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Target

Derivation

1. Initial program 30.2

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

3. Applied flip--30.0

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

5. Applied *-un-lft-identity30.0

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

6. Applied *-un-lft-identity30.0

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

7. Applied distribute-lft-out30.0

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

8. Applied *-un-lft-identity30.0

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

9. Applied times-frac30.0

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

10. Simplified30.0

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

11. Simplified0.2

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

13. Applied add-sqr-sqrt0.4

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

15. Applied pow1/20.4

    \[\leadsto 1 \cdot \frac{1}{\sqrt{\sqrt{1 + x} + \sqrt{x}} \cdot \color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\frac{1}{2}}}}\]

16. Applied pow1/20.4

    \[\leadsto 1 \cdot \frac{1}{\color{blue}{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\frac{1}{2}}} \cdot {\left(\sqrt{1 + x} + \sqrt{x}\right)}^{\frac{1}{2}}}\]

17. Applied pow-prod-down0.2

    \[\leadsto 1 \cdot \frac{1}{\color{blue}{{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\frac{1}{2}}}}\]

18. Applied pow-flip0.2

    \[\leadsto 1 \cdot \color{blue}{{\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\left(-\frac{1}{2}\right)}}\]

19. Simplified0.2

    \[\leadsto 1 \cdot {\left(\left(\sqrt{1 + x} + \sqrt{x}\right) \cdot \left(\sqrt{1 + x} + \sqrt{x}\right)\right)}^{\color{blue}{\frac{-1}{2}}}\]

20. Final simplification0.2

    \[\leadsto {\left(\left(\sqrt{x + 1} + \sqrt{x}\right) \cdot \left(\sqrt{x + 1} + \sqrt{x}\right)\right)}^{\frac{-1}{2}}\]

Runtime

Time bar (total: 16.4s)Debug logProfile

herbie shell --seed 2018254 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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