Henrywood and Agarwal, Equation (12)

Average Error: 26.4 → 16.8

Time: 55.8s

Precision: 64

Math

\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;M \le 6.910995874866271826055827219966866019596 \cdot 10^{-109}:\\ \;\;\;\;\sqrt[3]{\left(\left(\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(\sqrt[3]{\left(\left(\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \sqrt[3]{\left(\left(\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{h}{\sqrt[3]{\ell}}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r202136 = d;
        double r202137 = h;
        double r202138 = r202136 / r202137;
        double r202139 = 1.0;
        double r202140 = 2.0;
        double r202141 = r202139 / r202140;
        double r202142 = pow(r202138, r202141);
        double r202143 = l;
        double r202144 = r202136 / r202143;
        double r202145 = pow(r202144, r202141);
        double r202146 = r202142 * r202145;
        double r202147 = M;
        double r202148 = D;
        double r202149 = r202147 * r202148;
        double r202150 = r202140 * r202136;
        double r202151 = r202149 / r202150;
        double r202152 = pow(r202151, r202140);
        double r202153 = r202141 * r202152;
        double r202154 = r202137 / r202143;
        double r202155 = r202153 * r202154;
        double r202156 = r202139 - r202155;
        double r202157 = r202146 * r202156;
        return r202157;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r202158 = M;
        double r202159 = 6.910995874866272e-109;
        bool r202160 = r202158 <= r202159;
        double r202161 = 1.0;
        double r202162 = h;
        double r202163 = D;
        double r202164 = r202158 * r202163;
        double r202165 = d;
        double r202166 = 2.0;
        double r202167 = r202165 * r202166;
        double r202168 = r202164 / r202167;
        double r202169 = pow(r202168, r202166);
        double r202170 = r202161 / r202166;
        double r202171 = r202169 * r202170;
        double r202172 = r202162 * r202171;
        double r202173 = l;
        double r202174 = r202172 / r202173;
        double r202175 = r202161 - r202174;
        double r202176 = cbrt(r202165);
        double r202177 = r202176 / r202173;
        double r202178 = pow(r202177, r202170);
        double r202179 = r202175 * r202178;
        double r202180 = r202176 * r202176;
        double r202181 = pow(r202180, r202170);
        double r202182 = r202179 * r202181;
        double r202183 = cbrt(r202162);
        double r202184 = r202176 / r202183;
        double r202185 = pow(r202184, r202170);
        double r202186 = r202184 * r202184;
        double r202187 = pow(r202186, r202170);
        double r202188 = r202185 * r202187;
        double r202189 = r202182 * r202188;
        double r202190 = cbrt(r202189);
        double r202191 = r202190 * r202190;
        double r202192 = r202190 * r202191;
        double r202193 = r202163 / r202165;
        double r202194 = r202158 / r202166;
        double r202195 = r202193 * r202194;
        double r202196 = pow(r202195, r202166);
        double r202197 = r202196 * r202170;
        double r202198 = cbrt(r202173);
        double r202199 = r202198 * r202198;
        double r202200 = r202197 / r202199;
        double r202201 = r202162 / r202198;
        double r202202 = r202200 * r202201;
        double r202203 = r202161 - r202202;
        double r202204 = r202165 / r202173;
        double r202205 = pow(r202204, r202170);
        double r202206 = r202188 * r202205;
        double r202207 = r202203 * r202206;
        double r202208 = r202160 ? r202192 : r202207;
        return r202208;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 2 regimes

2. if M < 6.910995874866272e-109

   1. Initial program 25.7

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 26.0

      \[\leadsto \left({\left(\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   4. Applied add-cube-cbrt 26.1

      \[\leadsto \left({\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   5. Applied times-frac 26.1

      \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   6. Applied unpow-prod-down 20.8

      \[\leadsto \left(\color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   7. Simplified 20.8

      \[\leadsto \left(\left(\color{blue}{{\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 20.8

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\color{blue}{1 \cdot \ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   10. Applied add-cube-cbrt 21.0

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{1 \cdot \ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   11. Applied times-frac 21.0

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1} \cdot \frac{\sqrt[3]{d}}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   12. Applied unpow-prod-down 17.4

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   13. Simplified 17.4

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 17.6

       \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)} \cdot \sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\right) \cdot \sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}}\]

   16. Simplified 18.7

       \[\leadsto \color{blue}{\left(\sqrt[3]{\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \frac{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot h}{\ell}\right)\right)\right)} \cdot \sqrt[3]{\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \frac{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot h}{\ell}\right)\right)\right)}\right)} \cdot \sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)}\]

   17. Simplified 14.9

       \[\leadsto \left(\sqrt[3]{\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \frac{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot h}{\ell}\right)\right)\right)} \cdot \sqrt[3]{\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \frac{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot h}{\ell}\right)\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)} \cdot \left({\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(1 - \frac{\left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot h}{\ell}\right)\right)\right)}}\]

   if 6.910995874866272e-109 < M

   1. Initial program 28.3

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 28.5

      \[\leadsto \left({\left(\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   4. Applied add-cube-cbrt 28.6

      \[\leadsto \left({\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   5. Applied times-frac 28.6

      \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   6. Applied unpow-prod-down 22.9

      \[\leadsto \left(\color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]

   7. Simplified 22.9

      \[\leadsto \left(\left(\color{blue}{{\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 22.9

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right)\]

   10. Applied *-un-lft-identity 22.9

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{\color{blue}{1 \cdot h}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right)\]

   11. Applied times-frac 22.9

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{h}{\sqrt[3]{\ell}}\right)}\right)\]

   12. Applied associate-*r* 22.5

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{h}{\sqrt[3]{\ell}}}\right)\]

   13. Simplified 21.7

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

4. Final simplification 16.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \le 6.910995874866271826055827219966866019596 \cdot 10^{-109}:\\ \;\;\;\;\sqrt[3]{\left(\left(\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \left(\sqrt[3]{\left(\left(\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot \sqrt[3]{\left(\left(\left(1 - \frac{h \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right)}{\ell}\right) \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{{\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{h}{\sqrt[3]{\ell}}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))