Average Error: 13.9 → 9.2
Time: 33.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -5.513451714242142 \cdot 10^{+305}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.2945419865474642 \cdot 10^{-280}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]
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 -5.513451714242142 \cdot 10^{+305}:\\
\;\;\;\;w0\\

\mathbf{elif}\;\frac{h}{\ell} \le -1.2945419865474642 \cdot 10^{-280}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot 2}\right)}\\

\mathbf{else}:\\
\;\;\;\;w0\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r2566909 = w0;
        double r2566910 = 1.0;
        double r2566911 = M;
        double r2566912 = D;
        double r2566913 = r2566911 * r2566912;
        double r2566914 = 2.0;
        double r2566915 = d;
        double r2566916 = r2566914 * r2566915;
        double r2566917 = r2566913 / r2566916;
        double r2566918 = pow(r2566917, r2566914);
        double r2566919 = h;
        double r2566920 = l;
        double r2566921 = r2566919 / r2566920;
        double r2566922 = r2566918 * r2566921;
        double r2566923 = r2566910 - r2566922;
        double r2566924 = sqrt(r2566923);
        double r2566925 = r2566909 * r2566924;
        return r2566925;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r2566926 = h;
        double r2566927 = l;
        double r2566928 = r2566926 / r2566927;
        double r2566929 = -5.513451714242142e+305;
        bool r2566930 = r2566928 <= r2566929;
        double r2566931 = w0;
        double r2566932 = -1.2945419865474642e-280;
        bool r2566933 = r2566928 <= r2566932;
        double r2566934 = 1.0;
        double r2566935 = M;
        double r2566936 = D;
        double r2566937 = r2566935 * r2566936;
        double r2566938 = d;
        double r2566939 = 2.0;
        double r2566940 = r2566938 * r2566939;
        double r2566941 = r2566937 / r2566940;
        double r2566942 = r2566928 * r2566941;
        double r2566943 = r2566941 * r2566942;
        double r2566944 = r2566934 - r2566943;
        double r2566945 = sqrt(r2566944);
        double r2566946 = r2566931 * r2566945;
        double r2566947 = r2566933 ? r2566946 : r2566931;
        double r2566948 = r2566930 ? r2566931 : r2566947;
        return r2566948;
}

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) < -5.513451714242142e+305 or -1.2945419865474642e-280 < (/ h l)

    1. Initial program 14.2

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

    2. Simplified14.2

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

    3. Taylor expanded around 0 6.9

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

    if -5.513451714242142e+305 < (/ h l) < -1.2945419865474642e-280

    1. Initial program 13.7

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

    2. Simplified13.7

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

    4. Applied associate-*l*12.1

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -5.513451714242142 \cdot 10^{+305}:\\ \;\;\;\;w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.2945419865474642 \cdot 10^{-280}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{d \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))