Average Error: 18.3 → 1.4
Time: 1.3m
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(\left(\sqrt{\sqrt[3]{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|\right) \cdot c0\right) \cdot \left(\sqrt{\sqrt[3]{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\right)\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\left(\left(\sqrt{\sqrt[3]{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|\right) \cdot c0\right) \cdot \left(\sqrt{\sqrt[3]{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\right)
double f(double c0, double A, double V, double l) {
        double r7495018 = c0;
        double r7495019 = A;
        double r7495020 = V;
        double r7495021 = l;
        double r7495022 = r7495020 * r7495021;
        double r7495023 = r7495019 / r7495022;
        double r7495024 = sqrt(r7495023);
        double r7495025 = r7495018 * r7495024;
        return r7495025;
}
↓
double f(double c0, double A, double V, double l) {
        double r7495026 = 1.0;
        double r7495027 = l;
        double r7495028 = cbrt(r7495027);
        double r7495029 = r7495028 * r7495028;
        double r7495030 = r7495026 / r7495029;
        double r7495031 = cbrt(r7495030);
        double r7495032 = r7495031 * r7495031;
        double r7495033 = sqrt(r7495032);
        double r7495034 = A;
        double r7495035 = cbrt(r7495034);
        double r7495036 = V;
        double r7495037 = cbrt(r7495036);
        double r7495038 = r7495035 / r7495037;
        double r7495039 = fabs(r7495038);
        double r7495040 = r7495033 * r7495039;
        double r7495041 = c0;
        double r7495042 = r7495040 * r7495041;
        double r7495043 = sqrt(r7495031);
        double r7495044 = r7495038 / r7495028;
        double r7495045 = sqrt(r7495044);
        double r7495046 = r7495043 * r7495045;
        double r7495047 = r7495042 * r7495046;
        return r7495047;
}

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 18.3

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

  3. Applied add-cube-cbrt18.6

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

  4. Applied associate-*l*18.6

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

  5. Applied add-cube-cbrt18.8

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

  6. Applied times-frac15.7

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

  7. Applied sqrt-prod7.7

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

  8. Simplified6.1

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

  9. Simplified4.7

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

  11. Applied add-cube-cbrt4.8

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

  12. Applied *-un-lft-identity4.8

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

  13. Applied times-frac4.8

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

  14. Applied sqrt-prod2.2

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

  16. Applied add-cube-cbrt2.3

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

  17. Applied sqrt-prod2.3

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

  18. Applied associate-*l*2.3

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

  19. Applied associate-*r*2.3

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

  20. Applied associate-*r*1.4

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

  21. Final simplification1.4

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))