{ "capability_id": "math.root", "tool_id": "root", "engine": "root", "calculation_version": "1.0.6", "generated_at": "2026-10-28T03:01:57.875Z", "engine_build": "2026-10-26", "test_count": 24, "passed": 24, "failed": 0, "tests_summary": { "golden": { "passed": 10, "total": 10 }, "boundary": { "passed": 14, "total": 14 }, "property": { "passed": 0, "total": 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/root", "api": "https://www.calculatorx.com/api/v1/calc/root", "schema_input": "https://www.calculatorx.com/schemas/math.root.input.json", "schema_output": "https://www.calculatorx.com/schemas/math.root.output.json", "assumptions": [ "Real principal root only. This engine does not return complex values.", "Index n is a positive safe integer (1st, square, cube, 4th, …, nth root). Generalized powers a^(1/n) with non-integer, negative, or non-safe-integer n belong on the Exponent calculator.", "Even n requires a ≥ 0 and returns b ≥ 0. Odd n allows a < 0 (for example ∛(−8) = −2).", "n = 2 uses Math.sqrt. n = 3 uses Math.cbrt, returning an implementation-approximated binary64 result (including negatives). Other n use a^(1/n) for a ≥ 0, or −(|a|)^(1/n) for odd n and a < 0.", "The Square Root calculator is a different contract: √(x<0) returns the principal imaginary. This page rejects that case as NOT_REAL." ], "limitations": [ "Principal real nth root only. Even index requires a≥0 (NOT_REAL otherwise); odd-negative uses the real negative branch, distinct from complex principal powers.", "Strict decimal/scientific text up to512 characters; reject unknown keys, nonfinite, null, hex, comma, mismatching aliases, nonzero text rounded to zero. Exact positive safe-integer index; omitted index defaults to2, explicit blank/null rejects. Semantic cross-field and exact-text rules run after Schema shape validation.", "Binary64 identity for n=1, Math.sqrt for n=2, Math.cbrt for n=3, signed Math.pow for higher indices. Declared fixed references have≤64 ULP overall; existing path-specific O3 vectors retain0/2/64 ULP. Not a whole-domain guarantee or exact-decimal result.", "Relative condition number in radicand is1/n away from zero; derivative singular at zero for n>1. Input rounding, reciprocal-exponent error and root rounding are distinct. Raising rounded output to n can overflow or amplify error.", "12 significant digit display; exports keep full binary64 scalar, raw inputs and calculation version. Bounded real graph; typing computes locally, shared URLs and REST/MCP send inputs and may enter logs." ], "boundary_conditions": [ "Strict decimal/scientific text up to 512 characters; hex, commas and unknown keys reject. Input aliases must agree. Nonzero text rounded to zero returns NUMERIC_UNDERFLOW; explicit zero remains valid.", "Index text must represent an exact positive safe integer before binary64 rounding; for example 2.00000000000000001 is not a valid integer index.", "Missing or blank a returns MISSING_REQUIRED_INPUT. Omitted n defaults to 2; explicit blank or null n is rejected.", "Non-numeric, object, array, boolean, NaN, or Infinity inputs return INVALID_NUMBER.", "n that is not a positive safe integer (including n = 0, n < 0, non-integers such as 2.5, and n outside JavaScript's safe-integer range) returns INVALID_INDEX.", "a < 0 with even n returns NOT_REAL.", "a = 0 returns 0.", "Defensive guards exist for unexpected non-finite numerical outcomes (RESULT_OVERFLOW / RESULT_UNDERFLOW). Overflow or underflow is not expected for valid finite nth-root inputs." ], "sources": [ { "title": "NIST Digital Library of Mathematical Functions, Chapter 4", "section": "Logarithm, Exponential, Powers", "url": "https://dlmf.nist.gov/4", "supports": "nth root as the principal real value of a^(1/n) when it exists", "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 a^(1/n) for a > 0. Negative a is restricted here to odd integer n so the result stays the real odd root, not the complex principal power.", "kind": "reference", "publisher": "National Institute of Standards and Technology" }, { "title": "CalculatorX mathematical conventions", "section": "Real nth root; even roots of negatives are NOT_REAL; n is a positive safe integer", "supports": "Distinct from the Square Root calculator’s principal-imaginary contract for x < 0. Polynomial roots belong on Quadratic. Non-integer or non-safe-integer n is a generalized power (Exponent), not this nth-root contract. 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