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1Scan for outdated or missing drivers - takes under a minute2Clear out junk files and repair common Windows errors3Fix the driver behind crashes, sound loss and screen glitchesNeither method is universally better. Formal verification proves a precisely stated property within an explicit mathematical model and its assumptions. Numerical testing runs the implemented program on selected inputs and conditions, revealing concrete failures that may be invisible in an abstract model. For mathematical software, the defensible answer is usually complementary: formally prove high-value properties that can be specified precisely, then test the finite-precision implementation, numerical edge cases, and system interfaces.
What the two methods actually establish
The phrase checking mathematical software hides several different claims. A team may want to know that an algorithm is mathematically correct, that a computed result stays within an error bound, that a failure class is impossible, or that a production implementation behaves acceptably for representative workloads. No single technique supplies all of that evidence.
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Formal verification: a conditional proof
Formal verification starts with a formal specification: preconditions, postconditions, invariants, numerical bounds, or other properties written in a language precise enough to reason about. A model represents the program, algorithm, or hardware/software boundary. A proof then shows that the modeled behavior satisfies the stated property whenever the assumptions hold.
The result is stronger than observing a list of executions, but it is conditional. The proof does not establish that the specification expresses the real engineering objective, that every relevant environmental assumption is true, or that the implementation has been modeled faithfully. A proof can therefore be valid and still support the wrong claim.
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JPL guidance makes this limitation explicit: formal methods help only when designers understand what needs to be proved and have specifications that correspond to the system’s objectives. The specification is part of the assurance case, not paperwork that can be separated from it.
Numerical testing: evidence from executions
Numerical testing executes the actual implementation with selected inputs, data sets, machine conditions, and interfaces. The test oracle—an expected value, tolerance, invariant, reference implementation, or conservation law—determines whether each result is acceptable.
Tests can expose concrete defects in parsing, units, array indexing, memory management, compiler settings, parallel execution, exception handling, rounding, overflow, underflow, and integration. They can also reveal that a mathematically reasonable algorithm is unsuitable for a particular numerical regime.
Testing is bounded by what is exercised. Passing a finite set of tests supports the tested cases and conditions; it does not show that every untested input is safe. Test data and oracles can also be incomplete or wrong, so tests do not automatically establish the entire intended specification.
Why the real-number proof and the floating-point program can diverge
Many mathematical algorithms are first described over the real numbers. Production code uses finite-precision representations. Rounding changes intermediate values, cancellation can destroy significant digits, and exceptional ranges can produce overflow, underflow, infinities, or other behavior absent from the ideal model. A theorem about the real-number algorithm therefore does not automatically transfer to the implementation.
Goodloe, Muñoz, Kirchner, and Correnson state the point directly in their 2013 technical paper: “Therefore one cannot assert that theorems proven in the setting of the real numbers carry over to the implementation without additional arguments.” Those additional arguments may include a finite-precision model, a proved error bound, range restrictions, a stability argument, or a verified connection between abstract operations and machine operations.
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Testing complements that reasoning by running the real arithmetic. It can trigger a rounding boundary or exceptional value that the proof model omitted. It still samples only selected situations, so it cannot replace a general argument about all admissible values.
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Side-by-side comparison
| Decision axis | Formal verification | Numerical testing |
|---|---|---|
| Claim supported | A stated property is proved within a model and its assumptions. | Observed outcomes match the selected expectations for tested inputs and conditions. |
| Input coverage | Can generalize over modeled domains; proof complexity and scope limit what is practical. | Covers chosen examples, boundaries, random samples, and execution paths; untested cases remain. |
| Specification risk | A wrong, incomplete, or misaligned specification can yield a proof of the wrong claim. | Test oracles can also be wrong; tests may expose mismatches without establishing the full intended specification. |
| Floating-point behavior | Must reason about the implementation’s finite precision, not only the ideal real-number algorithm. | Runs the implementation and can expose observed rounding, overflow, underflow, and integration behavior for selected cases. |
| Typical expertise | Mathematical modeling, specification writing, proof effort, and specialized tool expertise. | Test design, numerical analysis, reference oracles, automation, and environment control. |
| Best role | Establish high-value invariants, bounds, and correctness properties that can be stated precisely. | Check executable behavior, concrete edge cases, numerical regimes, interfaces, and regressions. |
This is a qualitative synthesis of NASA and JPL guidance and technical publications, not a head-to-head benchmark. The available sources do not establish a universal cost ratio, defect-detection rate, or superiority claim.
Can formal verification prove numerical software is correct?
It can prove a carefully delimited notion of correctness. For example, a proof might show that an implementation preserves an invariant, returns a result satisfying a postcondition whenever its preconditions hold, or stays within a specified error bound for a defined range of inputs. If the model includes finite-precision operations and the proof covers the compiled or otherwise deployed artifact, the conclusion can be highly strong.
That conclusion remains conditional on four links:
- Objective to specification: the formal property must represent what users and system engineers actually require.
- Inputs to assumptions: documented ranges, units, formats, and environmental conditions must hold in operation.
- Algorithm to implementation: the modeled code must correspond to the code that runs, including arithmetic modes and exceptional behavior.
- Model to machine: any abstraction of finite precision, libraries, compilers, hardware, concurrency, or external services needs justification.
If one link is missing, the proof establishes only the narrower modeled claim. Saying that the software is simply “correct” overstates what the evidence shows.
Do you still need tests if software is formally verified?
Yes. A formally verified component still needs tests for behavior that the proof does not represent: actual floating-point execution, input conversion, file and network interfaces, configuration, performance-related limits, integration with unverified components, deployment assumptions, and regression protection during maintenance.
Testing is also a practical way to challenge the assumptions used in a proof. A boundary test may show that an advertised input range is violated in a real data pipeline. A comparison against an independent implementation can expose a translation error between the mathematical specification and production code. These findings do not invalidate a correctly scoped theorem; they show that the system being operated is broader than the theorem.
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NASA’s Software Engineering Handbook, SWE-028, says software verification and validation are performed in parallel with software development rather than only at the end. Its guidance treats testing, analysis, inspection, assessment, and other activities as selectable verification methods, not mutually exclusive alternatives.
What to decide before choosing a method
Start with the claim, not the tool. Write down what evidence the project needs and what failure would matter most.
Algorithmic correctness
If the central question is whether an algorithm preserves an invariant or returns a mathematically defined result under stated preconditions, formal verification is a strong candidate. Define the domain, termination expectations, exceptional cases, and postcondition before estimating proof effort.
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For numerical algorithms, specify the quantity being bounded, the norm or error measure, the allowable range, and whether the bound concerns exact arithmetic, finite precision, or both. A real-number convergence theorem is not a finite-precision error guarantee unless the connection is established.
Concrete failure modes
If the risk involves malformed input, unit conversion, overflow at a known threshold, a particular hardware instruction, a library call, or a service interface, executable tests are indispensable. Those behaviors may not exist in the abstract proof model.
Integration and operational behavior
System-level evidence usually requires tests, inspections, analysis, or demonstrations across component boundaries. A proof about one routine does not establish that callers supply valid units or that a downstream consumer interprets the result correctly.
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A practical combined workflow
- State the intended claim. Describe whether you need an invariant, an error bound, absence of a failure class, or acceptable behavior for representative workloads. Record what “acceptable” means numerically.
- Define the mathematical model. Specify input domains, units, initial conditions, exact or approximate arithmetic, rounding assumptions, exceptional values, termination conditions, and environmental dependencies.
- Partition the assurance case. Mark properties suitable for proof, behaviors that require execution, and assumptions that need inspection or independent evidence. Do not force an inherently operational question into a mathematical model.
- Prove the highest-value properties. Prioritize invariants, conservation laws, bounds, domain restrictions, and failure exclusions whose loss would create significant risk. Track every assumption as a reviewable item.
- Connect the proof to finite precision. Add rounding, overflow, underflow, range, and exceptional-value reasoning where relevant. If the proof remains over real numbers, label it as an algorithm-level result rather than an implementation guarantee.
- Build numerical tests around risk. Include nominal values, domain boundaries, just-inside and just-outside cases, zero and sign changes, extreme magnitudes, repeated operations, ill-conditioned inputs, exceptional values, and representative production data.
- Use independent oracles where possible. Compare with a high-precision calculation, an independently implemented method, a known identity, a conservation relation, or a tolerance justified by numerical analysis. Document why the oracle is trustworthy.
- Exercise interfaces and environments. Test parsing, units, serialization, array shapes, threading or parallel boundaries, library versions, configuration, error handling, and deployment hardware. These are common places where an abstract algorithm and a running system diverge.
- Preserve regression evidence. Automate tests that reproduce discovered failures and rerun them after changes. When the implementation, compiler, arithmetic mode, or platform changes, review which proof assumptions and tests need updating.
- Review the complete evidence against risk. NASA recommends tailoring verification plans to requirements, project risk, cost, schedule, and available resources. Record uncovered behaviors rather than implying that one successful technique provides universal assurance.
Designing numerical tests that add information
Boundary and partition tests
Partition the input domain into regions with different numerical behavior. Test exact boundaries, values immediately below and above them, and combinations of limits. For an algorithm that accepts a positive parameter, include zero, the smallest permitted positive value, a typical value, the largest permitted value, and invalid negatives if rejection is required.
Metamorphic and invariant checks
When an exact expected answer is difficult to compute, test relationships that should survive a controlled transformation. Examples include symmetry, monotonicity, conservation, idempotence, or agreement when an input is split and recombined. State the tolerance and the conditions under which the relation is expected to hold.
Reference comparisons
A slower, higher-precision, or independently written reference can expose implementation mistakes. Independence matters: copying the same algorithm, formula, or bug into the oracle gives a false sense of confirmation.
Failure-path tests
Deliberately exercise overflow, underflow, non-finite inputs, singular or ill-conditioned cases, invalid units, malformed records, and interrupted operations. Verify both the numerical result and the reported status, diagnostic, or fallback behavior.
What formal verification work must document
- The exact property proved and the version of the specification.
- Preconditions, invariants, postconditions, input ranges, and environmental assumptions.
- Whether arithmetic is modeled as real, fixed-point, floating-point, interval, or another representation.
- Which libraries, compiler transformations, hardware operations, concurrency effects, and external interfaces are inside the model.
- Proof obligations that were discharged automatically, reviewed manually, or left outside scope.
- The correspondence between verified source or intermediate representation and the deployed executable.
This record prevents a narrow theorem from being presented as a blanket certification of the whole product.
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“The proof passed, so testing is unnecessary.”
Why it fails: the proof may omit conversion, integration, finite-precision effects, or deployment assumptions.
Correction: retain executable tests for the implementation and its interfaces, and test every assumption that can be checked operationally.
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“The test suite is large, so the algorithm is proven.”
Why it fails: even broad finite coverage leaves untested inputs and paths.
Correction: use tests as evidence for selected conditions and add formal properties when a general claim is important and precisely expressible.
“A real-number theorem guarantees floating-point behavior.”
Why it fails: rounding, overflow, underflow, and exceptional ranges can alter the computation.
Correction: prove or analyze the finite-precision implementation, constrain ranges, establish an error bound, or clearly limit the claim to the ideal algorithm.
“A formal specification is automatically the right requirement.”
Why it fails: a formally precise statement can still omit a user need or encode the wrong units, tolerance, or failure policy.
Correction: review the specification with domain experts and trace it to system objectives before investing in extensive proof.
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Why it fails: NASA guidance defines verification and validation as a broader set of activities, including analysis, review, inspection, assessment, and testing.
Correction: select evidence methods for each requirement and risk instead of treating one activity as sufficient by default.
Cost, schedule, and expertise
There is no reliable universal price comparison in the available evidence. Formal work can demand substantial modeling, specification, proof maintenance, and specialist expertise. Testing may be easier to start, but meaningful numerical coverage, robust oracles, multiple platforms, and long-term regression suites also require significant design and execution effort.
Schedule pressure should not turn either method into a checkbox. A small, well-scoped proof of a critical invariant may provide more assurance than a large collection of low-information tests. Conversely, a short boundary test may find an integration defect that a carefully proved algorithm cannot see. Choose the least expensive evidence that credibly addresses each risk, then add independent evidence where the consequences justify it.
How to make the final decision
| If your primary question is… | Emphasize… | Still include… |
|---|---|---|
| Does the algorithm preserve a defined property for all modeled inputs? | Formal specification and proof. | Tests of implementation, assumptions, and interfaces. |
| What happens at numerical extremes on the target platform? | Finite-precision analysis and targeted execution tests. | Proof or analysis of range and error limits where feasible. |
| Does the product handle real data and integrations correctly? | End-to-end and failure-path testing. | Formal contracts for critical component boundaries. |
| Can a high-consequence failure class be ruled out? | Formal proof of the failure exclusion, with explicit assumptions. | Tests that challenge those assumptions and exercise the deployed system. |
The best answer to “which is better?” is therefore a risk-based assurance plan. Formal verification supplies general reasoning where the claim and model are precise. Numerical testing supplies observations of the program that actually runs. Used together, they cover different gaps instead of pretending that either one is universal.
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What the sources say about verification planning
NASA’s software verification planning guidance recommends tailoring activities to the project’s requirements, risks, cost, schedule, and available resources. Its software handbook also cautions against assuming that testing is the only verification method. A 1998 NASA Ames abstract describes mathematical verification as “a possibly useful orthogonal dimension to otherwise standard empirical testing,” which captures the practical relationship: proof adds a different kind of evidence rather than replacing execution-based evidence.
Those statements do not establish a required method, a defect rate, or a cost advantage for any particular project. Safety, regulatory, language, architecture, and deployment constraints can impose additional evidence beyond the approach described here.
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