Average Error: 14.0 → 9.1
Time: 59.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -2.459544082082623877126440069328057953432 \cdot 10^{-105} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 2.475593378518339803291364317838566404249 \cdot 10^{-226}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -\infty:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{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}}}\\

\mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -2.459544082082623877126440069328057953432 \cdot 10^{-105} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 2.475593378518339803291364317838566404249 \cdot 10^{-226}\right):\\
\;\;\;\;w0 \cdot \sqrt{1 - {\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)}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r184466 = w0;
        double r184467 = 1.0;
        double r184468 = M;
        double r184469 = D;
        double r184470 = r184468 * r184469;
        double r184471 = 2.0;
        double r184472 = d;
        double r184473 = r184471 * r184472;
        double r184474 = r184470 / r184473;
        double r184475 = pow(r184474, r184471);
        double r184476 = h;
        double r184477 = l;
        double r184478 = r184476 / r184477;
        double r184479 = r184475 * r184478;
        double r184480 = r184467 - r184479;
        double r184481 = sqrt(r184480);
        double r184482 = r184466 * r184481;
        return r184482;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r184483 = M;
        double r184484 = D;
        double r184485 = r184483 * r184484;
        double r184486 = 2.0;
        double r184487 = d;
        double r184488 = r184486 * r184487;
        double r184489 = r184485 / r184488;
        double r184490 = -inf.0;
        bool r184491 = r184489 <= r184490;
        double r184492 = w0;
        double r184493 = 1.0;
        double r184494 = r184488 / r184484;
        double r184495 = r184483 / r184494;
        double r184496 = pow(r184495, r184486);
        double r184497 = h;
        double r184498 = cbrt(r184497);
        double r184499 = r184498 * r184498;
        double r184500 = l;
        double r184501 = cbrt(r184500);
        double r184502 = r184501 * r184501;
        double r184503 = r184499 / r184502;
        double r184504 = r184496 * r184503;
        double r184505 = r184498 / r184501;
        double r184506 = r184504 * r184505;
        double r184507 = r184493 - r184506;
        double r184508 = sqrt(r184507);
        double r184509 = r184492 * r184508;
        double r184510 = -2.459544082082624e-105;
        bool r184511 = r184489 <= r184510;
        double r184512 = 2.4755933785183398e-226;
        bool r184513 = r184489 <= r184512;
        double r184514 = !r184513;
        bool r184515 = r184511 || r184514;
        double r184516 = 2.0;
        double r184517 = r184486 / r184516;
        double r184518 = pow(r184489, r184517);
        double r184519 = r184497 / r184500;
        double r184520 = r184518 * r184519;
        double r184521 = r184518 * r184520;
        double r184522 = r184493 - r184521;
        double r184523 = sqrt(r184522);
        double r184524 = r184492 * r184523;
        double r184525 = sqrt(r184493);
        double r184526 = r184492 * r184525;
        double r184527 = r184515 ? r184524 : r184526;
        double r184528 = r184491 ? r184509 : r184527;
        return r184528;
}

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 (/ (* M D) (* 2.0 d)) < -inf.0

    1. Initial program 64.0

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

    3. Applied add-cube-cbrt64.0

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

    4. Applied add-cube-cbrt64.0

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

    5. Applied times-frac64.0

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

    6. Applied associate-*r*64.0

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

    8. Applied associate-/l*56.3

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

    if -inf.0 < (/ (* M D) (* 2.0 d)) < -2.459544082082624e-105 or 2.4755933785183398e-226 < (/ (* M D) (* 2.0 d))

    1. Initial program 17.3

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

    3. Applied sqr-pow17.3

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

    4. Applied associate-*l*14.3

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

    if -2.459544082082624e-105 < (/ (* M D) (* 2.0 d)) < 2.4755933785183398e-226

    1. Initial program 7.5

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

    3. Applied add-cube-cbrt7.5

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

    4. Applied add-cube-cbrt7.5

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

    5. Applied times-frac7.5

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

    6. Applied associate-*r*2.2

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

    7. Taylor expanded around 0 1.0

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -2.459544082082623877126440069328057953432 \cdot 10^{-105} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 2.475593378518339803291364317838566404249 \cdot 10^{-226}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))