Average Error: 14.1 → 9.1
Time: 36.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le 3.582257138037586599318904097312066593376 \cdot 10^{122}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left({\left(\frac{1}{\frac{\frac{2 \cdot d}{D}}{M}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;\ell \le 6.489671467557263418113195619699661828203 \cdot 10^{210}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left({\left(\frac{1}{\frac{\frac{2 \cdot d}{D}}{M}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \end{array}\]
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 3.582257138037586599318904097312066593376 \cdot 10^{122}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left({\left(\frac{1}{\frac{\frac{2 \cdot d}{D}}{M}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\

\mathbf{elif}\;\ell \le 6.489671467557263418113195619699661828203 \cdot 10^{210}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r6057480 = w0;
        double r6057481 = 1.0;
        double r6057482 = M;
        double r6057483 = D;
        double r6057484 = r6057482 * r6057483;
        double r6057485 = 2.0;
        double r6057486 = d;
        double r6057487 = r6057485 * r6057486;
        double r6057488 = r6057484 / r6057487;
        double r6057489 = pow(r6057488, r6057485);
        double r6057490 = h;
        double r6057491 = l;
        double r6057492 = r6057490 / r6057491;
        double r6057493 = r6057489 * r6057492;
        double r6057494 = r6057481 - r6057493;
        double r6057495 = sqrt(r6057494);
        double r6057496 = r6057480 * r6057495;
        return r6057496;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r6057497 = l;
        double r6057498 = 3.5822571380375866e+122;
        bool r6057499 = r6057497 <= r6057498;
        double r6057500 = w0;
        double r6057501 = 1.0;
        double r6057502 = h;
        double r6057503 = cbrt(r6057502);
        double r6057504 = cbrt(r6057497);
        double r6057505 = r6057503 / r6057504;
        double r6057506 = 1.0;
        double r6057507 = 2.0;
        double r6057508 = d;
        double r6057509 = r6057507 * r6057508;
        double r6057510 = D;
        double r6057511 = r6057509 / r6057510;
        double r6057512 = M;
        double r6057513 = r6057511 / r6057512;
        double r6057514 = r6057506 / r6057513;
        double r6057515 = 2.0;
        double r6057516 = r6057507 / r6057515;
        double r6057517 = pow(r6057514, r6057516);
        double r6057518 = r6057512 / r6057511;
        double r6057519 = pow(r6057518, r6057516);
        double r6057520 = r6057519 * r6057505;
        double r6057521 = cbrt(r6057520);
        double r6057522 = r6057521 * r6057521;
        double r6057523 = r6057521 * r6057522;
        double r6057524 = r6057517 * r6057523;
        double r6057525 = r6057505 * r6057524;
        double r6057526 = r6057525 * r6057505;
        double r6057527 = r6057501 - r6057526;
        double r6057528 = sqrt(r6057527);
        double r6057529 = r6057500 * r6057528;
        double r6057530 = 6.489671467557263e+210;
        bool r6057531 = r6057497 <= r6057530;
        double r6057532 = r6057512 * r6057510;
        double r6057533 = r6057532 / r6057509;
        double r6057534 = pow(r6057533, r6057516);
        double r6057535 = r6057502 / r6057497;
        double r6057536 = r6057535 * r6057534;
        double r6057537 = r6057534 * r6057536;
        double r6057538 = r6057501 - r6057537;
        double r6057539 = sqrt(r6057538);
        double r6057540 = r6057500 * r6057539;
        double r6057541 = r6057531 ? r6057540 : r6057529;
        double r6057542 = r6057499 ? r6057529 : r6057541;
        return r6057542;
}

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 2 regimes

  2. if l < 3.5822571380375866e+122 or 6.489671467557263e+210 < l

    1. Initial program 14.5

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

    3. Applied add-cube-cbrt14.6

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

    4. Applied add-cube-cbrt14.6

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

    5. Applied times-frac14.6

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

    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. Simplified10.2

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

    9. Applied sqr-pow10.2

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

    10. Applied associate-*r*9.3

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

    12. Applied add-cube-cbrt9.3

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

    14. Applied clear-num9.3

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

    if 3.5822571380375866e+122 < l < 6.489671467557263e+210

    1. Initial program 9.2

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

    3. Applied sqr-pow9.2

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

    4. Applied associate-*l*6.7

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 3.582257138037586599318904097312066593376 \cdot 10^{122}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left({\left(\frac{1}{\frac{\frac{2 \cdot d}{D}}{M}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;\ell \le 6.489671467557263418113195619699661828203 \cdot 10^{210}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left({\left(\frac{1}{\frac{\frac{2 \cdot d}{D}}{M}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \left(\sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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))))))