{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-numerical-approximation-error",
  "semantic_family_id": "sf-numerical-approximation-error",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "數值近似與誤差"
  },
  "summary": {
    "zh-Hant": "絕對誤差是近似值與參照值之差的絕對值；相對誤差另要求參照值非零並除以其絕對值。"
  },
  "statement": {
    "zh-Hant": "給近似值 a 與參照值 x，絕對誤差為 |a-x|。若 x≠0，相對誤差為 |a-x|/|x|。容許誤差 τ≥0 時，|a-x|≤τ 才稱在該容許範圍內。"
  },
  "formula": {
    "tex": "e=A(a,x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>e</mi><mo>=</mo><mrow><mi>A</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "symbol",
        "name": "e",
        "glyph": "e"
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "b7bfe29d3312ffbe0ea75c298bf76d7dc566cc13b60f42280d1a50d2dfa078f4"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫使用精確有理數，不以二進位浮點顯示值冒充精確答案。"
    },
    {
      "id": "a2",
      "zh-Hant": "相對誤差只在參照值非零時求值；截斷誤差、捨入誤差與資料誤差不自動互換。"
    }
  ],
  "symbols": [
    {
      "token": "e",
      "role": "absolute_error",
      "role_zh": "絕對誤差",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "approximation",
      "role_zh": "近似值",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "reference_value",
      "role_zh": "參照值",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-real-number",
      "title_zh": "實數",
      "reason_zh": "近似值與參照值位於實數範圍。"
    },
    {
      "id": "mko-absolute-value",
      "title_zh": "絕對值",
      "reason_zh": "誤差大小使用絕對值。"
    },
    {
      "id": "mko-ratio-proportion",
      "title_zh": "比與比例",
      "reason_zh": "相對誤差是無因次比值。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "絕對誤差保留原量的尺度，相對誤差將其和非零參照量比較；二者回答不同問題。",
      "誤差界 |a-x|≤τ 等價於 a 位於 [x-τ,x+τ]；端點包含在容許範圍。",
      "誤差數值本身不說明來源。要區分離散化、截斷、捨入、量測與模型誤差，必須另有產生過程資料。"
    ],
    "common_misconception_zh": "參照值為零仍計算相對誤差，或把所有誤差都叫捨入誤差。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "精確有理核對只證指定數值與界；不評估演算法收斂或浮點實作。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入截斷與捨入來源、條件數、穩定性及收斂階。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-numerical-approximation-error.md 整理；2026-10-13 脈衝 R03（使用者手動接續 2026-10-20 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}