Average Error: 18.6 → 11.4
Time: 23.5s
Precision: 64
Internal Precision: 576
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(c0 \cdot \frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \sqrt{\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{A}{V}}}\]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 18.6

    \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]

  2. Initial simplification18.1

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

  4. Applied add-cube-cbrt18.5

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

  5. Applied sqrt-prod18.5

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

  6. Applied associate-*r*18.5

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

  8. Applied cbrt-div18.4

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

  9. Applied cbrt-div18.5

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

  10. Applied frac-times18.5

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

  11. Applied sqrt-div18.5

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

  12. Simplified18.5

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

  14. Applied div-inv18.5

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

  15. Applied cbrt-prod11.4

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

  16. Final simplification11.4

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

Runtime

Time bar (total: 23.5s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))