Calculator with Short-Circuit Evaluation — Solution¶
Setter idea: optional values carry deferred errors. Codex (GPT-6): complete parsing, safe integer evaluation, and the explicit-tree lazy alternative.
Establish the syntax tree's shape¶
Convert the expression to postfix tokens with an operator stack. Track whether an operand is expected to distinguish unary from binary signs. An incoming binary operator pops operators of higher or equal precedence. Unary operators associate rightward and do not pop equal-precedence operators. Parentheses form stack boundaries.
For ? , first reduce higher-precedence operations and completed conditionals at equal precedence, then push a question-marker delimiter. On : , pop until the matching question marker and replace it with a pending ternary operator. An unmatched question marker is never popped by precedence. This permits full expressions in middle branches while making later same-level conditionals reduce the previous completed conditional first. C++'s right-associative rule would be incorrect here.
Evaluate values with error flags¶
Each postfix stack entry stores (32-bit value, error flag) . Keep bit patterns in unsigned integers and convert to signed 64-bit values for comparisons and division. Implement wrapping through unsigned arithmetic. Mask shift counts to \(0,\ldots,31\) and implement signed right shift with an unsigned shift plus an explicit sign-fill mask. Check zero divisors before arithmetic and set an error flag instead of executing invalid division.
Ordinary operators propagate errors from either operand. Logical AND checks its left operand first: a valid zero returns a valid zero and ignores any right error. Logical OR is analogous. A conditional propagates an error in its condition; otherwise it returns only the selected branch's value and flag.
The implementation computes safe error flags even in unselected branches, but their errors do not affect the answer. This is valid because expressions have no side effects and dangerous arithmetic is always converted safely into a flag.
Correctness¶
Operator-stack reductions respect precedence, parentheses, and specified associativity, so the postfix sequence represents the required syntax tree. Induct on that tree: literals are exact; ordinary operations obey the integer rules and propagate errors; short-circuit nodes select only the required child result after checking their condition. Each value and observable error therefore agrees with actual short-circuit evaluation. The root is the answer.
Complexity and Contributions¶
Each token enters and leaves a stack constantly many times, giving \(O(n)\) time and space. Explicit stacks handle very deep 200,000-character expressions. The setter suggested optional values for deferred division errors. Codex (GPT-6) extended that idea to all operators, supplied integer semantics and left-associative conditionals, and verified std.cpp.
The alternative builds an explicit syntax tree and truly skips unselected branches.