diff --git a/Cargo.lock b/Cargo.lock
index c7dff8ec9..520486716 100644
--- a/Cargo.lock
+++ b/Cargo.lock
@@ -2967,6 +2967,7 @@ dependencies = [
  "dirs",
  "ecow",
  "env_proxy",
+ "fastrand",
  "flate2",
  "fontdb",
  "native-tls",
diff --git a/Cargo.toml b/Cargo.toml
index b225a5a0c..24a5168e4 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -55,6 +55,7 @@ ctrlc = "3.4.1"
 dirs = "6"
 ecow = { version = "0.2", features = ["serde"] }
 env_proxy = "0.4"
+fastrand = "2.3"
 flate2 = "1"
 fontdb = { version = "0.23", default-features = false }
 fs_extra = "1.3"
diff --git a/crates/typst-ide/src/analyze.rs b/crates/typst-ide/src/analyze.rs
index 7ee83e709..c493da81a 100644
--- a/crates/typst-ide/src/analyze.rs
+++ b/crates/typst-ide/src/analyze.rs
@@ -26,7 +26,7 @@ pub fn analyze_expr(
         ast::Expr::Str(v) => Value::Str(v.get().into()),
         _ => {
             if node.kind() == SyntaxKind::Contextual {
-                if let Some(child) = node.children().last() {
+                if let Some(child) = node.children().next_back() {
                     return analyze_expr(world, &child);
                 }
             }
diff --git a/crates/typst-kit/Cargo.toml b/crates/typst-kit/Cargo.toml
index 52aa407c3..e59127d71 100644
--- a/crates/typst-kit/Cargo.toml
+++ b/crates/typst-kit/Cargo.toml
@@ -19,6 +19,7 @@ typst-utils = { workspace = true }
 dirs = { workspace = true, optional = true }
 ecow = { workspace = true }
 env_proxy = { workspace = true, optional = true }
+fastrand = { workspace = true, optional = true }
 flate2 = { workspace = true, optional = true }
 fontdb = { workspace = true, optional = true }
 native-tls = { workspace = true, optional = true }
@@ -43,7 +44,7 @@ fonts = ["dep:fontdb", "fontdb/memmap", "fontdb/fontconfig"]
 downloads = ["dep:env_proxy", "dep:native-tls", "dep:ureq", "dep:openssl"]
 
 # Add package downloading utilities, implies `downloads`
-packages = ["downloads", "dep:dirs", "dep:flate2", "dep:tar"]
+packages = ["downloads", "dep:dirs", "dep:flate2", "dep:tar", "dep:fastrand"]
 
 # Embeds some fonts into the binary:
 # - For text: Libertinus Serif, New Computer Modern
diff --git a/crates/typst-kit/src/package.rs b/crates/typst-kit/src/package.rs
index 172d8740a..584ec83c0 100644
--- a/crates/typst-kit/src/package.rs
+++ b/crates/typst-kit/src/package.rs
@@ -1,6 +1,7 @@
 //! Download and unpack packages and package indices.
 
 use std::fs;
+use std::io;
 use std::path::{Path, PathBuf};
 
 use ecow::eco_format;
@@ -77,7 +78,8 @@ impl PackageStorage {
         self.package_path.as_deref()
     }
 
-    /// Make a package available in the on-disk.
+    /// Makes a package available on-disk and returns the path at which it is
+    /// located (will be either in the cache or package directory).
     pub fn prepare_package(
         &self,
         spec: &PackageSpec,
@@ -100,7 +102,7 @@ impl PackageStorage {
 
             // Download from network if it doesn't exist yet.
             if spec.namespace == DEFAULT_NAMESPACE {
-                self.download_package(spec, &dir, progress)?;
+                self.download_package(spec, cache_dir, progress)?;
                 if dir.exists() {
                     return Ok(dir);
                 }
@@ -110,7 +112,7 @@ impl PackageStorage {
         Err(PackageError::NotFound(spec.clone()))
     }
 
-    /// Try to determine the latest version of a package.
+    /// Tries to determine the latest version of a package.
     pub fn determine_latest_version(
         &self,
         spec: &VersionlessPackageSpec,
@@ -143,7 +145,7 @@ impl PackageStorage {
     }
 
     /// Download the package index. The result of this is cached for efficiency.
-    pub fn download_index(&self) -> StrResult<&[serde_json::Value]> {
+    fn download_index(&self) -> StrResult<&[serde_json::Value]> {
         self.index
             .get_or_try_init(|| {
                 let url = format!("{DEFAULT_REGISTRY}/{DEFAULT_NAMESPACE}/index.json");
@@ -164,10 +166,10 @@ impl PackageStorage {
     ///
     /// # Panics
     /// Panics if the package spec namespace isn't `DEFAULT_NAMESPACE`.
-    pub fn download_package(
+    fn download_package(
         &self,
         spec: &PackageSpec,
-        package_dir: &Path,
+        cache_dir: &Path,
         progress: &mut dyn Progress,
     ) -> PackageResult<()> {
         assert_eq!(spec.namespace, DEFAULT_NAMESPACE);
@@ -191,11 +193,52 @@ impl PackageStorage {
             }
         };
 
+        // The directory in which the package's version lives.
+        let base_dir = cache_dir.join(format!("{}/{}", spec.namespace, spec.name));
+
+        // The place at which the specific package version will live in the end.
+        let package_dir = base_dir.join(format!("{}", spec.version));
+
+        // To prevent multiple Typst instances from interferring, we download
+        // into a temporary directory first and then move this directory to
+        // its final destination.
+        //
+        // In the `rename` function's documentation it is stated:
+        // > This will not work if the new name is on a different mount point.
+        //
+        // By locating the temporary directory directly next to where the
+        // package directory will live, we are (trying our best) making sure
+        // that `tempdir` and `package_dir` are on the same mount point.
+        let tempdir = Tempdir::create(base_dir.join(format!(
+            ".tmp-{}-{}",
+            spec.version,
+            fastrand::u32(..),
+        )))
+        .map_err(|err| error("failed to create temporary package directory", err))?;
+
+        // Decompress the archive into the temporary directory.
         let decompressed = flate2::read::GzDecoder::new(data.as_slice());
-        tar::Archive::new(decompressed).unpack(package_dir).map_err(|err| {
-            fs::remove_dir_all(package_dir).ok();
-            PackageError::MalformedArchive(Some(eco_format!("{err}")))
-        })
+        tar::Archive::new(decompressed)
+            .unpack(&tempdir)
+            .map_err(|err| PackageError::MalformedArchive(Some(eco_format!("{err}"))))?;
+
+        // When trying to move (i.e., `rename`) the directory from one place to
+        // another and the target/destination directory is empty, then the
+        // operation will succeed (if it's atomic, or hardware doesn't fail, or
+        // power doesn't go off, etc.). If however the target directory is not
+        // empty, i.e., another instance already successfully moved the package,
+        // then we can safely ignore the `DirectoryNotEmpty` error.
+        //
+        // This means that we do not check the integrity of an existing moved
+        // package, just like we don't check the integrity if the package
+        // directory already existed in the first place. If situations with
+        // broken packages still occur even with the rename safeguard, we might
+        // consider more complex solutions like file locking or checksums.
+        match fs::rename(&tempdir, &package_dir) {
+            Ok(()) => Ok(()),
+            Err(err) if err.kind() == io::ErrorKind::DirectoryNotEmpty => Ok(()),
+            Err(err) => Err(error("failed to move downloaded package directory", err)),
+        }
     }
 }
 
@@ -207,6 +250,36 @@ struct MinimalPackageInfo {
     version: PackageVersion,
 }
 
+/// A temporary directory that is a automatically cleaned up.
+struct Tempdir(PathBuf);
+
+impl Tempdir {
+    /// Creates a directory at the path and auto-cleans it.
+    fn create(path: PathBuf) -> io::Result<Self> {
+        std::fs::create_dir_all(&path)?;
+        Ok(Self(path))
+    }
+}
+
+impl Drop for Tempdir {
+    fn drop(&mut self) {
+        _ = fs::remove_dir_all(&self.0);
+    }
+}
+
+impl AsRef<Path> for Tempdir {
+    fn as_ref(&self) -> &Path {
+        &self.0
+    }
+}
+
+/// Enriches an I/O error with a message and turns it into a
+/// `PackageError::Other`.
+#[cold]
+fn error(message: &str, err: io::Error) -> PackageError {
+    PackageError::Other(Some(eco_format!("{message}: {err}")))
+}
+
 #[cfg(test)]
 mod tests {
     use super::*;
diff --git a/crates/typst-layout/src/flow/compose.rs b/crates/typst-layout/src/flow/compose.rs
index 76af8f650..54dc487a3 100644
--- a/crates/typst-layout/src/flow/compose.rs
+++ b/crates/typst-layout/src/flow/compose.rs
@@ -115,7 +115,7 @@ impl<'a, 'b> Composer<'a, 'b, '_, '_> {
         let column_height = regions.size.y;
         let backlog: Vec<_> = std::iter::once(&column_height)
             .chain(regions.backlog)
-            .flat_map(|&h| std::iter::repeat(h).take(self.config.columns.count))
+            .flat_map(|&h| std::iter::repeat_n(h, self.config.columns.count))
             .skip(1)
             .collect();
 
diff --git a/crates/typst-layout/src/grid/layouter.rs b/crates/typst-layout/src/grid/layouter.rs
index af47ff72f..dc9e2238d 100644
--- a/crates/typst-layout/src/grid/layouter.rs
+++ b/crates/typst-layout/src/grid/layouter.rs
@@ -1469,7 +1469,7 @@ impl<'a> GridLayouter<'a> {
                 // last height is the one for the current region.
                 rowspan
                     .heights
-                    .extend(std::iter::repeat(Abs::zero()).take(amount_missing_heights));
+                    .extend(std::iter::repeat_n(Abs::zero(), amount_missing_heights));
 
                 // Ensure that, in this region, the rowspan will span at least
                 // this row.
diff --git a/crates/typst-layout/src/math/root.rs b/crates/typst-layout/src/math/root.rs
index a6b5c03d0..c7f41488e 100644
--- a/crates/typst-layout/src/math/root.rs
+++ b/crates/typst-layout/src/math/root.rs
@@ -85,14 +85,15 @@ pub fn layout_root(
         ascent.set_max(shift_up + index.ascent());
     }
 
-    let radicand_x = sqrt_offset + sqrt.width();
+    let sqrt_x = sqrt_offset.max(Abs::zero());
+    let radicand_x = sqrt_x + sqrt.width();
     let radicand_y = ascent - radicand.ascent();
     let width = radicand_x + radicand.width();
     let size = Size::new(width, ascent + descent);
 
     // The extra "- thickness" comes from the fact that the sqrt is placed
     // in `push_frame` with respect to its top, not its baseline.
-    let sqrt_pos = Point::new(sqrt_offset, radicand_y - gap - thickness);
+    let sqrt_pos = Point::new(sqrt_x, radicand_y - gap - thickness);
     let line_pos = Point::new(radicand_x, radicand_y - gap - (thickness / 2.0));
     let radicand_pos = Point::new(radicand_x, radicand_y);
 
@@ -100,7 +101,8 @@ pub fn layout_root(
     frame.set_baseline(ascent);
 
     if let Some(index) = index {
-        let index_pos = Point::new(kern_before, ascent - index.ascent() - shift_up);
+        let index_x = -sqrt_offset.min(Abs::zero()) + kern_before;
+        let index_pos = Point::new(index_x, ascent - index.ascent() - shift_up);
         frame.push_frame(index_pos, index);
     }
 
diff --git a/crates/typst-layout/src/math/stretch.rs b/crates/typst-layout/src/math/stretch.rs
index dafa8cbe8..f45035e27 100644
--- a/crates/typst-layout/src/math/stretch.rs
+++ b/crates/typst-layout/src/math/stretch.rs
@@ -302,6 +302,6 @@ fn assemble(
 fn parts(assembly: GlyphAssembly, repeat: usize) -> impl Iterator<Item = GlyphPart> + '_ {
     assembly.parts.into_iter().flat_map(move |part| {
         let count = if part.part_flags.extender() { repeat } else { 1 };
-        std::iter::repeat(part).take(count)
+        std::iter::repeat_n(part, count)
     })
 }
diff --git a/crates/typst-library/src/foundations/cast.rs b/crates/typst-library/src/foundations/cast.rs
index 38f409c67..73645491f 100644
--- a/crates/typst-library/src/foundations/cast.rs
+++ b/crates/typst-library/src/foundations/cast.rs
@@ -21,7 +21,7 @@ use crate::foundations::{
 ///
 /// Type casting works as follows:
 /// - [`Reflect for T`](Reflect) describes the possible Typst values for `T`
-///    (for documentation and autocomplete).
+///   (for documentation and autocomplete).
 /// - [`IntoValue for T`](IntoValue) is for conversion from `T -> Value`
 ///   (infallible)
 /// - [`FromValue for T`](FromValue) is for conversion from `Value -> T`
diff --git a/crates/typst-library/src/model/bibliography.rs b/crates/typst-library/src/model/bibliography.rs
index a391e5804..b11c61789 100644
--- a/crates/typst-library/src/model/bibliography.rs
+++ b/crates/typst-library/src/model/bibliography.rs
@@ -94,7 +94,7 @@ pub struct BibliographyElem {
     /// - A path string to load a bibliography file from the given path. For
     ///   more details about paths, see the [Paths section]($syntax/#paths).
     /// - Raw bytes from which the bibliography should be decoded.
-    /// - An array where each item is one the above.
+    /// - An array where each item is one of the above.
     #[required]
     #[parse(
         let sources = args.expect("sources")?;
diff --git a/crates/typst-library/src/model/numbering.rs b/crates/typst-library/src/model/numbering.rs
index 150506758..ada8a3965 100644
--- a/crates/typst-library/src/model/numbering.rs
+++ b/crates/typst-library/src/model/numbering.rs
@@ -394,7 +394,7 @@ impl NumberingKind {
                 const SYMBOLS: &[char] = &['*', '†', '‡', '§', '¶', '‖'];
                 let symbol = SYMBOLS[(n - 1) % SYMBOLS.len()];
                 let amount = ((n - 1) / SYMBOLS.len()) + 1;
-                std::iter::repeat(symbol).take(amount).collect()
+                std::iter::repeat_n(symbol, amount).collect()
             }
             Self::Hebrew => hebrew_numeral(n),
 
diff --git a/crates/typst-library/src/model/outline.rs b/crates/typst-library/src/model/outline.rs
index 7ceb530f8..489c375e6 100644
--- a/crates/typst-library/src/model/outline.rs
+++ b/crates/typst-library/src/model/outline.rs
@@ -388,7 +388,7 @@ pub struct OutlineEntry {
     /// space between the entry's body and the page number. When using show
     /// rules to override outline entries, it is thus recommended to wrap the
     /// fill in a [`box`] with fractional width, i.e.
-    /// `{box(width: 1fr, it.fill}`.
+    /// `{box(width: 1fr, it.fill)}`.
     ///
     /// When using [`repeat`], the [`gap`]($repeat.gap) property can be useful
     /// to tweak the visual weight of the fill.
diff --git a/crates/typst-library/src/text/raw.rs b/crates/typst-library/src/text/raw.rs
index 1ce8bfc61..d5c07424d 100644
--- a/crates/typst-library/src/text/raw.rs
+++ b/crates/typst-library/src/text/raw.rs
@@ -188,7 +188,7 @@ pub struct RawElem {
     /// - A path string to load a syntax file from the given path. For more
     ///   details about paths, see the [Paths section]($syntax/#paths).
     /// - Raw bytes from which the syntax should be decoded.
-    /// - An array where each item is one the above.
+    /// - An array where each item is one of the above.
     ///
     /// ````example
     /// #set raw(syntaxes: "SExpressions.sublime-syntax")
diff --git a/crates/typst-library/src/visualize/gradient.rs b/crates/typst-library/src/visualize/gradient.rs
index 1a723a9f5..d59175a4e 100644
--- a/crates/typst-library/src/visualize/gradient.rs
+++ b/crates/typst-library/src/visualize/gradient.rs
@@ -574,8 +574,7 @@ impl Gradient {
         }
 
         let n = repetitions.v;
-        let mut stops = std::iter::repeat(self.stops_ref())
-            .take(n)
+        let mut stops = std::iter::repeat_n(self.stops_ref(), n)
             .enumerate()
             .flat_map(|(i, stops)| {
                 let mut stops = stops
diff --git a/crates/typst-render/src/shape.rs b/crates/typst-render/src/shape.rs
index ba7ed6d89..9b50d5f1f 100644
--- a/crates/typst-render/src/shape.rs
+++ b/crates/typst-render/src/shape.rs
@@ -69,9 +69,11 @@ pub fn render_shape(canvas: &mut sk::Pixmap, state: State, shape: &Shape) -> Opt
             let dash = dash.as_ref().and_then(to_sk_dash_pattern);
 
             let bbox = shape.geometry.bbox_size();
-            let offset_bbox = (!matches!(shape.geometry, Geometry::Line(..)))
-                .then(|| offset_bounding_box(bbox, *thickness))
-                .unwrap_or(bbox);
+            let offset_bbox = if !matches!(shape.geometry, Geometry::Line(..)) {
+                offset_bounding_box(bbox, *thickness)
+            } else {
+                bbox
+            };
 
             let fill_transform =
                 (!matches!(shape.geometry, Geometry::Line(..))).then(|| {
diff --git a/docs/tutorial/1-writing.md b/docs/tutorial/1-writing.md
index 5a9fdd4f7..acc257830 100644
--- a/docs/tutorial/1-writing.md
+++ b/docs/tutorial/1-writing.md
@@ -172,7 +172,7 @@ nothing else.
 For example, the image function expects a path to an image file.
 It would not make sense to pass, e.g., a paragraph of text or another image as
 the image's path parameter. That's why only strings are allowed here.
-On the contrary, strings work wherever content is expected because text is a
+In contrast, strings work wherever content is expected because text is a
 valid kind of content.
 </div>
 
diff --git a/docs/tutorial/4-template.md b/docs/tutorial/4-template.md
index 209fa5546..7542cd6e4 100644
--- a/docs/tutorial/4-template.md
+++ b/docs/tutorial/4-template.md
@@ -44,7 +44,7 @@ I am #amazed(color: purple)[amazed]!
 Templates now work by wrapping our whole document in a custom function like
 `amazed`. But wrapping a whole document in a giant function call would be
 cumbersome! Instead, we can use an "everything" show rule to achieve the same
-with cleaner code. To write such a show rule, put a colon directly behind the
+with cleaner code. To write such a show rule, put a colon directly after the
 show keyword and then provide a function. This function is given the rest of the
 document as a parameter. The function can then do anything with this content.
 Since the `amazed` function can be called with a single content argument, we can
diff --git a/tests/ref/math-root-frame-size-index.png b/tests/ref/math-root-frame-size-index.png
new file mode 100644
index 000000000..41d4df2e9
Binary files /dev/null and b/tests/ref/math-root-frame-size-index.png differ
diff --git a/tests/suite/math/root.typ b/tests/suite/math/root.typ
index df339fa80..b70b4b9bf 100644
--- a/tests/suite/math/root.typ
+++ b/tests/suite/math/root.typ
@@ -44,3 +44,9 @@ $ root(2, x) quad
 $ √2^3 = sqrt(2^3) $
 $ √(x+y) quad ∛x quad ∜x $
 $ (√2+3) = (sqrt(2)+3) $
+
+--- math-root-frame-size-index ---
+// Test size of final frame when there is an index.
+$ a root(, 3)         & a root(., 3) \
+  a sqrt(3)           & a root(2, 3) \
+  a root(#h(-1em), 3) & a root(123, 3) $