Average Error: 18.8 → 13.2
Time: 15.7s
Precision: 64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;V \cdot \ell \le -2.4406467174030862 \cdot 10^{+269}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le -3.326542295985692 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \le -0.0:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le 3.222926338394294 \cdot 10^{+285}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array}\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;V \cdot \ell \le -2.4406467174030862 \cdot 10^{+269}:\\
\;\;\;\;\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\\

\mathbf{elif}\;V \cdot \ell \le -3.326542295985692 \cdot 10^{-282}:\\
\;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\

\mathbf{elif}\;V \cdot \ell \le -0.0:\\
\;\;\;\;\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\\

\mathbf{elif}\;V \cdot \ell \le 3.222926338394294 \cdot 10^{+285}:\\
\;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\

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

\end{array}
double f(double c0, double A, double V, double l) {
        double r1645498 = c0;
        double r1645499 = A;
        double r1645500 = V;
        double r1645501 = l;
        double r1645502 = r1645500 * r1645501;
        double r1645503 = r1645499 / r1645502;
        double r1645504 = sqrt(r1645503);
        double r1645505 = r1645498 * r1645504;
        return r1645505;
}
↓
double f(double c0, double A, double V, double l) {
        double r1645506 = V;
        double r1645507 = l;
        double r1645508 = r1645506 * r1645507;
        double r1645509 = -2.4406467174030862e+269;
        bool r1645510 = r1645508 <= r1645509;
        double r1645511 = c0;
        double r1645512 = A;
        double r1645513 = cbrt(r1645512);
        double r1645514 = r1645513 * r1645513;
        double r1645515 = r1645514 / r1645506;
        double r1645516 = sqrt(r1645515);
        double r1645517 = r1645511 * r1645516;
        double r1645518 = r1645513 / r1645507;
        double r1645519 = sqrt(r1645518);
        double r1645520 = r1645517 * r1645519;
        double r1645521 = -3.326542295985692e-282;
        bool r1645522 = r1645508 <= r1645521;
        double r1645523 = r1645512 / r1645508;
        double r1645524 = sqrt(r1645523);
        double r1645525 = r1645511 * r1645524;
        double r1645526 = -0.0;
        bool r1645527 = r1645508 <= r1645526;
        double r1645528 = 3.222926338394294e+285;
        bool r1645529 = r1645508 <= r1645528;
        double r1645530 = sqrt(r1645512);
        double r1645531 = sqrt(r1645508);
        double r1645532 = r1645530 / r1645531;
        double r1645533 = r1645532 * r1645511;
        double r1645534 = r1645512 / r1645506;
        double r1645535 = sqrt(r1645534);
        double r1645536 = r1645511 * r1645535;
        double r1645537 = sqrt(r1645507);
        double r1645538 = r1645536 / r1645537;
        double r1645539 = r1645529 ? r1645533 : r1645538;
        double r1645540 = r1645527 ? r1645520 : r1645539;
        double r1645541 = r1645522 ? r1645525 : r1645540;
        double r1645542 = r1645510 ? r1645520 : r1645541;
        return r1645542;
}

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 4 regimes

  2. if (* V l) < -2.4406467174030862e+269 or -3.326542295985692e-282 < (* V l) < -0.0

    1. Initial program 48.1

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

    3. Applied add-cube-cbrt48.2

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

    4. Applied times-frac30.6

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

    5. Applied sqrt-prod36.9

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

    6. Applied associate-*r*37.0

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

    if -2.4406467174030862e+269 < (* V l) < -3.326542295985692e-282

    1. Initial program 8.9

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

    3. Applied *-un-lft-identity8.9

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

    4. Applied times-frac15.7

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

    5. Taylor expanded around 0 8.9

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

    if -0.0 < (* V l) < 3.222926338394294e+285

    1. Initial program 9.5

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

    3. Applied sqrt-div0.7

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

    if 3.222926338394294e+285 < (* V l)

    1. Initial program 37.8

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

    3. Applied *-un-lft-identity37.8

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

    4. Applied times-frac23.0

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{V} \cdot \frac{A}{\ell}}}\]
    5. Using strategy rm

    6. Applied associate-*r/23.0

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

    7. Applied sqrt-div35.6

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

    8. Applied associate-*r/35.9

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

    9. Simplified35.9

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

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -2.4406467174030862 \cdot 10^{+269}:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le -3.326542295985692 \cdot 10^{-282}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;V \cdot \ell \le -0.0:\\ \;\;\;\;\left(c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{V}}\right) \cdot \sqrt{\frac{\sqrt[3]{A}}{\ell}}\\ \mathbf{elif}\;V \cdot \ell \le 3.222926338394294 \cdot 10^{+285}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0 \cdot \sqrt{\frac{A}{V}}}{\sqrt{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))