Average Error: 18.9 → 2.4
Time: 10.9s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)\right) \cdot c0\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{\sqrt[3]{A}}}{\sqrt[3]{\sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\right)\right) \cdot c0
double f(double c0, double A, double V, double l) {
        double r160027 = c0;
        double r160028 = A;
        double r160029 = V;
        double r160030 = l;
        double r160031 = r160029 * r160030;
        double r160032 = r160028 / r160031;
        double r160033 = sqrt(r160032);
        double r160034 = r160027 * r160033;
        return r160034;
}
↓
double f(double c0, double A, double V, double l) {
        double r160035 = A;
        double r160036 = cbrt(r160035);
        double r160037 = V;
        double r160038 = cbrt(r160037);
        double r160039 = r160036 / r160038;
        double r160040 = fabs(r160039);
        double r160041 = r160036 * r160036;
        double r160042 = cbrt(r160041);
        double r160043 = cbrt(r160038);
        double r160044 = r160043 * r160043;
        double r160045 = r160042 / r160044;
        double r160046 = l;
        double r160047 = cbrt(r160046);
        double r160048 = r160047 * r160047;
        double r160049 = r160045 / r160048;
        double r160050 = sqrt(r160049);
        double r160051 = cbrt(r160036);
        double r160052 = r160051 / r160043;
        double r160053 = r160052 / r160047;
        double r160054 = sqrt(r160053);
        double r160055 = r160050 * r160054;
        double r160056 = r160040 * r160055;
        double r160057 = c0;
        double r160058 = r160056 * r160057;
        return r160058;
}

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

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

  4. Applied sqrt-prod18.9

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

  5. Simplified18.9

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

  6. Simplified18.6

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

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

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

  9. Applied add-cube-cbrt19.0

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

  10. Applied add-cube-cbrt19.1

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

  11. Applied times-frac19.1

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

  12. Applied times-frac15.0

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

  13. Applied sqrt-prod6.6

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

  14. Simplified4.8

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

  16. Applied add-cube-cbrt5.0

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

  17. Applied add-cube-cbrt5.1

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

  18. Applied add-cube-cbrt5.1

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

  19. Applied cbrt-prod5.1

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

  20. Applied times-frac5.1

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

  21. Applied times-frac5.1

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

  22. Applied sqrt-prod2.4

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

  23. Final simplification2.4

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

Reproduce

herbie shell --seed 2020045 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))