Error

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. Applied simplify0.2

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

6. Applied add-sqr-sqrt0.4

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

8. Applied pow1/20.4

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

9. Applied pow1/20.4

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

10. Applied pow-prod-down0.2

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

11. Applied pow-flip0.2

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

13. Applied distribute-lft-in0.2

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

14. Applied simplify0.1

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

Runtime

Time bar (total: 19.7s)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(FPCore (x)
  :name "2sqrt (example 3.1)"

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

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