Average Error: 14.6 → 9.3
Time: 16.1s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\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 r176888 = w0;
        double r176889 = 1.0;
        double r176890 = M;
        double r176891 = D;
        double r176892 = r176890 * r176891;
        double r176893 = 2.0;
        double r176894 = d;
        double r176895 = r176893 * r176894;
        double r176896 = r176892 / r176895;
        double r176897 = pow(r176896, r176893);
        double r176898 = h;
        double r176899 = l;
        double r176900 = r176898 / r176899;
        double r176901 = r176897 * r176900;
        double r176902 = r176889 - r176901;
        double r176903 = sqrt(r176902);
        double r176904 = r176888 * r176903;
        return r176904;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r176905 = h;
        double r176906 = l;
        double r176907 = r176905 / r176906;
        double r176908 = -4.434769257142831e+294;
        bool r176909 = r176907 <= r176908;
        double r176910 = -3.0353158251021e-314;
        bool r176911 = r176907 <= r176910;
        double r176912 = !r176911;
        bool r176913 = r176909 || r176912;
        double r176914 = w0;
        double r176915 = 1.0;
        double r176916 = sqrt(r176915);
        double r176917 = r176914 * r176916;
        double r176918 = M;
        double r176919 = D;
        double r176920 = r176918 * r176919;
        double r176921 = 2.0;
        double r176922 = d;
        double r176923 = r176921 * r176922;
        double r176924 = r176920 / r176923;
        double r176925 = 2.0;
        double r176926 = r176921 / r176925;
        double r176927 = pow(r176924, r176926);
        double r176928 = r176927 * r176907;
        double r176929 = r176927 * r176928;
        double r176930 = r176915 - r176929;
        double r176931 = sqrt(r176930);
        double r176932 = r176914 * r176931;
        double r176933 = r176913 ? r176917 : r176932;
        return r176933;
}

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

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

    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

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

Reproduce

herbie shell --seed 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))))))