Henrywood and Agarwal, Equation (12)

Average Error: 25.5 → 12.8

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -7.41627020524322 \cdot 10^{-12}:\\ \;\;\;\;\left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot h\right) \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}{\ell \cdot 2}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 0.0:\\ \;\;\;\;\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \frac{-1}{2}\right)\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) + \left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot h\right) \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}{\ell \cdot 2}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r4207100 = d;
        double r4207101 = h;
        double r4207102 = r4207100 / r4207101;
        double r4207103 = 1.0;
        double r4207104 = 2.0;
        double r4207105 = r4207103 / r4207104;
        double r4207106 = pow(r4207102, r4207105);
        double r4207107 = l;
        double r4207108 = r4207100 / r4207107;
        double r4207109 = pow(r4207108, r4207105);
        double r4207110 = r4207106 * r4207109;
        double r4207111 = M;
        double r4207112 = D;
        double r4207113 = r4207111 * r4207112;
        double r4207114 = r4207104 * r4207100;
        double r4207115 = r4207113 / r4207114;
        double r4207116 = pow(r4207115, r4207104);
        double r4207117 = r4207105 * r4207116;
        double r4207118 = r4207101 / r4207107;
        double r4207119 = r4207117 * r4207118;
        double r4207120 = r4207103 - r4207119;
        double r4207121 = r4207110 * r4207120;
        return r4207121;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r4207122 = 1.0;
        double r4207123 = h;
        double r4207124 = l;
        double r4207125 = r4207123 / r4207124;
        double r4207126 = M;
        double r4207127 = D;
        double r4207128 = r4207126 * r4207127;
        double r4207129 = 2.0;
        double r4207130 = d;
        double r4207131 = r4207129 * r4207130;
        double r4207132 = r4207128 / r4207131;
        double r4207133 = pow(r4207132, r4207129);
        double r4207134 = 0.5;
        double r4207135 = r4207133 * r4207134;
        double r4207136 = r4207125 * r4207135;
        double r4207137 = r4207122 - r4207136;
        double r4207138 = r4207130 / r4207124;
        double r4207139 = pow(r4207138, r4207134);
        double r4207140 = r4207130 / r4207123;
        double r4207141 = pow(r4207140, r4207134);
        double r4207142 = r4207139 * r4207141;
        double r4207143 = r4207137 * r4207142;
        double r4207144 = -7.41627020524322e-12;
        bool r4207145 = r4207143 <= r4207144;
        double r4207146 = cbrt(r4207123);
        double r4207147 = r4207130 / r4207146;
        double r4207148 = sqrt(r4207147);
        double r4207149 = r4207146 * r4207146;
        double r4207150 = r4207122 / r4207149;
        double r4207151 = sqrt(r4207150);
        double r4207152 = r4207148 * r4207151;
        double r4207153 = cbrt(r4207124);
        double r4207154 = r4207122 / r4207153;
        double r4207155 = r4207154 / r4207153;
        double r4207156 = sqrt(r4207155);
        double r4207157 = r4207130 / r4207153;
        double r4207158 = sqrt(r4207157);
        double r4207159 = r4207156 * r4207158;
        double r4207160 = r4207152 * r4207159;
        double r4207161 = r4207127 / r4207130;
        double r4207162 = r4207129 / r4207161;
        double r4207163 = r4207126 / r4207162;
        double r4207164 = r4207163 * r4207123;
        double r4207165 = r4207164 * r4207163;
        double r4207166 = r4207124 * r4207129;
        double r4207167 = r4207165 / r4207166;
        double r4207168 = r4207122 - r4207167;
        double r4207169 = r4207160 * r4207168;
        double r4207170 = 0.0;
        bool r4207171 = r4207143 <= r4207170;
        double r4207172 = r4207163 * r4207163;
        double r4207173 = -0.5;
        double r4207174 = r4207172 * r4207173;
        double r4207175 = r4207125 * r4207174;
        double r4207176 = r4207175 * r4207159;
        double r4207177 = r4207152 * r4207176;
        double r4207178 = r4207177 + r4207160;
        double r4207179 = r4207171 ? r4207178 : r4207169;
        double r4207180 = r4207145 ? r4207169 : r4207179;
        return r4207180;
}

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 (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < -7.41627020524322e-12 or 0.0 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

   1. Initial program 24.4

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

   3. Applied add-cube-cbrt 24.7

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

   4. Applied *-un-lft-identity 24.7

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

   5. Applied times-frac 24.7

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

   6. Applied unpow-prod-down 19.7

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

   7. Simplified 19.7

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

   8. Simplified 19.7

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

   10. Applied add-cube-cbrt 19.8

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

   11. Applied *-un-lft-identity 19.8

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

   12. Applied times-frac 19.8

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

   13. Applied unpow-prod-down 18.2

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

   14. Simplified 18.2

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

   15. Simplified 18.2

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

   17. Applied associate-*l/ 18.2

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

   18. Applied frac-times 15.1

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

   19. Simplified 15.5

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

   21. Applied associate-*r* 13.4

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

   if -7.41627020524322e-12 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < 0.0

   1. Initial program 33.5

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

   3. Applied add-cube-cbrt 33.5

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

   4. Applied *-un-lft-identity 33.5

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

   5. Applied times-frac 33.5

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

   6. Applied unpow-prod-down 27.9

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

   7. Simplified 27.9

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

   8. Simplified 27.9

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

   10. Applied add-cube-cbrt 28.0

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

   11. Applied *-un-lft-identity 28.0

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

   12. Applied times-frac 28.0

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

   13. Applied unpow-prod-down 9.2

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

   14. Simplified 9.2

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

   15. Simplified 9.2

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

   17. Applied sub-neg 9.2

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

   18. Applied distribute-lft-in 9.2

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

   19. Simplified 8.4

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -7.41627020524322 \cdot 10^{-12}:\\ \;\;\;\;\left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot h\right) \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}{\ell \cdot 2}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 0.0:\\ \;\;\;\;\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\left(\frac{h}{\ell} \cdot \left(\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \frac{-1}{2}\right)\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) + \left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot h\right) \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}{\ell \cdot 2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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)))))