Error

Target

Derivation

1. Initial program 19.4

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

3. Applied frac-sub19.3

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

4. Simplified19.3

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

6. Applied flip--19.2

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

7. Applied associate-/l/19.2

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

8. Simplified0.7

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

10. Applied pow10.7

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

11. Applied pow10.7

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

12. Applied pow10.7

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

13. Applied pow-prod-down0.7

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

14. Applied pow-prod-down0.7

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

15. Simplified0.6

    \[\leadsto \frac{1}{{\color{blue}{\left((x \cdot \left(\sqrt{x + 1}\right) + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*\right)}}^{1}}\]
16. Using strategy rm

17. Applied add-cbrt-cube0.6

    \[\leadsto \frac{1}{{\left((x \cdot \color{blue}{\left(\sqrt[3]{\left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right) \cdot \sqrt{x + 1}}\right)} + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*\right)}^{1}}\]

18. Final simplification0.6

    \[\leadsto \frac{1}{(x \cdot \left(\sqrt[3]{\sqrt{x + 1} \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1}\right)}\right) + \left((\left(\sqrt{x}\right) \cdot x + \left(\sqrt{x}\right))_*\right))_*}\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (x)
  :name "2isqrt (example 3.6)"

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

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

Details

Time bar (total: 13.4s)Debug log