Fold what is known without inventing algebraic permission

A tensor's shape can be known while all its elements remain runtime values. Knowing that X has four columns allows shape arithmetic, but it does not justify precomputing XW when requests change X. A parameter may be intentionally frozen into a deployment artifact, supplied as a runtime input, or specialized with a checked value assumption. Each choice has a different update contract. If a model weight changes after capture, an artifact that embedded its old contents needs an appropriate invalidation or explicit immutable-artifact policy.

For a small real-valued expression f(x,c)=2(x+c), choosing c=3 leaves a residual expression depending on x. Evaluating the known c-dependent subcomputation is legitimate only under the chosen arithmetic semantics. A Python scalar captured as static can introduce a specialized-value condition, whereas a tensor scalar kept as an input remains dynamic. PyTorch 2.9 documents recompilations caused by changing scalar constants. The useful question is which values the artifact treats as fixed, not whether the source variable was given a suggestive name.

Under binary32 round-to-nearest, ties-to-even, the integer 2²⁴ is exactly representable but the next integer is not. Adding one to it rounds back to 2²⁴. Subtracting the original value then gives zero. Regrouping the same written numbers can instead produce one, so associativity over real numbers is not an unconditional floating-point transformation rule. Constant evaluation should reproduce the semantics of the original operations unless the IR explicitly grants a broader numerical contract.

Other familiar identities fail on special values or signs. Replacing x−x by zero is wrong for an infinite x under ordinary IEEE behavior because the original produces NaN. Replacing zero times x by a positive zero also needs to consider NaNs, infinities, and signed zero. LLVM 20.1.0 describes fast-math flags that permit particular otherwise unsafe transformations; flags are contractual permissions and assumptions, not a proof that every result remains bit-identical. Preserve the actual flags rather than silently importing them from an optimization wish list.

Test a partially evaluated artifact against its reference using both the fixed specialization and deliberate changes that should invalidate it. For numerical rewrites, include cancellation, very small magnitudes, large finite values, and special values when the public domain permits them. Keep unsupported or explicitly excluded values separate from supported inputs. A tolerance-based deployment contract can be legitimate, but it must define what deviation is allowed and does not automatically excuse dropped branches, incorrect state, or stale embedded parameters.

The function lab illustrates a changed fixed coefficient in a real analytic expression, not floating-point compiler folding. Reset, set both coefficient a=0.5 and position x=2, predict value two and slope two, then Play or Step. Reset and set both a=1,x=2 for value four and slope four. These fresh evaluations show why a residual specialized at one coefficient cannot silently stand in for another. The instrument uses its own numerical implementation and is not a binary32 rounding emulator or a compiler pass.