Average Error: 13.9 → 9.4
Time: 11.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.855277338734028038380413031509408290214 \cdot 10^{-289} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.933159505335491079472504373058255984824 \cdot 10^{291}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\sqrt[3]{\frac{h}{\ell}} \cdot \sqrt[3]{\frac{h}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{h}{\ell}}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.855277338734028038380413031509408290214 \cdot 10^{-289} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.933159505335491079472504373058255984824 \cdot 10^{291}\right):\\
\;\;\;\;w0 \cdot \sqrt{1}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r211290 = w0;
        double r211291 = 1.0;
        double r211292 = M;
        double r211293 = D;
        double r211294 = r211292 * r211293;
        double r211295 = 2.0;
        double r211296 = d;
        double r211297 = r211295 * r211296;
        double r211298 = r211294 / r211297;
        double r211299 = pow(r211298, r211295);
        double r211300 = h;
        double r211301 = l;
        double r211302 = r211300 / r211301;
        double r211303 = r211299 * r211302;
        double r211304 = r211291 - r211303;
        double r211305 = sqrt(r211304);
        double r211306 = r211290 * r211305;
        return r211306;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r211307 = M;
        double r211308 = D;
        double r211309 = r211307 * r211308;
        double r211310 = 2.0;
        double r211311 = d;
        double r211312 = r211310 * r211311;
        double r211313 = r211309 / r211312;
        double r211314 = pow(r211313, r211310);
        double r211315 = 1.855277338734028e-289;
        bool r211316 = r211314 <= r211315;
        double r211317 = 1.933159505335491e+291;
        bool r211318 = r211314 <= r211317;
        double r211319 = !r211318;
        bool r211320 = r211316 || r211319;
        double r211321 = w0;
        double r211322 = 1.0;
        double r211323 = sqrt(r211322);
        double r211324 = r211321 * r211323;
        double r211325 = h;
        double r211326 = l;
        double r211327 = r211325 / r211326;
        double r211328 = cbrt(r211327);
        double r211329 = r211328 * r211328;
        double r211330 = r211314 * r211329;
        double r211331 = r211330 * r211328;
        double r211332 = r211322 - r211331;
        double r211333 = sqrt(r211332);
        double r211334 = r211321 * r211333;
        double r211335 = r211320 ? r211324 : r211334;
        return r211335;
}

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 (pow (/ (* M D) (* 2.0 d)) 2.0) < 1.855277338734028e-289 or 1.933159505335491e+291 < (pow (/ (* M D) (* 2.0 d)) 2.0)

    1. Initial program 18.2

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

    3. Applied associate-*r/12.7

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

    5. Applied sqr-pow12.7

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

    6. Applied associate-*l*10.7

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

    7. Taylor expanded around 0 11.2

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1}}\]

    if 1.855277338734028e-289 < (pow (/ (* M D) (* 2.0 d)) 2.0) < 1.933159505335491e+291

    1. Initial program 6.0

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

    3. Applied add-cube-cbrt6.1

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

    4. Applied associate-*r*6.1

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.855277338734028038380413031509408290214 \cdot 10^{-289} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.933159505335491079472504373058255984824 \cdot 10^{291}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\sqrt[3]{\frac{h}{\ell}} \cdot \sqrt[3]{\frac{h}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{h}{\ell}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))))