sympy

Use SymPy's solveset for algebraic equations, linsolve for linear systems, nonlinsolve for nonlinear systems, and dsolve for differential equations. Define symbols first with symbols('x'), then call solveset(x**2 - 4, x) to get exact solutions like {-2, 2}.

Use SymPy when you need exact symbolic results such as sqrt(2), analytical derivatives, or closed-form equation solutions. Prefer NumPy or SciPy when floating-point approximations are sufficient, since numerical libraries are far faster for large-scale computation.

Use lambdify to convert a symbolic expression into a NumPy-compatible function, for example f = lambdify(x, expr, 'numpy'). This avoids slow loops of subs() and evalf() and lets you evaluate the expression over entire NumPy arrays efficiently.

Writing 0.5 * x creates a floating-point approximation instead of an exact rational. Use Rational(1, 2) or S(1)/2 to keep arithmetic exact, and call evalf() only when you explicitly need a numerical value.

Yes, sympy.utilities.codegen.codegen produces C or Fortran source from symbolic expressions, and autowrap compiles the result into a callable Python function. A C or Fortran compiler is required for autowrap and ufuncify workflows.

No, parse_expr uses eval internally and must never run on unsanitized input. Pass a local_dict of predefined symbols, restrict to standard_transformations, validate length and characters, and reject strings containing __, import, or assignment syntax.