Error

Target

Derivation

1. Initial program 29.9

   \[\sqrt{x + 1} - \sqrt{x}\]
2. Using strategy rm

3. Applied add-cube-cbrt29.9

   \[\leadsto \color{blue}{\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \sqrt[3]{\sqrt{x + 1}}} - \sqrt{x}\]

4. Applied fma-neg29.9

   \[\leadsto \color{blue}{(\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt{x + 1}}\right) + \left(-\sqrt{x}\right))_*}\]
5. Using strategy rm

6. Applied add-cube-cbrt29.9

   \[\leadsto (\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}}}\right) + \left(-\sqrt{x}\right))_*\]

7. Applied sqrt-prod29.9

   \[\leadsto (\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \left(\sqrt[3]{\color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}}}\right) + \left(-\sqrt{x}\right))_*\]

8. Applied cbrt-prod29.8

   \[\leadsto (\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{x + 1}}}\right)} + \left(-\sqrt{x}\right))_*\]
9. Using strategy rm

10. Applied add-sqr-sqrt29.8

    \[\leadsto (\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\color{blue}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}}}}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{x + 1}}}\right) + \left(-\sqrt{x}\right))_*\]

11. Applied cbrt-prod29.8

    \[\leadsto (\left(\sqrt[3]{\sqrt{x + 1}} \cdot \color{blue}{\left(\sqrt[3]{\sqrt{\sqrt{x + 1}}} \cdot \sqrt[3]{\sqrt{\sqrt{x + 1}}}\right)}\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{x + 1}}}\right) + \left(-\sqrt{x}\right))_*\]
12. Using strategy rm

13. Applied add-cbrt-cube29.8

    \[\leadsto (\left(\sqrt[3]{\sqrt{x + 1}} \cdot \left(\sqrt[3]{\sqrt{\sqrt{x + 1}}} \cdot \color{blue}{\sqrt[3]{\left(\sqrt[3]{\sqrt{\sqrt{x + 1}}} \cdot \sqrt[3]{\sqrt{\sqrt{x + 1}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{x + 1}}}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{x + 1}}}\right) + \left(-\sqrt{x}\right))_*\]

14. Final simplification29.8

    \[\leadsto (\left(\sqrt[3]{\sqrt{1 + x}} \cdot \left(\sqrt[3]{\left(\sqrt[3]{\sqrt{\sqrt{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}} \cdot \sqrt[3]{\sqrt{\sqrt{1 + x}}}\right)\right) \cdot \left(\sqrt[3]{\sqrt{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{1 + x}}}\right) + \left(-\sqrt{x}\right))_*\]

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

  :herbie-target
  (/ 1 (+ (sqrt (+ x 1)) (sqrt x)))

  (- (sqrt (+ x 1)) (sqrt x)))