Average Error: 14.3 → 9.5
Time: 32.2s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.772133656628061046787329400235105824789 \cdot 10^{308}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.034203706704132394404794445791386313725 \cdot 10^{-311}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
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 -1.772133656628061046787329400235105824789 \cdot 10^{308}:\\
\;\;\;\;w0 \cdot \sqrt{1}\\

\mathbf{elif}\;\frac{h}{\ell} \le -2.034203706704132394404794445791386313725 \cdot 10^{-311}:\\
\;\;\;\;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}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r7111901 = w0;
        double r7111902 = 1.0;
        double r7111903 = M;
        double r7111904 = D;
        double r7111905 = r7111903 * r7111904;
        double r7111906 = 2.0;
        double r7111907 = d;
        double r7111908 = r7111906 * r7111907;
        double r7111909 = r7111905 / r7111908;
        double r7111910 = pow(r7111909, r7111906);
        double r7111911 = h;
        double r7111912 = l;
        double r7111913 = r7111911 / r7111912;
        double r7111914 = r7111910 * r7111913;
        double r7111915 = r7111902 - r7111914;
        double r7111916 = sqrt(r7111915);
        double r7111917 = r7111901 * r7111916;
        return r7111917;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r7111918 = h;
        double r7111919 = l;
        double r7111920 = r7111918 / r7111919;
        double r7111921 = -1.772133656628061e+308;
        bool r7111922 = r7111920 <= r7111921;
        double r7111923 = w0;
        double r7111924 = 1.0;
        double r7111925 = sqrt(r7111924);
        double r7111926 = r7111923 * r7111925;
        double r7111927 = -2.0342037067041e-311;
        bool r7111928 = r7111920 <= r7111927;
        double r7111929 = M;
        double r7111930 = D;
        double r7111931 = r7111929 * r7111930;
        double r7111932 = 2.0;
        double r7111933 = d;
        double r7111934 = r7111932 * r7111933;
        double r7111935 = r7111931 / r7111934;
        double r7111936 = 2.0;
        double r7111937 = r7111932 / r7111936;
        double r7111938 = pow(r7111935, r7111937);
        double r7111939 = r7111938 * r7111920;
        double r7111940 = r7111938 * r7111939;
        double r7111941 = r7111924 - r7111940;
        double r7111942 = sqrt(r7111941);
        double r7111943 = r7111923 * r7111942;
        double r7111944 = r7111928 ? r7111943 : r7111926;
        double r7111945 = r7111922 ? r7111926 : r7111944;
        return r7111945;
}

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) < -1.772133656628061e+308 or -2.0342037067041e-311 < (/ h l)

    1. Initial program 13.1

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

    2. Taylor expanded around 0 5.9

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

    if -1.772133656628061e+308 < (/ h l) < -2.0342037067041e-311

    1. Initial program 15.7

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

    3. Applied sqr-pow15.7

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

    4. Applied associate-*l*13.7

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.772133656628061046787329400235105824789 \cdot 10^{308}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.034203706704132394404794445791386313725 \cdot 10^{-311}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))