{ "capability_id": "engineering.uncertainty.propagate", "tool_id": "uncertainty-propagate", "engine": "uncertainty-propagate", "calculation_version": "1.7.0", "generated_at": "2026-08-10T13:12:00.368Z", "engine_build": "2026-08-10", "test_count": 72, "passed": 72, "failed": 0, "tests_summary": { "golden": { "total": 10, "passed": 10 }, "boundary": { "total": 39, "passed": 39 }, "property": { "total": 23, "passed": 23 } }, "engine_tested": true, "expert_reviewed": true, "internally_reviewed": false, "review_kind": "expert", "source_checked": true, "reviewed_by": "CalculatorX engineering review", "reviewer_role": "metrology-engineer", "last_reviewed": "2026-08-10", "expert_review": { "status": "approved", "reviewer": { "name": "CalculatorX engineering review", "role": "metrology-engineer" }, "reviewed_at": "2026-08-10", "scope": [ "formula", "input semantics", "assumptions", "limitations", "sources", "tests" ], "kind": "expert" }, "internal_review": { "status": "not_applicable", "reviewer": null }, "page": "https://www.calculatorx.com/calc/engineering/uncertainty-propagate", "api": "https://www.calculatorx.com/api/v1/calc/uncertainty-propagate", "schema_input": "https://www.calculatorx.com/schemas/engineering.uncertainty.propagate.input.json", "schema_output": "https://www.calculatorx.com/schemas/engineering.uncertainty.propagate.output.json", "assumptions": [ "Default uncorrelated (rho=0); optional correlations when provided.", "Linear propagation assumes a local first-order approximation is adequate. For significant model nonlinearity, consider higher-order treatment (JCGM 100:2008/Amd.1:2026) or Monte Carlo propagation (JCGM 101:2008)." ], "limitations": [ "Linear (first-order) GUM; default uncorrelated (rho=0). Optional binary rho or sensitivity/budget correlations[]. For significant nonlinearity see JCGM 100 Amd.1:2026 / Monte Carlo (JCGM 101)", "budget mode is a named-component sensitivity API (Type A/B labels + distributions are metadata) — not a full interactive workspace UI / Monte Carlo", "Optional k, or confidence → Student-t k from νeff (missing/∞ νeff → normal quantile)", "contributions are diagonal (c·u)² fractions of u_c²; with correlations percents may not sum to 100%", "Product/quotient use absolute first-order forms; quotient rejects b=0 (DIVISION_BY_ZERO); u_rel omitted when y=0", "Optional unit_a/unit_b are labels only (sum/difference require match → UNIT_MISMATCH); not a full quantity/SI engine", "rho ∉ [-1,1] or confidence ∉ (0,1) → VALUE_OUT_OF_RANGE; k and confidence together → INVALID_INPUT; RSS rejects correlations" ], "boundary_conditions": [ "Missing required fields → MISSING_REQUIRED_INPUT", "Negative ua/ub → VALUE_MUST_BE_NON_NEGATIVE", "Unknown mode → INVALID_MODE", "Quotient with b=0 → DIVISION_BY_ZERO", "Product with a=0 or b=0 is allowed (absolute form); u_rel omitted when y=0", "sum/difference with unit_a ≠ unit_b → UNIT_MISMATCH", "rho ∉ [-1,1] → VALUE_OUT_OF_RANGE; k≤0 → VALUE_MUST_BE_POSITIVE", "RSS with correlations → INVALID_INPUT" ], "sources": [ "JCGM 100:2008 — Evaluation of measurement data (GUM) — Law of propagation of uncertainty — u_c² = Σ (∂f/∂xᵢ)² u²(xᵢ) for uncorrelated inputs (https://www.bipm.org/en/committees/jc/jcgm/publications)", "JCGM 100:2008/Amd.1:2026 — Nonlinearity in measurement models — Amendment 1 — First-order linear propagation may be inadequate when the measurement model is significantly nonlinear (https://www.bipm.org/en/committees/jc/jcgm/publications)", "JCGM GUM-6:2020 — Developing and using measurement models — Measurement modelling — Guidance on constructing measurement models for uncertainty evaluation (https://www.bipm.org/en/committees/jc/jcgm/publications)", "JCGM 101:2008 — Propagation of distributions using a Monte Carlo method — GUM Supplement 1 — Reference for Monte Carlo propagation when linear approximation is inadequate (out of scope for this calculator) (https://www.bipm.org/en/committees/jc/jcgm/publications)", "NIST Technical Note 1297 — Guidelines for evaluating and expressing uncertainty — Combined standard uncertainty practice; reporting of u_c (https://www.nist.gov/pml/nist-technical-note-1297)", "NIST Guide to the SI (SP 811) — Expression of uncertainty with SI quantities — Unit-consistent reporting of results and uncertainties (https://www.nist.gov/pml/special-publication-811)" ], "fingerprint": "sha256:32a709103df1467f4ef3e85aae6379024e3ebc82962f7207976f6bbf83ee748b", "tests": [ { "id": "sum-default", "ok": true, "kind": "golden", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2 }, "expect": { "y": 150, "u_c": 0.22360679774997896 }, "actual": { "mode": "sum", "y": 150, "u_c": 0.223606797749979, "u_rel": 0.00149071198499986, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 20, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 80, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 80 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.22, "y": 150, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "product-default", "ok": true, "kind": "golden", "inputs": { "mode": "product", "a": 10, "ua": 0.1, "b": 5, "ub": 0.05 }, "expect": { "y": 50, "u_c": 0.7071067811865476 }, "actual": { "mode": "product", "y": 50, "u_c": 0.7071067811865476, "u_rel": 0.01414213562373095, "a": 10, "b": 5, "ua": 0.1, "ub": 0.05, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 5, "u": 0.1, "variance": 0.25, "percent": 49.999999999999986, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 10, "u": 0.05, "variance": 0.25, "percent": 49.999999999999986, "type": null, "nu": null } ], "largest": { "i": 0, "label": "a", "percent": 49.999999999999986 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.71, "y": 50, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "sensitivity", "ok": true, "kind": "golden", "inputs": { "mode": "sensitivity", "coefficients": [ 2, 3 ], "uncertainties": [ 0.1, 0.2 ], "y": 10 }, "expect": { "u_c": 0.6324555320336759, "u_rel": 0.06324555320336758 }, "actual": { "mode": "sensitivity", "u_c": 0.632455532033676, "u_rel": 0.0632455532033676, "y": 10, "unit": null, "terms": 2, "correlations": [], "contributions": [ { "i": 0, "label": null, "c": 2, "u": 0.1, "variance": 0.04000000000000001, "percent": 9.999999999999998, "type": null, "nu": null }, { "i": 1, "label": null, "c": 3, "u": 0.2, "variance": 0.3600000000000001, "percent": 89.99999999999999, "type": null, "nu": null } ], "largest": { "i": 1, "label": null, "percent": 89.99999999999999 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.63, "y": 10, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "rss", "ok": true, "kind": "golden", "inputs": { "mode": "rss", "uncertainties": [ 0.3, 0.4 ] }, "expect": { "u_c": 0.5 }, "actual": { "mode": "rss", "u_c": 0.5, "y": null, "unit": null, "terms": 2, "contributions": [ { "i": 0, "label": null, "c": 1, "u": 0.3, "variance": 0.09, "percent": 36, "type": null, "nu": null }, { "i": 1, "label": null, "c": 1, "u": 0.4, "variance": 0.16000000000000003, "percent": 64.00000000000001, "type": null, "nu": null } ], "largest": { "i": 1, "label": null, "percent": 64.00000000000001 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.5, "y": null, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "product-a-zero", "ok": true, "kind": "golden", "inputs": { "mode": "product", "a": 0, "ua": 0.1, "b": 5, "ub": 0.05 }, "expect": { "y": 0, "u_c": 0.5 }, "actual": { "mode": "product", "y": 0, "u_c": 0.5, "u_rel": null, "a": 0, "b": 5, "ua": 0.1, "ub": 0.05, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 5, "u": 0.1, "variance": 0.25, "percent": 100, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 0, "u": 0.05, "variance": 0, "percent": 0, "type": null, "nu": null } ], "largest": { "i": 0, "label": "a", "percent": 100 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.5, "y": 0, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "quotient-b-zero", "ok": true, "kind": "boundary", "inputs": { "mode": "quotient", "a": 1, "ua": 0.1, "b": 0, "ub": 0.05 }, "expectError": "DIVISION_BY_ZERO", "code": "DIVISION_BY_ZERO", "error": "b must be non-zero for quotient" }, { "id": "missing-ua", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 1, "b": 2, "ub": 0.1 }, "expectError": "MISSING_REQUIRED_INPUT", "code": "MISSING_REQUIRED_INPUT", "error": "ua is required" }, { "id": "invalid-mode", "ok": true, "kind": "boundary", "inputs": { "mode": "montecarlo", "a": 1, "ua": 0.1, "b": 2, "ub": 0.1 }, "expectError": "INVALID_MODE", "code": "INVALID_MODE", "error": "unsupported uncertainty mode" }, { "id": "rho-sum", "ok": true, "kind": "golden", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "rho": 1 }, "expect": { "y": 150, "u_c": 0.3, "rho": 1 }, "actual": { "mode": "sum", "y": 150, "u_c": 0.30000000000000004, "u_rel": 0.0020000000000000005, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 1, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 11.11111111111111, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 44.44444444444444, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 44.44444444444444 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.3, "y": 150, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "k-expanded", "ok": true, "kind": "golden", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "k": 2 }, "expect": { "U": 0.4472135954999579, "k": 2 }, "actual": { "mode": "sum", "y": 150, "u_c": 0.223606797749979, "u_rel": 0.00149071198499986, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 20, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 80, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 80 }, "nu_eff": null, "nu_eff_infinite": false, "k": 2, "U": 0.447213595499958, "k_source": "user", "confidence": null, "reported": { "u_c": 0.22, "y": 150, "significant_digits": 2, "k": 2, "U": 0.45 } }, "tol": 1e-12 }, { "id": "nueff-binary", "ok": true, "kind": "golden", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "nua": 9, "nub": "inf" }, "expect": { "nu_eff": 225 }, "actual": { "mode": "sum", "y": 150, "u_c": 0.223606797749979, "u_rel": 0.00149071198499986, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 0, "nua": 9, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 20, "type": null, "nu": 9 }, { "i": 1, "label": "b", "c": 1, "u": 0.2, 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"a": 1, "ua": 0.1, "b": 0, "ub": 0.05 }, "expectError": "DIVISION_BY_ZERO", "code": "DIVISION_BY_ZERO", "error": "b must be non-zero for quotient" }, { "id": "boundary-sensitivity", "ok": true, "kind": "boundary", "inputs": { "mode": "sensitivity", "coefficients": [ 2, 3 ], "uncertainties": [ 0.1, 0.2 ], "y": 10 }, "expect": { "u_c": 0.6324555320336759, "u_rel": 0.06324555320336758 }, "actual": { "mode": "sensitivity", "u_c": 0.632455532033676, "u_rel": 0.0632455532033676, "y": 10, "unit": null, "terms": 2, "correlations": [], "contributions": [ { "i": 0, "label": null, "c": 2, "u": 0.1, "variance": 0.04000000000000001, "percent": 9.999999999999998, "type": null, "nu": null }, { "i": 1, "label": null, "c": 3, "u": 0.2, "variance": 0.3600000000000001, "percent": 89.99999999999999, "type": null, "nu": null } ], "largest": { "i": 1, "label": null, "percent": 89.99999999999999 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.63, "y": 10, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-rss", "ok": true, "kind": "boundary", "inputs": { "mode": "rss", "uncertainties": [ 0.3, 0.4 ] }, "expect": { "u_c": 0.5 }, "actual": { "mode": "rss", "u_c": 0.5, "y": null, "unit": null, "terms": 2, "contributions": [ { "i": 0, "label": null, "c": 1, "u": 0.3, "variance": 0.09, "percent": 36, "type": null, "nu": null }, { "i": 1, "label": null, "c": 1, "u": 0.4, "variance": 0.16000000000000003, "percent": 64.00000000000001, "type": null, "nu": null } ], "largest": { "i": 1, "label": null, "percent": 64.00000000000001 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.5, "y": null, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-all-zero-u", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 1, "ua": 0, "b": 1, "ub": 0 }, "expect": { "y": 2, "u_c": 0 }, "actual": { "mode": "sum", "y": 2, "u_c": 0, "u_rel": 0, "a": 1, "b": 1, "ua": 0, "ub": 0, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0, "variance": 0, "percent": 0, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 1, "u": 0, "variance": 0, "percent": 0, "type": null, "nu": null } ], "largest": { "i": 0, "label": "a", "percent": 0 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0, "y": 2, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-coeffs-length", "ok": true, "kind": "boundary", "inputs": { "mode": "sensitivity", "coefficients": [ 1, 2 ], "uncertainties": [ 0.1 ] }, "expectError": "INVALID_INPUT", "code": "INVALID_INPUT", "error": "coefficients must match uncertainties length" }, { "id": "boundary-empty-uncertainties", "ok": true, "kind": "boundary", "inputs": { "mode": "rss", "uncertainties": [] }, "expectError": "MISSING_REQUIRED_INPUT", "code": "MISSING_REQUIRED_INPUT", "error": "uncertainties array is required" }, { "id": "boundary-neg-ua", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 1, "ua": -0.1, "b": 2, "ub": 0.1 }, "expectError": "VALUE_MUST_BE_NON_NEGATIVE", "code": "VALUE_MUST_BE_NON_NEGATIVE", "error": "ua must be ≥ 0" }, { "id": "boundary-unit-mismatch", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "unit_a": "V", "unit_b": "A" }, "expectError": "UNIT_MISMATCH", "code": "UNIT_MISMATCH", "error": "unit_a and unit_b must match for sum/difference" }, { "id": "boundary-unit-sum", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "unit_a": "V", "unit_b": "V" }, "expect": { "y": 150, "u_c": 0.22360679774997896, "unit": "V" }, "actual": { "mode": "sum", "y": 150, "u_c": 0.223606797749979, "u_rel": 0.00149071198499986, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 0, "nua": null, "nub": null, "unit": "V", "unit_a": "V", "unit_b": "V", "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 20, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 80, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 80 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.22, "y": 150, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-unit-product", "ok": true, "kind": "boundary", "inputs": { "mode": "product", "a": 230, "ua": 1, "b": 12, "ub": 0.1, "unit_a": "V", "unit_b": "A" }, "expect": { "y": 2760, "unit": "V·A" }, "actual": { "mode": "product", "y": 2760, "u_c": 25.942243542145693, "u_rel": 0.0093993636022267, "a": 230, "b": 12, "ua": 1, "ub": 0.1, "rho": 0, "nua": null, "nub": null, "unit": "V·A", "unit_a": "V", "unit_b": "A", "contributions": [ { "i": 0, "label": "a", "c": 12, "u": 1, "variance": 144, "percent": 21.396731054977714, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 230, "u": 0.1, "variance": 529, "percent": 78.60326894502231, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 78.60326894502231 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 26, "y": 2760, "significant_digits": 2 } }, "tol": 1e-9 }, { "id": "boundary-missing-ua", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 1, "b": 2, "ub": 0.1 }, "expectError": "MISSING_REQUIRED_INPUT", "code": "MISSING_REQUIRED_INPUT", "error": "ua is required" }, { "id": "boundary-invalid-mode", "ok": true, "kind": "boundary", "inputs": { "mode": "montecarlo", "a": 1, "ua": 0.1, "b": 2, "ub": 0.1 }, "expectError": "INVALID_MODE", "code": "INVALID_MODE", "error": "unsupported uncertainty mode" }, { "id": "boundary-rho-sum", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "rho": 1 }, "expect": { "y": 150, "u_c": 0.3, "rho": 1 }, "actual": { "mode": "sum", "y": 150, "u_c": 0.30000000000000004, "u_rel": 0.0020000000000000005, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 1, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 11.11111111111111, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 44.44444444444444, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 44.44444444444444 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.3, "y": 150, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-rho-diff", "ok": true, "kind": "boundary", "inputs": { "mode": "difference", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "rho": 1 }, "expect": { "y": 50, "u_c": 0.1, "rho": 1 }, "actual": { "mode": "difference", "y": 50, "u_c": 0.1, "u_rel": 0.002, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 1, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 100, "type": null, "nu": null }, { "i": 1, "label": "b", "c": -1, "u": 0.2, "variance": 0.04000000000000001, "percent": 400, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 400 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.1, "y": 50, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-rho-out", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 1, "ua": 0.1, "b": 2, "ub": 0.1, "rho": 1.5 }, "expectError": "VALUE_OUT_OF_RANGE", "code": "VALUE_OUT_OF_RANGE", "error": "rho must be in [-1, 1]" }, { "id": "boundary-k-expanded", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "k": 2 }, "expect": { "U": 0.4472135954999579, "k": 2 }, "actual": { "mode": "sum", "y": 150, "u_c": 0.223606797749979, "u_rel": 0.00149071198499986, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 20, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 80, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 80 }, "nu_eff": null, "nu_eff_infinite": false, "k": 2, "U": 0.447213595499958, "k_source": "user", "confidence": null, "reported": { "u_c": 0.22, "y": 150, "significant_digits": 2, "k": 2, "U": 0.45 } }, "tol": 1e-12 }, { "id": "boundary-k-invalid", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 1, "ua": 0.1, "b": 2, "ub": 0.1, "k": 0 }, "expectError": "VALUE_MUST_BE_POSITIVE", "code": "VALUE_MUST_BE_POSITIVE", "error": "k must be > 0" }, { "id": "boundary-sens-corr", "ok": true, "kind": "boundary", "inputs": { "mode": "sensitivity", "coefficients": [ 1, 1 ], "uncertainties": [ 0.1, 0.2 ], "correlations": [ { "i": 0, "j": 1, "rho": 1 } ] }, "expect": { "u_c": 0.3 }, "actual": { "mode": "sensitivity", "u_c": 0.30000000000000004, "u_rel": null, "y": null, "unit": null, "terms": 2, "correlations": [ { "i": 0, "j": 1, "rho": 1 } ], "contributions": [ { "i": 0, "label": null, "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 11.11111111111111, "type": null, "nu": null }, { "i": 1, "label": null, "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 44.44444444444444, "type": null, "nu": null } ], "largest": { "i": 1, "label": null, "percent": 44.44444444444444 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.3, "y": null, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-rss-corr-reject", "ok": true, "kind": "boundary", "inputs": { "mode": "rss", "uncertainties": [ 0.1, 0.2 ], "correlations": [ { "i": 0, "j": 1, "rho": 0.5 } ] }, "expectError": "INVALID_INPUT", "code": "INVALID_INPUT", "error": "correlations require sensitivity mode" }, { "id": "boundary-contrib-sum", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2 }, "expect": { "u_c": 0.22360679774997896 }, "actual": { "mode": "sum", "y": 150, "u_c": 0.223606797749979, "u_rel": 0.00149071198499986, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 20, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 80, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 80 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.22, "y": 150, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-nueff-binary", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "nua": 9, "nub": "inf" }, "expect": { "nu_eff": 225 }, "actual": { "mode": "sum", "y": 150, "u_c": 0.223606797749979, "u_rel": 0.00149071198499986, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 0, "nua": 9, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 20, "type": null, "nu": 9 }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 80, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 80 }, "nu_eff": 225.00000000000003, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.22, "y": 150, "significant_digits": 2 } }, "tol": 1e-9 }, { "id": "boundary-nueff-sens", "ok": true, "kind": "boundary", "inputs": { "mode": "sensitivity", "coefficients": [ 2, 1 ], "uncertainties": [ 0.1, 0.2 ], "dof": [ 8, null ], "types": [ "A", "B" ], "labels": [ "rep", "cal" ] }, "expect": { "nu_eff": 32 }, "actual": { "mode": "sensitivity", "u_c": 0.28284271247461906, "u_rel": null, "y": null, "unit": null, "terms": 2, "correlations": [], "contributions": [ { "i": 0, "label": "rep", "c": 2, "u": 0.1, "variance": 0.04000000000000001, "percent": 49.99999999999999, "type": "A", "nu": 8 }, { "i": 1, "label": "cal", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 49.99999999999999, "type": "B", "nu": null } ], "largest": { "i": 0, "label": "rep", "percent": 49.99999999999999 }, "nu_eff": 32.000000000000014, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.28, "y": null, "significant_digits": 2 } }, "tol": 1e-9 }, { "id": "boundary-dof-length", "ok": true, "kind": "boundary", "inputs": { "mode": "rss", "uncertainties": [ 0.1, 0.2 ], "dof": [ 5 ] }, "expectError": "INVALID_INPUT", "code": "INVALID_INPUT", "error": "dof must match uncertainties length" }, { "id": "boundary-nua-alone", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 1, "ua": 0.1, "b": 2, "ub": 0.1, "nua": 10 }, "expectError": "INVALID_INPUT", "code": "INVALID_INPUT", "error": "nua and nub must both be set when using binary degrees of freedom" }, { "id": "boundary-student-t", "ok": true, "kind": "boundary", "inputs": { "mode": "rss", "uncertainties": [ 0.1 ], "dof": [ 9 ], "confidence": 0.95 }, "expect": { "nu_eff": 9, "k": 2.26215716274 }, "actual": { "mode": "rss", "u_c": 0.1, "y": null, "unit": null, "terms": 1, "contributions": [ { "i": 0, "label": null, "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 100, "type": null, "nu": 9 } ], "largest": { "i": 0, "label": null, "percent": 100 }, "nu_eff": 9, "nu_eff_infinite": false, "k": 2.2621571627981485, "U": 0.22621571627981485, "k_source": "student_t", "confidence": 0.95, "reported": { "u_c": 0.1, "y": null, "significant_digits": 2, "k": 2.2621571627981485, "U": 0.23 } }, "tol": 1e-8 }, { "id": "boundary-student-normal", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 100, "ua": 0.1, "b": 50, "ub": 0.2, "confidence": 0.95 }, "expect": { "k": 1.95996398454 }, "actual": { "mode": "sum", "y": 150, "u_c": 0.223606797749979, "u_rel": 0.00149071198499986, "a": 100, "b": 50, "ua": 0.1, "ub": 0.2, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 20, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 80, "type": null, "nu": null } ], "largest": { "i": 1, "label": "b", "percent": 80 }, "nu_eff": null, "nu_eff_infinite": false, "k": 1.959963986120195, "U": 0.43826127064162107, "k_source": "student_t", "confidence": 0.95, "reported": { "u_c": 0.22, "y": 150, "significant_digits": 2, "k": 1.959963986120195, "U": 0.44 } }, "tol": 1e-7 }, { "id": "boundary-k-and-confidence", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 1, "ua": 0.1, "b": 2, "ub": 0.1, "k": 2, "confidence": 0.95 }, "expectError": "INVALID_INPUT", "code": "INVALID_INPUT", "error": "provide either k or confidence, not both" }, { "id": "boundary-confidence-range", "ok": true, "kind": "boundary", "inputs": { "mode": "sum", "a": 1, "ua": 0.1, "b": 2, "ub": 0.1, "confidence": 1.5 }, "expectError": "VALUE_OUT_OF_RANGE", "code": "VALUE_OUT_OF_RANGE", "error": "confidence must be in (0, 1)" }, { "id": "boundary-budget", "ok": true, "kind": "boundary", "inputs": { "mode": "budget", "components": [ { "name": "Vref", "x": 5, "u": 0.002, "c": 1, "type": "B" }, { "name": "drift", "x": 0, "u": 0.001, "c": 1, "type": "B" } ] }, "expect": { "y": 5, "u_c": 0.0022360679774997894 }, "actual": { "mode": "budget", "u_c": 0.0022360679774997894, "u_rel": 0.0004472135954999579, "y": 5, "unit": null, "terms": 2, "correlations": [], "contributions": [ { "i": 0, "label": "Vref", "c": 1, "u": 0.002, "variance": 0.000004, "percent": 80.00000000000001, "type": "B", "nu": null, "distribution": null, "x": 5 }, { "i": 1, "label": "drift", "c": 1, "u": 0.001, "variance": 0.000001, "percent": 20.000000000000004, "type": "B", "nu": null, "distribution": null, "x": 0 } ], "largest": { "i": 0, "label": "Vref", "percent": 80.00000000000001 }, "components": [ { "name": "Vref", "u": 0.002, "c": 1, "type": "B", "nu": null, "distribution": null, "x": 5 }, { "name": "drift", "u": 0.001, "c": 1, "type": "B", "nu": null, "distribution": null, "x": 0 } ], "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.0022, "y": 5, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-budget-named-corr", "ok": true, "kind": "boundary", "inputs": { "mode": "budget", "components": [ { "name": "a", "u": 0.1, "c": 1 }, { "name": "b", "u": 0.2, "c": 1 } ], "correlations": [ { "i": "a", "j": "b", "rho": 1 } ] }, "expect": { "u_c": 0.3 }, "actual": { "mode": "budget", "u_c": 0.30000000000000004, "u_rel": null, "y": null, "unit": null, "terms": 2, "correlations": [ { "i": 0, "j": 1, "rho": 1, "i_name": "a", "j_name": "b" } ], "contributions": [ { "i": 0, "label": "a", "c": 1, "u": 0.1, "variance": 0.010000000000000002, "percent": 11.11111111111111, "type": null, "nu": null, "distribution": null, "x": null }, { "i": 1, "label": "b", "c": 1, "u": 0.2, "variance": 0.04000000000000001, "percent": 44.44444444444444, "type": null, "nu": null, "distribution": null, "x": null } ], "largest": { "i": 1, "label": "b", "percent": 44.44444444444444 }, "components": [ { "name": "a", "u": 0.1, "c": 1, "type": null, "nu": null, "distribution": null, "x": null }, { "name": "b", "u": 0.2, "c": 1, "type": null, "nu": null, "distribution": null, "x": null } ], "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.3, "y": null, "significant_digits": 2 } }, "tol": 1e-12 }, { "id": "boundary-budget-empty", "ok": true, "kind": "boundary", "inputs": { "mode": "budget", "components": [] }, "expectError": "MISSING_REQUIRED_INPUT", "code": "MISSING_REQUIRED_INPUT", "error": "components array is required" }, { "id": "boundary-budget-partial-x", "ok": true, "kind": "boundary", "inputs": { "mode": "budget", "components": [ { "name": "a", "x": 1, "u": 0.1 }, { "name": "b", "u": 0.1 } ] }, "expectError": "INVALID_INPUT", "code": "INVALID_INPUT", "error": "provide x/value for all components or none" }, { "id": "property-sum-y", "ok": true, "kind": "property", "detail": "y=a+b", "expect": 150, "actual": 150 }, { "id": "property-sum-u", "ok": true, "kind": "property", "detail": "u=√(ua²+ub²)", "expect": 0.22360679774997896, "actual": 0.223606797749979 }, { "id": "property-diff-u", "ok": true, "kind": "property", "detail": "diff shares RSS u", "expect": 0.223606797749979, "actual": 0.223606797749979 }, { "id": "property-product", "ok": true, "kind": "property", "detail": "product absolute first-order", "expect": { "y": 50, "u_c": 0.7071067811865476 }, "actual": { "mode": "product", "y": 50, "u_c": 0.7071067811865476, "u_rel": 0.01414213562373095, "a": 10, "b": 5, "ua": 0.1, "ub": 0.05, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 5, "u": 0.1, "variance": 0.25, "percent": 49.999999999999986, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 10, "u": 0.05, "variance": 0.25, "percent": 49.999999999999986, "type": null, "nu": null } ], "largest": { "i": 0, "label": "a", "percent": 49.999999999999986 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.71, "y": 50, "significant_digits": 2 } } }, { "id": "property-product-a-zero", "ok": true, "kind": "property", "detail": "product a=0 → y=0, u_c=|b|ua, u_rel null", "expect": { "y": 0, "u_c": 0.5, "u_rel": null }, "actual": { "mode": "product", "y": 0, "u_c": 0.5, "u_rel": null, "a": 0, "b": 5, "ua": 0.1, "ub": 0.05, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 5, "u": 0.1, "variance": 0.25, "percent": 100, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 0, "u": 0.05, "variance": 0, "percent": 0, "type": null, "nu": null } ], "largest": { "i": 0, "label": "a", "percent": 100 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.5, "y": 0, "significant_digits": 2 } } }, { "id": "property-quotient-a-zero", "ok": true, "kind": "property", "detail": "quotient a=0 → u_c=|ua/b|, u_rel null", "expect": { "y": 0, "u_c": 0.02, "u_rel": null }, "actual": { "mode": "quotient", "y": 0, "u_c": 0.020000000000000004, "u_rel": null, "a": 0, "b": 5, "ua": 0.1, "ub": 0.05, "rho": 0, "nua": null, "nub": null, "unit": null, "unit_a": null, "unit_b": null, "contributions": [ { "i": 0, "label": "a", "c": 0.2, "u": 0.1, "variance": 0.0004000000000000002, "percent": 100.00000000000001, "type": null, "nu": null }, { "i": 1, "label": "b", "c": 0, "u": 0.05, "variance": 0, "percent": 0, "type": null, "nu": null } ], "largest": { "i": 0, "label": "a", "percent": 100.00000000000001 }, "nu_eff": null, "nu_eff_infinite": false, "k": null, "U": null, "k_source": null, "confidence": null, "reported": { "u_c": 0.02, "y": 0, "significant_digits": 2 } } }, { "id": "property-quotient-b-zero", "ok": true, "kind": "property", "detail": "quotient b=0 → DIVISION_BY_ZERO", "expect": "DIVISION_BY_ZERO", "actual": true }, { "id": "property-sens", "ok": true, "kind": "property", "detail": "√Σ(c·u)²", "expect": 0.6324555320336759, "actual": 0.632455532033676 }, { "id": "property-rss", "ok": true, "kind": "property", "detail": "3-4-5 RSS", "expect": 0.5, "actual": 0.5 }, { "id": "property-zero-u", "ok": true, "kind": "property", "detail": "all ua=0 → u_c=0", "expect": 0, "actual": 0 }, { "id": "property-reported", "ok": true, "kind": "property", "detail": "reported u_c=0.22 (2 sig); raw retained", "expect": { "reported_u_c": 0.22, "raw": 0.22360679774997896 }, "actual": { "u_c": 0.22, "y": 150, "significant_digits": 2 } }, { "id": "property-unit-match", "ok": true, "kind": "property", "detail": "matching units → V", "expect": "V", "actual": "V" }, { "id": "property-unit-mismatch", "ok": true, "kind": "property", "detail": "V+A → UNIT_MISMATCH", "expect": "UNIT_MISMATCH", "actual": true }, { "id": "property-unit-product", "ok": true, "kind": "property", "detail": "product units → V·A", "expect": "V·A", "actual": "V·A" }, { "id": "property-rho-perfect", "ok": true, "kind": "property", "detail": "sum ρ=1 → ua+ub", "expect": 0.3, "actual": 0.30000000000000004 }, { "id": "property-rho-anti", "ok": true, "kind": "property", "detail": "sum ρ=−1 → |ua−ub|", "expect": 0.1, "actual": 0.1 }, { "id": "property-expanded-U", "ok": true, "kind": "property", "detail": "U=k·u_c with reported.U", "expect": { "U": 0.4472135954999579, "reported_U": 0.45 }, "actual": { "U": 0.447213595499958, "reported_U": 0.45 } }, { "id": "property-sens-rho0", "ok": true, "kind": "property", "detail": "sensitivity ρ=0 matches uncorrelated", "expect": 0.6324555320336759, "actual": 0.632455532033676 }, { "id": "property-contrib", "ok": true, "kind": "property", "detail": "ub dominates → 80%", "expect": { "p0": 20, "p1": 80, "largest": 1 }, "actual": { "p0": 20, "p1": 80, "largest": 1 } }, { "id": "property-nueff", "ok": true, "kind": "property", "detail": "WS νeff with νb→∞", "expect": 225, "actual": 225.00000000000003 }, { "id": "property-types", "ok": true, "kind": "property", "detail": "Type A/B + labels pass-through", "expect": { "type": "A", "label": "y" }, "actual": { "type": "A", "label": "y" } }, { "id": "property-student-t", "ok": true, "kind": "property", "detail": "Student-t k(ν=9, p=0.95)", "expect": { "k": 2.26215716274, "source": "student_t" }, "actual": { "k": 2.2621571627981485, "source": "student_t" } }, { "id": "property-budget", "ok": true, "kind": "property", "detail": "budget y=Σx, Vref largest", "expect": { "y": 5, "largest": "Vref" }, "actual": { "y": 5, "largest": "Vref" } } ], "tier": "expert_verified" }