What does this equation mean?

Numerical analysis has always run on an analogous verification principle, just usually informal: an error bound is only useful if it is derived from and checkable against a stated model of arithmetic. The classical example is the sensitivity of a linear system to input error, captured by the condition number of a matrix A : $\kappa(A) = \|A\| \, \|A^{-1}\|$. If $\kappa(A)$ is large, small errors in the input data or in floating-point rounding can produce large errors in the computed solution, independent of which algorithm is used to solve the system — a property of the problem, not the solver. Bringing formal verification to numerical software means proving that an implementation’s actual rounding behavior stays within a bound consistent with this kind of analysis, not just testing it empirically on sample inputs. That is a much harder and more labor-intensive proof target than a compiler’s semantic-preservation proof, because floating-point arithmetic is non-associative and the correctness statement has to quantify over rounding error explicitly rather than treating arithmetic as exact — which is precisely why, as of 2026, formally verified numerical libraries remain rare compared to verified compilers and kernels.