Henrywood and Agarwal, Equation (12)

Average Error: 25.9 → 11.5

Time: 1.1m

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 -2.452445300667083 \cdot 10^{+80}:\\ \;\;\;\;\left(1 - \left(\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\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 0.0:\\ \;\;\;\;\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(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}}\right)\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \left(\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\sqrt[3]{\ell} \cdot 2} \cdot \frac{\frac{M}{\frac{2 \cdot d}{D}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r6782633 = d;
        double r6782634 = h;
        double r6782635 = r6782633 / r6782634;
        double r6782636 = 1.0;
        double r6782637 = 2.0;
        double r6782638 = r6782636 / r6782637;
        double r6782639 = pow(r6782635, r6782638);
        double r6782640 = l;
        double r6782641 = r6782633 / r6782640;
        double r6782642 = pow(r6782641, r6782638);
        double r6782643 = r6782639 * r6782642;
        double r6782644 = M;
        double r6782645 = D;
        double r6782646 = r6782644 * r6782645;
        double r6782647 = r6782637 * r6782633;
        double r6782648 = r6782646 / r6782647;
        double r6782649 = pow(r6782648, r6782637);
        double r6782650 = r6782638 * r6782649;
        double r6782651 = r6782634 / r6782640;
        double r6782652 = r6782650 * r6782651;
        double r6782653 = r6782636 - r6782652;
        double r6782654 = r6782643 * r6782653;
        return r6782654;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r6782655 = 1.0;
        double r6782656 = h;
        double r6782657 = l;
        double r6782658 = r6782656 / r6782657;
        double r6782659 = M;
        double r6782660 = D;
        double r6782661 = r6782659 * r6782660;
        double r6782662 = 2.0;
        double r6782663 = d;
        double r6782664 = r6782662 * r6782663;
        double r6782665 = r6782661 / r6782664;
        double r6782666 = pow(r6782665, r6782662);
        double r6782667 = 0.5;
        double r6782668 = r6782666 * r6782667;
        double r6782669 = r6782658 * r6782668;
        double r6782670 = r6782655 - r6782669;
        double r6782671 = r6782663 / r6782657;
        double r6782672 = pow(r6782671, r6782667);
        double r6782673 = r6782663 / r6782656;
        double r6782674 = pow(r6782673, r6782667);
        double r6782675 = r6782672 * r6782674;
        double r6782676 = r6782670 * r6782675;
        double r6782677 = -2.452445300667083e+80;
        bool r6782678 = r6782676 <= r6782677;
        double r6782679 = cbrt(r6782656);
        double r6782680 = cbrt(r6782657);
        double r6782681 = r6782679 / r6782680;
        double r6782682 = r6782659 / r6782663;
        double r6782683 = r6782660 / r6782662;
        double r6782684 = r6782682 * r6782683;
        double r6782685 = r6782681 * r6782684;
        double r6782686 = r6782685 * r6782685;
        double r6782687 = r6782686 * r6782667;
        double r6782688 = r6782687 * r6782681;
        double r6782689 = r6782655 - r6782688;
        double r6782690 = r6782689 * r6782675;
        double r6782691 = 0.0;
        bool r6782692 = r6782676 <= r6782691;
        double r6782693 = r6782655 / r6782680;
        double r6782694 = r6782693 / r6782680;
        double r6782695 = sqrt(r6782694);
        double r6782696 = cbrt(r6782663);
        double r6782697 = cbrt(r6782680);
        double r6782698 = r6782696 / r6782697;
        double r6782699 = fabs(r6782698);
        double r6782700 = sqrt(r6782698);
        double r6782701 = r6782699 * r6782700;
        double r6782702 = r6782695 * r6782701;
        double r6782703 = r6782696 / r6782679;
        double r6782704 = sqrt(r6782703);
        double r6782705 = fabs(r6782703);
        double r6782706 = r6782704 * r6782705;
        double r6782707 = r6782702 * r6782706;
        double r6782708 = r6782670 * r6782707;
        double r6782709 = r6782663 / r6782680;
        double r6782710 = sqrt(r6782709);
        double r6782711 = r6782710 * r6782695;
        double r6782712 = r6782706 * r6782711;
        double r6782713 = r6782679 / r6782697;
        double r6782714 = r6782713 * r6782713;
        double r6782715 = r6782664 / r6782660;
        double r6782716 = r6782659 / r6782715;
        double r6782717 = r6782680 * r6782662;
        double r6782718 = r6782716 / r6782717;
        double r6782719 = r6782716 / r6782680;
        double r6782720 = r6782718 * r6782719;
        double r6782721 = r6782714 * r6782720;
        double r6782722 = r6782721 * r6782713;
        double r6782723 = r6782655 - r6782722;
        double r6782724 = r6782712 * r6782723;
        double r6782725 = r6782692 ? r6782708 : r6782724;
        double r6782726 = r6782678 ? r6782690 : r6782725;
        return r6782726;
}

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)))) < -2.452445300667083e+80

   1. Initial program 35.9

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

   3. Applied add-cube-cbrt 36.2

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

   4. Applied add-cube-cbrt 36.2

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

   5. Applied times-frac 36.2

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

   6. Applied associate-*r* 32.3

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

   7. Simplified 24.5

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

   if -2.452445300667083e+80 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < 0.0

   1. Initial program 26.5

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

   3. Applied add-cube-cbrt 26.5

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

   4. Applied add-cube-cbrt 26.6

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

   5. Applied times-frac 26.6

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

   6. Applied unpow-prod-down 21.5

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

   7. Simplified 21.5

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

   8. Simplified 21.5

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

   10. Applied add-cube-cbrt 21.6

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

   11. Applied *-un-lft-identity 21.6

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

   12. Applied times-frac 21.6

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

   13. Applied unpow-prod-down 7.7

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

   14. Simplified 7.7

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

   15. Simplified 7.7

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

   17. Applied add-cube-cbrt 7.8

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

   18. Applied add-cube-cbrt 7.9

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

   19. Applied times-frac 7.9

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

   20. Applied sqrt-prod 5.2

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

   21. Simplified 5.2

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

   if 0.0 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

   1. Initial program 24.0

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

   3. Applied add-cube-cbrt 24.4

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

   4. Applied add-cube-cbrt 24.5

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

   5. Applied times-frac 24.5

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

   6. Applied unpow-prod-down 18.8

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

   7. Simplified 17.8

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

   8. Simplified 17.8

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

   10. Applied add-cube-cbrt 17.9

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

   11. Applied *-un-lft-identity 17.9

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

   12. Applied times-frac 17.9

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

   13. Applied unpow-prod-down 15.9

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

   14. Simplified 15.9

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

   15. Simplified 15.9

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

   17. Applied add-cube-cbrt 15.9

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

   18. Applied *-un-lft-identity 15.9

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

   19. Applied times-frac 15.9

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

   20. Applied associate-*r* 13.6

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

   21. Simplified 13.4

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

   23. Applied add-cube-cbrt 13.4

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

   24. Applied add-cube-cbrt 13.4

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

   25. Applied times-frac 13.4

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

   26. Applied associate-*r* 12.2

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

   27. Simplified 10.4

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

4. Final simplification 11.5

   \[\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 -2.452445300667083 \cdot 10^{+80}:\\ \;\;\;\;\left(1 - \left(\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\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 0.0:\\ \;\;\;\;\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(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}}\right)\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \left(\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\sqrt[3]{\ell} \cdot 2} \cdot \frac{\frac{M}{\frac{2 \cdot d}{D}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))))