Average Error: 13.9 → 9.4
Time: 11.4s
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 r282294 = w0;
        double r282295 = 1.0;
        double r282296 = M;
        double r282297 = D;
        double r282298 = r282296 * r282297;
        double r282299 = 2.0;
        double r282300 = d;
        double r282301 = r282299 * r282300;
        double r282302 = r282298 / r282301;
        double r282303 = pow(r282302, r282299);
        double r282304 = h;
        double r282305 = l;
        double r282306 = r282304 / r282305;
        double r282307 = r282303 * r282306;
        double r282308 = r282295 - r282307;
        double r282309 = sqrt(r282308);
        double r282310 = r282294 * r282309;
        return r282310;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r282311 = M;
        double r282312 = D;
        double r282313 = r282311 * r282312;
        double r282314 = 2.0;
        double r282315 = d;
        double r282316 = r282314 * r282315;
        double r282317 = r282313 / r282316;
        double r282318 = pow(r282317, r282314);
        double r282319 = 1.855277338734028e-289;
        bool r282320 = r282318 <= r282319;
        double r282321 = 1.933159505335491e+291;
        bool r282322 = r282318 <= r282321;
        double r282323 = !r282322;
        bool r282324 = r282320 || r282323;
        double r282325 = w0;
        double r282326 = 1.0;
        double r282327 = sqrt(r282326);
        double r282328 = r282325 * r282327;
        double r282329 = h;
        double r282330 = l;
        double r282331 = r282329 / r282330;
        double r282332 = cbrt(r282331);
        double r282333 = r282332 * r282332;
        double r282334 = r282318 * r282333;
        double r282335 = r282334 * r282332;
        double r282336 = r282326 - r282335;
        double r282337 = sqrt(r282336);
        double r282338 = r282325 * r282337;
        double r282339 = r282324 ? r282328 : r282338;
        return r282339;
}

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. 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 
(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))))))