🧮 Math Engine

A deterministic calculation core. The model decides that something needs
computing and which mode; the actual math is done by calc.py (SymPy), which
returns exact results as JSON. This removes language-simulated arithmetic and
its carry/sign errors.

When to use

Trigger this skill when the user's message contains a computable math task,
e.g.:

• Exact arithmetic — 12345 * 67890, 2^64, factorial(20)
• Simplify / expand / factor — (a+b)^2, x^2 - 5x + 6
• Symbolic calculus — ∫ x² dx, d/dx sin(x)·x²
• Equation solving — x² − 5x + 6 = 0
• Matrices — 行列式 determinant, 矩阵乘法 multiplication, 逆矩阵 inverse,
  伴随矩阵 adjugate, 转置 transpose, 秩 rank, 行变换/简化阶梯形 RREF, eigenvalues

If the user writes an explicit force-call like \calc{...}, treat the contents
as the expression and call this skill unconditionally.

How to call

python3 <skill_dir>/calc.py "<expression>" --mode <MODE> [--var x] [--timeout 5]

MODE is one of:

mode	does	example expression
eval	exact value / 30-digit decimal	12345 * 67890
simplify	simplify (and expand) an expression	(a+b)^2
expand	expand only	(x+1)^3
factor	factor a polynomial	x^2 - 5x + 6
diff	differentiate / partial derivative	sin(x)*x^2, x^2*y (with --var)
integrate	indefinite integral (adds + C)	x^2
solve	solve an equation (= allowed)	x^2 - 5x + 6 = 0
matrix	free-form matrix arithmetic (* + - **, inverse via **-1, det()/trace()/transpose())	[[1,2],[3,4]]*[[5,6],[7,8]]
det	determinant 行列式	[[1,2],[3,4]]
inv	inverse 逆矩阵	[[1,2],[3,4]]
adjugate	adjugate 伴随矩阵	[[1,2],[3,4]]
rref	reduced row echelon 行变换/简化阶梯形	[[1,2,3],[4,5,6],[7,8,10]]
transpose	transpose 转置	[[1,2,3],[4,5,6]]
rank	rank 矩阵的秩	[[1,2],[2,4]]
eigenvals	eigenvalues 特征值 (dict: value→multiplicity)	[[2,0],[0,3]]
auto	matrix literal → matrix; = → solve; else simplify	(default)

Notes for the parser:

• Use ^ or ** for powers; 5x implicit multiplication is understood.
• Pass --var if the variable to operate on is ambiguous (defaults to x→y→z…).
• Partial derivatives use diff with --var:
  • --var y → ∂/∂y (partial w.r.t. y, others held constant)
  • --var x,y → mixed partial ∂²/∂x∂y (differentiate by x then y)
  • --var x --order 2 → second derivative ∂²/∂x²
• Write matrices as [[1,2],[3,4]] (rows of comma-separated entries). The
  single-matrix modes (det, inv, adjugate, rref, transpose, rank,
  eigenvals) expect exactly one matrix; matrix mode accepts several joined
  by operators, e.g. [[1,2],[3,4]] * [[5,6],[7,8]].

Reading the result

calc.py always prints one JSON object to stdout:

{"ok": true, "mode": "solve", "input": "x^2 - 5x + 6 = 0", "result": "[2, 3]", "latex": "...", "variable": "x"}

or, on failure:

{"ok": false, "error": "...", "input": "..."}

Parse the JSON, then phrase the answer naturally for the user. Use latex when
rendering math is appropriate. If ok is false, tell the user what went wrong
(bad syntax, timeout, disallowed input) — do not fall back to computing it
yourself.

Worked examples

User asks	Call	result
12345 × 67890	calc.py "12345 * 67890" --mode eval	838102050
expand (a+b)²	calc.py "(a+b)^2" --mode simplify	a**2 + 2*a*b + b**2
∫ x² dx	calc.py "x^2" --mode integrate	C + x**3/3
solve x²−5x+6=0	calc.py "x^2 - 5x + 6 = 0" --mode solve	[2, 3]
行列式 det A	calc.py "[[1,2],[3,4]]" --mode det	-2
伴随矩阵 adj A	calc.py "[[1,2],[3,4]]" --mode adjugate	[[4, -2], [-3, 1]]
矩阵乘法 A·B	calc.py "[[1,2],[3,4]]*[[5,6],[7,8]]" --mode matrix	[[19, 22], [43, 50]]
行变换 rref	calc.py "[[1,2,3],[4,5,6],[7,8,10]]" --mode rref	[[1,0,0],[0,1,0],[0,0,1]]
偏导 ∂/∂y (x²y+sin y)	calc.py "x^2*y + sin(y)" --mode diff --var y	x**2 + cos(y)
混合偏导 ∂²/∂x∂y	calc.py "x^2*y^3" --mode diff --var x,y	6*x*y**2
高阶 ∂²/∂x² (x⁴)	calc.py "x^4" --mode diff --var x --order 2	12*x**2