{
  "capability_id": "math.percent_error",
  "tool_id": "percent-error",
  "engine": "percent-error",
  "calculation_version": "1.1.4",
  "generated_at": "2026-09-16T22:03:43.775Z",
  "engine_build": "2026-09-10",
  "test_count": 14,
  "passed": 14,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 10,
      "passed": 10
    },
    "boundary": {
      "total": 4,
      "passed": 4
    },
    "property": {
      "total": 0,
      "passed": 0
    }
  },
  "engine_tested": true,
  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
  "reviewed_by": null,
  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
    "reviewer": null,
    "scope": [
      "formula",
      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
    ],
    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
  },
  "internal_review": {
    "status": "not_applicable",
    "reviewer": null
  },
  "page": "https://www.calculatorx.com/calc/math/percent-error",
  "api": "https://www.calculatorx.com/api/v1/calc/percent-error",
  "schema_input": "https://www.calculatorx.com/schemas/math.percent_error.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.percent_error.output.json",
  "assumptions": [
    "Interactive evaluation runs locally in the browser. Shared URLs and REST may evaluate the same versioned engine on the server (query SSR).",
    "Accepted value T ≠ 0.",
    "Default formula uses absolute value so % error ≥ 0.",
    "Exactly two fields are inputs. Explicit solve_for selects the output. If all three are supplied without solve_for, they are checked for consistency rather than overwritten."
  ],
  "limitations": [
    "T = 0 → TRUE_VALUE_ZERO",
    "Inconsistent triples may return status inconsistent",
    "Reverse solve edge cases → NO_REAL_SOLUTION / INFINITELY_MANY"
  ],
  "boundary_conditions": [
    "T = 0 → TRUE_VALUE_ZERO (undefined)",
    "Fewer than two inputs → MISSING_REQUIRED_INPUT",
    "Invalid solve_for → INVALID_SOLVE_FOR",
    "Negative P in absolute mode → INVALID_PERCENT",
    "O = 0, P = 100%, absolute, solve T → INFINITELY_MANY (any T ≠ 0)",
    "O = 0, P ≠ 100%, absolute, solve T → NO_REAL_SOLUTION",
    "O ≠ 0, P = 100%, absolute, solve T → T = O/2; the 1 − P/100 branch is omitted (not a full calculation error)",
    "O = 0, P = −100%, signed, solve T → INFINITELY_MANY (any T ≠ 0)",
    "O = 0, P ≠ −100%, signed, solve T → NO_REAL_SOLUTION",
    "O ≠ 0, P = −100%, signed, solve T → NO_REAL_SOLUTION",
    "Solving from P may yield two candidates (±) when absolute mode is on",
    "All three values filled without solve_for and not matching the formula → INCONSISTENT_INPUTS"
  ],
  "sources": [
    {
      "title": "JCGM 200:2012 — International vocabulary of metrology (VIM)",
      "section": "§2.16 measurement error",
      "url": "https://www.bipm.org/en/committees/jc/jcgm/publications",
      "supports": "Defines measurement error as measured quantity value minus a reference quantity value (ε = O − T). Does not define relative error — VIM §2.18 is measurement bias, not relative error. Does not prescribe a percent-error ×100% laboratory convention.",
      "kind": "reference",
      "accessed": "2026-08-28"
    },
    {
      "title": "NIST Dataplot PERCERR",
      "section": "Percent error of a true/accepted value vs an observed value",
      "url": "https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/percerr.htm",
      "supports": "Documents percent error from an accepted (true) value and an observed value, and notes that some conventions take the absolute value. Instantly readable page — not the frameset handbook root.",
      "kind": "reference",
      "accessed": "2026-08-28"
    },
    {
      "title": "NIST Dataplot RELERR",
      "section": "Relative error of a true/accepted value vs an observed value",
      "url": "https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/relerr.htm",
      "supports": "Documents relative error (Xt − Xo)/Xt. Notes that some sources take the absolute value, and some reverse numerator order — those change sign, not magnitude.",
      "kind": "reference",
      "accessed": "2026-08-28"
    },
    {
      "title": "CalculatorX laboratory convention",
      "section": "Absolute-value percent error, signed opt-in, and reverse-solve states",
      "supports": "Default % error = |O−T|/|T|×100% so the result is ≥ 0; signed (O−T)/T×100% is opt-in. Displayed absolute error is |ε|, the magnitude of VIM measurement error — not a separate VIM term. Absolute reverse-solves publish both ± candidates, omit a zero-denominator branch, and publish INFINITELY_MANY when every T ≠ 0 works. Signed P = −100% with O ≠ 0 is NO_REAL_SOLUTION.",
      "kind": "definition"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:783cd14e12cb9811044d6bc3fce9d6c1f6fd3cf96201ae81a3916d8ad4aad9f1",
  "tests": [
    {
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      "ok": true,
      "kind": "golden",
      "inputs": {
        "experimental": 101.5,
        "theoretical": 100
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      "expect": 1.5,
      "expectWarning": null,
      "actual": {
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        "value": 1.5,
        "observed": 101.5,
        "true": 100,
        "percent_error": 1.5,
        "absolute_error": 1.5,
        "relative_error": 0.015,
        "signed_percent_error": 1.5,
        "absolute": true,
        "direction": "observed_above",
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        "solution_type": "unique",
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          "significant_digits": 10,
          "trailing_zeros": "stripped"
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        "display": {
          "percent_error": "1.5%",
          "value": "1.5"
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      },
      "tol": 1e-9
    },
    {
      "id": "approx",
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      "inputs": {
        "experimental": 56.891,
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        "value": 8.7217417812505,
        "observed": 56.891,
        "true": 62.327,
        "percent_error": 8.7217417812505,
        "absolute_error": 5.436,
        "relative_error": 0.08721741781250501,
        "signed_percent_error": -8.7217417812505,
        "absolute": true,
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        "status": "success",
        "solution_type": "unique",
        "rounding": {
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        "display": {
          "percent_error": "8.72174%",
          "value": "8.721741781"
        }
      },
      "tol": 1e-12,
      "tol_type": "absolute"
    },
    {
      "id": "exact",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "experimental": 5,
        "theoretical": 5
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      "expect": 0,
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      "actual": {
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        "observed": 5,
        "true": 5,
        "percent_error": 0,
        "absolute_error": 0,
        "relative_error": 0,
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        "absolute": true,
        "direction": "equal",
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        "solution_type": "unique",
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          "significant_digits": 10,
          "trailing_zeros": "stripped"
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        "display": {
          "percent_error": "0%",
          "value": "0"
        }
      },
      "tol": 1e-9
    },
    {
      "id": "share-ot",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "observed": 98,
        "true": 100
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      "expect": {
        "percent_error": 2,
        "absolute_error": 2,
        "relative_error": 0.02
      },
      "expectWarning": null,
      "actual": {
        "solve_for": "p",
        "value": 2,
        "observed": 98,
        "true": 100,
        "percent_error": 2,
        "absolute_error": 2,
        "relative_error": 0.02,
        "signed_percent_error": -2,
        "absolute": true,
        "direction": "observed_below",
        "status": "success",
        "solution_type": "unique",
        "rounding": {
          "mode": "auto",
          "significant_digits": 10,
          "trailing_zeros": "stripped"
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        "display": {
          "percent_error": "2%",
          "value": "2"
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      },
      "tol": 1e-12
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    {
      "id": "signed",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "experimental": 7,
        "theoretical": 9,
        "abs": "0"
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      "expect": {
        "percent_error": -22.22222222222222
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      "actual": {
        "solve_for": "p",
        "value": -22.22222222222222,
        "observed": 7,
        "true": 9,
        "percent_error": -22.22222222222222,
        "absolute_error": 2,
        "relative_error": 0.2222222222222222,
        "signed_percent_error": -22.22222222222222,
        "absolute": false,
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        "status": "success",
        "solution_type": "unique",
        "rounding": {
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          "trailing_zeros": "stripped"
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        "display": {
          "percent_error": "-22.2222%",
          "value": "-22.22222222"
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      },
      "tol": 1e-9
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    {
      "id": "reverse-o-abs",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "true": 100,
        "percent": 2,
        "solve_for": "o",
        "abs": "1"
      },
      "expect": {
        "solve_for": "o",
        "value": 98,
        "candidates": [
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          102
        ],
        "solution_type": "two",
        "status": "success"
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      "actual": {
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        "value": 98,
        "candidates": [
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          102
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        "observed": null,
        "true": 100,
        "percent_error": 2,
        "absolute_error": 2,
        "relative_error": 0.02,
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        "direction": null,
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        "display": {
          "percent_error": "2%",
          "value": "98 or 102"
        }
      },
      "tol": 1e-12
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    {
      "id": "zero-true",
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      "kind": "boundary",
      "inputs": {
        "experimental": 1,
        "theoretical": 0
      },
      "expectError": "TRUE_VALUE_ZERO",
      "code": "TRUE_VALUE_ZERO",
      "error": "Accepted / reference value cannot be 0 (division by zero)."
    },
    {
      "id": "inconsistent",
      "ok": true,
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      "inputs": {
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        "percent": 20
      },
      "expect": {
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        "percent_error": 2
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        "value": 2,
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        "percent_error": 2,
        "entered_percent_error": 20,
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        "display": {
          "percent_error": "2%",
          "value": "2"
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        "message": "O=98 and T=100 imply a percent error of 2%, not 20%."
      },
      "tol": 1e-12
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    {
      "id": "infinite-t",
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      "inputs": {
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        "percent": 100,
        "solve_for": "t",
        "abs": "1"
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        "code": "INFINITELY_MANY",
        "constraint": "T != 0",
        "solution_type": "infinite"
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      "expectWarning": null,
      "actual": {
        "solve_for": "t",
        "value": null,
        "observed": 0,
        "true": null,
        "percent_error": 100,
        "absolute_error": null,
        "relative_error": 1,
        "signed_percent_error": null,
        "absolute": true,
        "direction": null,
        "status": "infinite",
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        "solution_type": "infinite",
        "constraint": "T != 0",
        "rounding": {
          "mode": "auto",
          "significant_digits": 10,
          "trailing_zeros": "stripped"
        },
        "display": {
          "percent_error": "100%",
          "value": "any T ≠ 0"
        },
        "message": "Any nonzero accepted/reference value T satisfies |0 − T| / |T| × 100% = 100%."
      },
      "tol": 1e-12
    },
    {
      "id": "no-sol-t",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "observed": 0,
        "percent": 50,
        "solve_for": "t",
        "abs": "1"
      },
      "expectError": "NO_REAL_SOLUTION",
      "code": "NO_REAL_SOLUTION",
      "error": "|O − T| / |T| is 100% for every T ≠ 0 when O = 0, so no accepted value yields this percent error."
    },
    {
      "id": "solve-t-p100",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "observed": 50,
        "percent": 100,
        "solve_for": "t",
        "abs": "1"
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      "expect": {
        "solve_for": "t",
        "value": 25,
        "true": 25,
        "solution_type": "unique"
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      "actual": {
        "solve_for": "t",
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        "observed": 50,
        "true": 25,
        "percent_error": 100,
        "absolute_error": 25,
        "relative_error": 1,
        "signed_percent_error": 100,
        "absolute": true,
        "direction": "observed_above",
        "status": "success",
        "solution_type": "unique",
        "omitted_branches": [
          {
            "branch": "T = O / (1 − P/100)",
            "code": "DENOMINATOR_ZERO"
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        ],
        "rounding": {
          "mode": "auto",
          "significant_digits": 10,
          "trailing_zeros": "stripped"
        },
        "display": {
          "percent_error": "100%",
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      },
      "tol": 1e-12
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    {
      "id": "infinite-t-signed",
      "ok": true,
      "kind": "golden",
      "inputs": {
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        "percent": -100,
        "solve_for": "t",
        "abs": "0"
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        "constraint": "T != 0",
        "solution_type": "infinite"
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      "actual": {
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        "true": null,
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        "constraint": "T != 0",
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          "trailing_zeros": "stripped"
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        "display": {
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          "value": "any T ≠ 0"
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      "tol": 1e-12
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      "code": "NO_REAL_SOLUTION",
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      "expectError": "NO_REAL_SOLUTION",
      "code": "NO_REAL_SOLUTION",
      "error": "With observed 0 and signed percent error, (0 − T) / T × 100% is −100% for every T ≠ 0."
    }
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