Average Error: 18.6 → 12.5
Time: 20.0s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell = -\infty:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A}}{\ell}} \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right)\\ \mathbf{elif}\;V \cdot \ell \le -4.6924056322212644 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \left(c0 \cdot \sqrt{\sqrt{\frac{\sqrt[3]{A}}{\sqrt[3]{V \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}}}\right)\\ \mathbf{elif}\;V \cdot \ell \le 5.41520176388353 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]
double f(double c0, double A, double V, double l) {
        double r9192182 = c0;
        double r9192183 = A;
        double r9192184 = V;
        double r9192185 = l;
        double r9192186 = r9192184 * r9192185;
        double r9192187 = r9192183 / r9192186;
        double r9192188 = sqrt(r9192187);
        double r9192189 = r9192182 * r9192188;
        return r9192189;
}
↓
double f(double c0, double A, double V, double l) {
        double r9192190 = V;
        double r9192191 = l;
        double r9192192 = r9192190 * r9192191;
        double r9192193 = -inf.0;
        bool r9192194 = r9192192 <= r9192193;
        double r9192195 = c0;
        double r9192196 = A;
        double r9192197 = cbrt(r9192196);
        double r9192198 = r9192197 / r9192191;
        double r9192199 = sqrt(r9192198);
        double r9192200 = r9192197 * r9192197;
        double r9192201 = r9192200 / r9192190;
        double r9192202 = sqrt(r9192201);
        double r9192203 = r9192199 * r9192202;
        double r9192204 = r9192195 * r9192203;
        double r9192205 = -4.6924056322212644e-245;
        bool r9192206 = r9192192 <= r9192205;
        double r9192207 = r9192196 / r9192192;
        double r9192208 = sqrt(r9192207);
        double r9192209 = sqrt(r9192208);
        double r9192210 = cbrt(r9192192);
        double r9192211 = r9192197 / r9192210;
        double r9192212 = sqrt(r9192211);
        double r9192213 = cbrt(r9192207);
        double r9192214 = r9192213 * r9192213;
        double r9192215 = sqrt(r9192214);
        double r9192216 = r9192212 * r9192215;
        double r9192217 = sqrt(r9192216);
        double r9192218 = r9192195 * r9192217;
        double r9192219 = r9192209 * r9192218;
        double r9192220 = 5.41520176388353e-310;
        bool r9192221 = r9192192 <= r9192220;
        double r9192222 = r9192196 / r9192190;
        double r9192223 = r9192222 / r9192191;
        double r9192224 = sqrt(r9192223);
        double r9192225 = r9192224 * r9192195;
        double r9192226 = sqrt(r9192196);
        double r9192227 = sqrt(r9192192);
        double r9192228 = r9192226 / r9192227;
        double r9192229 = r9192195 * r9192228;
        double r9192230 = r9192221 ? r9192225 : r9192229;
        double r9192231 = r9192206 ? r9192219 : r9192230;
        double r9192232 = r9192194 ? r9192204 : r9192231;
        return r9192232;
}


c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \cdot \ell = -\infty:\\
\;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A}}{\ell}} \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right)\\

\mathbf{elif}\;V \cdot \ell \le -4.6924056322212644 \cdot 10^{-245}:\\
\;\;\;\;\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \left(c0 \cdot \sqrt{\sqrt{\frac{\sqrt[3]{A}}{\sqrt[3]{V \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}}}\right)\\

\mathbf{elif}\;V \cdot \ell \le 5.41520176388353 \cdot 10^{-310}:\\
\;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\

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

\end{array}

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Derivation

  1. Split input into 4 regimes

  2. if (* V l) < -inf.0

    1. Initial program 39.2

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

    3. Applied add-cube-cbrt39.2

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

    4. Applied times-frac22.1

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

    5. Applied sqrt-prod34.0

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

    if -inf.0 < (* V l) < -4.6924056322212644e-245

    1. Initial program 8.6

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

    3. Applied add-sqr-sqrt8.6

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

    4. Applied sqrt-prod8.9

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

    5. Applied associate-*r*8.8

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

    7. Applied add-cube-cbrt8.9

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

    8. Applied sqrt-prod8.9

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

    10. Applied cbrt-div8.9

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

    if -4.6924056322212644e-245 < (* V l) < 5.41520176388353e-310

    1. Initial program 51.2

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

    3. Applied associate-/r*33.0

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

    if 5.41520176388353e-310 < (* V l)

    1. Initial program 14.7

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

    3. Applied sqrt-div6.5

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell = -\infty:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{A}}{\ell}} \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right)\\ \mathbf{elif}\;V \cdot \ell \le -4.6924056322212644 \cdot 10^{-245}:\\ \;\;\;\;\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \left(c0 \cdot \sqrt{\sqrt{\frac{\sqrt[3]{A}}{\sqrt[3]{V \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}}}\right)\\ \mathbf{elif}\;V \cdot \ell \le 5.41520176388353 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))