Average Error: 18.9 → 13.1
Time: 21.2s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -5.5339969382432195 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}}} \cdot \left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot c0\right)\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\ell}} \cdot c0\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \cdot \ell \le -5.5339969382432195 \cdot 10^{-300}:\\
\;\;\;\;\sqrt{\sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}}} \cdot \left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot c0\right)\\

\mathbf{elif}\;V \cdot \ell \le 0.0:\\
\;\;\;\;\left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\ell}} \cdot c0\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{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 r4164377 = c0;
        double r4164378 = A;
        double r4164379 = V;
        double r4164380 = l;
        double r4164381 = r4164379 * r4164380;
        double r4164382 = r4164378 / r4164381;
        double r4164383 = sqrt(r4164382);
        double r4164384 = r4164377 * r4164383;
        return r4164384;
}
↓
double f(double c0, double A, double V, double l) {
        double r4164385 = V;
        double r4164386 = l;
        double r4164387 = r4164385 * r4164386;
        double r4164388 = -5.5339969382432195e-300;
        bool r4164389 = r4164387 <= r4164388;
        double r4164390 = A;
        double r4164391 = r4164390 / r4164387;
        double r4164392 = cbrt(r4164391);
        double r4164393 = sqrt(r4164392);
        double r4164394 = r4164392 * r4164392;
        double r4164395 = sqrt(r4164394);
        double r4164396 = r4164393 * r4164395;
        double r4164397 = sqrt(r4164396);
        double r4164398 = sqrt(r4164391);
        double r4164399 = sqrt(r4164398);
        double r4164400 = c0;
        double r4164401 = r4164399 * r4164400;
        double r4164402 = r4164397 * r4164401;
        double r4164403 = 0.0;
        bool r4164404 = r4164387 <= r4164403;
        double r4164405 = cbrt(r4164390);
        double r4164406 = r4164405 * r4164405;
        double r4164407 = r4164406 / r4164386;
        double r4164408 = sqrt(r4164407);
        double r4164409 = r4164408 * r4164400;
        double r4164410 = r4164405 / r4164385;
        double r4164411 = sqrt(r4164410);
        double r4164412 = r4164409 * r4164411;
        double r4164413 = sqrt(r4164390);
        double r4164414 = sqrt(r4164387);
        double r4164415 = r4164413 / r4164414;
        double r4164416 = r4164400 * r4164415;
        double r4164417 = r4164404 ? r4164412 : r4164416;
        double r4164418 = r4164389 ? r4164402 : r4164417;
        return r4164418;
}

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) < -5.5339969382432195e-300

    1. Initial program 13.8

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

    3. Applied add-sqr-sqrt13.8

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

    4. Applied sqrt-prod14.0

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

    5. Applied associate-*r*14.0

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

    7. Applied add-cube-cbrt14.1

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

    8. Applied sqrt-prod14.1

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

    if -5.5339969382432195e-300 < (* V l) < 0.0

    1. Initial program 58.7

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

    2. Taylor expanded around -inf 58.7

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

    4. Applied add-cube-cbrt58.7

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

    5. Applied times-frac34.2

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

    6. Applied sqrt-prod36.9

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

    7. Applied associate-*r*37.1

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

    if 0.0 < (* V l)

    1. Initial program 15.9

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

    3. Applied sqrt-div7.4

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -5.5339969382432195 \cdot 10^{-300}:\\ \;\;\;\;\sqrt{\sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt[3]{\frac{A}{V \cdot \ell}} \cdot \sqrt[3]{\frac{A}{V \cdot \ell}}}} \cdot \left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot c0\right)\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\ell}} \cdot c0\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{V}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))