Average Error: 57.7 → 35.1
Time: 3.9m
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;M \le -3.3977465073006914 \cdot 10^{+140}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{d \cdot c0}{D \cdot w}}{\frac{h}{\frac{\frac{d \cdot c0}{w}}{D}}}}{2}\\ \mathbf{elif}\;M \le 3.6094771780886608:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{d \cdot c0}{D \cdot w}}{\frac{h}{\frac{\frac{d \cdot c0}{w}}{D}}}}{2}\\ \end{array}\]
\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 \le -3.3977465073006914 \cdot 10^{+140}:\\
\;\;\;\;\frac{2 \cdot \frac{\frac{d \cdot c0}{D \cdot w}}{\frac{h}{\frac{\frac{d \cdot c0}{w}}{D}}}}{2}\\

\mathbf{elif}\;M \le 3.6094771780886608:\\
\;\;\;\;0\\

\mathbf{else}:\\
\;\;\;\;\frac{2 \cdot \frac{\frac{d \cdot c0}{D \cdot w}}{\frac{h}{\frac{\frac{d \cdot c0}{w}}{D}}}}{2}\\

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r37777647 = c0;
        double r37777648 = 2.0;
        double r37777649 = w;
        double r37777650 = r37777648 * r37777649;
        double r37777651 = r37777647 / r37777650;
        double r37777652 = d;
        double r37777653 = r37777652 * r37777652;
        double r37777654 = r37777647 * r37777653;
        double r37777655 = h;
        double r37777656 = r37777649 * r37777655;
        double r37777657 = D;
        double r37777658 = r37777657 * r37777657;
        double r37777659 = r37777656 * r37777658;
        double r37777660 = r37777654 / r37777659;
        double r37777661 = r37777660 * r37777660;
        double r37777662 = M;
        double r37777663 = r37777662 * r37777662;
        double r37777664 = r37777661 - r37777663;
        double r37777665 = sqrt(r37777664);
        double r37777666 = r37777660 + r37777665;
        double r37777667 = r37777651 * r37777666;
        return r37777667;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r37777668 = M;
        double r37777669 = -3.3977465073006914e+140;
        bool r37777670 = r37777668 <= r37777669;
        double r37777671 = 2.0;
        double r37777672 = d;
        double r37777673 = c0;
        double r37777674 = r37777672 * r37777673;
        double r37777675 = D;
        double r37777676 = w;
        double r37777677 = r37777675 * r37777676;
        double r37777678 = r37777674 / r37777677;
        double r37777679 = h;
        double r37777680 = r37777674 / r37777676;
        double r37777681 = r37777680 / r37777675;
        double r37777682 = r37777679 / r37777681;
        double r37777683 = r37777678 / r37777682;
        double r37777684 = r37777671 * r37777683;
        double r37777685 = r37777684 / r37777671;
        double r37777686 = 3.6094771780886608;
        bool r37777687 = r37777668 <= r37777686;
        double r37777688 = 0.0;
        double r37777689 = r37777687 ? r37777688 : r37777685;
        double r37777690 = r37777670 ? r37777685 : r37777689;
        return r37777690;
}

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 < -3.3977465073006914e+140 or 3.6094771780886608 < M

    1. Initial program 61.6

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

    2. Simplified60.3

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

    3. Taylor expanded around 0 61.5

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

    4. Simplified52.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{d \cdot c0}{w \cdot D} \cdot \frac{d \cdot c0}{w \cdot D}}{h} \cdot 2}}{2}\]
    5. Using strategy rm

    6. Applied associate-/l*50.8

      \[\leadsto \frac{\color{blue}{\frac{\frac{d \cdot c0}{w \cdot D}}{\frac{h}{\frac{d \cdot c0}{w \cdot D}}}} \cdot 2}{2}\]
    7. Using strategy rm

    8. Applied associate-/r*51.2

      \[\leadsto \frac{\frac{\frac{d \cdot c0}{w \cdot D}}{\frac{h}{\color{blue}{\frac{\frac{d \cdot c0}{w}}{D}}}} \cdot 2}{2}\]

    if -3.3977465073006914e+140 < M < 3.6094771780886608

    1. Initial program 56.5

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

    2. Simplified48.8

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

    3. Taylor expanded around -inf 29.8

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

  4. Final simplification35.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \le -3.3977465073006914 \cdot 10^{+140}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{d \cdot c0}{D \cdot w}}{\frac{h}{\frac{\frac{d \cdot c0}{w}}{D}}}}{2}\\ \mathbf{elif}\;M \le 3.6094771780886608:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\frac{d \cdot c0}{D \cdot w}}{\frac{h}{\frac{\frac{d \cdot c0}{w}}{D}}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(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))))))