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Target

Derivation

1. Initial program 29.7

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

2. Initial simplification29.7

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

4. Applied flip--29.6

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

5. Taylor expanded around inf 0.2

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

7. Applied add-sqr-sqrt0.2

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

8. Applied sqrt-prod0.3

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

10. Applied *-un-lft-identity0.3

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

11. Applied associate-/r*0.3

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

12. Simplified0.2

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

13. Final simplification0.2

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

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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

Details

Time bar (total: 10.3s)Debug log