Henrywood and Agarwal, Equation (12)

Average Error: 26.0 → 16.4

Time: 51.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le 5.119782877590346 \cdot 10^{-133}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\frac{h}{\ell}\right), \left(\frac{-1}{2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot \frac{D \cdot M}{d \cdot 2}\right)\right), 1\right) \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \mathbf{elif}\;d \le 2.7672589156476904 \cdot 10^{+36}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \left(1 - \frac{\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\ell}}{d \cdot d} \cdot \frac{1}{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{d} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell}\right), \left(\frac{-1}{2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot \frac{D \cdot M}{d \cdot 2}\right)\right), 1\right)\right) \cdot \left(\sqrt{\sqrt[3]{d}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)}{\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{h}}}\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r2514589 = d;
        double r2514590 = h;
        double r2514591 = r2514589 / r2514590;
        double r2514592 = 1.0;
        double r2514593 = 2.0;
        double r2514594 = r2514592 / r2514593;
        double r2514595 = pow(r2514591, r2514594);
        double r2514596 = l;
        double r2514597 = r2514589 / r2514596;
        double r2514598 = pow(r2514597, r2514594);
        double r2514599 = r2514595 * r2514598;
        double r2514600 = M;
        double r2514601 = D;
        double r2514602 = r2514600 * r2514601;
        double r2514603 = r2514593 * r2514589;
        double r2514604 = r2514602 / r2514603;
        double r2514605 = pow(r2514604, r2514593);
        double r2514606 = r2514594 * r2514605;
        double r2514607 = r2514590 / r2514596;
        double r2514608 = r2514606 * r2514607;
        double r2514609 = r2514592 - r2514608;
        double r2514610 = r2514599 * r2514609;
        return r2514610;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r2514611 = d;
        double r2514612 = 5.119782877590346e-133;
        bool r2514613 = r2514611 <= r2514612;
        double r2514614 = h;
        double r2514615 = l;
        double r2514616 = r2514614 / r2514615;
        double r2514617 = -0.5;
        double r2514618 = D;
        double r2514619 = M;
        double r2514620 = r2514618 * r2514619;
        double r2514621 = 2.0;
        double r2514622 = r2514611 * r2514621;
        double r2514623 = r2514620 / r2514622;
        double r2514624 = r2514623 * r2514623;
        double r2514625 = r2514617 * r2514624;
        double r2514626 = 1.0;
        double r2514627 = fma(r2514616, r2514625, r2514626);
        double r2514628 = cbrt(r2514615);
        double r2514629 = r2514611 / r2514628;
        double r2514630 = sqrt(r2514629);
        double r2514631 = r2514627 * r2514630;
        double r2514632 = cbrt(r2514611);
        double r2514633 = cbrt(r2514614);
        double r2514634 = r2514632 / r2514633;
        double r2514635 = sqrt(r2514634);
        double r2514636 = fabs(r2514634);
        double r2514637 = r2514635 * r2514636;
        double r2514638 = r2514631 * r2514637;
        double r2514639 = r2514628 * r2514628;
        double r2514640 = sqrt(r2514639);
        double r2514641 = r2514638 / r2514640;
        double r2514642 = 2.7672589156476904e+36;
        bool r2514643 = r2514611 <= r2514642;
        double r2514644 = r2514626 / r2514639;
        double r2514645 = sqrt(r2514644);
        double r2514646 = r2514630 * r2514645;
        double r2514647 = r2514620 * r2514620;
        double r2514648 = r2514647 * r2514614;
        double r2514649 = r2514648 / r2514615;
        double r2514650 = r2514611 * r2514611;
        double r2514651 = r2514649 / r2514650;
        double r2514652 = 0.125;
        double r2514653 = r2514651 * r2514652;
        double r2514654 = r2514626 - r2514653;
        double r2514655 = r2514646 * r2514654;
        double r2514656 = r2514637 * r2514655;
        double r2514657 = sqrt(r2514611);
        double r2514658 = r2514657 * r2514645;
        double r2514659 = r2514658 * r2514627;
        double r2514660 = sqrt(r2514632);
        double r2514661 = r2514660 * r2514636;
        double r2514662 = r2514659 * r2514661;
        double r2514663 = sqrt(r2514628);
        double r2514664 = sqrt(r2514633);
        double r2514665 = r2514663 * r2514664;
        double r2514666 = r2514662 / r2514665;
        double r2514667 = r2514643 ? r2514656 : r2514666;
        double r2514668 = r2514613 ? r2514641 : r2514667;
        return r2514668;
}

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 d < 5.119782877590346e-133

   1. Initial program 28.7

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

   2. Simplified 28.7

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

   4. Applied add-cube-cbrt 28.9

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

   5. Applied add-cube-cbrt 29.0

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

   6. Applied times-frac 29.0

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

   7. Applied sqrt-prod 24.5

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

   8. Simplified 23.9

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

   10. Applied add-cube-cbrt 24.0

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

   11. Applied *-un-lft-identity 24.0

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

   12. Applied times-frac 24.0

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

   13. Applied sqrt-prod 19.8

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

   15. Applied associate-*r* 19.0

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

   17. Applied sqrt-div 19.0

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

   18. Applied associate-*l/ 18.9

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

   19. Applied associate-*r/ 19.0

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

   20. Applied associate-*l/ 19.7

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

   21. Simplified 19.7

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

   if 5.119782877590346e-133 < d < 2.7672589156476904e+36

   1. Initial program 18.9

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

   2. Simplified 18.9

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

   4. Applied add-cube-cbrt 19.2

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

   5. Applied add-cube-cbrt 19.4

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

   6. Applied times-frac 19.4

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

   7. Applied sqrt-prod 18.3

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

   8. Simplified 18.3

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

   10. Applied add-cube-cbrt 18.3

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

   11. Applied *-un-lft-identity 18.3

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

   12. Applied times-frac 18.3

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

   13. Applied sqrt-prod 15.2

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

   15. Applied associate-*r* 15.2

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

   16. Taylor expanded around inf 26.3

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

   17. Simplified 12.3

       \[\leadsto \left(\color{blue}{\left(1 - \frac{1}{8} \cdot \frac{\frac{\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h}{\ell}}{d \cdot d}\right)} \cdot \left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]

   if 2.7672589156476904e+36 < d

   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 23.1

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

   4. Applied add-cube-cbrt 23.3

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

   5. Applied add-cube-cbrt 23.5

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

   6. Applied times-frac 23.5

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

   7. Applied sqrt-prod 14.2

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

   8. Simplified 12.2

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

   10. Applied add-cube-cbrt 12.2

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

   11. Applied *-un-lft-identity 12.2

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

   12. Applied times-frac 12.3

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

   13. Applied sqrt-prod 9.4

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

   15. Applied associate-*r* 9.6

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

   17. Applied sqrt-div 9.6

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

   18. Applied associate-*r/ 9.6

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

   19. Applied sqrt-div 9.3

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

   20. Applied associate-*r/ 9.3

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

   21. Applied associate-*r/ 9.3

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

   22. Applied frac-times 9.5

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

4. Final simplification 16.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 5.119782877590346 \cdot 10^{-133}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\left(\frac{h}{\ell}\right), \left(\frac{-1}{2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot \frac{D \cdot M}{d \cdot 2}\right)\right), 1\right) \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \mathbf{elif}\;d \le 2.7672589156476904 \cdot 10^{+36}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \left(1 - \frac{\frac{\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h}{\ell}}{d \cdot d} \cdot \frac{1}{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\sqrt{d} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell}\right), \left(\frac{-1}{2} \cdot \left(\frac{D \cdot M}{d \cdot 2} \cdot \frac{D \cdot M}{d \cdot 2}\right)\right), 1\right)\right) \cdot \left(\sqrt{\sqrt[3]{d}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)}{\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{h}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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)))))