Average Error: 18.8 → 9.7
Time: 17.8s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.0028777413688825 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{V \cdot \ell}}} \cdot \left(\frac{\left|\sqrt[3]{A}\right|}{\sqrt{\sqrt[3]{V \cdot \ell} \cdot \sqrt[3]{V \cdot \ell}}} \cdot c0\right)\\ \mathbf{elif}\;V \cdot \ell \le 4.077916039006289 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \cdot \ell \le -1.0028777413688825 \cdot 10^{-276}:\\
\;\;\;\;\sqrt{\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{V \cdot \ell}}} \cdot \left(\frac{\left|\sqrt[3]{A}\right|}{\sqrt{\sqrt[3]{V \cdot \ell} \cdot \sqrt[3]{V \cdot \ell}}} \cdot c0\right)\\

\mathbf{elif}\;V \cdot \ell \le 4.077916039006289 \cdot 10^{-304}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\

\mathbf{else}:\\
\;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\

\end{array}
double f(double c0, double A, double V, double l) {
        double r2162142 = c0;
        double r2162143 = A;
        double r2162144 = V;
        double r2162145 = l;
        double r2162146 = r2162144 * r2162145;
        double r2162147 = r2162143 / r2162146;
        double r2162148 = sqrt(r2162147);
        double r2162149 = r2162142 * r2162148;
        return r2162149;
}
↓
double f(double c0, double A, double V, double l) {
        double r2162150 = V;
        double r2162151 = l;
        double r2162152 = r2162150 * r2162151;
        double r2162153 = -1.0028777413688825e-276;
        bool r2162154 = r2162152 <= r2162153;
        double r2162155 = A;
        double r2162156 = cbrt(r2162155);
        double r2162157 = 1.0;
        double r2162158 = r2162157 / r2162152;
        double r2162159 = cbrt(r2162158);
        double r2162160 = r2162156 * r2162159;
        double r2162161 = sqrt(r2162160);
        double r2162162 = fabs(r2162156);
        double r2162163 = cbrt(r2162152);
        double r2162164 = r2162163 * r2162163;
        double r2162165 = sqrt(r2162164);
        double r2162166 = r2162162 / r2162165;
        double r2162167 = c0;
        double r2162168 = r2162166 * r2162167;
        double r2162169 = r2162161 * r2162168;
        double r2162170 = 4.077916039006289e-304;
        bool r2162171 = r2162152 <= r2162170;
        double r2162172 = r2162155 / r2162151;
        double r2162173 = r2162157 / r2162150;
        double r2162174 = r2162172 * r2162173;
        double r2162175 = sqrt(r2162174);
        double r2162176 = r2162167 * r2162175;
        double r2162177 = sqrt(r2162155);
        double r2162178 = sqrt(r2162152);
        double r2162179 = r2162177 / r2162178;
        double r2162180 = r2162167 * r2162179;
        double r2162181 = r2162171 ? r2162176 : r2162180;
        double r2162182 = r2162154 ? r2162169 : r2162181;
        return r2162182;
}

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. Split input into 3 regimes

  2. if (* V l) < -1.0028777413688825e-276

    1. Initial program 14.5

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

    3. Applied add-cube-cbrt14.9

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

    4. Applied sqrt-prod14.9

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

    5. Applied associate-*r*14.9

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

    7. Applied cbrt-div14.9

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

    8. Applied cbrt-div14.9

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

    9. Applied frac-times14.9

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

    10. Applied sqrt-div14.9

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

    11. Simplified14.9

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

    13. Applied div-inv14.9

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

    14. Applied cbrt-prod7.4

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

    if -1.0028777413688825e-276 < (* V l) < 4.077916039006289e-304

    1. Initial program 53.8

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

    3. Applied *-un-lft-identity53.8

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

    4. Applied times-frac32.4

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

    if 4.077916039006289e-304 < (* V l)

    1. Initial program 14.8

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

    3. Applied sqrt-div6.7

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.0028777413688825 \cdot 10^{-276}:\\ \;\;\;\;\sqrt{\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{V \cdot \ell}}} \cdot \left(\frac{\left|\sqrt[3]{A}\right|}{\sqrt{\sqrt[3]{V \cdot \ell} \cdot \sqrt[3]{V \cdot \ell}}} \cdot c0\right)\\ \mathbf{elif}\;V \cdot \ell \le 4.077916039006289 \cdot 10^{-304}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{\ell} \cdot \frac{1}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))