Henrywood and Agarwal, Equation (12)

Average Error: 24.9 → 12.5

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;h \le 9.167067702253803 \cdot 10^{-79}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \left(\sqrt[3]{h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}} \cdot \left(\sqrt[3]{h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}} \cdot \sqrt[3]{h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}\right)\right)}{2 \cdot \ell}\right)\\ \mathbf{elif}\;h \le 6.989147569313924 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell}{h} \cdot \frac{2}{\frac{M}{2} \cdot \frac{D}{d}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \left(\sqrt{\sqrt[3]{d}} \cdot \sqrt{d}\right)\right)}{\sqrt{\sqrt[3]{h}} \cdot \left(\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left({\left(\frac{\sqrt{d}}{\sqrt{\ell}}\right)}^{\frac{1}{2}} \cdot {\left(\frac{\sqrt{d}}{\sqrt{\ell}}\right)}^{\frac{1}{2}}\right)\right) \cdot \left(1 - \frac{\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \left(h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)}{2 \cdot \ell}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r8938279 = d;
        double r8938280 = h;
        double r8938281 = r8938279 / r8938280;
        double r8938282 = 1.0;
        double r8938283 = 2.0;
        double r8938284 = r8938282 / r8938283;
        double r8938285 = pow(r8938281, r8938284);
        double r8938286 = l;
        double r8938287 = r8938279 / r8938286;
        double r8938288 = pow(r8938287, r8938284);
        double r8938289 = r8938285 * r8938288;
        double r8938290 = M;
        double r8938291 = D;
        double r8938292 = r8938290 * r8938291;
        double r8938293 = r8938283 * r8938279;
        double r8938294 = r8938292 / r8938293;
        double r8938295 = pow(r8938294, r8938283);
        double r8938296 = r8938284 * r8938295;
        double r8938297 = r8938280 / r8938286;
        double r8938298 = r8938296 * r8938297;
        double r8938299 = r8938282 - r8938298;
        double r8938300 = r8938289 * r8938299;
        return r8938300;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r8938301 = h;
        double r8938302 = 9.167067702253803e-79;
        bool r8938303 = r8938301 <= r8938302;
        double r8938304 = d;
        double r8938305 = cbrt(r8938304);
        double r8938306 = cbrt(r8938301);
        double r8938307 = r8938305 / r8938306;
        double r8938308 = fabs(r8938307);
        double r8938309 = sqrt(r8938307);
        double r8938310 = r8938308 * r8938309;
        double r8938311 = 1.0;
        double r8938312 = l;
        double r8938313 = cbrt(r8938312);
        double r8938314 = r8938313 * r8938313;
        double r8938315 = r8938311 / r8938314;
        double r8938316 = sqrt(r8938315);
        double r8938317 = r8938304 / r8938313;
        double r8938318 = sqrt(r8938317);
        double r8938319 = r8938316 * r8938318;
        double r8938320 = r8938310 * r8938319;
        double r8938321 = M;
        double r8938322 = 2.0;
        double r8938323 = D;
        double r8938324 = r8938323 / r8938304;
        double r8938325 = r8938322 / r8938324;
        double r8938326 = r8938321 / r8938325;
        double r8938327 = r8938301 * r8938326;
        double r8938328 = cbrt(r8938327);
        double r8938329 = r8938328 * r8938328;
        double r8938330 = r8938328 * r8938329;
        double r8938331 = r8938326 * r8938330;
        double r8938332 = r8938322 * r8938312;
        double r8938333 = r8938331 / r8938332;
        double r8938334 = r8938311 - r8938333;
        double r8938335 = r8938320 * r8938334;
        double r8938336 = 6.989147569313924e+191;
        bool r8938337 = r8938301 <= r8938336;
        double r8938338 = r8938321 / r8938322;
        double r8938339 = r8938338 * r8938324;
        double r8938340 = r8938312 / r8938301;
        double r8938341 = r8938322 / r8938339;
        double r8938342 = r8938340 * r8938341;
        double r8938343 = r8938339 / r8938342;
        double r8938344 = r8938311 - r8938343;
        double r8938345 = sqrt(r8938305);
        double r8938346 = sqrt(r8938304);
        double r8938347 = r8938345 * r8938346;
        double r8938348 = r8938308 * r8938347;
        double r8938349 = r8938344 * r8938348;
        double r8938350 = sqrt(r8938306);
        double r8938351 = sqrt(r8938314);
        double r8938352 = sqrt(r8938313);
        double r8938353 = r8938351 * r8938352;
        double r8938354 = r8938350 * r8938353;
        double r8938355 = r8938349 / r8938354;
        double r8938356 = sqrt(r8938312);
        double r8938357 = r8938346 / r8938356;
        double r8938358 = 0.5;
        double r8938359 = pow(r8938357, r8938358);
        double r8938360 = r8938359 * r8938359;
        double r8938361 = r8938310 * r8938360;
        double r8938362 = r8938326 * r8938327;
        double r8938363 = r8938362 / r8938332;
        double r8938364 = r8938311 - r8938363;
        double r8938365 = r8938361 * r8938364;
        double r8938366 = r8938337 ? r8938355 : r8938365;
        double r8938367 = r8938303 ? r8938335 : r8938366;
        return r8938367;
}

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 h < 9.167067702253803e-79

   1. Initial program 26.0

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

   3. Applied add-cube-cbrt 26.3

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

   4. Applied add-cube-cbrt 26.4

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

   5. Applied times-frac 26.4

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

   6. Applied unpow-prod-down 19.6

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

   7. Simplified 18.7

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

   8. Simplified 18.7

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

   10. Applied associate-*l/ 18.7

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

   11. Applied frac-times 18.0

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

   12. Simplified 16.8

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

   14. Applied add-cube-cbrt 16.9

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

   15. Applied *-un-lft-identity 16.9

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

   16. Applied times-frac 16.9

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

   17. Applied unpow-prod-down 12.0

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

   18. Simplified 12.0

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

   19. Simplified 12.0

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

   21. Applied add-cube-cbrt 12.0

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

   if 9.167067702253803e-79 < h < 6.989147569313924e+191

   1. Initial program 19.0

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

   3. Applied add-cube-cbrt 19.3

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

   4. Applied add-cube-cbrt 19.4

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

   5. Applied times-frac 19.4

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

   6. Applied unpow-prod-down 18.0

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

   7. Simplified 18.0

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

   8. Simplified 18.0

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

   10. Applied associate-*l/ 18.0

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

   11. Applied frac-times 17.9

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

   12. Simplified 17.5

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

   14. Applied add-cube-cbrt 17.6

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

   15. Applied *-un-lft-identity 17.6

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

   16. Applied times-frac 17.6

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

   17. Applied unpow-prod-down 12.6

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

   18. Simplified 12.6

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

   19. Simplified 12.6

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

   21. Applied sqrt-div 11.9

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

   22. Applied sqrt-div 11.9

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

   23. Applied frac-times 11.9

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

   24. Applied sqrt-div 11.9

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

   25. Applied associate-*r/ 11.9

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

   26. Applied frac-times 12.3

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

   27. Applied associate-*l/ 12.0

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

   28. Simplified 12.0

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

   if 6.989147569313924e+191 < h

   1. Initial program 31.9

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

   3. Applied add-cube-cbrt 32.1

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

   4. Applied add-cube-cbrt 32.2

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

   5. Applied times-frac 32.2

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

   6. Applied unpow-prod-down 30.2

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

   7. Simplified 30.2

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

   8. Simplified 30.2

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

   10. Applied associate-*l/ 30.2

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

   11. Applied frac-times 24.8

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

   12. Simplified 23.2

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

   14. Applied add-sqr-sqrt 23.2

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

   15. Applied add-sqr-sqrt 23.2

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

   16. Applied times-frac 23.2

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

   17. Applied unpow-prod-down 17.6

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

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \le 9.167067702253803 \cdot 10^{-79}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \frac{\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \left(\sqrt[3]{h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}} \cdot \left(\sqrt[3]{h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}} \cdot \sqrt[3]{h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}\right)\right)}{2 \cdot \ell}\right)\\ \mathbf{elif}\;h \le 6.989147569313924 \cdot 10^{+191}:\\ \;\;\;\;\frac{\left(1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\ell}{h} \cdot \frac{2}{\frac{M}{2} \cdot \frac{D}{d}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \left(\sqrt{\sqrt[3]{d}} \cdot \sqrt{d}\right)\right)}{\sqrt{\sqrt[3]{h}} \cdot \left(\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left({\left(\frac{\sqrt{d}}{\sqrt{\ell}}\right)}^{\frac{1}{2}} \cdot {\left(\frac{\sqrt{d}}{\sqrt{\ell}}\right)}^{\frac{1}{2}}\right)\right) \cdot \left(1 - \frac{\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \left(h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)}{2 \cdot \ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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)))))