Average Error: 18.9 → 1.1
Time: 16.0s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(\left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}\right| \cdot c0\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\left(\left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}\right| \cdot c0\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\ell}}}{\sqrt[3]{V}}}
double f(double c0, double A, double V, double l) {
        double r96066 = c0;
        double r96067 = A;
        double r96068 = V;
        double r96069 = l;
        double r96070 = r96068 * r96069;
        double r96071 = r96067 / r96070;
        double r96072 = sqrt(r96071);
        double r96073 = r96066 * r96072;
        return r96073;
}
↓
double f(double c0, double A, double V, double l) {
        double r96074 = A;
        double r96075 = cbrt(r96074);
        double r96076 = l;
        double r96077 = cbrt(r96076);
        double r96078 = r96075 / r96077;
        double r96079 = V;
        double r96080 = cbrt(r96079);
        double r96081 = r96078 / r96080;
        double r96082 = fabs(r96081);
        double r96083 = c0;
        double r96084 = r96082 * r96083;
        double r96085 = sqrt(r96081);
        double r96086 = r96084 * r96085;
        return r96086;
}

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 18.9

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

  3. Applied *-un-lft-identity18.9

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

  4. Applied times-frac18.9

    \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity18.9

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

  7. Applied associate-*l*18.9

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

  8. Simplified18.8

    \[\leadsto 1 \cdot \color{blue}{\left(c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\right)}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt19.2

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

  11. Applied add-cube-cbrt19.3

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

  12. Applied add-cube-cbrt19.4

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

  13. Applied times-frac19.4

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

  14. Applied times-frac15.2

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

  15. Applied sqrt-prod7.1

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

  16. Applied associate-*r*7.1

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

  17. Simplified1.1

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

  18. Final simplification1.1

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))