Average Error: 14.6 → 9.3
Time: 16.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\
\;\;\;\;\sqrt{1} \cdot w0\\

\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 r208846 = w0;
        double r208847 = 1.0;
        double r208848 = M;
        double r208849 = D;
        double r208850 = r208848 * r208849;
        double r208851 = 2.0;
        double r208852 = d;
        double r208853 = r208851 * r208852;
        double r208854 = r208850 / r208853;
        double r208855 = pow(r208854, r208851);
        double r208856 = h;
        double r208857 = l;
        double r208858 = r208856 / r208857;
        double r208859 = r208855 * r208858;
        double r208860 = r208847 - r208859;
        double r208861 = sqrt(r208860);
        double r208862 = r208846 * r208861;
        return r208862;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r208863 = h;
        double r208864 = l;
        double r208865 = r208863 / r208864;
        double r208866 = -4.434769257142831e+294;
        bool r208867 = r208865 <= r208866;
        double r208868 = -3.0353158251021e-314;
        bool r208869 = r208865 <= r208868;
        double r208870 = !r208869;
        bool r208871 = r208867 || r208870;
        double r208872 = 1.0;
        double r208873 = sqrt(r208872);
        double r208874 = w0;
        double r208875 = r208873 * r208874;
        double r208876 = M;
        double r208877 = D;
        double r208878 = r208876 * r208877;
        double r208879 = 2.0;
        double r208880 = d;
        double r208881 = r208879 * r208880;
        double r208882 = r208878 / r208881;
        double r208883 = 2.0;
        double r208884 = r208879 / r208883;
        double r208885 = pow(r208882, r208884);
        double r208886 = r208885 * r208865;
        double r208887 = r208885 * r208886;
        double r208888 = r208872 - r208887;
        double r208889 = sqrt(r208888);
        double r208890 = r208874 * r208889;
        double r208891 = r208871 ? r208875 : r208890;
        return r208891;
}

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) < -4.434769257142831e+294 or -3.0353158251021e-314 < (/ h l)

    1. Initial program 14.5

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

    2. Taylor expanded around 0 6.5

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

    if -4.434769257142831e+294 < (/ h l) < -3.0353158251021e-314

    1. Initial program 14.6

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

    3. Applied sqr-pow14.6

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

    4. Applied associate-*l*12.6

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))))))