Average Error: 59.4 → 53.0
Time: 55.4s
Precision: 64
\[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le 2.062340266333217399231822986622425403289 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)}} \cdot \sqrt{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)}} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \left(\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}}\right)}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;d \le 8.472196766396536195233884887397659818707 \cdot 10^{91}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{\frac{c0 \cdot 2}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{d \cdot d}}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{1}{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}}}{w} \cdot \frac{c0}{2}\\ \end{array}\]
\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}\;d \le 2.062340266333217399231822986622425403289 \cdot 10^{-69}:\\
\;\;\;\;\frac{\sqrt[3]{\sqrt{\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)}} \cdot \sqrt{\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 8.472196766396536195233884887397659818707 \cdot 10^{91}:\\
\;\;\;\;\frac{c0}{2} \cdot \frac{\frac{c0 \cdot 2}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{d \cdot d}}}{w}\\

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

\end{array}
double f(double c0, double w, double h, double D, double d, double M) {
        double r7334058 = c0;
        double r7334059 = 2.0;
        double r7334060 = w;
        double r7334061 = r7334059 * r7334060;
        double r7334062 = r7334058 / r7334061;
        double r7334063 = d;
        double r7334064 = r7334063 * r7334063;
        double r7334065 = r7334058 * r7334064;
        double r7334066 = h;
        double r7334067 = r7334060 * r7334066;
        double r7334068 = D;
        double r7334069 = r7334068 * r7334068;
        double r7334070 = r7334067 * r7334069;
        double r7334071 = r7334065 / r7334070;
        double r7334072 = r7334071 * r7334071;
        double r7334073 = M;
        double r7334074 = r7334073 * r7334073;
        double r7334075 = r7334072 - r7334074;
        double r7334076 = sqrt(r7334075);
        double r7334077 = r7334071 + r7334076;
        double r7334078 = r7334062 * r7334077;
        return r7334078;
}
↓
double f(double c0, double w, double h, double D, double d, double M) {
        double r7334079 = d;
        double r7334080 = 2.0623402663332174e-69;
        bool r7334081 = r7334079 <= r7334080;
        double r7334082 = M;
        double r7334083 = D;
        double r7334084 = r7334079 / r7334083;
        double r7334085 = c0;
        double r7334086 = r7334084 * r7334085;
        double r7334087 = r7334086 * r7334084;
        double r7334088 = w;
        double r7334089 = h;
        double r7334090 = r7334088 * r7334089;
        double r7334091 = r7334087 / r7334090;
        double r7334092 = r7334082 + r7334091;
        double r7334093 = r7334091 - r7334082;
        double r7334094 = r7334092 * r7334093;
        double r7334095 = sqrt(r7334094);
        double r7334096 = sqrt(r7334095);
        double r7334097 = r7334096 * r7334096;
        double r7334098 = r7334097 + r7334091;
        double r7334099 = cbrt(r7334098);
        double r7334100 = r7334095 + r7334091;
        double r7334101 = cbrt(r7334100);
        double r7334102 = r7334101 * r7334101;
        double r7334103 = r7334099 * r7334102;
        double r7334104 = r7334103 / r7334088;
        double r7334105 = 2.0;
        double r7334106 = r7334085 / r7334105;
        double r7334107 = r7334104 * r7334106;
        double r7334108 = 8.472196766396536e+91;
        bool r7334109 = r7334079 <= r7334108;
        double r7334110 = 2.0;
        double r7334111 = r7334085 * r7334110;
        double r7334112 = r7334083 * r7334090;
        double r7334113 = r7334083 * r7334112;
        double r7334114 = r7334079 * r7334079;
        double r7334115 = r7334113 / r7334114;
        double r7334116 = r7334111 / r7334115;
        double r7334117 = r7334116 / r7334088;
        double r7334118 = r7334106 * r7334117;
        double r7334119 = 1.0;
        double r7334120 = r7334090 / r7334087;
        double r7334121 = r7334119 / r7334120;
        double r7334122 = r7334095 + r7334121;
        double r7334123 = r7334122 / r7334088;
        double r7334124 = r7334123 * r7334106;
        double r7334125 = r7334109 ? r7334118 : r7334124;
        double r7334126 = r7334081 ? r7334107 : r7334125;
        return r7334126;
}

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 < 2.0623402663332174e-69

    1. Initial program 59.5

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

    2. Simplified51.9

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

    4. Applied add-cube-cbrt52.0

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

    6. Applied add-sqr-sqrt52.0

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

    if 2.0623402663332174e-69 < d < 8.472196766396536e+91

    1. Initial program 54.6

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

    2. Simplified52.1

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

    3. Taylor expanded around 0 55.4

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

    4. Simplified51.9

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

    if 8.472196766396536e+91 < d

    1. Initial program 62.1

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

    2. Simplified55.5

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

    4. Applied clear-num56.2

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

  4. Final simplification53.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 2.062340266333217399231822986622425403289 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)}} \cdot \sqrt{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)}} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \left(\sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}} \cdot \sqrt[3]{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}}\right)}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;d \le 8.472196766396536195233884887397659818707 \cdot 10^{91}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{\frac{c0 \cdot 2}{\frac{D \cdot \left(D \cdot \left(w \cdot h\right)\right)}{d \cdot d}}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(M + \frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h}\right) \cdot \left(\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M\right)} + \frac{1}{\frac{w \cdot h}{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}}}{w} \cdot \frac{c0}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))))