Average Error: 14.1 → 8.8
Time: 25.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.4589932569656996399740112599872681599 \cdot 10^{54}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.137897252452376107719566610395472751847 \cdot 10^{-223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \le -1.4589932569656996399740112599872681599 \cdot 10^{54}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -1.137897252452376107719566610395472751847 \cdot 10^{-223}:\\
\;\;\;\;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 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r130272 = w0;
        double r130273 = 1.0;
        double r130274 = M;
        double r130275 = D;
        double r130276 = r130274 * r130275;
        double r130277 = 2.0;
        double r130278 = d;
        double r130279 = r130277 * r130278;
        double r130280 = r130276 / r130279;
        double r130281 = pow(r130280, r130277);
        double r130282 = h;
        double r130283 = l;
        double r130284 = r130282 / r130283;
        double r130285 = r130281 * r130284;
        double r130286 = r130273 - r130285;
        double r130287 = sqrt(r130286);
        double r130288 = r130272 * r130287;
        return r130288;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r130289 = h;
        double r130290 = l;
        double r130291 = r130289 / r130290;
        double r130292 = -1.4589932569656996e+54;
        bool r130293 = r130291 <= r130292;
        double r130294 = w0;
        double r130295 = 1.0;
        double r130296 = M;
        double r130297 = 2.0;
        double r130298 = d;
        double r130299 = r130297 * r130298;
        double r130300 = D;
        double r130301 = r130299 / r130300;
        double r130302 = r130296 / r130301;
        double r130303 = 2.0;
        double r130304 = r130297 / r130303;
        double r130305 = pow(r130302, r130304);
        double r130306 = r130296 * r130300;
        double r130307 = 1.0;
        double r130308 = r130307 / r130299;
        double r130309 = r130306 * r130308;
        double r130310 = pow(r130309, r130304);
        double r130311 = r130310 * r130289;
        double r130312 = r130305 * r130311;
        double r130313 = r130312 / r130290;
        double r130314 = r130295 - r130313;
        double r130315 = sqrt(r130314);
        double r130316 = r130294 * r130315;
        double r130317 = -1.1378972524523761e-223;
        bool r130318 = r130291 <= r130317;
        double r130319 = r130306 / r130299;
        double r130320 = pow(r130319, r130304);
        double r130321 = r130320 * r130291;
        double r130322 = r130320 * r130321;
        double r130323 = r130295 - r130322;
        double r130324 = sqrt(r130323);
        double r130325 = r130294 * r130324;
        double r130326 = r130320 * r130311;
        double r130327 = r130326 / r130290;
        double r130328 = r130295 - r130327;
        double r130329 = sqrt(r130328);
        double r130330 = r130294 * r130329;
        double r130331 = r130318 ? r130325 : r130330;
        double r130332 = r130293 ? r130316 : r130331;
        return r130332;
}

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) < -1.4589932569656996e+54

    1. Initial program 28.1

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

    3. Applied associate-*r/18.8

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

    5. Applied sqr-pow18.8

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

    6. Applied associate-*l*17.7

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

    8. Applied div-inv17.7

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

    10. Applied associate-/l*18.3

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

    if -1.4589932569656996e+54 < (/ h l) < -1.1378972524523761e-223

    1. Initial program 13.5

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

    3. Applied sqr-pow13.5

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

    4. Applied associate-*l*11.2

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

    if -1.1378972524523761e-223 < (/ h l)

    1. Initial program 8.5

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

    3. Applied associate-*r/5.7

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

    5. Applied sqr-pow5.7

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

    6. Applied associate-*l*3.7

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

    8. Applied div-inv3.7

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.4589932569656996399740112599872681599 \cdot 10^{54}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.137897252452376107719566610395472751847 \cdot 10^{-223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))))