Henrywood and Agarwal, Equation (13)

Average Error: 58.2 → 34.8

Time: 2.9m

Precision: 64

• could not determine a ground truth for program body

  1. c0 = -4.660942649264086e+183
  2. w = -2.7366011941023535e-219
  3. h = -1.5822855227192185e-211
  4. D = 5.157363736082994e+82
  5. d = -1.7964296798570103e+233
  6. M = 1.6069807528843228e-193

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;w \le 5.1060876778360224 \cdot 10^{+44}:\\ \;\;\;\;\frac{0}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;w \le 1.3340810790915187 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} + \sqrt{\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M} \cdot \sqrt{\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} + M}}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;w \le 1.350659423435845 \cdot 10^{+163}:\\ \;\;\;\;\frac{0}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;w \le 1.0915833705181084 \cdot 10^{+193}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{\frac{\sqrt[3]{\sqrt{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M} \cdot \left(\sqrt{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M} \cdot \sqrt{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M}\right)} \cdot \left(\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M\right) + \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right)\right)}{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h} - \sqrt{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) + \left(\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M\right)}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{w} \cdot \frac{c0}{2}\\ \end{array}\]

C

double f(double c0, double w, double h, double D, double d, double M) {
        double r6073710 = c0;
        double r6073711 = 2.0;
        double r6073712 = w;
        double r6073713 = r6073711 * r6073712;
        double r6073714 = r6073710 / r6073713;
        double r6073715 = d;
        double r6073716 = r6073715 * r6073715;
        double r6073717 = r6073710 * r6073716;
        double r6073718 = h;
        double r6073719 = r6073712 * r6073718;
        double r6073720 = D;
        double r6073721 = r6073720 * r6073720;
        double r6073722 = r6073719 * r6073721;
        double r6073723 = r6073717 / r6073722;
        double r6073724 = r6073723 * r6073723;
        double r6073725 = M;
        double r6073726 = r6073725 * r6073725;
        double r6073727 = r6073724 - r6073726;
        double r6073728 = sqrt(r6073727);
        double r6073729 = r6073723 + r6073728;
        double r6073730 = r6073714 * r6073729;
        return r6073730;
}

↓

double f(double c0, double w, double h, double D, double d, double M) {
        double r6073731 = w;
        double r6073732 = 5.1060876778360224e+44;
        bool r6073733 = r6073731 <= r6073732;
        double r6073734 = 0.0;
        double r6073735 = r6073734 / r6073731;
        double r6073736 = c0;
        double r6073737 = 2.0;
        double r6073738 = r6073736 / r6073737;
        double r6073739 = r6073735 * r6073738;
        double r6073740 = 1.3340810790915187e+89;
        bool r6073741 = r6073731 <= r6073740;
        double r6073742 = d;
        double r6073743 = D;
        double r6073744 = r6073742 / r6073743;
        double r6073745 = r6073744 * r6073736;
        double r6073746 = r6073745 * r6073744;
        double r6073747 = h;
        double r6073748 = r6073731 * r6073747;
        double r6073749 = r6073746 / r6073748;
        double r6073750 = M;
        double r6073751 = r6073749 - r6073750;
        double r6073752 = sqrt(r6073751);
        double r6073753 = r6073749 + r6073750;
        double r6073754 = sqrt(r6073753);
        double r6073755 = r6073752 * r6073754;
        double r6073756 = r6073749 + r6073755;
        double r6073757 = r6073756 / r6073731;
        double r6073758 = r6073757 * r6073738;
        double r6073759 = 1.350659423435845e+163;
        bool r6073760 = r6073731 <= r6073759;
        double r6073761 = 1.0915833705181084e+193;
        bool r6073762 = r6073731 <= r6073761;
        double r6073763 = r6073731 / r6073744;
        double r6073764 = r6073736 / r6073763;
        double r6073765 = r6073744 / r6073747;
        double r6073766 = r6073764 * r6073765;
        double r6073767 = r6073766 * r6073766;
        double r6073768 = r6073750 * r6073750;
        double r6073769 = r6073767 - r6073768;
        double r6073770 = sqrt(r6073769);
        double r6073771 = r6073770 * r6073770;
        double r6073772 = r6073770 * r6073771;
        double r6073773 = cbrt(r6073772);
        double r6073774 = r6073773 * r6073769;
        double r6073775 = r6073766 * r6073767;
        double r6073776 = r6073774 + r6073775;
        double r6073777 = r6073766 - r6073770;
        double r6073778 = r6073777 * r6073766;
        double r6073779 = r6073778 + r6073769;
        double r6073780 = r6073776 / r6073779;
        double r6073781 = r6073780 / r6073731;
        double r6073782 = r6073738 * r6073781;
        double r6073783 = r6073762 ? r6073782 : r6073739;
        double r6073784 = r6073760 ? r6073739 : r6073783;
        double r6073785 = r6073741 ? r6073758 : r6073784;
        double r6073786 = r6073733 ? r6073739 : r6073785;
        return r6073786;
}

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

Derivation

1. Split input into 3 regimes

2. if w < 5.1060876778360224e+44 or 1.3340810790915187e+89 < w < 1.350659423435845e+163 or 1.0915833705181084e+193 < w

   1. Initial program 58.3

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

   2. Simplified 52.3

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

   4. Applied flip3-+ 55.6

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

   5. Simplified 56.2

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

   6. Simplified 54.3

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

   7. Taylor expanded around inf 33.9

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

   if 5.1060876778360224e+44 < w < 1.3340810790915187e+89

   1. Initial program 55.2

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

   2. Simplified 47.7

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

   4. Applied sqrt-prod 48.4

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

   if 1.350659423435845e+163 < w < 1.0915833705181084e+193

   1. Initial program 58.1

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

   2. Simplified 50.2

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

   4. Applied flip3-+ 52.5

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

   5. Simplified 53.1

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

   6. Simplified 48.8

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

   8. Applied add-cbrt-cube 50.1

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

4. Final simplification 34.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \le 5.1060876778360224 \cdot 10^{+44}:\\ \;\;\;\;\frac{0}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;w \le 1.3340810790915187 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} + \sqrt{\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} - M} \cdot \sqrt{\frac{\left(\frac{d}{D} \cdot c0\right) \cdot \frac{d}{D}}{w \cdot h} + M}}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;w \le 1.350659423435845 \cdot 10^{+163}:\\ \;\;\;\;\frac{0}{w} \cdot \frac{c0}{2}\\ \mathbf{elif}\;w \le 1.0915833705181084 \cdot 10^{+193}:\\ \;\;\;\;\frac{c0}{2} \cdot \frac{\frac{\sqrt[3]{\sqrt{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M} \cdot \left(\sqrt{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M} \cdot \sqrt{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M}\right)} \cdot \left(\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M\right) + \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right)\right)}{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h} - \sqrt{\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) + \left(\left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) \cdot \left(\frac{c0}{\frac{w}{\frac{d}{D}}} \cdot \frac{\frac{d}{D}}{h}\right) - M \cdot M\right)}}{w}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{w} \cdot \frac{c0}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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))))))