Average Error: 14.1 → 8.5
Time: 33.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le 3.379958001175248205895089975016318042271 \cdot 10^{162}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left({\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \le 2.502425320725228639287002195105093607169 \cdot 10^{212}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\ \mathbf{elif}\;\ell \le 2.380536168358356514766503762204427856526 \cdot 10^{290}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left({\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\ell \le 3.379958001175248205895089975016318042271 \cdot 10^{162}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left({\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)}\\

\mathbf{elif}\;\ell \le 2.502425320725228639287002195105093607169 \cdot 10^{212}:\\
\;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\

\mathbf{elif}\;\ell \le 2.380536168358356514766503762204427856526 \cdot 10^{290}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left({\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r6809469 = w0;
        double r6809470 = 1.0;
        double r6809471 = M;
        double r6809472 = D;
        double r6809473 = r6809471 * r6809472;
        double r6809474 = 2.0;
        double r6809475 = d;
        double r6809476 = r6809474 * r6809475;
        double r6809477 = r6809473 / r6809476;
        double r6809478 = pow(r6809477, r6809474);
        double r6809479 = h;
        double r6809480 = l;
        double r6809481 = r6809479 / r6809480;
        double r6809482 = r6809478 * r6809481;
        double r6809483 = r6809470 - r6809482;
        double r6809484 = sqrt(r6809483);
        double r6809485 = r6809469 * r6809484;
        return r6809485;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r6809486 = l;
        double r6809487 = 3.3799580011752482e+162;
        bool r6809488 = r6809486 <= r6809487;
        double r6809489 = w0;
        double r6809490 = 1.0;
        double r6809491 = h;
        double r6809492 = cbrt(r6809491);
        double r6809493 = cbrt(r6809486);
        double r6809494 = r6809492 / r6809493;
        double r6809495 = M;
        double r6809496 = d;
        double r6809497 = 2.0;
        double r6809498 = r6809496 * r6809497;
        double r6809499 = D;
        double r6809500 = r6809498 / r6809499;
        double r6809501 = r6809495 / r6809500;
        double r6809502 = 2.0;
        double r6809503 = r6809497 / r6809502;
        double r6809504 = pow(r6809501, r6809503);
        double r6809505 = r6809504 * r6809494;
        double r6809506 = cbrt(r6809505);
        double r6809507 = r6809506 * r6809506;
        double r6809508 = r6809506 * r6809507;
        double r6809509 = r6809505 * r6809508;
        double r6809510 = r6809494 * r6809509;
        double r6809511 = r6809490 - r6809510;
        double r6809512 = sqrt(r6809511);
        double r6809513 = r6809489 * r6809512;
        double r6809514 = 2.5024253207252286e+212;
        bool r6809515 = r6809486 <= r6809514;
        double r6809516 = r6809495 * r6809499;
        double r6809517 = r6809516 / r6809498;
        double r6809518 = pow(r6809517, r6809503);
        double r6809519 = r6809491 / r6809486;
        double r6809520 = r6809519 * r6809518;
        double r6809521 = r6809518 * r6809520;
        double r6809522 = r6809490 - r6809521;
        double r6809523 = sqrt(r6809522);
        double r6809524 = r6809523 * r6809489;
        double r6809525 = 2.3805361683583565e+290;
        bool r6809526 = r6809486 <= r6809525;
        double r6809527 = sqrt(r6809490);
        double r6809528 = r6809489 * r6809527;
        double r6809529 = r6809526 ? r6809513 : r6809528;
        double r6809530 = r6809515 ? r6809524 : r6809529;
        double r6809531 = r6809488 ? r6809513 : r6809530;
        return r6809531;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if l < 3.3799580011752482e+162 or 2.5024253207252286e+212 < l < 2.3805361683583565e+290

    1. Initial program 14.4

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt14.5

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\]

    4. Applied add-cube-cbrt14.5

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

    5. Applied times-frac14.5

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}}\]

    6. Applied associate-*r*11.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}\]

    7. Simplified10.9

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\]
    8. Using strategy rm

    9. Applied sqr-pow10.9

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \color{blue}{\left({\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\]

    10. Applied unswap-sqr8.6

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\]
    11. Using strategy rm

    12. Applied add-cube-cbrt8.6

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot {\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)}}\right)}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\]

    if 3.3799580011752482e+162 < l < 2.5024253207252286e+212

    1. Initial program 9.3

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
    2. Using strategy rm

    3. Applied sqr-pow9.3

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \frac{h}{\ell}}\]

    4. Applied associate-*l*7.1

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}}\]

    if 2.3805361683583565e+290 < l

    1. Initial program 9.3

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

    2. Taylor expanded around 0 9.2

      \[\leadsto \color{blue}{\sqrt{1} \cdot w0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 3.379958001175248205895089975016318042271 \cdot 10^{162}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left({\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)}\\ \mathbf{elif}\;\ell \le 2.502425320725228639287002195105093607169 \cdot 10^{212}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\ \mathbf{elif}\;\ell \le 2.380536168358356514766503762204427856526 \cdot 10^{290}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left({\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{d \cdot 2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))