Henrywood and Agarwal, Equation (12)

Average Error: 26.4 → 13.7

Time: 2.4m

Precision: 64

Math

\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(1.0 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot \frac{1.0}{2.0}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) = -\infty:\\ \;\;\;\;\left(1.0 - \left(h \cdot \left({\left(\frac{1}{\frac{2.0 \cdot d}{M \cdot D}}\right)}^{2.0} \cdot \frac{1.0}{2.0}\right)\right) \cdot \frac{1}{\ell}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right)\\ \mathbf{elif}\;\left(1.0 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot \frac{1.0}{2.0}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \le 0.0:\\ \;\;\;\;\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot \frac{1.0}{2.0}\right)\right)\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1.0 - \frac{\frac{1.0}{2.0} \cdot \left({\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot h\right)}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{1}{\sqrt[3]{\ell}}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r12410040 = d;
        double r12410041 = h;
        double r12410042 = r12410040 / r12410041;
        double r12410043 = 1.0;
        double r12410044 = 2.0;
        double r12410045 = r12410043 / r12410044;
        double r12410046 = pow(r12410042, r12410045);
        double r12410047 = l;
        double r12410048 = r12410040 / r12410047;
        double r12410049 = pow(r12410048, r12410045);
        double r12410050 = r12410046 * r12410049;
        double r12410051 = M;
        double r12410052 = D;
        double r12410053 = r12410051 * r12410052;
        double r12410054 = r12410044 * r12410040;
        double r12410055 = r12410053 / r12410054;
        double r12410056 = pow(r12410055, r12410044);
        double r12410057 = r12410045 * r12410056;
        double r12410058 = r12410041 / r12410047;
        double r12410059 = r12410057 * r12410058;
        double r12410060 = r12410043 - r12410059;
        double r12410061 = r12410050 * r12410060;
        return r12410061;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r12410062 = 1.0;
        double r12410063 = h;
        double r12410064 = l;
        double r12410065 = r12410063 / r12410064;
        double r12410066 = M;
        double r12410067 = D;
        double r12410068 = r12410066 * r12410067;
        double r12410069 = 2.0;
        double r12410070 = d;
        double r12410071 = r12410069 * r12410070;
        double r12410072 = r12410068 / r12410071;
        double r12410073 = pow(r12410072, r12410069);
        double r12410074 = r12410062 / r12410069;
        double r12410075 = r12410073 * r12410074;
        double r12410076 = r12410065 * r12410075;
        double r12410077 = r12410062 - r12410076;
        double r12410078 = r12410070 / r12410064;
        double r12410079 = pow(r12410078, r12410074);
        double r12410080 = r12410070 / r12410063;
        double r12410081 = pow(r12410080, r12410074);
        double r12410082 = r12410079 * r12410081;
        double r12410083 = r12410077 * r12410082;
        double r12410084 = -inf.0;
        bool r12410085 = r12410083 <= r12410084;
        double r12410086 = 1.0;
        double r12410087 = r12410071 / r12410068;
        double r12410088 = r12410086 / r12410087;
        double r12410089 = pow(r12410088, r12410069);
        double r12410090 = r12410089 * r12410074;
        double r12410091 = r12410063 * r12410090;
        double r12410092 = r12410086 / r12410064;
        double r12410093 = r12410091 * r12410092;
        double r12410094 = r12410062 - r12410093;
        double r12410095 = cbrt(r12410070);
        double r12410096 = cbrt(r12410064);
        double r12410097 = r12410095 / r12410096;
        double r12410098 = pow(r12410097, r12410074);
        double r12410099 = r12410097 * r12410097;
        double r12410100 = pow(r12410099, r12410074);
        double r12410101 = r12410098 * r12410100;
        double r12410102 = cbrt(r12410063);
        double r12410103 = r12410095 / r12410102;
        double r12410104 = pow(r12410103, r12410074);
        double r12410105 = r12410103 * r12410103;
        double r12410106 = pow(r12410105, r12410074);
        double r12410107 = r12410104 * r12410106;
        double r12410108 = r12410101 * r12410107;
        double r12410109 = r12410094 * r12410108;
        double r12410110 = 0.0;
        bool r12410111 = r12410083 <= r12410110;
        double r12410112 = r12410101 * r12410077;
        double r12410113 = r12410112 * r12410107;
        double r12410114 = r12410073 * r12410063;
        double r12410115 = r12410074 * r12410114;
        double r12410116 = r12410096 * r12410096;
        double r12410117 = r12410115 / r12410116;
        double r12410118 = r12410086 / r12410096;
        double r12410119 = r12410117 * r12410118;
        double r12410120 = r12410062 - r12410119;
        double r12410121 = r12410120 * r12410108;
        double r12410122 = r12410111 ? r12410113 : r12410121;
        double r12410123 = r12410085 ? r12410109 : r12410122;
        return r12410123;
}

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 3 regimes

2. if (* (* (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)))) < -inf.0

   1. Initial program 64.0

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 64.0

      \[\leadsto \left({\left(\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   4. Applied add-cube-cbrt 64.0

      \[\leadsto \left({\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   5. Applied times-frac 64.0

      \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   6. Applied unpow-prod-down 63.7

      \[\leadsto \left(\color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   7. Simplified 63.7

      \[\leadsto \left(\left(\color{blue}{{\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 63.7

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\left(\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   10. Applied add-cube-cbrt 63.7

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   11. Applied times-frac 63.7

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   12. Applied unpow-prod-down 62.3

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   13. Simplified 62.3

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(\color{blue}{{\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]
   14. Using strategy rm

   15. Applied div-inv 62.3

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}\right)\]

   16. Applied associate-*r* 52.5

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \color{blue}{\left(\left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot h\right) \cdot \frac{1}{\ell}}\right)\]
   17. Using strategy rm

   18. Applied clear-num 52.5

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \left(\left(\frac{1.0}{2.0} \cdot {\color{blue}{\left(\frac{1}{\frac{2.0 \cdot d}{M \cdot D}}\right)}}^{2.0}\right) \cdot h\right) \cdot \frac{1}{\ell}\right)\]

   if -inf.0 < (* (* (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)))) < 0.0

   1. Initial program 18.8

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 18.9

      \[\leadsto \left({\left(\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   4. Applied add-cube-cbrt 19.0

      \[\leadsto \left({\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   5. Applied times-frac 19.0

      \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   6. Applied unpow-prod-down 15.7

      \[\leadsto \left(\color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   7. Simplified 15.7

      \[\leadsto \left(\left(\color{blue}{{\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 15.8

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\left(\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   10. Applied add-cube-cbrt 15.9

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   11. Applied times-frac 15.9

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   12. Applied unpow-prod-down 5.6

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   13. Simplified 5.5

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(\color{blue}{{\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]
   14. Using strategy rm

   15. Applied associate-*l* 3.6

       \[\leadsto \color{blue}{\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\right)}\]

   if 0.0 < (* (* (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))))

   1. Initial program 24.6

      \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 24.9

      \[\leadsto \left({\left(\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   4. Applied add-cube-cbrt 25.0

      \[\leadsto \left({\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   5. Applied times-frac 25.0

      \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   6. Applied unpow-prod-down 18.4

      \[\leadsto \left(\color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   7. Simplified 18.4

      \[\leadsto \left(\left(\color{blue}{{\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 18.5

      \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\left(\frac{d}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   10. Applied add-cube-cbrt 18.7

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   11. Applied times-frac 18.7

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot {\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   12. Applied unpow-prod-down 16.2

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)}\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]

   13. Simplified 16.2

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(\color{blue}{{\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \frac{h}{\ell}\right)\]
   14. Using strategy rm

   15. Applied div-inv 16.2

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}\right)\]

   16. Applied associate-*r* 12.4

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \color{blue}{\left(\left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot h\right) \cdot \frac{1}{\ell}}\right)\]
   17. Using strategy rm

   18. Applied add-cube-cbrt 12.4

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \left(\left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot h\right) \cdot \frac{1}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right)\]

   19. Applied *-un-lft-identity 12.4

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \left(\left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot h\right) \cdot \frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right)\]

   20. Applied times-frac 12.4

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \left(\left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot h\right) \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)}\right)\]

   21. Applied associate-*r* 12.4

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \color{blue}{\left(\left(\left(\frac{1.0}{2.0} \cdot {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0}\right) \cdot h\right) \cdot \frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{1}{\sqrt[3]{\ell}}}\right)\]

   22. Simplified 12.4

       \[\leadsto \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right) \cdot \left(1.0 - \color{blue}{\frac{\frac{1.0}{2.0} \cdot \left({\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot h\right)}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{1}{\sqrt[3]{\ell}}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1.0 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot \frac{1.0}{2.0}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) = -\infty:\\ \;\;\;\;\left(1.0 - \left(h \cdot \left({\left(\frac{1}{\frac{2.0 \cdot d}{M \cdot D}}\right)}^{2.0} \cdot \frac{1.0}{2.0}\right)\right) \cdot \frac{1}{\ell}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right)\\ \mathbf{elif}\;\left(1.0 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot \frac{1.0}{2.0}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \le 0.0:\\ \;\;\;\;\left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left(1.0 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot \frac{1.0}{2.0}\right)\right)\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1.0 - \frac{\frac{1.0}{2.0} \cdot \left({\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot h\right)}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{1}{\sqrt[3]{\ell}}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1.0}{2.0}\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(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)))))