Henrywood and Agarwal, Equation (12)

Average Error: 25.8 → 16.3

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot D \le -467399.44507333206:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) - \frac{h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}{2} \cdot \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}{\ell}\\ \mathbf{elif}\;M \cdot D \le 1.697612410501425 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) - \frac{\left(\left(h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right)\right)}{2 \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) - \frac{\left(h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\left(\sqrt{\frac{d}{h}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right)}{2 \cdot \ell}\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r6925420 = d;
        double r6925421 = h;
        double r6925422 = r6925420 / r6925421;
        double r6925423 = 1.0;
        double r6925424 = 2.0;
        double r6925425 = r6925423 / r6925424;
        double r6925426 = pow(r6925422, r6925425);
        double r6925427 = l;
        double r6925428 = r6925420 / r6925427;
        double r6925429 = pow(r6925428, r6925425);
        double r6925430 = r6925426 * r6925429;
        double r6925431 = M;
        double r6925432 = D;
        double r6925433 = r6925431 * r6925432;
        double r6925434 = r6925424 * r6925420;
        double r6925435 = r6925433 / r6925434;
        double r6925436 = pow(r6925435, r6925424);
        double r6925437 = r6925425 * r6925436;
        double r6925438 = r6925421 / r6925427;
        double r6925439 = r6925437 * r6925438;
        double r6925440 = r6925423 - r6925439;
        double r6925441 = r6925430 * r6925440;
        return r6925441;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r6925442 = M;
        double r6925443 = D;
        double r6925444 = r6925442 * r6925443;
        double r6925445 = -467399.44507333206;
        bool r6925446 = r6925444 <= r6925445;
        double r6925447 = d;
        double r6925448 = h;
        double r6925449 = r6925447 / r6925448;
        double r6925450 = sqrt(r6925449);
        double r6925451 = cbrt(r6925447);
        double r6925452 = l;
        double r6925453 = cbrt(r6925452);
        double r6925454 = r6925451 / r6925453;
        double r6925455 = fabs(r6925454);
        double r6925456 = sqrt(r6925454);
        double r6925457 = r6925455 * r6925456;
        double r6925458 = r6925450 * r6925457;
        double r6925459 = 2.0;
        double r6925460 = r6925443 / r6925447;
        double r6925461 = r6925459 / r6925460;
        double r6925462 = r6925442 / r6925461;
        double r6925463 = r6925448 * r6925462;
        double r6925464 = r6925463 / r6925459;
        double r6925465 = r6925458 * r6925462;
        double r6925466 = r6925465 / r6925452;
        double r6925467 = r6925464 * r6925466;
        double r6925468 = r6925458 - r6925467;
        double r6925469 = 1.697612410501425e-175;
        bool r6925470 = r6925444 <= r6925469;
        double r6925471 = r6925463 * r6925462;
        double r6925472 = 1.0;
        double r6925473 = cbrt(r6925448);
        double r6925474 = r6925473 * r6925473;
        double r6925475 = r6925472 / r6925474;
        double r6925476 = sqrt(r6925475);
        double r6925477 = r6925447 / r6925473;
        double r6925478 = sqrt(r6925477);
        double r6925479 = r6925476 * r6925478;
        double r6925480 = r6925457 * r6925479;
        double r6925481 = r6925471 * r6925480;
        double r6925482 = r6925459 * r6925452;
        double r6925483 = r6925481 / r6925482;
        double r6925484 = r6925458 - r6925483;
        double r6925485 = r6925450 * r6925462;
        double r6925486 = r6925485 * r6925457;
        double r6925487 = r6925463 * r6925486;
        double r6925488 = r6925487 / r6925482;
        double r6925489 = r6925458 - r6925488;
        double r6925490 = r6925470 ? r6925484 : r6925489;
        double r6925491 = r6925446 ? r6925468 : r6925490;
        return r6925491;
}

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 (* M D) < -467399.44507333206

   1. Initial program 33.5

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

   2. Simplified 33.7

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

   4. Applied add-cube-cbrt 33.8

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

   5. Applied add-cube-cbrt 33.9

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

   6. Applied times-frac 33.9

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

   7. Applied sqrt-prod 33.2

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

   8. Simplified 33.1

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

   10. Applied add-cube-cbrt 33.2

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

   11. Applied add-cube-cbrt 33.2

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

   12. Applied times-frac 33.2

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

   13. Applied sqrt-prod 31.4

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

   14. Simplified 31.3

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

   16. Applied associate-*r* 26.9

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

   18. Applied times-frac 26.1

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

   if -467399.44507333206 < (* M D) < 1.697612410501425e-175

   1. Initial program 21.9

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

   2. Simplified 19.5

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

   4. Applied add-cube-cbrt 19.8

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

   5. Applied add-cube-cbrt 19.9

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

   6. Applied times-frac 19.9

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

   7. Applied sqrt-prod 16.6

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

   8. Simplified 16.2

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

   10. Applied add-cube-cbrt 16.3

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

   11. Applied add-cube-cbrt 16.3

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

   12. Applied times-frac 16.3

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

   13. Applied sqrt-prod 12.2

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

   14. Simplified 11.8

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

   16. Applied add-cube-cbrt 11.8

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

   17. Applied *-un-lft-identity 11.8

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

   18. Applied times-frac 11.8

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

   19. Applied sqrt-prod 10.6

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

   if 1.697612410501425e-175 < (* M D)

   1. Initial program 28.7

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

   2. Simplified 27.6

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

   4. Applied add-cube-cbrt 27.8

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

   5. Applied add-cube-cbrt 27.9

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

   6. Applied times-frac 27.9

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

   7. Applied sqrt-prod 26.9

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

   8. Simplified 26.7

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

   10. Applied add-cube-cbrt 26.8

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

   11. Applied add-cube-cbrt 26.8

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

   12. Applied times-frac 26.8

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

   13. Applied sqrt-prod 23.7

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

   14. Simplified 23.3

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

   16. Applied associate-*r* 21.1

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

   18. Applied associate-*l* 21.0

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

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \le -467399.44507333206:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) - \frac{h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}{2} \cdot \frac{\left(\sqrt{\frac{d}{h}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right) \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}{\ell}\\ \mathbf{elif}\;M \cdot D \le 1.697612410501425 \cdot 10^{-175}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) - \frac{\left(\left(h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right)\right)}{2 \cdot \ell}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) - \frac{\left(h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\left(\sqrt{\frac{d}{h}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right)}{2 \cdot \ell}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(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)))))