This trace follows the actual state transitions behind the companion How Bits Become Values. It describes a common execution path; implementation details can vary, so keep the contract separate from the mechanism.
Step 1: Identify width and type
Memory stores bit patterns; a type and an operation determine how a program interprets those bits. The same sequence can represent an integer, a floating-point value, an instruction, or encoded text. Understanding representation explains overflow, precision loss, and why serialization formats must specify byte order and field meaning.
Step 2: Decode the stored bit pattern
An unsigned w-bit integer represents values from zero through 2^w minus one. Two’s-complement signed integers reuse the same bit patterns with a different interpretation and arithmetic rules. Floating-point formats allocate bits to sign, exponent, and fraction, so many decimal fractions cannot be represented exactly.
Step 3: Apply the requested operation
Interpret the bits under the declared type before doing arithmetic; the same bit sequence has different values under signed, unsigned, and floating-point rules.
At this point, record the state that changed and check the invariant before advancing. If the operation repeats, make clear which values persist and which are recomputed.
Step 4: Check precision or overflow
Integer overflow behavior differs by language and type: some environments wrap, some trap, and some make signed overflow undefined. Floating-point rounding can make algebraic rearrangements change results. Treat external bytes as untrusted until length, encoding, and numeric range have been checked.
Step 5: Encode the result for output
Why can a program not safely assume that adding a small decimal fraction repeatedly produces the exact mathematical result? Give one way to represent money when exact decimal arithmetic matters.
The trace is complete when the result satisfies the stated contract. Compare this model with the concrete runtime or system you are studying before making a performance claim.