Henrywood and Agarwal, Equation (12)

Average Error: 26.0 → 13.4

Time: 2.1m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;h \le -2.217526380294814 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}}\right), \left(\frac{h}{\frac{\ell}{\frac{-1}{2} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{M \cdot D}{d \cdot 2}\right)}}\right), \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \mathbf{elif}\;h \le 5.988458482842814 \cdot 10^{-193}:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell}\right), \left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \left(\left(\sqrt[3]{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt[3]{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \sqrt[3]{\frac{M \cdot D}{d \cdot 2}}\right)\right)\right), 1\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\sqrt[3]{d}}\right) \cdot \mathsf{fma}\left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}}\right), \left(\frac{h}{\frac{\ell}{\frac{-1}{2} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{M \cdot D}{d \cdot 2}\right)}}\right), \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right)}{\sqrt{\sqrt[3]{h}} \cdot \left|\sqrt[3]{\ell}\right|}\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r41870507 = d;
        double r41870508 = h;
        double r41870509 = r41870507 / r41870508;
        double r41870510 = 1.0;
        double r41870511 = 2.0;
        double r41870512 = r41870510 / r41870511;
        double r41870513 = pow(r41870509, r41870512);
        double r41870514 = l;
        double r41870515 = r41870507 / r41870514;
        double r41870516 = pow(r41870515, r41870512);
        double r41870517 = r41870513 * r41870516;
        double r41870518 = M;
        double r41870519 = D;
        double r41870520 = r41870518 * r41870519;
        double r41870521 = r41870511 * r41870507;
        double r41870522 = r41870520 / r41870521;
        double r41870523 = pow(r41870522, r41870511);
        double r41870524 = r41870512 * r41870523;
        double r41870525 = r41870508 / r41870514;
        double r41870526 = r41870524 * r41870525;
        double r41870527 = r41870510 - r41870526;
        double r41870528 = r41870517 * r41870527;
        return r41870528;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r41870529 = h;
        double r41870530 = -2.217526380294814e-206;
        bool r41870531 = r41870529 <= r41870530;
        double r41870532 = d;
        double r41870533 = l;
        double r41870534 = cbrt(r41870533);
        double r41870535 = r41870532 / r41870534;
        double r41870536 = sqrt(r41870535);
        double r41870537 = -0.5;
        double r41870538 = M;
        double r41870539 = D;
        double r41870540 = r41870538 * r41870539;
        double r41870541 = 2.0;
        double r41870542 = r41870532 * r41870541;
        double r41870543 = r41870540 / r41870542;
        double r41870544 = r41870543 * r41870543;
        double r41870545 = r41870537 * r41870544;
        double r41870546 = r41870533 / r41870545;
        double r41870547 = r41870529 / r41870546;
        double r41870548 = fma(r41870536, r41870547, r41870536);
        double r41870549 = cbrt(r41870532);
        double r41870550 = cbrt(r41870529);
        double r41870551 = r41870549 / r41870550;
        double r41870552 = fabs(r41870551);
        double r41870553 = sqrt(r41870551);
        double r41870554 = r41870552 * r41870553;
        double r41870555 = r41870548 * r41870554;
        double r41870556 = r41870534 * r41870534;
        double r41870557 = sqrt(r41870556);
        double r41870558 = r41870555 / r41870557;
        double r41870559 = 5.988458482842814e-193;
        bool r41870560 = r41870529 <= r41870559;
        double r41870561 = 1.0;
        double r41870562 = r41870561 / r41870556;
        double r41870563 = sqrt(r41870562);
        double r41870564 = r41870563 * r41870536;
        double r41870565 = r41870529 / r41870533;
        double r41870566 = cbrt(r41870543);
        double r41870567 = r41870566 * r41870566;
        double r41870568 = r41870567 * r41870566;
        double r41870569 = r41870543 * r41870568;
        double r41870570 = r41870537 * r41870569;
        double r41870571 = fma(r41870565, r41870570, r41870561);
        double r41870572 = r41870564 * r41870571;
        double r41870573 = r41870572 * r41870554;
        double r41870574 = sqrt(r41870549);
        double r41870575 = r41870552 * r41870574;
        double r41870576 = r41870575 * r41870548;
        double r41870577 = sqrt(r41870550);
        double r41870578 = fabs(r41870534);
        double r41870579 = r41870577 * r41870578;
        double r41870580 = r41870576 / r41870579;
        double r41870581 = r41870560 ? r41870573 : r41870580;
        double r41870582 = r41870531 ? r41870558 : r41870581;
        return r41870582;
}

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 h < -2.217526380294814e-206

   1. Initial program 24.7

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

   2. Simplified 24.7

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

   4. Applied add-cube-cbrt 24.9

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

   5. Applied add-cube-cbrt 25.0

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

   6. Applied times-frac 25.0

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

   7. Applied sqrt-prod 21.6

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

   8. Simplified 21.6

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

   10. Applied add-cube-cbrt 21.6

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

   11. Applied *-un-lft-identity 21.6

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

   12. Applied times-frac 21.6

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

   13. Applied sqrt-prod 18.0

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

   15. Applied associate-*r* 17.3

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

   17. Applied sqrt-div 17.3

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

   18. Applied associate-*l/ 17.3

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

   19. Applied associate-*r/ 17.3

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

   20. Applied associate-*l/ 17.7

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

   21. Simplified 14.0

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

   if -2.217526380294814e-206 < h < 5.988458482842814e-193

   1. Initial program 34.5

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

   2. Simplified 34.5

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

   4. Applied add-cube-cbrt 34.8

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

   5. Applied add-cube-cbrt 34.9

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

   6. Applied times-frac 34.9

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

   7. Applied sqrt-prod 21.2

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

   8. Simplified 16.7

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

   10. Applied add-cube-cbrt 16.7

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

   11. Applied *-un-lft-identity 16.7

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

   12. Applied times-frac 16.8

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

   13. Applied sqrt-prod 12.5

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

   15. Applied associate-*r* 12.5

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

   17. Applied add-cube-cbrt 12.5

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

   if 5.988458482842814e-193 < h

   1. Initial program 23.8

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

   2. Simplified 23.8

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

   4. Applied add-cube-cbrt 24.1

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

   5. Applied add-cube-cbrt 24.2

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

   6. Applied times-frac 24.2

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

   7. Applied sqrt-prod 21.0

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

   8. Simplified 20.9

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

   10. Applied add-cube-cbrt 20.9

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

   11. Applied *-un-lft-identity 20.9

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

   12. Applied times-frac 20.9

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

   13. Applied sqrt-prod 17.4

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

   15. Applied associate-*r* 16.9

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

   17. Applied sqrt-div 16.9

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

   18. Applied associate-*r/ 16.9

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

   19. Applied sqrt-div 16.9

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

   20. Applied associate-*l/ 16.9

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

   21. Applied associate-*r/ 16.9

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

   22. Applied frac-times 17.1

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

   23. Simplified 13.1

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

   24. Simplified 13.1

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

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -2.217526380294814 \cdot 10^{-206}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}}\right), \left(\frac{h}{\frac{\ell}{\frac{-1}{2} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{M \cdot D}{d \cdot 2}\right)}}\right), \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\\ \mathbf{elif}\;h \le 5.988458482842814 \cdot 10^{-193}:\\ \;\;\;\;\left(\left(\sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{d}{\sqrt[3]{\ell}}}\right) \cdot \mathsf{fma}\left(\left(\frac{h}{\ell}\right), \left(\frac{-1}{2} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \left(\left(\sqrt[3]{\frac{M \cdot D}{d \cdot 2}} \cdot \sqrt[3]{\frac{M \cdot D}{d \cdot 2}}\right) \cdot \sqrt[3]{\frac{M \cdot D}{d \cdot 2}}\right)\right)\right), 1\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\sqrt[3]{d}}\right) \cdot \mathsf{fma}\left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}}\right), \left(\frac{h}{\frac{\ell}{\frac{-1}{2} \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{M \cdot D}{d \cdot 2}\right)}}\right), \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}}\right)\right)}{\sqrt{\sqrt[3]{h}} \cdot \left|\sqrt[3]{\ell}\right|}\\ \end{array}\]

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)))))