Average Error: 58.0 → 53.3
Time: 1.1m
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;c0 \le -5.885464112857827 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} + \sqrt{\left(\frac{\frac{d}{D}}{h} \cdot \frac{c0 \cdot \frac{d}{D}}{w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right)}}{2} \cdot \frac{c0}{w}\\ \mathbf{elif}\;c0 \le 1.1156496586195279 \cdot 10^{-69}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\log \left(e^{\sqrt{\frac{\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}{w} \cdot \frac{\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}{w} - M \cdot M} + \frac{\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}{w}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} + \sqrt{\left(\frac{\frac{d}{D}}{h} \cdot \frac{c0 \cdot \frac{d}{D}}{w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right)}}{2} \cdot \frac{c0}{w}\\ \end{array}\]
\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}\;c0 \le -5.885464112857827 \cdot 10^{-142}:\\
\;\;\;\;\frac{\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} + \sqrt{\left(\frac{\frac{d}{D}}{h} \cdot \frac{c0 \cdot \frac{d}{D}}{w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right)}}{2} \cdot \frac{c0}{w}\\

\mathbf{elif}\;c0 \le 1.1156496586195279 \cdot 10^{-69}:\\
\;\;\;\;\frac{c0}{w} \cdot \frac{\log \left(e^{\sqrt{\frac{\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}{w} \cdot \frac{\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}{w} - M \cdot M} + \frac{\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}{w}}\right)}{2}\\

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

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r4048109 = c0;
        double r4048110 = 2.0;
        double r4048111 = w;
        double r4048112 = r4048110 * r4048111;
        double r4048113 = r4048109 / r4048112;
        double r4048114 = d;
        double r4048115 = r4048114 * r4048114;
        double r4048116 = r4048109 * r4048115;
        double r4048117 = h;
        double r4048118 = r4048111 * r4048117;
        double r4048119 = D;
        double r4048120 = r4048119 * r4048119;
        double r4048121 = r4048118 * r4048120;
        double r4048122 = r4048116 / r4048121;
        double r4048123 = r4048122 * r4048122;
        double r4048124 = M;
        double r4048125 = r4048124 * r4048124;
        double r4048126 = r4048123 - r4048125;
        double r4048127 = sqrt(r4048126);
        double r4048128 = r4048122 + r4048127;
        double r4048129 = r4048113 * r4048128;
        return r4048129;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r4048130 = c0;
        double r4048131 = -5.885464112857827e-142;
        bool r4048132 = r4048130 <= r4048131;
        double r4048133 = d;
        double r4048134 = D;
        double r4048135 = r4048133 / r4048134;
        double r4048136 = r4048135 * r4048135;
        double r4048137 = h;
        double r4048138 = r4048136 / r4048137;
        double r4048139 = w;
        double r4048140 = r4048130 / r4048139;
        double r4048141 = r4048138 * r4048140;
        double r4048142 = r4048135 / r4048137;
        double r4048143 = r4048130 * r4048135;
        double r4048144 = r4048143 / r4048139;
        double r4048145 = r4048142 * r4048144;
        double r4048146 = M;
        double r4048147 = r4048145 + r4048146;
        double r4048148 = r4048141 - r4048146;
        double r4048149 = r4048147 * r4048148;
        double r4048150 = sqrt(r4048149);
        double r4048151 = r4048141 + r4048150;
        double r4048152 = 2.0;
        double r4048153 = r4048151 / r4048152;
        double r4048154 = r4048153 * r4048140;
        double r4048155 = 1.1156496586195279e-69;
        bool r4048156 = r4048130 <= r4048155;
        double r4048157 = r4048143 * r4048135;
        double r4048158 = r4048157 / r4048137;
        double r4048159 = r4048158 / r4048139;
        double r4048160 = r4048159 * r4048159;
        double r4048161 = r4048146 * r4048146;
        double r4048162 = r4048160 - r4048161;
        double r4048163 = sqrt(r4048162);
        double r4048164 = r4048163 + r4048159;
        double r4048165 = exp(r4048164);
        double r4048166 = log(r4048165);
        double r4048167 = r4048166 / r4048152;
        double r4048168 = r4048140 * r4048167;
        double r4048169 = r4048156 ? r4048168 : r4048154;
        double r4048170 = r4048132 ? r4048154 : r4048169;
        return r4048170;
}

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 < -5.885464112857827e-142 or 1.1156496586195279e-69 < c0

    1. Initial program 58.6

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

    2. Simplified52.9

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

    4. Applied *-un-lft-identity52.9

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

    5. Applied times-frac53.4

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

    6. Applied associate-*r*53.8

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

    7. Simplified54.2

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

    if -5.885464112857827e-142 < c0 < 1.1156496586195279e-69

    1. Initial program 56.7

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

    2. Simplified51.3

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

    4. Applied add-log-exp59.0

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

    5. Applied add-log-exp58.0

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

    6. Applied sum-log57.8

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

    7. Simplified51.5

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

  4. Final simplification53.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \le -5.885464112857827 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} + \sqrt{\left(\frac{\frac{d}{D}}{h} \cdot \frac{c0 \cdot \frac{d}{D}}{w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right)}}{2} \cdot \frac{c0}{w}\\ \mathbf{elif}\;c0 \le 1.1156496586195279 \cdot 10^{-69}:\\ \;\;\;\;\frac{c0}{w} \cdot \frac{\log \left(e^{\sqrt{\frac{\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}{w} \cdot \frac{\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}{w} - M \cdot M} + \frac{\frac{\left(c0 \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}{h}}{w}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} + \sqrt{\left(\frac{\frac{d}{D}}{h} \cdot \frac{c0 \cdot \frac{d}{D}}{w} + M\right) \cdot \left(\frac{\frac{d}{D} \cdot \frac{d}{D}}{h} \cdot \frac{c0}{w} - M\right)}}{2} \cdot \frac{c0}{w}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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))))))