feat: improve colors when using distance estimate

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

CLAUDE.md

@@ -31,10 +31,15 @@ cargo install wasm-bindgen-cli --version 0.2.128   # must match the wasm-bindgen
 python3 -m http.server -d dist 8080
 ```
 
-Native debug env vars (see `src/app.rs`, near the top of `FractalApp::new`):
+Native CLI flags (`src/cli.rs`, applied in `FractalApp::apply_cli`): `--kind`,
-`MANDEL_KIND`, `MANDEL_POWER`, `MANDEL_JULIA="re,im"`, `MANDEL_SHARE="<fragment>"`,
+`--power`, `--julia re,im`, `--share <fragment>`,
-`MANDEL_VIEW="re,im,half_height[,iterations]"`, `MANDEL_DE=1`,
+`--view re,im,half_height[,iterations]`, `--de`, `--buddhabrot`,
-`MANDEL_BUDDHABROT=1`, `MANDEL_EXPORT=1` (+ `MANDEL_EXPORT_PATH=out.png`).
+`--buddha-palette`, `--export` (+ `--export-path out.png`). `--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`, default 1920×1080) — implies `--export`'s
+save behavior without needing a GPU-backed window/event loop. Not yet
+supported with `--buddhabrot`. Run `mandelbrot --help` for the full list.
 
 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

Cargo.toml

@@ -14,6 +14,7 @@ png = "0.18.1"
 [target.'cfg(not(target_arch = "wasm32"))'.dependencies]
 env_logger = "0.11.11"
 clap = { version = "4.5.51", features = ["derive"] }
+pollster = "1.0.1"
 
 [target.'cfg(target_arch = "wasm32")'.dependencies]
 futures-channel = { version = "0.3.34", default-features = false, features = ["alloc", "std"] }

src/app.rs

@@ -2,16 +2,16 @@ use std::sync::{Arc, Mutex};
 
 use eframe::CreationContext;
 use eframe::egui_wgpu;
+#[cfg(target_arch = "wasm32")]
 use eframe::egui_wgpu::wgpu;
 
 #[cfg(not(target_arch = "wasm32"))]
 use crate::cli::Cli;
 use crate::fractal::{
     BuddhabrotCallback, BuddhabrotRenderer, BuddhabrotUniforms, ExportRender, FractalCallback,
-    FractalKind, FractalRenderer, MAX_REF_POINTS, ShareState, Uniforms,
+    FractalKind, FractalRenderer, MAX_REF_POINTS, ShareState, Uniforms, compute_reference,
+    compute_set_reference,
 };
-#[cfg(target_arch = "wasm32")]
-use crate::fractal::{compute_reference, compute_set_reference};
 use crate::lights::Light;
 use crate::view::{
     Big, DEFAULT_HALF_HEIGHT, ViewState, big_from_decimal_str, big_from_f64, big_to_decimal_str,
@@ -43,6 +43,7 @@ const BUDDHA_PALETTE_NAMES: &[&str] = &["Nebula", "Yellow", "Grayscale"];
 pub enum FractalMode {
     Mandelbrot,
     Julia,
+    Buddhabrot,
 }
 
 /// Selectable fractal formulas, with UI labels.
@@ -56,6 +57,7 @@ const KINDS: &[(FractalKind, &str)] = &[
     (FractalKind::Buffalo, "Buffalo"),
     (FractalKind::Phoenix, "Phoenix"),
     (FractalKind::Lambda, "Lambda"),
+    (FractalKind::ComplexMultibrot, "Complex Multibrot"),
 ];
 
 /// UI label for a fractal kind.
@@ -67,10 +69,30 @@ fn kind_label(kind: FractalKind) -> &'static str {
         .unwrap_or("Mandelbrot")
 }
 
+/// The iteration formula for a kind, in human-readable notation (mirrors the
+/// doc comments on `FractalKind`'s variants). `power` is only used by
+/// Multibrot; `complex_power` only by Complex Multibrot.
+fn kind_formula(kind: FractalKind, power: u32, complex_power: (f64, f64)) -> String {
+    match kind {
+        FractalKind::Mandelbrot => "z = z² + c".to_string(),
+        FractalKind::BurningShip => "z = (|Re(z)| + i|Im(z)|)² + c".to_string(),
+        FractalKind::Tricorn => "z = conj(z)² + c".to_string(),
+        FractalKind::Multibrot => format!("z = z^{power} + c"),
+        FractalKind::Celtic => "z = |Re(z²)| + i·Im(z²) + c".to_string(),
+        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::ComplexMultibrot => {
+            format!("z = z^({:.3}{:+.3}i) + c", complex_power.0, complex_power.1)
+        }
+    }
+}
+
 type JuliaPreset = (&'static str, f64, f64, u32, Option<(f64, f64)>);
 
 /// Nice-looking Julia constants offered as presets.
-const JULIA_PRESETS: [&[JuliaPreset]; FractalKind::Lambda as usize + 1] = [
+const JULIA_PRESETS: [&[JuliaPreset]; FractalKind::ComplexMultibrot as usize + 1] = [
     &[
         ("dendrite", -0.8, 0.156, 400, None),
         ("rabbit", -0.123, 0.745, 400, None),
@@ -89,6 +111,7 @@ const JULIA_PRESETS: [&[JuliaPreset]; FractalKind::Lambda as usize + 1] = [
         ("archipelago 2", -0.556, 0.253, 500, Some((-0.415, -0.267))),
     ],
     &[],
+    &[],
 ];
 
 type SetPreset = (
@@ -103,7 +126,7 @@ type SetPreset = (
 /// Curated beautiful locations offered as one-click presets.
 /// Each is `(name, center_re, center_im, half_height, iterations)`; the centers
 /// are decimals parsed at full precision so deep places stay sharp.
-const SET_PRESETS: [&[SetPreset]; FractalKind::Lambda as usize + 1] = [
+const SET_PRESETS: [&[SetPreset]; FractalKind::ComplexMultibrot as usize + 1] = [
     &[
         (
             "Seahorse Valley",
@@ -154,6 +177,7 @@ const SET_PRESETS: [&[SetPreset]; FractalKind::Lambda as usize + 1] = [
         Some((-0.9, -0.49)),
     )],
     &[],
+    &[],
 ];
 
 /// Parameters a reference orbit was (or will be) computed for. Used to decide
@@ -169,6 +193,7 @@ struct RequestKey {
     iter: u32,
     kind: FractalKind,
     power: u32,
+    complex_power: (f64, f64),
 }
 
 /// Shared state for an in-progress PNG export. The worker (a background thread
@@ -256,6 +281,8 @@ pub struct FractalApp {
     kind: FractalKind,
     /// Exponent for the Multibrot kind.
     power: u32,
+    /// Complex exponent for the Complex Multibrot kind (`z^power + c`).
+    complex_power: (f64, f64),
     julia_c: (f64, f64),
     /// Distortion constant `p` for the Phoenix kind (`z^2 + c + p·z_{n-1}`).
     phoenix_p: (f64, f64),
@@ -281,10 +308,6 @@ pub struct FractalApp {
     /// List of enabled lights in the world
     lights: Vec<Light>,
 
-    /// Render as a Buddhabrot (Monte-Carlo orbit-density histogram) instead of
-    /// the ordinary escape-time set. Plain f32 view — no deep zoom, no
-    /// perturbation/reference-orbit machinery (see `fractal::buddhabrot`).
-    buddhabrot: bool,
     /// Nested escape-iteration caps for the R/G/B histogram channels
     /// (Nebulabrot coloring); kept ordered r <= g <= b by the UI.
     buddha_r_cap: u32,
@@ -303,6 +326,11 @@ pub struct FractalApp {
     /// Whether the app is in fullscreen (browser Fullscreen API on web, viewport
     /// fullscreen on native). Kept in sync with the real state each frame.
     fullscreen: bool,
+    /// Whether the "Fractal Info" popup (formula/constants/zoom for the
+    /// current view) is open.
+    info_open: bool,
+    /// Whether the Help window (about + mouse/touch controls) is open.
+    help_open: bool,
     /// Time-based animation of colours / Julia c / Phoenix p / zoom.
     anim: AnimState,
 
@@ -395,6 +423,29 @@ impl FractalApp {
             guard.callback_resources.insert(buddhabrot_renderer);
         }
 
+        let mut app = Self::default_state();
+
+        // On the web, restore a shared view from the URL fragment (#...).
+        #[cfg(target_arch = "wasm32")]
+        if let Some(frag) = web_location_hash() {
+            if let Some(state) = ShareState::decode(&frag) {
+                app.apply_share(&state);
+            }
+        }
+
+        cc.egui_ctx.set_zoom_factor(1.1);
+
+        // Debug/testing hooks, driven by CLI flags.
+        #[cfg(not(target_arch = "wasm32"))]
+        app.apply_cli(Cli::parse());
+
+        app
+    }
+
+    /// Build the app's default state (no window, no GPU, no CLI applied yet).
+    /// Shared by the windowed app (`new`, which then layers CLI/share-link
+    /// overrides on top) and headless rendering.
+    pub(crate) fn default_state() -> Self {
         let view = ViewState::default();
         let ref_center_re = view.center_re.clone();
         let ref_center_im = view.center_im.clone();
@@ -404,11 +455,12 @@ impl FractalApp {
         let center_im_edit = big_to_decimal_str(&view.center_im, sig);
         let zoom_edit = format_magnification(view.magnification());
 
-        let mut app = Self {
+        Self {
             view,
             mode: FractalMode::Mandelbrot,
             kind: FractalKind::Mandelbrot,
             power: 3,
+            complex_power: (2.0, 0.5),
             julia_c: (-0.8, 0.156),
             phoenix_p: (-0.5, 0.0),
             lambda_l: (-0.5, 0.0),
@@ -422,7 +474,6 @@ impl FractalApp {
             de_coloring: false,
             shadow: false,
             lights: vec![Light::default()],
-            buddhabrot: false,
             buddha_r_cap: 50,
             buddha_g_cap: 500,
             buddha_b_cap: 2000,
@@ -431,6 +482,8 @@ impl FractalApp {
             buddha_accumulate: true,
             controls_open: true,
             fullscreen: false,
+            info_open: false,
+            help_open: false,
             anim: AnimState::default(),
             fps: 0.0,
             fps_frames: 0,
@@ -455,63 +508,62 @@ impl FractalApp {
             center_im_edit,
             zoom_edit,
             zoom_edited: false,
-        };
-
-        // On the web, restore a shared view from the URL fragment (#...).
-        #[cfg(target_arch = "wasm32")]
-        if let Some(frag) = web_location_hash() {
-            if let Some(state) = ShareState::decode(&frag) {
-                app.apply_share(&state);
-            }
         }
+    }
 
-        // Debug/testing hooks, driven by CLI flags.
+    /// Apply native CLI flags on top of the default state: fractal kind/mode,
-        #[cfg(not(target_arch = "wasm32"))]
+    /// a restored share link or explicit view, coloring toggles, and export
-        {
+    /// options. Shared by the windowed app and headless rendering.
-            let cli = Cli::parse();
+    #[cfg(not(target_arch = "wasm32"))]
-
+    pub(crate) fn apply_cli(&mut self, cli: Cli) {
-            if let Some(k) = cli.kind {
+        if let Some(k) = cli.kind {
-                app.kind = k.into();
+            self.kind = k.into();
-                if let Some(p) = cli.power {
+            if let Some(p) = cli.power {
-                    app.power = p.clamp(2, 8);
+                self.power = p.clamp(2, 8);
-                }
-                app.view = Self::default_view_for(app.mode, app.kind);
             }
-            if let Some(jc) = cli.julia {
+            if let Some(cp) = &cli.complex_power {
-                let p: Vec<&str> = jc.split(',').collect();
+                let p: Vec<&str> = cp.split(',').collect();
                 if let (Some(Ok(re)), Some(Ok(im))) = (
                     p.first().map(|s| s.trim().parse::<f64>()),
                     p.get(1).map(|s| s.trim().parse::<f64>()),
                 ) {
-                    app.mode = FractalMode::Julia;
+                    self.complex_power = (re, im);
-                    app.julia_c = (re, im);
-                    app.view = Self::default_view_for(FractalMode::Julia, app.kind);
                 }
             }
-            if let Some(frag) = cli.share
+            self.view = Self::default_view_for(self.mode, self.kind);
-                && let Some(state) = ShareState::decode(&frag)
+        }
-            {
+        if let Some(jc) = cli.julia {
-                app.apply_share(&state);
+            let p: Vec<&str> = jc.split(',').collect();
-            }
+            if let (Some(Ok(re)), Some(Ok(im))) = (
-            if let Some(spec) = cli.view {
+                p.first().map(|s| s.trim().parse::<f64>()),
-                app.apply_view_spec(&spec);
+                p.get(1).map(|s| s.trim().parse::<f64>()),
-            }
+            ) {
-            if cli.de {
+                self.mode = FractalMode::Julia;
-                app.de_coloring = true;
+                self.julia_c = (re, im);
-            }
+                self.view = Self::default_view_for(FractalMode::Julia, self.kind);
-            if cli.buddhabrot {
-                app.buddhabrot = true;
-            }
-            if let Some(p) = cli.buddha_palette {
-                app.buddha_palette = p.min(BUDDHA_PALETTE_NAMES.len() as u32 - 1);
-            }
-            app.export_path = cli.export_path;
-            if cli.export {
-                app.export_requested = true;
             }
         }
-
+        if let Some(frag) = cli.share
-        app
+            && let Some(state) = ShareState::decode(&frag)
+        {
+            self.apply_share(&state);
+        }
+        if let Some(spec) = cli.view {
+            self.apply_view_spec(&spec);
+        }
+        if cli.de {
+            self.de_coloring = true;
+        }
+        if cli.buddhabrot {
+            self.mode = FractalMode::Buddhabrot;
+        }
+        if let Some(p) = cli.buddha_palette {
+            self.buddha_palette = p.min(BUDDHA_PALETTE_NAMES.len() as u32 - 1);
+        }
+        self.export_path = cli.export_path;
+        if cli.export {
+            self.export_requested = true;
+        }
     }
 
     /// Apply a view spec "re,im,half_height[,iterations]" (re/im are decimal,
@@ -583,6 +635,7 @@ impl FractalApp {
             julia_c: self.julia_c,
             phoenix_p: self.phoenix_p,
             lambda_l: self.lambda_l,
+            complex_power: self.complex_power,
             color_scale: self.color_scale,
             color_offset: self.color_offset,
             palette: self.palette,
@@ -602,6 +655,7 @@ impl FractalApp {
         self.julia_c = s.julia_c;
         self.phoenix_p = s.phoenix_p;
         self.lambda_l = s.lambda_l;
+        self.complex_power = s.complex_power;
         self.color_scale = s.color_scale;
         self.color_offset = s.color_offset;
         self.palette = (s.palette as usize).min(PALETTE_NAMES.len() - 1) as u32;
@@ -653,6 +707,7 @@ impl FractalApp {
             FractalKind::Buffalo => (-0.5, -0.5, 1.5),
             FractalKind::Phoenix => (0.0, 0.0, 1.6),
             FractalKind::Lambda => (0.0, 0.0, 1.6),
+            FractalKind::ComplexMultibrot => (0.0, 0.0, 1.5),
         };
         ViewState::with_center(big_from_f64(cr, 53), big_from_f64(ci, 53), hh)
     }
@@ -669,6 +724,7 @@ impl FractalApp {
             iter: self.max_iterations,
             kind: self.kind,
             power: self.power,
+            complex_power: self.complex_power,
         }
     }
 
@@ -694,6 +750,7 @@ impl FractalApp {
             || key.iter != self.max_iterations
             || key.kind != self.kind
             || key.power != self.power
+            || key.complex_power != self.complex_power
         {
             return true;
         }
@@ -726,6 +783,14 @@ impl FractalApp {
         self.generation = self.generation.wrapping_add(1);
     }
 
+    /// The current reference orbit, as uploaded to the GPU. Used by headless
+    /// rendering to build its own `ExportRender` without going through
+    /// `egui_wgpu`'s callback machinery.
+    #[cfg(not(target_arch = "wasm32"))]
+    pub(crate) fn reference_points(&self) -> &[[f32; 2]] {
+        &self.reference
+    }
+
     /// Recompute the reference orbit when needed. Native: dispatch to a worker
     /// thread and pick up completed results. Web: compute inline.
     fn ensure_reference(&mut self) {
@@ -754,6 +819,7 @@ impl FractalApp {
                     power: key.power,
                     phoenix_p: key.phoenix_p,
                     lambda_l: key.lambda_l,
+                    complex_power: key.complex_power,
                 });
                 self.pending = true;
             }
@@ -773,6 +839,7 @@ impl FractalApp {
                         key.power,
                         key.phoenix_p,
                         key.lambda_l,
+                        key.complex_power,
                     )
                 } else {
                     compute_set_reference(
@@ -784,6 +851,7 @@ impl FractalApp {
                         key.power,
                         key.phoenix_p,
                         key.lambda_l,
+                        key.complex_power,
                     )
                 };
                 self.apply_reference(
@@ -804,7 +872,65 @@ impl FractalApp {
         }
     }
 
-    fn make_uniforms(&self, aspect: f64) -> Uniforms {
+    /// Compute the reference orbit for the current view synchronously, on the
+    /// calling thread — unlike `ensure_reference`, which dispatches to the
+    /// native worker (or, on wasm, computes inline but still runs once per
+    /// frame poll). Used by headless rendering, which has no frame loop to
+    /// poll a background result on and only ever needs one reference.
+    #[cfg(not(target_arch = "wasm32"))]
+    pub(crate) fn compute_reference_blocking(&mut self) {
+        if self.auto_iterations {
+            self.max_iterations = self.auto_iteration_count();
+        }
+        let mut key = self.current_key();
+        let precision = self.view.precision_bits();
+        let max_iter = key.iter.min(MAX_REF_POINTS as u32 - 1);
+
+        // Lambda in Set mode has a static fractal centered at origin.
+        if key.kind == FractalKind::Lambda && !key.julia {
+            key.center_re = big_from_f64(0.0, precision);
+            key.center_im = big_from_f64(0.0, precision);
+        }
+
+        let points = if key.julia {
+            let jr = big_from_f64(key.julia_c.0, precision);
+            let ji = big_from_f64(key.julia_c.1, precision);
+            compute_reference(
+                &key.center_re,
+                &key.center_im,
+                &jr,
+                &ji,
+                max_iter,
+                precision,
+                key.kind,
+                key.power,
+                key.phoenix_p,
+                key.lambda_l,
+                key.complex_power,
+            )
+        } else {
+            compute_set_reference(
+                &key.center_re,
+                &key.center_im,
+                max_iter,
+                precision,
+                key.kind,
+                key.power,
+                key.phoenix_p,
+                key.lambda_l,
+                key.complex_power,
+            )
+        };
+        self.apply_reference(
+            points,
+            key.center_re.clone(),
+            key.center_im.clone(),
+            key.half_height,
+        );
+        self.last_request = Some(key);
+    }
+
+    pub(crate) fn make_uniforms(&self, aspect: f64) -> Uniforms {
         let (span_x, span_y) = self.view.span(aspect);
         Uniforms {
             span: [span_x as f32, span_y as f32],
@@ -822,6 +948,7 @@ impl FractalApp {
             dc_offset: self.dc_offset(),
             phoenix_p: [self.phoenix_p.0 as f32, self.phoenix_p.1 as f32],
             lambda_l: [self.lambda_l.0 as f32, self.lambda_l.1 as f32],
+            complex_power: [self.complex_power.0 as f32, self.complex_power.1 as f32],
             de_coloring: (self.de_coloring | self.shadow) as u32,
             shadow: self.shadow as u32,
             _pad: [0; _],
@@ -842,6 +969,7 @@ impl FractalApp {
             aspect: aspect as f32,
             phoenix_p: [self.phoenix_p.0 as f32, self.phoenix_p.1 as f32],
             lambda_l: [self.lambda_l.0 as f32, self.lambda_l.1 as f32],
+            complex_power: [self.complex_power.0 as f32, self.complex_power.1 as f32],
             bailout_sq: BAILOUT_SQ,
             kind: self.kind as u32,
             power: self.power,
@@ -855,7 +983,7 @@ impl FractalApp {
             height: 0,          // set by the callback from size_px
             total_samples: 0.0, // tracked by the renderer across frames
             palette: self.buddha_palette,
-            _pad: [0; 3],
+            _pad0: 0,
         }
     }
 
@@ -866,7 +994,7 @@ impl FractalApp {
         if self.export.is_some() {
             return; // one export at a time
         }
-        if self.buddhabrot {
+        if self.mode == FractalMode::Buddhabrot {
             self.status = Some("PNG export isn't available in Buddhabrot mode yet".into());
             return;
         }
@@ -905,9 +1033,6 @@ impl FractalApp {
         self.status = None;
         self.export = Some(Arc::clone(&shared));
 
-        // Progress budget: rendering fills [0, RENDER_END], encoding the rest.
-        const RENDER_END: f32 = 0.6;
-
         #[cfg(not(target_arch = "wasm32"))]
         {
             let name = self
@@ -926,51 +1051,11 @@ impl FractalApp {
                     uniforms,
                     reference.as_slice(),
                 );
-
+                let sh = Arc::clone(&shared);
-                // Render the image tile by tile, waiting for each so progress
+                let png =
-                // reflects real GPU work.
+                    crate::fractal::export_to_png_blocking(&device, &queue, &er, |phase, f| {
-                for t in 0..er.tiles {
+                        set_progress(&sh, phase, f)
-                    er.render_tile(&device, &queue, t);
-                    let _ = device.poll(wgpu::PollType::Wait {
-                        submission_index: None,
-                        timeout: None,
                     });
-                    let done = (t + 1) as f32 / er.tiles as f32;
-                    set_progress(&shared, "Rendering", RENDER_END * done);
-                }
-                er.copy_to_readback(&device, &queue);
-
-                // Wait for the copy, then read the mapped bytes.
-                let (tx, rx) = std::sync::mpsc::channel();
-                er.readback()
-                    .slice(..)
-                    .map_async(wgpu::MapMode::Read, move |res| {
-                        let _ = tx.send(res);
-                    });
-                let _ = device.poll(wgpu::PollType::Wait {
-                    submission_index: None,
-                    timeout: None,
-                });
-                let _ = rx.recv();
-
-                set_progress(&shared, "Encoding", RENDER_END);
-                let png = {
-                    let data = er
-                        .readback()
-                        .slice(..)
-                        .get_mapped_range()
-                        .expect("map readback buffer");
-                    let sh = Arc::clone(&shared);
-                    crate::fractal::encode_png_with_progress(
-                        &data,
-                        er.width,
-                        er.height,
-                        er.padded_bpr,
-                        er.swap_rb,
-                        |f| set_progress(&sh, "Encoding", RENDER_END + (0.97 - RENDER_END) * f),
-                    )
-                };
-                er.readback().unmap();
 
                 set_progress(&shared, "Saving", 0.98);
                 let result = std::fs::write(&name, &png)
@@ -981,6 +1066,8 @@ impl FractalApp {
         }
         #[cfg(target_arch = "wasm32")]
         {
+            // Progress budget: rendering fills [0, RENDER_END], encoding the rest.
+            const RENDER_END: f32 = 0.6;
             wasm_bindgen_futures::spawn_local(async move {
                 let er = ExportRender::new(
                     &device,
@@ -1093,6 +1180,13 @@ impl FractalApp {
                                 self.fullscreen = !self.fullscreen;
                                 self.apply_fullscreen(ui.ctx());
                             }
+                            if ui
+                                .button("Help")
+                                .on_hover_text("About this app, and mouse/touch controls")
+                                .clicked()
+                            {
+                                self.help_open = !self.help_open;
+                            }
                             // FPS readout. Monospace + fixed width so the number
                             // changing doesn't jitter the button row.
                             ui.add(
@@ -1111,6 +1205,180 @@ impl FractalApp {
             });
     }
 
+    /// Floating bottom-left overlay: a single button that toggles the
+    /// "Fractal Info" window. Kept separate from `overlay_buttons` (top-left)
+    /// so it stays out of the way of the panel toggle / fullscreen controls,
+    /// but is still reachable even when the controls panel is collapsed.
+    fn info_button(&mut self, ui: &mut egui::Ui) {
+        egui::Area::new(egui::Id::new("info_button"))
+            .anchor(egui::Align2::LEFT_BOTTOM, egui::vec2(8.0, -8.0))
+            .show(ui.ctx(), |ui| {
+                egui::Frame::popup(ui.style())
+                    .shadow(egui::Shadow::NONE)
+                    .show(ui, |ui| {
+                        if ui
+                            .button("Fractal infos")
+                            .on_hover_text("Show details about the current fractal")
+                            .clicked()
+                        {
+                            self.info_open = !self.info_open;
+                        }
+                    });
+            });
+    }
+
+    /// Window with details about what's currently on screen: formula, active
+    /// per-kind constants, zoom depth, iteration count. Reads live state, so
+    /// it stays correct as the user pans/zooms/switches kinds.
+    fn info_window(&mut self, ctx: &egui::Context) {
+        let mut open = self.info_open;
+        egui::Window::new("Fractal Info")
+            .id(egui::Id::new("info_window"))
+            .open(&mut open)
+            .collapsible(false)
+            .resizable(false)
+            .anchor(egui::Align2::LEFT_BOTTOM, egui::vec2(8.0, -44.0))
+            .show(ctx, |ui| {
+                ui.label(
+                    egui::RichText::new(kind_label(self.kind))
+                        .strong()
+                        .heading(),
+                );
+                let mode_label = match self.mode {
+                    FractalMode::Mandelbrot => {
+                        "Mandelbrot mode — parameter space (c varies per pixel, z₀ = 0)"
+                    }
+                    FractalMode::Julia => {
+                        "Julia mode — dynamical plane for a fixed c (z₀ varies per pixel)"
+                    }
+                    FractalMode::Buddhabrot => "Buddhabrot mode — orbit density (random c, z₀ = 0)",
+                };
+                ui.label(mode_label);
+                ui.separator();
+
+                ui.label(format!(
+                    "formula: {}",
+                    kind_formula(self.kind, self.power, self.complex_power)
+                ));
+                if self.mode == FractalMode::Julia {
+                    ui.label(format!("c = {:.6} {:+.6}i", self.julia_c.0, self.julia_c.1));
+                }
+                if self.kind == FractalKind::Phoenix {
+                    ui.label(format!(
+                        "p = {:.6} {:+.6}i",
+                        self.phoenix_p.0, self.phoenix_p.1
+                    ));
+                }
+                if self.kind == FractalKind::Lambda {
+                    ui.label(format!(
+                        "λ = {:.6} {:+.6}i",
+                        self.lambda_l.0, self.lambda_l.1
+                    ));
+                }
+                if self.kind == FractalKind::ComplexMultibrot {
+                    ui.label(format!(
+                        "power = {:.6} {:+.6}i",
+                        self.complex_power.0, self.complex_power.1
+                    ));
+                }
+                ui.separator();
+
+                ui.label(self.kind.description());
+            });
+        self.info_open = open;
+    }
+
+    /// Help window: what the app does, plus a reference for mouse/touch and
+    /// keyboard controls.
+    fn help_window(&mut self, ctx: &egui::Context) {
+        let mut open = self.help_open;
+        egui::Window::new("Help")
+            .id(egui::Id::new("help_window"))
+            .open(&mut open)
+            .collapsible(false)
+            .default_width(360.0)
+            .show(ctx, |ui| {
+                egui::ScrollArea::vertical()
+                    .max_height(480.0)
+                    .show(ui, |ui| {
+                        ui.heading("About");
+                        ui.label(
+                            "A deep-zoom fractal explorer. It renders the Mandelbrot set \
+                             and several related fractals (Burning Ship, Tricorn, \
+                             Multibrot, Complex Multibrot, Celtic, Perpendicular, Buffalo, \
+                             Phoenix, Lambda).",
+                        );
+                        ui.add_space(4.0);
+                        ui.label(
+                            "Each fractals can be rendered in different modes: \n\
+                             • Mandelbrot mode fixes z₀=0 and then for each pixel, set c as it's position \
+                             in the complex plane. \n\
+                             • Julia mode fixes c and instead varies the \
+                             starting point z₀ across the plane. \n\
+                             • Buddhabrot mode switches to a different, Monte-Carlo rendering of orbit density \
+                             instead of the ordinary escape-time set.",
+                        );
+                        ui.separator();
+
+                        ui.heading("Mouse & touch");
+                        egui::Grid::new("help_mouse_grid")
+                            .num_columns(2)
+                            .spacing([12.0, 6.0])
+                            .show(ui, |ui| {
+                                ui.label("Drag");
+                                ui.label("Pan the view");
+                                ui.end_row();
+                                ui.label("Scroll / trackpad");
+                                ui.label("Zoom toward the cursor");
+                                ui.end_row();
+                                ui.label("Pinch (touch)");
+                                ui.label("Zoom toward the gesture center");
+                                ui.end_row();
+                                ui.label("Two-finger drag (touch)");
+                                ui.label("Pan the view");
+                                ui.end_row();
+                            });
+                        ui.separator();
+
+                        ui.heading("Keyboard");
+                        egui::Grid::new("help_keyboard_grid")
+                            .num_columns(2)
+                            .spacing([12.0, 6.0])
+                            .show(ui, |ui| {
+                                ui.label("Arrow keys");
+                                ui.label("Pan the view");
+                                ui.end_row();
+                                ui.label("Z / S");
+                                ui.label("Zoom in / out toward the center");
+                                ui.end_row();
+                                ui.label("+ / -");
+                                ui.label("Increase / decrease iterations");
+                                ui.end_row();
+                                ui.label("R");
+                                ui.label("Reset to the default view");
+                                ui.end_row();
+                                ui.label("H");
+                                ui.label("Toggle this Help window");
+                                ui.end_row();
+                                ui.label("I");
+                                ui.label("Toggle the Info window");
+                                ui.end_row();
+                                ui.label("A");
+                                ui.label("Toggle antialiasing (2×2)");
+                                ui.end_row();
+                            });
+                        ui.separator();
+
+                        ui.heading("Tips");
+                        ui.label(
+                            "• \"Copy link\" (in the panel) encodes the exact view so it \
+                             can be reopened later or sent to someone else.",
+                        );
+                    });
+            });
+        self.help_open = open;
+    }
+
     /// Push the desired fullscreen state to the platform.
     #[cfg(not(target_arch = "wasm32"))]
     fn apply_fullscreen(&mut self, ctx: &egui::Context) {
@@ -1233,6 +1501,7 @@ impl FractalApp {
     fn controls_ui(&mut self, ui: &mut egui::Ui) {
         ui.heading("Fractal Explorer");
         ui.separator();
+        ui.add_space(4.);
 
         // Fractal formula. Switching kinds jumps to a sensible default view,
         // since interesting regions differ between fractals.
@@ -1279,20 +1548,42 @@ impl FractalApp {
                 ui.label("i");
             });
         }
+        if self.kind == FractalKind::ComplexMultibrot {
+            ui.horizontal(|ui| {
+                ui.label("power =");
+                ui.add(
+                    egui::DragValue::new(&mut self.complex_power.0)
+                        .speed(0.01)
+                        .range(-8.0..=8.0),
+                );
+                ui.add(
+                    egui::DragValue::new(&mut self.complex_power.1)
+                        .speed(0.01)
+                        .range(-8.0..=8.0),
+                );
+                ui.label("i");
+            });
+        }
         if self.kind != prev_kind {
             self.view = Self::default_view_for(self.mode, self.kind);
         }
 
-        ui.checkbox(&mut self.buddhabrot, "Buddhabrot")
+        ui.horizontal(|ui| {
-            .on_hover_text(
+            ui.radio_value(&mut self.mode, FractalMode::Mandelbrot, "Set");
-                "Monte-Carlo density of escaping orbits instead of the ordinary \
+            ui.radio_value(&mut self.mode, FractalMode::Julia, "Julia");
+            ui.radio_value(&mut self.mode, FractalMode::Buddhabrot, "Buddhabrot")
+                .on_hover_text(
+                    "Monte-Carlo density of escaping orbits instead of the ordinary \
                  escape-time set. Plain f32 view (no deep zoom); the image \
                  progressively sharpens while the view stays still.",
-            );
+                );
+        });
 
-        if self.buddhabrot {
+        if self.mode == FractalMode::Buddhabrot {
             self.buddhabrot_ui(ui);
+            ui.add_space(4.);
             ui.separator();
+            ui.add_space(4.);
             if ui.button("Reset view").clicked() {
                 self.view = Self::default_view_for(self.mode, self.kind);
             }
@@ -1301,11 +1592,6 @@ impl FractalApp {
             return;
         }
 
-        ui.horizontal(|ui| {
-            ui.radio_value(&mut self.mode, FractalMode::Mandelbrot, "Set");
-            ui.radio_value(&mut self.mode, FractalMode::Julia, "Julia");
-        });
-
         if self.mode == FractalMode::Julia && self.kind != FractalKind::Lambda {
             ui.horizontal(|ui| {
                 ui.label("c =");
@@ -1323,6 +1609,7 @@ impl FractalApp {
             });
 
             if !JULIA_PRESETS[self.kind as usize].is_empty() {
+                ui.label("places:");
                 ui.horizontal_wrapped(|ui| {
                     for &(name, re, im, iterations, phoenix) in JULIA_PRESETS[self.kind as usize] {
                         if ui.small_button(name).clicked() {
@@ -1353,7 +1640,9 @@ impl FractalApp {
             });
         }
 
+        ui.add_space(4.);
         ui.separator();
+        ui.add_space(4.);
         ui.checkbox(&mut self.auto_iterations, "Auto iterations")
             .on_hover_text("Scale the iteration count with zoom depth so deep zooms stay sharp.");
         if self.auto_iterations {
@@ -1407,7 +1696,9 @@ impl FractalApp {
                     });
                 });
         }
+        ui.add_space(4.);
         ui.separator();
+        ui.add_space(4.);
         ui.checkbox(&mut self.antialias, "Antialiasing (2×2)")
             .on_hover_text("Supersample each pixel for smoother edges (~4× slower).");
         if !self.shadow {
@@ -1498,7 +1789,9 @@ impl FractalApp {
             }
         });
 
+        ui.add_space(4.);
         ui.separator();
+        ui.add_space(4.);
         // Editable center coordinates. Shown at full precision; parsed
         // losslessly on commit (Enter or focus loss). While a field is focused
         // we leave the user's text alone; otherwise we refresh it from the live
@@ -1574,7 +1867,9 @@ impl FractalApp {
             ui.colored_label(egui::Color32::LIGHT_YELLOW, "computing reference…");
         }
 
+        ui.add_space(4.);
         ui.separator();
+        ui.add_space(4.);
         let exporting = self.export.is_some();
         ui.horizontal(|ui| {
             if ui.button("Copy link").clicked() {
@@ -1594,10 +1889,11 @@ impl FractalApp {
             ui.add(
                 egui::DragValue::new(&mut self.export_scale)
                     .range(1.0..=16.0)
-                    .speed(0.5),
+                    .speed(0.25)
+                    .custom_formatter(|x, _| format!("x{:.1}", x)),
             );
             ui.label(format!(
-                "→ {}×{}",
+                "= {}×{}",
                 (self.last_size_px.x * self.export_scale) as u32,
                 (self.last_size_px.y * self.export_scale) as u32,
             ));
@@ -1616,7 +1912,9 @@ impl FractalApp {
             ui.small(status);
         }
 
+        ui.add_space(4.);
         ui.separator();
+        ui.add_space(4.);
         if ui.button("Reset view").clicked() {
             self.view = Self::default_view_for(self.mode, self.kind);
         }
@@ -1726,7 +2024,85 @@ impl FractalApp {
             ui.ctx().request_repaint();
         }
 
-        if self.buddhabrot {
+        // Keyboard: arrows pan, z/s zoom in/out, +/- adjust iterations, R
+        // resets the view, H/I toggle the Help/Info windows. Skipped while a
+        // text field (e.g. the center/zoom edit boxes) has focus.
+        if !ui.ctx().egui_wants_keyboard_input() {
+            let dt = ui.input(|i| i.stable_dt as f64).clamp(0.0, 0.1);
+            let (left, right, up, down, zoom_in, zoom_out) = ui.input(|i| {
+                (
+                    i.key_down(egui::Key::ArrowLeft),
+                    i.key_down(egui::Key::ArrowRight),
+                    i.key_down(egui::Key::ArrowUp),
+                    i.key_down(egui::Key::ArrowDown),
+                    i.key_down(egui::Key::Z),
+                    i.key_down(egui::Key::S),
+                )
+            });
+
+            // Pixels/sec pan speed — matches a brisk mouse drag regardless of
+            // frame rate. See `pan_pixels`'s screen-space (+x right, +y down)
+            // convention: Right/Down pan the *camera* right/down, which is
+            // the opposite delta sign from a drag that would show the same
+            // content (a drag grabs the canvas; these keys move the camera).
+            const PAN_SPEED_PX: f64 = 700.0;
+            let mut dx = 0.0;
+            let mut dy = 0.0;
+            if left {
+                dx += PAN_SPEED_PX * dt;
+            }
+            if right {
+                dx -= PAN_SPEED_PX * dt;
+            }
+            if down {
+                dy += PAN_SPEED_PX * dt;
+            }
+            if up {
+                dy -= PAN_SPEED_PX * dt;
+            }
+            if dx != 0.0 || dy != 0.0 {
+                self.view.pan_pixels(dx, dy, height_px);
+                interacted = true;
+            }
+
+            // e-folds/sec, same scale as the auto-zoom animation.
+            const ZOOM_SPEED: f64 = 1.0;
+            if zoom_in != zoom_out {
+                let rate = if zoom_in { ZOOM_SPEED } else { -ZOOM_SPEED };
+                let factor = (-rate * dt).exp();
+                self.view.zoom_at_pixel(0.0, 0.0, height_px, factor);
+                interacted = true;
+            }
+            if left || right || up || down || zoom_in || zoom_out {
+                ui.ctx().request_repaint();
+            }
+
+            if ui.input(|i| i.key_pressed(egui::Key::R)) {
+                self.view = Self::default_view_for(self.mode, self.kind);
+                interacted = true;
+            }
+            if ui.input(|i| i.key_pressed(egui::Key::H)) {
+                self.help_open = !self.help_open;
+            }
+            if ui.input(|i| i.key_pressed(egui::Key::I)) {
+                self.info_open = !self.info_open;
+            }
+            if ui.input(|i| i.key_pressed(egui::Key::A)) {
+                self.antialias = !self.antialias;
+            }
+            if ui.input(|i| i.key_pressed(egui::Key::Plus) || i.key_pressed(egui::Key::Equals)) {
+                self.auto_iterations = false;
+                self.max_iterations = ((self.max_iterations as f64 * 1.25).round() as u32)
+                    .clamp(32, MAX_REF_POINTS as u32 - 1);
+            }
+            if ui.input(|i| i.key_pressed(egui::Key::Minus)) {
+                self.auto_iterations = false;
+                self.max_iterations = ((self.max_iterations as f64 / 1.25).round() as u32)
+                    .clamp(32, MAX_REF_POINTS as u32 - 1);
+            }
+        }
+
+        if self.mode == FractalMode::Buddhabrot {
             // No reference orbit / perturbation machinery: iterate directly in
             // f32 from the live view. Progressive accumulation means this
             // needs its own continuous repaint, separate from the escape-time
@@ -1844,6 +2220,9 @@ impl eframe::App for FractalApp {
         // Floating overlay, always reachable (even when the panel is collapsed):
         // toggle the panel and toggle fullscreen. Essential on a phone.
         self.overlay_buttons(ui);
+        self.info_button(ui);
+        self.info_window(ui.ctx());
+        self.help_window(ui.ctx());
 
         if std::mem::take(&mut self.export_requested) {
             self.do_export(frame);
@@ -1872,7 +2251,7 @@ fn finish_export(shared: &Arc<Mutex<ExportShared>>, result: Result<String, Strin
 }
 
 #[cfg(not(target_arch = "wasm32"))]
-fn unix_timestamp() -> u64 {
+pub(crate) fn unix_timestamp() -> u64 {
     std::time::SystemTime::now()
         .duration_since(std::time::UNIX_EPOCH)
         .map(|d| d.as_secs())

src/cli.rs

@@ -17,6 +17,10 @@ pub struct Cli {
     #[arg(long)]
     pub power: Option<u32>,
 
+    /// Complex exponent for the Complex Multibrot kind (z -> z^power + c).
+    #[arg(long, value_name = "RE,IM")]
+    pub complex_power: Option<String>,
+
     /// Start in Julia mode with this seed constant.
     #[arg(long, value_name = "RE,IM")]
     pub julia: Option<String>,
@@ -45,9 +49,23 @@ pub struct Cli {
     #[arg(long)]
     pub export: bool,
 
-    /// Output path for --export (default: fractal-<timestamp>.png).
+    /// Output path for --export/--headless (default: fractal-<timestamp>.png).
     #[arg(long, value_name = "PATH")]
     pub export_path: Option<String>,
+
+    /// 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. Not yet supported with --buddhabrot.
+    #[arg(long)]
+    pub headless: bool,
+
+    /// Output image width in pixels (--headless only).
+    #[arg(long, value_name = "PX", default_value_t = 1920)]
+    pub width: u32,
+
+    /// Output image height in pixels (--headless only).
+    #[arg(long, value_name = "PX", default_value_t = 1080)]
+    pub height: u32,
 }
 
 #[derive(Copy, Clone, Debug, ValueEnum)]
@@ -65,6 +83,8 @@ pub enum KindArg {
     Buffalo,
     Phoenix,
     Lambda,
+    #[value(alias = "cmulti")]
+    ComplexMultibrot,
 }
 
 impl From<KindArg> for FractalKind {
@@ -79,6 +99,7 @@ impl From<KindArg> for FractalKind {
             KindArg::Buffalo => FractalKind::Buffalo,
             KindArg::Phoenix => FractalKind::Phoenix,
             KindArg::Lambda => FractalKind::Lambda,
+            KindArg::ComplexMultibrot => FractalKind::ComplexMultibrot,
         }
     }
 }

src/fractal/buddhabrot.rs

@@ -46,7 +46,11 @@ pub struct BuddhabrotUniforms {
     /// (yellow core, blue halo), 2 = grayscale. Display-only, like `exposure`
     /// — excluded from `ContentKey` so changing it doesn't reset accumulation.
     pub palette: u32,
-    pub _pad: [u32; 3],
+    /// Padding so `complex_power` (a vec2, 8-byte aligned in the shader)
+    /// starts on an 8-byte boundary.
+    pub _pad0: u32,
+    /// Complex exponent for the Complex Multibrot kind; ignored by other kinds.
+    pub complex_power: [f32; 2],
 }
 
 /// The subset of `BuddhabrotUniforms` that determines the *content* of the
@@ -62,6 +66,7 @@ struct ContentKey {
     bailout_sq: f32,
     kind: u32,
     power: u32,
+    complex_power: [f32; 2],
     r_cap: u32,
     g_cap: u32,
     b_cap: u32,
@@ -78,6 +83,7 @@ impl From<&BuddhabrotUniforms> for ContentKey {
             bailout_sq: u.bailout_sq,
             kind: u.kind,
             power: u.power,
+            complex_power: u.complex_power,
             r_cap: u.r_cap,
             g_cap: u.g_cap,
             b_cap: u.b_cap,

src/fractal/mod.rs

@@ -8,8 +8,9 @@ pub mod share;
 
 pub use buddhabrot::{BuddhabrotCallback, BuddhabrotRenderer, BuddhabrotUniforms};
 pub use reference::{FractalKind, compute_reference, compute_set_reference};
-pub use renderer::{
+#[cfg(target_arch = "wasm32")]
-    ExportRender, FractalCallback, FractalRenderer, MAX_REF_POINTS, Uniforms,
+pub use renderer::encode_png_with_progress;
-    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};
 pub use share::ShareState;

src/fractal/reference.rs

@@ -35,6 +35,30 @@ pub enum FractalKind {
     Phoenix = 7,
     /// `z -> lambda·z(1 - z)` (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)`.
+    ComplexMultibrot = 9,
+}
+
+impl FractalKind {
+    pub fn description(&self) -> &str {
+        match self {
+            FractalKind::Mandelbrot => {
+                "The Mandelbrot set is the most famous fractal set, obtained with the simplest escape-time formula. This set represents all Julia fractals: each points of the Mandelbrot set is related to a specific Julia fractal."
+            }
+            FractalKind::BurningShip => {
+                "A variation of the famous Mandelbrot set, using absolute values on the real and imaginary part of each iterations."
+            }
+            FractalKind::Tricorn => "The Tricorn set is obtained using the same formula as the Mandelbrot set, taking the complex conjugate of the previous iteration.",
+            FractalKind::Multibrot => "Multibrot use the same formula as the Mandelbrot set, with a bigger exposant.",
+            FractalKind::Celtic => "",
+            FractalKind::Perpendicular => "",
+            FractalKind::Buffalo => "",
+            FractalKind::Phoenix => "",
+            FractalKind::Lambda => "",
+            FractalKind::ComplexMultibrot => "Like Multibrot, but the exponent itself is a complex number instead of a plain integer, via z^p = exp(p·ln z).",
+        }
+    }
 }
 
 /// Reference orbit escapes once |Z|^2 exceeds this. Kept larger than the pixel
@@ -57,6 +81,7 @@ pub fn compute_reference(
     power: u32,
     phoenix_p: (f64, f64),
     lambda_l: (f64, f64),
+    complex_power: (f64, f64),
 ) -> Vec<[f32; 2]> {
     let cr = c_re.clone().with_precision(precision).value();
     let ci = c_im.clone().with_precision(precision).value();
@@ -72,6 +97,9 @@ pub fn compute_reference(
     // Lambda distortion constant `l` (a small fixed complex number).
     let lr = big_from_f64(lambda_l.0, precision);
     let li = big_from_f64(lambda_l.1, precision);
+    // Complex Multibrot exponent (a fixed complex number).
+    let cpow_re = big_from_f64(complex_power.0, precision);
+    let cpow_im = big_from_f64(complex_power.1, precision);
 
     let mut points: Vec<[f32; 2]> = Vec::with_capacity(max_iter as usize + 1);
 
@@ -142,6 +170,10 @@ pub fn compute_reference(
                 let lzi = &lr * &zi + &li * &zr;
                 (&lzr * &re2 - &lzi * &im2, re2 * lzi + lzr * im2)
             }
+            FractalKind::ComplexMultibrot => {
+                let (pr, pi) = complex_pow_complex(&zr, &zi, &cpow_re, &cpow_im, precision);
+                (pr + &cr, pi + &ci)
+            }
         };
 
         // Shift the previous iterate (only the Phoenix arm reads it).
@@ -178,6 +210,32 @@ fn complex_pow(zr: &Big, zi: &Big, power: u32, precision: usize) -> (Big, Big) {
     (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`.
+/// `z = 0` is special-cased to `0` (the formula's `ln(0)` would otherwise
+/// 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) {
+        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_im = (pr * &theta + pi * &ln_r).with_precision(precision).value();
+    let mag = exp_re.exp();
+    let (sin_a, cos_a) = exp_im.sin_cos();
+    (&mag * &cos_a, &mag * &sin_a)
+}
+
 /// Convenience: parameter-plane ("Mandelbrot-set") reference (`z0 = 0`,
 /// `c = center`) for any `kind`.
 #[allow(clippy::too_many_arguments)]
@@ -190,10 +248,21 @@ pub fn compute_set_reference(
     power: u32,
     phoenix_p: (f64, f64),
     lambda_l: (f64, f64),
+    complex_power: (f64, f64),
 ) -> Vec<[f32; 2]> {
     let zero = big_zero(precision);
     compute_reference(
-        &zero, &zero, center_re, center_im, max_iter, precision, kind, power, phoenix_p, lambda_l,
+        &zero,
+        &zero,
+        center_re,
+        center_im,
+        max_iter,
+        precision,
+        kind,
+        power,
+        phoenix_p,
+        lambda_l,
+        complex_power,
     )
 }
 
@@ -216,6 +285,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         // Independent naive f64 orbit.
@@ -255,6 +325,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
         assert_eq!(points.len(), 501, "interior orbit should not escape");
     }
@@ -273,6 +344,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-1.75_f64, -0.03_f64);
@@ -302,6 +374,7 @@ mod tests {
             3,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (0.3_f64, 0.2_f64);
@@ -337,6 +410,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (mut zr, mut zi) = (0.15_f64, -0.1_f64);
@@ -366,6 +440,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-0.6_f64, 0.4_f64);
@@ -396,6 +471,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-0.7_f64, -0.2_f64);
@@ -426,6 +502,7 @@ mod tests {
             2,
             (0.0, 0.0),
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (c_re, c_im) = (-1.2_f64, -0.35_f64);
@@ -448,8 +525,17 @@ mod tests {
         let cr = Big::try_from(0.5667_f64).unwrap();
         let ci = Big::try_from(0.0_f64).unwrap();
         let p = (-0.5_f64, 0.0_f64);
-        let points =
+        let points = compute_set_reference(
-            compute_set_reference(&cr, &ci, 60, 200, FractalKind::Phoenix, 2, p, (0.0, 0.0));
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Phoenix,
+            2,
+            p,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         let (c_re, c_im) = (0.5667_f64, 0.0_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
@@ -469,4 +555,50 @@ mod tests {
             zi = nzi;
         }
     }
+
+    /// Complex Multibrot (power 2.5 + 0.3i) reference matches a naive f64
+    /// 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 ci = Big::try_from(-0.2_f64).unwrap();
+        let power = (2.5_f64, 0.3_f64);
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::ComplexMultibrot,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+            power,
+        );
+
+        // Naive f64 complex power via z^p = exp(p * ln z), ln z = ln|z| + i*arg(z).
+        fn naive_cpow(zr: f64, zi: f64, pr: f64, pi: f64) -> (f64, f64) {
+            if zr == 0.0 && zi == 0.0 {
+                return (0.0, 0.0);
+            }
+            let ln_r = 0.5 * (zr * zr + zi * zi).ln();
+            let theta = zi.atan2(zr);
+            let exp_re = pr * ln_r - pi * theta;
+            let exp_im = pr * theta + pi * ln_r;
+            let mag = exp_re.exp();
+            (mag * exp_im.cos(), mag * exp_im.sin())
+        }
+
+        let (c_re, c_im) = (0.1_f64, -0.2_f64);
+        let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
+        for point in &points {
+            let tol = 1e-4 * (1.0 + zr.abs().max(zi.abs()));
+            assert!((point[0] as f64 - zr).abs() < tol, "re: {point:?} vs {zr}");
+            assert!((point[1] as f64 - zi).abs() < tol, "im: {point:?} vs {zi}");
+            let (pr, pi) = naive_cpow(zr, zi, power.0, power.1);
+            let nzr = pr + c_re;
+            let nzi = pi + c_im;
+            zr = nzr;
+            zi = nzi;
+        }
+    }
 }

src/fractal/renderer.rs

@@ -37,6 +37,7 @@ fn geom_differs(a: &Uniforms, b: &Uniforms) -> bool {
         || a.aa_level != b.aa_level
         || a.kind != b.kind
         || a.power != b.power
+        || a.complex_power != b.complex_power
         || a.dc_offset != b.dc_offset
         || a.phoenix_p != b.phoenix_p
         || a.de_coloring != b.de_coloring
@@ -87,6 +88,9 @@ pub struct Uniforms {
     /// Distortion constant `l` for the Lambda map (`l·z(1 - z)`);
     /// ignored by other kinds.
     pub lambda_l: [f32; 2],
+    /// Complex exponent for the Complex Multibrot kind (`z^power + c`);
+    /// ignored by other kinds.
+    pub complex_power: [f32; 2],
     /// 0 = escape-time coloring, 1 = distance-estimation shading.
     pub de_coloring: u32,
     // 0 = classic colors, 1 = shadows
@@ -757,6 +761,59 @@ impl ExportRender {
     }
 }
 
+/// Render `er` tile by tile (blocking on the GPU after each tile so progress
+/// reflects real work), read it back, and encode the result as PNG bytes.
+/// Blocks the calling thread throughout, so it's only for native targets:
+/// the UI export path runs it on a background thread, headless rendering
+/// runs it directly since it has no frame loop to share a thread with.
+#[cfg(not(target_arch = "wasm32"))]
+pub fn export_to_png_blocking(
+    device: &wgpu::Device,
+    queue: &wgpu::Queue,
+    er: &ExportRender,
+    mut on_progress: impl FnMut(&'static str, f32),
+) -> Vec<u8> {
+    // Progress budget: rendering fills [0, RENDER_END], encoding the rest.
+    const RENDER_END: f32 = 0.6;
+
+    for t in 0..er.tiles {
+        er.render_tile(device, queue, t);
+        let _ = device.poll(wgpu::PollType::Wait {
+            submission_index: None,
+            timeout: None,
+        });
+        let done = (t + 1) as f32 / er.tiles as f32;
+        on_progress("Rendering", RENDER_END * done);
+    }
+    er.copy_to_readback(device, queue);
+
+    let (tx, rx) = std::sync::mpsc::channel();
+    er.readback()
+        .slice(..)
+        .map_async(wgpu::MapMode::Read, move |res| {
+            let _ = tx.send(res);
+        });
+    let _ = device.poll(wgpu::PollType::Wait {
+        submission_index: None,
+        timeout: None,
+    });
+    let _ = rx.recv();
+
+    on_progress("Encoding", RENDER_END);
+    let png = {
+        let data = er
+            .readback()
+            .slice(..)
+            .get_mapped_range()
+            .expect("map readback buffer");
+        encode_png_with_progress(&data, er.width, er.height, er.padded_bpr, er.swap_rb, |f| {
+            on_progress("Encoding", RENDER_END + (0.97 - RENDER_END) * f)
+        })
+    };
+    er.readback().unmap();
+    png
+}
+
 /// Convert a padded BGRA/RGBA readback into tightly-packed RGBA8 and encode it
 /// as PNG bytes, reporting progress in `[0, 1]` via `on_progress` as rows are
 /// streamed to the compressor (encoding is the slow, subdividable phase).

src/fractal/share.rs

@@ -25,6 +25,8 @@ pub struct ShareState {
     pub phoenix_p: (f64, f64),
     /// Distortion constant for the Lambda kind (ignored by others).
     pub lambda_l: (f64, f64),
+    /// Complex exponent for the Complex Multibrot kind (ignored by others).
+    pub complex_power: (f64, f64),
     pub color_scale: f32,
     pub color_offset: f32,
     /// Palette index (`palette_id` in the shader).
@@ -49,6 +51,7 @@ impl ShareState {
                 FractalKind::Buffalo => "buffalo",
                 FractalKind::Phoenix => "phoenix",
                 FractalKind::Lambda => "lambda",
+                FractalKind::ComplexMultibrot => "cmulti",
             }
         ));
         s.push_str(&format!("&pw={}", self.power));
@@ -60,8 +63,12 @@ impl ShareState {
         s.push_str(&format!("&px={}&py={}", self.phoenix_p.0, self.phoenix_p.1));
         s.push_str(&format!("&lx={}&ly={}", self.lambda_l.0, self.lambda_l.1));
         s.push_str(&format!(
-            "&cs={}&co={}&pal={}",
+            "&cpr={}&cpi={}",
-            self.color_scale, self.color_offset, self.palette
+            self.complex_power.0, self.complex_power.1
+        ));
+        s.push_str(&format!(
+            "&cs={}&co={}&pal={}&spal={}",
+            self.color_scale, self.color_offset, self.palette, self.shadow_palette
         ));
         s
     }
@@ -89,6 +96,7 @@ impl ShareState {
                     "buffalo" => FractalKind::Buffalo,
                     "phoenix" => FractalKind::Phoenix,
                     "lambda" => FractalKind::Lambda,
+                    "cmulti" => FractalKind::ComplexMultibrot,
                     _ => FractalKind::Mandelbrot,
                 })
                 .unwrap_or(FractalKind::Mandelbrot),
@@ -109,6 +117,10 @@ impl ShareState {
                 map.get("lx").and_then(|s| s.parse().ok()).unwrap_or(-0.5),
                 map.get("ly").and_then(|s| s.parse().ok()).unwrap_or(0.0),
             ),
+            complex_power: (
+                map.get("cpr").and_then(|s| s.parse().ok()).unwrap_or(2.0),
+                map.get("cpi").and_then(|s| s.parse().ok()).unwrap_or(0.0),
+            ),
             color_scale: map.get("cs").and_then(|s| s.parse().ok()).unwrap_or(0.02),
             color_offset: map.get("co").and_then(|s| s.parse().ok()).unwrap_or(0.0),
             palette: map.get("pal").and_then(|s| s.parse().ok()).unwrap_or(0),
@@ -134,6 +146,7 @@ mod tests {
             julia_c: (-0.123, 0.745),
             phoenix_p: (-0.5, 0.1),
             lambda_l: (-0.5, 0.0),
+            complex_power: (2.5, 0.3),
             color_scale: 0.02,
             color_offset: 0.25,
             palette: 3,
@@ -149,6 +162,7 @@ mod tests {
         assert_eq!(d.iterations, s.iterations);
         assert_eq!(d.julia_c, s.julia_c);
         assert_eq!(d.phoenix_p, s.phoenix_p);
+        assert_eq!(d.complex_power, s.complex_power);
         assert_eq!(d.palette, s.palette);
         assert_eq!(d.shadow_palette, s.shadow_palette);
     }

src/headless.rs

@@ -0,0 +1,82 @@
+// Headless PNG rendering: parse the CLI, build the exact same view/state the
+// windowed app would from it, then render straight to a file. No window, no
+// 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.
+
+use eframe::egui_wgpu::wgpu;
+
+use crate::app::{FractalApp, unix_timestamp};
+use crate::cli::Cli;
+use crate::fractal::{ExportRender, FractalRenderer, export_to_png_blocking};
+
+/// Cap on the output image dimension (px), to stay within GPU texture limits.
+const MAX_DIM: u32 = 8192 * 16;
+
+pub fn run(cli: Cli) -> Result<(), String> {
+    if cli.buddhabrot {
+        return Err("headless mode doesn't support --buddhabrot yet".into());
+    }
+
+    let width = cli.width.clamp(16, MAX_DIM);
+    let height = cli.height.clamp(16, MAX_DIM);
+    let export_path = cli
+        .export_path
+        .clone()
+        .unwrap_or_else(|| format!("fractal-{}.png", unix_timestamp()));
+
+    let mut app = FractalApp::default_state();
+    app.apply_cli(cli);
+
+    eprintln!("computing reference orbit…");
+    app.compute_reference_blocking();
+
+    let (device, queue) = pollster::block_on(request_device())?;
+    let format = wgpu::TextureFormat::Rgba8UnormSrgb;
+    let renderer = FractalRenderer::new(&device, format);
+    let (pipeline, bind_group_layout, format) = renderer.export_handles();
+
+    let uniforms = app.make_uniforms(width as f64 / height as f64);
+    let er = ExportRender::new(
+        &device,
+        &queue,
+        pipeline,
+        &bind_group_layout,
+        format,
+        width,
+        height,
+        uniforms,
+        app.reference_points(),
+    );
+
+    eprintln!("rendering {width}×{height}…");
+    let png = export_to_png_blocking(&device, &queue, &er, |phase, fraction| {
+        eprint!("\r{phase} {:>3.0}%", fraction * 100.0);
+    });
+    eprintln!();
+
+    std::fs::write(&export_path, &png).map_err(|e| format!("save failed: {e}"))?;
+    println!("saved {export_path} ({width}×{height})");
+    Ok(())
+}
+
+/// Set up a wgpu device with no surface/window attached, matching the limits
+/// `main::wgpu_options` requests for the windowed app (the fractal fragment
+/// shader needs storage buffers, which downlevel/WebGL-style limits disallow).
+async fn request_device() -> Result<(wgpu::Device, wgpu::Queue), String> {
+    let instance = wgpu::Instance::default();
+    let adapter = instance
+        .request_adapter(&wgpu::RequestAdapterOptions::default())
+        .await
+        .map_err(|e| format!("no compatible GPU adapter: {e}"))?;
+    adapter
+        .request_device(&wgpu::DeviceDescriptor {
+            label: Some("headless fractal device"),
+            required_features: wgpu::Features::empty(),
+            required_limits: adapter.limits(),
+            ..Default::default()
+        })
+        .await
+        .map_err(|e| format!("failed to create device: {e}"))
+}

src/main.rs

@@ -16,6 +16,8 @@ mod view;
 #[cfg(not(target_arch = "wasm32"))]
 mod cli;
 #[cfg(not(target_arch = "wasm32"))]
+mod headless;
+#[cfg(not(target_arch = "wasm32"))]
 mod worker;
 
 use app::FractalApp;
@@ -49,11 +51,24 @@ fn wgpu_options() -> eframe::egui_wgpu::WgpuConfiguration {
 
 #[cfg(not(target_arch = "wasm32"))]
 fn main() -> eframe::Result {
+    use clap::Parser as _;
+
     env_logger::builder()
         .filter_level(log::LevelFilter::Info)
         .parse_default_env()
         .init();
 
+    let cli = cli::Cli::parse();
+    if cli.headless {
+        return match headless::run(cli) {
+            Ok(()) => Ok(()),
+            Err(e) => {
+                eprintln!("error: {e}");
+                std::process::exit(1);
+            }
+        };
+    }
+
     let native_options = eframe::NativeOptions {
         renderer: eframe::Renderer::Wgpu,
         wgpu_options: wgpu_options(),

src/shaders/buddhabrot.wgsl

@@ -47,14 +47,15 @@ struct Uniforms {
     // Tonemap colour style: 0 = classic (R/G/B = raw caps), 1 = nebula
     // (yellow core, blue halo), 2 = grayscale.
     palette: u32,
-    // Padding to a 16-byte multiple. NOT vec3<u32> — that type aligns to 16
+    // Padding so `complex_power` (a vec2, 8-byte aligned) starts on an
-    // bytes in WGSL (unlike Rust's `[u32; 3]`, which aligns to 4), which
+    // 8-byte boundary. NOT vec3<u32> — that type aligns to 16 bytes in WGSL
-    // silently added 32 bytes instead of 16 and mismatched the Rust struct's
+    // (unlike Rust's `[u32; 3]`, which aligns to 4), which silently added 32
-    // size (a wgpu validation error at dispatch time: "size 96 where the
+    // bytes instead of 16 and mismatched the Rust struct's size (a wgpu
-    // shader expects 112").
+    // validation error at dispatch time: "size 96 where the shader expects
+    // 112").
     _pad0: u32,
-    _pad1: u32,
+    // Complex exponent for the Complex Multibrot kind; unused by other kinds.
-    _pad2: u32,
+    complex_power: vec2<f32>,
 };
 
 const PALETTE_NEBULA: u32 = 0u;
@@ -70,6 +71,7 @@ const KIND_PERPENDICULAR: u32 = 5u;
 const KIND_BUFFALO: u32 = 6u;
 const KIND_PHOENIX: u32 = 7u;
 const KIND_LAMBDA: u32 = 8u;
+const KIND_COMPLEX_MULTIBROT: u32 = 9u;
 
 @group(0) @binding(0) var<uniform> u: Uniforms;
 // Compute pass: read-write atomic histogram (3 planes of width*height, R/G/B).
@@ -103,6 +105,21 @@ fn complex_pow(z: vec2<f32>, p: u32) -> vec2<f32> {
     return r;
 }
 
+// z^p for a complex exponent p, via the principal branch z^p = exp(p * ln z),
+// ln z = ln|z| + i*arg(z). z = 0 maps to 0 (the correct limit for the
+// Re(p) > 0 region the UI exposes; ln(0) would otherwise be -inf).
+fn cpow(z: vec2<f32>, p: vec2<f32>) -> vec2<f32> {
+    let r2 = dot(z, z);
+    if r2 < 1e-30 {
+        return vec2<f32>(0.0, 0.0);
+    }
+    let ln_r = 0.5 * log(r2);
+    let theta = atan2(z.y, z.x);
+    let mag = exp(p.x * ln_r - p.y * theta);
+    let ang = p.x * theta + p.y * ln_r;
+    return mag * vec2<f32>(cos(ang), sin(ang));
+}
+
 // One iteration step z_n -> z_{n+1} for the current kind. `zp` is the
 // previous iterate (z_{n-1}), used only by the Phoenix two-term recurrence.
 // Must match `FractalKind` in reference.rs (the direct, non-perturbative form
@@ -126,6 +143,8 @@ fn advance(z: vec2<f32>, zp: vec2<f32>, c: vec2<f32>) -> vec2<f32> {
     } else if u.kind == KIND_LAMBDA {
         // l * z * (1 - z); c is unused (see file doc comment above).
         return cmul(u.lambda_l, cmul(z, vec2<f32>(1.0 - z.x, -z.y)));
+    } else if u.kind == KIND_COMPLEX_MULTIBROT {
+        return cpow(z, u.complex_power) + c;
     }
     return vec2<f32>(z.x * z.x - z.y * z.y, 2.0 * z.x * z.y) + c; // Mandelbrot
 }

src/shaders/colorize.wgsl

@@ -27,6 +27,7 @@ struct Uniforms {
     dc_offset: vec2<f32>,
     phoenix_p: vec2<f32>,
     lambda_l: vec2<f32>,
+    complex_power: vec2<f32>,
     de_coloring: u32,
     shadow: u32,
 };
@@ -159,7 +160,7 @@ fn fs_main(@builtin(position) pos: vec4<f32>) -> @location(0) vec4<f32> {
         let interior_frac = d.b;
 
         let t = fract(ci * u.color_scale + u.color_offset);
-        var col = palette(u.palette_id, t) * de;
+        var col = palette(u.palette_id, t) * sqrt(de);
         // Anti-alias the set boundary: fade toward black by the fraction of the
         // pixel's sub-samples that landed in the interior.
         col = col * (1.0 - interior_frac);

src/shaders/mandelbrot.wgsl

@@ -34,6 +34,9 @@ struct Uniforms {
     // Distortion constant l for the Lambda map (l*z(1 - z_{n-1})); unused
     // by other kinds.
     lambda_l: vec2<f32>,
+    // Complex exponent for the Complex Multibrot kind (z^power + c); unused
+    // by other kinds.
+    complex_power: vec2<f32>,
     // 0 = escape-time coloring, 1 = distance-estimation shading.
     de_coloring: u32,
     // 0 = classic colors, 1 = shadows
@@ -49,6 +52,7 @@ const KIND_PERPENDICULAR: u32 = 5u;
 const KIND_BUFFALO: u32 = 6u;
 const KIND_PHOENIX: u32 = 7u;
 const KIND_LAMBDA: u32 = 8u;
+const KIND_COMPLEX_MULTIBROT: u32 = 9u;
 
 @group(0) @binding(0) var<uniform> u: Uniforms;
 @group(0) @binding(1) var<storage, read> ref_orbit: array<vec2<f32>>;
@@ -84,6 +88,27 @@ fn conj(a: vec2<f32>) -> vec2<f32> {
     return vec2<f32>(a.x, -a.y);
 }
 
+// Complex division a / b.
+fn cdiv(a: vec2<f32>, b: vec2<f32>) -> vec2<f32> {
+    let d = dot(b, b);
+    return vec2<f32>(a.x * b.x + a.y * b.y, a.y * b.x - a.x * b.y) / d;
+}
+
+// z^p for a complex exponent p, via the principal branch z^p = exp(p * ln z),
+// ln z = ln|z| + i*arg(z). z = 0 maps to 0 (the correct limit for the
+// Re(p) > 0 region the UI exposes; ln(0) would otherwise be -inf).
+fn cpow(z: vec2<f32>, p: vec2<f32>) -> vec2<f32> {
+    let r2 = dot(z, z);
+    if r2 < 1e-30 {
+        return vec2<f32>(0.0, 0.0);
+    }
+    let ln_r = 0.5 * log(r2);
+    let theta = atan2(z.y, z.x);
+    let mag = exp(p.x * ln_r - p.y * theta);
+    let ang = p.x * theta + p.y * ln_r;
+    return mag * vec2<f32>(cos(ang), sin(ang));
+}
+
 // |c + d| - |c|, evaluated exactly (no catastrophic cancellation even when the
 // sum crosses zero). This is what makes the Burning Ship delta correct through
 // the sign flips that happen all along the axes, where the ship's detail lives.
@@ -123,6 +148,38 @@ fn multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: u32) -> vec2<f32> {
     return acc;
 }
 
+// Number of terms kept in `complex_multibrot_delta`'s series. Truncation, not
+// exactness: unlike `multibrot_delta` (a finite binomial sum for an integer
+// power), a complex power has no finite expansion, so this converges rather
+// than terminates. Fine as long as perturbation's usual invariant (|e| << |z|,
+// kept true by rebasing) holds, since each extra term is O(w^k) smaller.
+const COMPLEX_MULTIBROT_TERMS: u32 = 16u;
+
+// Perturbation delta for z -> z^p with a complex p: (Z+e)^p - Z^p.
+// = Z^p * ((1+w)^p - 1), w = e/Z, expanded as a Taylor series in w (never
+// forming 1+w, which would round tiny w away in f32 — the same reason
+// `multibrot_delta` never forms Z+e directly). Series: (1+w)^p - 1 =
+// sum_{k=1}^N C(p,k) w^k, with the complex binomial coefficient built up
+// incrementally: C(p,k) = C(p,k-1) * (p-(k-1)) / k.
+//
+// Z ~ 0 (the reference start, X_0 = 0 for Mandelbrot) makes w singular; there
+// (0+e)^p - 0^p = e^p exactly, so that case is handled directly via `cpow`.
+fn complex_multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: vec2<f32>) -> vec2<f32> {
+    if dot(z, z) < 1e-20 {
+        return cpow(e, p);
+    }
+    let w = cdiv(e, z);
+    var wk = vec2<f32>(1.0, 0.0); // w^0
+    var coef = vec2<f32>(1.0, 0.0); // C(p,0)
+    var acc = vec2<f32>(0.0, 0.0);
+    for (var k: u32 = 1u; k <= COMPLEX_MULTIBROT_TERMS; k = k + 1u) {
+        coef = cdiv(cmul(coef, p - vec2<f32>(f32(k - 1u), 0.0)), vec2<f32>(f32(k), 0.0));
+        wk = cmul(wk, w);
+        acc = acc + cmul(coef, wk);
+    }
+    return cmul(cpow(z, p), acc);
+}
+
 // One perturbation step of the current fractal's delta: e -> f(Z+e) - f(Z),
 // where `z` is the reference orbit value X_m. `step_add` (dc) is added by the
 // caller. Must match `FractalKind` on the CPU side.
@@ -163,6 +220,8 @@ fn advance_delta(z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
         // Lambda map: z^{n+1} = λ·z·(1-z). Delta: e = λ·e·(1-2z-e).
         let one_minus_2z_minus_e = vec2<f32>(1.0 - 2.0 * z.x - e.x, -2.0 * z.y - e.y);
         return cmul(u.lambda_l, cmul(e, one_minus_2z_minus_e));
+    } else if u.kind == KIND_COMPLEX_MULTIBROT {
+        return complex_multibrot_delta(z, e, u.complex_power);
     }
     return 2.0 * cmul(z, e) + cmul(e, e); // Mandelbrot (and Phoenix square part)
 }
@@ -183,6 +242,9 @@ fn fprime(z: vec2<f32>) -> vec2<f32> {
     } else if u.kind == KIND_LAMBDA {
         // Lambda: f'(z) = λ·(1-2z).
         return cmul(u.lambda_l, vec2<f32>(1.0 - 2.0 * z.x, -2.0 * z.y));
+    } else if u.kind == KIND_COMPLEX_MULTIBROT {
+        // f'(z) = p * z^(p-1).
+        return cmul(u.complex_power, cpow(z, u.complex_power - vec2<f32>(1.0, 0.0)));
     }
     return 2.0 * z;
 }

src/worker.rs

@@ -26,6 +26,8 @@ pub struct RefRequest {
     pub phoenix_p: (f64, f64),
     /// Distortion constant for the Lambda map (ignored by other kinds).
     pub lambda_l: (f64, f64),
+    /// Complex exponent for the Complex Multibrot kind (ignored by other kinds).
+    pub complex_power: (f64, f64),
 }
 
 pub struct RefResult {
@@ -107,6 +109,7 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.power,
             req.phoenix_p,
             req.lambda_l,
+            req.complex_power,
         )
     } else {
         compute_set_reference(
@@ -118,6 +121,7 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.power,
             req.phoenix_p,
             req.lambda_l,
+            req.complex_power,
         )
     }
 }