Henrywood and Agarwal, Equation (12)

Average Error: 25.6 → 16.1

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot D \le -7.879313011708416 \cdot 10^{+37}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 9.498414440691377 \cdot 10^{-42}:\\ \;\;\;\;\left(1 - \frac{\frac{h}{\sqrt[3]{\ell}} \cdot \left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{1}{2} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(D \cdot \frac{M}{2}\right)\right)\right)\right)}{d}\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 3.662838660216526 \cdot 10^{+105}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\left(D \cdot \frac{M}{2}\right) \cdot \left(D \cdot \frac{M}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\left(\left(\sqrt[3]{\ell} \cdot d\right) \cdot \left(\sqrt[3]{\ell} \cdot d\right)\right) \cdot \sqrt[3]{\ell}}\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r3854235 = d;
        double r3854236 = h;
        double r3854237 = r3854235 / r3854236;
        double r3854238 = 1.0;
        double r3854239 = 2.0;
        double r3854240 = r3854238 / r3854239;
        double r3854241 = pow(r3854237, r3854240);
        double r3854242 = l;
        double r3854243 = r3854235 / r3854242;
        double r3854244 = pow(r3854243, r3854240);
        double r3854245 = r3854241 * r3854244;
        double r3854246 = M;
        double r3854247 = D;
        double r3854248 = r3854246 * r3854247;
        double r3854249 = r3854239 * r3854235;
        double r3854250 = r3854248 / r3854249;
        double r3854251 = pow(r3854250, r3854239);
        double r3854252 = r3854240 * r3854251;
        double r3854253 = r3854236 / r3854242;
        double r3854254 = r3854252 * r3854253;
        double r3854255 = r3854238 - r3854254;
        double r3854256 = r3854245 * r3854255;
        return r3854256;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r3854257 = M;
        double r3854258 = D;
        double r3854259 = r3854257 * r3854258;
        double r3854260 = -7.879313011708416e+37;
        bool r3854261 = r3854259 <= r3854260;
        double r3854262 = d;
        double r3854263 = cbrt(r3854262);
        double r3854264 = l;
        double r3854265 = r3854263 / r3854264;
        double r3854266 = sqrt(r3854265);
        double r3854267 = h;
        double r3854268 = cbrt(r3854267);
        double r3854269 = r3854262 / r3854268;
        double r3854270 = sqrt(r3854269);
        double r3854271 = 1.0;
        double r3854272 = r3854268 * r3854268;
        double r3854273 = r3854271 / r3854272;
        double r3854274 = sqrt(r3854273);
        double r3854275 = r3854270 * r3854274;
        double r3854276 = fabs(r3854263);
        double r3854277 = r3854275 * r3854276;
        double r3854278 = r3854266 * r3854277;
        double r3854279 = r3854267 / r3854264;
        double r3854280 = 2.0;
        double r3854281 = r3854257 / r3854280;
        double r3854282 = r3854258 / r3854262;
        double r3854283 = r3854281 * r3854282;
        double r3854284 = r3854283 * r3854283;
        double r3854285 = 0.5;
        double r3854286 = r3854284 * r3854285;
        double r3854287 = r3854279 * r3854286;
        double r3854288 = r3854271 - r3854287;
        double r3854289 = r3854278 * r3854288;
        double r3854290 = 9.498414440691377e-42;
        bool r3854291 = r3854259 <= r3854290;
        double r3854292 = cbrt(r3854264);
        double r3854293 = r3854267 / r3854292;
        double r3854294 = r3854292 * r3854292;
        double r3854295 = r3854271 / r3854294;
        double r3854296 = r3854258 * r3854281;
        double r3854297 = r3854283 * r3854296;
        double r3854298 = r3854285 * r3854297;
        double r3854299 = r3854295 * r3854298;
        double r3854300 = r3854293 * r3854299;
        double r3854301 = r3854300 / r3854262;
        double r3854302 = r3854271 - r3854301;
        double r3854303 = r3854263 * r3854263;
        double r3854304 = sqrt(r3854303);
        double r3854305 = r3854304 * r3854266;
        double r3854306 = r3854275 * r3854305;
        double r3854307 = r3854302 * r3854306;
        double r3854308 = 3.662838660216526e+105;
        bool r3854309 = r3854259 <= r3854308;
        double r3854310 = r3854296 * r3854296;
        double r3854311 = r3854310 * r3854285;
        double r3854312 = r3854311 * r3854267;
        double r3854313 = r3854292 * r3854262;
        double r3854314 = r3854313 * r3854313;
        double r3854315 = r3854314 * r3854292;
        double r3854316 = r3854312 / r3854315;
        double r3854317 = r3854271 - r3854316;
        double r3854318 = r3854317 * r3854306;
        double r3854319 = r3854280 * r3854262;
        double r3854320 = r3854319 / r3854259;
        double r3854321 = r3854271 / r3854320;
        double r3854322 = pow(r3854321, r3854280);
        double r3854323 = r3854285 * r3854322;
        double r3854324 = r3854279 * r3854323;
        double r3854325 = r3854271 - r3854324;
        double r3854326 = r3854262 / r3854267;
        double r3854327 = pow(r3854326, r3854285);
        double r3854328 = r3854327 * r3854305;
        double r3854329 = r3854325 * r3854328;
        double r3854330 = r3854309 ? r3854318 : r3854329;
        double r3854331 = r3854291 ? r3854307 : r3854330;
        double r3854332 = r3854261 ? r3854289 : r3854331;
        return r3854332;
}

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 4 regimes

2. if (* M D) < -7.879313011708416e+37

   1. Initial program 34.2

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

   3. Applied *-un-lft-identity 34.2

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

   4. Applied add-cube-cbrt 34.4

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

   5. Applied times-frac 34.4

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

   6. Applied unpow-prod-down 33.2

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

   7. Simplified 33.2

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

   8. Simplified 33.2

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

   10. Applied add-cube-cbrt 33.3

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

   11. Applied *-un-lft-identity 33.3

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

   12. Applied times-frac 33.3

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

   13. Applied unpow-prod-down 27.4

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

   14. Simplified 27.4

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

   15. Simplified 27.4

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

   17. Applied pow1 27.4

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

   18. Applied pow1 27.4

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

   19. Applied pow1 27.4

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

   20. Applied pow1 27.4

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

   21. Applied pow-prod-down 27.4

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

   22. Applied pow-prod-down 27.4

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

   23. Applied pow-prod-down 27.4

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

   24. Simplified 26.3

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

   if -7.879313011708416e+37 < (* M D) < 9.498414440691377e-42

   1. Initial program 22.3

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

   3. Applied *-un-lft-identity 22.3

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

   4. Applied add-cube-cbrt 22.6

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

   5. Applied times-frac 22.6

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

   6. Applied unpow-prod-down 18.1

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

   7. Simplified 18.1

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

   8. Simplified 18.1

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

   10. Applied add-cube-cbrt 18.2

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

   11. Applied *-un-lft-identity 18.2

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

   12. Applied times-frac 18.2

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

   13. Applied unpow-prod-down 13.9

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

   14. Simplified 13.9

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

   15. Simplified 13.9

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

   17. Applied add-cube-cbrt 13.9

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

   18. Applied *-un-lft-identity 13.9

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

   19. Applied times-frac 13.9

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

   20. Applied associate-*r* 11.1

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

   21. Simplified 11.5

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

   23. Applied associate-*r/ 11.7

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

   24. Applied associate-*l/ 12.1

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

   25. Applied associate-*r/ 12.1

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

   26. Applied associate-*l/ 11.9

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

   27. Applied associate-*l/ 11.3

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

   if 9.498414440691377e-42 < (* M D) < 3.662838660216526e+105

   1. Initial program 29.8

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

   3. Applied *-un-lft-identity 29.8

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

   4. Applied add-cube-cbrt 30.0

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

   5. Applied times-frac 30.0

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

   6. Applied unpow-prod-down 27.4

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

   7. Simplified 27.4

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

   8. Simplified 27.4

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

   10. Applied add-cube-cbrt 27.5

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

   11. Applied *-un-lft-identity 27.5

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

   12. Applied times-frac 27.5

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

   13. Applied unpow-prod-down 22.8

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

   14. Simplified 22.8

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

   15. Simplified 22.8

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

   17. Applied add-cube-cbrt 22.8

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

   18. Applied *-un-lft-identity 22.8

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

   19. Applied times-frac 22.8

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

   20. Applied associate-*r* 22.0

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

   21. Simplified 23.6

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

   23. Applied associate-*r/ 23.6

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

   24. Applied associate-*r/ 22.0

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

   25. Applied frac-times 22.6

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

   26. Applied associate-*r/ 22.6

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

   27. Applied frac-times 20.6

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

   28. Applied frac-times 23.1

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

   29. Simplified 23.1

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

   30. Simplified 21.6

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

   if 3.662838660216526e+105 < (* M D)

   1. Initial program 36.1

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

   3. Applied *-un-lft-identity 36.1

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

   4. Applied add-cube-cbrt 36.3

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

   5. Applied times-frac 36.3

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

   6. Applied unpow-prod-down 35.5

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

   7. Simplified 35.5

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

   8. Simplified 35.5

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

   10. Applied clear-num 35.5

       \[\leadsto \left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot \left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}}^{2}\right) \cdot \frac{h}{\ell}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \le -7.879313011708416 \cdot 10^{+37}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 9.498414440691377 \cdot 10^{-42}:\\ \;\;\;\;\left(1 - \frac{\frac{h}{\sqrt[3]{\ell}} \cdot \left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{1}{2} \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(D \cdot \frac{M}{2}\right)\right)\right)\right)}{d}\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 3.662838660216526 \cdot 10^{+105}:\\ \;\;\;\;\left(1 - \frac{\left(\left(\left(D \cdot \frac{M}{2}\right) \cdot \left(D \cdot \frac{M}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot h}{\left(\left(\sqrt[3]{\ell} \cdot d\right) \cdot \left(\sqrt[3]{\ell} \cdot d\right)\right) \cdot \sqrt[3]{\ell}}\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left(\frac{1}{2} \cdot {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2}\right)\right) \cdot \left({\left(\frac{d}{h}\right)}^{\frac{1}{2}} \cdot \left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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)))))