Average Error: 14.6 → 10.7
Time: 11.2s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot d \le -4.9978883920778511 \cdot 10^{159}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{elif}\;2 \cdot d \le -8.4818635748686101 \cdot 10^{-74}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(0.5 \cdot \left(\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{h} \cdot \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;2 \cdot d \le -4.9978883920778511 \cdot 10^{159}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\

\mathbf{elif}\;2 \cdot d \le -8.4818635748686101 \cdot 10^{-74}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(0.5 \cdot \left(\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)\right)}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r267448 = w0;
        double r267449 = 1.0;
        double r267450 = M;
        double r267451 = D;
        double r267452 = r267450 * r267451;
        double r267453 = 2.0;
        double r267454 = d;
        double r267455 = r267453 * r267454;
        double r267456 = r267452 / r267455;
        double r267457 = pow(r267456, r267453);
        double r267458 = h;
        double r267459 = l;
        double r267460 = r267458 / r267459;
        double r267461 = r267457 * r267460;
        double r267462 = r267449 - r267461;
        double r267463 = sqrt(r267462);
        double r267464 = r267448 * r267463;
        return r267464;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r267465 = 2.0;
        double r267466 = d;
        double r267467 = r267465 * r267466;
        double r267468 = -4.997888392077851e+159;
        bool r267469 = r267467 <= r267468;
        double r267470 = w0;
        double r267471 = 1.0;
        double r267472 = M;
        double r267473 = r267472 / r267465;
        double r267474 = D;
        double r267475 = r267474 / r267466;
        double r267476 = r267473 * r267475;
        double r267477 = pow(r267476, r267465);
        double r267478 = h;
        double r267479 = l;
        double r267480 = r267478 / r267479;
        double r267481 = r267477 * r267480;
        double r267482 = r267471 - r267481;
        double r267483 = sqrt(r267482);
        double r267484 = r267470 * r267483;
        double r267485 = -8.48186357486861e-74;
        bool r267486 = r267467 <= r267485;
        double r267487 = r267472 * r267474;
        double r267488 = r267487 / r267467;
        double r267489 = 2.0;
        double r267490 = r267465 / r267489;
        double r267491 = pow(r267488, r267490);
        double r267492 = 0.5;
        double r267493 = r267478 * r267487;
        double r267494 = r267466 * r267479;
        double r267495 = r267493 / r267494;
        double r267496 = 1.0;
        double r267497 = -1.0;
        double r267498 = pow(r267497, r267465);
        double r267499 = r267496 / r267498;
        double r267500 = pow(r267499, r267471);
        double r267501 = r267495 * r267500;
        double r267502 = r267492 * r267501;
        double r267503 = r267491 * r267502;
        double r267504 = r267471 - r267503;
        double r267505 = sqrt(r267504);
        double r267506 = r267470 * r267505;
        double r267507 = cbrt(r267478);
        double r267508 = r267507 * r267491;
        double r267509 = r267507 * r267508;
        double r267510 = r267507 / r267479;
        double r267511 = r267509 * r267510;
        double r267512 = r267491 * r267511;
        double r267513 = r267471 - r267512;
        double r267514 = sqrt(r267513);
        double r267515 = r267470 * r267514;
        double r267516 = r267486 ? r267506 : r267515;
        double r267517 = r267469 ? r267484 : r267516;
        return r267517;
}

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 (* 2.0 d) < -4.997888392077851e+159

    1. Initial program 10.5

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

    3. Applied times-frac8.9

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

    if -4.997888392077851e+159 < (* 2.0 d) < -8.48186357486861e-74

    1. Initial program 12.3

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

    3. Applied sqr-pow12.3

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

    4. Applied associate-*l*10.9

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

    5. Taylor expanded around -inf 8.6

      \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(0.5 \cdot \left(\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)\right)}}\]

    if -8.48186357486861e-74 < (* 2.0 d)

    1. Initial program 16.5

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

    3. Applied sqr-pow16.5

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

    4. Applied associate-*l*14.6

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

    6. Applied *-un-lft-identity14.6

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

    7. Applied add-cube-cbrt14.6

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

    8. Applied times-frac14.6

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

    9. Applied associate-*r*11.8

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

    10. Simplified11.8

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

    12. Applied associate-*l*11.8

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \le -4.9978883920778511 \cdot 10^{159}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{elif}\;2 \cdot d \le -8.4818635748686101 \cdot 10^{-74}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(0.5 \cdot \left(\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{h} \cdot \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))))))