fix: bracket a fraction only where the slash would split it

`\frac{n(n+1)}{2}` came out as `(n(n+1))/2` and the quadratic formula
as a wall of parentheses, because the test was "is this operand a
single token" rather than "would the slash change what this means".

Brackets now appear in exactly two cases: a sum, where the slash would
otherwise divide only the last term, and a denominator that is more
than one term, where `1/2a` is read two ways. `n(n+1)/2`, `√π/2` and
`mc²/√(1 − v²/c²)` need nothing, while `1 / (2a)` and
`(−b ± √(b² − 4ac)) / (2a)` keep their brackets. A script is read
through rather than counted, since a sum above the line is unambiguous
by construction.

The slash is spaced only when something beside it is bracketed, which
is where the brackets would otherwise crowd it.

README.md

@@ -275,6 +275,8 @@ Two facts about the renderer decided the design. A paragraph is a `TextEdit` wit
 
 So this is a linearisation rather than a typesetter. `\frac{a}{b}` becomes `a/b`, `\sqrt{x+1}` becomes `√(x+1)`, `\sum_{i=1}^{n}` becomes `∑ᵢ₌₁ⁿ`, and only what Unicode has no form for falls back to a tag, so `x^{a/b}` becomes `x<sup>a/b</sup>`. `$...$`, `\(...\)`, `$$...$$` or `\[...\]` on a line of its own, and a fence labelled `math`, are all read.
 
+Brackets appear only where the slash would otherwise change what the formula means. `\frac{n(n+1)}{2}` is `n(n+1)/2` and `\frac{\sqrt{\pi}}{2}` is `√π/2`, because one product either side of a slash needs nothing, while `\frac{1}{2a}` is `1 / (2a)` because `1/2a` is read two ways, and a sum is always bracketed so the slash cannot divide only its last term: `(−b ± √(b² − 4ac)) / (2a)`. The space around the slash shows up only when something beside it is bracketed, which is where the brackets would otherwise crowd it.
+
 The one guard is worth knowing about. A lone `$` pair counts as maths only when the inside looks like maths, because `x=$a$b` and `cd $A; echo $B` are not formulas, and turning them into one is the single way this feature can make an answer worse than the dollar signs it replaced. `$5 and $10` is left exactly as written.
 
 ### Custom HTML

src/__tests__/latex.test.ts

@@ -24,10 +24,35 @@ describe("renderLatex", () => {
     expect(renderLatex("\\int_{0}^{\\infty}")).toBe("∫₀<sup>∞</sup>");
   });
 
-  it("linearises a fraction and brackets only what needs it", () => {
+  it("linearises a fraction and brackets only what the slash would split", () => {
     expect(renderLatex("\\frac{1}{2}")).toBe("1/2");
     expect(renderLatex("\\frac{a}{b}")).toBe("a/b");
-    expect(renderLatex("\\frac{a+b}{c}")).toBe("(a+b)/c");
+    expect(renderLatex("\\frac{a+b}{c}")).toBe("(a+b) / c");
+  });
+
+  it("leaves a numerator that is one product unbracketed", () => {
+    // `n(n+1)/2` is unambiguous. Asking whether the operand was a single token
+    // instead produced `(n(n+1))/2`, which is a wall of parentheses.
+    expect(renderLatex("\\frac{n(n+1)}{2}")).toBe("n(n+1)/2");
+    expect(renderLatex("\\frac{\\sqrt{\\pi}}{2}")).toBe("√π/2");
+    expect(renderLatex("E = \\frac{mc^2}{\\sqrt{1 - v^2/c^2}}")).toBe("E = mc²/√(1 − v²/c²)");
+  });
+
+  it("brackets a denominator of more than one term, because 1/2a is read two ways", () => {
+    expect(renderLatex("\\frac{1}{2a}")).toBe("1 / (2a)");
+    expect(renderLatex("\\frac{\\pi}{2}")).toBe("π/2");
+  });
+
+  it("brackets a sum so the slash cannot divide only its last term", () => {
+    expect(renderLatex("\\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}")).toBe("(−b ± √(b² − 4ac)) / (2a)");
+    expect(renderLatex("\\frac{\\sin x}{x}")).toBe("(sin x) / x");
+  });
+
+  it("reads through a script rather than counting it as a split", () => {
+    // The sum is above the line by construction, so it needs no brackets. A
+    // script with no Unicode form is what puts a tag in the way of the scan.
+    expect(renderLatex("\\frac{x^{a/b}}{y}")).toBe("x<sup>a/b</sup>/y");
+    expect(renderLatex("\\frac{x^{a+b}}{y}")).toBe("xᵃ⁺ᵇ/y");
   });
 
   it("renders a root, using the root sign for the degree it has", () => {

src/lib/latex.ts

@@ -611,18 +611,86 @@ function subscript(rendered: string): string {
   return mapScript(rendered, SUBSCRIPTS) ?? `<sub>${rendered}</sub>`;
 }
 
+/**
+ * Characters that split a fraction when a formula is read left to right.
+ */
+const OPERAND_SPLIT = /[+−±∓]/;
+
+/** Prefixes that apply to what follows them, so they are not an operand of their own. */
+const PREFIX_OPERATORS = new Set(["√", "∛", "∜"]);
+
+/**
+ * Whether a rendered operand needs brackets beside the fraction slash.
+ *
+ * The rendered text is what a reader sees, so that is what this looks at. Two
+ * cases need brackets and nothing else does: a sum, because the slash would
+ * otherwise divide only its last term, and a denominator that is more than one
+ * thing, because `1/2a` is read as `1/(2a)` by some readers and `(1/2)a` by
+ * others.
+ *
+ * Asking whether an operand is a single token instead, which is what this used
+ * to do, brackets almost everything: `n(n+1)/2` came out as `(n(n+1))/2` and the
+ * quadratic formula as a wall of parentheses.
+ */
+function needsFractionBrackets(rendered: string, side: "numerator" | "denominator"): boolean {
+  // A script is skipped rather than read, because `x^{a+b}/y` is already
+  // unambiguous: the sum is above the line by construction.
+  const plain = rendered
+    .replace(/<(?:sup|sub)>[\s\S]*?<\/(?:sup|sub)>/g, "")
+    .replace(/<[^>]*>/g, "")
+    .trim();
+
+  let depth = 0;
+  let operands = 0;
+
+  for (let index = 0; index < plain.length; index += 1) {
+    const char = plain[index];
+
+    if (char === "(" || char === "[" || char === "{") {
+      depth += 1;
+      continue;
+    }
+    if (char === ")" || char === "]" || char === "}") {
+      depth = Math.max(0, depth - 1);
+      continue;
+    }
+    if (depth > 0) continue;
+
+    // A space at the top level is function application or a product, and either
+    // one is clearer bracketed: `(sin x) / x` rather than `sin x/x`.
+    if (char === " ") return true;
+
+    if (OPERAND_SPLIT.test(char)) {
+      // A leading sign belongs to the term rather than splitting the fraction.
+      if (index !== 0) return true;
+      continue;
+    }
+
+    // `√π` is one thing; `2π` is two multiplied together.
+    if (PREFIX_OPERATORS.has(char)) continue;
+
+    operands += 1;
+  }
+
+  return side === "denominator" && operands > 1;
+}
+
 /**
  * `\frac{a}{b}` as a slash, with brackets only where they change the meaning.
  *
  * A stacked fraction is the thing this cannot do, and `1/2` is what a reader
- * reconstructs from it anyway. `(a+b)/2` needs the brackets; `\alpha/2` does not.
+ * reconstructs from it anyway. The space around the slash appears only when
+ * something beside it is bracketed, where the brackets would otherwise crowd it.
  */
 function renderFraction(numerator: string, denominator: string, display: boolean): string {
   const top = convert(numerator, display);
   const bottom = convert(denominator, display);
 
-  const bracket = (rendered: string, body: string) => (isAtomic(body) ? rendered : `(${rendered})`);
-  return `${bracket(top, numerator)}/${bracket(bottom, denominator)}`;
+  const bracketedTop = needsFractionBrackets(top, "numerator");
+  const bracketedBottom = needsFractionBrackets(bottom, "denominator");
+  const slash = bracketedTop || bracketedBottom ? " / " : "/";
+
+  return `${bracketedTop ? `(${top})` : top}${slash}${bracketedBottom ? `(${bottom})` : bottom}`;
 }
 
 /** `\sqrt{x}` and `\sqrt[3]{x}`, where only a cube and a fourth root have a glyph. */