Error

Target

Derivation

1. Initial program 30.0

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

3. Applied add-sqr-sqrt30.0

   \[\leadsto \sqrt{x + 1} - \color{blue}{\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}}}\]

4. Applied add-cube-cbrt30.0

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

5. Applied sqrt-prod30.0

   \[\leadsto \color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}} - \sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}}\]

6. Applied prod-diff30.0

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

7. Simplified30.0

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

8. Simplified30.0

   \[\leadsto (\left(\left|\sqrt[3]{1 + x}\right|\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt{x}\right))_* + \color{blue}{0}\]
9. Using strategy rm

10. Applied add-exp-log29.9

    \[\leadsto \color{blue}{e^{\log \left((\left(\left|\sqrt[3]{1 + x}\right|\right) \cdot \left(\sqrt{\sqrt[3]{1 + x}}\right) + \left(-\sqrt{x}\right))_*\right)}} + 0\]
11. Using strategy rm

12. Applied add-sqr-sqrt29.8

    \[\leadsto e^{\log \left((\left(\left|\sqrt[3]{1 + x}\right|\right) \cdot \left(\sqrt{\sqrt[3]{\color{blue}{\sqrt{1 + x} \cdot \sqrt{1 + x}}}}\right) + \left(-\sqrt{x}\right))_*\right)} + 0\]

13. Applied cbrt-prod29.8

    \[\leadsto e^{\log \left((\left(\left|\sqrt[3]{1 + x}\right|\right) \cdot \left(\sqrt{\color{blue}{\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}}}\right) + \left(-\sqrt{x}\right))_*\right)} + 0\]
14. Using strategy rm

15. Applied pow129.8

    \[\leadsto e^{\log \color{blue}{\left({\left((\left(\left|\sqrt[3]{1 + x}\right|\right) \cdot \left(\sqrt{\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}}\right) + \left(-\sqrt{x}\right))_*\right)}^{1}\right)}} + 0\]

16. Applied log-pow29.8

    \[\leadsto e^{\color{blue}{1 \cdot \log \left((\left(\left|\sqrt[3]{1 + x}\right|\right) \cdot \left(\sqrt{\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}}\right) + \left(-\sqrt{x}\right))_*\right)}} + 0\]

17. Applied exp-prod29.8

    \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left((\left(\left|\sqrt[3]{1 + x}\right|\right) \cdot \left(\sqrt{\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}}\right) + \left(-\sqrt{x}\right))_*\right)\right)}} + 0\]

18. Simplified29.8

    \[\leadsto {\color{blue}{e}}^{\left(\log \left((\left(\left|\sqrt[3]{1 + x}\right|\right) \cdot \left(\sqrt{\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}}\right) + \left(-\sqrt{x}\right))_*\right)\right)} + 0\]

19. Final simplification29.8

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

Runtime

Time bar (total: 14.4s)Debug logProfile

herbie shell --seed 2018249 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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