Henrywood and Agarwal, Equation (12)

Average Error: 25.7 → 13.9

Time: 1.9m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 9.238858072207594 \cdot 10^{-116}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} - \frac{\left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \frac{\frac{M}{d}}{\frac{2}{D}}\right) \cdot \frac{\frac{M}{d}}{\frac{2}{D}}}{2 \cdot \frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r7596497 = d;
        double r7596498 = h;
        double r7596499 = r7596497 / r7596498;
        double r7596500 = 1.0;
        double r7596501 = 2.0;
        double r7596502 = r7596500 / r7596501;
        double r7596503 = pow(r7596499, r7596502);
        double r7596504 = l;
        double r7596505 = r7596497 / r7596504;
        double r7596506 = pow(r7596505, r7596502);
        double r7596507 = r7596503 * r7596506;
        double r7596508 = M;
        double r7596509 = D;
        double r7596510 = r7596508 * r7596509;
        double r7596511 = r7596501 * r7596497;
        double r7596512 = r7596510 / r7596511;
        double r7596513 = pow(r7596512, r7596501);
        double r7596514 = r7596502 * r7596513;
        double r7596515 = r7596498 / r7596504;
        double r7596516 = r7596514 * r7596515;
        double r7596517 = r7596500 - r7596516;
        double r7596518 = r7596507 * r7596517;
        return r7596518;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r7596519 = 1.0;
        double r7596520 = h;
        double r7596521 = l;
        double r7596522 = r7596520 / r7596521;
        double r7596523 = M;
        double r7596524 = D;
        double r7596525 = r7596523 * r7596524;
        double r7596526 = 2.0;
        double r7596527 = d;
        double r7596528 = r7596526 * r7596527;
        double r7596529 = r7596525 / r7596528;
        double r7596530 = pow(r7596529, r7596526);
        double r7596531 = 0.5;
        double r7596532 = r7596530 * r7596531;
        double r7596533 = r7596522 * r7596532;
        double r7596534 = r7596519 - r7596533;
        double r7596535 = r7596527 / r7596521;
        double r7596536 = pow(r7596535, r7596531);
        double r7596537 = r7596527 / r7596520;
        double r7596538 = pow(r7596537, r7596531);
        double r7596539 = r7596536 * r7596538;
        double r7596540 = r7596534 * r7596539;
        double r7596541 = 9.238858072207594e-116;
        bool r7596542 = r7596540 <= r7596541;
        double r7596543 = cbrt(r7596521);
        double r7596544 = r7596527 / r7596543;
        double r7596545 = sqrt(r7596544);
        double r7596546 = r7596543 * r7596543;
        double r7596547 = r7596519 / r7596546;
        double r7596548 = sqrt(r7596547);
        double r7596549 = r7596545 * r7596548;
        double r7596550 = r7596523 / r7596527;
        double r7596551 = r7596526 / r7596524;
        double r7596552 = r7596550 / r7596551;
        double r7596553 = r7596549 * r7596552;
        double r7596554 = r7596553 * r7596552;
        double r7596555 = r7596521 / r7596520;
        double r7596556 = r7596526 * r7596555;
        double r7596557 = r7596554 / r7596556;
        double r7596558 = r7596549 - r7596557;
        double r7596559 = cbrt(r7596527);
        double r7596560 = cbrt(r7596520);
        double r7596561 = r7596559 / r7596560;
        double r7596562 = sqrt(r7596561);
        double r7596563 = r7596561 * r7596561;
        double r7596564 = sqrt(r7596563);
        double r7596565 = r7596562 * r7596564;
        double r7596566 = r7596558 * r7596565;
        double r7596567 = r7596565 * r7596549;
        double r7596568 = r7596542 ? r7596566 : r7596567;
        return r7596568;
}

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)))) < 9.238858072207594e-116

   1. Initial program 25.0

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

   2. Simplified 25.5

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

   4. Applied add-cube-cbrt 25.7

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

   5. Applied add-cube-cbrt 25.8

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

   6. Applied times-frac 25.8

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

   7. Applied sqrt-prod 23.2

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

   8. Simplified 23.2

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

   10. Applied add-cube-cbrt 23.3

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

   11. Applied *-un-lft-identity 23.3

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

   12. Applied times-frac 23.3

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

   13. Applied sqrt-prod 17.7

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

   15. Applied add-cube-cbrt 17.7

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

   16. Applied *-un-lft-identity 17.7

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

   17. Applied times-frac 17.7

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

   18. Applied sqrt-prod 16.5

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

   20. Applied associate-*l* 14.1

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

   if 9.238858072207594e-116 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

   1. Initial program 26.1

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

   2. Simplified 26.0

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

   4. Applied add-cube-cbrt 26.3

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

   5. Applied add-cube-cbrt 26.4

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

   6. Applied times-frac 26.4

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

   7. Applied sqrt-prod 19.4

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

   8. Simplified 19.4

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

   10. Applied add-cube-cbrt 19.5

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

   11. Applied *-un-lft-identity 19.5

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

   12. Applied times-frac 19.5

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

   13. Applied sqrt-prod 19.5

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

   14. Taylor expanded around 0 13.8

       \[\leadsto \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}} - \color{blue}{0}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 9.238858072207594 \cdot 10^{-116}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} - \frac{\left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \frac{\frac{M}{d}}{\frac{2}{D}}\right) \cdot \frac{\frac{M}{d}}{\frac{2}{D}}}{2 \cdot \frac{\ell}{h}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(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)))))