Henrywood and Agarwal, Equation (9a)

Average Error: 14.0 → 7.5

Time: 8.7m

Precision: 64

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

↓

\[w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}} \cdot \frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{1}{\frac{1}{\sqrt[3]{d} \cdot \sqrt[3]{d}}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{M}}{\frac{2}{\frac{D}{\sqrt[3]{d}}}}\right)\right)}\]

C

double f(double w0, double M, double D, double h, double l, double d) {
        double r28383483 = w0;
        double r28383484 = 1.0;
        double r28383485 = M;
        double r28383486 = D;
        double r28383487 = r28383485 * r28383486;
        double r28383488 = 2.0;
        double r28383489 = d;
        double r28383490 = r28383488 * r28383489;
        double r28383491 = r28383487 / r28383490;
        double r28383492 = pow(r28383491, r28383488);
        double r28383493 = h;
        double r28383494 = l;
        double r28383495 = r28383493 / r28383494;
        double r28383496 = r28383492 * r28383495;
        double r28383497 = r28383484 - r28383496;
        double r28383498 = sqrt(r28383497);
        double r28383499 = r28383483 * r28383498;
        return r28383499;
}

↓

double f(double w0, double M, double D, double h, double l, double d) {
        double r28383500 = w0;
        double r28383501 = 1.0;
        double r28383502 = h;
        double r28383503 = cbrt(r28383502);
        double r28383504 = l;
        double r28383505 = cbrt(r28383504);
        double r28383506 = r28383503 / r28383505;
        double r28383507 = M;
        double r28383508 = 2.0;
        double r28383509 = D;
        double r28383510 = cbrt(r28383509);
        double r28383511 = d;
        double r28383512 = cbrt(r28383511);
        double r28383513 = r28383510 / r28383512;
        double r28383514 = r28383508 / r28383513;
        double r28383515 = r28383507 / r28383514;
        double r28383516 = r28383513 * r28383513;
        double r28383517 = r28383503 * r28383516;
        double r28383518 = r28383517 / r28383505;
        double r28383519 = r28383515 * r28383518;
        double r28383520 = cbrt(r28383507);
        double r28383521 = r28383520 * r28383520;
        double r28383522 = r28383512 * r28383512;
        double r28383523 = r28383501 / r28383522;
        double r28383524 = r28383501 / r28383523;
        double r28383525 = r28383521 / r28383524;
        double r28383526 = r28383525 * r28383506;
        double r28383527 = r28383509 / r28383512;
        double r28383528 = r28383508 / r28383527;
        double r28383529 = r28383520 / r28383528;
        double r28383530 = r28383526 * r28383529;
        double r28383531 = r28383519 * r28383530;
        double r28383532 = r28383506 * r28383531;
        double r28383533 = r28383501 - r28383532;
        double r28383534 = sqrt(r28383533);
        double r28383535 = r28383500 * r28383534;
        return r28383535;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

1. Initial program 14.0

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

2. Simplified 14.0

   \[\leadsto \color{blue}{\sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \cdot w0}\]
3. Using strategy rm

4. Applied add-cube-cbrt 14.0

   \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \cdot w0\]

5. Applied add-cube-cbrt 14.0

   \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot w0\]

6. Applied times-frac 14.0

   \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}} \cdot w0\]

7. Applied associate-*r* 10.7

   \[\leadsto \sqrt{1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}} \cdot w0\]

8. Simplified 8.3

   \[\leadsto \sqrt{1 - \color{blue}{\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]
9. Using strategy rm

10. Applied add-cube-cbrt 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

11. Applied add-cube-cbrt 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\color{blue}{\left(\sqrt[3]{D} \cdot \sqrt[3]{D}\right) \cdot \sqrt[3]{D}}}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

12. Applied times-frac 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\color{blue}{\frac{\sqrt[3]{D} \cdot \sqrt[3]{D}}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

13. Applied *-un-lft-identity 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{\color{blue}{1 \cdot 2}}{\frac{\sqrt[3]{D} \cdot \sqrt[3]{D}}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

14. Applied times-frac 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\color{blue}{\frac{1}{\frac{\sqrt[3]{D} \cdot \sqrt[3]{D}}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

15. Applied *-un-lft-identity 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\color{blue}{1 \cdot M}}{\frac{1}{\frac{\sqrt[3]{D} \cdot \sqrt[3]{D}}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

16. Applied times-frac 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \color{blue}{\left(\frac{1}{\frac{1}{\frac{\sqrt[3]{D} \cdot \sqrt[3]{D}}{\sqrt[3]{d} \cdot \sqrt[3]{d}}}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

17. Applied associate-*r* 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \color{blue}{\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{1}{\frac{1}{\frac{\sqrt[3]{D} \cdot \sqrt[3]{D}}{\sqrt[3]{d} \cdot \sqrt[3]{d}}}}\right) \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

18. Simplified 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \left(\color{blue}{\frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]
19. Using strategy rm

20. Applied add-cube-cbrt 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

21. Applied *-un-lft-identity 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\color{blue}{1 \cdot D}}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

22. Applied times-frac 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\color{blue}{\frac{1}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{D}{\sqrt[3]{d}}}}}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

23. Applied *-un-lft-identity 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{\color{blue}{1 \cdot 2}}{\frac{1}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{D}{\sqrt[3]{d}}}}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

24. Applied times-frac 8.2

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\color{blue}{\frac{1}{\frac{1}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{2}{\frac{D}{\sqrt[3]{d}}}}}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

25. Applied add-cube-cbrt 8.2

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\color{blue}{\left(\sqrt[3]{M} \cdot \sqrt[3]{M}\right) \cdot \sqrt[3]{M}}}{\frac{1}{\frac{1}{\sqrt[3]{d} \cdot \sqrt[3]{d}}} \cdot \frac{2}{\frac{D}{\sqrt[3]{d}}}}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

26. Applied times-frac 7.6

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \color{blue}{\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{1}{\frac{1}{\sqrt[3]{d} \cdot \sqrt[3]{d}}}} \cdot \frac{\sqrt[3]{M}}{\frac{2}{\frac{D}{\sqrt[3]{d}}}}\right)}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

27. Applied associate-*r* 7.5

    \[\leadsto \sqrt{1 - \left(\color{blue}{\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{1}{\frac{1}{\sqrt[3]{d} \cdot \sqrt[3]{d}}}}\right) \cdot \frac{\sqrt[3]{M}}{\frac{2}{\frac{D}{\sqrt[3]{d}}}}\right)} \cdot \left(\frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]

28. Final simplification 7.5

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{M}{\frac{2}{\frac{\sqrt[3]{D}}{\sqrt[3]{d}}}} \cdot \frac{\sqrt[3]{h} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{1}{\frac{1}{\sqrt[3]{d} \cdot \sqrt[3]{d}}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{M}}{\frac{2}{\frac{D}{\sqrt[3]{d}}}}\right)\right)}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))