Error

Target

Derivation

1. Initial program 19.5

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

2. Initial simplification19.5

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

4. Applied frac-sub19.5

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

5. Simplified19.5

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

7. Applied flip--19.4

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

8. Applied associate-/l/19.4

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

9. Simplified0.8

   \[\leadsto \frac{\color{blue}{1}}{\left(\sqrt{x} \cdot \sqrt{x + 1}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)}\]
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 *-un-lft-identity0.4

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

14. Applied *-un-lft-identity0.4

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

15. Applied distribute-lft-out0.4

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

16. Applied *-un-lft-identity0.4

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

17. Applied times-frac0.4

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

18. Simplified0.4

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

19. Simplified0.3

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

20. Final simplification0.3

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

Reproduce

herbie shell --seed 2018360 +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)))))

Details

Time bar (total: 18.3s)Debug log