Henrywood and Agarwal, Equation (12)

Average Error: 26.0 → 13.4

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -1.761651198141392 \cdot 10^{-55}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} - \frac{h \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right)}{\frac{\ell \cdot 2}{\frac{D}{2} \cdot M} \cdot d}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 5.2855475699966966 \cdot 10^{+84}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| - \left(\frac{\left(\frac{M}{d} \cdot \frac{D}{2}\right) \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}{\ell \cdot 2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} - \frac{h \cdot \left(\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|\right) \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right)}{\frac{\ell \cdot 2}{\frac{M}{d} \cdot \frac{D}{2}}}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r8031413 = d;
        double r8031414 = h;
        double r8031415 = r8031413 / r8031414;
        double r8031416 = 1.0;
        double r8031417 = 2.0;
        double r8031418 = r8031416 / r8031417;
        double r8031419 = pow(r8031415, r8031418);
        double r8031420 = l;
        double r8031421 = r8031413 / r8031420;
        double r8031422 = pow(r8031421, r8031418);
        double r8031423 = r8031419 * r8031422;
        double r8031424 = M;
        double r8031425 = D;
        double r8031426 = r8031424 * r8031425;
        double r8031427 = r8031417 * r8031413;
        double r8031428 = r8031426 / r8031427;
        double r8031429 = pow(r8031428, r8031417);
        double r8031430 = r8031418 * r8031429;
        double r8031431 = r8031414 / r8031420;
        double r8031432 = r8031430 * r8031431;
        double r8031433 = r8031416 - r8031432;
        double r8031434 = r8031423 * r8031433;
        return r8031434;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r8031435 = 1.0;
        double r8031436 = h;
        double r8031437 = l;
        double r8031438 = r8031436 / r8031437;
        double r8031439 = M;
        double r8031440 = D;
        double r8031441 = r8031439 * r8031440;
        double r8031442 = 2.0;
        double r8031443 = d;
        double r8031444 = r8031442 * r8031443;
        double r8031445 = r8031441 / r8031444;
        double r8031446 = pow(r8031445, r8031442);
        double r8031447 = 0.5;
        double r8031448 = r8031446 * r8031447;
        double r8031449 = r8031438 * r8031448;
        double r8031450 = r8031435 - r8031449;
        double r8031451 = r8031443 / r8031437;
        double r8031452 = pow(r8031451, r8031447);
        double r8031453 = r8031443 / r8031436;
        double r8031454 = pow(r8031453, r8031447);
        double r8031455 = r8031452 * r8031454;
        double r8031456 = r8031450 * r8031455;
        double r8031457 = -1.761651198141392e-55;
        bool r8031458 = r8031456 <= r8031457;
        double r8031459 = sqrt(r8031451);
        double r8031460 = r8031439 / r8031443;
        double r8031461 = r8031440 / r8031442;
        double r8031462 = r8031460 * r8031461;
        double r8031463 = r8031459 * r8031462;
        double r8031464 = r8031436 * r8031463;
        double r8031465 = r8031437 * r8031442;
        double r8031466 = r8031461 * r8031439;
        double r8031467 = r8031465 / r8031466;
        double r8031468 = r8031467 * r8031443;
        double r8031469 = r8031464 / r8031468;
        double r8031470 = r8031459 - r8031469;
        double r8031471 = cbrt(r8031443);
        double r8031472 = cbrt(r8031436);
        double r8031473 = r8031471 / r8031472;
        double r8031474 = fabs(r8031473);
        double r8031475 = sqrt(r8031473);
        double r8031476 = r8031474 * r8031475;
        double r8031477 = r8031470 * r8031476;
        double r8031478 = 5.2855475699966966e+84;
        bool r8031479 = r8031456 <= r8031478;
        double r8031480 = cbrt(r8031437);
        double r8031481 = r8031471 / r8031480;
        double r8031482 = sqrt(r8031481);
        double r8031483 = fabs(r8031481);
        double r8031484 = r8031482 * r8031483;
        double r8031485 = r8031462 * r8031462;
        double r8031486 = r8031485 / r8031465;
        double r8031487 = r8031486 * r8031459;
        double r8031488 = r8031487 * r8031436;
        double r8031489 = r8031484 - r8031488;
        double r8031490 = sqrt(r8031453);
        double r8031491 = r8031489 * r8031490;
        double r8031492 = r8031484 * r8031462;
        double r8031493 = r8031436 * r8031492;
        double r8031494 = r8031465 / r8031462;
        double r8031495 = r8031493 / r8031494;
        double r8031496 = r8031459 - r8031495;
        double r8031497 = r8031476 * r8031496;
        double r8031498 = r8031479 ? r8031491 : r8031497;
        double r8031499 = r8031458 ? r8031477 : r8031498;
        return r8031499;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 3 regimes

2. if (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < -1.761651198141392e-55

   1. Initial program 27.8

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

   2. Simplified 30.9

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

   4. Applied add-cube-cbrt 31.0

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

   5. Applied add-cube-cbrt 31.1

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

   6. Applied times-frac 31.1

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

   7. Applied sqrt-prod 30.5

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

   8. Simplified 30.5

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

   10. Applied associate-/l* 25.0

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

   12. Applied associate-*r/ 21.9

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

   13. Applied associate-*l/ 17.2

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

   15. Applied associate-*l/ 19.3

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

   16. Applied associate-/r/ 19.3

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

   if -1.761651198141392e-55 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < 5.2855475699966966e+84

   1. Initial program 10.6

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

   2. Simplified 11.2

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

   4. Applied add-cube-cbrt 11.5

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

   5. Applied add-cube-cbrt 11.7

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

   6. Applied times-frac 11.7

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

   7. Applied sqrt-prod 6.6

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

   8. Simplified 6.1

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

   if 5.2855475699966966e+84 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

   1. Initial program 40.7

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

   2. Simplified 38.2

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

   4. Applied add-cube-cbrt 38.4

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

   5. Applied add-cube-cbrt 38.5

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

   6. Applied times-frac 38.5

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

   7. Applied sqrt-prod 26.9

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

   8. Simplified 24.8

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

   10. Applied associate-/l* 22.6

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

   12. Applied associate-*r/ 22.2

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

   13. Applied associate-*l/ 21.9

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

   15. Applied add-cube-cbrt 21.9

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

   16. Applied add-cube-cbrt 22.0

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

   17. Applied times-frac 22.0

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

   18. Applied sqrt-prod 18.9

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

   19. Simplified 18.3

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -1.761651198141392 \cdot 10^{-55}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\ell}} - \frac{h \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right)}{\frac{\ell \cdot 2}{\frac{D}{2} \cdot M} \cdot d}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 5.2855475699966966 \cdot 10^{+84}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| - \left(\frac{\left(\frac{M}{d} \cdot \frac{D}{2}\right) \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}{\ell \cdot 2} \cdot \sqrt{\frac{d}{\ell}}\right) \cdot h\right) \cdot \sqrt{\frac{d}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{d}{\ell}} - \frac{h \cdot \left(\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|\right) \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right)}{\frac{\ell \cdot 2}{\frac{M}{d} \cdot \frac{D}{2}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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)))))