Error

Try it out

Target

Derivation

1. Initial program 29.4

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

2. Initial simplification29.4

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

4. Applied flip--29.3

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

6. Applied *-un-lft-identity29.3

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

7. Applied add-sqr-sqrt29.3

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

8. Applied times-frac29.3

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

9. Simplified29.2

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

10. Simplified0.2

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

11. Final simplification0.2

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

Runtime

Time bar (total: 18.3s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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