Average Error: 18.9 → 1.3
Time: 32.5s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{V}} \cdot \left(\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}\right)}}{\sqrt[3]{\ell}}} \cdot \left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell} \cdot \sqrt[3]{V}}\right|\right)\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{\sqrt[3]{V}} \cdot \left(\sqrt[3]{\sqrt[3]{V}} \cdot \sqrt[3]{\sqrt[3]{V}}\right)}}{\sqrt[3]{\ell}}} \cdot \left(c0 \cdot \left|\frac{\sqrt[3]{A}}{\sqrt[3]{\ell} \cdot \sqrt[3]{V}}\right|\right)
double f(double c0, double A, double V, double l) {
        double r46440092 = c0;
        double r46440093 = A;
        double r46440094 = V;
        double r46440095 = l;
        double r46440096 = r46440094 * r46440095;
        double r46440097 = r46440093 / r46440096;
        double r46440098 = sqrt(r46440097);
        double r46440099 = r46440092 * r46440098;
        return r46440099;
}
↓
double f(double c0, double A, double V, double l) {
        double r46440100 = A;
        double r46440101 = cbrt(r46440100);
        double r46440102 = V;
        double r46440103 = cbrt(r46440102);
        double r46440104 = cbrt(r46440103);
        double r46440105 = r46440104 * r46440104;
        double r46440106 = r46440104 * r46440105;
        double r46440107 = r46440101 / r46440106;
        double r46440108 = l;
        double r46440109 = cbrt(r46440108);
        double r46440110 = r46440107 / r46440109;
        double r46440111 = sqrt(r46440110);
        double r46440112 = c0;
        double r46440113 = r46440109 * r46440103;
        double r46440114 = r46440101 / r46440113;
        double r46440115 = fabs(r46440114);
        double r46440116 = r46440112 * r46440115;
        double r46440117 = r46440111 * r46440116;
        return r46440117;
}

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 associate-/r*19.2

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

  5. Applied add-cube-cbrt19.5

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

  6. Applied add-cube-cbrt19.6

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

  7. Applied add-cube-cbrt19.7

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

  8. Applied times-frac19.7

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

  9. Applied times-frac15.6

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

  10. Applied sqrt-prod7.2

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

  11. Applied associate-*r*7.2

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

  12. Simplified1.1

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

  14. Applied add-cube-cbrt1.3

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

  15. Final simplification1.3

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))