Average Error: 14.4 → 8.9
Time: 11.2s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -6.595620386253778118443402069571269976276 \cdot 10^{-220}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{\sqrt{1}}{\sqrt[3]{\frac{2 \cdot d}{M \cdot D}} \cdot \sqrt[3]{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot \left({\left(\frac{1}{\sqrt[3]{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot h\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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 -6.595620386253778118443402069571269976276 \cdot 10^{-220}\right):\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{\sqrt{1}}{\sqrt[3]{\frac{2 \cdot d}{M \cdot D}} \cdot \sqrt[3]{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot \left({\left(\frac{1}{\sqrt[3]{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot h\right)}{\ell}}\\

\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 r270006 = w0;
        double r270007 = 1.0;
        double r270008 = M;
        double r270009 = D;
        double r270010 = r270008 * r270009;
        double r270011 = 2.0;
        double r270012 = d;
        double r270013 = r270011 * r270012;
        double r270014 = r270010 / r270013;
        double r270015 = pow(r270014, r270011);
        double r270016 = h;
        double r270017 = l;
        double r270018 = r270016 / r270017;
        double r270019 = r270015 * r270018;
        double r270020 = r270007 - r270019;
        double r270021 = sqrt(r270020);
        double r270022 = r270006 * r270021;
        return r270022;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r270023 = h;
        double r270024 = l;
        double r270025 = r270023 / r270024;
        double r270026 = -inf.0;
        bool r270027 = r270025 <= r270026;
        double r270028 = -6.595620386253778e-220;
        bool r270029 = r270025 <= r270028;
        double r270030 = !r270029;
        bool r270031 = r270027 || r270030;
        double r270032 = w0;
        double r270033 = 1.0;
        double r270034 = 1.0;
        double r270035 = sqrt(r270034);
        double r270036 = 2.0;
        double r270037 = d;
        double r270038 = r270036 * r270037;
        double r270039 = M;
        double r270040 = D;
        double r270041 = r270039 * r270040;
        double r270042 = r270038 / r270041;
        double r270043 = cbrt(r270042);
        double r270044 = r270043 * r270043;
        double r270045 = r270035 / r270044;
        double r270046 = pow(r270045, r270036);
        double r270047 = r270034 / r270043;
        double r270048 = pow(r270047, r270036);
        double r270049 = r270048 * r270023;
        double r270050 = r270046 * r270049;
        double r270051 = r270050 / r270024;
        double r270052 = r270033 - r270051;
        double r270053 = sqrt(r270052);
        double r270054 = r270032 * r270053;
        double r270055 = r270041 / r270038;
        double r270056 = 2.0;
        double r270057 = r270036 / r270056;
        double r270058 = pow(r270055, r270057);
        double r270059 = r270058 * r270025;
        double r270060 = r270058 * r270059;
        double r270061 = r270033 - r270060;
        double r270062 = sqrt(r270061);
        double r270063 = r270032 * r270062;
        double r270064 = r270031 ? r270054 : r270063;
        return r270064;
}

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 -6.595620386253778e-220 < (/ h l)

    1. Initial program 14.9

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

    3. Applied associate-*r/8.0

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

    5. Applied clear-num8.0

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

    7. Applied add-cube-cbrt8.1

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

    8. Applied add-sqr-sqrt8.1

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

    9. Applied times-frac8.1

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

    10. Applied unpow-prod-down8.1

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

    11. Applied associate-*l*6.3

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

    12. Simplified6.3

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

    if -inf.0 < (/ h l) < -6.595620386253778e-220

    1. Initial program 13.7

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

    3. Applied sqr-pow13.7

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

    4. Applied associate-*l*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 simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -6.595620386253778118443402069571269976276 \cdot 10^{-220}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{\sqrt{1}}{\sqrt[3]{\frac{2 \cdot d}{M \cdot D}} \cdot \sqrt[3]{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot \left({\left(\frac{1}{\sqrt[3]{\frac{2 \cdot d}{M \cdot D}}}\right)}^{2} \cdot h\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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))))))