Henrywood and Agarwal, Equation (12)

Average Error: 25.6 → 8.9

Time: 2.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\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) = -\infty:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} - \frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left(\frac{\left(M \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right) \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}}{2} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}{\frac{2}{\frac{D}{d} \cdot \frac{M}{2}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\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 -8.712621141433588 \cdot 10^{-39}:\\ \;\;\;\;\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)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} - \frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left(\frac{\left(M \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right) \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}}{2} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}{\frac{2}{\frac{D}{d} \cdot \frac{M}{2}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r9741402 = d;
        double r9741403 = h;
        double r9741404 = r9741402 / r9741403;
        double r9741405 = 1.0;
        double r9741406 = 2.0;
        double r9741407 = r9741405 / r9741406;
        double r9741408 = pow(r9741404, r9741407);
        double r9741409 = l;
        double r9741410 = r9741402 / r9741409;
        double r9741411 = pow(r9741410, r9741407);
        double r9741412 = r9741408 * r9741411;
        double r9741413 = M;
        double r9741414 = D;
        double r9741415 = r9741413 * r9741414;
        double r9741416 = r9741406 * r9741402;
        double r9741417 = r9741415 / r9741416;
        double r9741418 = pow(r9741417, r9741406);
        double r9741419 = r9741407 * r9741418;
        double r9741420 = r9741403 / r9741409;
        double r9741421 = r9741419 * r9741420;
        double r9741422 = r9741405 - r9741421;
        double r9741423 = r9741412 * r9741422;
        return r9741423;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r9741424 = 1.0;
        double r9741425 = h;
        double r9741426 = l;
        double r9741427 = r9741425 / r9741426;
        double r9741428 = M;
        double r9741429 = D;
        double r9741430 = r9741428 * r9741429;
        double r9741431 = 2.0;
        double r9741432 = d;
        double r9741433 = r9741431 * r9741432;
        double r9741434 = r9741430 / r9741433;
        double r9741435 = pow(r9741434, r9741431);
        double r9741436 = 0.5;
        double r9741437 = r9741435 * r9741436;
        double r9741438 = r9741427 * r9741437;
        double r9741439 = r9741424 - r9741438;
        double r9741440 = r9741432 / r9741426;
        double r9741441 = pow(r9741440, r9741436);
        double r9741442 = r9741432 / r9741425;
        double r9741443 = pow(r9741442, r9741436);
        double r9741444 = r9741441 * r9741443;
        double r9741445 = r9741439 * r9741444;
        double r9741446 = -inf.0;
        bool r9741447 = r9741445 <= r9741446;
        double r9741448 = cbrt(r9741432);
        double r9741449 = cbrt(r9741426);
        double r9741450 = r9741448 / r9741449;
        double r9741451 = fabs(r9741450);
        double r9741452 = sqrt(r9741450);
        double r9741453 = r9741451 * r9741452;
        double r9741454 = r9741450 * r9741450;
        double r9741455 = sqrt(r9741454);
        double r9741456 = cbrt(r9741429);
        double r9741457 = r9741456 / r9741448;
        double r9741458 = r9741428 * r9741457;
        double r9741459 = r9741458 * r9741457;
        double r9741460 = r9741459 / r9741431;
        double r9741461 = r9741460 * r9741457;
        double r9741462 = r9741455 * r9741461;
        double r9741463 = cbrt(r9741425);
        double r9741464 = r9741449 / r9741463;
        double r9741465 = r9741464 * r9741464;
        double r9741466 = r9741462 / r9741465;
        double r9741467 = r9741429 / r9741432;
        double r9741468 = r9741428 / r9741431;
        double r9741469 = r9741467 * r9741468;
        double r9741470 = r9741431 / r9741469;
        double r9741471 = r9741452 / r9741470;
        double r9741472 = r9741471 / r9741464;
        double r9741473 = r9741466 * r9741472;
        double r9741474 = r9741453 - r9741473;
        double r9741475 = r9741448 / r9741463;
        double r9741476 = sqrt(r9741475);
        double r9741477 = fabs(r9741475);
        double r9741478 = r9741476 * r9741477;
        double r9741479 = r9741474 * r9741478;
        double r9741480 = -8.712621141433588e-39;
        bool r9741481 = r9741445 <= r9741480;
        double r9741482 = r9741481 ? r9741445 : r9741479;
        double r9741483 = r9741447 ? r9741479 : r9741482;
        return r9741483;
}

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 2 regimes

2. if (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < -inf.0 or -8.712621141433588e-39 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

   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 27.4

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

   4. Applied add-cube-cbrt 27.5

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

   5. Applied add-cube-cbrt 27.5

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

   6. Applied times-frac 27.5

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

   7. Applied *-un-lft-identity 27.5

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

   8. Applied times-frac 27.3

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

   9. Applied add-cube-cbrt 27.3

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

   10. Applied add-cube-cbrt 27.3

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

   11. Applied times-frac 27.3

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

   12. Applied sqrt-prod 27.0

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

   13. Applied times-frac 25.8

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

   14. Applied times-frac 22.6

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

   15. Simplified 22.6

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

   17. Applied add-cube-cbrt 22.9

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

   18. Applied add-cube-cbrt 23.0

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

   19. Applied times-frac 23.0

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

   20. Applied sqrt-prod 15.8

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

   21. Simplified 14.8

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

   23. Applied add-cube-cbrt 14.9

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

   24. Applied add-cube-cbrt 15.1

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

   25. Applied times-frac 15.1

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

   26. Applied sqrt-prod 9.7

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

   27. Simplified 9.6

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

   29. Applied add-cube-cbrt 9.5

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

   30. Applied add-cube-cbrt 9.6

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

   31. Applied times-frac 9.6

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

   32. Applied associate-*r* 9.6

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

   33. Simplified 9.6

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

   if -inf.0 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < -8.712621141433588e-39

   1. Initial program 1.0

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

4. Final simplification 8.9

   \[\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) = -\infty:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} - \frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left(\frac{\left(M \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right) \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}}{2} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}{\frac{2}{\frac{D}{d} \cdot \frac{M}{2}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\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 -8.712621141433588 \cdot 10^{-39}:\\ \;\;\;\;\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)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} - \frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left(\frac{\left(M \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right) \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}}{2} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}}{\frac{2}{\frac{D}{d} \cdot \frac{M}{2}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))