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Target

Derivation

1. Initial program 29.8

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

3. Applied flip--29.7

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

5. Applied *-un-lft-identity29.7

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

6. Applied add-cube-cbrt29.7

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

7. Applied times-frac29.7

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

8. Simplified29.7

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

9. Simplified0.2

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

10. Final simplification0.2

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018216 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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