Henrywood and Agarwal, Equation (12)

Average Error: 26.6 → 11.7

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \le -2.038789370416465276496488997117639381253 \cdot 10^{-193} \lor \neg \left(M \le 8.150978219323250257332845802976006349903 \cdot 10^{-41}\right):\\ \;\;\;\;\left(\left(1 - \left(\left(h \cdot \frac{\sqrt[3]{{\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\frac{\ell}{\sqrt[3]{{\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{\left(\frac{2}{2}\right)}}}}\right) \cdot {\left(\frac{M}{\frac{d}{D} \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{1}{2}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left({\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{1}{2} \cdot \left(h \cdot \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r252593 = d;
        double r252594 = h;
        double r252595 = r252593 / r252594;
        double r252596 = 1.0;
        double r252597 = 2.0;
        double r252598 = r252596 / r252597;
        double r252599 = pow(r252595, r252598);
        double r252600 = l;
        double r252601 = r252593 / r252600;
        double r252602 = pow(r252601, r252598);
        double r252603 = r252599 * r252602;
        double r252604 = M;
        double r252605 = D;
        double r252606 = r252604 * r252605;
        double r252607 = r252597 * r252593;
        double r252608 = r252606 / r252607;
        double r252609 = pow(r252608, r252597);
        double r252610 = r252598 * r252609;
        double r252611 = r252594 / r252600;
        double r252612 = r252610 * r252611;
        double r252613 = r252596 - r252612;
        double r252614 = r252603 * r252613;
        return r252614;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r252615 = M;
        double r252616 = -2.0387893704164653e-193;
        bool r252617 = r252615 <= r252616;
        double r252618 = 8.15097821932325e-41;
        bool r252619 = r252615 <= r252618;
        double r252620 = !r252619;
        bool r252621 = r252617 || r252620;
        double r252622 = 1.0;
        double r252623 = h;
        double r252624 = d;
        double r252625 = D;
        double r252626 = r252624 / r252625;
        double r252627 = r252615 / r252626;
        double r252628 = 2.0;
        double r252629 = r252627 / r252628;
        double r252630 = 2.0;
        double r252631 = r252628 / r252630;
        double r252632 = pow(r252629, r252631);
        double r252633 = cbrt(r252632);
        double r252634 = r252633 * r252633;
        double r252635 = l;
        double r252636 = r252635 / r252633;
        double r252637 = r252634 / r252636;
        double r252638 = r252623 * r252637;
        double r252639 = r252626 * r252628;
        double r252640 = r252615 / r252639;
        double r252641 = pow(r252640, r252631);
        double r252642 = r252638 * r252641;
        double r252643 = r252622 / r252628;
        double r252644 = r252642 * r252643;
        double r252645 = r252622 - r252644;
        double r252646 = cbrt(r252624);
        double r252647 = r252646 * r252646;
        double r252648 = cbrt(r252635);
        double r252649 = r252648 * r252648;
        double r252650 = r252647 / r252649;
        double r252651 = pow(r252650, r252643);
        double r252652 = r252646 / r252648;
        double r252653 = pow(r252652, r252643);
        double r252654 = r252651 * r252653;
        double r252655 = r252645 * r252654;
        double r252656 = cbrt(r252623);
        double r252657 = r252624 / r252656;
        double r252658 = pow(r252657, r252643);
        double r252659 = 1.0;
        double r252660 = r252659 / r252656;
        double r252661 = r252660 / r252656;
        double r252662 = pow(r252661, r252643);
        double r252663 = r252658 * r252662;
        double r252664 = r252655 * r252663;
        double r252665 = r252628 * r252624;
        double r252666 = r252615 * r252625;
        double r252667 = r252665 / r252666;
        double r252668 = r252659 / r252667;
        double r252669 = pow(r252668, r252628);
        double r252670 = r252669 / r252635;
        double r252671 = r252623 * r252670;
        double r252672 = r252643 * r252671;
        double r252673 = r252622 - r252672;
        double r252674 = r252654 * r252673;
        double r252675 = r252663 * r252674;
        double r252676 = r252621 ? r252664 : r252675;
        return r252676;
}

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 M < -2.0387893704164653e-193 or 8.15097821932325e-41 < M

   1. Initial program 29.0

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

   2. Simplified 28.3

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

   4. Applied add-cube-cbrt 28.5

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

   5. Applied add-cube-cbrt 28.6

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

   6. Applied times-frac 28.6

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

   7. Applied unpow-prod-down 23.4

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

   9. Applied add-cube-cbrt 23.5

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

   10. Applied *-un-lft-identity 23.5

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

   11. Applied times-frac 23.5

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

   12. Applied unpow-prod-down 17.7

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

   13. Simplified 17.7

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

   15. Applied *-un-lft-identity 17.7

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

   16. Applied sqr-pow 17.7

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

   17. Applied times-frac 14.7

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

   18. Applied associate-*l* 12.8

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

   19. Simplified 12.8

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

   21. Applied add-cube-cbrt 12.9

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

   22. Applied associate-/l* 12.9

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

   if -2.0387893704164653e-193 < M < 8.15097821932325e-41

   1. Initial program 23.1

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

   2. Simplified 22.5

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

   4. Applied add-cube-cbrt 22.8

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

   5. Applied add-cube-cbrt 22.9

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

   6. Applied times-frac 22.9

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

   7. Applied unpow-prod-down 17.0

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

   9. Applied add-cube-cbrt 17.0

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

   10. Applied *-un-lft-identity 17.0

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

   11. Applied times-frac 17.1

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

   12. Applied unpow-prod-down 11.2

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

   13. Simplified 11.1

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

   15. Applied clear-num 11.2

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

   16. Simplified 9.8

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

4. Final simplification 11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \le -2.038789370416465276496488997117639381253 \cdot 10^{-193} \lor \neg \left(M \le 8.150978219323250257332845802976006349903 \cdot 10^{-41}\right):\\ \;\;\;\;\left(\left(1 - \left(\left(h \cdot \frac{\sqrt[3]{{\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{\left(\frac{2}{2}\right)}}}{\frac{\ell}{\sqrt[3]{{\left(\frac{\frac{M}{\frac{d}{D}}}{2}\right)}^{\left(\frac{2}{2}\right)}}}}\right) \cdot {\left(\frac{M}{\frac{d}{D} \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{1}{2}\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left({\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{1}{2} \cdot \left(h \cdot \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{2}}{\ell}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))