Average Error: 13.9 → 9.4
Time: 11.7s
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 r282151 = w0;
        double r282152 = 1.0;
        double r282153 = M;
        double r282154 = D;
        double r282155 = r282153 * r282154;
        double r282156 = 2.0;
        double r282157 = d;
        double r282158 = r282156 * r282157;
        double r282159 = r282155 / r282158;
        double r282160 = pow(r282159, r282156);
        double r282161 = h;
        double r282162 = l;
        double r282163 = r282161 / r282162;
        double r282164 = r282160 * r282163;
        double r282165 = r282152 - r282164;
        double r282166 = sqrt(r282165);
        double r282167 = r282151 * r282166;
        return r282167;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r282168 = M;
        double r282169 = D;
        double r282170 = r282168 * r282169;
        double r282171 = 2.0;
        double r282172 = d;
        double r282173 = r282171 * r282172;
        double r282174 = r282170 / r282173;
        double r282175 = pow(r282174, r282171);
        double r282176 = 1.855277338734028e-289;
        bool r282177 = r282175 <= r282176;
        double r282178 = 1.933159505335491e+291;
        bool r282179 = r282175 <= r282178;
        double r282180 = !r282179;
        bool r282181 = r282177 || r282180;
        double r282182 = w0;
        double r282183 = 1.0;
        double r282184 = sqrt(r282183);
        double r282185 = r282182 * r282184;
        double r282186 = h;
        double r282187 = l;
        double r282188 = r282186 / r282187;
        double r282189 = cbrt(r282188);
        double r282190 = r282189 * r282189;
        double r282191 = r282175 * r282190;
        double r282192 = r282191 * r282189;
        double r282193 = r282183 - r282192;
        double r282194 = sqrt(r282193);
        double r282195 = r282182 * r282194;
        double r282196 = r282181 ? r282185 : r282195;
        return r282196;
}

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 
(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))))))