{
  "capability_id": "math.square_root",
  "tool_id": "square-root",
  "engine": "square-root",
  "calculation_version": "1.0.3",
  "generated_at": "2026-09-16T22:03:43.777Z",
  "engine_build": "2026-09-10",
  "test_count": 18,
  "passed": 18,
  "failed": 0,
  "tests_summary": {
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      "total": 11,
      "passed": 11
    },
    "boundary": {
      "total": 7,
      "passed": 7
    },
    "property": {
      "total": 0,
      "passed": 0
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  "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/square-root",
  "api": "https://www.calculatorx.com/api/v1/calc/square-root",
  "schema_input": "https://www.calculatorx.com/schemas/math.square_root.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.square_root.output.json",
  "assumptions": [
    "Principal branch: nonnegative real root when x ≥ 0.",
    "Unlike Log / Ln, this engine does not reject x < 0. It returns the principal imaginary √|x| i, not a real error.",
    "The other real root for x > 0 is −√x; the other imaginary root for x < 0 is −√|x| i. The published value is the principal one."
  ],
  "limitations": [
    "x ≥ 0 → principal = √x, imag = 0",
    "x < 0 → real = 0, imag = √|x|, principal = null (unlike math.root NOT_REAL)",
    "Missing x → MISSING_REQUIRED_INPUT"
  ],
  "boundary_conditions": [
    "Missing x returns MISSING_REQUIRED_INPUT.",
    "Non-numeric, object, array, boolean, NaN, or Infinity inputs return INVALID_NUMBER.",
    "√0 = 0.",
    "x < 0 → imag = √(−x), principal = null (not NaN).",
    "IEEE-754 overflow of √x returns RESULT_OVERFLOW, not Infinity."
  ],
  "sources": [
    {
      "title": "NIST Digital Library of Mathematical Functions, Chapter 4",
      "section": "Logarithm, Exponential, Powers",
      "url": "https://dlmf.nist.gov/4",
      "supports": "Square root as the principal value of x^(1/2)",
      "kind": "reference"
    },
    {
      "title": "NIST DLMF §4.2(iv)",
      "section": "Powers: principal value z^a = exp(a ln z)",
      "url": "https://dlmf.nist.gov/4.2",
      "supports": "Principal square root is the principal branch of z^(1/2); nonnegative for real x > 0. The branch cut is the negative real axis; this engine reports the principal imaginary on that cut as √|x| i.",
      "kind": "reference"
    },
    {
      "title": "CalculatorX mathematical conventions",
      "section": "Principal real root; principal imaginary for x < 0",
      "supports": "Matches the on-page contract. Cube and nth roots are the Root calculator, not this engine.",
      "kind": "definition"
    }
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  "methods": [],
  "fingerprint": "sha256:01a95214633a22c075abfe506188cc7661a6d663cf51ab4754c55d1807615663",
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      },
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        "imag": 0
      },
      "expectWarning": null,
      "actual": {
        "real": 12,
        "imag": 0,
        "principal": 12
      },
      "tol": 0
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        "x": 81
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        "principal": 9
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      "expectWarning": null,
      "actual": {
        "real": 9,
        "imag": 0,
        "principal": 9
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      "tol": 0
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        "real": 1.4142135623730951,
        "imag": 0,
        "principal": 1.4142135623730951
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      "tol": 1e-12
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      "tol": 1e-12
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        "imag": 0
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      "tol": 0
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        "x": 0.25
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      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "Enter a finite radicand x."
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      "total": 18,
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