Henrywood and Agarwal, Equation (12)

Average Error: 25.4 → 16.6

Time: 50.5s

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r2799348 = d;
        double r2799349 = h;
        double r2799350 = r2799348 / r2799349;
        double r2799351 = 1.0;
        double r2799352 = 2.0;
        double r2799353 = r2799351 / r2799352;
        double r2799354 = pow(r2799350, r2799353);
        double r2799355 = l;
        double r2799356 = r2799348 / r2799355;
        double r2799357 = pow(r2799356, r2799353);
        double r2799358 = r2799354 * r2799357;
        double r2799359 = M;
        double r2799360 = D;
        double r2799361 = r2799359 * r2799360;
        double r2799362 = r2799352 * r2799348;
        double r2799363 = r2799361 / r2799362;
        double r2799364 = pow(r2799363, r2799352);
        double r2799365 = r2799353 * r2799364;
        double r2799366 = r2799349 / r2799355;
        double r2799367 = r2799365 * r2799366;
        double r2799368 = r2799351 - r2799367;
        double r2799369 = r2799358 * r2799368;
        return r2799369;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r2799370 = d;
        double r2799371 = cbrt(r2799370);
        double r2799372 = l;
        double r2799373 = cbrt(r2799372);
        double r2799374 = r2799371 / r2799373;
        double r2799375 = fabs(r2799374);
        double r2799376 = sqrt(r2799374);
        double r2799377 = r2799375 * r2799376;
        double r2799378 = 1.0;
        double r2799379 = h;
        double r2799380 = cbrt(r2799379);
        double r2799381 = r2799380 * r2799380;
        double r2799382 = r2799378 / r2799381;
        double r2799383 = sqrt(r2799382);
        double r2799384 = r2799370 / r2799380;
        double r2799385 = sqrt(r2799384);
        double r2799386 = r2799383 * r2799385;
        double r2799387 = r2799377 * r2799386;
        double r2799388 = -0.5;
        double r2799389 = D;
        double r2799390 = 2.0;
        double r2799391 = r2799389 / r2799390;
        double r2799392 = M;
        double r2799393 = r2799392 / r2799370;
        double r2799394 = r2799391 * r2799393;
        double r2799395 = r2799380 / r2799373;
        double r2799396 = r2799394 * r2799395;
        double r2799397 = r2799396 * r2799396;
        double r2799398 = r2799397 * r2799395;
        double r2799399 = r2799388 * r2799398;
        double r2799400 = r2799370 / r2799373;
        double r2799401 = sqrt(r2799400);
        double r2799402 = r2799373 * r2799373;
        double r2799403 = r2799378 / r2799402;
        double r2799404 = sqrt(r2799403);
        double r2799405 = r2799401 * r2799404;
        double r2799406 = r2799370 / r2799379;
        double r2799407 = sqrt(r2799406);
        double r2799408 = r2799405 * r2799407;
        double r2799409 = fma(r2799387, r2799399, r2799408);
        return r2799409;
}

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 25.4

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

2. Simplified 25.7

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

4. Applied add-cube-cbrt 25.9

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

5. Applied *-un-lft-identity 25.9

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

6. Applied times-frac 25.9

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

7. Applied sqrt-prod 24.0

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

9. Applied add-cube-cbrt 24.0

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

10. Applied add-cube-cbrt 24.0

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

11. Applied times-frac 24.0

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

12. Applied associate-*r* 22.4

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

13. Simplified 20.8

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

15. Applied add-cube-cbrt 20.9

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

16. Applied add-cube-cbrt 20.9

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

17. Applied times-frac 20.9

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

18. Applied sqrt-prod 17.5

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

19. Simplified 17.4

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

21. Applied add-cube-cbrt 17.3

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

22. Applied *-un-lft-identity 17.3

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

23. Applied times-frac 17.4

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

24. Applied sqrt-prod 16.6

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

25. Final simplification 16.6

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

Reproduce

herbie shell --seed 2019154 +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)))))