Error

Derivation

1. Initial program 19.0

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
2. Using strategy rm

3. Applied add-cube-cbrt19.4

   \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\left(\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \sqrt[3]{V}\right)} \cdot \ell}}\]

4. Applied associate-*l*19.4

   \[\leadsto c0 \cdot \sqrt{\frac{A}{\color{blue}{\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \left(\sqrt[3]{V} \cdot \ell\right)}}}\]

5. Applied add-cube-cbrt19.5

   \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A}}}{\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \left(\sqrt[3]{V} \cdot \ell\right)}}\]

6. Applied times-frac16.2

   \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \ell}}}\]

7. Applied sqrt-prod7.9

   \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}} \cdot \sqrt{\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \ell}}\right)}\]

8. Simplified6.3

   \[\leadsto c0 \cdot \left(\color{blue}{\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|} \cdot \sqrt{\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \ell}}\right)\]
9. Using strategy rm

10. Applied *-un-lft-identity6.3

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{\sqrt[3]{V} \cdot \color{blue}{\left(1 \cdot \ell\right)}}}\right)\]

11. Applied associate-*r*6.3

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\frac{\sqrt[3]{A}}{\color{blue}{\left(\sqrt[3]{V} \cdot 1\right) \cdot \ell}}}\right)\]

12. Applied add-cube-cbrt6.5

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \sqrt[3]{\sqrt[3]{A}}}}{\left(\sqrt[3]{V} \cdot 1\right) \cdot \ell}}\right)\]

13. Applied times-frac5.4

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V} \cdot 1} \cdot \frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}}\right)\]

14. Simplified5.4

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{A}}\right)} \cdot \frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)\]
15. Using strategy rm

16. Applied add-cube-cbrt5.5

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\left(\color{blue}{\left(\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right)} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)\]

17. Applied associate-*l*5.5

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\color{blue}{\left(\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right)\right)} \cdot \frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)\]

18. Applied associate-*l*5.6

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right) \cdot \left(\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}\right)}}\right)\]

19. Applied sqrt-prod4.1

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \color{blue}{\left(\sqrt{\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}} \cdot \sqrt{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)}\right)\]

20. Simplified4.1

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\color{blue}{\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right|} \cdot \sqrt{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{A}}}{\ell}}\right)\right)\]
21. Using strategy rm

22. Applied add-cube-cbrt4.1

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \sqrt{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \sqrt[3]{\sqrt[3]{A}}}}}{\ell}}\right)\right)\]

23. Applied cbrt-prod4.1

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \sqrt{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\color{blue}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}{\ell}}\right)\right)\]

24. Applied associate-/l*4.2

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \sqrt{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \color{blue}{\frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}}{\frac{\ell}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}}}\right)\right)\]

25. Applied add-cube-cbrt4.2

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \sqrt{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \sqrt[3]{V}}}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}}{\frac{\ell}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}}\right)\right)\]

26. Applied cbrt-prod4.2

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \sqrt{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\color{blue}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}}{\frac{\ell}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}}\right)\right)\]

27. Applied associate-/r*4.2

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \sqrt{\left(\sqrt[3]{\color{blue}{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}{\sqrt[3]{\sqrt[3]{V}}}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}}{\frac{\ell}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}}\right)\right)\]

28. Applied cbrt-div4.2

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \sqrt{\left(\color{blue}{\frac{\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{V}}}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}}{\frac{\ell}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}}\right)\right)\]

29. Applied associate-*l/4.2

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}} \cdot \sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{V}}}}} \cdot \frac{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}}{\frac{\ell}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}}\right)\right)\]

30. Applied frac-times4.2

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \sqrt{\color{blue}{\frac{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}}{\sqrt[3]{\sqrt[3]{\sqrt[3]{V}}} \cdot \frac{\ell}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}}}\right)\right)\]

31. Applied sqrt-div2.8

    \[\leadsto c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \color{blue}{\frac{\sqrt{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}}}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{V}}} \cdot \frac{\ell}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}}}\right)\right)\]

32. Final simplification2.8

    \[\leadsto \left(\left(\left|\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}\right| \cdot \frac{\sqrt{\left(\sqrt[3]{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}} \cdot \sqrt[3]{\sqrt[3]{A}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}}}{\sqrt{\sqrt[3]{\sqrt[3]{\sqrt[3]{V}}} \cdot \frac{\ell}{\sqrt[3]{\sqrt[3]{\sqrt[3]{A}}}}}}\right) \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|\right) \cdot c0\]

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))