Error

Target

Derivation

1. Initial program 20.5

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

3. Applied frac-sub20.4

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

4. Applied simplify20.4

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

6. Applied flip--20.2

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

7. Applied simplify0.4

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

9. Applied add-sqr-sqrt0.5

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

10. Applied times-frac0.5

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

12. Applied add-cube-cbrt0.5

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

13. Applied sqrt-prod0.5

    \[\leadsto \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x}} \cdot \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\color{blue}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}}}\]

14. Applied *-un-lft-identity0.5

    \[\leadsto \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x}} \cdot \frac{\sqrt{\color{blue}{1 \cdot \frac{1}{\sqrt{x + 1} + \sqrt{x}}}}}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}}\]

15. Applied sqrt-prod0.5

    \[\leadsto \frac{\sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}{\sqrt{x}} \cdot \frac{\color{blue}{\sqrt{1} \cdot \sqrt{\frac{1}{\sqrt{x + 1} + \sqrt{x}}}}}{\sqrt{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt{\sqrt[3]{x + 1}}}\]

16. Applied times-frac0.5

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

17. Applied simplify0.5

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

Runtime

Time bar (total: 44.4s)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' 
(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)))))