# 數值誤差密度：residual、forward error與conditioning基礎 · 脈衝 R05 本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。 本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。 batch-20261015-advance-20261119-r05-numerical-approximation-error-foundations · 2026-10-15 · 100 題 欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。 [欄序與來源綁定資料](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json) ## 1. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":-4,"intercept":1,"slope":-3}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-3, 1, -4, 1/3, 13, 13/3, 13, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-3, 1, -4, 1/3, 13, 13/3, 13, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0001/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 2. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":-3,"intercept":2,"slope":-2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-2, 2, -3, 1, 8, 4, 8, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-2, 2, -3, 1, 8, 4, 8, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0002/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 3. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":-2,"intercept":3,"slope":-1}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-1, 3, -2, 3, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-1, 3, -2, 3, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0003/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 4. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":-1,"intercept":4,"slope":2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(2, 4, -1, -2, 2, 1, 2, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(2, 4, -1, -2, 2, 1, 2, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0004/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 5. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":0,"intercept":5,"slope":1}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(1, 5, 0, -5, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(1, 5, 0, -5, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0005/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 6. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":1,"intercept":6,"slope":2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(2, 6, 1, -3, 8, 4, 8, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(2, 6, 1, -3, 8, 4, 8, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0006/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 7. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":2,"intercept":7,"slope":3}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(3, 7, 2, -7/3, 13, 13/3, 13, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(3, 7, 2, -7/3, 13, 13/3, 13, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0007/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 8. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":3,"intercept":8,"slope":-3}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-3, 8, 3, 8/3, 1, 1/3, 1, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-3, 8, 3, 8/3, 1, 1/3, 1, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0008/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 9. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":4,"intercept":9,"slope":-2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-2, 9, 4, 9/2, 1, 1/2, 1, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-2, 9, 4, 9/2, 1, 1/2, 1, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0009/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 10. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":5,"intercept":10,"slope":-1}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-1, 10, 5, 10, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-1, 10, 5, 10, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0010/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 11. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":6,"intercept":11,"slope":2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(2, 11, 6, -11/2, 23, 23/2, 23, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(2, 11, 6, -11/2, 23, 23/2, 23, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0011/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 12. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":7,"intercept":12,"slope":1}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(1, 12, 7, -12, 19, 19, 19, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(1, 12, 7, -12, 19, 19, 19, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0012/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 13. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":8,"intercept":13,"slope":2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(2, 13, 8, -13/2, 29, 29/2, 29, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(2, 13, 8, -13/2, 29, 29/2, 29, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0013/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 14. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":9,"intercept":14,"slope":3}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(3, 14, 9, -14/3, 41, 41/3, 41, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(3, 14, 9, -14/3, 41, 41/3, 41, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0014/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 15. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":10,"intercept":15,"slope":-3}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-3, 15, 10, 5, 15, 5, 15, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-3, 15, 10, 5, 15, 5, 15, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0015/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 16. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":11,"intercept":16,"slope":-2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-2, 16, 11, 8, 6, 3, 6, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-2, 16, 11, 8, 6, 3, 6, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0016/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 17. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":12,"intercept":17,"slope":-1}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-1, 17, 12, 17, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-1, 17, 12, 17, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0017/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 18. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":13,"intercept":18,"slope":2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(2, 18, 13, -9, 44, 22, 44, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(2, 18, 13, -9, 44, 22, 44, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0018/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 19. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":14,"intercept":19,"slope":1}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(1, 19, 14, -19, 33, 33, 33, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(1, 19, 14, -19, 33, 33, 33, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0019/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 20. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":15,"intercept":20,"slope":2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(2, 20, 15, -10, 50, 25, 50, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(2, 20, 15, -10, 50, 25, 50, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0020/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 21. 數值近似誤差 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 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(3, 21, 16, -7, 69, 23, 69, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(3, 21, 16, -7, 69, 23, 69, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0021/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 22. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":17,"intercept":22,"slope":-3}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-3, 22, 17, 22/3, 29, 29/3, 29, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-3, 22, 17, 22/3, 29, 29/3, 29, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0022/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 23. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":18,"intercept":23,"slope":-2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-2, 23, 18, 23/2, 13, 13/2, 13, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-2, 23, 18, 23/2, 13, 13/2, 13, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0023/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 24. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":19,"intercept":24,"slope":-1}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(-1, 24, 19, 24, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(-1, 24, 19, 24, 5, 5, 5, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0024/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 25. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-residual-forward；明示資料={"approximation":20,"intercept":25,"slope":2}；請依 [slope, intercept, approximation, exact_root, absolute_residual, forward_error, slope_scaled_forward_error, scale_identity, quantity_distinction] 回答。 查看答案與解釋(2, 25, 20, -25/2, 65, 65/2, 65, 是, residual_and_forward_error_are_distinct_quantities) Wave6第十三層核對=(2, 25, 20, -25/2, 65, 65/2, 65, 是, residual_and_forward_error_are_distinct_quantities)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0025/) 數值近似與誤差 · qf-mod50-numerr-affine-residual-forward · candidate ## 26. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[1,3],"slope":-3}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-3, 1/3, 1, affine_output_absolute_error_bound) Wave6第十三層核對=(-3, 1/3, 1, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0026/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 27. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[2,3],"slope":-2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-2, 2/3, 4/3, affine_output_absolute_error_bound) Wave6第十三層核對=(-2, 2/3, 4/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0027/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 28. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[3,3],"slope":-1}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-1, 1, 1, affine_output_absolute_error_bound) Wave6第十三層核對=(-1, 1, 1, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0028/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 29. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[4,3],"slope":2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(2, 4/3, 8/3, affine_output_absolute_error_bound) Wave6第十三層核對=(2, 4/3, 8/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0029/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 30. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[5,3],"slope":1}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(1, 5/3, 5/3, affine_output_absolute_error_bound) Wave6第十三層核對=(1, 5/3, 5/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0030/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 31. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[6,3],"slope":2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(2, 2, 4, affine_output_absolute_error_bound) Wave6第十三層核對=(2, 2, 4, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0031/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 32. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[7,3],"slope":3}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(3, 7/3, 7, affine_output_absolute_error_bound) Wave6第十三層核對=(3, 7/3, 7, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0032/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 33. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[8,3],"slope":-3}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-3, 8/3, 8, affine_output_absolute_error_bound) Wave6第十三層核對=(-3, 8/3, 8, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0033/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 34. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[9,3],"slope":-2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-2, 3, 6, affine_output_absolute_error_bound) Wave6第十三層核對=(-2, 3, 6, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0034/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 35. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[10,3],"slope":-1}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-1, 10/3, 10/3, affine_output_absolute_error_bound) Wave6第十三層核對=(-1, 10/3, 10/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0035/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 36. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[11,3],"slope":2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(2, 11/3, 22/3, affine_output_absolute_error_bound) Wave6第十三層核對=(2, 11/3, 22/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0036/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 37. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[12,3],"slope":1}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(1, 4, 4, affine_output_absolute_error_bound) Wave6第十三層核對=(1, 4, 4, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0037/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 38. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[13,3],"slope":2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(2, 13/3, 26/3, affine_output_absolute_error_bound) Wave6第十三層核對=(2, 13/3, 26/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0038/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 39. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[14,3],"slope":3}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(3, 14/3, 14, affine_output_absolute_error_bound) Wave6第十三層核對=(3, 14/3, 14, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0039/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 40. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[15,3],"slope":-3}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-3, 5, 15, affine_output_absolute_error_bound) Wave6第十三層核對=(-3, 5, 15, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0040/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 41. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[16,3],"slope":-2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-2, 16/3, 32/3, affine_output_absolute_error_bound) Wave6第十三層核對=(-2, 16/3, 32/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0041/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 42. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[17,3],"slope":-1}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-1, 17/3, 17/3, affine_output_absolute_error_bound) Wave6第十三層核對=(-1, 17/3, 17/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0042/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 43. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[18,3],"slope":2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(2, 6, 12, affine_output_absolute_error_bound) Wave6第十三層核對=(2, 6, 12, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0043/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 44. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[19,3],"slope":1}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(1, 19/3, 19/3, affine_output_absolute_error_bound) Wave6第十三層核對=(1, 19/3, 19/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0044/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 45. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[20,3],"slope":2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(2, 20/3, 40/3, affine_output_absolute_error_bound) Wave6第十三層核對=(2, 20/3, 40/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0045/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 46. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[21,3],"slope":3}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(3, 7, 21, affine_output_absolute_error_bound) Wave6第十三層核對=(3, 7, 21, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0046/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 47. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[22,3],"slope":-3}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-3, 22/3, 22, affine_output_absolute_error_bound) Wave6第十三層核對=(-3, 22/3, 22, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0047/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 48. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[23,3],"slope":-2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-2, 23/3, 46/3, affine_output_absolute_error_bound) Wave6第十三層核對=(-2, 23/3, 46/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0048/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 49. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[24,3],"slope":-1}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(-1, 8, 8, affine_output_absolute_error_bound) Wave6第十三層核對=(-1, 8, 8, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0049/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 50. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-affine-propagation；明示資料={"input_error_bound":[25,3],"slope":2}；請依 [slope, input_absolute_error_bound, output_absolute_error_bound, scope] 回答。 查看答案與解釋(2, 25/3, 50/3, affine_output_absolute_error_bound) Wave6第十三層核對=(2, 25/3, 50/3, affine_output_absolute_error_bound)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0050/) 數值近似與誤差 · qf-mod50-numerr-affine-propagation · candidate ## 51. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":1,"slope_a":1,"slope_b":2}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(1, 1, 1, 2, 1/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(1, 1, 1, 2, 1/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0051/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 52. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":2,"slope_a":1,"slope_b":3}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(2, 1, 2, 3, 2/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(2, 1, 2, 3, 2/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0052/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 53. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":3,"slope_a":1,"slope_b":4}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(3, 1, 3, 4, 3/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(3, 1, 3, 4, 3/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0053/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 54. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":4,"slope_a":1,"slope_b":5}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(4, 1, 4, 5, 4/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(4, 1, 4, 5, 4/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0054/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 55. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":5,"slope_a":1,"slope_b":6}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(5, 1, 5, 6, 5/6, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(5, 1, 5, 6, 5/6, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0055/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 56. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":6,"slope_a":1,"slope_b":2}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(6, 1, 6, 2, 3, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(6, 1, 6, 2, 3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0056/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 57. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":7,"slope_a":1,"slope_b":3}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(7, 1, 7, 3, 7/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(7, 1, 7, 3, 7/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0057/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 58. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":8,"slope_a":1,"slope_b":4}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(8, 1, 8, 4, 2, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(8, 1, 8, 4, 2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0058/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 59. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":9,"slope_a":1,"slope_b":5}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(9, 1, 9, 5, 9/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(9, 1, 9, 5, 9/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0059/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 60. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":10,"slope_a":1,"slope_b":6}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(10, 1, 10, 6, 5/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(10, 1, 10, 6, 5/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0060/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 61. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":11,"slope_a":1,"slope_b":2}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(11, 1, 11, 2, 11/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(11, 1, 11, 2, 11/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0061/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 62. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":12,"slope_a":1,"slope_b":3}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(12, 1, 12, 3, 4, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(12, 1, 12, 3, 4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0062/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 63. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":13,"slope_a":1,"slope_b":4}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(13, 1, 13, 4, 13/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(13, 1, 13, 4, 13/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0063/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 64. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":14,"slope_a":1,"slope_b":5}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(14, 1, 14, 5, 14/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(14, 1, 14, 5, 14/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0064/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 65. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":15,"slope_a":1,"slope_b":6}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(15, 1, 15, 6, 5/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(15, 1, 15, 6, 5/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0065/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 66. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":16,"slope_a":1,"slope_b":2}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(16, 1, 16, 2, 8, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(16, 1, 16, 2, 8, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0066/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 67. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":17,"slope_a":1,"slope_b":3}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(17, 1, 17, 3, 17/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(17, 1, 17, 3, 17/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0067/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 68. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":18,"slope_a":1,"slope_b":4}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(18, 1, 18, 4, 9/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(18, 1, 18, 4, 9/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0068/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 69. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":19,"slope_a":1,"slope_b":5}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(19, 1, 19, 5, 19/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(19, 1, 19, 5, 19/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0069/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 70. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":20,"slope_a":1,"slope_b":6}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(20, 1, 20, 6, 10/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(20, 1, 20, 6, 10/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0070/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 71. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":21,"slope_a":1,"slope_b":2}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(21, 1, 21, 2, 21/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(21, 1, 21, 2, 21/2, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0071/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 72. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":22,"slope_a":1,"slope_b":3}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(22, 1, 22, 3, 22/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(22, 1, 22, 3, 22/3, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0072/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 73. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":23,"slope_a":1,"slope_b":4}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(23, 1, 23, 4, 23/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(23, 1, 23, 4, 23/4, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0073/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 74. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":24,"slope_a":1,"slope_b":5}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(24, 1, 24, 5, 24/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(24, 1, 24, 5, 24/5, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0074/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 75. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-conditioning-witness；明示資料={"residual":25,"slope_a":1,"slope_b":6}；請依 [absolute_residual, first_slope, first_forward_error, second_slope, second_forward_error, forward_errors_differ, conditioning_scope] 回答。 查看答案與解釋(25, 1, 25, 6, 25/6, 是, same_residual_can_have_different_forward_error_under_different_conditioning) Wave6第十三層核對=(25, 1, 25, 6, 25/6, 是, same_residual_can_have_different_forward_error_under_different_conditioning)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0075/) 數值近似與誤差 · qf-mod50-numerr-conditioning-witness · candidate ## 76. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={"mode":"zero_reference","model":{"approximation":1}}；請依 [mode, approximation, reference_description, relative_error_status, relative_error_defined_claim] 回答。 四種mode為5/5/5/7欄；strict與inclusive兩mode雖同欄數，等號處結果相反。reference=0時relative error未定義。 查看答案與解釋(zero_reference, 1, reference=0, relative_error=undefined, 否) Wave6第十三層核對=(zero_reference, 1, reference=0, relative_error=undefined, 否)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0076/) 數值近似與誤差 · qf-mod50-numerr-boundary · candidate ## 77. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={"mode":"inclusive_tolerance","model":{"error":[2,2],"tolerance":[2,2]}}；請依 [mode, absolute_error, tolerance, inclusive_test_result, equality_convention] 回答。 四種mode為5/5/5/7欄；strict與inclusive兩mode雖同欄數，等號處結果相反。reference=0時relative error未定義。 查看答案與解釋(inclusive_tolerance, 1, 1, 是, error<=tolerance_is_true_at_equality) Wave6第十三層核對=(inclusive_tolerance, 1, 1, 是, error<=tolerance_is_true_at_equality)；依完整明示資料精確重算；affine residual=abs(slope)*forward error，兩者不可混同；zero reference relative error未定義；strict與inclusive tolerance在等號處不同。 [本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-pulse-20261119-r05-0077/) 數值近似與誤差 · qf-mod50-numerr-boundary · candidate ## 78. 數值近似誤差 exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={"mode":"strict_tolerance","model":{"error":[3,2],"tolerance":[3,2]}}；請依 [mode, absolute_error, tolerance, strict_test_result, equality_convention] 回答。 四種mode為5/5/5/7欄；strict與inclusive兩mode雖同欄數，等號處結果相反。reference=0時relative error未定義。 查看答案與解釋(strict_tolerance, 3/2, 3/2, 否, error