Average Error: 14.6 → 10.0
Time: 11.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.0393769549213093 \cdot 10^{257} \lor \neg \left(\frac{h}{\ell} \le 4.30477040146389894 \cdot 10^{222}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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 -4.0393769549213093 \cdot 10^{257} \lor \neg \left(\frac{h}{\ell} \le 4.30477040146389894 \cdot 10^{222}\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 r270984 = w0;
        double r270985 = 1.0;
        double r270986 = M;
        double r270987 = D;
        double r270988 = r270986 * r270987;
        double r270989 = 2.0;
        double r270990 = d;
        double r270991 = r270989 * r270990;
        double r270992 = r270988 / r270991;
        double r270993 = pow(r270992, r270989);
        double r270994 = h;
        double r270995 = l;
        double r270996 = r270994 / r270995;
        double r270997 = r270993 * r270996;
        double r270998 = r270985 - r270997;
        double r270999 = sqrt(r270998);
        double r271000 = r270984 * r270999;
        return r271000;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r271001 = h;
        double r271002 = l;
        double r271003 = r271001 / r271002;
        double r271004 = -4.039376954921309e+257;
        bool r271005 = r271003 <= r271004;
        double r271006 = 4.304770401463899e+222;
        bool r271007 = r271003 <= r271006;
        double r271008 = !r271007;
        bool r271009 = r271005 || r271008;
        double r271010 = w0;
        double r271011 = 1.0;
        double r271012 = sqrt(r271011);
        double r271013 = r271010 * r271012;
        double r271014 = M;
        double r271015 = D;
        double r271016 = r271014 * r271015;
        double r271017 = 2.0;
        double r271018 = d;
        double r271019 = r271017 * r271018;
        double r271020 = r271016 / r271019;
        double r271021 = 2.0;
        double r271022 = r271017 / r271021;
        double r271023 = pow(r271020, r271022);
        double r271024 = r271023 * r271003;
        double r271025 = r271023 * r271024;
        double r271026 = r271011 - r271025;
        double r271027 = sqrt(r271026);
        double r271028 = r271010 * r271027;
        double r271029 = r271009 ? r271013 : r271028;
        return r271029;
}

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) < -4.039376954921309e+257 or 4.304770401463899e+222 < (/ h l)

    1. Initial program 45.3

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

    3. Applied associate-*r/15.5

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

    4. Taylor expanded around 0 21.6

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

    if -4.039376954921309e+257 < (/ h l) < 4.304770401463899e+222

    1. Initial program 10.0

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

    3. Applied sqr-pow10.0

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

    4. Applied associate-*l*8.2

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.0393769549213093 \cdot 10^{257} \lor \neg \left(\frac{h}{\ell} \le 4.30477040146389894 \cdot 10^{222}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))))))