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  "engine": "arctan",
  "calculation_version": "1.0.9",
  "generated_at": "2026-10-19T06:49:24.323Z",
  "engine_build": "2026-10-15",
  "test_count": 10,
  "passed": 10,
  "failed": 0,
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      "passed": 7,
      "total": 7
    },
    "boundary": {
      "passed": 3,
      "total": 3
    },
    "property": {
      "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,
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    "reviewer": null,
    "scope": [
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      "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
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  "page": "https://www.calculatorx.com/calc/math/arctan",
  "api": "https://www.calculatorx.com/api/v1/calc/arctan",
  "schema_input": "https://www.calculatorx.com/schemas/math.arctan.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.arctan.output.json",
  "assumptions": [
    "Real inverse tangent with principal branch (−π/2, π/2) (NIST DLMF §4.23(ii)). Infinity is not accepted as an input.",
    "Opposite/adjacent in the UI is a right-triangle length convenience (opposite ≥ 0, adjacent > 0, same unit). It is not atan2. The published engine remains single-argument arctan(x). Use Decimal for a signed ratio, and Tan for the forward map.",
    "When x is exactly a listed common value (0, ±1/√3, ±1, ±√3), the engine returns the exact algebraic angle such as 45° = π/4. Nearby floats are not snapped."
  ],
  "limitations": [
    "Missing x → MISSING_REQUIRED_INPUT",
    "Non-finite → INVALID_NUMBER",
    "Near-common-value hint is display metadata; it does not replace the canonical deg/rad or assert mathematical exactness.",
    "Raw input accepts finite JSON numbers or decimal strings; normalized inputs contain numeric x only. Nonzero text rounded to zero is NUMERIC_UNDERFLOW.",
    "Unknown inputs are UNKNOWN_INPUT; x/xInput conflicts are AMBIGUOUS_INPUT.",
    "Legacy engine test counts are separate from the CVP Manifest suite. CVP identity is included; production status requires the linked report matched by version, revision and digest."
  ],
  "boundary_conditions": [
    "Decimal strings are accepted and normalized to binary64 x. Nonzero decimal text that rounds to zero returns NUMERIC_UNDERFLOW; explicit zero remains valid.",
    "REST/MCP accept only x and xInput; unknown fields return UNKNOWN_INPUT and conflicting aliases return AMBIGUOUS_INPUT.",
    "Missing x returns MISSING_REQUIRED_INPUT.",
    "Non-numeric, object, array, boolean, NaN, or Infinity inputs return INVALID_NUMBER.",
    "arctan(0) = 0°; arctan(1) = 45°; arctan(√3) = 60°; arctan(−1) = −45°.",
    "Large finite |x| approaches ±90°. The mathematical principal range is the open interval (−90°, 90°). IEEE-754 binary64 results may round to ±90° for extremely large finite inputs (for example 1e20 and Number.MAX_VALUE). When the result rounds to ±90°, tan(θ) cannot recover x and verification.passed may be false."
  ],
  "sources": [
    {
      "title": "NIST Digital Library of Mathematical Functions, Chapter 4",
      "section": "Elementary Functions",
      "url": "https://dlmf.nist.gov/4",
      "supports": "Inverse trigonometric functions as elementary inverses of sine, cosine, and tangent",
      "kind": "reference"
    },
    {
      "title": "NIST DLMF §4.23",
      "section": "Inverse trigonometric functions; principal values",
      "url": "https://dlmf.nist.gov/4.23",
      "supports": "Principal arctan on the real line with range (−π/2, π/2); degree output is θ × 180/π",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
    },
    {
      "title": "CalculatorX mathematical conventions",
      "section": "Principal branch; exact-value recognition; ratio mode is UI-only",
      "supports": "Published engine is arctan(x) of one real argument (not atan2).",
      "kind": "definition"
    }
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  "methods": [],
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  "tier": "engine_tested",
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      "total": 10,
      "golden": 7,
      "boundary": 3,
      "property": 0
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