Average Error: 14.0 → 8.9
Time: 28.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -6.051505276988730809064682225880124571638 \cdot 10^{-311}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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} = -\infty \lor \neg \left(\frac{h}{\ell} \le -6.051505276988730809064682225880124571638 \cdot 10^{-311}\right):\\
\;\;\;\;w0 \cdot \sqrt{1}\\

\mathbf{else}:\\
\;\;\;\;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)}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r181846 = w0;
        double r181847 = 1.0;
        double r181848 = M;
        double r181849 = D;
        double r181850 = r181848 * r181849;
        double r181851 = 2.0;
        double r181852 = d;
        double r181853 = r181851 * r181852;
        double r181854 = r181850 / r181853;
        double r181855 = pow(r181854, r181851);
        double r181856 = h;
        double r181857 = l;
        double r181858 = r181856 / r181857;
        double r181859 = r181855 * r181858;
        double r181860 = r181847 - r181859;
        double r181861 = sqrt(r181860);
        double r181862 = r181846 * r181861;
        return r181862;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r181863 = h;
        double r181864 = l;
        double r181865 = r181863 / r181864;
        double r181866 = -inf.0;
        bool r181867 = r181865 <= r181866;
        double r181868 = -6.0515052769887e-311;
        bool r181869 = r181865 <= r181868;
        double r181870 = !r181869;
        bool r181871 = r181867 || r181870;
        double r181872 = w0;
        double r181873 = 1.0;
        double r181874 = sqrt(r181873);
        double r181875 = r181872 * r181874;
        double r181876 = M;
        double r181877 = D;
        double r181878 = r181876 * r181877;
        double r181879 = 2.0;
        double r181880 = d;
        double r181881 = r181879 * r181880;
        double r181882 = r181878 / r181881;
        double r181883 = 2.0;
        double r181884 = r181879 / r181883;
        double r181885 = pow(r181882, r181884);
        double r181886 = r181885 * r181865;
        double r181887 = r181885 * r181886;
        double r181888 = r181873 - r181887;
        double r181889 = sqrt(r181888);
        double r181890 = r181872 * r181889;
        double r181891 = r181871 ? r181875 : r181890;
        return r181891;
}

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) < -inf.0 or -6.0515052769887e-311 < (/ h l)

    1. Initial program 13.6

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

    2. Taylor expanded around 0 5.8

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

    if -inf.0 < (/ h l) < -6.0515052769887e-311

    1. Initial program 14.4

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

    3. Applied sqr-pow14.4

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

    4. Applied associate-*l*12.4

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -6.051505276988730809064682225880124571638 \cdot 10^{-311}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))))