Average Error: 58.0 → 51.9
Time: 3.5m
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;M \cdot M \le 3.410750937078938 \cdot 10^{-272}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}} \cdot \sqrt[3]{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + \frac{c0}{\frac{h}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{w}}}\right) \cdot \frac{c0}{w}}{2}\\ \end{array}\]
\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)
↓
\begin{array}{l}
\mathbf{if}\;M \cdot M \le 3.410750937078938 \cdot 10^{-272}:\\
\;\;\;\;\frac{\frac{c0}{w} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}} \cdot \sqrt[3]{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}}\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + \frac{c0}{\frac{h}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{w}}}\right) \cdot \frac{c0}{w}}{2}\\

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r39886857 = c0;
        double r39886858 = 2.0;
        double r39886859 = w;
        double r39886860 = r39886858 * r39886859;
        double r39886861 = r39886857 / r39886860;
        double r39886862 = d;
        double r39886863 = r39886862 * r39886862;
        double r39886864 = r39886857 * r39886863;
        double r39886865 = h;
        double r39886866 = r39886859 * r39886865;
        double r39886867 = D;
        double r39886868 = r39886867 * r39886867;
        double r39886869 = r39886866 * r39886868;
        double r39886870 = r39886864 / r39886869;
        double r39886871 = r39886870 * r39886870;
        double r39886872 = M;
        double r39886873 = r39886872 * r39886872;
        double r39886874 = r39886871 - r39886873;
        double r39886875 = sqrt(r39886874);
        double r39886876 = r39886870 + r39886875;
        double r39886877 = r39886861 * r39886876;
        return r39886877;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r39886878 = M;
        double r39886879 = r39886878 * r39886878;
        double r39886880 = 3.410750937078938e-272;
        bool r39886881 = r39886879 <= r39886880;
        double r39886882 = c0;
        double r39886883 = w;
        double r39886884 = r39886882 / r39886883;
        double r39886885 = d;
        double r39886886 = D;
        double r39886887 = r39886885 / r39886886;
        double r39886888 = r39886887 * r39886887;
        double r39886889 = r39886888 * r39886884;
        double r39886890 = h;
        double r39886891 = r39886889 / r39886890;
        double r39886892 = r39886891 - r39886878;
        double r39886893 = r39886891 + r39886878;
        double r39886894 = r39886892 * r39886893;
        double r39886895 = sqrt(r39886894);
        double r39886896 = r39886895 + r39886891;
        double r39886897 = cbrt(r39886896);
        double r39886898 = r39886897 * r39886897;
        double r39886899 = r39886897 * r39886898;
        double r39886900 = r39886884 * r39886899;
        double r39886901 = 2.0;
        double r39886902 = r39886900 / r39886901;
        double r39886903 = r39886888 / r39886883;
        double r39886904 = r39886890 / r39886903;
        double r39886905 = r39886882 / r39886904;
        double r39886906 = r39886891 + r39886905;
        double r39886907 = r39886906 * r39886884;
        double r39886908 = r39886907 / r39886901;
        double r39886909 = r39886881 ? r39886902 : r39886908;
        return r39886909;
}

Error

Bits error versus c0

Bits error versus w

Bits error versus h

Bits error versus D

Bits error versus d

Bits error versus M

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* M M) < 3.410750937078938e-272

    1. Initial program 54.5

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Simplified44.5

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(M + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \left(\frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h} - M\right)} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \frac{c0}{w}}{2}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt44.6

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\sqrt{\left(M + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \left(\frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h} - M\right)} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \left(\frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h} - M\right)} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}}\right) \cdot \sqrt[3]{\sqrt{\left(M + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \left(\frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h} - M\right)} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}}\right)} \cdot \frac{c0}{w}}{2}\]

    if 3.410750937078938e-272 < (* M M)

    1. Initial program 60.4

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

    2. Simplified57.7

      \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(M + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \left(\frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h} - M\right)} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \frac{c0}{w}}{2}}\]

    3. Taylor expanded around 0 60.3

      \[\leadsto \frac{\left(\color{blue}{\frac{c0 \cdot {d}^{2}}{w \cdot \left({D}^{2} \cdot h\right)}} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \frac{c0}{w}}{2}\]

    4. Simplified56.9

      \[\leadsto \frac{\left(\color{blue}{\frac{c0}{\frac{h}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{w}}}} + \frac{\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)}{h}\right) \cdot \frac{c0}{w}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification51.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot M \le 3.410750937078938 \cdot 10^{-272}:\\ \;\;\;\;\frac{\frac{c0}{w} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}} \cdot \left(\sqrt[3]{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}} \cdot \sqrt[3]{\sqrt{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} - M\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + M\right)} + \frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}}{h} + \frac{c0}{\frac{h}{\frac{\frac{d}{D} \cdot \frac{d}{D}}{w}}}\right) \cdot \frac{c0}{w}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))