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Target

Derivation

1. Initial program 19.7

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

3. Applied flip--19.8

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

4. Applied simplify19.8

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

6. Applied frac-sub19.2

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

7. Applied simplify5.5

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

9. Applied *-un-lft-identity5.5

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

10. Applied associate-/r*5.5

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

11. Applied simplify5.1

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

13. Applied *-un-lft-identity5.1

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

14. Applied add-sqr-sqrt5.1

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

15. Applied add-sqr-sqrt5.2

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

16. Applied times-frac5.2

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

17. Applied times-frac0.5

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

18. Applied simplify0.5

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

19. Applied simplify0.5

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018167 
(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)))))