Average Error: 18.5 → 1.3
Time: 19.8s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\left(c0 \cdot \left|\frac{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}} \cdot \sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right|\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}
double f(double c0, double A, double V, double l) {
        double r1875757 = c0;
        double r1875758 = A;
        double r1875759 = V;
        double r1875760 = l;
        double r1875761 = r1875759 * r1875760;
        double r1875762 = r1875758 / r1875761;
        double r1875763 = sqrt(r1875762);
        double r1875764 = r1875757 * r1875763;
        return r1875764;
}
↓
double f(double c0, double A, double V, double l) {
        double r1875765 = c0;
        double r1875766 = A;
        double r1875767 = cbrt(r1875766);
        double r1875768 = r1875767 * r1875767;
        double r1875769 = cbrt(r1875768);
        double r1875770 = cbrt(r1875767);
        double r1875771 = r1875769 * r1875770;
        double r1875772 = V;
        double r1875773 = cbrt(r1875772);
        double r1875774 = r1875771 / r1875773;
        double r1875775 = l;
        double r1875776 = cbrt(r1875775);
        double r1875777 = r1875774 / r1875776;
        double r1875778 = fabs(r1875777);
        double r1875779 = r1875765 * r1875778;
        double r1875780 = r1875767 / r1875773;
        double r1875781 = r1875780 / r1875776;
        double r1875782 = sqrt(r1875781);
        double r1875783 = r1875779 * r1875782;
        return r1875783;
}

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. Initial program 18.5

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

  3. Applied associate-/r*18.5

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

  5. Applied add-cube-cbrt18.8

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

  6. Applied add-cube-cbrt18.9

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

  7. Applied add-cube-cbrt19.0

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

  8. Applied times-frac19.0

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

  9. Applied times-frac14.9

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

  10. Applied sqrt-prod7.0

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

  11. Applied associate-*r*7.0

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

  12. Simplified1.1

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

  14. Applied add-cube-cbrt1.2

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

  15. Applied cbrt-prod1.3

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

  16. Final simplification1.3

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))