Error

Try it out

Target

Derivation

1. Initial program 29.5

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

2. Initial simplification29.5

   \[\leadsto \sqrt{1 + x} - \sqrt{x}\]
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. Taylor expanded around 0 0.2

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

7. Applied add-sqr-sqrt0.4

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

8. Applied associate-/r*0.3

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

10. Applied pow1/20.3

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

11. Applied pow-flip0.3

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

12. Simplified0.3

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

13. Final simplification0.3

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

Runtime

Time bar (total: 15.9s)Debug logProfile

herbie shell --seed 2018355 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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