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Target

Derivation

1. Initial program 29.9

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

3. Applied add-cbrt-cube29.9

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

5. Applied flip--29.9

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

6. Applied flip--29.9

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

7. Applied flip--29.7

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

8. Applied frac-times29.7

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

9. Applied frac-times29.7

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

10. Applied cbrt-div29.7

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

11. Simplified10.3

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

13. Applied pow310.3

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

14. Applied rem-cbrt-cube0.2

    \[\leadsto \frac{1}{\color{blue}{\sqrt{x + 1} + \sqrt{x}}}\]
15. Using strategy rm

16. Applied add-sqr-sqrt0.3

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

17. Final simplification0.3

    \[\leadsto \frac{1}{\sqrt{\sqrt{x + 1}} \cdot \sqrt{\sqrt{x + 1}} + \sqrt{x}}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018255 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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