Average Error: 18.9 → 15.2
Time: 23.2s
Precision: 64
Internal Precision: 128
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le 5.86570255508252 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\sqrt[3]{\frac{\frac{A}{V}}{\ell}} \cdot \left(\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{\frac{A}{V}}\right)}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \left(c0 \cdot \sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{A}{V}}\right) \cdot c0\\ \end{array}\]

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if l < 5.86570255508252e-309

    1. Initial program 18.9

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

    2. Initial simplification18.9

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

    4. Applied add-sqr-sqrt19.1

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

    5. Applied associate-*r*19.1

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

    7. Applied add-cube-cbrt19.1

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

    9. Applied cbrt-div19.1

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

    10. Applied cbrt-div19.1

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

    11. Applied frac-times19.1

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

    12. Applied associate-*l/19.1

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

    13. Applied sqrt-div19.1

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

    15. Applied div-inv19.1

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

    16. Applied cbrt-prod19.1

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

    if 5.86570255508252e-309 < l

    1. Initial program 18.8

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

    2. Initial simplification18.9

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

    4. Applied div-inv18.9

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}}\]

    5. Applied sqrt-prod11.4

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{A}{V}} \cdot \sqrt{\frac{1}{\ell}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 5.86570255508252 \cdot 10^{-309}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\sqrt[3]{\frac{\frac{A}{V}}{\ell}} \cdot \left(\left(\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{\frac{A}{V}}\right)}}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \left(c0 \cdot \sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{\ell}} \cdot \sqrt{\frac{A}{V}}\right) \cdot c0\\ \end{array}\]

Runtime

Time bar (total: 23.2s)Debug logProfile

herbie shell --seed 2018349 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))