fix: remove zoom limits

CLAUDE.md

@@ -11,8 +11,9 @@ and every pixel is rendered on the GPU as a cheap `f32` delta from it, with
 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), reaching ~10³⁰⁰× (the `f64` limit of
-`half_height`, `view::MIN_HALF_HEIGHT`). Runs
+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).
 
@@ -94,8 +95,13 @@ 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`
-  type alias), pixel scale stays `f64` (so zoom is clamped at
-  `MIN_HALF_HEIGHT` = 1e-300). Precision (bits) scales with zoom depth
+  type alias). 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

src/app.rs

@@ -19,7 +19,7 @@ use crate::lights::{Light, gpu_lights};
 use crate::view::parse_half_height_spec;
 use crate::view::parse_re_im_spec;
 use crate::view::{
-    Big, DEFAULT_HALF_HEIGHT, MIN_HALF_HEIGHT, ViewState, big_from_decimal_str, big_from_f64,
+    Big, DEFAULT_HALF_HEIGHT, MAX_PRECISION_BITS, Scale, ViewState, big_from_decimal_str, big_from_f64,
     big_to_decimal_str, deep_scale_exp, interpolate_view, needs_deep, parse_view_spec,
     precision_for,
 };
@@ -206,7 +206,7 @@ impl RefJob {
 struct RequestKey {
     center_re: Big,
     center_im: Big,
-    half_height: f64,
+    half_height: Scale,
     julia: bool,
     julia_c: (f64, f64),
     phoenix_p: (f64, f64),
@@ -556,7 +556,7 @@ pub struct FractalApp {
     /// slightly from the live view; the shader compensates via `dc_offset`).
     ref_center_re: Big,
     ref_center_im: Big,
-    ref_half_height: f64,
+    ref_half_height: Scale,
     /// Kind and kind-switch morph the current `reference` was computed with.
     /// The shader iterates with these (not the live kind/morph) so its delta
     /// formula always matches the orbit, even while the worker lags a frame
@@ -614,8 +614,8 @@ fn sig_digits_for(bits: usize) -> usize {
 }
 
 /// Format a magnification for the editable field (compact scientific).
-fn format_zoom(m: f64) -> String {
-    format!("{m:.4e}")
+fn format_zoom(m: Scale) -> String {
+    format!("{m:.4}")
 }
 
 /// Precision (bits) to parse a typed center at: at least what the current zoom
@@ -624,7 +624,7 @@ fn format_zoom(m: f64) -> String {
 fn parse_bits_for(s: &str, min_bits: usize) -> usize {
     let digits = s.chars().filter(char::is_ascii_digit).count();
     let from_input = (digits as f64 * std::f64::consts::LOG2_10).ceil() as usize + 16;
-    min_bits.max(from_input).min(2048)
+    min_bits.max(from_input).min(MAX_PRECISION_BITS)
 }
 
 impl FractalApp {
@@ -746,7 +746,7 @@ impl FractalApp {
         if let Some(k) = cli.kind {
             self.kind = k.into();
             if let Some(p) = cli.power {
-                self.power = p.clamp(2, 8);
+                self.power = p.clamp(2, 20);
             }
             self.view = Self::default_view_for(self.mode, self.kind);
         }
@@ -979,6 +979,7 @@ impl FractalApp {
     /// Jump to a preset Mandelbrot location: decimal center (parsed at the
     /// precision the zoom needs), half-height, and a fitting iteration count.
     fn go_to_place(&mut self, re: &str, im: &str, half_height: f64, iterations: u32) {
+        let half_height = Scale::from_f64(half_height);
         let bits = precision_for(half_height);
         if let (Some(cre), Some(cim)) = (
             big_from_decimal_str(re, bits),
@@ -997,7 +998,7 @@ impl FractalApp {
     /// `auto_iterations` is on. Grows roughly linearly with zoom decades so deep
     /// zooms keep enough iterations to stay sharp instead of banding.
     fn auto_iteration_count(&self) -> u32 {
-        let decades = self.view.magnification().log10().max(0.0);
+        let decades = self.view.magnification_log10().max(0.0);
         let iters = 400.0 + 900.0 * decades;
         (iters.round() as u32).clamp(200, MAX_REF_POINTS as u32 - 1)
     }
@@ -1033,7 +1034,7 @@ impl FractalApp {
             FractalMode::Mandelbrot
         };
         self.kind = s.kind;
-        self.power = s.power.clamp(2, 8);
+        self.power = s.power.clamp(2, 200);
         self.julia_c = s.julia_c;
         self.phoenix_p = s.phoenix_p;
         self.lambda_l = s.lambda_l;
@@ -1077,10 +1078,10 @@ impl FractalApp {
     /// each kind's interesting region.
     fn default_view_for(mode: FractalMode, kind: FractalKind) -> ViewState {
         if mode == FractalMode::Julia {
-            return ViewState::with_center(big_from_f64(0.0, 53), big_from_f64(0.0, 53), 1.5);
+            return ViewState::with_center(big_from_f64(0.0, 53), big_from_f64(0.0, 53), Scale::from_f64(1.5));
         }
         let (cr, ci, hh) = kind.default_set_view();
-        ViewState::with_center(big_from_f64(cr, 53), big_from_f64(ci, 53), hh)
+        ViewState::with_center(big_from_f64(cr, 53), big_from_f64(ci, 53), Scale::from_f64(hh))
     }
 
     /// The request key for the current state. Its `iter` is the reference
@@ -1104,10 +1105,14 @@ impl FractalApp {
     }
 
     /// Distance (complex units) the live view center has drifted from `key`.
+    /// Distance of the live center from `key`'s, in units of the live
+    /// half-height (measured at that scale, so it works past f64's range).
     fn drift_from(&self, key: &RequestKey) -> f64 {
-        let dre = (&self.view.center_re - &key.center_re).to_f64().value();
-        let dim = (&self.view.center_im - &key.center_im).to_f64().value();
-        (dre * dre + dim * dim).sqrt()
+        let hh = self.view.half_height;
+        let k = -hh.exponent() as isize;
+        let dre = ((&self.view.center_re - &key.center_re) << k).to_f64().value();
+        let dim = ((&self.view.center_im - &key.center_im) << k).to_f64().value();
+        (dre * dre + dim * dim).sqrt() / hh.scaled_f64(-hh.exponent())
     }
 
     /// Whether the reference should be (re)computed: parameters changed, or the
@@ -1143,19 +1148,22 @@ impl FractalApp {
             && self.morph.is_none()
         {
             // But still recompute on significant zoom changes for precision
-            let ratio = self.view.half_height / key.half_height;
+            let ratio = self.view.half_height.ratio(key.half_height);
             return !(0.5..=2.0).contains(&ratio);
         }
-        let ratio = self.view.half_height / key.half_height;
-        self.drift_from(key) > 0.5 * self.view.half_height || !(0.5..=2.0).contains(&ratio)
+        let ratio = self.view.half_height.ratio(key.half_height);
+        self.drift_from(key) > 0.5 || !(0.5..=2.0).contains(&ratio)
     }
 
-    /// Complex offset of the live view center from the reference center.
-    fn dc_offset(&self) -> (f64, f64) {
-        let dre = (&self.view.center_re - &self.ref_center_re)
+    /// Complex offset of the live view center from the reference center, in
+    /// units of `2^scale_exp` (shifted exactly in `Big`, so it doesn't
+    /// underflow f64 at deep zooms).
+    fn dc_offset(&self, scale_exp: i32) -> (f64, f64) {
+        let k = -scale_exp as isize;
+        let dre = ((&self.view.center_re - &self.ref_center_re) << k)
             .to_f64()
             .value();
-        let dim = (&self.view.center_im - &self.ref_center_im)
+        let dim = ((&self.view.center_im - &self.ref_center_im) << k)
             .to_f64()
             .value();
         (dre, dim)
@@ -1178,7 +1186,7 @@ impl FractalApp {
         points: RefOrbit,
         cre: Big,
         cim: Big,
-        hh: f64,
+        hh: Scale,
         kind: FractalKind,
         morph: Option<(FractalKind, f32)>,
     ) {
@@ -1404,10 +1412,12 @@ impl FractalApp {
         let mode = self.effective_rendering_mode();
         // Deep views upload the geometry pre-multiplied by 2^-E (exact).
         let scale_exp = self.scale_exp(height_px);
-        let inv_scale = 2f64.powi(-scale_exp);
-        let (dc_re, dc_im) = self.dc_offset();
+        let (dc_re, dc_im) = self.dc_offset(scale_exp);
         Uniforms {
-            span: [(span_x * inv_scale) as f32, (span_y * inv_scale) as f32],
+            span: [
+                span_x.scaled_f64(-scale_exp) as f32,
+                span_y.scaled_f64(-scale_exp) as f32,
+            ],
             max_iter: self.max_iterations.min(MAX_REF_POINTS as u32 - 1),
             ref_len: self.reference.len() as u32,
             color_offset: self.color_offset,
@@ -1420,7 +1430,7 @@ impl FractalApp {
             kind: self.ref_kind.unwrap_or(self.kind) as u32,
             power: self.power,
             morph_from: self.ref_morph.map_or(0, |(k, _)| k as u32),
-            dc_offset: [(dc_re * inv_scale) as f32, (dc_im * inv_scale) as f32],
+            dc_offset: [dc_re as f32, dc_im 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],
@@ -1450,7 +1460,7 @@ impl FractalApp {
         ];
         BuddhabrotUniforms {
             center,
-            half_height: self.view.half_height as f32,
+            half_height: self.view.half_height.to_f64() as f32,
             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],
@@ -2016,11 +2026,10 @@ impl FractalApp {
             }
         }
         if self.anim.zoom && self.anim.zoom_speed != 0.0 {
-            let min_hh = DEFAULT_HALF_HEIGHT * 1.0e-26; // practical f32-perturbation depth
-            let max_hh = DEFAULT_HALF_HEIGHT * 4.0;
+            let max_hh = Scale::from_f64(DEFAULT_HALF_HEIGHT * 4.0);
             let factor = (-(self.anim.zoom_speed as f64) * dt).exp();
-            let target = (self.view.half_height * factor).clamp(min_hh, max_hh);
-            let f = target / self.view.half_height;
+            let target = self.view.half_height.mul_f64(factor).clamp(Scale::MIN, max_hh);
+            let f = target.ratio(self.view.half_height);
             if (f - 1.0).abs() > 1.0e-9 {
                 self.view
                     .zoom_at_pixel(0.0, 0.0, self.last_size_px.y.max(1.0) as f64, f);
@@ -2453,11 +2462,9 @@ impl FractalApp {
         }
         if zoom_resp.lost_focus() {
             if self.zoom_edited
-                && let Ok(hh) = self.zoom_edit.trim().parse::<f64>()
-                && hh > 0.0
-                && hh.is_finite()
+                && let Ok(hh) = self.zoom_edit.parse::<Scale>()
             {
-                self.view.half_height = hh.max(MIN_HALF_HEIGHT);
+                self.view.half_height = hh;
                 self.view.sync_precision();
             }
             self.zoom_edited = false;
@@ -2560,7 +2567,7 @@ impl FractalApp {
             });
         ui.checkbox(&mut self.buddha_accumulate, "Keep sampling")
             .on_hover_text("Dispatch a fresh batch of random samples every frame.");
-        if self.view.magnification() > 1.0e5 {
+        if self.view.magnification_log10() > 5.0 {
             ui.colored_label(
                 egui::Color32::LIGHT_YELLOW,
                 "deep zoom isn't supported here (f32 precision only)",

src/fractal/reference.rs

@@ -380,7 +380,7 @@ fn step(
         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 < Big::ZERO {
                 ci + ((zr * zi) << 1)
             } else {
                 ci - ((zr * zi) << 1)
@@ -422,10 +422,10 @@ 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.
+/// Absolute value of a `Big`. The sign comes from the `Big` itself: through
+/// f64, anything below ~1e-308 reads as ±0 and would keep its sign.
 fn big_abs(x: Big) -> Big {
-    if x.to_f64().value() < 0.0 { -x } else { x }
+    if x < Big::ZERO { -x } else { x }
 }
 
 /// `(zr + i zi)^power` by repeated complex multiply at `precision` bits.
@@ -442,10 +442,10 @@ 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)`.
+/// `true` if `x` is exactly zero (special-cases `ln(0)`). Not via f64,
+/// which flushes values below ~1e-308 to zero.
 fn is_big_zero(x: &Big) -> bool {
-    x.to_f64().value() == 0.0
+    x.repr().significand().is_zero()
 }
 
 /// `(zr + i zi)^(pr + i pi)` for a complex exponent, via the principal branch
@@ -504,6 +504,17 @@ 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::try_from(1.0_f64).unwrap().with_precision(64).value() >> 5000;
+        assert!(!is_big_zero(&tiny));
+        assert!(is_big_zero(&big_zero(64)));
+        assert_eq!(big_abs(-tiny.clone()), tiny);
+        assert_eq!(big_abs(tiny.clone()), tiny);
+    }
+
     /// The high-precision reference must agree with a plain f64 iteration for a
     /// shallow point (where f64 is accurate).
     #[test]

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/view.rs

@@ -1,11 +1,12 @@
 //! Camera / view state over the complex plane.
 //!
 //! The center is stored in arbitrary precision (`FBig`) — this is what lets us
-//! zoom far past f64's ~1e13x limit. The pixel *scale* stays `f64`, which
-//! bounds zoom at ~10^300x (`MIN_HALF_HEIGHT`). Only the center needs the
-//! extra digits. 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.
+//! zoom far past f64's ~1e13x limit. The pixel *scale* is a [`Scale`]: an
+//! f64 mantissa with its own `i32` binary exponent, so it isn't bound by
+//! 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;
 
@@ -18,9 +19,6 @@ 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;
 
-/// Smallest half-height the view can zoom to: f64's range (the pixel scale,
-/// and the rescaled GPU uniforms, are computed in f64).
-pub const MIN_HALF_HEIGHT: f64 = 1e-300;
 
 /// 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
@@ -34,62 +32,266 @@ 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: f64, height_px: f64) -> bool {
-    2.0 * half_height / height_px.max(1.0) < 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: f64) -> i32 {
-    let e = half_height.max(MIN_HALF_HEIGHT).log2().floor() as i32;
+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$e}"),
+                None => write!(f, "{x:e}"),
+            };
+        }
+        // Out of f64's range: round the exact decimal expansion instead.
+        let sig = f.precision().map_or(17, |p| p + 1);
+        let dec = self.to_big().to_decimal().value().with_precision(sig).value();
+        let repr = dec.repr();
+        let digits = repr.significand().to_string();
+        let digits = digits.trim_end_matches('0');
+        let digits = if digits.is_empty() { "0" } else { digits };
+        // value = significand · 10^exponent; move the point after the first digit.
+        let exp10 = repr.exponent() + repr.significand().to_string().len() as isize - 1;
+        let (head, tail) = digits.split_at(1);
+        let tail = match f.precision() {
+            Some(p) => format!("{tail:0<p$}"),
+            None => tail.to_string(),
+        };
+        if tail.is_empty() {
+            write!(f, "{head}e{exp10}")
+        } else {
+            write!(f, "{head}.{tail}e{exp10}")
+        }
+    }
+}
+
+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 dec = DBig::from_str(s).map_err(|_| ())?;
+        let bin: Big = dec.with_base_and_precision::<2>(64).value();
+        if bin < Big::ZERO {
+            return Err(());
+        }
+        let repr = bin.repr();
+        let digits = repr.digits();
+        if digits == 0 {
+            return Err(());
+        }
+        let top = repr.exponent() + digits as isize - 1;
+        let m = (bin.clone() >> top).to_f64().value();
+        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
-/// this; the cap just prevents pathological allocation).
-const MAX_PRECISION_BITS: usize = 2048;
+/// Upper bound on center precision: what `Scale::MIN` needs. Only a guard
+/// against pathological input; the reference orbit is impractically slow
+/// 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) {
-        let h = self.half_height * 2.0;
-        (h * aspect, h)
+    pub fn span(&self, aspect: f64) -> (Scale, Scale) {
+        let h = self.half_height.mul_f64(2.0);
+        (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 {
-        (self.half_height * 2.0) / height_px
+    pub fn complex_per_pixel(&self, height_px: f64) -> Scale {
+        self.half_height.mul_f64(2.0 / height_px)
     }
 
-    /// Current magnification relative to the default view.
-    pub fn magnification(&self) -> f64 {
-        DEFAULT_HALF_HEIGHT / self.half_height
+    /// log10 of the current magnification relative to the default view.
+    pub fn magnification_log10(&self) -> f64 {
+        DEFAULT_HALF_HEIGHT.log10() - self.half_height.log10()
     }
 
     /// Current zoom level.
-    pub fn zoom(&self) -> f64 {
+    pub fn zoom(&self) -> Scale {
         self.half_height
     }
 
@@ -116,8 +318,8 @@ 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_im = &self.center_im - &big_from_f64(dy * cpp, bits);
+        self.center_re = &self.center_re - &cpp.big_times(dx, bits);
+        self.center_im = &self.center_im - &cpp.big_times(dy, bits);
     }
 
     /// Zoom by `factor` (<1 zooms in) keeping the complex point currently under
@@ -130,18 +332,18 @@ 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);
-        self.center_re = &self.center_re + &big_from_f64(off_x * k, bits);
-        self.center_im = &self.center_im + &big_from_f64(off_y * k, bits);
-        self.half_height = (self.half_height * factor).max(MIN_HALF_HEIGHT);
+        let k = 1.0 - factor;
+        self.center_re = &self.center_re + &cpp.big_times(off_x * k, bits);
+        self.center_im = &self.center_im + &cpp.big_times(off_y * k, bits);
+        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,
-            half_height: half_height.max(MIN_HALF_HEIGHT),
+            half_height,
         };
         v.sync_precision();
         v
@@ -164,11 +366,7 @@ 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()?;
-    if !(half_height > 0.0 && half_height.is_finite()) {
-        return None;
-    }
-    let half_height = half_height.max(MIN_HALF_HEIGHT);
+    let half_height = parse_half_height_spec(parts[2])?;
     let bits = precision_for(half_height);
     let re = big_from_decimal_str(parts[0], bits)?;
     let im = big_from_decimal_str(parts[1], bits)?;
@@ -179,12 +377,8 @@ 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> {
-    let half_height = spec.trim().parse::<f64>().ok()?;
-    if !(half_height > 0.0 && half_height.is_finite()) {
-        return None;
-    }
-    Some(half_height.max(MIN_HALF_HEIGHT))
+pub fn parse_half_height_spec(spec: &str) -> Option<Scale> {
+    spec.parse::<Scale>().ok()
 }
 /// Parse a "re,im" spec (re/im decimal, parsed at
 /// full precision) into a view. Shared by
@@ -213,16 +407,32 @@ 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 half_height = from.half_height * q.powf(t);
-    let bits = precision_for(half_height);
-    let g = if (q - 1.0).abs() < 1e-12 {
-        1.0 - t
+    let (l0, l1) = (from.half_height.log2(), to.half_height.log2());
+    let d = l1 - l0;
+    let half_height = if t <= 0.0 {
+        from.half_height
+    } else if t >= 1.0 {
+        to.half_height
     } else {
-        (q.powf(t) - q) / (1.0 - q)
+        Scale::from_log2(l0 + t * d)
+    };
+    let bits = precision_for(half_height);
+    let ln2 = core::f64::consts::LN_2;
+    let g_big = if d.abs() < 1e-12 {
+        big_from_f64(1.0 - t, bits)
+    } 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 g_big = big_from_f64(g, bits);
     let re0 = from.center_re.clone().with_precision(bits).value();
     let im0 = from.center_im.clone().with_precision(bits).value();
     let re1 = to.center_re.clone().with_precision(bits).value();
@@ -248,14 +458,10 @@ pub fn big_to_decimal_str(x: &Big, sig_digits: usize) -> String {
 }
 
 /// 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() {
-        (-half_height.log2()).ceil().max(0.0) as usize
-    } else {
-        0
-    };
+    let zoom_bits = (-half_height.log2()).ceil().max(0.0) as usize;
     (zoom_bits + GUARD_BITS).clamp(53, MAX_PRECISION_BITS)
 }
 
@@ -271,6 +477,10 @@ pub fn big_from_f64(x: f64, bits: usize) -> Big {
 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 im: f64 = v.center_im.to_decimal().value().to_f64().value();
@@ -279,12 +489,12 @@ mod tests {
 
     #[test]
     fn interpolate_view_hits_exact_endpoints() {
-        let bits = precision_for(1.0);
-        let from = ViewState::with_center(big_from_f64(-0.5, bits), big_from_f64(0.0, bits), 1.5);
+        let bits = precision_for(sc(1.0));
+        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(0.1013, precision_for(1e-20)),
-            1e-20,
+            big_from_f64(-0.7515, precision_for(sc(1e-20))),
+            big_from_f64(0.1013, precision_for(sc(1e-20))),
+            sc(1e-20),
         );
 
         let start = interpolate_view(&from, &to, 0.0);
@@ -304,12 +514,12 @@ mod tests {
     /// instead stay roughly bounded throughout.
     #[test]
     fn interpolate_view_keeps_target_offset_bounded() {
-        let bits = precision_for(1.0);
-        let from = ViewState::with_center(big_from_f64(-0.5, bits), big_from_f64(0.0, bits), 1.5);
+        let bits = precision_for(sc(1.0));
+        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(0.1013, precision_for(1e-20)),
-            1e-20,
+            big_from_f64(-0.7515, precision_for(sc(1e-20))),
+            big_from_f64(0.1013, precision_for(sc(1e-20))),
+            sc(1e-20),
         );
         let (to_re, to_im) = re_im_f64(&to);
 
@@ -318,7 +528,7 @@ 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={})",
@@ -326,4 +536,74 @@ 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().value();
+            let dim = ((&mid.center_im - &to.center_im) << k).to_f64().value();
+            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

@@ -10,12 +10,12 @@ use std::sync::mpsc::{Receiver, Sender, TryRecvError, channel};
 use std::thread;
 
 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,
@@ -35,7 +35,7 @@ pub struct RefRequest {
 pub struct RefResult {
     pub center_re: Big,
     pub center_im: Big,
-    pub half_height: f64,
+    pub half_height: Scale,
     pub points: RefOrbit,
     /// The kind and morph `points` was computed with (echoed from the
     /// request).