Henrywood and Agarwal, Equation (12)

Average Error: 25.8 → 15.6

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le 1.2924863083363923 \cdot 10^{-78}:\\ \;\;\;\;\left(\sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{h}} \cdot \sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right)\right) \cdot \left(1 - \frac{\frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}{2} \cdot h}{\ell}\right)} \cdot \sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{h}} \cdot \sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right)\right) \cdot \left(1 - \frac{\frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}{2} \cdot h}{\ell}\right)}\right) \cdot \sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{h}} \cdot \sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right)\right) \cdot \left(1 - \frac{\frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}{2} \cdot h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{h}{\ell} \cdot \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}{2}\right) \cdot \left(\left(\sqrt{\sqrt[3]{d}} \cdot \left|\sqrt[3]{d}\right|\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right)}{\sqrt{h}}\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r7076077 = d;
        double r7076078 = h;
        double r7076079 = r7076077 / r7076078;
        double r7076080 = 1.0;
        double r7076081 = 2.0;
        double r7076082 = r7076080 / r7076081;
        double r7076083 = pow(r7076079, r7076082);
        double r7076084 = l;
        double r7076085 = r7076077 / r7076084;
        double r7076086 = pow(r7076085, r7076082);
        double r7076087 = r7076083 * r7076086;
        double r7076088 = M;
        double r7076089 = D;
        double r7076090 = r7076088 * r7076089;
        double r7076091 = r7076081 * r7076077;
        double r7076092 = r7076090 / r7076091;
        double r7076093 = pow(r7076092, r7076081);
        double r7076094 = r7076082 * r7076093;
        double r7076095 = r7076078 / r7076084;
        double r7076096 = r7076094 * r7076095;
        double r7076097 = r7076080 - r7076096;
        double r7076098 = r7076087 * r7076097;
        return r7076098;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r7076099 = l;
        double r7076100 = 1.2924863083363923e-78;
        bool r7076101 = r7076099 <= r7076100;
        double r7076102 = d;
        double r7076103 = cbrt(r7076102);
        double r7076104 = cbrt(r7076099);
        double r7076105 = r7076103 / r7076104;
        double r7076106 = r7076105 * r7076105;
        double r7076107 = 0.5;
        double r7076108 = pow(r7076106, r7076107);
        double r7076109 = pow(r7076105, r7076107);
        double r7076110 = r7076108 * r7076109;
        double r7076111 = h;
        double r7076112 = r7076103 / r7076111;
        double r7076113 = sqrt(r7076112);
        double r7076114 = r7076103 * r7076103;
        double r7076115 = sqrt(r7076114);
        double r7076116 = r7076113 * r7076115;
        double r7076117 = r7076110 * r7076116;
        double r7076118 = 1.0;
        double r7076119 = D;
        double r7076120 = r7076119 / r7076102;
        double r7076121 = M;
        double r7076122 = 2.0;
        double r7076123 = r7076121 / r7076122;
        double r7076124 = r7076120 * r7076123;
        double r7076125 = r7076124 * r7076124;
        double r7076126 = r7076125 / r7076122;
        double r7076127 = r7076126 * r7076111;
        double r7076128 = r7076127 / r7076099;
        double r7076129 = r7076118 - r7076128;
        double r7076130 = r7076117 * r7076129;
        double r7076131 = cbrt(r7076130);
        double r7076132 = r7076131 * r7076131;
        double r7076133 = r7076132 * r7076131;
        double r7076134 = r7076111 / r7076099;
        double r7076135 = r7076134 * r7076126;
        double r7076136 = r7076118 - r7076135;
        double r7076137 = sqrt(r7076103);
        double r7076138 = fabs(r7076103);
        double r7076139 = r7076137 * r7076138;
        double r7076140 = sqrt(r7076106);
        double r7076141 = sqrt(r7076105);
        double r7076142 = r7076140 * r7076141;
        double r7076143 = r7076139 * r7076142;
        double r7076144 = r7076136 * r7076143;
        double r7076145 = sqrt(r7076111);
        double r7076146 = r7076144 / r7076145;
        double r7076147 = r7076101 ? r7076133 : r7076146;
        return r7076147;
}

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 l < 1.2924863083363923e-78

   1. Initial program 26.8

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

   3. Applied *-un-lft-identity 26.8

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

   4. Applied add-cube-cbrt 27.1

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

   5. Applied times-frac 27.1

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

   6. Applied unpow-prod-down 23.4

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

   7. Simplified 23.4

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

   8. Simplified 23.4

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

   10. Applied add-cube-cbrt 23.5

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

   11. Applied add-cube-cbrt 23.7

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

   12. Applied times-frac 23.7

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

   13. Applied unpow-prod-down 19.4

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

   14. Simplified 19.4

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

   16. Applied associate-*r/ 14.9

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

   17. Simplified 15.2

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

   19. Applied add-cube-cbrt 15.4

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

   if 1.2924863083363923e-78 < l

   1. Initial program 24.3

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

   3. Applied *-un-lft-identity 24.3

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

   4. Applied add-cube-cbrt 24.6

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

   5. Applied times-frac 24.6

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

   6. Applied unpow-prod-down 19.7

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

   7. Simplified 19.7

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

   8. Simplified 19.7

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

   10. Applied add-cube-cbrt 19.8

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

   11. Applied add-cube-cbrt 20.0

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

   12. Applied times-frac 20.0

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

   13. Applied unpow-prod-down 16.8

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

   14. Simplified 16.8

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

   16. Applied sqrt-div 15.1

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

   17. Applied associate-*r/ 15.1

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

   18. Applied associate-*l/ 15.8

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

   19. Applied associate-*l/ 15.5

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

   20. Simplified 15.8

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

4. Final simplification 15.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 1.2924863083363923 \cdot 10^{-78}:\\ \;\;\;\;\left(\sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{h}} \cdot \sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right)\right) \cdot \left(1 - \frac{\frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}{2} \cdot h}{\ell}\right)} \cdot \sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{h}} \cdot \sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right)\right) \cdot \left(1 - \frac{\frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}{2} \cdot h}{\ell}\right)}\right) \cdot \sqrt[3]{\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{h}} \cdot \sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right)\right) \cdot \left(1 - \frac{\frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}{2} \cdot h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{h}{\ell} \cdot \frac{\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}{2}\right) \cdot \left(\left(\sqrt{\sqrt[3]{d}} \cdot \left|\sqrt[3]{d}\right|\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right)}{\sqrt{h}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))