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Target

Derivation

1. Initial program 19.1

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

2. Initial simplification19.1

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

4. Applied flip--19.1

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

6. Applied frac-times24.1

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

7. Applied frac-times19.2

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

8. Applied frac-sub19.0

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

9. Simplified5.3

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

10. Simplified5.2

    \[\leadsto \frac{\frac{1}{\color{blue}{(x \cdot x + x)_*}}}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}\]
11. Using strategy rm

12. Applied div-inv5.2

    \[\leadsto \color{blue}{\frac{1}{(x \cdot x + x)_*} \cdot \frac{1}{\frac{1}{\sqrt{x}} + \frac{1}{\sqrt{x + 1}}}}\]
13. Using strategy rm

14. Applied pow15.2

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

15. Applied pow15.2

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

16. Applied pow-prod-down5.2

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

17. Simplified0.3

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

18. Final simplification0.3

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

Runtime

Time bar (total: 52.1s)Debug logProfile

herbie shell --seed 2018348 +o rules:numerics
(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)))))