Average Error: 59.6 → 32.1
Time: 57.6s
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \cdot \frac{c0}{w \cdot 2} \le 1.884987459885323114731507884787749259003 \cdot 10^{277}:\\ \;\;\;\;\left(\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{0}{w}\\ \end{array}\]
\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}\;\left(\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \cdot \frac{c0}{w \cdot 2} \le 1.884987459885323114731507884787749259003 \cdot 10^{277}:\\
\;\;\;\;\left(\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \cdot \frac{c0}{w \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c0}{2} \cdot \frac{0}{w}\\

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r6031876 = c0;
        double r6031877 = 2.0;
        double r6031878 = w;
        double r6031879 = r6031877 * r6031878;
        double r6031880 = r6031876 / r6031879;
        double r6031881 = d;
        double r6031882 = r6031881 * r6031881;
        double r6031883 = r6031876 * r6031882;
        double r6031884 = h;
        double r6031885 = r6031878 * r6031884;
        double r6031886 = D;
        double r6031887 = r6031886 * r6031886;
        double r6031888 = r6031885 * r6031887;
        double r6031889 = r6031883 / r6031888;
        double r6031890 = r6031889 * r6031889;
        double r6031891 = M;
        double r6031892 = r6031891 * r6031891;
        double r6031893 = r6031890 - r6031892;
        double r6031894 = sqrt(r6031893);
        double r6031895 = r6031889 + r6031894;
        double r6031896 = r6031880 * r6031895;
        return r6031896;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r6031897 = d;
        double r6031898 = r6031897 * r6031897;
        double r6031899 = c0;
        double r6031900 = r6031898 * r6031899;
        double r6031901 = D;
        double r6031902 = r6031901 * r6031901;
        double r6031903 = w;
        double r6031904 = h;
        double r6031905 = r6031903 * r6031904;
        double r6031906 = r6031902 * r6031905;
        double r6031907 = r6031900 / r6031906;
        double r6031908 = r6031907 * r6031907;
        double r6031909 = M;
        double r6031910 = r6031909 * r6031909;
        double r6031911 = r6031908 - r6031910;
        double r6031912 = sqrt(r6031911);
        double r6031913 = r6031907 + r6031912;
        double r6031914 = 2.0;
        double r6031915 = r6031903 * r6031914;
        double r6031916 = r6031899 / r6031915;
        double r6031917 = r6031913 * r6031916;
        double r6031918 = 1.884987459885323e+277;
        bool r6031919 = r6031917 <= r6031918;
        double r6031920 = r6031899 / r6031914;
        double r6031921 = 0.0;
        double r6031922 = r6031921 / r6031903;
        double r6031923 = r6031920 * r6031922;
        double r6031924 = r6031919 ? r6031917 : r6031923;
        return r6031924;
}

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 (* (/ c0 (* 2.0 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))))) < 1.884987459885323e+277

    1. Initial program 36.0

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

    if 1.884987459885323e+277 < (* (/ c0 (* 2.0 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)))))

    1. Initial program 64.0

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

    2. Simplified57.3

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

    3. Taylor expanded around inf 31.4

      \[\leadsto \frac{c0}{2} \cdot \frac{\color{blue}{0}}{w}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification32.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \cdot \frac{c0}{w \cdot 2} \le 1.884987459885323114731507884787749259003 \cdot 10^{277}:\\ \;\;\;\;\left(\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} + \sqrt{\frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} \cdot \frac{\left(d \cdot d\right) \cdot c0}{\left(D \cdot D\right) \cdot \left(w \cdot h\right)} - M \cdot M}\right) \cdot \frac{c0}{w \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{0}{w}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2.0 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))))))