第 33971 題：數值近似誤差

原題與已知條件

exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":16,"intercept":21,"slope":3}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":16,"intercept":21,"slope":3}；請依 exact_ordered_tuple_declared_by_interface 回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-numerical-approximation-error

逐步推導

source roles = [qf-mod50-numerr-affine-residual-forward, [approximation, domain, intercept, operation, slope]]

derived = [numerr-affine-residual-forward, Exact_numerical_error_density]

source roles = [error ORIGINAL slope, 100000000, 3, 1]

derived = [[3, 1], 3]

error ORIGINAL slope ORIGINAL numerator over denominator := div(3, 1)

error ORIGINAL slope ORIGINAL numerator over denominator = 3

source roles = [error ORIGINAL intercept, 100000000, 21, 1]

derived = [[21, 1], 21]

error ORIGINAL intercept ORIGINAL numerator over denominator := div(21, 1)

error ORIGINAL intercept ORIGINAL numerator over denominator = 21

source roles = [error ORIGINAL approximation, 100000000, 16, 1]

derived = [[16, 1], 16]

error ORIGINAL approximation ORIGINAL numerator over denominator := div(16, 1)

error ORIGINAL approximation ORIGINAL numerator over denominator = 16

source roles = [Affine exact root SOURCE naive div, 100000000, -21, 3]

derived = [[-7, 1], -7]

Affine exact root SOURCE naive div ORIGINAL numerator over denominator := div(-21, 3)

Affine exact root SOURCE naive div ORIGINAL numerator over denominator = -7

Affine exact root ORIGINAL rational div := div(-21, 3)

Affine exact root ORIGINAL rational div = -7

source roles = [Affine residual original slope times approximation SOURCE naive mul, 100000000, 48, 1]

derived = [[48, 1], 48]

Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator := div(48, 1)

Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator = 48

Affine residual original slope times approximation ORIGINAL rational mul := mul(3, 16)

Affine residual original slope times approximation ORIGINAL rational mul = 48

source roles = [Affine residual add original intercept SOURCE naive sum, 100000000, 69, 1]

derived = [[69, 1], 69]

Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator := div(69, 1)

Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator = 69

Affine residual add original intercept ORIGINAL rational sum := sum(48, 21)

Affine residual add original intercept ORIGINAL rational sum = 69

source roles = [Affine forward original approximation minus exact root SOURCE naive sub, 100000000, 23, 1]

derived = [[23, 1], 23]

Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator := div(23, 1)

Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator = 23

Affine forward original approximation minus exact root ORIGINAL rational sub := sub(16, -7)

Affine forward original approximation minus exact root ORIGINAL rational sub = 23

source roles = [Affine absolute slope times forward error SOURCE naive mul, 100000000, 69, 1]

derived = [[69, 1], 69]

Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator := div(69, 1)

Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator = 69

Affine absolute slope times forward error ORIGINAL rational mul := mul(3, 23)

Affine absolute slope times forward error ORIGINAL rational mul = 69

source roles = [3, 21, 16]

derived = [-7, 69, 69, 23, 23, 69, 是]

逐步反向核對

• S023 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S022 · premise_and_result_audit — 以精確有理數重算 Affine absolute slope times forward error ORIGINAL rational mul，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Affine absolute slope times forward error SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S020 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S019 · premise_and_result_audit — 以精確有理數重算 Affine forward original approximation minus exact root ORIGINAL rational sub，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Affine forward original approximation minus exact root SOURCE naive sub ORIGINAL numerator over denominator，不以小數近似替代。
• S017 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S016 · premise_and_result_audit — 以精確有理數重算 Affine residual add original intercept ORIGINAL rational sum，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Affine residual add original intercept SOURCE naive sum ORIGINAL numerator over denominator，不以小數近似替代。
• S014 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S013 · premise_and_result_audit — 以精確有理數重算 Affine residual original slope times approximation ORIGINAL rational mul，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Affine residual original slope times approximation SOURCE naive mul ORIGINAL numerator over denominator，不以小數近似替代。
• S011 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S010 · premise_and_result_audit — 以精確有理數重算 Affine exact root ORIGINAL rational div，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Affine exact root SOURCE naive div ORIGINAL numerator over denominator，不以小數近似替代。
• S008 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S007 · premise_and_result_audit — 以精確有理數重算 error ORIGINAL approximation ORIGINAL numerator over denominator，不以小數近似替代。
• S006 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S005 · premise_and_result_audit — 以精確有理數重算 error ORIGINAL intercept ORIGINAL numerator over denominator，不以小數近似替代。
• S004 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S003 · premise_and_result_audit — 以精確有理數重算 error ORIGINAL slope ORIGINAL numerator over denominator，不以小數近似替代。
• S002 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S001 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。

結論

(3, 21, 16, -7, 69, 23, 69, 是, residual_and_forward_error_are_distinct_quantities)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。