Average Error: 13.6 → 9.2
Time: 27.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.459510325973658786992790942635709262941 \cdot 10^{-131} \lor \neg \left(\frac{h}{\ell} \le -0.0\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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 -2.459510325973658786992790942635709262941 \cdot 10^{-131} \lor \neg \left(\frac{h}{\ell} \le -0.0\right):\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;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)}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r138363 = w0;
        double r138364 = 1.0;
        double r138365 = M;
        double r138366 = D;
        double r138367 = r138365 * r138366;
        double r138368 = 2.0;
        double r138369 = d;
        double r138370 = r138368 * r138369;
        double r138371 = r138367 / r138370;
        double r138372 = pow(r138371, r138368);
        double r138373 = h;
        double r138374 = l;
        double r138375 = r138373 / r138374;
        double r138376 = r138372 * r138375;
        double r138377 = r138364 - r138376;
        double r138378 = sqrt(r138377);
        double r138379 = r138363 * r138378;
        return r138379;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r138380 = h;
        double r138381 = l;
        double r138382 = r138380 / r138381;
        double r138383 = -2.459510325973659e-131;
        bool r138384 = r138382 <= r138383;
        double r138385 = -0.0;
        bool r138386 = r138382 <= r138385;
        double r138387 = !r138386;
        bool r138388 = r138384 || r138387;
        double r138389 = w0;
        double r138390 = 1.0;
        double r138391 = M;
        double r138392 = 2.0;
        double r138393 = d;
        double r138394 = r138392 * r138393;
        double r138395 = D;
        double r138396 = r138394 / r138395;
        double r138397 = r138391 / r138396;
        double r138398 = pow(r138397, r138392);
        double r138399 = r138398 * r138380;
        double r138400 = 1.0;
        double r138401 = r138400 / r138381;
        double r138402 = r138399 * r138401;
        double r138403 = r138390 - r138402;
        double r138404 = sqrt(r138403);
        double r138405 = r138389 * r138404;
        double r138406 = r138391 * r138395;
        double r138407 = r138406 / r138394;
        double r138408 = 2.0;
        double r138409 = r138392 / r138408;
        double r138410 = pow(r138407, r138409);
        double r138411 = r138410 * r138382;
        double r138412 = r138410 * r138411;
        double r138413 = r138390 - r138412;
        double r138414 = sqrt(r138413);
        double r138415 = r138389 * r138414;
        double r138416 = r138388 ? r138405 : r138415;
        return r138416;
}

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 (/ h l) < -2.459510325973659e-131 or -0.0 < (/ h l)

    1. Initial program 13.0

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

    3. Applied div-inv13.0

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

    4. Applied associate-*r*9.2

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

    6. Applied associate-/l*9.1

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

    if -2.459510325973659e-131 < (/ h l) < -0.0

    1. Initial program 15.7

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

    3. Applied sqr-pow15.7

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

    4. Applied associate-*l*11.0

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.459510325973658786992790942635709262941 \cdot 10^{-131} \lor \neg \left(\frac{h}{\ell} \le -0.0\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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))))))