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‎.gitignore‎

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doc/
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# Auto-generated API documentation
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src/api.md
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TODO.md
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REVIEW.md
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PLAYGROUND.md
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build/
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dist/
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stage/

‎CHANGELOG.md‎

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Note: OEIS lookups require network access to oeis.org.
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- **Multi-Index Sequences**: Define sequences with multiple indices like Pascal's
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triangle P_{n,k} or grid-based recurrences:
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```javascript
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// Pascal's Triangle: P_{n,k} = P_{n-1,k-1} + P_{n-1,k}
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ce.declareSequence('P', {
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variables: ['n', 'k'],
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base: { 'n,0': 1, 'n,n': 1 }, // Pattern-based base cases
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recurrence: 'P_{n-1,k-1} + P_{n-1,k}',
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domain: { n: { min: 0 }, k: { min: 0 } },
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constraints: 'k <= n', // k must not exceed n
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});
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ce.parse('P_{5,2}').evaluate(); // → 10
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ce.parse('P_{10,5}').evaluate(); // → 252
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```
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Features:
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- Multiple index variables with `variables: ['n', 'k']`
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- Pattern-based base cases: `'n,0'` matches any (n, 0), `'n,n'` matches diagonal
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- Per-variable domain constraints
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- Constraint expressions (e.g., `'k <= n'`)
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- Composite key memoization (e.g., `'5,2'`)
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- Full introspection support with `isMultiIndex` flag
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Pattern matching for base cases:
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- Exact values: `'0,0'` matches only (0, 0)
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- Wildcards: `'n,0'` matches any value for n with k=0
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- Equality: `'n,n'` matches when both indices are equal
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- Priority: exact matches are checked before patterns
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#### Special Functions
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- **Special Function Definitions**: Added type signatures for special mathematical
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guard against pathological expressions. All recursive calls now track depth
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and gracefully return `undefined` if the limit is exceeded.
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### Bug Fixes
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- **Equation Equivalence in `isEqual()`** (Issue #275): Two equations are now
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correctly recognized as equivalent if they have the same solution set:
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```javascript
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ce.parse('2x+1=0').isEqual(ce.parse('x=-1/2')); // → true
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ce.parse('3x+1=0').isEqual(ce.parse('6x+2=0')); // → true
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```
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The implementation uses a sampling-based approach to check if the ratio of
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(LHS₁-RHS₁) to (LHS₂-RHS₂) is a non-zero constant, which indicates equivalent
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solution sets.
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## 0.33.0 _2026-01-30_
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### Bug Fixes

‎PLAYGROUND.md‎

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# Playground Test Results
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This document summarizes the status of various test snippets from
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`test/playground.ts`.
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## Summary
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| Category | Fixed | Issues Remaining |
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| ----------------- | ----- | ---------------- |
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| Parsing | 14 | 6 |
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| Simplification | 18 | 0 |
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| Evaluation | 17 | 1 |
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| Solve | 4 | 1 |
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| Matrix Operations | 6 | 0 |
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| Pattern Matching | 0 | 2 |
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| Formatting | 3 | 0 |
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| Other | 4 | 1 |
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---
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## Parsing
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### Working Correctly
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| Test | Line | Result |
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| ----------------------------------------------- | ------- | ------------------------------------------------ |
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| `\textcolor{red}{y + 1}` in expression | 6 | Parses with `Annotated` wrapper |
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| Sum with multiple bounds `\sum_{n=0,m=4}^{4,8}` | 8 | Correctly parses as multiple `Limits` |
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| `f()` and `f(x)` parsing | 99-103 | All variants parse correctly |
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| Plus/minus `21\pm1` | 79-80 | Parses as `PlusMinus`, evaluates to `Tuple` |
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| Non-canonical parsing `12+(-2)` | 196-197 | Correctly preserves structure |
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| `x_\text{a}` subscript | 226 | Parses as `"x_a"` |
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| `f'` derivative notation | 613 | Parses as `["Derivative","f"]` |
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| `x_{1,2}` subscript with comma | 615-616 | Parses as `Subscript(x, 1 2)` |
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| Sum with subscript `\sum_{n,m} k_{n,m}` | 455 | Correctly uses `Subscript`, not `At` |
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| Integral LaTeX output | 618-619 | Correct LaTeX: `\int_{0}^{1}\!x^2\, \mathrm{d}x` |
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| `\sin(-23\pi/12)` | 499 | Parses and evaluates correctly |
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| Error handling for `(`, `(3+x`, `)` | 668-674 | Correct error messages |
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| `\operatorname{HorizontalScaling}(3)` | 628-629 | Parses as `["HorizontalScaling", 3]` |
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| Knuth's interval `(a..b)` | 731 | Parses as `["Range", "a", "b"]` |
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### Issues Remaining
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| Test | Line | Expected | Actual | Notes |
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| ------------------------------------- | ------- | ---------------------- | ------------------------------------- | ----------------------------------- |
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| `\textcolor{red}{=}` | 11-12 | Styled equals | `Error('expected-closing-delimiter')` | Cannot style delimiters |
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| Double integral parsing | 27-31 | Nested integrals | Complex nested structure | Parses but N() times out |
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| `\mathrm{x+\alpha}` | 188 | Symbol with operators | `"xplusalpha"` (loses plus) | Characters merged |
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| `\mathrm{\oplus}` | 189 | Valid symbol | Error: invalid-first-char | Cannot use operators in mathrm |
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| `\gamma(2, 1)` | 430 | `Gamma(2, 1)` function | `"EulerGamma" * (2, 1)` | Should be incomplete gamma |
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| `\sin\left(x\right.` | 645 | Graceful handling | Multiple errors | Missing matchfix handling |
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---
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## Simplification
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### Working Correctly
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| Test | Line | Input | Result |
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| ---------------------------- | ------- | ---------------------- | -------------- |
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| Arithmetic | 277 | `-1234 - 5678` | `-6912` |
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| Sqrt addition | 278 | `2\sqrt{3}+\sqrt{1+2}` | `3sqrt(3)` |
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| cos(30°) | 281 | `\cos(30\degree)` | `sqrt(3)/2` |
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| Abs of pi | 295-298 | `\|-\pi\|` | `pi` |
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| Nested abs | 269 | `\|\|a\| + 3\|` | `\|a\| + 3` |
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| Multiply simplify | 517 | `2\times3xxx` | `6x^3` |
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| Trig identity | 529 | `1+4\sin(\pi/10)` | `sqrt(5)` |
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| Sqrt division | 650-658 | `\sqrt{15}/\sqrt{3}` | `sqrt(5)` |
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| `x+x` | 244 | `x+x` | `2x` |
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| `e^x e^{-x}` | 301-304 | `e^x e^{-x}` | `1` |
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| `\log_4(x^2)` | 307-310 | `\log_4(x^2)` | `2 log_4(x)` |
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| `\log_4(x^{7/4})` | 319-322 | `\log_4(x^{7/4})` | `7/4 log_4(x)` |
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| `\sin(\infty)` | 313-316 | `\sin(\infty)` | `NaN` |
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| `\tanh(\infty)` | 402-404 | `\tanh(\infty)` | `1` |
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| `\sqrt{x^2}` with `x > 0` | 257-259 | `x` | `x` |
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| `\sqrt[4]{x^4}` with `x > 0` | 257-259 | `x` | `x` |
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| `\cos(5\pi+k)` | 588-590 | `-cos(k)` | `-cos(k)` |
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| `a \times (c+d)` expand | 561-565 | `ac + ad` | `ac + ad` |
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### Notes
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- **Distribution/Expansion**: `simplify()` does NOT automatically distribute. Use
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`.expand()` to distribute: `a*(c+d)` → `ac + ad`. This is by design.
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- **Assumptions**: Simplifications that depend on assumptions (like `x > 0`) now
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work correctly. Use `ce.assume()` before simplifying.
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---
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## Evaluation
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### Working Correctly
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| Test | Line | Input | Result |
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| ------------------- | ------- | ---------------------------- | --------------------------------- |
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| Precision 30 digits | 69-73 | `\pi.N()` | `3.14159265358979323846264338328` |
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| Type of fraction | 75 | `3/4` type | `finite_rational` |
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| Replace | 58-66 | Note: use `.subs()` instead | Returns `3` (by design) |
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| `\sin(\pi^2)` | 181 | Evaluate | `-0.430301217...` |
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| Solve `x^2-1=0` | 201-206 | Solutions | `['1', '-1']` |
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| Variance functions | 208-210 | Statistics | Correct values |
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| Cube root `-1` | 228 | `(-1)^{1/3}` | `-1` |
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| Polynomial expand | 431-434 | `(2x^2+3x+1)(2x+1)` | `4x^3 + 8x^2 + 5x + 1` |
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| Floating point | 437-438 | `(0+1.1-1.1)(0+1/4-1/4)` | `0` |
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| Definite integral | 513 | `\int_0^1 x^2 dx` | `1/3` |
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| Bigint sqrt | 505-507 | Large number | `3513640562152.025...` |
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| Numerical integral | 44 | `\int_0^1 \sin(x) dx` `.N()` | `0.4598 ± 0.0001` |
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| Subscript fn call | 222-223 | `f_\text{a}(5)` with assign | `6` (evaluates correctly) |
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| Symbolic factorial | 342 | `(n-1)!` | `(n - 1)!` (stays symbolic) |
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### Issues Remaining
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| Test | Line | Expected | Actual | Notes |
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| -------------------- | ------- | --------------- | ------ | --------------------------------- |
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| `.replace()` bug | 61-72 | `2*x + b` | `2` | Single-char symbols auto-wildcard |
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**Note:** `.replace({match: 'a', replace: 2})` on `a*x + b` returns `2` instead of `2*x + b`.
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The bug is in `parseRulePart` (rules.ts:350) which auto-converts all single-character symbols
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to wildcards. So `'a'` becomes `'_a'`, matching ANY expression rather than the literal symbol `a`.
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See TODO.md #23.
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### Expected Behavior (Not Bugs)
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| Test | Line | Behavior | Workaround |
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| ------------------------------- | ------- | --------------------------- | -------------------------------- |
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| `D(\sin(x), x)` via LaTeX | 52-56 | Parses `D` as user symbol | Use `ce.box(['D', ...])` instead |
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| Power `.value` | 229-230 | `undefined` before evaluate | Use `.evaluate().value` instead |
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**Notes:**
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- In LaTeX, `D` is parsed as a predicate/symbol. For derivatives, use `ce.box(['D', expr, var])`
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or LaTeX notation like `\frac{d}{dx}`.
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- `.value` returns the numeric value only for already-evaluated expressions. For symbolic
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expressions like `Power(2, 3)`, call `.evaluate()` first.
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- For simple variable substitution, use `.subs()` not `.replace()`. The `.replace()` method
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is for pattern-based rule replacement.
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---
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## Solve
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### Working Correctly
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| Test | Line | Equation | Solutions |
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| -------------- | ------- | --------- | --------- |
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| `2x=\sqrt{5x}` | 156-158 | Quadratic | `5/4, 0` |
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| `x^2-1=0` | 201-206 | Quadratic | `1, -1` |
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| `5x=0` | 497-500 | Linear | `0` |
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| `x=\sqrt{5}` | 502-504 | Identity | `sqrt(5)` |
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### Issues Remaining
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| Test | Line | Expected | Actual | Notes |
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| ------------------------- | ---- | -------- | ------- | ---------------------------------------- |
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| `2x+1=0` isEqual `x=-1/2` | — | `true` | `false` | `isEqual` should recognize equivalent equations |
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**Note:** `isEqual` is for mathematical equality (vs `isSame` for structural equality).
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For equations, `isEqual` should check if `(LHS1-RHS1)/(LHS2-RHS2)` simplifies to a
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non-zero constant, indicating the same solution set. See TODO.md #22.
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---
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## Matrix Operations
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### Working Correctly
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| Test | Line | Input | Result |
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| ----------------------- | ------- | ---------------------------------- | ------------------- |
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| `Shape(A)` | 532-534 | 2x2 numeric matrix | `(2, 2)` |
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| `Rank(A)` | 532-534 | 2x2 numeric matrix | `2` |
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| `Flatten(A)` | 532-534 | `[[1,2],[3,4]]` | `[1,2,3,4]` |
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| `Transpose(A)` | 532-534 | `[[1,2],[3,4]]` | `[[1,3],[2,4]]` |
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| `Determinant(A)` | 532-534 | Numeric matrix | `-2` |
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| `Determinant(X)` symbolic | 536-549 | `[[a,b],[c,d]]` | `-b*c + a*d` |
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### Notes
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All matrix operations now work correctly, including symbolic matrices assigned via `ce.assign()`.
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---
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## Pattern Matching
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### Issues Remaining
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| Test | Line | Expected | Actual | Notes |
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| --------------------- | ------- | ------------------ | ------ | --------------------------- |
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| Match with variation | 135-148 | Substitution found | `null` | Match `0` against `_a*x` with `a=0` variation |
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| Complex pattern match | 153-165 | Substitution | `null` | Match `2x-√5√x` against complex Add pattern |
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**Note:** Pattern matching with `useVariations: true` has known limitations. The system
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doesn't fully handle all algebraic variations (like matching `0` as `0*x`).
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---
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## Formatting / Serialization
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### Working Correctly
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| Test | Line | Input | Result |
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| ------------------------- | ------- | -------------------------------- | ------------------------------ |
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| Scientific notation | 40-44 | `1000` | `1\cdot10^{3}` |
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| Fraction canonical | 298 | `\frac{2}{-3222233}+\frac{1}{3}` | Uses `Rational` not `Subtract` |
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| `1/(2\sqrt{3})` canonical | 617 | Rationalized | `\frac{\sqrt{3}}{6}` |
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### Expected Behavior
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| Test | Line | Behavior | Notes |
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| -------------------- | ---- | --------------- | --------------------------------------- |
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| `3\times3` canonical | 59 | Returns `3 * 3` | `.simplify()` returns `9`. Converting to `3^2` would be a separate optimization. |
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---
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## Other Features
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### Working Correctly
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| Test | Line | Feature | Result |
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| ------------- | ------- | ---------------------------- | --------------------------------- |
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| Filter | 573-577 | `Filter([1,2,3,4,5], IsOdd)` | `[1,3,5]` |
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| Expand | 467-475 | `4x(3x+2)-5(5x-4)` | `12x^2 - 17x + 20` |
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| Negate i | 405-406 | `-i` | `-i` |
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| Hold | 232-234 | `Add(1, Hold(2))` | `1 + Hold(2)` |
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### Issues Remaining
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| Test | Line | Expected | Actual | Notes |
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| ------------------------- | ------- | ---------------- | --------------------------------- | ----------------------------- |
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| `List(Filter).evaluate()` | 581-583 | Evaluated filter | Filter not evaluated inside List | Nested evaluation issue |
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### Expected Behavior
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| Test | Line | Behavior | Notes |
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| ----------------- | ------- | ------------------ | -------------------------------------------------- |
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| Sum with data | 175-186 | `[50, 130]` | Element-wise multiplication is correct; use explicit indexing for dot product |
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| `Floor(Cos(n))` | 271-273 | `floor(cos(n))` | `n` is unknown, so expression stays symbolic |
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---
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## Parsing Edge Cases (Error Handling)
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These tests verify error handling for malformed input:
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| Input | Line | Status |
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| ------------------------- | ---- | ------------------------------------ |
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| `x__+1` | 510 | Returns string `"x__"` + 1 (not At) |
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| `\operatorname{a?#_!}` | 160 | Error: invalid-symbol |
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| `(a, b; c, d, ;; n ,, m)` | 677 | Parses as nested Tuples with Nothing |
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| `\operatorname{$invalid}` | 680 | Error for invalid identifier |
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---
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## Notes
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1. **Double/complex integrals**: The playground includes double integrals and
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complex integration tests that timeout during numerical evaluation.
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2. **Assumptions**: Simplifications that depend on assumptions (like `x > 0`)
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now work correctly. Use `ce.assume(ce.parse('x > 0'))` before simplifying.
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3. **Matrix operations**: Matrix-related functions (Shape, Rank, Flatten,
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Transpose, Determinant, Inverse, Trace) now work correctly for both numeric
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and symbolic matrices.
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4. **Pattern matching**: The pattern matching system has issues with variations
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and complex patterns.
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5. **D operator syntax**: The `D` operator must be used via `ce.box(['D', ...])`,
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not LaTeX parsing. In LaTeX, `D` is parsed as a user symbol.
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6. **Trig periodicity**: Trigonometric functions now reduce arguments by their
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period (e.g., `cos(5π + k)` simplifies to `-cos(k)`).

‎package-lock.json‎

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‎package.json‎

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"jest-silent-reporter": "^0.6.0",
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"open": "^10.1.2",
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"prettier": "^3.6.2",
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"prettier-2": "npm:prettier@^2",
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"prettier-2": "npm:prettier@^2.8.8",
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"serve-http": "^1.0.7",
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"ts-jest": "^29.4.0",
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"ts-node": "^10.9.2",

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