Average Error: 19.3 → 1.2
Time: 27.6s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\left(\left|\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}\right| \cdot c0\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}{\sqrt[3]{\ell}}}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\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]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}}{\sqrt[3]{\ell}}}
double f(double c0, double A, double V, double l) {
        double r16445964 = c0;
        double r16445965 = A;
        double r16445966 = V;
        double r16445967 = l;
        double r16445968 = r16445966 * r16445967;
        double r16445969 = r16445965 / r16445968;
        double r16445970 = sqrt(r16445969);
        double r16445971 = r16445964 * r16445970;
        return r16445971;
}
↓
double f(double c0, double A, double V, double l) {
        double r16445972 = A;
        double r16445973 = cbrt(r16445972);
        double r16445974 = V;
        double r16445975 = cbrt(r16445974);
        double r16445976 = r16445973 / r16445975;
        double r16445977 = l;
        double r16445978 = cbrt(r16445977);
        double r16445979 = r16445976 / r16445978;
        double r16445980 = fabs(r16445979);
        double r16445981 = c0;
        double r16445982 = r16445980 * r16445981;
        double r16445983 = cbrt(r16445975);
        double r16445984 = r16445975 * r16445975;
        double r16445985 = cbrt(r16445984);
        double r16445986 = r16445983 * r16445985;
        double r16445987 = r16445973 / r16445986;
        double r16445988 = r16445987 / r16445978;
        double r16445989 = sqrt(r16445988);
        double r16445990 = r16445982 * r16445989;
        return r16445990;
}

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 19.3

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

  3. Applied associate-/r*19.5

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

  5. Applied add-cube-cbrt19.8

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

  6. Applied add-cube-cbrt20.0

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

  7. Applied add-cube-cbrt20.0

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

  8. Applied times-frac20.0

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

  9. Applied times-frac15.8

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

  10. Applied sqrt-prod7.6

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

  11. Simplified2.3

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

  13. Applied associate-*r*1.2

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

  15. Applied add-cube-cbrt1.2

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

  16. Applied cbrt-prod1.2

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

  17. Final simplification1.2

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

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))