Average Error: 59.1 → 52.4
Time: 51.1s
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le 1.294498908037386754246307415653783576682 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \left(\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}}\right)}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;d \le 7.635350550318417612119703934768856347212 \cdot 10^{145}:\\ \;\;\;\;\frac{\frac{c0 \cdot 2}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{d \cdot d}}}{w} \cdot \frac{c0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{1}{\left(\sqrt[3]{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}} \cdot \sqrt[3]{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}}\right) \cdot \sqrt[3]{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}}}}{w}\\ \end{array}\]
\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}\;d \le 1.294498908037386754246307415653783576682 \cdot 10^{-101}:\\
\;\;\;\;\frac{\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \left(\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}}\right)}{w} \cdot \frac{c0}{2}\\

\mathbf{elif}\;d \le 7.635350550318417612119703934768856347212 \cdot 10^{145}:\\
\;\;\;\;\frac{\frac{c0 \cdot 2}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{d \cdot d}}}{w} \cdot \frac{c0}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c0}{2} \cdot \frac{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{1}{\left(\sqrt[3]{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}} \cdot \sqrt[3]{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}}\right) \cdot \sqrt[3]{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}}}}{w}\\

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r7659079 = c0;
        double r7659080 = 2.0;
        double r7659081 = w;
        double r7659082 = r7659080 * r7659081;
        double r7659083 = r7659079 / r7659082;
        double r7659084 = d;
        double r7659085 = r7659084 * r7659084;
        double r7659086 = r7659079 * r7659085;
        double r7659087 = h;
        double r7659088 = r7659081 * r7659087;
        double r7659089 = D;
        double r7659090 = r7659089 * r7659089;
        double r7659091 = r7659088 * r7659090;
        double r7659092 = r7659086 / r7659091;
        double r7659093 = r7659092 * r7659092;
        double r7659094 = M;
        double r7659095 = r7659094 * r7659094;
        double r7659096 = r7659093 - r7659095;
        double r7659097 = sqrt(r7659096);
        double r7659098 = r7659092 + r7659097;
        double r7659099 = r7659083 * r7659098;
        return r7659099;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r7659100 = d;
        double r7659101 = 1.2944989080373868e-101;
        bool r7659102 = r7659100 <= r7659101;
        double r7659103 = M;
        double r7659104 = D;
        double r7659105 = r7659100 / r7659104;
        double r7659106 = c0;
        double r7659107 = r7659105 * r7659106;
        double r7659108 = r7659107 * r7659105;
        double r7659109 = w;
        double r7659110 = h;
        double r7659111 = r7659109 * r7659110;
        double r7659112 = r7659108 / r7659111;
        double r7659113 = r7659103 + r7659112;
        double r7659114 = r7659112 - r7659103;
        double r7659115 = r7659113 * r7659114;
        double r7659116 = sqrt(r7659115);
        double r7659117 = r7659116 + r7659112;
        double r7659118 = cbrt(r7659117);
        double r7659119 = r7659118 * r7659118;
        double r7659120 = r7659118 * r7659119;
        double r7659121 = r7659120 / r7659109;
        double r7659122 = 2.0;
        double r7659123 = r7659106 / r7659122;
        double r7659124 = r7659121 * r7659123;
        double r7659125 = 7.635350550318418e+145;
        bool r7659126 = r7659100 <= r7659125;
        double r7659127 = 2.0;
        double r7659128 = r7659106 * r7659127;
        double r7659129 = r7659104 * r7659111;
        double r7659130 = r7659104 * r7659129;
        double r7659131 = r7659100 * r7659100;
        double r7659132 = r7659130 / r7659131;
        double r7659133 = r7659128 / r7659132;
        double r7659134 = r7659133 / r7659109;
        double r7659135 = r7659134 * r7659123;
        double r7659136 = 1.0;
        double r7659137 = r7659111 / r7659108;
        double r7659138 = cbrt(r7659137);
        double r7659139 = r7659138 * r7659138;
        double r7659140 = r7659139 * r7659138;
        double r7659141 = r7659136 / r7659140;
        double r7659142 = r7659116 + r7659141;
        double r7659143 = r7659142 / r7659109;
        double r7659144 = r7659123 * r7659143;
        double r7659145 = r7659126 ? r7659135 : r7659144;
        double r7659146 = r7659102 ? r7659124 : r7659145;
        return r7659146;
}

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 3 regimes

  2. if d < 1.2944989080373868e-101

    1. Initial program 59.0

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

    2. Simplified50.8

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

    4. Applied add-cube-cbrt50.9

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

    if 1.2944989080373868e-101 < d < 7.635350550318418e+145

    1. Initial program 55.1

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

    2. Simplified52.0

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

    3. Taylor expanded around 0 55.3

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

    4. Simplified51.4

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

    if 7.635350550318418e+145 < d

    1. Initial program 63.7

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

    2. Simplified56.5

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

    4. Applied clear-num57.1

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

    6. Applied add-cube-cbrt57.7

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

  4. Final simplification52.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 1.294498908037386754246307415653783576682 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \left(\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}}\right)}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;d \le 7.635350550318417612119703934768856347212 \cdot 10^{145}:\\ \;\;\;\;\frac{\frac{c0 \cdot 2}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{d \cdot d}}}{w} \cdot \frac{c0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{1}{\left(\sqrt[3]{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}} \cdot \sqrt[3]{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}}\right) \cdot \sqrt[3]{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}}}}{w}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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))))))