How do you evaluate a postfix expression using a Stack?

We are building a simple command-line calculator that takes space-separated postfix strings like '3 4 + 2 *'. Could you walk me through how you'd use a stack to parse and evaluate this, and show me how you'd handle the order of operands during subtraction or division?

Imagine we run your postfix evaluator on the input '5 0 /'. How would you write the evaluation logic to catch this division-by-zero error and return a clean error state instead of crashing the application?

We're adding a feature to a spreadsheet app that evaluates user-defined formulas. Users are complaining that when they enter invalid formulas, the app either freezes or throws unhelpful stack underflow errors. How would you refactor a standard stack-based evaluator to validate the expression on the fly and return precise error messages like 'Missing operand at position X'?

Our legacy postfix evaluator only handles single-digit integers. We need to upgrade it to support multi-digit numbers, decimals, and variable names (like 'x y +'). How would you design the tokenizer and stack evaluator to handle these token types safely?

We are building a high-throughput IoT telemetry rules engine that evaluates millions of incoming postfix-like threshold expressions per second. Memory allocation is our primary bottleneck. How would you optimize a stack-based evaluator to run with zero-allocation or minimal garbage collection overhead in this pipeline?

Instead of just evaluating the expression to a single value, we now need to compile these postfix expressions into an Abstract Syntax Tree (AST) for optimization before execution. How would you adapt your stack-based approach to output a tree structure, and how would you handle operator precedence if we transition to infix inputs later?

Our platform allows enterprise clients to write custom mathematical rules using a domain-specific language (DSL) that compiles to postfix. Over time, we need to support custom plugins/functions (e.g., 'USER_AGE() 18 >'). How would you design an extensible execution engine architecture that allows other engineering teams to register custom operators and functions without modifying the core stack-based evaluation engine?

We are migrating a legacy financial calculation engine from a synchronous, stack-based VM to a distributed, parallelized execution model to handle massive datasets. What are the architectural limitations of a stack-based evaluation model in a distributed environment, and how would you design the transition to a register-based or dataflow-graph execution model?