Average Error: 14.5 → 10.0
Time: 32.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.510341556579411067921188404923562874146 \cdot 10^{201}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -8.676239891926242557896224383038014757216 \cdot 10^{-146}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \le -1.510341556579411067921188404923562874146 \cdot 10^{201}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)\right) \cdot \frac{1}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -8.676239891926242557896224383038014757216 \cdot 10^{-146}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r163389 = w0;
        double r163390 = 1.0;
        double r163391 = M;
        double r163392 = D;
        double r163393 = r163391 * r163392;
        double r163394 = 2.0;
        double r163395 = d;
        double r163396 = r163394 * r163395;
        double r163397 = r163393 / r163396;
        double r163398 = pow(r163397, r163394);
        double r163399 = h;
        double r163400 = l;
        double r163401 = r163399 / r163400;
        double r163402 = r163398 * r163401;
        double r163403 = r163390 - r163402;
        double r163404 = sqrt(r163403);
        double r163405 = r163389 * r163404;
        return r163405;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r163406 = h;
        double r163407 = l;
        double r163408 = r163406 / r163407;
        double r163409 = -1.510341556579411e+201;
        bool r163410 = r163408 <= r163409;
        double r163411 = w0;
        double r163412 = 1.0;
        double r163413 = M;
        double r163414 = D;
        double r163415 = r163413 * r163414;
        double r163416 = 2.0;
        double r163417 = d;
        double r163418 = r163416 * r163417;
        double r163419 = r163415 / r163418;
        double r163420 = 2.0;
        double r163421 = r163416 / r163420;
        double r163422 = pow(r163419, r163421);
        double r163423 = r163413 / r163416;
        double r163424 = pow(r163423, r163421);
        double r163425 = r163414 / r163417;
        double r163426 = pow(r163425, r163421);
        double r163427 = r163426 * r163406;
        double r163428 = r163424 * r163427;
        double r163429 = r163422 * r163428;
        double r163430 = 1.0;
        double r163431 = r163430 / r163407;
        double r163432 = r163429 * r163431;
        double r163433 = r163412 - r163432;
        double r163434 = sqrt(r163433);
        double r163435 = r163411 * r163434;
        double r163436 = -8.676239891926243e-146;
        bool r163437 = r163408 <= r163436;
        double r163438 = r163418 / r163414;
        double r163439 = r163413 / r163438;
        double r163440 = pow(r163439, r163416);
        double r163441 = r163440 * r163408;
        double r163442 = r163412 - r163441;
        double r163443 = sqrt(r163442);
        double r163444 = r163411 * r163443;
        double r163445 = sqrt(r163412);
        double r163446 = r163411 * r163445;
        double r163447 = r163437 ? r163444 : r163446;
        double r163448 = r163410 ? r163435 : r163447;
        return r163448;
}

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 (/ h l) < -1.510341556579411e+201

    1. Initial program 40.9

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

    3. Applied div-inv40.9

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

    4. Applied associate-*r*21.6

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

    6. Applied sqr-pow21.6

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

    7. Applied associate-*l*20.3

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

    9. Applied times-frac21.2

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

    10. Applied unpow-prod-down21.2

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

    11. Applied associate-*l*23.7

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

    if -1.510341556579411e+201 < (/ h l) < -8.676239891926243e-146

    1. Initial program 13.2

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

    3. Applied associate-/l*12.9

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

    if -8.676239891926243e-146 < (/ h l)

    1. Initial program 9.9

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

    2. Taylor expanded around 0 5.9

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.510341556579411067921188404923562874146 \cdot 10^{201}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -8.676239891926242557896224383038014757216 \cdot 10^{-146}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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))))))