        (Self::new(s, self.prec), Self::new(c, self.prec))
+    }
+
+    /// `sig` significant decimal digits (rounded to nearest).
+    pub fn to_decimal_parts(&self, sig: usize) -> DecimalParts {
+        if self.f.is_zero() {
+            return DecimalParts::new(false, "", 0);
+        }
+        let mut options = ToSciOptions::default();
+        options.set_precision(sig as u64);
+        let s = self.f.to_sci_with_options(options).to_string();
+        // `[-]int[.frac][e±N]`, positional or scientific depending on size.
+        let (negative, s) = match s.strip_prefix('-') {
+            Some(rest) => (true, rest),
+            None => (false, s.as_str()),
+        };
+        let (mantissa, exp) = match s.split_once(['e', 'E']) {
+            Some((m, e)) => (m, e.trim_start_matches('+').parse::<isize>().unwrap_or(0)),
+            None => (s, 0),
+        };
+        let (int, frac) = mantissa.split_once('.').unwrap_or((mantissa, ""));
+        let all = format!("{int}{frac}");
+        let digits = all.trim_start_matches('0');
+        let leading_zeros = (all.len() - digits.len()) as isize;
+        // 0.digits · 10^exp10 = int.frac · 10^exp.
+        let exp10 = int.len() as isize - leading_zeros + exp;
+        DecimalParts::new(negative, digits, exp10)
+    }
+
+    pub(super) fn add_ref(&self, rhs: &Big) -> Big {
+        let p = self.prec.max(rhs.prec);
+        Big::new(self.f.add_prec_ref_ref(&rhs.f, p).0, p)
+    }
+
+    pub(super) fn sub_ref(&self, rhs: &Big) -> Big {
+        let p = self.prec.max(rhs.prec);
+        Big::new(self.f.sub_prec_ref_ref(&rhs.f, p).0, p)
+    }
+
+    pub(super) fn mul_ref(&self, rhs: &Big) -> Big {
+        let p = self.prec.max(rhs.prec);
+        Big::new(self.f.mul_prec_ref_ref(&rhs.f, p).0, p)
+    }
+
+    pub(super) fn negated(self) -> Big {
+        Big::new(-self.f, self.prec)
+    }
+
+    /// `self · 2^k`, exact (malachite's exponent range is ±2^30, far past
+    /// what zoom depth needs).
+    pub(super) fn mul_pow2(self, k: isize) -> Big {
+        Big::new(self.f << k as i64, self.prec)
+    }
+}

src/camera.rs

@@ -2,6 +2,10 @@ use std::f32::consts::{PI, TAU};
 
 use glam::Vec3;
 
+/// Pitch is kept just short of straight up/down so the view never flips past
+/// the pole (and the raymarcher's rays always have `z > 0`).
+const PITCH_LIMIT: f32 = PI / 2.0 - 0.01;
+
 #[derive(Default, Clone)]
 pub struct Camera {
     pub position: glam::Vec3,
@@ -35,16 +39,29 @@ impl Camera {
     /// Yaw is wrapped to [-π, π) so the 2D <-> 3D transition (which scales
     /// yaw by `t`) always unwinds the short way instead of every past turn.
     pub fn rotate(&mut self, dyaw: f32, dpitch: f32) {
-        const PITCH_LIMIT: f32 = PI / 2.0 - 0.01;
         self.yaw = (self.yaw + dyaw + PI).rem_euclid(TAU) - PI;
         self.pitch = (self.pitch + dpitch).clamp(-PITCH_LIMIT, PITCH_LIMIT);
     }
 
-    pub fn orthographic(&self, t: f32) -> glam::Mat4 {
+    /// Set yaw/pitch (radians) outright. Pitch is clamped as in `rotate`,
+    /// but yaw is left unwrapped: only a camera that stays in 3D (headless
+    /// animation, which interpolates yaw across several turns) should use
+    /// this; `rotate(0., 0.)` afterwards wraps it for the 2D <-> 3D transition.
+    #[cfg(not(target_arch = "wasm32"))]
+    pub fn set_angles(&mut self, yaw: f32, pitch: f32) {
+        self.yaw = yaw;
+        self.pitch = pitch.clamp(-PITCH_LIMIT, PITCH_LIMIT);
+    }
+
+    /// `render_scale` is the 3D texture's per-axis scale (see
+    /// `FractalApp::render_scale_3d`): the full 3D camera (`t = 1`) zooms in
+    /// by the same factor, so one texel still covers one screen pixel and the
+    /// extra texels become terrain beyond the screen edges.
+    pub fn orthographic(&self, t: f32, render_scale: f32) -> glam::Mat4 {
         let yaw = self.yaw * t;
         let pitch = self.pitch * t;
 
-        let zoom = 0.5 * (1. + t);
+        let zoom = 0.5 * (1. + t * (render_scale - 1.));
         // Orbit pivot: the center of the fractal texture, which the raymarcher's
         // `sdf` lays out over world x ∈ [0, aspect], y ∈ [0, 1] on the z = 0 plane.
         let view = glam::Mat4::from_translation(Vec3::new(0.5 * self.aspect_ratio, 0.5, 0.))

src/cli.rs

@@ -25,7 +25,7 @@ pub struct Cli {
     #[arg(long)]
     pub rendering_kind: Option<RenderingKindArg>,
 
-    /// Exponent for the Multibrot kind (z -> z^power + c), clamped to [2, 8].
+    /// Exponent for the Multibrot kind (z -> z^power + c), clamped to [2, 20].
     #[arg(long)]
     pub power: Option<u32>,
 
@@ -61,10 +61,23 @@ pub struct Cli {
     #[arg(long("zoom"), short('z'))]
     pub half_height: Option<String>,
 
+    /// 3D camera yaw in degrees (with --rendering-kind 3d).
+    #[arg(long, value_name = "DEG", allow_hyphen_values = true)]
+    pub yaw: Option<f32>,
+
+    /// 3D camera pitch in degrees (with --rendering-kind 3d); negative tilts
+    /// the view down toward the fractal. Clamped short of ±90.
+    #[arg(long, value_name = "DEG", allow_hyphen_values = true)]
+    pub pitch: Option<f32>,
+
     /// Enable distance-estimation shading.
     #[arg(long)]
     pub de: bool,
 
+    /// Enable 2×2 antialiasing (supersampling; ~4× slower).
+    #[arg(long)]
+    pub antialias: bool,
+
     /// Coloring palette index.
     #[arg(long, value_name = "INDEX")]
     pub palette: Option<u32>,
@@ -72,7 +85,9 @@ pub struct Cli {
     /// Output path for --headless (default: fractal-<timestamp>.png). When
     /// animating (--to-view/--to-share), this is a directory of
     /// frame-00001.png, frame-00002.png, ... instead (default:
-    /// frames-<timestamp>/).
+    /// frames-<timestamp>/). "-" writes to stdout: the PNG for a single
+    /// image, or raw RGBA8 frames in order for an animation, to pipe into
+    /// `ffmpeg -f rawvideo -pix_fmt rgba -s WxH -r FPS -i - ...`.
     #[arg(long, value_name = "PATH")]
     pub export_path: Option<String>,
 
@@ -93,6 +108,49 @@ pub struct Cli {
     #[arg(long)]
     pub to_iterations: Option<u32>,
 
+    /// End Julia constant for an animation: c is interpolated from --julia
+    /// to this over the frames.
+    #[arg(long, value_name = "RE,IM", allow_hyphen_values = true)]
+    pub to_julia: Option<String>,
+
+    /// End Phoenix constant p for an animation (from --phoenix-p).
+    #[arg(long, value_name = "RE,IM", allow_hyphen_values = true)]
+    pub to_phoenix_p: Option<String>,
+
+    /// End Lambda constant λ for an animation (from --lambda-l).
+    #[arg(long, value_name = "RE,IM", allow_hyphen_values = true)]
+    pub to_lambda_l: Option<String>,
+
+    /// End Complex Multibrot exponent for an animation (from
+    /// --complex-power). Shorthand for --to-complex-power-re +
+    /// --to-complex-power-im.
+    #[arg(long, value_name = "RE,IM", allow_hyphen_values = true)]
+    pub to_complex_power: Option<String>,
+
+    /// End real part of the Complex Multibrot exponent for an animation;
+    /// the imaginary part stays put unless --to-complex-power-im is given.
+    #[arg(long, value_name = "RE", allow_hyphen_values = true)]
+    pub to_complex_power_re: Option<f64>,
+
+    /// End imaginary part of the Complex Multibrot exponent for an animation;
+    /// the real part stays put unless --to-complex-power-re is given.
+    #[arg(long, value_name = "IM", allow_hyphen_values = true)]
+    pub to_complex_power_im: Option<f64>,
+
+    /// End 3D camera yaw for an animation, in degrees (from --yaw). Not
+    /// wrapped: --yaw 0 --to-yaw 720 orbits twice.
+    #[arg(long, value_name = "DEG", allow_hyphen_values = true)]
+    pub to_yaw: Option<f32>,
+
+    /// End 3D camera pitch for an animation, in degrees (from --pitch).
+    #[arg(long, value_name = "DEG", allow_hyphen_values = true)]
+    pub to_pitch: Option<f32>,
+
+    /// Morph the iteration formula from the start kind (--kind) to this one
+    /// over the animation. The camera is unaffected (use --to-view for that).
+    #[arg(long, value_enum)]
+    pub to_kind: Option<KindArg>,
+
     /// Number of frames to render for an animation. Alternative to --fps +
     /// --duration.
     #[arg(long, value_name = "N")]
@@ -113,10 +171,21 @@ pub struct Cli {
     #[arg(long)]
     pub linear: bool,
 
+    /// Split the animation into N equal parts (use with --shard) to render
+    /// it across several runs. Each part keeps the whole animation's timing
+    /// and easing, so the clips join seamlessly.
+    #[arg(long, value_name = "N")]
+    pub shards: Option<u32>,
+
+    /// Which part of --shards to render, 1-based. PNG frames keep their
+    /// global numbering, so all shards can share one --export-path directory.
+    #[arg(long, value_name = "K")]
+    pub shard: Option<u32>,
+
     /// Run without opening a window: render the current view to a PNG and
     /// exit. Combine with --kind/--julia/--share/--view etc. to pick what to
-    /// render, or --to-view/--to-share to render an animation instead of a
+    /// render, or any --to-* flag (--to-view, --to-julia, --to-kind, ...) to
-    /// single frame. Not yet supported with --buddhabrot.
+    /// render an animation instead of a single frame. Not yet supported with --buddhabrot.
     #[arg(long)]
     pub headless: bool,
 

src/fractal/buddhabrot.rs

@@ -5,7 +5,8 @@
 
 use std::collections::HashMap;
 
-use eframe::egui_wgpu::{self, wgpu};
+#[cfg(feature = "gui")]
+use eframe::egui_wgpu;
 
 /// Random samples dispatched per accumulating frame. Chosen so a frame stays
 /// interactive on a modest GPU even when most samples run the full `b_cap`
@@ -125,7 +126,9 @@ pub struct BuddhabrotRenderer {
 
 impl BuddhabrotRenderer {
     pub fn new(device: &wgpu::Device, target_format: wgpu::TextureFormat) -> Self {
-        let shader = device.create_shader_module(wgpu::ShaderModuleDescriptor {
+        let shader = unsafe {
+            device.create_shader_module_trusted(
+                wgpu::ShaderModuleDescriptor {
                     label: Some("buddhabrot"),
                     source: wgpu::ShaderSource::Wgsl(
                         concat!(
@@ -134,7 +137,10 @@ impl BuddhabrotRenderer {
                         )
                         .into(),
                     ),
-        });
+                },
+                wgpu::ShaderRuntimeChecks::unchecked(),
+            )
+        };
 
         let uniform_buffer = device.create_buffer(&wgpu::BufferDescriptor {
             label: Some("buddhabrot uniforms"),
@@ -336,6 +342,7 @@ pub struct BuddhabrotCallback {
     pub size_px: [u32; 2],
 }
 
+#[cfg(feature = "gui")]
 impl egui_wgpu::CallbackTrait for BuddhabrotCallback {
     fn prepare(
         &self,

src/fractal/kind.rs

@@ -16,7 +16,7 @@ pub enum FractalKind {
     BurningShip = 1,
     /// `z -> conj(z)^2 + c` (the Mandelbar).
     Tricorn = 2,
-    /// `z -> z^power + c` (power >= 2).
+    /// `z -> z^power + c` (integer power in [2, 20]).
     Multibrot = 3,
     /// `z -> |Re(z^2)| + i·Im(z^2) + c` (abs on the real output of the square).
     Celtic = 4,
@@ -26,7 +26,7 @@ pub enum FractalKind {
     Buffalo = 6,
     /// `z -> z^2 + c + p·z_{n-1}` (two-term recurrence; `p` is `phoenix_p`).
     Phoenix = 7,
-    /// `z -> lambda·z(1 - z)` (logistic map).
+    /// `z -> lambda·z(1 - z) + c` (logistic map).
     Lambda = 8,
     /// `z -> z^power + c`, where `power` is a complex constant (the
     /// `complex_power` argument), via the principal branch `z^p = exp(p·ln z)`.
@@ -104,7 +104,7 @@ impl FractalKind {
             FractalKind::Perpendicular => "z = (x² − y²) − 2x|y|i + c".to_string(),
             FractalKind::Buffalo => "z = |Re(z²)| − i|Im(z²)| + c".to_string(),
             FractalKind::Phoenix => "z = z² + c + p·z_prev".to_string(),
-            FractalKind::Lambda => "z = λ·z(1 − z)".to_string(),
+            FractalKind::Lambda => "z = λ·z(1 − z) + c".to_string(),
             FractalKind::ComplexMultibrot => {
                 format!("z = z^({:.3}{:+.3}i) + c", complex_power.0, complex_power.1)
             }

src/fractal/mod.rs

@@ -9,10 +9,12 @@ pub mod share;
 
 pub use buddhabrot::{BuddhabrotCallback, BuddhabrotRenderer, BuddhabrotUniforms};
 pub use kind::FractalKind;
-pub use reference::{compute_reference, compute_set_reference};
+pub use reference::{RefOrbit, compute_reference, compute_set_reference};
+#[cfg(not(target_arch = "wasm32"))]
+pub use renderer::PipelineKey;
 #[cfg(target_arch = "wasm32")]
 pub use renderer::encode_png_with_progress;
-#[cfg(not(target_arch = "wasm32"))]
-pub use renderer::export_to_png_blocking;
 pub use renderer::{ExportRender, FractalCallback, FractalRenderer, MAX_REF_POINTS, Uniforms};
+#[cfg(not(target_arch = "wasm32"))]
+pub use renderer::{encode_png, export_to_png_blocking, render_readback_blocking, unpad_rgba};
 pub use share::ShareState;

src/fractal/reference.rs

@@ -1,7 +1,7 @@
 //! High-precision reference-orbit computation for perturbation rendering.
 //!
 //! We iterate the fractal's formula `Z_{n+1} = f(Z_n, C)` at high precision
-//! (`dashu-float`), storing each `Z_n` as an `f32` pair. Every pixel is then
+//! ([`Big`]: `rug` natively, pure Rust on the web), storing each `Z_n` as an `f32` pair. Every pixel is then
 //! rendered on the GPU as a small `f32` delta from this orbit — that is what
 //! makes deep zoom cheap. See `shaders/mandelbrot.wgsl` for the delta side; the
 //! delta formula there must match the orbit formula here.
@@ -24,7 +24,7 @@ use crate::view::{Big, big_from_f64};
 const REFERENCE_ESCAPE_SQ: f64 = 1.0e10;
 
 /// Up to this working precision (bits) the orbit is iterated in plain `f64`
-/// instead of `FBig` — orders of magnitude faster, which matters most on the
+/// instead of `Big` — orders of magnitude faster, which matters most on the
 /// web (where the reference is computed inline on the UI thread).
 ///
 /// `precision_for` asks for `zoom_bits + 48` guard bits, but the GPU only
@@ -39,6 +39,81 @@ const REFERENCE_ESCAPE_SQ: f64 = 1.0e10;
 ///   below 0.1% of a pixel.
 const F64_MAX_PRECISION: usize = 80;
 
+/// Orbit points with a magnitude below `2^TINY_LOG2` are stored normalized
+/// (mantissa + exponent, see [`RefOrbit::exps`]): f32's smallest normal is
+/// ~2^-126, and the GPU's deep (rescaled) phase needs these points' exact
+/// value to decide rebasing. The margin keeps a few mantissa bits clear of
+/// the subnormal range for the smaller component.
+const TINY_LOG2: i32 = -100;
+
+/// A reference orbit as uploaded to the GPU.
+#[derive(Clone, Debug, Default, PartialEq)]
+pub struct RefOrbit {
+    /// `Z_n` as f32 pairs. For points with a non-zero `exps[n]`, a mantissa
+    /// instead: the true value is `points[n] * 2^exps[n]`.
+    pub points: Vec<[f32; 2]>,
+    /// Per-point binary exponent (same length as `points`). Non-zero only for
+    /// points too small for f32's exponent range (see [`TINY_LOG2`]); only
+    /// the deep shader pipeline reads it, so [`Self::has_scaled`] forces it.
+    pub exps: Vec<i32>,
+}
+
+impl RefOrbit {
+    fn with_capacity(n: usize) -> Self {
+        // Most orbits escape long before `max_iter`; don't reserve hundreds of
+        // MB up front for a multi-million iteration request.
+        let n = n.min(1 << 17);
+        Self {
+            points: Vec::with_capacity(n),
+            exps: Vec::with_capacity(n),
+        }
+    }
+
+    fn push(&mut self, point: [f32; 2]) {
+        self.points.push(point);
+        self.exps.push(0);
+    }
+
+    /// Whether any point is stored as mantissa + exponent, i.e. the orbit
+    /// can only be read by the deep pipeline.
+    pub fn has_scaled(&self) -> bool {
+        self.exps.iter().any(|&e| e != 0)
+    }
+}
+
+impl core::ops::Deref for RefOrbit {
+    type Target = [[f32; 2]];
+    fn deref(&self) -> &Self::Target {
+        &self.points
+    }
+}
+
+impl<'a> IntoIterator for &'a RefOrbit {
+    type Item = &'a [f32; 2];
+    type IntoIter = core::slice::Iter<'a, [f32; 2]>;
+    fn into_iter(self) -> Self::IntoIter {
+        self.points.iter()
+    }
+}
+
+/// Store `(zr, zi)` into `orbit`, as plain f32 unless its magnitude is below
+/// `2^TINY_LOG2`, in which case both components share an exponent `k` and
+/// the stored mantissa `Z * 2^-k` has its larger component in `[0.5, 1)`.
+fn push_big_point(orbit: &mut RefOrbit, zr: &Big, zi: &Big) {
+    let (lr, li) = (zr.log2_floor(), zi.log2_floor());
+    let top = lr.max(li);
+    match top {
+        Some(top) if top < TINY_LOG2 as isize => {
+            let k = top + 1;
+            let mr = (zr.clone() << -k).to_f64() as f32;
+            let mi = (zi.clone() << -k).to_f64() as f32;
+            orbit.points.push([mr, mi]);
+            orbit.exps.push(k as i32);
+        }
+        _ => orbit.push([zr.to_f64() as f32, zi.to_f64() as f32]),
+    }
+}
+
 /// Compute the reference orbit `Z_0..Z_{len-1}` where `Z_0 = z0` and
 /// `Z_{n+1} = f(Z_n, c)` for the given `kind` (and `power`, for Multibrot), up
 /// to `max_iter` steps at `precision` bits. Each entry is `[re, im]` in f32.
@@ -59,64 +134,22 @@ pub fn compute_reference(
     lambda_l: (f64, f64),
     complex_power: (f64, f64),
     morph: Option<(FractalKind, f64)>,
-) -> Vec<[f32; 2]> {
+) -> RefOrbit {
-    compute_reference_inner(
-        z0_re,
-        z0_im,
-        c_re,
-        c_im,
-        max_iter,
-        precision,
-        kind,
-        power,
-        phoenix_p,
-        lambda_l,
-        complex_power,
-        morph,
-        false,
-    )
-}
-
-/// [`compute_reference`] plus `set_plane` (see [`StepConsts::set_plane`]):
-/// picks the `f64` fast path or the `FBig` path by precision.
-#[allow(clippy::too_many_arguments)]
-fn compute_reference_inner(
-    z0_re: &Big,
-    z0_im: &Big,
-    c_re: &Big,
-    c_im: &Big,
-    max_iter: u32,
-    precision: usize,
-    kind: FractalKind,
-    power: u32,
-    phoenix_p: (f64, f64),
-    lambda_l: (f64, f64),
-    complex_power: (f64, f64),
-    morph: Option<(FractalKind, f64)>,
-    set_plane: bool,
-) -> Vec<[f32; 2]> {
     // A zero-weight morph is just the plain kind; skip the second formula.
     let morph = morph.filter(|&(_, w)| w != 0.0);
     if precision <= F64_MAX_PRECISION {
         let k = StepConstsF64 {
-            c: (c_re.to_f64().value(), c_im.to_f64().value()),
+            c: (c_re.to_f64(), c_im.to_f64()),
             p: phoenix_p,
             l: lambda_l,
             cpow: complex_power,
             power,
-            set_plane,
         };
-        return compute_reference_f64(
+        return compute_reference_f64((z0_re.to_f64(), z0_im.to_f64()), max_iter, kind, &k, morph);
-            (z0_re.to_f64().value(), z0_im.to_f64().value()),
-            max_iter,
-            kind,
-            &k,
-            morph,
-        );
     }
     let k = StepConsts {
-        cr: c_re.clone().with_precision(precision).value(),
+        cr: c_re.clone().with_precision(precision),
-        ci: c_im.clone().with_precision(precision).value(),
+        ci: c_im.clone().with_precision(precision),
         pr: big_from_f64(phoenix_p.0, precision),
         pi: big_from_f64(phoenix_p.1, precision),
         lr: big_from_f64(lambda_l.0, precision),
@@ -125,7 +158,6 @@ fn compute_reference_inner(
         cpow_im: big_from_f64(complex_power.1, precision),
         power,
         precision,
-        set_plane,
     };
     compute_reference_big(z0_re, z0_im, max_iter, kind, &k, morph)
 }
@@ -137,7 +169,6 @@ struct StepConstsF64 {
     l: (f64, f64),
     cpow: (f64, f64),
     power: u32,
-    set_plane: bool,
 }
 
 /// [`compute_reference`]'s fast path for shallow views (see
@@ -148,12 +179,12 @@ fn compute_reference_f64(
     kind: FractalKind,
     k: &StepConstsF64,
     morph: Option<(FractalKind, f64)>,
-) -> Vec<[f32; 2]> {
+) -> RefOrbit {
     let (mut zr, mut zi) = z0;
     // Previous iterate, for the Phoenix two-term recurrence (Y_{-1} = 0).
     let mut prev = (0.0f64, 0.0f64);
 
-    let mut points: Vec<[f32; 2]> = Vec::with_capacity(max_iter as usize + 1);
+    let mut points = RefOrbit::with_capacity(max_iter as usize + 1);
     for _ in 0..=max_iter {
         points.push([zr as f32, zi as f32]);
         if zr * zr + zi * zi > REFERENCE_ESCAPE_SQ {
@@ -204,16 +235,11 @@ fn step_f64(
             )
         }
         FractalKind::Lambda => {
-            // λ·z(1 - z) (+ c on the parameter plane).
+            // λ·z(1 - z) + c.
             let (lr, li) = k.l;
             let (re2, im2) = (1.0 - zr, -zi);
             let (lzr, lzi) = (lr * zr - li * zi, lr * zi + li * zr);
-            let (re, im) = (lzr * re2 - lzi * im2, re2 * lzi + lzr * im2);
+            (lzr * re2 - lzi * im2 + cr, re2 * lzi + lzr * im2 + ci)
-            if k.set_plane {
-                (re + cr, im + ci)
-            } else {
-                (re, im)
-            }
         }
         FractalKind::ComplexMultibrot => {
             let (pr, pi) = complex_pow_complex_f64(zr, zi, k.cpow.0, k.cpow.1);
@@ -250,12 +276,9 @@ struct StepConsts {
     cpow_im: Big,
     power: u32,
     precision: usize,
-    /// Parameter plane: the GPU adds `dc` every step for every kind, so the
-    /// Lambda map (which has no `c` of its own) is `λ·z(1 - z) + c` there.
-    set_plane: bool,
 }
 
-/// [`compute_reference`] at arbitrary precision (`FBig`), for deep views.
+/// [`compute_reference`] at arbitrary precision ([`Big`]), for deep views.
 fn compute_reference_big(
     z0_re: &Big,
     z0_im: &Big,
@@ -263,23 +286,22 @@ fn compute_reference_big(
     kind: FractalKind,
     k: &StepConsts,
     morph: Option<(FractalKind, f64)>,
-) -> Vec<[f32; 2]> {
+) -> RefOrbit {
     let precision = k.precision;
     let morph = morph.map(|(from, w)| (from, big_from_f64(w, precision)));
 
-    let mut zr = z0_re.clone().with_precision(precision).value();
+    let mut zr = z0_re.clone().with_precision(precision);
-    let mut zi = z0_im.clone().with_precision(precision).value();
+    let mut zi = z0_im.clone().with_precision(precision);
     // Previous iterate, for the Phoenix two-term recurrence (Y_{-1} = 0).
-    let mut zr_prev = big_zero(precision);
+    let mut zr_prev = Big::zero(precision);
-    let mut zi_prev = big_zero(precision);
+    let mut zi_prev = Big::zero(precision);
 
-    let mut points: Vec<[f32; 2]> = Vec::with_capacity(max_iter as usize + 1);
+    let mut points = RefOrbit::with_capacity(max_iter as usize + 1);
 
     for _ in 0..=max_iter {
-        let fr = zr.to_f64().value() as f32;
+        push_big_point(&mut points, &zr, &zi);
-        let fi = zi.to_f64().value() as f32;
-        points.push([fr, fi]);
 
+        let [fr, fi] = *points.points.last().unwrap();
         let mag = (fr as f64) * (fr as f64) + (fi as f64) * (fi as f64);
         if mag > REFERENCE_ESCAPE_SQ {
             break;
@@ -296,8 +318,8 @@ fn compute_reference_big(
         // Shift the previous iterate (only the Phoenix arm reads it).
         zr_prev = zr;
         zi_prev = zi;
-        zr = new_zr.with_precision(precision).value();
+        zr = new_zr.with_precision(precision);
-        zi = new_zi.with_precision(precision).value();
+        zi = new_zi.with_precision(precision);
     }
 
     points
@@ -325,7 +347,7 @@ fn step(
         FractalKind::BurningShip => {
             // (|zr| + i|zi|)^2 = (zr^2 - zi^2) + 2|zr zi| i.
             let re = &zr.sqr() - &zi.sqr() + cr;
-            let im = big_abs((zr * zi) << 1) + ci;
+            let im = ((zr * zi) << 1).abs() + ci;
             (re, im)
         }
         FractalKind::Tricorn => {
@@ -340,14 +362,14 @@ fn step(
         }
         FractalKind::Celtic => {
             // |Re(z^2)| + i·Im(z^2): abs the real output of the square.
-            let re = big_abs(&zr.sqr() - &zi.sqr()) + cr;
+            let re = (&zr.sqr() - &zi.sqr()).abs() + cr;
             let im = ((zr * zi) << 1) + ci;
             (re, im)
         }
         FractalKind::Perpendicular => {
             // (x^2 - y^2) - 2·x·|y| i: abs the imaginary input.
             let re = &zr.sqr() - &zi.sqr() + cr;
-            let im = if zi.to_f64().value() < 0.0 {
+            let im = if zi.is_negative() {
                 ci + ((zr * zi) << 1)
             } else {
                 ci - ((zr * zi) << 1)
@@ -356,8 +378,8 @@ fn step(
         }
         FractalKind::Buffalo => {
             // |Re(z^2)| - |Im(z^2)| i: abs both outputs.
-            let re = big_abs(&zr.sqr() - &zi.sqr()) + cr;
+            let re = (&zr.sqr() - &zi.sqr()).abs() + cr;
-            let im = ci - big_abs((zr * zi) << 1);
+            let im = ci - ((zr * zi) << 1).abs();
             (re, im)
         }
         FractalKind::Phoenix => {
@@ -369,18 +391,14 @@ fn step(
             (re2 + cr + pzr, im2 + ci + pzi)
         }
         FractalKind::Lambda => {
-            // λ·z(1 - z): logistic map (+ c on the parameter plane).
+            // λ·z(1 - z) + c: logistic map plus the usual additive `c`.
-            let re2 = 1 - zr;
+            let re2 = Big::from_f64(1.0, k.precision) - zr;
             let im2 = -zi;
             let lzr = &k.lr * zr - &k.li * zi;
             let lzi = &k.lr * zi + &k.li * zr;
             let re = &lzr * &re2 - &lzi * &im2;
             let im = re2 * lzi + lzr * im2;
-            if k.set_plane {
             (re + cr, im + ci)
-            } else {
-                (re, im)
-            }
         }
         FractalKind::ComplexMultibrot => {
             let (pr, pi) = complex_pow_complex(zr, zi, &k.cpow_re, &k.cpow_im, k.precision);
@@ -389,36 +407,20 @@ fn step(
     }
 }
 
-fn big_zero(precision: usize) -> Big {
-    Big::from(0i32).with_precision(precision).value()
-}
-
-/// Absolute value of a `Big`. The sign check via f64 is exact except for values
-/// so tiny that |x| ≈ x either way — negligible against the f32 orbit storage.
-fn big_abs(x: Big) -> Big {
-    if x.to_f64().value() < 0.0 { -x } else { x }
-}
-
 /// `(zr + i zi)^power` by repeated complex multiply at `precision` bits.
 fn complex_pow(zr: &Big, zi: &Big, power: u32, precision: usize) -> (Big, Big) {
-    let mut rr = Big::from(1i32).with_precision(precision).value();
+    let mut rr = Big::from_f64(1.0, precision);
-    let mut ri = big_zero(precision);
+    let mut ri = Big::zero(precision);
     for _ in 0..power {
         // (rr + i ri)(zr + i zi) = (rr zr - ri zi) + (rr zi + ri zr) i.
-        let nr = (&rr * zr - &ri * zi).with_precision(precision).value();
+        let nr = (&rr * zr - &ri * zi).with_precision(precision);
-        let ni = (&rr * zi + &ri * zr).with_precision(precision).value();
+        let ni = (&rr * zi + &ri * zr).with_precision(precision);
         rr = nr;
         ri = ni;
     }
     (rr, ri)
 }
 
-/// `true` if `x` is (numerically) zero. The f64 check is exact for a true
-/// zero; only matters here to special-case `ln(0)`.
-fn is_big_zero(x: &Big) -> bool {
-    x.to_f64().value() == 0.0
-}
-
 /// `(zr + i zi)^(pr + i pi)` for a complex exponent, via the principal branch
 /// `z^p = exp(p·ln z)` where `ln z = ln|z| + i·arg(z)`. Used by
 /// `ComplexMultibrot`; must be kept in sync with the shader's `cpow`.
@@ -426,14 +428,14 @@ fn is_big_zero(x: &Big) -> bool {
 /// panic; this is the correct limit for the `Re(p) > 0` region the UI
 /// exposes).
 fn complex_pow_complex(zr: &Big, zi: &Big, pr: &Big, pi: &Big, precision: usize) -> (Big, Big) {
-    if is_big_zero(zr) && is_big_zero(zi) {
+    if zr.is_zero() && zi.is_zero() {
-        return (big_zero(precision), big_zero(precision));
+        return (Big::zero(precision), Big::zero(precision));
     }
     let r2 = &zr.sqr() + &zi.sqr();
     let ln_r = r2.ln() >> 1; // 0.5 * ln(r2) = ln(sqrt(r2)); exact halving.
     let theta = zi.atan2(zr);
-    let exp_re = (pr * &ln_r - pi * &theta).with_precision(precision).value();
+    let exp_re = (pr * &ln_r - pi * &theta).with_precision(precision);
-    let exp_im = (pr * &theta + pi * &ln_r).with_precision(precision).value();
+    let exp_im = (pr * &theta + pi * &ln_r).with_precision(precision);
     let mag = exp_re.exp();
     let (sin_a, cos_a) = exp_im.sin_cos();
     (&mag * &cos_a, &mag * &sin_a)
@@ -453,9 +455,9 @@ pub fn compute_set_reference(
     lambda_l: (f64, f64),
     complex_power: (f64, f64),
     morph: Option<(FractalKind, f64)>,
-) -> Vec<[f32; 2]> {
+) -> RefOrbit {
-    let zero = big_zero(precision);
+    let zero = Big::zero(precision);
-    compute_reference_inner(
+    compute_reference(
         &zero,
         &zero,
         center_re,
@@ -468,7 +470,6 @@ pub fn compute_set_reference(
         lambda_l,
         complex_power,
         morph,
-        true,
     )
 }
 
@@ -476,12 +477,23 @@ pub fn compute_set_reference(
 mod tests {
     use super::*;
 
+    /// Sign and zero tests must hold far below f64's range, where
+    /// `to_f64` reads as ±0.
+    #[test]
+    fn sign_and_zero_below_f64_range() {
+        let tiny = Big::from_f64(1.0, 64) >> 5000;
+        assert!(!tiny.is_zero());
+        assert!(Big::zero(64).is_zero());
+        assert_eq!((-tiny.clone()).abs(), tiny);
+        assert_eq!(tiny.clone().abs(), tiny);
+    }
+
     /// The high-precision reference must agree with a plain f64 iteration for a
     /// shallow point (where f64 is accurate).
     #[test]
     fn reference_matches_naive_f64() {
-        let cr = Big::try_from(-0.75_f64).unwrap();
+        let cr = Big::from_f64(-0.75, 53);
-        let ci = Big::try_from(0.1_f64).unwrap();
+        let ci = Big::from_f64(0.1, 53);
         let points = compute_set_reference(
             &cr,
             &ci,
@@ -525,13 +537,17 @@ mod tests {
     fn f64_fast_path_matches_big() {
         let bits_fast = F64_MAX_PRECISION;
         let bits_big = F64_MAX_PRECISION + 64;
-        for kind in FractalKind::ALL {
+        let cases = FractalKind::ALL
+            .into_iter()
+            .map(|kind| (kind, 3))
+            .chain([(FractalKind::Multibrot, 20)]); // highest supported power
+        for (kind, power) in cases {
             for julia in [false, true] {
                 for morph in [None, Some((FractalKind::Phoenix, 0.3))] {
                     let run = |bits: usize| {
                         let (a, b) = (big_from_f64(-0.3, bits), big_from_f64(0.2, bits));
                         let (jr, ji) = (big_from_f64(-0.4, bits), big_from_f64(0.55, bits));
-                        let args = (60, bits, kind, 3, (0.1, -0.2), (0.9, 0.3), (2.3, 0.4));
+                        let args = (60, bits, kind, power, (0.1, -0.2), (0.9, 0.3), (2.3, 0.4));
                         if julia {
                             compute_reference(
                                 &a, &b, &jr, &ji, args.0, args.1, args.2, args.3, args.4, args.5,
@@ -544,7 +560,7 @@ mod tests {
                             )
                         }
                     };
-                    let ctx = format!("{kind:?} julia={julia} morph={morph:?}");
+                    let ctx = format!("{kind:?} power={power} julia={julia} morph={morph:?}");
                     let (fast, big) = (run(bits_fast), run(bits_big));
                     assert_eq!(fast.len(), big.len(), "{ctx}: length");
                     for (i, (f, b)) in fast.iter().zip(&big).enumerate() {
@@ -561,11 +577,57 @@ mod tests {
         }
     }
 
+    /// Orbit points below f32's range are stored as a normalized mantissa
+    /// plus exponent (for the deep GPU phase); every other point stays a
+    /// plain f32 with exponent 0.
+    #[test]
+    fn tiny_points_are_stored_normalized() {
+        let bits = 400;
+        // c = -1 + δ: X_2 = c(c + 1) = -δ + δ², far below f32's range.
+        let delta = 1e-45_f64;
+        let cr = big_from_f64(-1.0, bits) + big_from_f64(delta, bits);
+        let ci = big_from_f64(0.0, bits);
+        let orbit = compute_set_reference(
+            &cr,
+            &ci,
+            3,
+            bits,
+            FractalKind::Mandelbrot,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            (0.0, 0.0),
+            None,
+        );
+        assert_eq!(orbit.exps.len(), orbit.points.len());
+        assert!(orbit.has_scaled());
+        assert_eq!(&orbit.exps[..2], &[0, 0], "X_0 = 0 and X_1 = c are plain");
+        let [mr, mi] = orbit.points[2];
+        let k = orbit.exps[2];
+        assert!(k < TINY_LOG2, "exponent {k}");
+        assert!(
+            (0.5..1.0).contains(&mr.abs()),
+            "mantissa {mr} not normalized"
+        );
+        assert_eq!(mi, 0.0);
+        let x2 = mr as f64 * 2f64.powi(k);
+        assert!(
+            (x2 + delta).abs() < 1e-6 * delta,
+            "X_2 = {x2}, expected {}",
+            -delta
+        );
+
+        // A shallow orbit stays entirely plain.
+        let plain = set_ref(-0.75, 0.1, FractalKind::Mandelbrot, None);
+        assert!(!plain.has_scaled());
+        assert!(plain.exps.iter().all(|&e| e == 0));
+    }
+
     /// A point inside the main cardioid never escapes: full-length orbit.
     #[test]
     fn interior_orbit_runs_full_length() {
-        let cr = Big::try_from(-0.2_f64).unwrap();
+        let cr = Big::from_f64(-0.2, 53);
-        let ci = Big::try_from(0.0_f64).unwrap();
+        let ci = Big::from_f64(0.0, 53);
         let points = compute_set_reference(
             &cr,
             &ci,
@@ -584,8 +646,8 @@ mod tests {
     /// Burning Ship reference matches a naive f64 iteration of the same formula.
     #[test]
     fn burning_ship_reference_matches_naive_f64() {
-        let cr = Big::try_from(-1.75_f64).unwrap();
+        let cr = Big::from_f64(-1.75, 53);
-        let ci = Big::try_from(-0.03_f64).unwrap();
+        let ci = Big::from_f64(-0.03, 53);
         let points = compute_set_reference(
             &cr,
             &ci,
@@ -615,8 +677,8 @@ mod tests {
     /// Multibrot (power 3) reference matches a naive f64 cube iteration.
     #[test]
     fn multibrot3_reference_matches_naive_f64() {
-        let cr = Big::try_from(0.3_f64).unwrap();
+        let cr = Big::from_f64(0.3, 53);
-        let ci = Big::try_from(0.2_f64).unwrap();
+        let ci = Big::from_f64(0.2, 53);
         let points = compute_set_reference(
             &cr,
             &ci,
@@ -648,10 +710,10 @@ mod tests {
     /// Julia orbit (fixed c, z0 = center) matches a naive f64 iteration.
     #[test]
     fn julia_reference_matches_naive_f64() {
-        let z0_re = Big::try_from(0.15_f64).unwrap();
+        let z0_re = Big::from_f64(0.15, 53);
-        let z0_im = Big::try_from(-0.1_f64).unwrap();
+        let z0_im = Big::from_f64(-0.1, 53);
-        let c_re = Big::try_from(-0.8_f64).unwrap();
+        let c_re = Big::from_f64(-0.8, 53);
-        let c_im = Big::try_from(0.156_f64).unwrap();
+        let c_im = Big::from_f64(0.156, 53);
         let points = compute_reference(
             &z0_re,
             &z0_im,
@@ -680,11 +742,43 @@ mod tests {
         }
     }
 
+    /// Lambda Julia orbit adds the Julia `c`: `z -> λ·z(1 - z) + c`.
+    #[test]
+    fn lambda_julia_reference_matches_naive_f64() {
+        let (lr, li) = (-0.5_f64, 0.2_f64);
+        let (cr, ci) = (0.1_f64, -0.3_f64);
+        let points = compute_reference(
+            &Big::from_f64(0.2, 53),
+            &Big::from_f64(0.1, 53),
+            &Big::from_f64(cr, 53),
+            &Big::from_f64(ci, 53),
+            60,
+            200,
+            FractalKind::Lambda,
+            2,
+            (0.0, 0.0),
+            (lr, li),
+            (0.0, 0.0),
+            None,
+        );
+
+        let (mut zr, mut zi) = (0.2_f64, 0.1_f64);
+        for point in &points {
+            let tol = 1e-4 * (1.0 + zr.abs().max(zi.abs()));
+            assert!((point[0] as f64 - zr).abs() < tol, "{point:?} vs {zr}");
+            assert!((point[1] as f64 - zi).abs() < tol, "{point:?} vs {zi}");
+            let (lzr, lzi) = (lr * zr - li * zi, lr * zi + li * zr);
+            let (ar, ai) = (1.0 - zr, -zi);
+            zr = lzr * ar - lzi * ai + cr;
+            zi = lzr * ai + lzi * ar + ci;
+        }
+    }
+
     /// Celtic reference matches a naive f64 iteration: real = |x^2 - y^2| + cr.
     #[test]
     fn celtic_reference_matches_naive_f64() {
-        let cr = Big::try_from(-0.6_f64).unwrap();
+        let cr = Big::from_f64(-0.6, 53);
-        let ci = Big::try_from(0.4_f64).unwrap();
+        let ci = Big::from_f64(0.4, 53);
         let points = compute_set_reference(
             &cr,
             &ci,
@@ -715,8 +809,8 @@ mod tests {
     /// real = x^2 - y^2 + cr, imag = -2·x·|y| + ci.
     #[test]
     fn perpendicular_reference_matches_naive_f64() {
-        let cr = Big::try_from(-0.7_f64).unwrap();
+        let cr = Big::from_f64(-0.7, 53);
-        let ci = Big::try_from(-0.2_f64).unwrap();
+        let ci = Big::from_f64(-0.2, 53);
         let points = compute_set_reference(
             &cr,
             &ci,
@@ -747,8 +841,8 @@ mod tests {
     /// real = |x^2 - y^2| + cr, imag = -|2·x·y| + ci.
     #[test]
     fn buffalo_reference_matches_naive_f64() {
-        let cr = Big::try_from(-1.2_f64).unwrap();
+        let cr = Big::from_f64(-1.2, 53);
-        let ci = Big::try_from(-0.35_f64).unwrap();
+        let ci = Big::from_f64(-0.35, 53);
         let points = compute_set_reference(
             &cr,
             &ci,
@@ -779,8 +873,8 @@ mod tests {
     /// `z_{n+1} = z_n^2 + c + p·z_{n-1}` (z_0 = 0, z_{-1} = 0).
     #[test]
     fn phoenix_reference_matches_naive_f64() {
-        let cr = Big::try_from(0.5667_f64).unwrap();
+        let cr = Big::from_f64(0.5667, 53);
-        let ci = Big::try_from(0.0_f64).unwrap();
+        let ci = Big::from_f64(0.0, 53);
         let p = (-0.5_f64, 0.0_f64);
         let points = compute_set_reference(
             &cr,
@@ -818,8 +912,8 @@ mod tests {
     /// iteration of `z^p = exp(p·ln z)`.
     #[test]
     fn complex_multibrot_reference_matches_naive_f64() {
-        let cr = Big::try_from(0.1_f64).unwrap();
+        let cr = Big::from_f64(0.1, 53);
-        let ci = Big::try_from(-0.2_f64).unwrap();
+        let ci = Big::from_f64(-0.2, 53);
         let power = (2.5_f64, 0.3_f64);
         let points = compute_set_reference(
             &cr,
@@ -861,15 +955,10 @@ mod tests {
         }
     }
 
-    fn set_ref(
+    fn set_ref(cr: f64, ci: f64, kind: FractalKind, morph: Option<(FractalKind, f64)>) -> RefOrbit {
-        cr: f64,
-        ci: f64,
-        kind: FractalKind,
-        morph: Option<(FractalKind, f64)>,
-    ) -> Vec<[f32; 2]> {
         compute_set_reference(
-            &Big::try_from(cr).unwrap(),
+            &Big::from_f64(cr, 53),
-            &Big::try_from(ci).unwrap(),
+            &Big::from_f64(ci, 53),
             60,
             200,
             kind,

src/fractal/share.rs

@@ -2,13 +2,14 @@
 //! iterations, Julia constant, coloring) as a compact URL fragment so deep-zoom
 //! locations can be shared or bookmarked.
 //!
-//! Format: `m=m&f=<str>&re=<dec>&im=<dec>&hh=<f64>&it=<u32>&cs=<f32>&co=<f32>` with
+//! Format: `m=m&f=<str>&re=<dec>&im=<dec>&hh=<sci>&it=<u32>&cs=<f32>&co=<f32>` with
 //! `m=j&jr=<f64>&ji=<f64>` added for Julia. `re`/`im` are full-precision decimal
-//! strings.
+//! strings; `hh` is a `Scale` in scientific notation (any exponent).
 
 use std::collections::HashMap;
 
 use crate::fractal::FractalKind;
+use crate::view::Scale;
 
 #[derive(Clone, Debug)]
 pub struct ShareState {
@@ -18,7 +19,7 @@ pub struct ShareState {
     pub power: u32,
     pub center_re: String,
     pub center_im: String,
-    pub half_height: f64,
+    pub half_height: Scale,
     pub iterations: u32,
     pub julia_c: (f64, f64),
     /// Distortion constant for the Phoenix kind (ignored by others).
@@ -115,7 +116,7 @@ mod tests {
             power: 5,
             center_re: "-0.743643887037158704752191506114774".into(),
             center_im: "0.131825904205311970493132056385139".into(),
-            half_height: 1.5e-20,
+            half_height: Scale::from_f64(1.5e-20),
             iterations: 4000,
             julia_c: (-0.123, 0.745),
             phoenix_p: (-0.5, 0.1),
@@ -146,5 +147,15 @@ mod tests {
         let d = ShareState::decode("#m=m&re=0.0&im=0.0&hh=1.25&it=256").unwrap();
         assert!(!d.julia);
         assert_eq!(d.iterations, 256);
+        assert_eq!(d.half_height, Scale::from_f64(1.25));
+    }
+
+    /// Zooms past f64's range survive a round trip.
+    #[test]
+    fn round_trip_past_f64_range() {
+        let d = ShareState::decode("#m=m&re=0.0&im=0.0&hh=1.5e-1234&it=256").unwrap();
+        assert_eq!(d.half_height, "1.5e-1234".parse().unwrap());
+        let d2 = ShareState::decode(&d.encode()).unwrap();
+        assert_eq!(d2.half_height, d.half_height);
     }
 }

src/headless.rs

@@ -3,13 +3,21 @@
 // event loop, no worker-thread debounce (nothing to debounce for a one-shot
 // render); it just creates its own wgpu device, computes the reference orbit
 // once, and renders through the same `ExportRender` path the "Export PNG"
-// button uses.
+// button uses. `--export-path -` writes to stdout instead: the PNG for a
+// single image, or a raw RGBA8 video stream for an animation (for piping
+// into ffmpeg).
 
-use eframe::egui_wgpu::wgpu;
+use std::collections::{BTreeMap, HashMap};
+use std::io::{IsTerminal, Write};
+use std::sync::atomic::{AtomicBool, AtomicUsize, Ordering};
+use std::sync::{Mutex, mpsc};
 
-use crate::app::{FractalApp, unix_timestamp};
+use crate::app::{FractalApp, RefJob, parse_complex_pair, unix_timestamp};
 use crate::cli::Cli;
-use crate::fractal::{ExportRender, FractalRenderer, ShareState, export_to_png_blocking};
+use crate::fractal::{
+    ExportRender, FractalKind, FractalRenderer, PipelineKey, ShareState, encode_png,
+    export_to_png_blocking, render_readback_blocking, unpad_rgba,
+};
 use crate::view::{
     ViewState, big_from_decimal_str, interpolate_f64, interpolate_view, parse_view_spec,
     precision_for,
@@ -18,6 +26,21 @@ use crate::view::{
 /// Cap on the output image dimension (px), to stay within GPU texture limits.
 const MAX_DIM: u32 = 8192 * 16;
 
+/// `--export-path` value meaning "write to stdout".
+const STDOUT_PATH: &str = "-";
+
+/// Refuse to dump binary image data onto a terminal.
+fn check_stdout_piped() -> Result<(), String> {
+    if std::io::stdout().is_terminal() {
+        return Err(
+            "--export-path - writes binary data to stdout; pipe it somewhere \
+                    (e.g. `| ffmpeg ...`)"
+                .into(),
+        );
+    }
+    Ok(())
+}
+
 pub fn run(cli: Cli) -> Result<(), String> {
     if cli.buddhabrot {
         return Err("headless mode doesn't support --buddhabrot yet".into());
@@ -28,32 +51,22 @@ pub fn run(cli: Cli) -> Result<(), String> {
 
     // These drive the animation path below; grab them before `apply_cli`
     // consumes `cli` to build the start state.
-    let to_view = cli.to_view.clone();
+    let targets = AnimTargets::from_cli(&cli)?;
-    let to_share = cli.to_share.clone();
-    let to_iterations = cli.to_iterations;
-    let frames_arg = cli.frames;
-    let fps = cli.fps;
-    let duration = cli.duration;
-    let linear = cli.linear;
     let export_path = cli.export_path.clone();
+    if export_path.as_deref() == Some(STDOUT_PATH) {
+        check_stdout_piped()?;
+    }
+
+    if targets.shard.is_some() && !targets.any() {
+        return Err("--shard/--shards only apply to animations (give a --to-* target)".into());
+    }
 
     let mut app = FractalApp::default_state();
     app.apply_cli(cli);
+    app.set_output_size(width, height);
 
-    if to_view.is_some() || to_share.is_some() {
+    if targets.any() {
-        return run_animation(
+        return run_animation(app, targets, width, height, export_path);
-            app,
-            to_view,
-            to_share,
-            to_iterations,
-            frames_arg,
-            fps,
-            duration,
-            linear,
-            width,
-            height,
-            export_path,
-        );
     }
 
     let export_path = export_path.unwrap_or_else(|| format!("fractal-{}.png", unix_timestamp()));
@@ -64,15 +77,13 @@ pub fn run(cli: Cli) -> Result<(), String> {
     let (device, queue) = pollster::block_on(request_device())?;
     let format = wgpu::TextureFormat::Bgra8Unorm;
     let renderer = FractalRenderer::new(&device, format);
-    let uniforms = app.make_uniforms(width as f64 / height as f64);
+    let uniforms = app.make_uniforms(width as f64 / height as f64, height as f64);
-    let (pipeline, bind_group_layout, format) = renderer.export_handles(&device, &uniforms);
+    let handles = renderer.export_handles(&device, &uniforms);
 
     let er = ExportRender::new(
         &device,
         &queue,
-        pipeline,
+        &handles,
-        &bind_group_layout,
-        format,
         width,
         height,
         uniforms,
@@ -86,48 +97,140 @@ pub fn run(cli: Cli) -> Result<(), String> {
     });
     eprintln!();
 
+    if export_path == STDOUT_PATH {
+        let mut out = std::io::stdout().lock();
+        out.write_all(&png)
+            .and_then(|()| out.flush())
+            .map_err(|e| format!("writing to stdout failed: {e}"))?;
+        eprintln!("wrote PNG to stdout ({width}×{height})");
+    } else {
         std::fs::write(&export_path, &png).map_err(|e| format!("save failed: {e}"))?;
         println!("saved {export_path} ({width}×{height})");
+    }
     Ok(())
 }
 
-/// Render a sequence of frames sweeping the camera from the app's current
+/// The `--to-*` end state of a headless animation, plus its pacing. Each
-/// (start) view to an end view, for feeding into ffmpeg. Everything other
+/// target is optional; anything left unset stays at its start value.
-/// than the view (kind, colors, iteration cap policy, ...) stays fixed at
+struct AnimTargets {
-/// whatever `apply_cli` set up for the start; only the camera moves.
-#[allow(clippy::too_many_arguments)]
-fn run_animation(
-    mut app: FractalApp,
     to_view: Option<String>,
     to_share: Option<String>,
-    mut to_iterations: Option<u32>,
+    to_iterations: Option<u32>,
-    frames_arg: Option<u32>,
+    to_julia: Option<(f64, f64)>,
+    to_phoenix_p: Option<(f64, f64)>,
+    to_lambda_l: Option<(f64, f64)>,
+    /// Complex Multibrot exponent, per component (either may move alone).
+    to_cpow_re: Option<f64>,
+    to_cpow_im: Option<f64>,
+    to_kind: Option<FractalKind>,
+    /// 3D camera, degrees.
+    to_yaw: Option<f32>,
+    to_pitch: Option<f32>,
+    frames: Option<u32>,
     fps: f64,
     duration: Option<f64>,
     linear: bool,
+    /// `--shard K --shards N`: render only the K-th (1-based) of N parts.
+    shard: Option<(u32, u32)>,
+}
+
+impl AnimTargets {
+    fn from_cli(cli: &Cli) -> Result<Self, String> {
+        let pair = |flag: &str, v: &Option<String>| -> Result<Option<(f64, f64)>, String> {
+            v.as_deref()
+                .map(|s| parse_complex_pair(s).ok_or_else(|| format!("invalid --{flag}: {s}")))
+                .transpose()
+        };
+        let to_cpow = pair("to-complex-power", &cli.to_complex_power)?;
+        let shard = match (cli.shard, cli.shards) {
+            (None, None) | (None, Some(1)) => None,
+            (Some(k), Some(n)) if (1..=n).contains(&k) => Some((k, n)),
+            (Some(k), Some(n)) => {
+                return Err(format!(
+                    "--shard {k} is out of range 1..={n} (--shards {n})"
+                ));
+            }
+            _ => return Err("--shard and --shards must be given together".into()),
+        };
+        Ok(Self {
+            to_view: cli.to_view.clone(),
+            to_share: cli.to_share.clone(),
+            to_iterations: cli.to_iterations,
+            to_julia: pair("to-julia", &cli.to_julia)?,
+            to_phoenix_p: pair("to-phoenix-p", &cli.to_phoenix_p)?,
+            to_lambda_l: pair("to-lambda-l", &cli.to_lambda_l)?,
+            to_cpow_re: cli.to_complex_power_re.or(to_cpow.map(|p| p.0)),
+            to_cpow_im: cli.to_complex_power_im.or(to_cpow.map(|p| p.1)),
+            to_kind: cli.to_kind.map(Into::into),
+            to_yaw: cli.to_yaw,
+            to_pitch: cli.to_pitch,
+            frames: cli.frames,
+            fps: cli.fps,
+            duration: cli.duration,
+            linear: cli.linear,
+            shard,
+        })
+    }
+
+    /// Whether any end state was given, i.e. this is an animation.
+    fn any(&self) -> bool {
+        self.to_view.is_some()
+            || self.to_share.is_some()
+            || self.to_iterations.is_some()
+            || self.to_julia.is_some()
+            || self.to_phoenix_p.is_some()
+            || self.to_lambda_l.is_some()
+            || self.to_cpow_re.is_some()
+            || self.to_cpow_im.is_some()
+            || self.to_kind.is_some()
+            || self.to_yaw.is_some()
+            || self.to_pitch.is_some()
+    }
+}
+
+/// Render a sequence of frames interpolating from the app's current (start)
+/// state to `targets`, for feeding into ffmpeg: the camera, iteration count,
+/// per-kind constants (c, p, λ, complex power) and, through a kind morph,
+/// the iteration formula, and the 3D camera angles. Everything else (colors, ...) stays fixed at
+/// whatever `apply_cli` set up for the start. With `--export-path -`, frames
+/// are streamed in order to stdout as raw RGBA8 (for ffmpeg's `rawvideo`
+/// demuxer) instead of being written as PNGs.
+fn run_animation(
+    mut app: FractalApp,
+    targets: AnimTargets,
     width: u32,
     height: u32,
     export_path: Option<String>,
 ) -> Result<(), String> {
-    let frames = match frames_arg {
+    let fps = targets.fps;
+    let frames = match targets.frames {
         Some(n) => n,
         None => {
-            let dur = duration.ok_or("animation needs --frames, or --duration (with --fps)")?;
+            let dur = targets
+                .duration
+                .ok_or("animation needs --frames, or --duration (with --fps)")?;
             ((fps * dur).round() as u32).max(2)
         }
     };
     if frames < 2 {
         return Err("animation needs at least 2 frames".into());
     }
-
+    // Frames are still timed against the whole animation (`apply_frame` takes
-    let (to, to_iterations_share) =
+    // the global index); a shard only picks which of them this run renders.
-        parse_animation_target(to_view.as_deref(), to_share.as_deref())?;
+    let range = match targets.shard {
-    if to_iterations.is_none()
+        Some((_, n)) if n > frames => {
-        && let Some(to_iterations_share) = to_iterations_share
+            return Err(format!("--shards {n} is more than the {frames} frames"));
-    {
-        to_iterations = Some(to_iterations_share);
         }
+        Some((k, n)) => shard_range(frames, k, n),
+        None => 0..frames,
+    };
+    let (first, count) = (range.start, range.len());
+
     let from = app.view_state().clone();
+    let (to, to_iterations_share) =
+        parse_animation_target(targets.to_view.as_deref(), targets.to_share.as_deref())?
+            .unwrap_or_else(|| (from.clone(), None));
+    let to_iterations = targets.to_iterations.or(to_iterations_share);
     let from_iterations = app.max_iterations();
     if to_iterations.is_none() {
         // Iteration count auto-scales with zoom depth per frame, the same way it
@@ -135,77 +238,281 @@ fn run_animation(
         app.set_auto_iterations(true);
     }
 
+    let from_consts = app.constants();
+    let [c0, p0, l0, cp0] = from_consts;
+    let to_consts = [
+        targets.to_julia.unwrap_or(c0),
+        targets.to_phoenix_p.unwrap_or(p0),
+        targets.to_lambda_l.unwrap_or(l0),
+        (
+            targets.to_cpow_re.unwrap_or(cp0.0),
+            targets.to_cpow_im.unwrap_or(cp0.1),
+        ),
+    ];
+    let from_kind = app.kind();
+    let to_kind = targets.to_kind.unwrap_or(from_kind);
+    let (yaw0, pitch0) = app.camera_angles();
+    let yaw1 = targets.to_yaw.map_or(yaw0, f32::to_radians);
+    let pitch1 = targets.to_pitch.map_or(pitch0, f32::to_radians);
+
+    let stream = export_path.as_deref() == Some(STDOUT_PATH);
     let out_dir = export_path.unwrap_or_else(|| format!("frames-{}", unix_timestamp()));
-    std::fs::create_dir_all(&out_dir).map_err(|e| format!("failed to create {out_dir}: {e}"))?;
+    if stream {
+        eprintln!(
+            "streaming raw video to stdout; ffmpeg input: \
+             -f rawvideo -pix_fmt rgba -s {width}x{height} -r {fps} -i -"
+        );
+    } else {
+        std::fs::create_dir_all(&out_dir)
+            .map_err(|e| format!("failed to create {out_dir}: {e}"))?;
+    }
 
-    let (device, queue) = pollster::block_on(request_device())?;
+    // Everything about frame `i` is a pure function of its `t`, so the app can
-    let format = wgpu::TextureFormat::Bgra8Unorm;
+    // be put into any frame's state at any time, in any order.
-    let renderer = FractalRenderer::new(&device, format);
+    let apply_frame = |app: &mut FractalApp, i: u32| {
-    // Only the camera animates, so the shader specialization (kind, Julia,
-    // DE) is the same for every frame.
-    let (pipeline, bind_group_layout, format) =
-        renderer.export_handles(&device, &app.make_uniforms(width as f64 / height as f64));
-
-    for i in 0..frames {
         let raw_t = i as f64 / (frames - 1) as f64;
-        let t = if linear { raw_t } else { smoothstep(raw_t) };
+        let t = if targets.linear {
+            raw_t
+        } else {
+            smoothstep(raw_t)
+        };
         if let Some(to) = to_iterations {
             app.set_max_iterations(
                 interpolate_f64(from_iterations as f64, to as f64, t).round() as u32,
             );
         }
         app.set_view(interpolate_view(&from, &to, t));
+        app.set_constants(std::array::from_fn(|k| {
+            (
+                interpolate_f64(from_consts[k].0, to_consts[k].0, t),
+                interpolate_f64(from_consts[k].1, to_consts[k].1, t),
+            )
+        }));
+        app.set_kind_morph(from_kind, to_kind, t);
+        app.set_camera_angles(
+            interpolate_f64(yaw0 as f64, yaw1 as f64, t) as f32,
+            interpolate_f64(pitch0 as f64, pitch1 as f64, t) as f32,
+        );
+    };
 
-        eprintln!("[{:>4}/{frames}] computing reference orbit…", i + 1);
+    // Snapshot every frame's reference-orbit job up front (cheap: just the
-        app.compute_reference_blocking();
+    // parameters), so the orbits themselves can be computed in parallel.
+    // `jobs[j]` is global frame `first + j`; the pipeline below works in
+    // local indices `j`, so the stdout writer's ordering is per shard.
+    let jobs: Vec<RefJob> = range
+        .clone()
+        .map(|i| {
+            apply_frame(&mut app, i);
+            app.reference_job()
+        })
+        .collect();
 
-        let uniforms = app.make_uniforms(width as f64 / height as f64);
+    let (device, queue) = pollster::block_on(request_device())?;
+    let format = wgpu::TextureFormat::Bgra8Unorm;
+    let renderer = FractalRenderer::new(&device, format);
+    let aspect = width as f64 / height as f64;
+
+    // Three-stage pipeline, connected by bounded channels (which also cap
+    // memory): `threads` workers compute reference orbits (CPU, the expensive
+    // part at deep zoom) → this thread renders each frame on the GPU → `threads`
+    // workers PNG-encode and write frames. Frames flow through out of order
+    // (at most ~`threads` apart); each is written under its own index.
+    //
+    // When streaming, the last stage instead unpads frames to raw RGBA and a
+    // single writer thread puts them back in order before writing to stdout.
+    // Its reorder buffer can't apply backpressure (blocking it while waiting
+    // for frame `k` could stall the pipeline before `k` gets through), so the
+    // orbit workers bound it instead: they don't start a frame more than
+    // `window` ahead of the last one written.
+    let threads = std::thread::available_parallelism().map_or(4, |n| n.get());
+    let window = threads * 4;
+    let next_job = AtomicUsize::new(0);
+    let saved = AtomicUsize::new(0);
+    let failed = AtomicBool::new(false);
+    let error: Mutex<Option<String>> = Mutex::new(None);
+    let fail = |e: String| {
+        failed.store(true, Ordering::Relaxed);
+        error.lock().unwrap().get_or_insert(e);
+    };
+
+    if let Some((k, n)) = targets.shard {
+        eprintln!(
+            "shard {k}/{n}: frames {}–{} of {frames}",
+            range.start + 1,
+            range.end
+        );
+    }
+    eprintln!("rendering {count} frames ({width}×{height}) on {threads} threads…");
+    let (png_tx, png_rx) = mpsc::sync_channel::<(usize, Vec<u8>, u32, bool)>(threads * 2);
+    let png_rx = Mutex::new(png_rx);
+    let (raw_tx, raw_rx) = mpsc::sync_channel::<(usize, Vec<u8>)>(threads * 2);
+    std::thread::scope(|scope| {
+        let (ref_tx, ref_rx) = mpsc::sync_channel::<(usize, crate::fractal::RefOrbit)>(threads * 2);
+        for _ in 0..threads {
+            let ref_tx = ref_tx.clone();
+            let (jobs, next_job, saved, failed) = (&jobs, &next_job, &saved, &failed);
+            scope.spawn(move || {
+                loop {
+                    let i = next_job.fetch_add(1, Ordering::Relaxed);
+                    if i >= jobs.len() || failed.load(Ordering::Relaxed) {
+                        break;
+                    }
+                    while stream
+                        && i >= saved.load(Ordering::Relaxed) + window
+                        && !failed.load(Ordering::Relaxed)
+                    {
+                        std::thread::sleep(std::time::Duration::from_millis(2));
+                    }
+                    if ref_tx.send((i, jobs[i].compute())).is_err() {
+                        break;
+                    }
+                }
+            });
+        }
+        drop(ref_tx);
+
+        if stream {
+            let (saved, fail) = (&saved, &fail);
+            scope.spawn(move || {
+                let mut out = std::io::stdout().lock();
+                let mut pending = BTreeMap::new();
+                let mut next = 0;
+                for (i, raw) in raw_rx.iter() {
+                    pending.insert(i, raw);
+                    while let Some(raw) = pending.remove(&next) {
+                        if let Err(e) = out.write_all(&raw) {
+                            fail(format!("writing to stdout failed: {e}"));
+                            return;
+                        }
+                        next += 1;
+                        saved.store(next, Ordering::Relaxed);
+                        eprint!("\r[{next:>4}/{count}] streamed");
+                    }
+                }
+                if let Err(e) = out.flush() {
+                    fail(format!("writing to stdout failed: {e}"));
+                }
+            });
+        } else {
+            drop(raw_rx);
+        }
+
+        for _ in 0..threads {
+            let raw_tx = raw_tx.clone();
+            let (png_rx, out_dir, saved, failed, fail) =
+                (&png_rx, &out_dir, &saved, &failed, &fail);
+            scope.spawn(move || {
+                loop {
+                    // Hold the lock only for the receive, not the encode.
+                    let Ok((i, padded, bpr, swap_rb)) = png_rx.lock().unwrap().recv() else {
+                        break;
+                    };
+                    // After a failure, keep draining (without work) until the
+                    // GPU stage hangs up, so it can't block on a full channel.
+                    if failed.load(Ordering::Relaxed) {
+                        continue;
+                    }
+                    if stream {
+                        let raw = unpad_rgba(&padded, width, height, bpr, swap_rb);
+                        // Only fails once the writer has failed and hung up.
+                        let _ = raw_tx.send((i, raw));
+                        continue;
+                    }
+                    let png =
+                        encode_png(&padded, width, height, bpr, swap_rb, png::Compression::Fast);
+                    let path = format!("{out_dir}/frame-{:05}.png", first as usize + i + 1);
+                    if let Err(e) = std::fs::write(&path, &png) {
+                        fail(format!("save failed: {e}"));
+                        continue;
+                    }
+                    let done = saved.fetch_add(1, Ordering::Relaxed) + 1;
+                    eprint!("\r[{done:>4}/{count}] saved");
+                }
+            });
+        }
+
+        // GPU stage, on this thread (it owns the app and the device). The
+        // shader specialization (kind, Julia, DE, morph) can change between
+        // frames during a kind morph; build each pipeline once.
+        let mut pipelines = HashMap::new();
+        for (i, points) in ref_rx.iter() {
+            if failed.load(Ordering::Relaxed) {
+                break;
+            }
+            apply_frame(&mut app, first + i as u32);
+            app.finish_reference(jobs[i].clone(), points);
+
+            let uniforms = app.make_uniforms(aspect, height as f64);
+            let handles = pipelines
+                .entry(PipelineKey::from_uniforms(&uniforms))
+                .or_insert_with(|| renderer.export_handles(&device, &uniforms));
             let er = ExportRender::new(
                 &device,
                 &queue,
-            pipeline.clone(),
+                handles,
-            &bind_group_layout,
-            format,
                 width,
                 height,
                 uniforms,
                 app.reference_points(),
                 app.lights(),
             );
-
+            let padded = render_readback_blocking(&device, &queue, &er);
-        let png = export_to_png_blocking(&device, &queue, &er, |phase, fraction| {
+            if png_tx.send((i, padded, er.padded_bpr, er.swap_rb)).is_err() {
-            eprint!(
+                break;
-                "\r[{:>4}/{frames}] {phase} {:>3.0}%",
+            }
-                i + 1,
+        }
-                fraction * 100.0
+        // Dropping the channel ends lets the workers drain and exit.
-            );
+        drop(raw_tx);
+        drop(png_tx);
+        drop(ref_rx);
     });
     eprintln!();
 
-        let path = format!("{out_dir}/frame-{:05}.png", i + 1);
+    if let Some(e) = error.into_inner().unwrap() {
-        std::fs::write(&path, &png).map_err(|e| format!("save failed: {e}"))?;
+        return Err(e);
+    }
+    let saved = saved.into_inner();
+    if saved != count {
+        return Err(format!("only {saved} of {count} frames were rendered"));
     }
 
-    println!("saved {frames} frames to {out_dir}/ ({width}×{height})");
+    if stream {
+        eprintln!("streamed {count} frames ({width}×{height})");
+        return Ok(());
+    }
+    println!("saved {count} frames to {out_dir}/ ({width}×{height})");
+    if targets.shard.is_some() {
+        println!(
+            "tip: once every shard is rendered into {out_dir}/, they form the full sequence; \
+             this shard alone: ffmpeg -framerate {fps} -start_number {} -i {out_dir}/frame-%05d.png \
+             -frames:v {count} -c:v libx264 -pix_fmt yuv420p out.mp4",
+            range.start + 1
+        );
+    } else {
         println!(
             "tip: ffmpeg -framerate {fps} -i {out_dir}/frame-%05d.png -c:v libx264 -pix_fmt yuv420p out.mp4"
         );
+    }
     Ok(())
 }
 
-/// Parse `--to-view`/`--to-share` (exactly one must be set) into the end
+/// Parse `--to-view`/`--to-share` (at most one is used) into the end view
-/// view of an animation. Only position/zoom/iterations are pulled from a
+/// of an animation, or `None` if neither is set (the camera stays put). Only position/zoom/iterations are pulled from a
 /// share fragment — the rest of its state (kind, colors, ...) is ignored, so
 /// pasting a link from the app doesn't unexpectedly change the fractal kind
 /// mid-animation.
 fn parse_animation_target(
     to_view: Option<&str>,
     to_share: Option<&str>,
-) -> Result<(ViewState, Option<u32>), String> {
+) -> Result<Option<(ViewState, Option<u32>)>, String> {
     if let Some(spec) = to_view {
-        return parse_view_spec(spec).ok_or_else(|| format!("invalid --to-view spec: {spec}"));
+        return parse_view_spec(spec)
+            .map(Some)
+            .ok_or_else(|| format!("invalid --to-view spec: {spec}"));
     }
-    let frag = to_share.expect("run_animation only called with one of to_view/to_share set");
+    let Some(frag) = to_share else {
+        return Ok(None);
+    };
     let state =
         ShareState::decode(frag).ok_or_else(|| format!("invalid --to-share fragment: {frag}"))?;
     let bits = precision_for(state.half_height);
@@ -213,10 +520,18 @@ fn parse_animation_target(
         big_from_decimal_str(&state.center_re, bits).ok_or("invalid --to-share center (re)")?;
     let im =
         big_from_decimal_str(&state.center_im, bits).ok_or("invalid --to-share center (im)")?;
-    Ok((
+    Ok(Some((
         ViewState::with_center(re, im, state.half_height),
         Some(state.iterations),
-    ))
+    )))
+}
+
+/// Global frame indices of shard `shard` (1-based) out of `shards` equal
+/// parts of a `frames`-frame animation. Consecutive shards tile `0..frames`
+/// with no gap or overlap.
+fn shard_range(frames: u32, shard: u32, shards: u32) -> std::ops::Range<u32> {
+    let bound = |k: u32| (k as u64 * frames as u64 / shards as u64) as u32;
+    bound(shard - 1)..bound(shard)
 }
 
 /// Ease-in/ease-out pacing: slow at both ends, fast through the middle.
@@ -243,3 +558,30 @@ async fn request_device() -> Result<(wgpu::Device, wgpu::Queue), String> {
         .await
         .map_err(|e| format!("failed to create device: {e}"))
 }
+
+#[cfg(test)]
+mod tests {
+    use super::shard_range;
+
+    #[test]
+    fn shards_tile_all_frames() {
+        for frames in [2, 3, 10, 97, 1000, u32::MAX] {
+            for shards in [1, 2, 3, 7, 10] {
+                if shards > frames {
+                    continue;
+                }
+                let mut next = 0;
+                for k in 1..=shards {
+                    let r = shard_range(frames, k, shards);
+                    assert_eq!(
+                        r.start, next,
+                        "gap/overlap at shard {k}/{shards} of {frames}"
+                    );
+                    assert!(!r.is_empty(), "empty shard {k}/{shards} of {frames}");
+                    next = r.end;
+                }
+                assert_eq!(next, frames);
+            }
+        }
+    }
+}

src/lights.rs

@@ -1,7 +1,9 @@
 use std::f32::consts::PI;
 
 use bytemuck::{Pod, Zeroable};
-use egui::{Color32, Ui};
+use ecolor::Color32;
+#[cfg(feature = "gui")]
+use egui::Ui;
 
 /// Maximum number of simultaneous lights.
 pub const MAX_LIGHT_COUNT: usize = 16;
@@ -27,6 +29,7 @@ impl Default for Light {
 }
 
 impl Light {
+    #[cfg(feature = "gui")]
     pub fn widget(&mut self, ui: &mut Ui) -> bool {
         let formater = |v, _| format!("{}°", ((v * 180. / std::f64::consts::PI) as u32));
         let parser = |s: &str| {

src/main.rs

@@ -1,6 +1,8 @@
 // Without these, rust fails to infer Send/Sync trait impls
 // Probably caused by the new trait solver
 #![recursion_limit = "256"]
+// Without the `gui` feature, UI-only state and helpers go unused.
+#![cfg_attr(not(feature = "gui"), allow(dead_code, unused_imports))]
 
 // Fractal Explorer — Rust + wgpu + egui + WGSL deep-zoom Mandelbrot.
 //
@@ -9,6 +11,7 @@
 // and calls the wasm `main`, which boots eframe onto the page's <canvas>.
 
 mod app;
+mod bignum;
 mod camera;
 mod fractal;
 mod lights;
@@ -21,8 +24,17 @@ mod headless;
 #[cfg(not(target_arch = "wasm32"))]
 mod worker;
 
+#[cfg(all(target_arch = "wasm32", not(feature = "gui")))]
+compile_error!("the web build needs the `gui` feature");
+
+#[cfg(feature = "gui")]
 use app::FractalApp;
 
+#[cfg(all(feature = "gui", not(target_arch = "wasm32")))]
+type MainResult = eframe::Result;
+#[cfg(all(not(feature = "gui"), not(target_arch = "wasm32")))]
+type MainResult = Result<(), String>;
+
 /// wgpu configuration for eframe. The fractal fragment shader reads the
 /// reference orbit from a **storage buffer**, so the device must allow storage
 /// buffers in the fragment stage. eframe's default requests WebGL2-downlevel
@@ -30,6 +42,7 @@ use app::FractalApp;
 ///   * request the adapter's real limits (which include storage buffers), and
 ///   * force the WebGPU backend on the web (WebGL2 can't do storage buffers at
 ///     all) — failing cleanly on browsers without WebGPU, per the design.
+#[cfg(feature = "gui")]
 fn wgpu_options() -> eframe::egui_wgpu::WgpuConfiguration {
     use eframe::egui_wgpu::{WgpuSetup, wgpu};
 
@@ -51,7 +64,7 @@ fn wgpu_options() -> eframe::egui_wgpu::WgpuConfiguration {
 }
 
 #[cfg(not(target_arch = "wasm32"))]
-fn main() -> eframe::Result {
+fn main() -> MainResult {
     use clap::Parser as _;
 
     env_logger::builder()
@@ -70,6 +83,13 @@ fn main() -> eframe::Result {
         };
     }
 
+    #[cfg(not(feature = "gui"))]
+    {
+        eprintln!("error: built without the \"gui\" feature; only --headless is supported");
+        std::process::exit(1);
+    }
+
+    #[cfg(feature = "gui")]
     let native_options = eframe::NativeOptions {
         renderer: eframe::Renderer::Wgpu,
         wgpu_options: wgpu_options(),
@@ -80,6 +100,7 @@ fn main() -> eframe::Result {
         ..Default::default()
     };
 
+    #[cfg(feature = "gui")]
     eframe::run_native(
         "Fractal Explorer",
         native_options,

src/shaders/buddhabrot.wgsl

@@ -106,7 +106,7 @@ fn advance(z: vec2<f32>, zp: vec2<f32>, c: vec2<f32>) -> vec2<f32> {
     } else if KIND == KIND_TRICORN {
         return vec2<f32>(z.x * z.x - z.y * z.y, -2.0 * z.x * z.y) + c;
     } else if KIND == KIND_MULTIBROT {
-        return complex_pow(z, clamp(u.power, 2u, 8u)) + c;
+        return complex_pow(z, clamp(u.power, 2u, MULTIBROT_MAX_POWER)) + c;
     } else if KIND == KIND_CELTIC {
         return vec2<f32>(abs(z.x * z.x - z.y * z.y), 2.0 * z.x * z.y) + c;
     } else if KIND == KIND_PERPENDICULAR {

src/shaders/colorize.wgsl

@@ -150,19 +150,45 @@ fn ray_marching(pos: vec4<f32>) -> vec4<f32> {
     // About 1/50 of a texel at typical sizes: tighter only adds steps
     // without visibly moving the hit.
     let dist_threshold = 0.00001;
+    // Rays grazing the exponential slope see a tiny `dist` for many steps in
+    // a row and would crawl along it until the step budget runs out. Force a
+    // step of at least half a texel (the height field is nearest-sampled, so
+    // nothing finer exists), and bisect back if that lands inside the solid.
+    let min_step = 0.5 * k.inv_size_y;
     var hit = false;
+    var t_prev = t;
     for (var i = 0u; i < 100u; i++) {
         let dist = sdf(start + t * ray_dir, k);
         if dist < dist_threshold {
             hit = true;
+            if dist < 0. {
+                // Overshot: t_prev is outside, t inside. Refine the crossing.
+                var lo = t_prev;
+                var hi = t;
+                for (var j = 0u; j < 8u; j++) {
+                    let mid = 0.5 * (lo + hi);
+                    if sdf(start + mid * ray_dir, k) < dist_threshold {
+                        hi = mid;
+                    } else {
+                        lo = mid;
+                    }
+                }
+                t = hi;
+            }
             break;
         }
-        t += dist;
+        t_prev = t;
+        t += max(dist, min_step);
         // Past the footprint's far edge: nothing left to hit.
         if t >= t_leave {
             break;
         }
     }
+    // Out of steps while still over the footprint: the ray is skimming the
+    // surface, so shade where it got to rather than reporting a miss.
+    if !hit && t < t_leave {
+        hit = true;
+    }
     // Shade outside the loop, so its registers don't weigh on the march.
     if !hit {
         return RAY_MISS;

src/shaders/common.wgsl

@@ -43,6 +43,10 @@ const KIND_MANDELBROT: u32 = 0u;
 const KIND_BURNING_SHIP: u32 = 1u;
 const KIND_TRICORN: u32 = 2u;
 const KIND_MULTIBROT: u32 = 3u;
+// Highest Multibrot power (the UI/CLI/share-link clamp in app.rs matches).
+// `bailout_sq` in app.rs shrinks the bailout with the power so z^p stays a
+// finite f32.
+const MULTIBROT_MAX_POWER: u32 = 20u;
 const KIND_CELTIC: u32 = 4u;
 const KIND_PERPENDICULAR: u32 = 5u;
 const KIND_BUFFALO: u32 = 6u;

src/shaders/iterate_uniforms.wgsl

@@ -48,6 +48,10 @@ struct Uniforms {
     // Kind-switch morph weight: each step is (1 - w)*f_kind + w*f_morph_from;
     // 0 = no morph. Only read by MORPH pipelines (see mandelbrot.wgsl).
     morph_w: f32,
+    // Deep views: binary exponent E of the view scale. `span` and `dc_offset`
+    // are uploaded multiplied by 2^-E so they stay in f32's range; 0 = not
+    // deep (plain f32 values). Only read by DEEP pipelines (mandelbrot.wgsl).
+    scale_exp: i32,
     // Complex binomial coefficients C(complex_power, k) for k = 1..16, two per
     // vec4 (k odd in .xy, k even in .zw), for the Complex Multibrot delta
     // series. Precomputed on the CPU since they only depend on the power.
@@ -164,7 +168,7 @@ fn shadow_color(normal: vec3<f32>, ci: f32) -> vec3<f32> {
         color = filmic(color, 4.2);
     } else if u.shadow_palette_id == 3u {
         // No DE darkening as in `classic_color`: in shadow/3D modes the DE
-        // is a height (clamped to 1000, not 1), and the lighting already
+        // is a height (unclamped, not capped at 1), and the lighting already
         // shows the relief.
         let t = fract(ci * u.color_scale + u.color_offset);
         let ambient = 0.25;

src/shaders/lipschitz.wgsl

@@ -0,0 +1,121 @@
+// Distance field for the shadow and 3D views of kinds whose map is
+// discontinuous (Complex Multibrot's principal-branch z^p). The per-pixel
+// DE, |z|·ln|z|/|dz| = G/|G'|, is only a distance when the potential G is
+// continuous. Across a branch-cut preimage G jumps (|z| jumps by
+// e^(2π·Im p)), each side's DE describes the set as continued on its own
+// branch: seams in shadow mode, walls in the raymarched terrain. A true
+// distance to the set is continuous, so this rebuilds one from the pixels that are unambiguously
+// next to the set ("seeds": interior texels, and exterior texels whose DE is
+// below SEED_DE, i.e. within about a pixel of it):
+//     h(x) = min over seeds y of  DE(y) + DIST_SLOPE·|x - y|
+// computed by jump flooding: `fs_seed` marks seeds, `fs_jump` runs with
+// `step` halving from about half the texture size down to 1 (plus one more
+// step-1 pass), each texel keeping the best seed among itself and its 8
+// neighbours at ±step, and `fs_compose` writes the result, eased by
+// `ease_distance`, as the data texture's G channel. Seeds are carried as coordinates, so the distance is
+// Euclidean rather than an 8-direction chamfer approximation.
+//
+// The set only counts where it's in view: next to the image edge, set just
+// outside it is missed and the height there reads a bit high.
+//
+// Texel layout of the seed textures: (seed x, seed y, seed DE, 1), or all 0
+// for "no seed yet". R (palette parameter) and B (interior fraction) of the
+// data texture pass through untouched.
+
+struct LipschitzStep {
+    step: i32,
+    _pad0: i32,
+    _pad1: i32,
+    _pad2: i32,
+};
+
+@group(0) @binding(0) var data_tex: texture_2d<f32>;
+@group(0) @binding(1) var seed_tex: texture_2d<f32>;
+@group(0) @binding(2) var<uniform> ls: LipschitzStep;
+
+// DE units per texel of a true distance: DE is in units of `pixel_size`,
+// the texel's (|dx| + |dy|) footprint, i.e. sqrt(2) texels.
+const DIST_SLOPE: f32 = 0.70710678;
+// Exterior texels with a DE below this (in DE units) are seeds.
+const SEED_DE: f32 = 1.0;
+
+@vertex
+fn vs_main(@builtin(vertex_index) idx: u32) -> @builtin(position) vec4<f32> {
+    return vec4<f32>(fullscreen_triangle_pos(idx), 0.0, 1.0);
+}
+
+@fragment
+fn fs_seed(@builtin(position) pos: vec4<f32>) -> @location(0) vec4<f32> {
+    let p = vec2<i32>(pos.xy);
+    let d = textureLoad(data_tex, p, 0);
+    if d.b != 0.0 {
+        return vec4<f32>(vec2<f32>(p), 0.0, 1.0);
+    }
+    if d.g < SEED_DE {
+        return vec4<f32>(vec2<f32>(p), d.g, 1.0);
+    }
+    return vec4<f32>(0.0);
+}
+
+// Height at `p` through seed `s` (a seed texel), or +inf for "no seed".
+fn through(p: vec2<i32>, s: vec4<f32>) -> f32 {
+    if s.a == 0.0 {
+        return 3.0e38;
+    }
+    return s.z + DIST_SLOPE * distance(vec2<f32>(p), s.xy);
+}
+
+@fragment
+fn fs_jump(@builtin(position) pos: vec4<f32>) -> @location(0) vec4<f32> {
+    let p = vec2<i32>(pos.xy);
+    let hi = vec2<i32>(textureDimensions(seed_tex)) - vec2<i32>(1, 1);
+    let st = ls.step;
+    var best = textureLoad(seed_tex, p, 0);
+    var best_h = through(p, best);
+    for (var dy = -1; dy <= 1; dy++) {
+        for (var dx = -1; dx <= 1; dx++) {
+            if dx == 0 && dy == 0 {
+                continue;
+            }
+            let q = p + st * vec2<i32>(dx, dy);
+            if any(q < vec2<i32>(0, 0)) || any(q > hi) {
+                continue;
+            }
+            let s = textureLoad(seed_tex, q, 0);
+            let h = through(p, s);
+            if h < best_h {
+                best = s;
+                best_h = h;
+            }
+        }
+    }
+    return best;
+}
+
+// Scale of `ease_distance`, in screen heights.
+const EASE_K: f32 = 0.02;
+
+// Concave easing of the distance field `h` (DE units, `size_y` texels
+// tall): K·ln(1 + x/K) in screen heights, the unit `sdf` in colorize.wgsl
+// reads. Slope 1 at the set, flattening far from it, so the terrain rises
+// steeply out of the set and then levels off. The slope never exceeds 1,
+// so the field stays a valid (conservative) distance for the sphere tracer.
+fn ease_distance(h: f32, size_y: f32) -> f32 {
+    return EASE_K * log(1.0 + h / (size_y * EASE_K)) * size_y;
+}
+
+@fragment
+fn fs_compose(@builtin(position) pos: vec4<f32>) -> @location(0) vec4<f32> {
+    let p = vec2<i32>(pos.xy);
+    let d = textureLoad(data_tex, p, 0);
+    if d.b != 0.0 {
+        return d;
+    }
+    // Not min(h, DE): that keeps the too-low side of every jump, so the
+    // walls come back as shards.
+    let h = through(p, textureLoad(seed_tex, p, 0));
+    if h >= 3.0e38 {
+        return d; // no set anywhere in view: keep the DE
+    }
+    return vec4<f32>(d.r, ease_distance(h, f32(textureDimensions(data_tex).y)), d.b, d.a);
+}

src/shaders/mandelbrot.wgsl

@@ -11,6 +11,11 @@
 // true value |y| drops below the delta |e|, or the reference runs out, we reset
 // the reference index to 0 and carry the full value as the new delta (valid
 // because X_0 = 0).
+//
+// Past ~1e30 zoom the deltas themselves leave f32's exponent range (smallest
+// normal ~1.2e-38), so `DEEP` pipelines start each pixel in a rescaled form,
+// e = w * 2^s with an f32 mantissa `w` and an i32 exponent `s`, and hand over
+// to the plain f32 loop once the delta is big enough (see `iterate_sample`).
 
 @group(0) @binding(0) var<uniform> u: Uniforms;
 @group(0) @binding(1) var<storage, read> ref_orbit: array<vec2<f32>>;
@@ -20,6 +25,11 @@
 // Only read by the adaptive-AA refine pass (`fs_refine`): the 1-sample-per-
 // pixel data texture written by `fs_data`, which decides where to supersample.
 @group(1) @binding(0) var coarse_tex: texture_2d<f32>;
+// Per-point binary exponents of the reference orbit (`RefOrbit::exps`): the
+// true X[m] is ref_orbit[m] * 2^ref_exp[m]. Non-zero only for points below
+// f32's range, which only deep references contain; only `DEEP` pipelines read
+// it (see `ref_at`).
+@group(0) @binding(3) var<storage, read> ref_exp: array<i32>;
 
 // Pipeline-overridable specialization constants, set per pipeline from the
 // uniforms' `kind` / `is_julia` / `de_coloring` (see `PipelineKey` in
@@ -35,6 +45,10 @@ override DE: bool = false;
 // second kind, `u.morph_from`. That one is a runtime value (it only lives for
 // the length of the animation), so only MORPH pipelines pay for its branches.
 override MORPH: bool = false;
+// Deep view (`u.scale_exp != 0`): the per-pixel offset `dc`, the pixel size
+// and `u.span` / `u.dc_offset` are all in units of 2^scale_exp, and each pixel
+// starts in the rescaled deep phase (see `iterate_sample`).
+override DEEP: bool = false;
 
 struct VsOut {
     @builtin(position) pos: vec4<f32>,
@@ -82,14 +96,18 @@ fn diffabs(c: f32, d: f32) -> f32 {
 // tiny, but that only perturbs `s` by a relative f32 epsilon, and the result
 // is `e * s`, so the delta keeps full relative precision.
 fn multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: u32) -> vec2<f32> {
-    let y = z + e;
+    return cmul(e, multibrot_sum(z, z + e, p));
+}
+
+// The sum in `multibrot_delta`, sum_{k=0}^{p-1} y^k Z^{p-1-k} with y = Z+e.
+fn multibrot_sum(z: vec2<f32>, y: vec2<f32>, p: u32) -> vec2<f32> {
     var s = vec2<f32>(1.0, 0.0);
     var zj = vec2<f32>(1.0, 0.0);
     for (var j: u32 = 1u; j < p; j = j + 1u) {
         zj = cmul(zj, z); // Z^j
         s = cmul(s, y) + zj;
     }
-    return cmul(e, s);
+    return s;
 }
 
 // Maximum number of terms in `complex_multibrot_delta`'s series (matches the
@@ -121,11 +139,21 @@ fn cm_coef(k: u32) -> vec2<f32> {
 // dynamics, not a deep-zoom edge case — and the series above would diverge.
 // But forming Z+e directly is numerically safe exactly there (e isn't many
 // orders of magnitude smaller than Z), so fall back to a plain subtraction.
+//
+// The series is also wrong when Z -> Z+e crosses the principal branch cut
+// (negative real axis) that `cpow` and the CPU reference use. There it
+// continues Z's branch, but the true map jumps by a factor e^{2πip}. Which
+// pixels crossed then depended on the reference's position, so whole disks
+// flipped branch while panning. With |w| < 0.5, arg(1+w) is within ±30°, so an
+// Im sign flip with Re(Z) < 0 is exactly a cut crossing. The direct form is
+// fine there: the true delta across the cut is large, not tiny.
 fn complex_multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: vec2<f32>) -> vec2<f32> {
+    let y = z + e;
+    let crosses_cut = z.x < 0.0 && ((z.y < 0.0) != (y.y < 0.0));
     // |w|^2 = |e|^2 / |Z|^2; inf or nan (Z ~ 0, or both ~ 0) correctly fails
     // the `< 0.25` test below and falls through to the direct branch.
     let w2 = dot(e, e) / dot(z, z);
-    if w2 < 0.25 {
+    if w2 < 0.25 && !crosses_cut {
         let w = cdiv(e, z);
         var wk = w; // w^1
         var acc = vec2<f32>(0.0, 0.0);
@@ -138,7 +166,7 @@ fn complex_multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: vec2<f32>) -> vec2<f32
         }
         return cmul(cpow(z, p), acc);
     }
-    return cpow(z + e, p) - cpow(z, p);
+    return cpow(y, p) - cpow(z, p);
 }
 
 // One perturbation step of `kind`'s delta: e -> f(Z+e) - f(Z), where `z` is
@@ -159,7 +187,7 @@ fn advance_delta_kind(kind: u32, z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
         let ce = conj(e);
         return 2.0 * cmul(cz, ce) + cmul(ce, ce);
     } else if kind == KIND_MULTIBROT {
-        return multibrot_delta(z, e, clamp(u.power, 2u, 8u));
+        return multibrot_delta(z, e, clamp(u.power, 2u, MULTIBROT_MAX_POWER));
     } else if kind == KIND_CELTIC {
         // z^2 delta split: sq.x = delta of Re(z^2), sq.y = delta of Im(z^2).
         // Celtic abs the real output, so |Re(z^2)| delta = diffabs(Re(Z^2), sq.x).
@@ -194,7 +222,7 @@ fn advance_delta_kind(kind: u32, z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
 // enough to de-speckle filaments.
 fn fprime_kind(kind: u32, z: vec2<f32>) -> vec2<f32> {
     if kind == KIND_MULTIBROT {
-        let p = clamp(u.power, 2u, 8u);
+        let p = clamp(u.power, 2u, MULTIBROT_MAX_POWER);
         var zk = z; // Z^1
         for (var k: u32 = 2u; k < p; k = k + 1u) {
             zk = cmul(zk, z); // -> Z^{p-1}
@@ -232,6 +260,324 @@ fn fprime(z: vec2<f32>) -> vec2<f32> {
     return d;
 }
 
+// ---- Deep (rescaled) phase helpers -------------------------------------
+//
+// A deep value is an f32 mantissa times 2^exponent. All scaling is by exact
+// powers of two, so it never rounds.
+
+const LN2: f32 = 0.6931471805599453;
+// Where `ldexp_sat` saturates. Only ever compared against small values
+// (see `diffabs_scaled`), and far enough from f32's max that doubling it,
+// or squaring a value of the size it's compared with, stays finite.
+const LDEXP_SAT: f32 = 1.2676506e30; // 2^100
+
+// x * 2^k for any k. WGSL's `ldexp` is only defined for exponents inside
+// f32's range, so go through `frexp`: results below the smallest normal
+// flush to 0, results above 2^100 saturate to ±2^100.
+fn ldexp_sat(x: f32, k: i32) -> f32 {
+    if x == 0.0 {
+        return 0.0;
+    }
+    let f = frexp(x);
+    let ex = f.exp + k;
+    if ex > 100 {
+        return select(-LDEXP_SAT, LDEXP_SAT, x > 0.0);
+    }
+    if ex < -125 {
+        return 0.0;
+    }
+    return ldexp(f.fract, ex);
+}
+
+fn ldexp2_sat(v: vec2<f32>, k: i32) -> vec2<f32> {
+    return vec2<f32>(ldexp_sat(v.x, k), ldexp_sat(v.y, k));
+}
+
+// A complex number as mantissa * 2^e, with the mantissa's larger component in
+// [0.5, 1) (or exactly 0, with e = 0).
+struct Fe {
+    m: vec2<f32>,
+    e: i32,
+};
+
+fn fe_make(v: vec2<f32>, e: i32) -> Fe {
+    let a = max(abs(v.x), abs(v.y));
+    if a == 0.0 {
+        return Fe(vec2<f32>(0.0, 0.0), 0);
+    }
+    let k = frexp(a).exp;
+    return Fe(ldexp2_sat(v, -k), e + k);
+}
+
+// Reference point X[m] as f32 (points stored normalized flush to 0 here;
+// only deep references have any, see `ref_exp`).
+fn ref_at(m: u32) -> vec2<f32> {
+    let x = ref_orbit[m];
+    if DEEP {
+        let k = ref_exp[m];
+        if k != 0 {
+            return ldexp2_sat(x, k);
+        }
+    }
+    return x;
+}
+
+// X[m] with its full exponent range (deep phase only).
+fn ref_fe(m: u32) -> Fe {
+    return fe_make(ref_orbit[m], ref_exp[m]);
+}
+
+// Complex log of m * 2^e (m != 0).
+fn clog_fe(m: vec2<f32>, e: i32) -> vec2<f32> {
+    return vec2<f32>(0.5 * log(dot(m, m)) + f32(e) * LN2, atan2(m.y, m.x));
+}
+
+fn cexp(a: vec2<f32>) -> vec2<f32> {
+    return exp(a.x) * vec2<f32>(cos(a.y), sin(a.y));
+}
+
+// diffabs(c, 2^s * d) / 2^s = diffabs(c / 2^s, d): diffabs is positively
+// homogeneous. Saturating c / 2^s is harmless: once it dwarfs |d| the result
+// is just ±d.
+fn diffabs_scaled(c: f32, d: f32, s: i32) -> f32 {
+    return diffabs(ldexp_sat(c, -s), d);
+}
+
+// Sum of two `Fe`s, at the larger one's exponent.
+fn fe_add(a: Fe, b: Fe) -> Fe {
+    if a.m.x == 0.0 && a.m.y == 0.0 {
+        return b;
+    }
+    if b.m.x == 0.0 && b.m.y == 0.0 {
+        return a;
+    }
+    let e = max(a.e, b.e);
+    return fe_make(ldexp2_sat(a.m, a.e - e) + ldexp2_sat(b.m, b.e - e), e);
+}
+
+// One deep step's delta, (f(X + e) - f(X)) / 2^t: the mantissa `w` and its
+// exponent `t`. `t` is the input scale s except next to the critical point,
+// where the linear part of the step vanishes and the result is ~e^2, far
+// below the input's scale.
+struct DeepStep {
+    w: vec2<f32>,
+    t: i32,
+};
+
+// The per-kind formula of a deep step: (f(X + e) - f(X)) / 2^t for the
+// delta e = w * 2^s, given X measured in units of 2^u (`x` = X / 2^u) and
+// `sc` = 2^se, se = s - u, the delta's scale in those units. The result's
+// scale is t = s + (p-1)·u for a degree-p kind, since every kind here but
+// Lambda is p-homogeneous in (X, e) jointly (`diffabs_scaled` rescales the
+// fold-point comparisons the same way). Usually u = 0 (x = X, sc = 2^s,
+// t = s); see `deep_step_kind` for when it isn't. Lambda always gets u = 0.
+// Every kind needs an arm here too.
+fn advance_delta_scaled_kind(kind: u32, x: vec2<f32>, w: vec2<f32>, sc: f32, se: i32) -> vec2<f32> {
+    if kind == KIND_BURNING_SHIP {
+        let base = 2.0 * cmul(x, w) + sc * cmul(w, w);
+        let dp = x.x * w.y + x.y * w.x + sc * w.x * w.y;
+        return vec2<f32>(base.x, 2.0 * diffabs_scaled(x.x * x.y, dp, se));
+    } else if kind == KIND_TRICORN {
+        let cx = conj(x);
+        let cw = conj(w);
+        return 2.0 * cmul(cx, cw) + sc * cmul(cw, cw);
+    } else if kind == KIND_MULTIBROT {
+        return cmul(w, multibrot_sum(x, x + sc * w, clamp(u.power, 2u, MULTIBROT_MAX_POWER)));
+    } else if kind == KIND_CELTIC {
+        let sq = 2.0 * cmul(x, w) + sc * cmul(w, w);
+        return vec2<f32>(diffabs_scaled(x.x * x.x - x.y * x.y, sq.x, se), sq.y);
+    } else if kind == KIND_BUFFALO {
+        let sq = 2.0 * cmul(x, w) + sc * cmul(w, w);
+        return vec2<f32>(diffabs_scaled(x.x * x.x - x.y * x.y, sq.x, se),
+                         -diffabs_scaled(2.0 * x.x * x.y, sq.y, se));
+    } else if kind == KIND_PERPENDICULAR {
+        let sq = 2.0 * cmul(x, w) + sc * cmul(w, w);
+        let da = diffabs_scaled(x.y, w.y, se); // (|Y + ey| - |Y|) / 2^s
+        let abs_yf = abs(x.y) + sc * da;       // |Y + ey| * 2^(s-t)
+        return vec2<f32>(sq.x, -2.0 * (x.x * da + w.x * abs_yf));
+    } else if kind == KIND_LAMBDA {
+        let t = vec2<f32>(1.0 - 2.0 * x.x - sc * w.x, -2.0 * x.y - sc * w.y);
+        return cmul(u.lambda_l, cmul(w, t));
+    }
+    return 2.0 * cmul(x, w) + sc * cmul(w, w); // Mandelbrot (and Phoenix square part)
+}
+
+// Below this exponent a reference point counts as next to the critical point
+// 0 (see `deep_step_kind`). Above it, the e^2 terms 2^s * w^2 can only flush
+// to 0 when they are below 2^-50 of the linear ones.
+const DEEP_X_NEAR_LOG2: i32 = -60;
+
+// One deep step of `kind` for the delta e = w * 2^s from X (`x` as f32, `xf`
+// at full range). Usually the input scale is kept (t = s). But when X is
+// tiny (next to the critical point, e.g. at a minibrot's period), the linear
+// term vanishes and the step's value is ~e^p, which would flush to 0 at
+// scale s. There every z^p-like kind is p-homogeneous in (X, e) jointly, so
+// both are measured in units of 2^u (u = the larger one's exponent) and the
+// result lands at t = s + (p-1)·u (`ue` below).
+fn deep_step_kind(kind: u32, x: vec2<f32>, xf: Fe, w: vec2<f32>, sc: f32, s: i32) -> DeepStep {
+    if kind == KIND_COMPLEX_MULTIBROT {
+        return complex_multibrot_step(xf, w, s);
+    }
+    let x_zero = xf.m.x == 0.0 && xf.m.y == 0.0;
+    // Lambda's critical point is 1/2 and its step has a constant linear
+    // term (λ·e), so it never needs this.
+    if kind == KIND_LAMBDA || (!x_zero && xf.e >= DEEP_X_NEAR_LOG2) {
+        return DeepStep(advance_delta_scaled_kind(kind, x, w, sc, s), s);
+    }
+    let kw = deep_log2(w, vec2<f32>(0.0, 0.0));
+    if kw == DEEP_ZERO {
+        return DeepStep(w, s); // e = 0: f(X) - f(X)
+    }
+    var ue = s + kw;
+    if !x_zero {
+        ue = max(ue, xf.e);
+    }
+    var deg = 2;
+    if kind == KIND_MULTIBROT {
+        deg = i32(clamp(u.power, 2u, MULTIBROT_MAX_POWER));
+    }
+    let xk = ldexp2_sat(xf.m, xf.e - ue);
+    let se = s - ue;
+    let dw = advance_delta_scaled_kind(kind, xk, w, ldexp_sat(1.0, se), se);
+    return DeepStep(dw, s + (deg - 1) * ue);
+}
+
+// Scaled `complex_multibrot_delta` with X as a full-range `Fe`. Same series
+// as the f32 version, rewritten as X^(p-1) * w * sum_k C(p,k) r^(k-1)
+// (r = e/X) so nothing is formed at the delta's true scale; X^(p-1)'s own
+// exponent goes into the result's `t`. When |e/X| >= 0.5, X is itself tiny
+// (|X| <= 2|e|), so both terms of the direct form are taken in log space at
+// the larger one's scale. Branch-cut crossings leave the deep phase before
+// stepping (`deep_cut_crossing`).
+fn complex_multibrot_step(xf: Fe, w: vec2<f32>, s: i32) -> DeepStep {
+    let p = u.complex_power;
+    let wf = fe_make(w, s);
+    if wf.m.x == 0.0 && wf.m.y == 0.0 {
+        return DeepStep(vec2<f32>(0.0, 0.0), s);
+    }
+    if xf.m.x == 0.0 && xf.m.y == 0.0 {
+        // e^p.
+        let l = cmul(p, clog_fe(wf.m, wf.e));
+        let k = i32(floor(l.x / LN2));
+        return DeepStep(cexp(l - vec2<f32>(f32(k) * LN2, 0.0)), k);
+    }
+    let r = ldexp2_sat(cdiv(wf.m, xf.m), wf.e - xf.e);
+    if dot(r, r) < 0.25 {
+        var acc = cm_coef(1u);
+        var rk = r; // r^(k-1)
+        for (var k: u32 = 2u; k <= COMPLEX_MULTIBROT_TERMS; k = k + 1u) {
+            acc = acc + cmul(cm_coef(k), rk);
+            rk = cmul(rk, r);
+            if dot(rk, rk) < 1e-18 * dot(acc, acc) {
+                break;
+            }
+        }
+        // X^(p-1) = cexp(l) = cexp(l - k·ln2) * 2^k.
+        let l = cmul(p - vec2<f32>(1.0, 0.0), clog_fe(xf.m, xf.e));
+        let k = i32(floor(l.x / LN2));
+        let x_pm1 = cexp(l - vec2<f32>(f32(k) * LN2, 0.0));
+        return DeepStep(cmul(cmul(w, x_pm1), acc), s + k);
+    }
+    // (X + e)^p - X^p, both in log space (X + e may be exactly 0).
+    let xs = ldexp2_sat(xf.m, xf.e - s);
+    let yf = fe_make(xs + w, s);
+    let lb = cmul(p, clog_fe(xf.m, xf.e));
+    var k = i32(floor(lb.x / LN2));
+    var la = vec2<f32>(0.0, 0.0);
+    let y_zero = yf.m.x == 0.0 && yf.m.y == 0.0;
+    if !y_zero {
+        la = cmul(p, clog_fe(yf.m, yf.e));
+        k = max(k, i32(floor(la.x / LN2)));
+    }
+    let kl = vec2<f32>(f32(k) * LN2, 0.0);
+    var ya = vec2<f32>(0.0, 0.0);
+    if !y_zero {
+        ya = cexp(la - kl);
+    }
+    return DeepStep(ya - cexp(lb - kl), k);
+}
+
+// Deep-phase twin of `advance_delta` (same morph blend, at the larger of the
+// two kinds' output scales).
+fn advance_delta_scaled(x: vec2<f32>, xf: Fe, w: vec2<f32>, sc: f32, s: i32) -> DeepStep {
+    let a = deep_step_kind(KIND, x, xf, w, sc, s);
+    if MORPH {
+        let b = deep_step_kind(u.morph_from, x, xf, w, sc, s);
+        let t = max(a.t, b.t);
+        return DeepStep(mix(ldexp2_sat(a.w, a.t - t), ldexp2_sat(b.w, b.t - t), u.morph_w), t);
+    }
+    return a;
+}
+
+// Whether this step would take Complex Multibrot's X + e across the branch
+// cut (see `complex_multibrot_delta`). The delta then jumps to the size of X,
+// so the deep phase ends and the f32 loop takes the step.
+fn deep_cut_crossing(xf: Fe, w: vec2<f32>, s: i32) -> bool {
+    let cm = KIND == KIND_COMPLEX_MULTIBROT || (MORPH && u.morph_from == KIND_COMPLEX_MULTIBROT);
+    if !cm {
+        return false;
+    }
+    let yn = xf.m + ldexp2_sat(w, s - xf.e); // (X + e) / 2^xe
+    return xf.m.x < 0.0 && ((xf.m.y < 0.0) != (yn.y < 0.0));
+}
+
+// Binary exponent of the largest component of a pair of complex mantissas
+// (`frexp` convention: |x| < 2^k), or DEEP_ZERO when both are exactly 0.
+const DEEP_ZERO: i32 = -100000;
+fn deep_log2(a: vec2<f32>, b: vec2<f32>) -> i32 {
+    let m = max(max(abs(a.x), abs(a.y)), max(abs(b.x), abs(b.y)));
+    if m == 0.0 {
+        return DEEP_ZERO;
+    }
+    return frexp(m).exp;
+}
+
+// f'(y) at the full value y = X + w * 2^s, as an `Fe`. Plain `fprime` in f32
+// unless y is below f32's comfortable range, which only happens next to the
+// critical point 0 (after a rebase), where X is itself tiny: then y is
+// formed in the 2^s-scaled domain and f' ~ p·y^(p-1) keeps its exponent.
+fn deep_fprime(xf: Fe, w: vec2<f32>, s: i32, yt: vec2<f32>) -> Fe {
+    if MORPH || max(abs(yt.x), abs(yt.y)) >= DEEP_TINY {
+        return Fe(fprime(yt), 0);
+    }
+    let yf = fe_make(ldexp2_sat(xf.m, xf.e - s) + w, s);
+    if yf.m.x == 0.0 && yf.m.y == 0.0 {
+        return Fe(fprime(vec2<f32>(0.0, 0.0)), 0);
+    }
+    if KIND == KIND_MULTIBROT {
+        let p = clamp(u.power, 2u, MULTIBROT_MAX_POWER);
+        var ym = yf.m; // m^(p-1)
+        for (var k: u32 = 2u; k < p; k = k + 1u) {
+            ym = cmul(ym, yf.m);
+        }
+        return fe_make(f32(p) * ym, yf.e * i32(p - 1u));
+    } else if KIND == KIND_LAMBDA {
+        return Fe(fprime(vec2<f32>(0.0, 0.0)), 0); // λ(1 - 2y) ~ λ
+    } else if KIND == KIND_COMPLEX_MULTIBROT {
+        // p·y^(p-1) = p·exp(l), l = (p-1)·ln y; keep exp(Re l)'s exponent.
+        let p = u.complex_power;
+        let l = cmul(p - vec2<f32>(1.0, 0.0), clog_fe(yf.m, yf.e));
+        let k = i32(floor(l.x / LN2));
+        return fe_make(cmul(p, cexp(vec2<f32>(l.x - f32(k) * LN2, l.y))), k);
+    }
+    // z^2-like kinds: 2y (|f'| = |2y| for the abs variants too, see fprime).
+    return Fe(2.0 * yf.m, yf.e);
+}
+
+// The deep phase hands over to the f32 loop once the delta's magnitude
+// reaches 2^DEEP_EXIT_LOG2: by then |e|^2 is still a normal f32 and the
+// pixel offset dc (< 2^-99 on deep views) is below f32 rounding of e. The DE
+// derivative only has to be a comfortably normal f32 (DEEP_EXIT_DZ_LOG2).
+const DEEP_EXIT_LOG2: i32 = -48;
+const DEEP_EXIT_DZ_LOG2: i32 = -100;
+// Below this, a full value y goes through `deep_fprime`'s extended path.
+const DEEP_TINY: f32 = 7.888609e-31; // 2^-100
+// The mantissas are renormalized once their exponent drifts past ±this.
+const DEEP_RENORM_LOG2: i32 = 16;
+// A reference point can only matter for rebasing when it's within this many
+// binades above the delta's scale (|w| < 2^DEEP_RENORM_LOG2).
+const DEEP_NEAR_LOG2: i32 = 24;
+
 // Weight of the Phoenix kind's p*z_{n-1} term in the current map: 1 for plain
 // Phoenix, its morph share while switching to/from Phoenix, else 0.
 fn phoenix_weight() -> f32 {
@@ -245,10 +591,19 @@ fn phoenix_weight() -> f32 {
     return w;
 }
 
-// Periodicity (interior) detection, Brent-style: the full orbit value is
+// Periodicity (interior) detection, Brent-style: windows end at iterations
-// saved at iterations PERIOD_FIRST_CHECK, 2x that, 4x ..., and every later
+// PERIOD_FIRST_CHECK, 2x that, 4x ..., and at each window end the window's
-// iterate is compared against the last saved one. Returning within
+// iterate closest to the critical point is saved. Every later iterate is
-// PERIOD_EPS2 (relative, squared) means the orbit has closed a cycle.
+// compared against the last saved one. Returning within PERIOD_EPS2
+// (relative, squared) means the orbit has closed a cycle.
+//
+// Why the closest-to-critical iterate rather than the one at the window end:
+// near a deep minibrot, an orbit follows the minibrot's cycle with a
+// deviation at the minibrot's own scale. At an arbitrary phase |z| ~ 1, so
+// that deviation is far below the relative tolerance and *any* nearby
+// exterior orbit "returned" (a large black disk around a minibrot at
+// ~1e-13 zoom). At the phase nearest the critical point, z itself is at the
+// minibrot's scale, so the relative tolerance measures the actual return.
 //
 // A close return alone isn't trusted. A pixel just *outside* the set (at a
 // minibrot's edge, or a cusp) can shadow a cycle for thousands of iterations
@@ -290,6 +645,15 @@ fn periodic_enabled() -> bool {
     return true;
 }
 
+// Critical point of the current map (where f' = 0), the periodicity save
+// point's reference: 0 for every z^p-like kind, 1/2 for Lambda's λz(1-z).
+fn critical_point() -> vec2<f32> {
+    if KIND == KIND_LAMBDA {
+        return vec2<f32>(0.5, 0.0);
+    }
+    return vec2<f32>(0.0, 0.0);
+}
+
 // Escape data for one sample: `ci` is the (color-independent) palette parameter,
 // `de` the distance-estimate darkening factor in [0,1], `escaped` false for the
 // interior of the set. Splitting iteration from coloring lets a colour change be
@@ -303,13 +667,14 @@ struct Sample {
 // Perturbation iterate a single sample. `offset` is the per-pixel offset in
 // complex units. For Mandelbrot it is the c-plane offset added every step (delta
 // starts at 0); for Julia it is the z-plane offset that seeds the initial delta
-// (c is fixed, so nothing is added per step).
+// (c is fixed, so nothing is added per step). In `DEEP` pipelines both
+// `offset` and `px` are in units of 2^u.scale_exp.
 fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
     // Loop invariants, read once instead of on every iteration.
     let max_iter = u.max_iter;
     let bailout_sq = u.bailout_sq;
     let ref_len = u.ref_len;
-    let z0 = ref_orbit[0]; // reference start (0 for Mandelbrot, center for Julia)
+    let z0 = ref_at(0u); // reference start (0 for Mandelbrot, center for Julia)
 
     // Main cardioid / period-2 bulb bypass: those points never escape, so skip
     // iterating them (they'd otherwise all burn the full max_iter). `offset` is
@@ -317,7 +682,7 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
     // orbit itself, since X_1 = X_0^2 + C_ref = C_ref. That's only f32-accurate,
     // so skip the test once a pixel is smaller than that error (deep zoom),
     // where it could misclassify pixels right at the boundary.
-    if KIND == KIND_MANDELBROT && !MORPH && !IS_JULIA && ref_len > 1u && px > 1e-6 {
+    if KIND == KIND_MANDELBROT && !MORPH && !IS_JULIA && !DEEP && ref_len > 1u && px > 1e-6 {
         let c = ref_orbit[1] + offset;
         let xq = c.x - 0.25;
         let q = xq * xq + c.y * c.y;
@@ -333,6 +698,8 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
     // seeds the delta and nothing is added per step.
     var step_add = offset;
     var e = vec2<f32>(0.0, 0.0);
+    // Pixel size in complex units (`px` is pre-scaled in DEEP pipelines).
+    var px_t = px;
     // Orbit derivative for distance estimation, pre-multiplied by the pixel
     // size `px`. For the set plane it is px·d/dc (starts at 0, gains +px each
     // step); for Julia it is px·d/dz0 (starts at px). The raw derivative grows
@@ -343,7 +710,7 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
     if IS_JULIA {
         step_add = vec2<f32>(0.0, 0.0);
         e = offset;
-        dzs = vec2<f32>(px, 0.0);
+        dzs = vec2<f32>(px_t, 0.0);
     }
     // Previous-iterate state for the Phoenix two-term recurrence (delta of
     // y_{n-1}, and its scaled derivative for DE). Both start at 0 (y_{-1} = 0).
@@ -368,9 +735,254 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
     var check_at = PERIOD_FIRST_CHECK;
     var period_hit = false;
     var period_streak = 0u;
+    // This window's save candidate: its iterate closest to the critical
+    // point, that distance squared, and the |f'|^2 product since it.
+    let crit = critical_point();
+    var z_cand = z;
+    var cand_d2 = 3.0e38;
+    var mult2_cand = 1.0;
+
+    // Deep phase (DEEP pipelines only): iterate the delta as e = w * 2^sx,
+    // with `w` an f32 mantissa kept near 1 by renormalizing and `sx` an i32
+    // exponent, while |e| is too small for f32 (it starts at the pixel offset,
+    // ~2^scale_exp). The DE derivative is linear in the same way and carried
+    // as dzs = v * 2^sv, with its own exponent: near the critical point (after
+    // a rebase) f' is tiny, so dzs and e can drift far apart. Once both are
+    // big enough (or a rebase makes the delta large), the state is converted
+    // to plain f32 and the loop below carries on from the same n / m.
+    if DEEP {
+        let scale_e = u.scale_exp;
+        var sx = scale_e;
+        var sv = scale_e;
+        var sc = ldexp_sat(1.0, sx);
+        var w = vec2<f32>(0.0, 0.0);
+        var v = vec2<f32>(0.0, 0.0);
+        var w_prev = vec2<f32>(0.0, 0.0);
+        var v_prev = vec2<f32>(0.0, 0.0);
+        // Per-step additions (dc and px for the set plane), in units of 2^sx
+        // and 2^sv.
+        var d = offset;
+        var pd = px;
+        if IS_JULIA {
+            w = offset;
+            v = vec2<f32>(px, 0.0);
+            d = vec2<f32>(0.0, 0.0);
+            pd = 0.0;
+        }
+        let z0f = ref_fe(0u);
+        var xf = z0f;
+        var xt = z0;
+        var xf_old = xf;
+        var xt_old = xt;
+        var w_old = w;
+        var rebase_exit = false;
+        var cut_exit = false;
+        loop {
+            // Full value y = X + e as f32 (e flushes to 0 when negligible).
+            let yt = xt + ldexp2_sat(w, sx);
+            if dot(yt, yt) > bailout_sq {
+                escaped = true;
+                break;
+            }
+            if n >= max_iter {
+                break;
+            }
+            if deep_cut_crossing(xf, w, sx) {
+                cut_exit = true;
+                break;
+            }
+
+            if DE {
+                let fp = deep_fprime(xf, w, sx, yt);
+                if fp.e == 0 {
+                    var v_new = cmul(fp.m, v);
+                    if !IS_JULIA {
+                        v_new.x = v_new.x + pd;
+                    }
+                    if phoenix_w > 0.0 {
+                        v_new = v_new + phoenix_w * cmul(u.phoenix_p, v_prev);
+                        v_prev = v;
+                    }
+                    v = v_new;
+                } else {
+                    // f' is below f32's range (next to the critical point):
+                    // sum the terms at the largest one's scale and move
+                    // there, like the delta below.
+                    var acc = fe_make(cmul(fp.m, v), fp.e + sv);
+                    if !IS_JULIA {
+                        acc = fe_add(acc, fe_make(vec2<f32>(px, 0.0), scale_e));
+                    }
+                    if phoenix_w > 0.0 {
+                        acc = fe_add(acc, fe_make(phoenix_w * cmul(u.phoenix_p, v_prev), sv));
+                    }
+                    if acc.m.x == 0.0 && acc.m.y == 0.0 {
+                        if phoenix_w > 0.0 {
+                            v_prev = v;
+                        }
+                        v = acc.m;
+                    } else {
+                        if phoenix_w > 0.0 {
+                            v_prev = ldexp2_sat(v, sv - acc.e);
+                        }
+                        v = acc.m;
+                        sv = acc.e;
+                        if !IS_JULIA {
+                            pd = ldexp_sat(px, scale_e - sv);
+                        }
+                    }
+                }
+            }
+            w_old = w;
+            let st = advance_delta_scaled(xt, xf, w, sc, sx);
+            if st.t == sx {
+                w = st.w + d;
+                if phoenix_w > 0.0 {
+                    w = w + phoenix_w * cmul(u.phoenix_p, w_prev);
+                    w_prev = w_old;
+                }
+            } else {
+                // The step's value is at another scale (next to the
+                // critical point it is ~e^2, far below 2^sx): add dc and the
+                // Phoenix term at the largest addend's scale and move there.
+                var acc = fe_make(st.w, st.t);
+                if !IS_JULIA {
+                    acc = fe_add(acc, fe_make(offset, scale_e));
+                }
+                if phoenix_w > 0.0 {
+                    acc = fe_add(acc, fe_make(phoenix_w * cmul(u.phoenix_p, w_prev), sx));
+                }
+                if acc.m.x == 0.0 && acc.m.y == 0.0 {
+                    w = acc.m;
+                    if phoenix_w > 0.0 {
+                        w_prev = w_old;
+                    }
+                } else {
+                    // w_old (the pre-step delta) is still needed at the new
+                    // scale: it's the next previous delta, and a rebase reads it.
+                    w_old = ldexp2_sat(w_old, sx - acc.e);
+                    if phoenix_w > 0.0 {
+                        w_prev = w_old;
+                    }
+                    w = acc.m;
+                    sx = acc.e;
+                    sc = ldexp_sat(1.0, sx);
+                    if !IS_JULIA {
+                        d = ldexp2_sat(offset, scale_e - sx);
+                    }
+                }
+            }
+            m = m + 1u;
+            n = n + 1u;
+
+            if m >= ref_len {
+                escaped = true; // see the f32 loop's reference-exhausted case
+                break;
+            }
+            xf_old = xf;
+            xt_old = xt;
+            xf = ref_fe(m);
+            xt = ldexp2_sat(xf.m, xf.e);
+
+            // Rebase test |X + e| < |e|, in units of 2^sx. Only possible when
+            // |X| is within a few binades of |e| (|w| < 2^DEEP_RENORM_LOG2).
+            if xf.e - sx < DEEP_NEAR_LOG2 {
+                let q = ldexp2_sat(xf.m, xf.e - sx) + w;
+                if dot(q, q) < dot(w, w) {
+                    // The new delta y - X[0] stays tiny only if X[0] is
+                    // (always, for the set plane), and for Phoenix only if the
+                    // previous full value y_{n-1} (its new previous delta) is.
+                    let z0_small = (z0f.m.x == 0.0 && z0f.m.y == 0.0)
+                        || z0f.e - sx < DEEP_NEAR_LOG2;
+                    let prev_small = phoenix_w == 0.0 || xf_old.e - sx < DEEP_NEAR_LOG2;
+                    if !(z0_small && prev_small) {
+                        rebase_exit = true;
+                        break;
+                    }
+                    if phoenix_w > 0.0 {
+                        w_prev = ldexp2_sat(xf_old.m, xf_old.e - sx) + w_old;
+                    }
+                    w = q - ldexp2_sat(z0f.m, z0f.e - sx);
+                    xf = z0f;
+                    xt = z0;
+                    m = 0u;
+                }
+            }
+
+            // Leave once both the delta and the derivative fit in f32 (an
+            // exactly-zero one always does); otherwise renormalize the
+            // mantissas when they drift (exact: powers of two only).
+            let kw = deep_log2(w, w_prev);
+            let kv = deep_log2(v, v_prev);
+            let w_ok = kw == DEEP_ZERO || sx + kw > DEEP_EXIT_LOG2;
+            let v_ok = !DE || kv == DEEP_ZERO || sv + kv > DEEP_EXIT_DZ_LOG2;
+            if w_ok && v_ok {
+                break;
+            }
+            if kw != DEEP_ZERO && abs(kw) > DEEP_RENORM_LOG2 {
+                w = ldexp2_sat(w, -kw);
+                w_prev = ldexp2_sat(w_prev, -kw);
+                sx = sx + kw;
+                sc = ldexp_sat(1.0, sx);
+                if !IS_JULIA {
+                    d = ldexp2_sat(offset, scale_e - sx);
+                }
+            }
+            if DE && kv != DEEP_ZERO && abs(kv) > DEEP_RENORM_LOG2 {
+                v = ldexp2_sat(v, -kv);
+                v_prev = ldexp2_sat(v_prev, -kv);
+                sv = sv + kv;
+                if !IS_JULIA {
+                    pd = ldexp_sat(px, scale_e - sv);
+                }
+            }
+        }
+
+        // Hand over to the f32 loop. dc and px may flush to 0 here: they
+        // are below f32 rounding of the (now large enough) delta and
+        // derivative.
+        px_t = ldexp_sat(px, scale_e);
+        if !IS_JULIA {
+            step_add = ldexp2_sat(offset, scale_e);
+        }
+        e = ldexp2_sat(w, sx);
+        e_prev = ldexp2_sat(w_prev, sx);
+        dzs = ldexp2_sat(v, sv);
+        dzs_prev = ldexp2_sat(v_prev, sv);
+        xm = xt;
+        if rebase_exit {
+            // Rebase in plain f32: the new delta (y - X[0], and for Phoenix
+            // the previous full value) is no longer tiny.
+            if phoenix_w > 0.0 {
+                e_prev = xt_old + ldexp2_sat(w_old, sx);
+            }
+            e = (xt + e) - z0;
+            xm = z0;
+            m = 0u;
+        }
+        if cut_exit && e.y == 0.0 && w.y != 0.0 {
+            // Keep the side of the cut the pixel is on even if e.y flushed
+            // to 0: that decides the branch in the next (f32) step.
+            e.y = select(-1.17549435e-38, 1.17549435e-38, w.y > 0.0);
+        }
+        z = xm + e;
+        z2 = dot(z, z);
+        // Restart periodicity detection from here (check_at must stay ahead
+        // of n, or no window would ever close). Nothing is saved until the
+        // first window closes: `z` here is at an arbitrary phase, and the
+        // pixel still shadows the reference (exactly periodic when it's a
+        // minibrot nucleus), so comparing against it would flag exterior
+        // pixels as interior (see PERIOD_FIRST_CHECK). The sentinel is far
+        // outside the bailout radius, so no return can match it.
+        z_saved = vec2<f32>(1e18, 1e18);
+        z_cand = z;
+        while check_at <= n {
+            check_at = check_at * 2u;
+        }
+    }
 
     loop {
-        if z2 > bailout_sq {
+        // (`escaped` may already be set by the deep phase.)
+        if escaped || z2 > bailout_sq {
             escaped = true;
             break;
         }
@@ -387,12 +999,14 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
             fp = fprime(z);
         }
         if periodic {
-            mult2 = mult2 * dot(fp, fp);
+            let fp2 = dot(fp, fp);
+            mult2 = mult2 * fp2;
+            mult2_cand = mult2_cand * fp2;
         }
         if DE {
             var dzs_new = cmul(fp, dzs);
             if !IS_JULIA {
-                dzs_new.x = dzs_new.x + px;
+                dzs_new.x = dzs_new.x + px_t;
             }
             if phoenix_w > 0.0 {
                 dzs_new = dzs_new + phoenix_w * cmul(u.phoenix_p, dzs_prev);
@@ -421,7 +1035,7 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
             escaped = true;
             break;
         }
-        xm = ref_orbit[m];
+        xm = ref_at(m);
         z = xm + e;
         z2 = dot(z, z);
         if z2 < dot(e, e) {
@@ -449,13 +1063,20 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
                     break;
                 }
             }
+            let dc2 = dot(z - crit, z - crit);
+            if dc2 < cand_d2 {
+                cand_d2 = dc2;
+                z_cand = z;
+                mult2_cand = 1.0;
+            }
             if n == check_at {
                 if !period_hit {
                     period_streak = 0u;
                 }
                 period_hit = false;
-                z_saved = z;
+                z_saved = z_cand;
-                mult2 = 1.0;
+                mult2 = mult2_cand;
+                cand_d2 = 3.0e38;
                 check_at = check_at * 2u;
             }
         }
@@ -492,14 +1113,51 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
         // filaments stay crisp instead of aliasing into speckle. If |dzs|
         // overflowed (far sub-pixel from the set), de -> 0 and the boundary
         // simply reads as dark, which is the correct limit.
+        // Shadow/3D use DE as a height field, so it must stay unclamped: any
+        // cap flattens everything farther than that from the set into a
+        // uniform plateau (a visible circle around the set when zoomed out).
+        // 3D saturates heights smoothly itself (`sdf` in colorize.wgsl).
         let zmag = sqrt(max(z2, 1.0));
         let dzmag = sqrt(max(dot(dzs_esc, dzs_esc), 1e-30));
-        let max_de = select(1.0, 1000.0, u.shadow != 0u);
+        // |z|·ln|z|/|dz| is G/|G'| (G the potential, ln|z|/2^n). Far from the
-        de = clamp(zmag * log(zmag) / dzmag, 0.0, max_de);
+        // set G ~ ln r, so it grows like r·ln r rather than r: zoomed far out,
+        // the 3D cone got steeper with every zoom step, down to a texel-wide
+        // needle cut off above the plateau. Replacing G by 2(1 - e^(-G/2))
+        // leaves it unchanged near the set (G -> 0, and ~0.8x at the edge of
+        // the default view) but caps it at 2, so far away DE grows like 2r
+        // and the cone stops narrowing. G underflows to 0 past ~150
+        // iterations, where the factor is 1 anyway. G divides by the map's
+        // degree d per step, not 2: with 2^-n, pixels either side of a band
+        // boundary got G off by d/2, a DE seam on every band for d != 2.
+        let g = log(zmag) * pow(escape_degree(), -f32(n));
+        let far = select(1.0, 2.0 * (1.0 - exp(-g / 2.0)) / g, g > 1e-4);
+        let max_de = select(1.0, 1e30, u.shadow != 0u);
+        de = clamp(zmag * log(zmag) / dzmag * far, 0.0, max_de);
     }
     return Sample(ci, de, true);
 }
 
+// Degree d of the current map at infinity (|f(z)| ~ |z|^d), which sets how
+// fast the potential G = ln|z_n| / d^n shrinks per step. Complex Multibrot's
+// |z^p| = |z|^Re(p)·e^(-Im(p)·arg z) grows like |z|^Re(p) (arg is bounded).
+// A morph blend is dominated by the higher degree.
+fn kind_degree(kind: u32) -> f32 {
+    if kind == KIND_MULTIBROT {
+        return f32(clamp(u.power, 2u, MULTIBROT_MAX_POWER));
+    } else if kind == KIND_COMPLEX_MULTIBROT {
+        return max(u.complex_power.x, 1.0);
+    }
+    return 2.0;
+}
+
+fn escape_degree() -> f32 {
+    let d = kind_degree(KIND);
+    if MORPH {
+        return max(d, kind_degree(u.morph_from));
+    }
+    return d;
+}
+
 // 1 / ln(2), for the smooth iteration count's log2(ln|z| / ln 2).
 const INV_LN2: f32 = 1.4426950408889634;
 
@@ -576,7 +1234,7 @@ fn fs_data(in: VsOut) -> @location(0) vec4<f32> {
 // 4-neighbour's 1-spp sample differs from its own by more than this. `ci`
 // steps are palette-phase steps of `ci * color_scale` (color_scale <= 1 in the
 // UI), so 0.02 keeps anything visibly banded; DE is compared relative to its
-// own magnitude (it's in pixels, up to 1000 for shadow/3D height fields).
+// own magnitude (it's in pixels, unbounded for shadow/3D height fields).
 const AA_CI_EPS: f32 = 0.02;
 const AA_DE_EPS: f32 = 0.1;
 

src/view.rs

@@ -1,65 +1,303 @@
 //! Camera / view state over the complex plane.
 //!
-//! The center is stored in arbitrary precision (`FBig`) — this is what lets us
+//! The center is stored in arbitrary precision ([`Big`]) — this is what lets us
-//! zoom far past f64's ~1e13x limit. The pixel *scale* stays `f64`: even at
+//! zoom far past f64's ~1e13x limit. The pixel *scale* is a [`Scale`]: an
-//! 10^30x zoom the scale is ~1e-33, comfortably inside f64's range. Only the
+//! f64 mantissa with its own `i32` binary exponent, so it isn't bound by
-//! center needs the extra digits.
+//! f64's ~1e-308 range either (the floor, `Scale::MIN`, only keeps the GPU's
+//! i32 exponent arithmetic far from overflow). Once a pixel is smaller than
+//! `DEEP_PIXEL_SIZE` the GPU switches to rescaled deltas (see `needs_deep`),
+//! since f32 alone bottoms out near 1e-38.
 
 use core::str::FromStr;
 
-use dashu_float::round::mode::HalfAway;
+pub use crate::bignum::Big;
-use dashu_float::{DBig, FBig};
-
-/// Arbitrary-precision binary float (base 2, round-half-away). One coordinate.
-pub type Big = FBig<HalfAway, 2>;
 
 /// Half-height (complex units) of the default view; also the zoom-1 reference.
 pub const DEFAULT_HALF_HEIGHT: f64 = 1.25;
 
+/// Below this pixel size (complex units per pixel) the GPU renders with the
+/// deep pipeline, whose per-pixel deltas start out as an f32 mantissa times
+/// `2^scale_exp`. Plain f32 stays exact as long as the smallest per-pixel
+/// offsets (a quarter pixel, for the AA grid) are normal floats (>= 2^-126),
+/// i.e. down to 2^-124 per pixel. Measured: pixel-identical to the deep path
+/// down to 2^-124 (no AA), first errors at 2^-126, all black by 2^-136. The
+/// deep path is slower, so the switch is as late as that allows, with two
+/// binades of margin.
+pub const DEEP_PIXEL_SIZE: f64 = 1.0 / (1u128 << 122) as f64; // 2^-122
+
+/// Whether a view rendered `height_px` pixels tall needs the deep pipeline
+/// (see `DEEP_PIXEL_SIZE`).
+pub fn needs_deep(half_height: Scale, height_px: f64) -> bool {
+    half_height.mul_f64(2.0 / height_px.max(1.0)) < Scale::from_f64(DEEP_PIXEL_SIZE)
+}
+
+/// Binary exponent `E` of the deep view scale: `floor(log2(half_height))`,
+/// so the rescaled span is in `[2, 4)`. Never 0, which means "not deep"
+/// (see `Uniforms::scale_exp`).
+pub fn deep_scale_exp(half_height: Scale) -> i32 {
+    let e = half_height.exponent();
+    if e == 0 { -1 } else { e }
+}
+
+/// `x * 2^k` for any `k`, saturating to 0 / infinity like the true value
+/// would (`powi` alone overflows at 2^±1024 even when the product fits).
+fn ldexp(mut x: f64, mut k: i32) -> f64 {
+    while k > 1000 {
+        x *= 2f64.powi(1000);
+        k -= 1000;
+        if x.is_infinite() || x == 0.0 {
+            return x;
+        }
+    }
+    while k < -1000 {
+        x *= 2f64.powi(-1000);
+        k += 1000;
+        if x == 0.0 || x.is_infinite() {
+            return x;
+        }
+    }
+    x * 2f64.powi(k)
+}
+
+/// A positive real with f64 precision and an `i32` binary exponent:
+/// `m · 2^e`, `m` in `[1, 2)`. The view's half-height (and the pixel size
+/// derived from it) is one of these, so zoom isn't bound by f64's range.
+#[derive(Clone, Copy, Debug, PartialEq)]
+pub struct Scale {
+    m: f64,
+    e: i32,
+}
+
+impl Scale {
+    /// Deepest scale the view can reach: 2^-(2^20) (about 1e-315653). Far
+    /// past anything a reference orbit can practically be computed for; it
+    /// only keeps the shader's i32 exponent sums (scale × degree) from
+    /// overflowing.
+    pub const MIN: Scale = Scale {
+        m: 1.0,
+        e: -(1 << 20),
+    };
+
+    /// `m · 2^e`, normalized. Non-positive or NaN input gives `MIN`.
+    pub fn from_parts(m: f64, e: i32) -> Self {
+        if m.is_nan() || m <= 0.0 {
+            return Self::MIN;
+        }
+        if m.is_infinite() {
+            return Scale {
+                m: 1.0,
+                e: i32::MAX / 2,
+            };
+        }
+        // Bring m into [1, 2) through its own binary exponent (exact).
+        let k = m.log2().floor() as i32;
+        let mut m = ldexp(m, -k);
+        let mut e = e.saturating_add(k);
+        // log2 can round across a power of two.
+        if m >= 2.0 {
+            m /= 2.0;
+            e = e.saturating_add(1);
+        } else if m < 1.0 {
+            m *= 2.0;
+            e = e.saturating_sub(1);
+        }
+        Scale { m, e }.max(Self::MIN)
+    }
+
+    pub fn from_f64(x: f64) -> Self {
+        Self::from_parts(x, 0)
+    }
+
+    /// `2^l`.
+    pub fn from_log2(l: f64) -> Self {
+        let e = l.floor();
+        Self::from_parts((l - e).exp2(), e as i32)
+    }
+
+    /// The value as an f64 (0 or infinity outside its range).
+    pub fn to_f64(self) -> f64 {
+        ldexp(self.m, self.e)
+    }
+
+    /// `self · 2^k` as an f64: the value in units of `2^-k`.
+    pub fn scaled_f64(self, k: i32) -> f64 {
+        ldexp(self.m, self.e.saturating_add(k))
+    }
+
+    /// `floor(log2(self))`.
+    pub fn exponent(self) -> i32 {
+        self.e
+    }
+
+    pub fn log2(self) -> f64 {
+        self.m.log2() + self.e as f64
+    }
+
+    pub fn log10(self) -> f64 {
+        self.log2() * core::f64::consts::LOG10_2
+    }
+
+    /// `self · f` (`f > 0`).
+    pub fn mul_f64(self, f: f64) -> Self {
+        Self::from_parts(self.m * f, self.e)
+    }
+
+    /// `self / other`, as an f64.
+    pub fn ratio(self, other: Scale) -> f64 {
+        ldexp(self.m / other.m, self.e.saturating_sub(other.e))
+    }
+
+    pub fn max(self, other: Scale) -> Self {
+        if other > self { other } else { self }
+    }
+
+    pub fn min(self, other: Scale) -> Self {
+        if other < self { other } else { self }
+    }
+
+    pub fn clamp(self, lo: Scale, hi: Scale) -> Self {
+        self.max(lo).min(hi)
+    }
+
+    /// `f · self` as an exact `Big` at `bits` of precision (`f` any f64).
+    pub fn big_times(self, f: f64, bits: usize) -> Big {
+        big_from_f64(f * self.m, bits) << self.e as isize
+    }
+
+    /// Exact binary value as a `Big`.
+    fn to_big(self) -> Big {
+        big_from_f64(self.m, 53) << self.e as isize
+    }
+}
+
+impl PartialOrd for Scale {
+    fn partial_cmp(&self, other: &Self) -> Option<core::cmp::Ordering> {
+        // Normalized and positive: the exponent decides, then the mantissa.
+        Some(self.e.cmp(&other.e).then(self.m.partial_cmp(&other.m)?))
+    }
+}
+
+impl core::fmt::Display for Scale {
+    /// Scientific notation, `1.5e-20` / `3.7e-4000`. The precision flag
+    /// (`{:.4}`) sets mantissa digits after the point; without it, enough
+    /// digits to parse back to the same value.
+    fn fmt(&self, f: &mut core::fmt::Formatter<'_>) -> core::fmt::Result {
+        let x = self.to_f64();
+        if x.is_normal() {
+            return match f.precision() {
+                Some(p) => write!(f, "{x:.p$e}"),
+                None => write!(f, "{x:e}"),
+            };
+        }
+        // Out of f64's range: round the exact decimal expansion instead.
+        let big = self.to_big();
+        let sci = |sig: usize, pad: Option<usize>| {
+            let parts = big.to_decimal_parts(sig);
+            let digits = if parts.digits.is_empty() {
+                "0"
+            } else {
+                &parts.digits
+            };
+            // value = 0.digits · 10^exp10; move the point after the first digit.
+            let exp10 = parts.exp10 - 1;
+            let (head, tail) = digits.split_at(1);
+            let tail = match pad {
+                Some(p) => format!("{tail:0<p$}"),
+                None => tail.to_string(),
+            };
+            if tail.is_empty() {
+                format!("{head}e{exp10}")
+            } else {
+                format!("{head}.{tail}e{exp10}")
+            }
+        };
+        match f.precision() {
+            Some(p) => f.write_str(&sci(p + 1, Some(p))),
+            None => {
+                // Like `{:e}` on f64: the shortest string that parses back.
+                let s = (1..17)
+                    .map(|sig| sci(sig, None))
+                    .find(|s| s.parse::<Scale>() == Ok(*self))
+                    .unwrap_or_else(|| sci(17, None));
+                f.write_str(&s)
+            }
+        }
+    }
+}
+
+impl FromStr for Scale {
+    type Err = ();
+
+    /// Parses any positive decimal (`1.25`, `1.5e-20`, `3.7e-4000`), rounding
+    /// to the nearest f64 mantissa. Values past `MIN` clamp to it.
+    fn from_str(s: &str) -> Result<Self, ()> {
+        let s = s.trim();
+        if let Ok(x) = s.parse::<f64>()
+            && x.is_normal()
+        {
+            return if x > 0.0 {
+                Ok(Self::from_f64(x))
+            } else {
+                Err(())
+            };
+        }
+        // Too small (or large) for f64: go through an exact decimal.
+        let bin = Big::from_decimal_str(s, 64).ok_or(())?;
+        if bin.is_negative() {
+            return Err(());
+        }
+        let top = bin.log2_floor().ok_or(())?;
+        let m = (bin >> top).to_f64();
+        let e = top.clamp(i32::MIN as isize, i32::MAX as isize) as i32;
+        Ok(Self::from_parts(m, e))
+    }
+}
+
 /// Guard bits added on top of the zoom-dictated precision.
 const GUARD_BITS: usize = 48;
-/// Upper bound on center precision (f32 GPU perturbation degrades long before
+/// Upper bound on center precision: what `Scale::MIN` needs. Only a guard
-/// this; the cap just prevents pathological allocation).
+/// against pathological input; the reference orbit is impractically slow
-const MAX_PRECISION_BITS: usize = 2048;
+/// long before this.
+pub const MAX_PRECISION_BITS: usize = (1 << 20) + GUARD_BITS;
 
 #[derive(Clone, Debug)]
 pub struct ViewState {
     pub center_re: Big,
     pub center_im: Big,
     /// Half the view height in complex-plane units. Zooming in shrinks this.
-    pub half_height: f64,
+    pub half_height: Scale,
 }
 
 impl Default for ViewState {
     fn default() -> Self {
-        let bits = precision_for(DEFAULT_HALF_HEIGHT);
+        let bits = precision_for(Scale::from_f64(DEFAULT_HALF_HEIGHT));
         Self {
             center_re: big_from_f64(-0.5, bits),
             center_im: big_from_f64(0.0, bits),
-            half_height: DEFAULT_HALF_HEIGHT,
+            half_height: Scale::from_f64(DEFAULT_HALF_HEIGHT),
         }
     }
 }
 
 impl ViewState {
     /// Complex-plane span (width, height) for the given pixel aspect ratio.
-    pub fn span(&self, aspect: f64) -> (f64, f64) {
+    pub fn span(&self, aspect: f64) -> (Scale, Scale) {
-        let h = self.half_height * 2.0;
+        let h = self.half_height.mul_f64(2.0);
-        (h * aspect, h)
+        (h.mul_f64(aspect), h)
     }
 
     /// Complex-plane units per pixel, given the viewport height in pixels.
-    pub fn complex_per_pixel(&self, height_px: f64) -> f64 {
+    pub fn complex_per_pixel(&self, height_px: f64) -> Scale {
-        (self.half_height * 2.0) / height_px
+        self.half_height.mul_f64(2.0 / height_px)
     }
 
-    /// Current magnification relative to the default view.
+    /// log10 of the current magnification relative to the default view.
-    pub fn magnification(&self) -> f64 {
+    pub fn magnification_log10(&self) -> f64 {
-        DEFAULT_HALF_HEIGHT / self.half_height
+        DEFAULT_HALF_HEIGHT.log10() - self.half_height.log10()
     }
 
     /// Current zoom level.
-    pub fn zoom(&self) -> f64 {
+    pub fn zoom(&self) -> Scale {
         self.half_height
     }
 
@@ -73,10 +311,10 @@ impl ViewState {
     pub fn sync_precision(&mut self) {
         let bits = self.precision_bits();
         if self.center_re.precision() < bits {
-            self.center_re = self.center_re.clone().with_precision(bits).value();
+            self.center_re = self.center_re.clone().with_precision(bits);
         }
         if self.center_im.precision() < bits {
-            self.center_im = self.center_im.clone().with_precision(bits).value();
+            self.center_im = self.center_im.clone().with_precision(bits);
         }
     }
 
@@ -86,8 +324,8 @@ impl ViewState {
         let cpp = self.complex_per_pixel(height_px);
         let bits = self.precision_bits();
         // Grab-and-drag: moving the mouse right shows content to the left.
-        self.center_re = &self.center_re - &big_from_f64(dx * cpp, bits);
+        self.center_re = &self.center_re - &cpp.big_times(dx, bits);
-        self.center_im = &self.center_im - &big_from_f64(dy * cpp, bits);
+        self.center_im = &self.center_im - &cpp.big_times(dy, bits);
     }
 
     /// Zoom by `factor` (<1 zooms in) keeping the complex point currently under
@@ -100,14 +338,14 @@ impl ViewState {
         // The cursor's complex offset from the center is (off * cpp). Keeping it
         // fixed while scaling the view by `factor` moves the center by
         // off * cpp * (1 - factor). (Derivation: new_c = fixed + (c-fixed)*f.)
-        let k = cpp * (1.0 - factor);
+        let k = 1.0 - factor;
-        self.center_re = &self.center_re + &big_from_f64(off_x * k, bits);
+        self.center_re = &self.center_re + &cpp.big_times(off_x * k, bits);
-        self.center_im = &self.center_im + &big_from_f64(off_y * k, bits);
+        self.center_im = &self.center_im + &cpp.big_times(off_y * k, bits);
-        self.half_height *= factor;
+        self.half_height = self.half_height.mul_f64(factor);
     }
 
     /// Build a view from full-precision center coordinates and a half-height.
-    pub fn with_center(center_re: Big, center_im: Big, half_height: f64) -> Self {
+    pub fn with_center(center_re: Big, center_im: Big, half_height: Scale) -> Self {
         let mut v = Self {
             center_re,
             center_im,
@@ -121,8 +359,7 @@ impl ViewState {
 /// Parse a decimal string (any number of digits) losslessly into a `Big` with at
 /// least `bits` of precision. Used for share links and debug view specs.
 pub fn big_from_decimal_str(s: &str, bits: usize) -> Option<Big> {
-    let dec = DBig::from_str(s.trim()).ok()?;
+    Big::from_decimal_str(s, bits)
-    Some(dec.with_base_and_precision::<2>(bits.max(53)).value())
 }
 
 /// Parse a "re,im,half_height[,iterations]" spec (re/im decimal, parsed at
@@ -134,10 +371,7 @@ pub fn parse_view_spec(spec: &str) -> Option<(ViewState, Option<u32>)> {
     if parts.len() < 3 {
         return None;
     }
-    let half_height = parts[2].trim().parse::<f64>().ok()?;
+    let half_height = parse_half_height_spec(parts[2])?;
-    if !(half_height > 0.0 && half_height.is_finite()) {
-        return None;
-    }
     let bits = precision_for(half_height);
     let re = big_from_decimal_str(parts[0], bits)?;
     let im = big_from_decimal_str(parts[1], bits)?;
@@ -148,12 +382,8 @@ pub fn parse_view_spec(spec: &str) -> Option<(ViewState, Option<u32>)> {
 /// Parse a half_height spec. Shared by
 /// `FractalApp::apply_half_height_spec` (the `--zoom` CLI flag) and headless
 /// animation's `--to-zoom`.
-pub fn parse_half_height_spec(spec: &str) -> Option<f64> {
+pub fn parse_half_height_spec(spec: &str) -> Option<Scale> {
-    let half_height = spec.trim().parse::<f64>().ok()?;
+    spec.parse::<Scale>().ok()
-    if !(half_height > 0.0 && half_height.is_finite()) {
-        return None;
-    }
-    Some(half_height)
 }
 /// Parse a "re,im" spec (re/im decimal, parsed at
 /// full precision) into a view. Shared by
@@ -182,77 +412,87 @@ pub fn parse_re_im_spec(spec: &str, bits: usize) -> Option<(Big, Big)> {
 /// offset/half_height ratio roughly constant, i.e. the target's on-screen
 /// position steady) but is shifted so it lands on exactly 1 at `t = 0` and
 /// exactly 0 at `t = 1`.
+///
+/// With `d = log2(q)`, `g = q^t · (1 - q^(1-t)) / (1 - q)`, all in `Scale`
+/// / `expm1` form: zooming in by more than f64's range, `q` (and `q^t`)
+/// underflow, yet `g · (from - to)` must keep tracking the half-height.
 pub fn interpolate_view(from: &ViewState, to: &ViewState, t: f64) -> ViewState {
-    let q = to.half_height / from.half_height;
+    let (l0, l1) = (from.half_height.log2(), to.half_height.log2());
-    let half_height = from.half_height * q.powf(t);
+    let d = l1 - l0;
-    let bits = precision_for(half_height);
+    let half_height = if t <= 0.0 {
-    let g = if (q - 1.0).abs() < 1e-12 {
+        from.half_height
-        1.0 - t
+    } else if t >= 1.0 {
+        to.half_height
     } else {
-        (q.powf(t) - q) / (1.0 - q)
+        Scale::from_log2(l0 + t * d)
     };
-    let g_big = big_from_f64(g, bits);
+    let bits = precision_for(half_height);
-    let re0 = from.center_re.clone().with_precision(bits).value();
+    let ln2 = core::f64::consts::LN_2;
-    let im0 = from.center_im.clone().with_precision(bits).value();
+    let g_big = if d.abs() < 1e-12 {
-    let re1 = to.center_re.clone().with_precision(bits).value();
+        big_from_f64(1.0 - t, bits)
-    let im1 = to.center_im.clone().with_precision(bits).value();
+    } else if d < 0.0 {
+        // Zooming in: q^t may be far below f64's range, keep it as a Scale.
+        let f = ((1.0 - t) * d * ln2).exp_m1() / (d * ln2).exp_m1();
+        Scale::from_log2(t * d).big_times(f, bits)
+    } else {
+        // Zooming out: g = (1 - q^(t-1)) / (1 - q^-1), every term bounded.
+        big_from_f64(((t - 1.0) * d * ln2).exp_m1() / (-d * ln2).exp_m1(), bits)
+    };
+    let re0 = from.center_re.clone().with_precision(bits);
+    let im0 = from.center_im.clone().with_precision(bits);
+    let re1 = to.center_re.clone().with_precision(bits);
+    let im1 = to.center_im.clone().with_precision(bits);
     let center_re = &re1 + &(&(&re0 - &re1) * &g_big);
     let center_im = &im1 + &(&(&im0 - &im1) * &g_big);
     ViewState::with_center(center_re, center_im, half_height)
 }
 
+#[cfg(not(target_arch = "wasm32"))]
 pub fn interpolate_f64(from: f64, to: f64, t: f64) -> f64 {
     from + (to - from) * t
 }
 
 /// Render a `Big` as a decimal string with `sig_digits` significant digits.
 pub fn big_to_decimal_str(x: &Big, sig_digits: usize) -> String {
-    let dec = x
+    x.to_decimal_string(sig_digits)
-        .to_decimal()
-        .value()
-        .with_precision(sig_digits.max(1))
-        .value();
-    format!("{dec}")
 }
 
 /// Precision (bits) needed to resolve the center at a given half-height.
-pub fn precision_for(half_height: f64) -> usize {
+pub fn precision_for(half_height: Scale) -> usize {
     // We need enough bits to distinguish points a pixel apart, i.e. roughly
     // log2(1 / half_height) significant bits, plus a guard margin.
-    let zoom_bits = if half_height > 0.0 && half_height.is_finite() {
+    let zoom_bits = (-half_height.log2()).ceil().max(0.0) as usize;
-        (-half_height.log2()).ceil().max(0.0) as usize
-    } else {
-        0
-    };
     (zoom_bits + GUARD_BITS).clamp(53, MAX_PRECISION_BITS)
 }
 
-/// Build an `FBig` from an f64 with an explicit precision context.
+/// Build a `Big` from an f64 with an explicit precision.
 pub fn big_from_f64(x: f64, bits: usize) -> Big {
-    Big::try_from(x)
+    Big::from_f64(x, bits)
-        .unwrap_or_default()
-        .with_precision(bits)
-        .value()
 }
 
 #[cfg(test)]
 mod tests {
     use super::*;
 
+    fn sc(x: f64) -> Scale {
+        Scale::from_f64(x)
+    }
+
     fn re_im_f64(v: &ViewState) -> (f64, f64) {
-        let re: f64 = v.center_re.to_decimal().value().to_f64().value();
+        let re: f64 = v.center_re.to_f64();
-        let im: f64 = v.center_im.to_decimal().value().to_f64().value();
+        let im: f64 = v.center_im.to_f64();
         (re, im)
     }
 
     #[test]
     fn interpolate_view_hits_exact_endpoints() {
-        let bits = precision_for(1.0);
+        let bits = precision_for(sc(1.0));
-        let from = ViewState::with_center(big_from_f64(-0.5, bits), big_from_f64(0.0, bits), 1.5);
+        let from =
+            ViewState::with_center(big_from_f64(-0.5, bits), big_from_f64(0.0, bits), sc(1.5));
         let to = ViewState::with_center(
-            big_from_f64(-0.7515, precision_for(1e-20)),
+            big_from_f64(-0.7515, precision_for(sc(1e-20))),
-            big_from_f64(0.1013, precision_for(1e-20)),
+            big_from_f64(0.1013, precision_for(sc(1e-20))),
-            1e-20,
+            sc(1e-20),
         );
 
         let start = interpolate_view(&from, &to, 0.0);
@@ -272,12 +512,13 @@ mod tests {
     /// instead stay roughly bounded throughout.
     #[test]
     fn interpolate_view_keeps_target_offset_bounded() {
-        let bits = precision_for(1.0);
+        let bits = precision_for(sc(1.0));
-        let from = ViewState::with_center(big_from_f64(-0.5, bits), big_from_f64(0.0, bits), 1.5);
+        let from =
+            ViewState::with_center(big_from_f64(-0.5, bits), big_from_f64(0.0, bits), sc(1.5));
         let to = ViewState::with_center(
-            big_from_f64(-0.7515, precision_for(1e-20)),
+            big_from_f64(-0.7515, precision_for(sc(1e-20))),
-            big_from_f64(0.1013, precision_for(1e-20)),
+            big_from_f64(0.1013, precision_for(sc(1e-20))),
-            1e-20,
+            sc(1e-20),
         );
         let (to_re, to_im) = re_im_f64(&to);
 
@@ -286,7 +527,7 @@ mod tests {
             let mid = interpolate_view(&from, &to, t);
             let (re, im) = re_im_f64(&mid);
             let offset = ((re - to_re).powi(2) + (im - to_im).powi(2)).sqrt();
-            let ratio = offset / mid.half_height;
+            let ratio = offset / mid.half_height.to_f64();
             assert!(
                 ratio < 10.0,
                 "t={t}: offset/half_height ratio {ratio} blew up (offset={offset}, half_height={})",
@@ -294,4 +535,77 @@ mod tests {
             );
         }
     }
+
+    #[test]
+    fn scale_parse_display_round_trip() {
+        for s in [
+            "1.25",
+            "1e-20",
+            "1.5e-20",
+            "3.7e-4000",
+            "1e-400",
+            "9.99999e-310",
+        ] {
+            let a: Scale = s.parse().unwrap();
+            let b: Scale = a.to_string().parse().unwrap();
+            assert_eq!(a, b, "{s} -> {a}");
+        }
+        assert_eq!("1.25".parse::<Scale>().unwrap().to_f64(), 1.25);
+        assert_eq!(sc(1.5e-20).to_string(), "1.5e-20");
+        let deep: Scale = "3.7e-4000".parse().unwrap();
+        assert_eq!(deep.to_string(), "3.7e-4000");
+        assert_eq!(format!("{deep:.2}"), "3.70e-4000");
+        assert!((deep.log10() - (3.7f64.log10() - 4000.0)).abs() < 1e-9);
+        assert!("0".parse::<Scale>().is_err());
+        assert!("-1e-500".parse::<Scale>().is_err());
+        assert!("abc".parse::<Scale>().is_err());
+    }
+
+    #[test]
+    fn scale_arithmetic() {
+        let a: Scale = "1e-1000".parse().unwrap();
+        let b = a.mul_f64(0.25);
+        assert!((b.ratio(a) - 0.25).abs() < 1e-15);
+        assert!(b < a && a > b);
+        assert_eq!(a.mul_f64(3.0).mul_f64(1.0 / 3.0).exponent(), a.exponent());
+        assert_eq!(sc(1.0).exponent(), 0);
+        assert_eq!(sc(0.75).exponent(), -1);
+        assert_eq!(sc(4.0).scaled_f64(-2), 1.0);
+        assert_eq!(
+            a.scaled_f64(-a.exponent()),
+            a.mul_f64(1.0).scaled_f64(-a.exponent())
+        );
+        assert!((1.0..2.0).contains(&a.scaled_f64(-a.exponent())));
+        assert_eq!(a.to_f64(), 0.0);
+        assert_eq!(Scale::MIN.mul_f64(0.5), Scale::MIN);
+        let p = precision_for(a);
+        assert!((3322 + 48..=3323 + 48).contains(&p), "{p}");
+    }
+
+    /// Past f64's range, the center must still land on the target at the
+    /// same geometric pace as the half-height.
+    #[test]
+    fn interpolate_view_past_f64_range() {
+        let from = ViewState::with_center(big_from_f64(-0.5, 64), big_from_f64(0.0, 64), sc(1.5));
+        let hh: Scale = "1e-1000".parse().unwrap();
+        let bits = precision_for(hh);
+        let to =
+            ViewState::with_center(big_from_f64(-0.7515, bits), big_from_f64(0.1013, bits), hh);
+        let end = interpolate_view(&from, &to, 1.0);
+        assert_eq!(end.half_height, hh);
+        assert_eq!(re_im_f64(&end), re_im_f64(&to));
+        let mut prev = from.half_height;
+        for i in 1..20 {
+            let t = i as f64 / 20.0;
+            let mid = interpolate_view(&from, &to, t);
+            assert!(mid.half_height < prev);
+            prev = mid.half_height;
+            // Offset from the target, in units of the view's half-height.
+            let k = -mid.half_height.exponent() as isize;
+            let dre = ((&mid.center_re - &to.center_re) << k).to_f64();
+            let dim = ((&mid.center_im - &to.center_im) << k).to_f64();
+            let ratio = (dre * dre + dim * dim).sqrt() / mid.half_height.scaled_f64(k as i32);
+            assert!(ratio > 0.01 && ratio < 10.0, "t={t}: ratio {ratio}");
+        }
+    }
 }

src/worker.rs

@@ -1,7 +1,7 @@
 //! Native background worker for reference-orbit computation.
 //!
 //! At deep zoom the high-precision reference can take many milliseconds (tens of
-//! thousands of `FBig` iterations), which would stutter the UI if done inline.
+//! thousands of `Big` iterations), which would stutter the UI if done inline.
 //! This runs it on a thread and coalesces bursts of requests (e.g. during a
 //! drag) down to the most recent one. On the web we compute inline instead
 //! (browsers need a Web Worker for threads); see `app.rs`.
@@ -9,13 +9,13 @@
 use std::sync::mpsc::{Receiver, Sender, TryRecvError, channel};
 use std::thread;
 
-use crate::fractal::{FractalKind, compute_reference, compute_set_reference};
+use crate::fractal::{FractalKind, RefOrbit, compute_reference, compute_set_reference};
-use crate::view::{Big, big_from_f64};
+use crate::view::{Big, Scale, big_from_f64};
 
 pub struct RefRequest {
     pub center_re: Big,
     pub center_im: Big,
-    pub half_height: f64,
+    pub half_height: Scale,
     pub julia: bool,
     pub julia_c: (f64, f64),
     pub max_iter: u32,
@@ -35,8 +35,8 @@ pub struct RefRequest {
 pub struct RefResult {
     pub center_re: Big,
     pub center_im: Big,
-    pub half_height: f64,
+    pub half_height: Scale,
-    pub points: Vec<[f32; 2]>,
+    pub points: RefOrbit,
     /// The kind and morph `points` was computed with (echoed from the
     /// request).
     pub kind: FractalKind,
@@ -102,7 +102,7 @@ impl RefWorker {
     }
 }
 
-fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
+fn compute(req: &RefRequest) -> RefOrbit {
     if req.julia {
         let jr = big_from_f64(req.julia_c.0, req.precision);
         let ji = big_from_f64(req.julia_c.1, req.precision);

tests/shader_valid.rs

@@ -77,7 +77,7 @@ fn mandelbrot_shader_is_valid() {
 }
 
 /// Every specialization renderer.rs can build (`PipelineKey`: kind × Julia ×
-/// DE × morph), for every fragment entry point.
+/// DE × morph × deep), for every fragment entry point.
 #[test]
 fn mandelbrot_shader_specializations_compile() {
     let (module, info) = validate("mandelbrot.wgsl", MANDELBROT_SRC);
@@ -85,11 +85,13 @@ fn mandelbrot_shader_specializations_compile() {
         for julia in [0.0, 1.0] {
             for de in [0.0, 1.0] {
                 for morph in [0.0, 1.0] {
+                    for deep in [0.0, 1.0] {
                         let constants = [
                             ("KIND", kind as f64),
                             ("IS_JULIA", julia),
                             ("DE", de),
                             ("MORPH", morph),
+                            ("DEEP", deep),
                         ];
                         for entry in ["fs_data", "fs_refine", "fs_color"] {
                             specialize(
@@ -106,6 +108,7 @@ fn mandelbrot_shader_specializations_compile() {
             }
         }
     }
+}
 
 #[test]
 fn colorize_shader_is_valid() {
@@ -119,6 +122,17 @@ fn colorize_shader_is_valid() {
     );
 }
 
+#[test]
+fn lipschitz_shader_is_valid() {
+    validate(
+        "lipschitz.wgsl",
+        concat!(
+            include_str!("../src/shaders/common.wgsl"),
+            include_str!("../src/shaders/lipschitz.wgsl"),
+        ),
+    );
+}
+
 #[test]
 fn blit_shader_is_valid() {
     validate(

tools/bench/.gitignore

@@ -0,0 +1,2 @@
+results/
+.worktrees/

tools/bench/README.md

@@ -0,0 +1,48 @@
+# Benchmarks
+
+`bench.sh` times `mandelbrot --headless` on a fixed set of views with
+[hyperfine](https://github.com/sharkdp/hyperfine). That's the whole real
+pipeline: the reference orbit on the CPU, then the GPU render, readback and PNG
+encode. It builds the headless-only binary (`--no-default-features`) itself.
+
+```sh
+tools/bench/bench.sh --rev HEAD            # did my uncommitted edits help? (vs last commit)
+tools/bench/bench.sh --rev main seahorse-1e300   # one scenario vs another branch
+tools/bench/bench.sh --quick               # all scenarios, 960×540, 3 runs
+tools/bench/bench.sh                       # all scenarios, 1920×1080, 5 runs
+tools/bench/bench.sh --list
+```
+
+`--rev REF` checks REF out into `.worktrees/<sha>` and builds it there, with its
+own target dir, so your `target/` stays warm. Each scenario then runs both
+binaries back to back in one hyperfine call. The line to read is
+`working tree ran 1.23 ± 0.04 times faster than main (…)`. A ratio within about
+2σ of 1.00 is noise.
+
+Results (hyperfine JSON and markdown per scenario) go to
+`results/<sha>[-dirty]/`, or `results/<rev>-vs-<sha>/` with `--rev`.
+Both `results/` and `.worktrees/` are git-ignored. To clean up:
+`rm -rf tools/bench/.worktrees && git worktree prune`.
+
+## Scenarios
+
+| name | mostly measures |
+|---|---|
+| `shallow` | GPU, plain f32 path, f64 orbit fast path |
+| `seahorse-1e30` | GPU f32 perturbation from a `Big` orbit |
+| `seahorse-1e100` | the `DEEP` shader pipeline |
+| `elephant-1e100-aa-de` | 2×2 antialiasing + DE shading |
+| `burning-ship` | a non-holomorphic kind |
+| `3d` | the raymarched 3D export path |
+| `anim-zoom` | 24-frame zoom (⅓ size): the parallel orbit, GPU and PNG pipeline |
+| `seahorse-1e300` *(slow)* | ~270k iterations per pixel on the deep path, ~40 s per run even with `--quick`. It's GPU-bound: the orbit itself is under a second. Only runs when named or with `--all` |
+
+Views come from `tools/deep-zoom/LOCATIONS.md`. Each wall time includes process
+start, device creation and shader compilation, so a small change to a GPU-bound
+scenario shows up diluted.
+
+## Tips
+
+- Keep the machine idle and on AC power. Other GPU apps (browsers, video) add a lot of noise.
+- Use `--quick` while you iterate and a full run to confirm.
+- Timings are only comparable on the same machine and GPU.

tools/bench/bench.sh

@@ -0,0 +1,128 @@
+#!/usr/bin/env bash
+# Benchmark `mandelbrot --headless` on a fixed set of scenarios with hyperfine.
+#
+# Usage:
+#   bench.sh [--quick] [--runs N] [--rev REF] [--all] [--list] [scenario...]
+#
+#   (default)   time the working tree; results go to results/<sha>[-dirty]/
+#   --rev REF   also build REF (in a git worktree) and time both binaries side
+#               by side; hyperfine prints "X ± σ times faster" per scenario
+#   --quick     960×540 and 3 runs instead of 1920×1080 and 5
+#   --runs N    timed runs per command (after 1 warmup run)
+#   --all       include the slow scenarios
+#   --list      print the scenario names and exit
+#
+# With no scenario names, all but the slow ones run. See README.md.
+
+set -euo pipefail
+
+here=$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)
+root=$(git -C "$here" rev-parse --show-toplevel)
+
+# Views from tools/deep-zoom/LOCATIONS.md.
+SEAHORSE_40='-0.77568376800905379745347652613832924487504096622022,0.13646736829469012473375311880735014411233827361594'
+SEAHORSE_101='-0.77568376800905379746948350393474104572824650461579342765769586746577714101553375163935420167637213972439444483,0.13646736829469012473327440961784876519777211159246218448277041663416833530840062457397927487282583495378053495'
+SEAHORSE_300='-0.775683768009053797469483503934741045728246504615800394794952001400666017222819589894883159368782255028252296689050972508942991790804637379375064948276511044167168394677127466362472192356953918438738697548775738756167744421444715134058363508346750264411321139485834882343906172015818621107808449401958194265462,0.136467368294690124733274409617848765197772111592467696407800195400835665974815500875448611244178656625706823059759836009902469392551108149827753781145338163746214967880988693412636771323356061132825037923307975359146666033691080243216637006179374407866237575096209504710178751614036006531433101633993532313930'
+ELEPHANT_100='0.2920738619144539179997175176922057340804018914331633820622644536373359023331882946705674169275530423410817910,0.01577091721139011808246902823033833768260048863232968822435568049949556139801802963232184866659058960379114747'
+
+# name -> headless args (no size or --export-path; those are added per run).
+# Order matters for display, hence the separate list. SLOW ones (tens of
+# seconds per run) only run when named or with --all.
+SCENARIOS=(shallow seahorse-1e30 seahorse-1e100 elephant-1e100-aa-de burning-ship 3d anim-zoom)
+SLOW=(seahorse-1e300)
+declare -A ARGS=(
+    [shallow]="--view=-0.75,0,1.5"
+    [seahorse-1e30]="--view=$SEAHORSE_40,1e-30"
+    [seahorse-1e100]="--view=$SEAHORSE_101,3.4e-101"
+    [seahorse-1e300]="--view=$SEAHORSE_300,5.5e-300"
+    [elephant-1e100-aa-de]="--view=$ELEPHANT_100,1e-75 --antialias --de"
+    [burning-ship]="--kind burning-ship"
+    [3d]="--rendering-kind 3d --de --pitch -40 --view=-0.75,0,1.5"
+    [anim-zoom]="--view=-0.75,0,1.5 --to-view=$SEAHORSE_40,1e-30 --frames 24"
+)
+
+quick=0
+runs=
+rev=
+picked=()
+while (($#)); do
+    case $1 in
+        --quick) quick=1 ;;
+        --runs) runs=$2; shift ;;
+        --rev) rev=$2; shift ;;
+        --all) picked+=("${SCENARIOS[@]}" "${SLOW[@]}") ;;
+        --list) printf '%s\n' "${SCENARIOS[@]}"; printf '%s (slow)\n' "${SLOW[@]}"; exit 0 ;;
+        -h|--help) sed -n '2,16s/^# \{0,1\}//p' "$0"; exit 0 ;;
+        -*) echo "unknown option: $1" >&2; exit 2 ;;
+        *)
+            [[ -v ARGS[$1] ]] || { echo "unknown scenario: $1 (try --list)" >&2; exit 2; }
+            picked+=("$1") ;;
+    esac
+    shift
+done
+((${#picked[@]})) || picked=("${SCENARIOS[@]}")
+
+if ((quick)); then
+    width=960 height=540 runs=${runs:-3}
+else
+    width=1920 height=1080 runs=${runs:-5}
+fi
+
+command -v hyperfine >/dev/null || { echo "hyperfine not found (https://github.com/sharkdp/hyperfine)" >&2; exit 1; }
+
+# Headless-only binary: no eframe/egui, faster to build, same render path.
+build() { cargo build --release --no-default-features --quiet --manifest-path "$1/Cargo.toml"; }
+
+echo "building working tree…" >&2
+build "$root"
+new_bin="$root/target/release/mandelbrot"
+
+if [[ -n $rev ]]; then
+    rev_sha=$(git -C "$root" rev-parse --short "$rev^{commit}")
+    wt="$here/.worktrees/$rev_sha"
+    [[ -d $wt ]] || git -C "$root" worktree add --detach --quiet "$wt" "$rev_sha"
+    echo "building $rev ($rev_sha)…" >&2
+    # Separate target dir so switching revisions doesn't invalidate target/.
+    CARGO_TARGET_DIR="$here/.worktrees/target" build "$wt"
+    old_bin="$here/.worktrees/target/release/mandelbrot"
+fi
+
+tmp=$(mktemp -d)
+trap 'rm -rf "$tmp"' EXIT
+
+sha=$(git -C "$root" rev-parse --short HEAD)
+git -C "$root" diff --quiet HEAD -- . ':!tools/bench' || sha+=-dirty
+out="$here/results/${rev:+$rev_sha-vs-}$sha"
+mkdir -p "$out"
+
+# Full headless command line for scenario $2 with binary $1.
+command_for() {
+    local bin=$1 name=$2 w=$width h=$height dest=$tmp/$2.png
+    if [[ $name == anim-* ]]; then
+        # Animation: a directory of frames, at a smaller size.
+        w=$((width / 3)) h=$((height / 3)) dest=$tmp/$name
+    fi
+    echo "$bin --headless --width $w --height $h --export-path $dest ${ARGS[$name]}"
+}
+
+for name in "${picked[@]}"; do
+    echo >&2
+    echo "=== $name ===" >&2
+    if [[ -n $rev ]]; then
+        cmds=(-n "$rev ($rev_sha)" "$(command_for "$old_bin" "$name")"
+              -n "working tree" "$(command_for "$new_bin" "$name")")
+    else
+        cmds=(-n "$name" "$(command_for "$new_bin" "$name")")
+    fi
+    hyperfine --warmup 1 --runs "$runs" --style basic \
+        --export-json "$out/$name.json" --export-markdown "$out/$name.md" \
+        "${cmds[@]}"
+done
+
+echo
+echo "## Summary (${width}×${height}, $runs runs; results in ${out#"$root"/}/)"
+for name in "${picked[@]}"; do
+    echo
+    echo "### $name"
+    cat "$out/$name.md"
+done

tools/deep-zoom/LOCATIONS.md

@@ -0,0 +1,167 @@
+# Deep-zoom locations
+
+Beautiful deep Mandelbrot locations, all computed with `find_deep.py` and checked with a
+`--headless` render. Every one is an exact minibrot nucleus (Newton's method at
+`2·depth + 60` digits), not a coordinate copied from somewhere, so the digits are right all
+the way down.
+
+## How to use them
+
+Each location has two views on the same centre:
+
+- **Minibrot:** half-height ≈ 3× the minibrot's size. A tiny copy of the whole set sits in
+  concentric colour bands, and there are more bands the deeper it is.
+- **Embedded Julia set:** half-height ≈ size^0.75, about ¾ of the way down in log-zoom.
+  This is where each location looks unique. The minibrots all look alike.
+
+```sh
+cargo run --release -- --view=RE,IM,HALF_HEIGHT        # interactive
+cargo run --release -- --headless --view=RE,IM,HALF_HEIGHT --export-path shot.png
+# zoom video from the full set down to a minibrot, piped to ffmpeg (see Makefile):
+cargo run --release -- --headless --view=-0.75,0,1.5 --to-view=RE,IM,HALF_HEIGHT \
+    --fps 60 --duration 60 --export-path - | make hevc-nvenc
+```
+
+Pass negative coordinates as `--view=-0.77...` (with the `=`): clap would read `--view -0.77...` as a flag.
+Iterations scale with depth automatically (about 90k at 10⁻¹⁰⁰ and 270k at 10⁻³⁰⁰), which is
+enough for these periods.
+
+## Infinite spirals (Misiurewicz points)
+
+These are the anchor points. Zoom straight into one and the same spiral repeats forever,
+at any depth. Every minibrot below sits just beside one of them.
+
+| Region | Point | Preperiod, period | Looks like |
+|---|---|---|---|
+| Seahorse valley | `-0.7756837680090537974694835039347410457282465046158` `0.13646736829469012473327440961784876519777211159247` | 24, 2 | Two-armed spirals of seahorses, repeating at every depth. |
+| Elephant valley | `0.29207386191445391799971751769220573408040189143315` `0.015770917211390118082469028230338337682600488632339` | 32, 3 | Three-armed spirals of elephant trunks. |
+| Top antenna | `-0.10110375797210418675706890936004723524958119917843` `0.95629400795235409290719434118824484558046591980423` | 38, 4 | Four-armed branch points on the upper antenna, with thin dendrites instead of spirals. |
+
+## Seahorse valley
+
+### Seahorse valley, minibrot at 3.7e-41 (period 1034)
+
+The embedded Julia set is a pale double spiral: two S-shaped arms of tiny seahorses wind into a bright centre, with a third lace band sweeping around them. The best one for a short video, and it renders in seconds.
+
+```sh
+# embedded Julia set
+--view=-0.77568376800905379745347652613832924487504096622022,0.13646736829469012473375311880735014411233827361594,1e-30
+# minibrot
+--view=-0.77568376800905379745347652613832924487504096622022,0.13646736829469012473375311880735014411233827361594,1.1e-40
+```
+
+### Seahorse valley, minibrot at 1.14e-101 (period 2868)
+
+At the Julia depth there's a dark blue field with a glowing core and lace threads crossing it.
+
+```sh
+# embedded Julia set
+--view=-0.77568376800905379746948350393474104572824650461579342765769586746577714101553375163935420167637213972439444483,0.13646736829469012473327440961784876519777211159246218448277041663416833530840062457397927487282583495378053495,1e-76
+# minibrot
+--view=-0.77568376800905379746948350393474104572824650461579342765769586746577714101553375163935420167637213972439444483,0.13646736829469012473327440961784876519777211159246218448277041663416833530840062457397927487282583495378053495,3.4e-101
+```
+
+### Seahorse valley, minibrot at 1.37e-200 (period 5895)
+
+Two filaments cross in an X over a warm brown background, with a small dark spiral where they meet.
+
+```sh
+# embedded Julia set
+--view=-0.77568376800905379746948350393474104572824650461580039479495200140066601722281958989488315936878225494657853849784368247991666558116221905228489656259693587754973261632614186930985854124920967267440737850057557,0.13646736829469012473327440961784876519777211159246769640780019540083566597481550087544861124417865661202932094564677695883049019713653069907950852152823160218141353878979871215911794286224832089496659451498185,1e-150
+# minibrot
+--view=-0.77568376800905379746948350393474104572824650461580039479495200140066601722281958989488315936878225494657853849784368247991666558116221905228489656259693587754973261632614186930985854124920967267440737850057557,0.13646736829469012473327440961784876519777211159246769640780019540083566597481550087544861124417865661202932094564677695883049019713653069907950852152823160218141353878979871215911794286224832089496659451498185,4.1e-200
+```
+
+### Seahorse valley, minibrot at 1.83e-300 (period 8922)
+
+Faint cream threads, and a spiral wound so tight it reads as a point. Period 8922, so the minibrot needs a long render.
+
+```sh
+# embedded Julia set
+--view=-0.775683768009053797469483503934741045728246504615800394794952001400666017222819589894883159368782255028252296689050972508942991790804637379375064948276511044167168394677127466362472192356953918438738697548775738756167744421444715134058363508346750264411321139485834882343906172015818621107808449401958194265462,0.136467368294690124733274409617848765197772111592467696407800195400835665974815500875448611244178656625706823059759836009902469392551108149827753781145338163746214967880988693412636771323356061132825037923307975359146666033691080243216637006179374407866237575096209504710178751614036006531433101633993532313930,1e-225
+# minibrot
+--view=-0.775683768009053797469483503934741045728246504615800394794952001400666017222819589894883159368782255028252296689050972508942991790804637379375064948276511044167168394677127466362472192356953918438738697548775738756167744421444715134058363508346750264411321139485834882343906172015818621107808449401958194265462,0.136467368294690124733274409617848765197772111592467696407800195400835665974815500875448611244178656625706823059759836009902469392551108149827753781145338163746214967880988693412636771323356061132825037923307975359146666033691080243216637006179374407866237575096209504710178751614036006531433101633993532313930,5.5e-300
+```
+
+## Elephant valley
+
+### Elephant valley, minibrot at 1.29e-100 (period 782)
+
+A cream-coloured double S-spiral, like the seahorse one but with fatter, beaded arms.
+
+```sh
+# embedded Julia set
+--view=0.2920738619144539179997175176922057340804018914331633820622644536373359023331882946705674169275530423410817910,0.01577091721139011808246902823033833768260048863232968822435568049949556139801802963232184866659058960379114747,1e-75
+# minibrot
+--view=0.2920738619144539179997175176922057340804018914331633820622644536373359023331882946705674169275530423410817910,0.01577091721139011808246902823033833768260048863232968822435568049949556139801802963232184866659058960379114747,3.9e-100
+```
+
+### Elephant valley, minibrot at 1.43e-199 (period 1585)
+
+A single huge spiral arm made of blue beads curling across the whole frame.
+
+```sh
+# embedded Julia set
+--view=0.2920738619144539179997175176922057340804018914331538397786950546981063169810374784665382516408545756138262706218874924432355556243165250541700374639889797148498919340003273815669351046658761147565052417000965,0.01577091721139011808246902823033833768260048863233880759961760295375606839216565332458201188913912642954367433579737643167232896069795830226759356080847862646789481028238105648099709502522309228500630759855757,1e-149
+# minibrot
+--view=0.2920738619144539179997175176922057340804018914331538397786950546981063169810374784665382516408545756138262706218874924432355556243165250541700374639889797148498919340003273815669351046658761147565052417000965,0.01577091721139011808246902823033833768260048863233880759961760295375606839216565332458201188913912642954367433579737643167232896069795830226759356080847862646789481028238105648099709502522309228500630759855757,4.3e-199
+```
+
+### Elephant valley, minibrot at 3.82e-300 (period 2395)
+
+A spiral galaxy: several beaded arms around a pinwheel centre. One of the prettiest frames here.
+
+```sh
+# embedded Julia set
+--view=0.292073861914453917999717517692205734080401891433153839778695054698106316981037478466538251640854575589843066619974864800862332115945960921998657430606126553202998390050225413256445451049563878137878839982267258218894836912449161037745627504826537881261626243871678392881757578278459983842286413975991353319739,0.0157709172113901180824690282303383376826004886323388075996176029537560683921656533245820118891391263013273917522825544701901299061542128984468742890308365571942081484001405437827548345359998503228665872823816067983725089244499770271352539008763406229496083105636482420467114597267096082878888045800333807911146,1e-225
+# minibrot
+--view=0.292073861914453917999717517692205734080401891433153839778695054698106316981037478466538251640854575589843066619974864800862332115945960921998657430606126553202998390050225413256445451049563878137878839982267258218894836912449161037745627504826537881261626243871678392881757578278459983842286413975991353319739,0.0157709172113901180824690282303383376826004886323388075996176029537560683921656533245820118891391263013273917522825544701901299061542128984468742890308365571942081484001405437827548345359998503228665872823816067983725089244499770271352539008763406229496083105636482420467114597267096082878888045800333807911146,1.1e-299
+```
+
+## Top antenna
+
+### Top antenna, minibrot at 6.11e-101 (period 245)
+
+A dark branching dendrite like a frost cross, with four-way junctions.
+
+```sh
+# embedded Julia set
+--view=-0.10110375797210418675706890936004723524958119917842627217950669362738928440088634862532432662296073501093701563,0.95629400795235409290719434118824484558046591980423293884325556254155313535666457147658005358708674409029391915,1e-75
+# minibrot
+--view=-0.10110375797210418675706890936004723524958119917842627217950669362738928440088634862532432662296073501093701563,0.95629400795235409290719434118824484558046591980423293884325556254155313535666457147658005358708674409029391915,1.8e-100
+```
+
+### Top antenna, minibrot at 5.28e-200 (period 473)
+
+A chain of six-armed snowflakes linked by thin filaments.
+
+```sh
+# embedded Julia set
+--view=-0.10110375797210418675706890936004723524958119917842839398761868801597328463328677019373309617158827470368429189213012997868613724029746440432852247903869013157556141380437202406412221853553594880182186959159214,0.95629400795235409290719434118824484558046591980423399388630140415938010604675760901574969597486831722998038561988250849596064882005832681021199867736493316924923123916964900533995659228499205141785254396348248,1e-149
+# minibrot
+--view=-0.10110375797210418675706890936004723524958119917842839398761868801597328463328677019373309617158827470368429189213012997868613724029746440432852247903869013157556141380437202406412221853553594880182186959159214,0.95629400795235409290719434118824484558046591980423399388630140415938010604675760901574969597486831722998038561988250849596064882005832681021199867736493316924923123916964900533995659228499205141785254396348248,1.6e-199
+```
+
+### Top antenna, minibrot at 8.34e-301 (period 705)
+
+A single twelve-armed star, very symmetric. The antenna minibrots have short periods (245-705), so these are the fastest deep renders.
+
+```sh
+# embedded Julia set
+--view=-0.1011037579721041867570689093600472352495811991784283939876186880159732846332867701937330961715882746718394428627914481009391257007785408536971203165750666758077852384490279260184029373211737264310068845597273991782305778295187471341711226626762421910644338164121042775094463557363347706110614218633830535086184,0.9562940079523540929071943411882448455804659198042339938863014041593801060467576090157496959748683171680339080646917869452781885502668216669916185990922330667568398963038228431738928191227884090004548615469303071899797403388188723954061899156655653387543351937028936674153437083087070219108726088551230849901498,1e-225
+# minibrot
+--view=-0.1011037579721041867570689093600472352495811991784283939876186880159732846332867701937330961715882746718394428627914481009391257007785408536971203165750666758077852384490279260184029373211737264310068845597273991782305778295187471341711226626762421910644338164121042775094463557363347706110614218633830535086184,0.9562940079523540929071943411882448455804659198042339938863014041593801060467576090157496959748683171680339080646917869452781885502668216669916185990922330667568398963038228431738928191227884090004548615469303071899797403388188723954061899156655653387543351937028936674153437083087070219108726088551230849901498,2.5e-300
+```
+
+## Finding more
+
+```sh
+./find_deep.py --list                  # known regions
+./find_deep.py seahorse 250            # minibrot of size ~1e-500 near the seahorse point
+./find_deep.py seahorse 100 0.2,0.7    # a different minibrot at the same depth
+./find_deep.py -0.1011,0.9563,38,4 80  # any Misiurewicz point, given as re,im,k,p
+```
+
+A disk of radius 10⁻ᴰ around a Misiurewicz point holds a minibrot of size about 10⁻²ᴰ.
+Newton sometimes fails to converge (elephant at depth 100 with the default offset did). The
+script detects this from the size and asks for a different offset. Needs `mpmath`.

tools/deep-zoom/find_deep.py

@@ -0,0 +1,145 @@
+#!/usr/bin/env python3
+"""Find deep Mandelbrot minibrots near a Misiurewicz point.
+
+Misiurewicz points (preperiodic c, f^(k+p)(0) = f^k(0)) have self-similar
+spirals at every depth. A disk of radius r = 10^-D around one contains a
+minibrot of size ~r^2, found by:
+
+  1. the ball-period method: the first n where the disk's image covers 0
+     gives the period p of a nucleus inside it;
+  2. Newton's method on f^p(0) = 0 at high precision for the nucleus;
+  3. the standard size estimate (Heiland-Allen) for its scale.
+
+Zooming at the nucleus shows spirals, then an embedded Julia set (around
+size^0.75), then the minibrot (half-height ~3x size).
+
+Usage:
+  find_deep.py REGION DEPTH [OFFSET_RE,OFFSET_IM]
+  find_deep.py --list
+
+REGION is a name from REGIONS or "re,im,k,p" for any Misiurewicz point.
+OFFSET (default 0.6,0.3, in units of r) picks a different minibrot at the
+same depth. If the printed size is far below 10^-(2*DEPTH), Newton didn't
+converge: try another offset.
+
+Needs mpmath (pip install mpmath).
+"""
+
+import sys
+
+from mpmath import log10, mp, mpc, mpf
+
+# name: (approximate seed, preperiod k, period p). The exact point is
+# refined by Newton at the requested precision.
+REGIONS = {
+    "seahorse": (("-0.77568377", "0.13646737"), 24, 2),
+    "elephant": (("0.2925", "0.0149"), 32, 3),
+    "antenna": (("-0.1011", "0.9563"), 38, 4),
+}
+
+
+def converged(d):
+    return abs(d) < mpf(10) ** (-mp.dps + 20)
+
+
+def misiurewicz(c, k, p, steps=200):
+    """Newton on f^(k+p)(0) - f^k(0) = 0."""
+    for _ in range(steps):
+        z = dz = mpc(0)
+        zk = dzk = None
+        for i in range(1, k + p + 1):
+            dz = 2 * z * dz + 1
+            z = z * z + c
+            if i == k:
+                zk, dzk = z, dz
+        d = (z - zk) / (dz - dzk)
+        c -= d
+        if converged(d):
+            break
+    return c
+
+
+def period_in_ball(c, r, maxit=1_000_000):
+    """First n where the disk of radius r around c maps onto a disk containing 0."""
+    z = dz = mpc(0)
+    for n in range(1, maxit):
+        dz = 2 * z * dz + 1
+        z = z * z + c
+        if abs(z) < abs(dz) * r:
+            return n
+        if abs(z) > 4:
+            return None
+    return None
+
+
+def nucleus(c, p, steps=200):
+    """Newton on f^p(0) = 0."""
+    for _ in range(steps):
+        z = dz = mpc(0)
+        for _ in range(p):
+            dz = 2 * z * dz + 1
+            z = z * z + c
+        d = z / dz
+        c -= d
+        if converged(d):
+            break
+    return c
+
+
+def size(c, p):
+    """Approximate size of the minibrot with nucleus c and period p."""
+    z = mpc(0)
+    l = b = mpc(1)
+    for _ in range(1, p):
+        z = z * z + c
+        l = 2 * z * l
+        b = b + 1 / l
+    return abs(1 / (b * l * l))
+
+
+def main(argv):
+    if len(argv) >= 1 and argv[0] == "--list":
+        for name, ((re, im), k, p) in REGIONS.items():
+            print(f"{name:10} ~{re}{'+' if not im.startswith('-') else ''}{im}i  M({k},{p})")
+        return 0
+    if len(argv) not in (2, 3):
+        print(__doc__, file=sys.stderr)
+        return 2
+
+    region, depth = argv[0], int(argv[1])
+    if region in REGIONS:
+        (re, im), k, p = REGIONS[region]
+    else:
+        re, im, k, p = region.split(",")
+        k, p = int(k), int(p)
+    off_re, off_im = argv[2].split(",") if len(argv) == 3 else ("0.6", "0.3")
+
+    mp.dps = 2 * depth + 60
+    m = misiurewicz(mpc(re, im), k, p)
+    r = mpf(10) ** (-depth)
+    c0 = m + r * mpc(off_re, off_im)
+    period = period_in_ball(c0, r)
+    if period is None:
+        print("no period found (the disk escapes)", file=sys.stderr)
+        return 1
+    n = nucleus(c0, period)
+    s = size(n, period)
+    if s < mpf(10) ** (-2 * depth - 10):
+        print(f"size {mp.nstr(s, 3)} is implausibly small: Newton didn't converge, "
+              "try another offset", file=sys.stderr)
+        return 1
+
+    digits = int(-log10(s)) + 10
+    log_s = float(log10(s))
+    re_s = mp.nstr(n.real, digits, strip_zeros=False)
+    im_s = mp.nstr(n.imag, digits, strip_zeros=False)
+    print(f"# {region} depth={depth} period={period} size={mp.nstr(s, 3)}")
+    print("# minibrot:")
+    print(f"--view={re_s},{im_s},{mp.nstr(3 * s, 2)}")
+    print(f"# embedded Julia set on the way down:")
+    print(f"--view={re_s},{im_s},1e{round(0.75 * log_s)}")
+    return 0
+
+
+if __name__ == "__main__":
+    sys.exit(main(sys.argv[1:]))

tools/encode/.gitignore

@@ -0,0 +1,3 @@
+clips/
+work/
+results/

tools/encode/configs/speed.txt

@@ -0,0 +1,8 @@
+# Pass 1: which presets fit the 3-5 fps budget at 4K on this machine.
+x265-fast      | -pix_fmt yuv420p10le -c:v libx265 -profile:v main10 -preset fast   -crf 18 -x265-params log-level=error:aq-mode=3:no-sao=1
+x265-medium    | -pix_fmt yuv420p10le -c:v libx265 -profile:v main10 -preset medium -crf 18 -x265-params log-level=error:aq-mode=3:no-sao=1
+x265-slow      | -pix_fmt yuv420p10le -c:v libx265 -profile:v main10 -preset slow   -crf 18 -x265-params log-level=error:aq-mode=3:no-sao=1
+av1-p5         | -pix_fmt yuv420p10le -c:v libsvtav1 -preset 5 -crf 26 -svtav1-params film-grain=0
+av1-p6         | -pix_fmt yuv420p10le -c:v libsvtav1 -preset 6 -crf 26 -svtav1-params film-grain=0
+av1-p7         | -pix_fmt yuv420p10le -c:v libsvtav1 -preset 7 -crf 26 -svtav1-params film-grain=0
+av1-p8         | -pix_fmt yuv420p10le -c:v libsvtav1 -preset 8 -crf 26 -svtav1-params film-grain=0

tools/encode/make_clips.sh

@@ -0,0 +1,66 @@
+#!/usr/bin/env bash
+# Render the lossless 4K60 test clips that sweep.sh encodes.
+#
+# Each clip goes through the same RGBA -> bt709 tv-range 10-bit conversion as
+# the Makefile, then into lossless FFV1. The clips are therefore exactly what the
+# encoders see, so VMAF against them measures the encoder alone.
+#
+#   tools/encode/make_clips.sh [--size WxH] [--seconds S] [--force] [clip...]
+set -euo pipefail
+
+here=$(cd "$(dirname "$0")" && pwd)
+root=$(cd "$here/../.." && pwd)
+out="$here/clips"
+size=3840x2160
+seconds=4
+fps=60
+force=0
+only=()
+
+while [[ $# -gt 0 ]]; do
+    case $1 in
+        --size) size=$2; shift 2 ;;
+        --seconds) seconds=$2; shift 2 ;;
+        --force) force=1; shift ;;
+        -h|--help) sed -n '2,9p' "$0"; exit 0 ;;
+        *) only+=("$1"); shift ;;
+    esac
+done
+
+# Seahorse valley, minibrot at 3.7e-41 (tools/deep-zoom/LOCATIONS.md).
+SH_RE=-0.77568376800905379745347652613832924487504096622022
+SH_IM=0.13646736829469012473375311880735014411233827361594
+
+# name | mandelbrot flags. Zoom speeds bracket a typical 60 s dive (~0.7 decade/s).
+clips=(
+    # Worst case: dense filaments, 1.5 decades/s of scaling motion.
+    "filaments|--view=$SH_RE,$SH_IM,1e-8 --to-view=$SH_RE,$SH_IM,3e-15 --linear"
+    # Smooth concentric bands closing in on the minibrot: banding/blocking risk.
+    "bands|--view=$SH_RE,$SH_IM,1e-38 --to-view=$SH_RE,$SH_IM,1.1e-40 --linear"
+    # Embedded Julia set, slow drift: fine static texture, psy/detail retention.
+    "julia|--view=$SH_RE,$SH_IM,2e-30 --to-view=$SH_RE,$SH_IM,1e-30 --linear"
+)
+
+[[ -x $root/target/release/mandelbrot ]] || cargo build --release --manifest-path "$root/Cargo.toml"
+mkdir -p "$out"
+w=${size%x*}
+h=${size#*x}
+
+for entry in "${clips[@]}"; do
+    name=${entry%%|*}
+    flags=${entry#*|}
+    if [[ ${#only[@]} -gt 0 && ! " ${only[*]} " =~ " $name " ]]; then continue; fi
+    dst="$out/$name.mkv"
+    if [[ -f $dst && $force = 0 ]]; then echo "skip $name (exists, --force to redo)"; continue; fi
+    echo "render $name ($size, ${seconds}s @ ${fps}fps)"
+    # shellcheck disable=SC2086
+    "$root/target/release/mandelbrot" --headless --antialias $flags \
+        --width "$w" --height "$h" --fps "$fps" --duration "$seconds" --export-path - |
+        ffmpeg -hide_banner -loglevel error -y \
+            -f rawvideo -pix_fmt rgba -s "$size" -framerate "$fps" -i - \
+            -vf "scale=out_color_matrix=bt709:out_range=tv:flags=accurate_rnd+full_chroma_int+bitexact,format=yuv420p10le" \
+            -c:v ffv1 -level 3 -slices 16 -g 1 \
+            -colorspace bt709 -color_primaries bt709 -color_trc bt709 -color_range tv \
+            "$dst.tmp.mkv"
+    mv "$dst.tmp.mkv" "$dst"
+done

tools/encode/sweep.sh

@@ -0,0 +1,87 @@
+#!/usr/bin/env bash
+# Encode every test clip with every config, then score each encode.
+#
+#   tools/encode/sweep.sh [--clips a,b] [--keep] [--tag NAME] CONFIG_FILE...
+#
+# A config line is `name | ffmpeg output args` (blank lines and # comments are
+# skipped). The args go between `-i clip.mkv` and the output file, e.g.
+#   x265-crf18 | -c:v libx265 -preset medium -crf 18 -x265-params aq-mode=3
+# Each row of results/<tag>.tsv has encode fps, bitrate, VMAF 4K (mean, 1st
+# percentile, min), VMAF-NEG mean, CAMBI banding (mean, max) and PSNR-Y.
+set -euo pipefail
+
+here=$(cd "$(dirname "$0")" && pwd)
+clipdir="$here/clips"
+model=/usr/share/model
+export SVT_LOG=1  # errors only
+clips=""
+keep=0
+tag=$(date +%Y%m%d-%H%M%S)
+configs=()
+
+while [[ $# -gt 0 ]]; do
+    case $1 in
+        --clips) clips=$2; shift 2 ;;
+        --keep) keep=1; shift ;;
+        --tag) tag=$2; shift 2 ;;
+        -h|--help) sed -n '2,11p' "$0"; exit 0 ;;
+        *) configs+=("$1"); shift ;;
+    esac
+done
+[[ ${#configs[@]} -gt 0 ]] || { echo "usage: $0 [--clips a,b] [--keep] [--tag NAME] CONFIG_FILE..." >&2; exit 1; }
+
+if [[ -z $clips ]]; then
+    clips=$(cd "$clipdir" && ls ./*.mkv | sed 's|^\./||; s|\.mkv$||' | paste -sd,)
+fi
+[[ -n $clips ]] || { echo "no clips in $clipdir: run make_clips.sh first" >&2; exit 1; }
+
+mkdir -p "$here/results" "$here/work"
+tsv="$here/results/$tag.tsv"
+[[ -f $tsv ]] || printf 'clip\tconfig\tfps\tmbps\tvmaf\tvmaf_p1\tvmaf_min\tvmaf_neg\tcambi\tcambi_max\tpsnr_y\n' > "$tsv"
+
+IFS=, read -ra clip_list <<< "$clips"
+for cfg in "${configs[@]}"; do
+    while IFS= read -r line || [[ -n $line ]]; do
+        [[ $line =~ ^[[:space:]]*(#|$) ]] && continue
+        name=$(sed 's/[[:space:]]*|.*//' <<< "$line")
+        args=$(sed 's/^[^|]*|[[:space:]]*//' <<< "$line")
+        for clip in "${clip_list[@]}"; do
+            ref="$clipdir/$clip.mkv"
+            enc="$here/work/$clip--$name.mp4"
+            log="$here/work/$clip--$name.json"
+            frames=$(ffprobe -v error -count_packets -select_streams v:0 \
+                -show_entries stream=nb_read_packets -of csv=p=0 "$ref")
+            printf '%-10s %-40s ' "$clip" "$name"
+
+            t0=$(date +%s.%N)
+            # shellcheck disable=SC2086
+            ffmpeg -hide_banner -loglevel error -y -i "$ref" $args -an "$enc" < /dev/null
+            t1=$(date +%s.%N)
+
+            # 4K model for both VMAF variants; CAMBI flags banding, which VMAF
+            # barely sees and which is the main risk on smooth fractal gradients.
+            ffmpeg -hide_banner -loglevel error -i "$enc" -i "$ref" -lavfi \
+                "[0:v]setpts=PTS-STARTPTS[d];[1:v]setpts=PTS-STARTPTS[r];[d][r]libvmaf=log_fmt=json:log_path=$log:n_threads=$(nproc):n_subsample=2:model='path=$model/vmaf_4k_v0.6.1.json\\:name=vmaf|path=$model/vmaf_4k_v0.6.1neg.json\\:name=neg':feature='name=cambi|name=psnr'" \
+                -f null - < /dev/null
+
+            row=$(python3 - "$log" "$enc" "$frames" "$t0" "$t1" <<'PY'
+import json, os, sys
+log, enc, frames, t0, t1 = sys.argv[1], sys.argv[2], int(sys.argv[3]), float(sys.argv[4]), float(sys.argv[5])
+fr = json.load(open(log))["frames"]
+col = lambda k: sorted(f["metrics"][k] for f in fr)
+v = col("vmaf")
+p1 = v[max(0, int(0.01 * len(v)) - 1)] if len(v) >= 100 else v[0]
+mean = lambda xs: sum(xs) / len(xs)
+cambi = col("cambi")
+fps = frames / (t1 - t0)
+mbps = os.path.getsize(enc) * 8 / (frames / 60) / 1e6
+print(f"{fps:.2f}\t{mbps:.1f}\t{mean(v):.2f}\t{p1:.2f}\t{v[0]:.2f}\t{mean(col('neg')):.2f}\t{mean(cambi):.2f}\t{cambi[-1]:.2f}\t{mean(col('psnr_y')):.2f}")
+PY
+)
+            printf '%s\t%s\t%s\n' "$clip" "$name" "$row" >> "$tsv"
+            awk -F'\t' '{printf "%6s fps %7s Mb/s  vmaf %s p1 %s min %s neg %s  cambi %s/%s  psnr %s\n",$1,$2,$3,$4,$5,$6,$7,$8,$9}' <<< "$row"
+            [[ $keep = 1 ]] || rm -f "$enc"
+        done
+    done < "$cfg"
+done
+echo "results: $tsv"