Error

Target

Derivation

1. Initial program 29.8

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

3. Applied add-cube-cbrt30.1

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

4. Applied add-cube-cbrt30.0

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

5. Applied sqrt-prod30.0

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

6. Applied prod-diff30.0

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

7. Simplified29.9

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

8. Simplified29.8

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

10. Applied add-exp-log29.7

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

12. Applied add-cube-cbrt29.7

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

14. Applied log1p-expm1-u29.7

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

15. Final simplification29.7

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

Runtime

Time bar (total: 16.4s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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