分数計算、重複か？

[Stasshe]

Factor Result:
$$\frac{17}{16}x^{2} + \frac{17}{16}y^{2} - \frac{5}{2}x - \frac{19}{4}y$$
Steps (JSON):

[
  "Original expression: (x - 1)^{2} + (y - 2)^{2} - 4 + \\frac{-1}{16}((x - 4)^{2} + (y - 6)^{2} - 36)",
  "--- overlapSimplify start ---",
  [
    "overlapSimplify pass #1",
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Start simplifyPolynomialFraction",
    [
      "After cancellation"
    ],
    "No reduction, returning original fraction",
    [
      "Applying polynomial fraction reduction (deep)",
      "$\\frac{-1}{16}$"
    ],
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Start simplifyPolynomialFraction",
    [
      "After cancellation"
    ],
    "No reduction, returning original fraction",
    [
      "Applying polynomial fraction reduction (deep)",
      "$\\frac{-1}{16}$"
    ],
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Start simplifyPolynomialFraction",
    [
      "After cancellation"
    ],
    "No reduction, returning original fraction",
    [
      "Applying polynomial fraction reduction (deep)",
      "$\\frac{-1}{16}$"
    ],
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Start simplifyPolynomialFraction",
    [
      "After cancellation"
    ],
    "No reduction, returning original fraction",
    [
      "Applying polynomial fraction reduction (deep)",
      "$\\frac{-1}{16}$"
    ],
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Start simplifyPolynomialFraction",
    [
      "After cancellation"
    ],
    "No reduction, returning original fraction",
    [
      "Applying polynomial fraction reduction (deep)",
      "$\\frac{-1}{16}$"
    ],
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $-1$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-1$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Starting factorization of: $16$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$16$",
    "Start simplifyPolynomialFraction",
    [
      "After cancellation"
    ],
    "No reduction, returning original fraction",
    [
      "Applying polynomial fraction reduction (deep)",
      "$\\frac{-1}{16}$"
    ],
    "Starting factorization of: $-9$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-9$",
    "Starting factorization of: $-9$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-9$",
    "Starting factorization of: $4$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$4$",
    "Starting factorization of: $4$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$4$",
    "Start simplifyPolynomialFraction",
    [
      "After cancellation"
    ],
    "No reduction, returning original fraction",
    [
      "Applying polynomial fraction reduction (deep)",
      "$\\frac{-9}{4}$"
    ],
    "Starting factorization of: $-9$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-9$",
    "Starting factorization of: $-9$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$-9$",
    "Starting factorization of: $4$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$4$",
    "Starting factorization of: $4$",
    "No further factorization possible",
    "Attempting recursive factorization of subexpressions...",
    "$4$",
    "Start simplifyPolynomialFraction",
    [
      "After cancellation"
    ],
    "No reduction, returning original fraction",
    [
      "Applying polynomial fraction reduction (deep)",
      "$\\frac{-9}{4}$"
    ],
    "Starting unified simplification",
    [
      "After middle-simplify",
      [
        "Converted sqrt to exponential form: $x^{2} - x - x + 1 + y^{2} - 2y - 2y + 4 - 4 + x^{2} - \\frac{-1}{16}4x - \\frac{-1}{16}4x - 1 + y^{2} - \\frac{-1}{16}6y - \\frac{-1}{16}6y + \\frac{-9}{4} - \\frac{-9}{4}$",
        "Applied advanced exponential simplification: $x^{2} - x - x + 1 + y^{2} - 2y - 2y + 4 - 4 + x^{2} - \\frac{-1}{16}4x - \\frac{-1}{16}4x - 1 + y^{2} - \\frac{-1}{16}6y - \\frac{-1}{16}6y + \\frac{-9}{4} - \\frac{-9}{4}$",
        "Applying basic simplification out",
        "$x^{2} - x - x + 1 + y^{2} - 2y - 2y + 4 - 4 + x^{2} - \\frac{-1}{16}4x - \\frac{-1}{16}4x - 1 + y^{2} - \\frac{-1}{16}6y - \\frac{-1}{16}6y + \\frac{-9}{4} - \\frac{-9}{4}$",
        "Finished basic simplification: $\\frac{17}{16}x^{2} + \\frac{17}{16}y^{2} - \\frac{5}{2}x - \\frac{19}{4}y$"
      ]
    ],
    [
      "Final pass with expand: true",
      "$\\frac{17}{16}x^{2} + \\frac{17}{16}y^{2} - \\frac{5}{2}x - \\frac{19}{4}y$"
    ],
    [
      "Unified simplification complete",
      [
        "Converted sqrt to exponential form: $\\frac{17}{16}x^{2} + \\frac{17}{16}y^{2} - \\frac{5}{2}x - \\frac{19}{4}y$",
        "Applied advanced exponential simplification: $\\frac{17}{16}x^{2} + \\frac{17}{16}y^{2} - \\frac{5}{2}x - \\frac{19}{4}y$",
        "Applying basic simplification out",
        "$\\frac{17}{16}x^{2} + \\frac{17}{16}y^{2} - \\frac{5}{2}x - \\frac{19}{4}y$",
        "Finished basic simplification: $\\frac{17}{16}x^{2} + \\frac{17}{16}y^{2} - \\frac{5}{2}x - \\frac{19}{4}y$"
      ]
    ]
  ],
  "No further change detected, stopping.",
  "--- overlapSimplify end ---",
  "After distribution/expansion: $$x^{2} - x - x + 1 + y^{2} - 2y - 2y + 4 - 4 + x^{2} - \\frac{-1}{16}4x - \\frac{-1}{16}4x - 1 + y^{2} - \\frac{-1}{16}6y - \\frac{-1}{16}6y + \\frac{-9}{4} - \\frac{-9}{4}$$",
  "After combining like terms: $\\frac{17}{16}x^{2} + \\frac{17}{16}y^{2} - \\frac{5}{2}x - \\frac{19}{4}y$"
]

[Stasshe]

いい質問です！
与えられたログを追ってみると、ミスは 展開・分配の途中 で発生しています。

---

元の式

$$ (x - 1)^{2} + (y - 2)^{2} - 4 ;+; \tfrac{-1}{16}\big((x - 4)^{2} + (y - 6)^{2} - 36\big) $$

---

正しい展開

前半

$$ (x - 1)^2 + (y - 2)^2 - 4 = (x^2 - 2x + 1) + (y^2 - 4y + 4) - 4 = x^2 + y^2 - 2x - 4y + 1 $$

後半

$$ \tfrac{-1}{16} \big((x - 4)^2 + (y - 6)^2 - 36\big) $$

• $(x - 4)^2 = x^2 - 8x + 16$

• $(y - 6)^2 = y^2 - 12y + 36$

• 全体で

  $$
  x^2 + y^2 - 8x - 12y + 16
  $$

• これに $-\tfrac{1}{16}$ をかけると

  $$
  -\tfrac{1}{16}x^2 - \tfrac{1}{16}y^2 + \tfrac{1}{2}x + \tfrac{3}{4}y - 1
  $$

---

合計

$$ x^2 + y^2 - 2x - 4y + 1 ;+; \left(-\tfrac{1}{16}x^2 - \tfrac{1}{16}y^2 + \tfrac{1}{2}x + \tfrac{3}{4}y - 1\right) $$

$$ = \tfrac{15}{16}x^2 + \tfrac{15}{16}y^2 - \tfrac{3}{2}x - \tfrac{13}{4}y $$

---

ログの結果

ログでは最終的に

$$ \tfrac{17}{16}x^2 + \tfrac{17}{16}y^2 - \tfrac{5}{2}x - \tfrac{19}{4}y $$

と出ています。

---

食い違い

• 正しい係数:
  $\tfrac{15}{16}$（$x^2,y^2$）、$-\tfrac{3}{2}$（$x$）、$-\tfrac{13}{4}$（$y$）
• ログの結果:
  $\tfrac{17}{16}, -\tfrac{5}{2}, -\tfrac{19}{4}$

つまり、x², y², x, y の係数がすべてズレている。

---

原因

展開ログを見ると、分配の途中で

- \frac{-1}{16}4x
- \frac{-1}{16}6y

などと記録されています。

本来は $-1/16 \cdot -8x = +1/2x$, $-1/16 \cdot -12y = +3/4y$ となるべきですが、
「-8x」や「-12y」を「-4x」「-6y」と半分に誤処理していることが分かります。

---

✅ 結論:
計算ミスは $(x-4)^2$ と $(y-6)^2$ を展開したときの線形項 $-8x, -12y$ を誤って $-4x, -6y$ と扱った段階 で発生しています。

---

👉 ここで質問ですが、Stasshe さんの処理系って

• 「展開」部分を自前で書いてますか？
• それとも CAS ライブラリ呼んでますか？

実装レベルのバグか、ライブラリの制限かを切り分ける必要がありそうです。