Henrywood and Agarwal, Equation (12)

Average Error: 26.4 → 13.7

Time: 2.2m

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 r8061168 = d;
        double r8061169 = h;
        double r8061170 = r8061168 / r8061169;
        double r8061171 = 1.0;
        double r8061172 = 2.0;
        double r8061173 = r8061171 / r8061172;
        double r8061174 = pow(r8061170, r8061173);
        double r8061175 = l;
        double r8061176 = r8061168 / r8061175;
        double r8061177 = pow(r8061176, r8061173);
        double r8061178 = r8061174 * r8061177;
        double r8061179 = M;
        double r8061180 = D;
        double r8061181 = r8061179 * r8061180;
        double r8061182 = r8061172 * r8061168;
        double r8061183 = r8061181 / r8061182;
        double r8061184 = pow(r8061183, r8061172);
        double r8061185 = r8061173 * r8061184;
        double r8061186 = r8061169 / r8061175;
        double r8061187 = r8061185 * r8061186;
        double r8061188 = r8061171 - r8061187;
        double r8061189 = r8061178 * r8061188;
        return r8061189;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r8061190 = 1.0;
        double r8061191 = h;
        double r8061192 = l;
        double r8061193 = r8061191 / r8061192;
        double r8061194 = M;
        double r8061195 = D;
        double r8061196 = r8061194 * r8061195;
        double r8061197 = 2.0;
        double r8061198 = d;
        double r8061199 = r8061197 * r8061198;
        double r8061200 = r8061196 / r8061199;
        double r8061201 = pow(r8061200, r8061197);
        double r8061202 = r8061190 / r8061197;
        double r8061203 = r8061201 * r8061202;
        double r8061204 = r8061193 * r8061203;
        double r8061205 = r8061190 - r8061204;
        double r8061206 = r8061198 / r8061192;
        double r8061207 = pow(r8061206, r8061202);
        double r8061208 = r8061198 / r8061191;
        double r8061209 = pow(r8061208, r8061202);
        double r8061210 = r8061207 * r8061209;
        double r8061211 = r8061205 * r8061210;
        double r8061212 = -inf.0;
        bool r8061213 = r8061211 <= r8061212;
        double r8061214 = 1.0;
        double r8061215 = r8061199 / r8061196;
        double r8061216 = r8061214 / r8061215;
        double r8061217 = pow(r8061216, r8061197);
        double r8061218 = r8061217 * r8061202;
        double r8061219 = r8061191 * r8061218;
        double r8061220 = r8061214 / r8061192;
        double r8061221 = r8061219 * r8061220;
        double r8061222 = r8061190 - r8061221;
        double r8061223 = cbrt(r8061198);
        double r8061224 = cbrt(r8061192);
        double r8061225 = r8061223 / r8061224;
        double r8061226 = pow(r8061225, r8061202);
        double r8061227 = r8061225 * r8061225;
        double r8061228 = pow(r8061227, r8061202);
        double r8061229 = r8061226 * r8061228;
        double r8061230 = cbrt(r8061191);
        double r8061231 = r8061223 / r8061230;
        double r8061232 = pow(r8061231, r8061202);
        double r8061233 = r8061231 * r8061231;
        double r8061234 = pow(r8061233, r8061202);
        double r8061235 = r8061232 * r8061234;
        double r8061236 = r8061229 * r8061235;
        double r8061237 = r8061222 * r8061236;
        double r8061238 = 0.0;
        bool r8061239 = r8061211 <= r8061238;
        double r8061240 = r8061229 * r8061205;
        double r8061241 = r8061240 * r8061235;
        double r8061242 = r8061201 * r8061191;
        double r8061243 = r8061202 * r8061242;
        double r8061244 = r8061224 * r8061224;
        double r8061245 = r8061243 / r8061244;
        double r8061246 = r8061214 / r8061224;
        double r8061247 = r8061245 * r8061246;
        double r8061248 = r8061190 - r8061247;
        double r8061249 = r8061248 * r8061236;
        double r8061250 = r8061239 ? r8061241 : r8061249;
        double r8061251 = r8061213 ? r8061237 : r8061250;
        return r8061251;
}

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 +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)))))