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‎SIMPLIFY.md‎

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# Remaining Skipped Simplification Tests
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There are **42 skipped tests** remaining in `test/compute-engine/simplify.test.ts`
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There are **19 skipped tests** remaining in `test/compute-engine/simplify.test.ts`
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(down from 93 originally). This document lists only the items that still need
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resolution.
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---
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## 1. Logarithm Rules (14 tests)
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## 1. Logarithm Rules (3 tests)
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### 1a. Power rule for general log bases
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### 1a. ~~Power rule for general log bases~~ DONE
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| Line | Test | Expected |
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| ---- | --------------------- | -------------------- |
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| 578 | `log_3(x^sqrt(2))` | `sqrt(2)*log_3(x)` |
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| 587 | `log_4(x^{7/4})` | `7/4*log_4(x)` |
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`log_c(x^n) → n*log_c(x)` — already works for irrational and non-integer
14+
rational exponents. Even exponents now correctly produce `log_c(|x|)`.
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The `ln(x^n) → n*ln(x)` rule works but the analogous `log_c(x^n) → n*log_c(x)`
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does not fire for arbitrary base `c`.
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### 1b. ~~Ln cancellation~~ MOSTLY DONE
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### 1b. Power rule with even exponent (absolute value needed)
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`ln(sqrt(x))-ln(x)/2 = 0` and `ln(3)+ln(1/3) = 0` now pass.
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`ln(p/q) → ln(p) - ln(q)` rule added for positive rationals.
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| Line | Test | Expected |
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| ---- | ----------------- | -------------------- |
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| 583 | `log_4(x^2)` | `2*log_4(\|x\|)` |
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| 585 | `log_4(x^{2/3})` | `2/3*log_4(\|x\|)` |
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When the exponent makes `x^n ≥ 0`, the result should use `|x|`.
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### 1c. Change of base / ratio
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| Line | Test | Expected |
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| ---- | --------------- | -------- |
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| 283 | `ln(9)/ln(3)` | `2` |
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Needs: `ln(a)/ln(b) → log_b(a)`, then evaluate when both are known constants.
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### 1d. Ln cancellation / collection
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| Line | Test | Expected |
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| ---- | ------------------------------- | ---------------- |
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| 531 | `ln(x^{2/3}) - 4/3*ln(x)` | `-2/3*ln(x)` |
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| 536 | `ln(pi^{2/3}) - 1/3*ln(pi)` | `1/3*ln(pi)` |
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| 541 | `ln(sqrt(x)) - ln(x)/2` | `0` |
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| 543 | `ln(3) + ln(1/3)` | `0` |
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Lines 531/541 may have wrong expected values — verify LaTeX parsing.
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Line 543 may involve floating-point error (`ln(1/3)` is not exact `-ln(3)`).
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### 1e. Exponential of log difference
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| Line | Test | Expected |
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| ---- | ------------------- | ---------------- |
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| 553 | `e^{ln(x) - y^2}` | `x / e^{y^2}` |
55-
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The existing `e^{ln(x) + f}` rule doesn't handle arbitrary non-ln addends.
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### 1f. Log of quotient involving e
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### 1c. Log of quotient involving e
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| Line | Test | Expected |
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| ---- | ------------------------ | ----------------- |
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| 564 | `ln((x+1)/e^{2x})` | `ln(x+1) - 2x` |
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The `ln(a/b)` rule exists but may not fire in this context.
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### 1g. Reciprocal base
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| Line | Test | Expected |
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| ---- | --------------- | -------------- |
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| 574 | `log_{1/2}(x)` | `-log_2(x)` |
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| 498 | `ln((x+1)/e^{2x})` | `ln(x+1) - 2x` |
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New rule: `log_{1/b}(x) = -log_b(x)`.
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Operand simplification expands the fraction before the log quotient rule fires.
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Deep ordering issue.
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### 1h. Mixed log identities with arbitrary base
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### 1d. Mixed log product identity
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| Line | Test | Expected |
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| ---- | -------------------- | ------------------- |
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| 611 | `log_c(c^x * y)` | `x + log_c(y)` |
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| 613 | `log_c(c^x / y)` | `x - log_c(y)` |
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| 615 | `log_c(y / c^x)` | `log_c(y) - x` |
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| 629 | `log_c(a) * ln(a)` | `ln(c)` |
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Generalize the `ln(e^x * y)` family to `log_c(c^x * y)`.
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---
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## 2. Absolute Value Identities (5 tests)
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| 548 | `log_c(a) * ln(a)` | `ln(c)` |
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| Line | Test | Expected |
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| ---- | -------------------------- | ------------- |
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| 647 | `\|x\|^{4/3}` | `x^{4/3}` |
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| 649 | `\|xy\| - \|x\|*\|y\|` | `0` |
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| 654 | `\|2/x\| - 1/\|x\|` | `1/\|x\|` |
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| 656 | `\|1/x\| - 1/\|x\|` | `0` |
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| 658 | `\|x\|\|y\| - \|xy\|` | `0` |
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Missing rules:
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- `|a*b| → |a|*|b|` (multiplicative identity)
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- `|a/b| → |a|/|b|`
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- `|x|^{p/q} → x^{p/q}` when the result is always non-negative (needs care
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with branch conventions)
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---
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## 3. Powers with Negative Bases / Odd Roots (2 tests)
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| Line | Test | Expected | Notes |
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| ---- | -------------- | -------------- | ----------------------------------- |
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| 452 | `(-x)^{3/4}` | `x^{3/4}` | **Questionable** — complex for x>0 |
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| 454 | `cbrt(-2)` | `-cbrt(2)` | Sign extraction for odd roots |
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For `cbrt(-a) = -cbrt(a)`: add rule `Root(n, -a) → -Root(n, a)` when `n` is
113-
odd and `a ≥ 0`.
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**NOTE**: This test is mathematically wrong. `log_c(a)*ln(a) = ln(a)²/ln(c)`,
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not `ln(c)`. Likely intended: `log_a(c)*ln(a) = ln(c)`.
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For `(-x)^{3/4}`: review domain — this is complex for positive `x`. May be a
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test bug.
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### 1e. ~~Change of base / ratio~~ DONE
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---
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## 4. Double Powers (1 test)
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| Line | Test | Expected |
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| ---- | --------------- | -------- |
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| 146 | `(x^3)^{1/3}` | `x` |
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### 1f. ~~Reciprocal base~~ DONE
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126-
`(x^n)^{1/n} → x` when `n` is odd. The existing rule requires `baseNonNeg ||
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innerIsOddInteger`, but the combination `3 * 1/3 = 1` may not simplify to
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`x^1 → x`. Needs debugging.
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### 1g. ~~Mixed log identities~~ DONE (except product)
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---
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## 5. Power of Quotient (1 test)
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| Line | Test | Expected |
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| ---- | ----------------- | ----------------- |
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| 485 | `x/(pi/y)^3` | `x*y^3/pi^3` |
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`(a/b)^n → a^n/b^n` may not fire when the powered quotient is in a denominator.
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---
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## 6. Power Combination with Symbolic Exponents (1 test)
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| Line | Test | Expected |
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| ---- | --------------------- | ---------------------- |
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| 492 | `x^{sqrt(2)} / x^3` | `x^{sqrt(2) - 3}` |
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## 2. Absolute Value (0 remaining)
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148-
The `x^a * x^b → x^{a+b}` rule works for numeric exponents but not symbolic.
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All abs tests now pass:
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- `|x|^{4/3} = x^{4/3}` — fixed by handling Rational exponents in
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`simplifyAbsPower`
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- `|xy| = |x||y|` — multiplicative identity added
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- `|x/y| = |x|/|y|` — quotient identity added
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---
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## 7. Root Simplification (2 tests)
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## 3. Powers and Roots (3 tests)
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| Line | Test | Expected |
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| ---- | --------------- | -------------- |
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| 499 | `root4(16*b^4)` | `2\|b\|` |
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| 503 | `root4(x^6)` | `sqrt(x^3)` |
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Factor numeric coefficients out of roots and reduce `root(n, x^m) → x^{m/n}`
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to simplest radical form.
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| Line | Test | Expected | Notes |
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| ---- | -------------- | -------------- | ----------------------------------- |
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| 404 | `(-x)^{3/4}` | `x^{3/4}` | **Wrong test** — complex for x > 0 |
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| 441 | `x^{sqrt(2)}/x^3` | `x^{sqrt(2)-3}` | sqrt(2).sub(3) evaluates to float |
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| 447 | `root4(16b^4)` | `2\|b\|` | Factor numeric coefficients from roots |
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---
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## 8. Common Denominator (2 tests)
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## 4. Common Denominator (2 tests)
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| Line | Test | Expected |
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| ---- | ----------------- | ---------------- |
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| 513 | `1/(x+1) - 1/x` | `-1/(x^2+x)` |
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| 515 | `1/x - 1/(x+1)` | `1/(x^2+x)` |
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| 458 | `1/(x+1) - 1/x` | `-1/(x^2+x)` |
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| 460 | `1/x - 1/(x+1)` | `1/(x^2+x)` |
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Requires finding a common denominator for fractions with polynomial
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denominators — a significant new capability.
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---
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## 9. Multi-Variable Expansion (1 test)
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## 5. Multi-Variable Expansion (1 test)
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| Line | Test | Expected |
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| ---- | ----------------------- | ---------------- |
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| 522 | `2*(x+h)^2 - 2*x^2` | `4xh + 2h^2` |
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| 466 | `2*(x+h)^2 - 2*x^2` | `4xh + 2h^2` |
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Single-variable `(x+1)^2 - x^2 = 2x+1` works. Debug why two variables fail.
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Single-variable `(x+1)^2 - x^2 = 2x+1` works. Multi-variable expansion
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(`(x+h)^2`) does not expand.
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---
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## 10. Inverse Hyperbolic ↔ Logarithm Rewrites (6 tests)
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## 6. Float / Mixed Arithmetic (2 tests)
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| Line | Test | Expected |
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| ---- | ------------------------------ | ------------- |
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| 968 | `1/2*ln((x+1)/(x-1))` | `arccoth(x)` |
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| 973 | `ln(x + sqrt(x^2+1))` | `arsinh(x)` |
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| 978 | `ln(x + sqrt(x^2-1))` | `arcosh(x)` |
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| 983 | `1/2*ln((1+x)/(1-x))` | `artanh(x)` |
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| 988 | `ln((1+sqrt(1-x^2))/x)` | `arsech(x)` |
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| 993 | `ln(1/x + sqrt(1/x^2+1))` | `arcsch(x)` |
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| Line | Test | Expected |
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| ---- | ---------------- | ---------------------------------- |
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| 43 | `sqrt(3.1)` | `1.76068168616590091458` (decimal) |
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| 58 | `sqrt(3) + 0.3` | `2.0320508075688772` (decimal) |
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Complex structural pattern-matching. Low priority / niche.
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`simplify()` should trigger numeric evaluation when floats are present.
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19999
---
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## 11. Inverse Trig Rewrite (1 test)
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## 7. Inequality Simplification (1 test)
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| Line | Test | Expected |
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| ---- | ----------------------------- | ------------- |
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| 1001 | `arctan(x/sqrt(1-x^2))` | `arcsin(x)` |
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| Line | Test | Expected |
104+
| ---- | -------------------------- | ------------- |
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| 113 | `(2*pi + 2*pi*e) < 4*pi` | `1 + e < 2` |
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Structural pattern-matching rewrite.
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Extend inequality GCD-factor-out to handle sums with common factors.
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209109
---
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## 12. Float / Mixed Arithmetic (2 tests)
111+
## 8. Inverse Hyperbolic ↔ Logarithm Rewrites (6 tests)
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| Line | Test | Expected |
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| ---- | ---------------- | ---------------------------------- |
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| 43 | `sqrt(3.1)` | `1.76068168616590091458` (decimal) |
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| 58 | `sqrt(3) + 0.3` | `2.0320508075688772` (decimal) |
113+
| Line | Test | Expected |
114+
| ---- | ------------------------------ | ------------- |
115+
| 822 | `1/2*ln((x+1)/(x-1))` | `arccoth(x)` |
116+
| 827 | `ln(x + sqrt(x^2+1))` | `arsinh(x)` |
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| 829 | `ln(x + sqrt(x^2-1))` | `arcosh(x)` |
118+
| 831 | `1/2*ln((1+x)/(1-x))` | `artanh(x)` |
119+
| 833 | `ln((1+sqrt(1-x^2))/x)` | `arsech(x)` |
120+
| 835 | `ln(1/x + sqrt(1/x^2+1))` | `arcsch(x)` |
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`simplify()` should trigger numeric evaluation when floats are present.
122+
Complex structural pattern-matching. Low priority / niche.
219123

220124
---
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## 13. Inequality Simplification (1 test)
126+
## 9. Inverse Trig Rewrite (1 test)
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224-
| Line | Test | Expected |
225-
| ---- | -------------------------- | ------------- |
226-
| 125 | `(2*pi + 2*pi*e) < 4*pi` | `1 + e < 2` |
128+
| Line | Test | Expected |
129+
| ---- | ----------------------------- | ------------- |
130+
| 843 | `arctan(x/sqrt(1-x^2))` | `arcsin(x)` |
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Extend inequality GCD-factor-out to handle sums with common factors.
132+
Structural pattern-matching rewrite.
229133

230134
---
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232-
## 14. Fu Trig Simplification — Phase 14 (1 test)
136+
## 10. Fu Trig Simplification — Phase 14 (1 test)
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234138
| Line | Test | Expected |
235139
| ---- | ----------------------------------------------- | ---------------------- |
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| 1446 | `1 - (1/4)*sin^2(2x) - sin^2(y) - cos^4(x)` | `sin(x+y)*sin(x-y)` |
140+
| 1279 | `1 - (1/4)*sin^2(2x) - sin^2(y) - cos^4(x)` | `sin(x+y)*sin(x-y)` |
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238142
Requires extending the Fu algorithm implementation.
239143

240144
---
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242146
## Priority Order
243147

244-
### Medium effort, broadly useful
245-
246-
1. **Log power rule for arbitrary base** (1a) — generalize existing ln rule
247-
2. **Reciprocal log base** (1g) — new rule `log_{1/b}(x) = -log_b(x)`
248-
3. **Abs multiplicative identity** (2) — `|ab| = |a||b|`
249-
4. **cbrt(-a) = -cbrt(a)** (3) — sign extraction for odd roots
250-
5. **Ln cancellation** (1d) — verify parsing, extend collection
251-
6. **Mixed log identities** (1h) — generalize ln(e^x*y) to log_c(c^x*y)
252-
253148
### Medium effort, moderate value
254149

255-
7. **Double powers** (4) — debug (x^3)^{1/3} = x
256-
8. **Power of quotient** (5) — (a/b)^n in denominator
257-
9. **Symbolic exponent combination** (6) — x^a / x^b with symbolic a, b
258-
10. **Root simplification** (7) — factor coefficients out of radicals
259-
11. **Exp of log difference** (1e) — e^{ln(x) - f}
260-
12. **Log of quotient** (1f) — ln((x+1)/e^{2x})
261-
13. **Change of base ratio** (1c) — ln(a)/ln(b)
150+
1. **Root factoring** (3) — `root4(16b^4) = 2|b|`
151+
2. **Multi-variable expansion** (5) — `2*(x+h)^2 - 2*x^2`
152+
3. **Log of quotient** (1c) — `ln((x+1)/e^{2x})`
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263154
### High effort / niche
264155

265-
14. **Common denominator** (8) — partial fraction / rational expression
266-
15. **Multi-variable expansion** (9) — debug two-variable case
267-
16. **Float arithmetic** (12) — N() integration in simplify
268-
17. **Inverse hyp ↔ log rewrites** (10) — complex pattern matching
269-
18. **Inverse trig rewrite** (11) — structural pattern matching
270-
19. **Inequality GCD** (13) — extend to sums
271-
20. **Fu Phase 14** (14) — advanced trig
156+
4. **Float arithmetic** (6) — N() integration in simplify
157+
5. **Common denominator** (4) — partial fraction / rational expression
158+
6. **Symbolic exponent** (3) — `x^{sqrt(2)}/x^3` (float loss)
159+
7. **Inequality GCD** (7) — extend to sums
160+
8. **Inverse hyp ↔ log rewrites** (8) — complex pattern matching
161+
9. **Inverse trig rewrite** (9) — structural pattern matching
162+
10. **Fu Phase 14** (10) — advanced trig
163+
164+
### Bug / Questionable
165+
166+
- `(-x)^{3/4} = x^{3/4}` — complex for real x > 0, test likely wrong
167+
- `log_c(a)*ln(a) = ln(c)` — mathematically incorrect expected value

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