Average Error: 59.6 → 33.9
Time: 51.9s
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le -6.236251574640033082104111451559545918058 \cdot 10^{-177}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 1.879814245604766896508697725116661664417 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{1}{h} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}\right) + \sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right) - M \cdot M}}{2} \cdot \frac{c0}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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}\;d \le -6.236251574640033082104111451559545918058 \cdot 10^{-177}:\\
\;\;\;\;0\\

\mathbf{elif}\;d \le 1.879814245604766896508697725116661664417 \cdot 10^{-151}:\\
\;\;\;\;\frac{\frac{1}{h} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}\right) + \sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right) - M \cdot M}}{2} \cdot \frac{c0}{w}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r5152663 = c0;
        double r5152664 = 2.0;
        double r5152665 = w;
        double r5152666 = r5152664 * r5152665;
        double r5152667 = r5152663 / r5152666;
        double r5152668 = d;
        double r5152669 = r5152668 * r5152668;
        double r5152670 = r5152663 * r5152669;
        double r5152671 = h;
        double r5152672 = r5152665 * r5152671;
        double r5152673 = D;
        double r5152674 = r5152673 * r5152673;
        double r5152675 = r5152672 * r5152674;
        double r5152676 = r5152670 / r5152675;
        double r5152677 = r5152676 * r5152676;
        double r5152678 = M;
        double r5152679 = r5152678 * r5152678;
        double r5152680 = r5152677 - r5152679;
        double r5152681 = sqrt(r5152680);
        double r5152682 = r5152676 + r5152681;
        double r5152683 = r5152667 * r5152682;
        return r5152683;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r5152684 = d;
        double r5152685 = -6.236251574640033e-177;
        bool r5152686 = r5152684 <= r5152685;
        double r5152687 = 0.0;
        double r5152688 = 1.879814245604767e-151;
        bool r5152689 = r5152684 <= r5152688;
        double r5152690 = 1.0;
        double r5152691 = h;
        double r5152692 = r5152690 / r5152691;
        double r5152693 = D;
        double r5152694 = r5152684 / r5152693;
        double r5152695 = r5152694 * r5152694;
        double r5152696 = c0;
        double r5152697 = w;
        double r5152698 = r5152696 / r5152697;
        double r5152699 = r5152695 * r5152698;
        double r5152700 = r5152692 * r5152699;
        double r5152701 = r5152695 / r5152691;
        double r5152702 = r5152701 * r5152698;
        double r5152703 = r5152702 * r5152702;
        double r5152704 = M;
        double r5152705 = r5152704 * r5152704;
        double r5152706 = r5152703 - r5152705;
        double r5152707 = sqrt(r5152706);
        double r5152708 = r5152700 + r5152707;
        double r5152709 = 2.0;
        double r5152710 = r5152708 / r5152709;
        double r5152711 = r5152710 * r5152698;
        double r5152712 = r5152689 ? r5152711 : r5152687;
        double r5152713 = r5152686 ? r5152687 : r5152712;
        return r5152713;
}

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 d < -6.236251574640033e-177 or 1.879814245604767e-151 < d

    1. Initial program 59.3

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

    2. Simplified55.0

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

    3. Taylor expanded around inf 34.6

      \[\leadsto \frac{c0}{w} \cdot \frac{\color{blue}{0}}{2}\]

    4. Taylor expanded around 0 32.7

      \[\leadsto \color{blue}{0}\]

    if -6.236251574640033e-177 < d < 1.879814245604767e-151

    1. Initial program 62.7

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

    2. Simplified42.7

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

    4. Applied div-inv44.1

      \[\leadsto \frac{c0}{w} \cdot \frac{\sqrt{\left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right) - M \cdot M} + \frac{c0}{w} \cdot \color{blue}{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{1}{h}\right)}}{2}\]

    5. Applied associate-*r*45.2

      \[\leadsto \frac{c0}{w} \cdot \frac{\sqrt{\left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{w} \cdot \frac{\frac{d}{D} \cdot \frac{d}{D}}{h}\right) - M \cdot M} + \color{blue}{\left(\frac{c0}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \frac{1}{h}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification33.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -6.236251574640033082104111451559545918058 \cdot 10^{-177}:\\ \;\;\;\;0\\ \mathbf{elif}\;d \le 1.879814245604766896508697725116661664417 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{1}{h} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{c0}{w}\right) + \sqrt{\left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w}\right) - M \cdot M}}{2} \cdot \frac{c0}{w}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +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))))))