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Target

Derivation

1. Initial program 20.2

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\]

2. Initial simplification20.2

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

4. Applied frac-sub20.2

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

5. Simplified20.2

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

7. Applied flip--19.9

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

8. Applied associate-/l/19.9

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

9. Simplified0.7

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

11. Applied associate-/r*0.4

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

13. Applied add-sqr-sqrt0.4

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

14. Final simplification0.4

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

Runtime

Time bar (total: 16.0s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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