Henrywood and Agarwal, Equation (12)

Average Error: 25.3 → 16.1

Time: 44.5s

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r5787442 = d;
        double r5787443 = h;
        double r5787444 = r5787442 / r5787443;
        double r5787445 = 1.0;
        double r5787446 = 2.0;
        double r5787447 = r5787445 / r5787446;
        double r5787448 = pow(r5787444, r5787447);
        double r5787449 = l;
        double r5787450 = r5787442 / r5787449;
        double r5787451 = pow(r5787450, r5787447);
        double r5787452 = r5787448 * r5787451;
        double r5787453 = M;
        double r5787454 = D;
        double r5787455 = r5787453 * r5787454;
        double r5787456 = r5787446 * r5787442;
        double r5787457 = r5787455 / r5787456;
        double r5787458 = pow(r5787457, r5787446);
        double r5787459 = r5787447 * r5787458;
        double r5787460 = r5787443 / r5787449;
        double r5787461 = r5787459 * r5787460;
        double r5787462 = r5787445 - r5787461;
        double r5787463 = r5787452 * r5787462;
        return r5787463;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r5787464 = d;
        double r5787465 = cbrt(r5787464);
        double r5787466 = l;
        double r5787467 = cbrt(r5787466);
        double r5787468 = r5787465 / r5787467;
        double r5787469 = fabs(r5787468);
        double r5787470 = sqrt(r5787468);
        double r5787471 = r5787469 * r5787470;
        double r5787472 = h;
        double r5787473 = r5787464 / r5787472;
        double r5787474 = sqrt(r5787473);
        double r5787475 = r5787471 * r5787474;
        double r5787476 = -0.5;
        double r5787477 = M;
        double r5787478 = 2.0;
        double r5787479 = r5787478 * r5787464;
        double r5787480 = D;
        double r5787481 = r5787479 / r5787480;
        double r5787482 = r5787477 / r5787481;
        double r5787483 = cbrt(r5787472);
        double r5787484 = r5787483 / r5787467;
        double r5787485 = r5787482 * r5787484;
        double r5787486 = r5787484 * r5787485;
        double r5787487 = r5787486 * r5787484;
        double r5787488 = r5787482 * r5787487;
        double r5787489 = r5787476 * r5787488;
        double r5787490 = 1.0;
        double r5787491 = r5787483 * r5787483;
        double r5787492 = r5787490 / r5787491;
        double r5787493 = sqrt(r5787492);
        double r5787494 = r5787464 / r5787483;
        double r5787495 = sqrt(r5787494);
        double r5787496 = r5787493 * r5787495;
        double r5787497 = r5787496 * r5787471;
        double r5787498 = fma(r5787475, r5787489, r5787497);
        return r5787498;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Initial program 25.3

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

2. Simplified 24.4

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

4. Applied add-cube-cbrt 24.6

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

5. Applied add-cube-cbrt 24.7

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

6. Applied times-frac 24.7

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

7. Applied sqrt-prod 22.6

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

8. Simplified 22.4

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

10. Applied add-cube-cbrt 22.4

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

11. Applied add-cube-cbrt 22.4

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

12. Applied times-frac 22.4

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

13. Applied sqrt-prod 20.3

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

14. Simplified 20.3

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

16. Applied add-cube-cbrt 20.3

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

17. Applied add-cube-cbrt 20.3

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

18. Applied times-frac 20.3

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

19. Applied associate-*r* 17.3

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

20. Simplified 16.4

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

22. Applied add-cube-cbrt 16.5

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

23. Applied *-un-lft-identity 16.5

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

24. Applied times-frac 16.5

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

25. Applied sqrt-prod 16.1

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

26. Final simplification 16.1

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

Reproduce

herbie shell --seed 2019164 +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)))))