Henrywood and Agarwal, Equation (12)

Average Error: 25.6 → 13.2

Time: 1.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot D \le -2.1021038359939756 \cdot 10^{+79}:\\ \;\;\;\;\left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 1.3396067446246392 \cdot 10^{-79}:\\ \;\;\;\;\left(1 - \frac{\sqrt[3]{\sqrt[3]{h}}}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(\sqrt[3]{\sqrt[3]{h}} \cdot \frac{M \cdot \sqrt[3]{h}}{\frac{d}{\frac{D}{2}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{h}} \cdot \frac{M \cdot \sqrt[3]{h}}{\frac{d}{\frac{D}{2}}}\right)\right)\right)\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 1.5623144389475482 \cdot 10^{+166}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\frac{d}{\sqrt[3]{\ell}}}} \cdot \left(\sqrt[3]{\sqrt{\frac{d}{\sqrt[3]{\ell}}}} \cdot \sqrt[3]{\sqrt{\frac{d}{\sqrt[3]{\ell}}}}\right)\right) \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(\left(\frac{M}{2 \cdot d} \cdot D\right) \cdot \left(\left(\frac{M}{2 \cdot d} \cdot D\right) \cdot \frac{h}{\ell}\right)\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r8731212 = d;
        double r8731213 = h;
        double r8731214 = r8731212 / r8731213;
        double r8731215 = 1.0;
        double r8731216 = 2.0;
        double r8731217 = r8731215 / r8731216;
        double r8731218 = pow(r8731214, r8731217);
        double r8731219 = l;
        double r8731220 = r8731212 / r8731219;
        double r8731221 = pow(r8731220, r8731217);
        double r8731222 = r8731218 * r8731221;
        double r8731223 = M;
        double r8731224 = D;
        double r8731225 = r8731223 * r8731224;
        double r8731226 = r8731216 * r8731212;
        double r8731227 = r8731225 / r8731226;
        double r8731228 = pow(r8731227, r8731216);
        double r8731229 = r8731217 * r8731228;
        double r8731230 = r8731213 / r8731219;
        double r8731231 = r8731229 * r8731230;
        double r8731232 = r8731215 - r8731231;
        double r8731233 = r8731222 * r8731232;
        return r8731233;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r8731234 = M;
        double r8731235 = D;
        double r8731236 = r8731234 * r8731235;
        double r8731237 = -2.1021038359939756e+79;
        bool r8731238 = r8731236 <= r8731237;
        double r8731239 = 1.0;
        double r8731240 = 0.5;
        double r8731241 = h;
        double r8731242 = cbrt(r8731241);
        double r8731243 = l;
        double r8731244 = cbrt(r8731243);
        double r8731245 = r8731242 / r8731244;
        double r8731246 = 2.0;
        double r8731247 = d;
        double r8731248 = r8731235 / r8731247;
        double r8731249 = r8731246 / r8731248;
        double r8731250 = r8731234 / r8731249;
        double r8731251 = r8731245 * r8731250;
        double r8731252 = r8731251 * r8731251;
        double r8731253 = r8731240 * r8731252;
        double r8731254 = r8731253 * r8731245;
        double r8731255 = r8731239 - r8731254;
        double r8731256 = r8731247 / r8731243;
        double r8731257 = pow(r8731256, r8731240);
        double r8731258 = cbrt(r8731247);
        double r8731259 = r8731258 / r8731242;
        double r8731260 = fabs(r8731259);
        double r8731261 = sqrt(r8731259);
        double r8731262 = r8731260 * r8731261;
        double r8731263 = r8731257 * r8731262;
        double r8731264 = r8731255 * r8731263;
        double r8731265 = 1.3396067446246392e-79;
        bool r8731266 = r8731236 <= r8731265;
        double r8731267 = cbrt(r8731242);
        double r8731268 = r8731267 / r8731243;
        double r8731269 = r8731234 * r8731242;
        double r8731270 = r8731235 / r8731246;
        double r8731271 = r8731247 / r8731270;
        double r8731272 = r8731269 / r8731271;
        double r8731273 = r8731267 * r8731272;
        double r8731274 = r8731273 * r8731273;
        double r8731275 = r8731240 * r8731274;
        double r8731276 = r8731268 * r8731275;
        double r8731277 = r8731239 - r8731276;
        double r8731278 = r8731247 / r8731244;
        double r8731279 = sqrt(r8731278);
        double r8731280 = r8731239 / r8731244;
        double r8731281 = r8731280 / r8731244;
        double r8731282 = sqrt(r8731281);
        double r8731283 = r8731279 * r8731282;
        double r8731284 = r8731283 * r8731262;
        double r8731285 = r8731277 * r8731284;
        double r8731286 = 1.5623144389475482e+166;
        bool r8731287 = r8731236 <= r8731286;
        double r8731288 = cbrt(r8731279);
        double r8731289 = r8731288 * r8731288;
        double r8731290 = r8731288 * r8731289;
        double r8731291 = r8731290 * r8731282;
        double r8731292 = r8731262 * r8731291;
        double r8731293 = r8731241 / r8731243;
        double r8731294 = r8731246 * r8731247;
        double r8731295 = r8731236 / r8731294;
        double r8731296 = pow(r8731295, r8731246);
        double r8731297 = r8731296 * r8731240;
        double r8731298 = r8731293 * r8731297;
        double r8731299 = r8731239 - r8731298;
        double r8731300 = r8731292 * r8731299;
        double r8731301 = r8731234 / r8731294;
        double r8731302 = r8731301 * r8731235;
        double r8731303 = r8731302 * r8731293;
        double r8731304 = r8731302 * r8731303;
        double r8731305 = r8731240 * r8731304;
        double r8731306 = r8731239 - r8731305;
        double r8731307 = r8731247 / r8731241;
        double r8731308 = pow(r8731307, r8731240);
        double r8731309 = r8731257 * r8731308;
        double r8731310 = r8731306 * r8731309;
        double r8731311 = r8731287 ? r8731300 : r8731310;
        double r8731312 = r8731266 ? r8731285 : r8731311;
        double r8731313 = r8731238 ? r8731264 : r8731312;
        return r8731313;
}

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) < -2.1021038359939756e+79

   1. Initial program 35.4

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

   3. Applied add-cube-cbrt 35.6

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

   4. Applied add-cube-cbrt 35.7

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

   5. Applied times-frac 35.7

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

   6. Applied unpow-prod-down 29.4

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

   7. Simplified 28.0

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

   8. Simplified 28.0

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

   10. Applied add-cube-cbrt 28.0

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

   11. Applied add-cube-cbrt 28.0

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

   12. Applied times-frac 28.0

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

   13. Applied associate-*r* 27.7

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

   14. Simplified 18.3

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

   if -2.1021038359939756e+79 < (* M D) < 1.3396067446246392e-79

   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 add-cube-cbrt 22.6

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

   4. Applied add-cube-cbrt 22.7

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

   5. Applied times-frac 22.7

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

   6. Applied unpow-prod-down 17.9

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

   7. Simplified 17.4

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

   8. Simplified 17.4

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

   10. Applied add-cube-cbrt 17.5

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

   11. Applied *-un-lft-identity 17.5

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

   12. Applied times-frac 17.4

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

   13. Applied unpow-prod-down 13.0

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

   14. Simplified 13.0

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

   15. Simplified 13.0

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

   17. Applied *-un-lft-identity 13.0

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

   18. Applied add-cube-cbrt 13.0

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

   19. Applied times-frac 13.0

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

   20. Applied associate-*r* 9.8

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

   21. Simplified 9.7

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

   23. Applied *-un-lft-identity 9.7

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

   24. Applied add-cube-cbrt 9.8

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

   25. Applied times-frac 9.8

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

   26. Applied associate-*r* 9.1

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

   27. Simplified 9.2

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

   if 1.3396067446246392e-79 < (* M D) < 1.5623144389475482e+166

   1. Initial program 29.5

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

   3. Applied add-cube-cbrt 29.7

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

   4. Applied add-cube-cbrt 29.8

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

   5. Applied times-frac 29.8

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

   6. Applied unpow-prod-down 24.7

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

   7. Simplified 24.1

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

   8. Simplified 24.1

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

   10. Applied add-cube-cbrt 24.2

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

   11. Applied *-un-lft-identity 24.2

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

   12. Applied times-frac 24.2

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

   13. Applied unpow-prod-down 20.4

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

   14. Simplified 20.4

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

   15. Simplified 20.4

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

   17. Applied add-cube-cbrt 20.7

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

   if 1.5623144389475482e+166 < (* M D)

   1. Initial program 40.3

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

   3. Applied associate-*l* 40.3

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

   4. Simplified 33.0

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \le -2.1021038359939756 \cdot 10^{+79}:\\ \;\;\;\;\left(1 - \left(\frac{1}{2} \cdot \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 1.3396067446246392 \cdot 10^{-79}:\\ \;\;\;\;\left(1 - \frac{\sqrt[3]{\sqrt[3]{h}}}{\ell} \cdot \left(\frac{1}{2} \cdot \left(\left(\sqrt[3]{\sqrt[3]{h}} \cdot \frac{M \cdot \sqrt[3]{h}}{\frac{d}{\frac{D}{2}}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{h}} \cdot \frac{M \cdot \sqrt[3]{h}}{\frac{d}{\frac{D}{2}}}\right)\right)\right)\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 1.5623144389475482 \cdot 10^{+166}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left(\sqrt[3]{\sqrt{\frac{d}{\sqrt[3]{\ell}}}} \cdot \left(\sqrt[3]{\sqrt{\frac{d}{\sqrt[3]{\ell}}}} \cdot \sqrt[3]{\sqrt{\frac{d}{\sqrt[3]{\ell}}}}\right)\right) \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(\left(\frac{M}{2 \cdot d} \cdot D\right) \cdot \left(\left(\frac{M}{2 \cdot d} \cdot D\right) \cdot \frac{h}{\ell}\right)\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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)))))