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Neither method is universally better. Formal verification proves a precisely stated property within a formal model and its assumptions. Numerical testing runs the implemented program on selected inputs and conditions, exposing concrete failures that an abstract model may omit. For mathematical software, the most defensible practice is usually complementary: prove the high-value properties that can be specified precisely, then test the finite-precision implementation, edge cases, and integrations that the proof does not cover.
What each method actually tells you
Formal verification proves a stated claim
Formal verification starts with a specification: a mathematical description of permitted inputs, required outputs, invariants, safety conditions, or error bounds. A proof then shows that the modeled program satisfies that claim for every behavior represented by the model.
That result is stronger than observing a collection of successful runs, but it is conditional. The proof depends on the specification being complete and aligned with the system’s purpose, on assumptions about the environment being true, on the model representing the implementation, and on the proof process and tools being used correctly. A formally verified program can therefore be correct with respect to the wrong property.
Numerical testing observes executions
Numerical testing executes the actual implementation with chosen values and records whether outputs, errors, convergence behavior, and interactions match an oracle or an expected range. Tests can use representative workloads, boundary values, random samples, exceptional values, and deliberately difficult numerical regimes.
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A passing test establishes evidence for the input and conditions that were exercised. It does not establish that every untested input is safe, that every execution path was taken, or that the test oracle expresses the complete intended behavior. Test suites can also contain incorrect expectations, so testing is evidence rather than a universal proof.
Verification is broader than proof or testing
NASA treats software verification and validation (V&V) as a set of activities that can include analysis, review, inspection, assessment, demonstration, and testing. Choosing between a theorem prover and a test runner is therefore only one part of an assurance plan. The appropriate combination depends on the requirement, risk, schedule, available expertise, and resources.
Formal verification and numerical testing compared
| Decision axis | Formal verification | Numerical testing |
|---|---|---|
| Claim supported | A stated property is proved within a model and its assumptions. | Observed outcomes match expectations for tested inputs and conditions. |
| Input coverage | Can generalize over modeled input domains; proof complexity and scope limit what is practical. | Covers selected examples, boundaries, random samples, and execution paths; untested cases remain. |
| Specification risk | An incomplete or misaligned specification can yield a valid proof of the wrong claim. | Test oracles can also be wrong; tests may reveal mismatches without establishing the full specification. |
| Floating-point behavior | Must reason about the implementation’s finite precision, not only an ideal real-number algorithm. | Runs the implementation and can expose observed rounding, overflow, underflow, and integration behavior for selected cases. |
| Cost and expertise | Usually requires mathematical modeling, specification skill, proof effort, and tool expertise. | Can start quickly and remain useful throughout development, while broad coverage may require substantial test design and execution. |
| Best role | Establish high-value invariants, bounds, and correctness properties that can be stated precisely. | Check executable behavior, concrete numerical regimes, edge cases, interfaces, and regressions. |
This is a qualitative comparison, not a measured cost or defect-detection benchmark. Available sources do not establish a universal percentage advantage, cost ratio, or failure-detection rate for either method.
What formal verification can prove for mathematical software
Algorithmic properties
A proof can establish that an algorithm preserves an invariant, terminates under stated conditions, returns a mathematically specified result, or maintains a bound on an intermediate quantity. For a numerical routine, the claim might concern stability conditions, a residual bound, monotonicity, conservation, or the absence of a class of modeled failures.
Generalization over a domain
Once the model and proof are sound, the result can apply to an entire modeled input domain rather than only to the examples selected by a test author. That generalization is the main reason formal methods add evidence that empirical testing cannot provide by itself.
What the proof does not automatically establish
- That the specification captures the user’s real objective.
- That the assumptions about hardware, libraries, inputs, concurrency, or the operating environment hold in deployment.
- That the implementation matches the abstract algorithm.
- That a theorem proved over exact real numbers also holds for finite-precision machine arithmetic.
- That external interfaces, file formats, configuration, performance limits, or integration behavior are correct unless they are included in the model.
JPL guidance emphasizes that formal methods help only when designers understand what must be proved and write specifications that correspond to the system’s objectives. Proof scope should therefore be documented alongside the theorem, not treated as an implicit promise about the whole product.
Why real-number mathematics does not settle floating-point correctness
The abstraction gap
Many mathematical algorithms are first described over real numbers. Production code normally uses finite-precision formats, whose operations can round, overflow, underflow, produce signed zeros or exceptional values, and behave differently across hardware or compiler settings. These effects can change branches, convergence, ordering, and final results.
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As Alwyn E. Goodloe, César Muñoz, Florent Kirchner, and Loïc Correnson state in a 2013 paper: “Therefore one cannot assert that theorems proven in the setting of the real numbers carry over to the implementation without additional arguments.”
What an adequate proof needs
A proof that targets deployed numerical code must connect the mathematical model to the implementation. Depending on the claim, that may require a floating-point error model, rounding analysis, range arguments, treatment of exceptional values, assumptions about evaluation order, or a verified correspondence between the algorithm and code.
For example, proving that an exact real-number iteration converges does not by itself prove that a floating-point implementation reaches the desired tolerance, avoids overflow, or makes progress after rounding. Those are separate obligations unless they are included in the formal specification and proof.
Why tests remain important here
Running the actual binary can reveal rounding behavior, overflow and underflow, library interactions, compiler effects, and platform-specific differences for selected cases. Tests do not replace a general numerical error argument, but they observe phenomena that an idealized model may have excluded.
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How specification risk affects both methods
A proof can be perfectly wrong for the project
If the formal specification omits a required input class, uses an unrealistic precondition, or defines the wrong output relation, a proof can be flawless and still irrelevant to the system’s objective. The same issue appears when a model silently assumes exact arithmetic while the implementation is finite precision.
A test oracle can encode the wrong answer
Tests depend on expected values, tolerances, invariants, or a reference implementation. Those oracles can be incomplete or incorrect. A test suite may pass because it repeats the same mistaken assumption as the code under test, or because its tolerance is too generous to expose a meaningful error.
Make the intended claim explicit
Before choosing a method, write down:
- The mathematical result or behavior users require.
- The input domain, including units, ranges, exceptional values, and invalid-input policy.
- The arithmetic model and relevant environmental assumptions.
- Acceptable error, convergence, performance, and resource limits.
- Which failures are safety-critical, financially significant, or merely inconvenient.
This statement becomes the reference for deciding which parts can be proved, which require executable tests, and which need inspection or analysis.
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What numerical testing can reveal that a proof model may miss
Boundary and exceptional values
Exercise minimum and maximum representable values, zero and signed zero where relevant, subnormal values, infinities, NaNs, empty inputs, repeated values, ill-conditioned data, and values just inside and outside documented limits. These cases can trigger branches and arithmetic behavior absent from ordinary examples.
Representative numerical regimes
Include well-conditioned and poorly conditioned problems, widely different scales, nearly equal quantities that invite cancellation, and workloads that stress iteration limits or memory. The expected result may need to be expressed as an error bound, residual, interval, or invariant rather than exact equality.
Integration and operational behavior
Test the interfaces around the numerical kernel: parsing, units, serialization, configuration, threading, error reporting, dependency versions, and downstream consumers. A theorem about a function’s mathematics does not automatically cover these boundaries.
Regression evidence
Keep failing examples as permanent regression tests after a defect is fixed. Tests can also confirm that a refactoring, compiler change, hardware migration, or dependency update has not altered observed behavior.
How to plan a complementary assurance strategy
- Classify the claim. Decide whether you need algorithmic correctness, an error bound, absence of a failure class, or expected behavior for representative workloads.
- Define the model and domain. Record arithmetic, input ranges, environmental assumptions, units, and exceptional-value behavior.
- Partition the obligations. Mark claims suitable for formal proof, claims best observed in the executable implementation, and claims requiring review, inspection, analysis, or demonstration.
- Prove the highest-value properties. Prioritize invariants, bounds, safety conditions, and mathematical relations whose failure would have serious consequences and whose specifications can be made precise.
- Connect proof to code. Check that the verified algorithm, data representation, compiler assumptions, libraries, and arithmetic model correspond to the released implementation.
- Design numerical tests. Cover boundaries, representative regimes, exceptional values, randomized samples where useful, and every important interface.
- Use independent oracles when possible. Compare against a trusted analytical result, a higher-precision calculation, interval or arbitrary-precision arithmetic, conservation laws, or an independently implemented method.
- Run tests continuously. NASA’s Software Engineering Handbook, SWE-028, says: “Software V&V is performed in parallel with software development, not at the conclusion of the development effort.” Integrate tests and review evidence as the code and requirements evolve.
- Review assumptions and failures. When a test fails, determine whether the implementation, specification, oracle, environment, or proof model is wrong. When a proof is difficult, reconsider whether the property is stated at the right abstraction level rather than weakening it silently.
- Document the evidence boundary. Record what was proved, what was tested, under which conditions, and which behaviors remain outside the assurance case.
Choosing the emphasis by risk and project constraints
| Project situation | Evidence to emphasize | Reason |
|---|---|---|
| A small, well-specified mathematical kernel with high consequence if wrong | Formal invariants, bounds, and a code-to-model argument, plus targeted implementation tests | The property may be precise enough to prove, while finite-precision and integration behavior still need observation. |
| Rapidly changing exploratory numerical code | Executable tests, diagnostics, reviews, and gradually formalized high-value properties | Requirements and interfaces may change faster than a complete proof can be maintained. |
| Code dominated by platform, I/O, or third-party behavior | Integration, boundary, fault-injection, and regression testing, with proofs for isolated logic where practical | The important behavior may lie outside a tractable mathematical model. |
| Safety- or mission-critical numerical computation | A tailored combination of proof, numerical analysis, testing, inspection, and independent review | The assurance case must address both general properties and concrete implementation hazards. |
| Limited proof expertise or schedule | Risk-based testing and analysis first, while reserving formal work for the most valuable claims | NASA guidance calls for verification plans that reflect requirements, risk, cost, schedule, and available resources. |
This table is a planning aid, not a mandatory standard. A project’s language, architecture, safety case, regulatory obligations, and contractual requirements can impose additional evidence.
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Common mistakes and how to correct them
“The theorem proves the product is correct.”
Problem: The theorem may cover only an abstract routine or a restricted domain.
Correction: State the exact theorem, preconditions, arithmetic model, implementation correspondence, and excluded behavior. Add tests for the remaining executable and integration obligations.
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“The test suite passes, so every input is safe.”
Problem: Passing tests say nothing directly about untested inputs and paths.
Correction: Expand boundary and randomized coverage, use properties that apply across many inputs, and prove general claims where the risk justifies the effort.
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“A real-number error bound applies to floating point.”
Problem: Rounding, overflow, underflow, and exceptional values can invalidate the transfer.
Correction: Add finite-precision reasoning and executable tests that target the ranges and operations where the assumptions may fail.
“Formal methods and tests compete for the same job.”
Problem: The methods produce different kinds of evidence.
Correction: Assign each obligation to the method that can address it most directly, then look for independent evidence on high-risk claims.
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Problem: Late discovery of ambiguous requirements or untestable properties is expensive.
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Correction: Define claims, assumptions, proof targets, test oracles, and review activities alongside development. NASA explicitly describes V&V as parallel with software development.
Can formal verification prove numerical software is correct?
It can prove a carefully defined correctness property of numerical software, including properties involving finite-precision arithmetic, when the specification, model, implementation relationship, assumptions, and proof are all adequate. It cannot, by the word “verified” alone, prove that the software meets an unstated user objective, behaves correctly outside the modeled environment, or implements an ideal real-number theorem without a finite-precision argument.
The useful question is therefore “correct with respect to which claim, domain, arithmetic model, and environment?” A precise answer to that question determines whether the proof is sufficient for one obligation or only one part of the assurance case.
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Yes. Tests remain valuable for the compiled implementation, floating-point behavior, exceptional ranges, interfaces, dependencies, configuration, and integration paths that may not be represented in the proof. They also provide concrete regression evidence after changes. The tests should be designed around obligations outside the proof as well as around high-risk assumptions that deserve independent confirmation.
The practical verdict
Formal verification is the stronger tool for general claims that can be stated precisely and modeled faithfully. Numerical testing is the indispensable tool for observing the implementation under real arithmetic, real interfaces, and selected operating conditions. Neither establishes the other’s evidence. For mathematical software, use proof to cover what testing cannot generalize over, and use numerical tests to challenge the assumptions, arithmetic, boundaries, and integrations that a proof may abstract away.
NASA Ames researchers Michael Lowry, Mark Boyd, and Deepak Kulkarni described mathematical verification as “a possibly useful orthogonal dimension to otherwise standard empirical testing.” That is the most practical answer to the comparison: build an assurance case in which the two methods address different failure modes, with the balance set by the project’s claims and risks.
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