Error

Target

Derivation

1. Initial program 29.8

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

5. Applied add-cube-cbrt29.8

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

6. Applied sqrt-prod29.8

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

7. Applied cbrt-prod29.8

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

9. Applied flip--29.7

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

10. Taylor expanded around 0 0.4

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

11. Final simplification0.4

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

Reproduce

herbie shell --seed 2019053 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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