Henrywood and Agarwal, Equation (12)

Average Error: 26.4 → 11.5

Time: 1.0m

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r214472 = d;
        double r214473 = h;
        double r214474 = r214472 / r214473;
        double r214475 = 1.0;
        double r214476 = 2.0;
        double r214477 = r214475 / r214476;
        double r214478 = pow(r214474, r214477);
        double r214479 = l;
        double r214480 = r214472 / r214479;
        double r214481 = pow(r214480, r214477);
        double r214482 = r214478 * r214481;
        double r214483 = M;
        double r214484 = D;
        double r214485 = r214483 * r214484;
        double r214486 = r214476 * r214472;
        double r214487 = r214485 / r214486;
        double r214488 = pow(r214487, r214476);
        double r214489 = r214477 * r214488;
        double r214490 = r214473 / r214479;
        double r214491 = r214489 * r214490;
        double r214492 = r214475 - r214491;
        double r214493 = r214482 * r214492;
        return r214493;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r214494 = d;
        double r214495 = cbrt(r214494);
        double r214496 = r214495 * r214495;
        double r214497 = h;
        double r214498 = cbrt(r214497);
        double r214499 = r214498 * r214498;
        double r214500 = r214496 / r214499;
        double r214501 = 1.0;
        double r214502 = 2.0;
        double r214503 = r214501 / r214502;
        double r214504 = pow(r214500, r214503);
        double r214505 = r214495 / r214498;
        double r214506 = pow(r214505, r214503);
        double r214507 = r214504 * r214506;
        double r214508 = l;
        double r214509 = cbrt(r214508);
        double r214510 = r214509 * r214509;
        double r214511 = r214496 / r214510;
        double r214512 = pow(r214511, r214503);
        double r214513 = r214495 / r214509;
        double r214514 = pow(r214513, r214503);
        double r214515 = r214512 * r214514;
        double r214516 = M;
        double r214517 = D;
        double r214518 = r214494 / r214517;
        double r214519 = r214502 * r214518;
        double r214520 = r214516 / r214519;
        double r214521 = 2.0;
        double r214522 = r214502 / r214521;
        double r214523 = pow(r214520, r214522);
        double r214524 = r214502 * r214494;
        double r214525 = r214524 / r214517;
        double r214526 = r214516 / r214525;
        double r214527 = pow(r214526, r214522);
        double r214528 = 1.0;
        double r214529 = r214528 / r214508;
        double r214530 = r214527 * r214529;
        double r214531 = r214497 * r214530;
        double r214532 = r214523 * r214531;
        double r214533 = r214503 * r214532;
        double r214534 = r214501 - r214533;
        double r214535 = r214515 * r214534;
        double r214536 = r214507 * r214535;
        return r214536;
}

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 26.4

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

2. Simplified 25.8

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

4. Applied add-cube-cbrt 26.1

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

5. Applied add-cube-cbrt 26.2

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

6. Applied times-frac 26.2

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

7. Applied unpow-prod-down 20.9

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

9. Applied add-cube-cbrt 21.0

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

10. Applied add-cube-cbrt 21.2

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

11. Applied times-frac 21.2

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

12. Applied unpow-prod-down 14.8

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

14. Applied *-un-lft-identity 14.8

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

15. Applied sqr-pow 14.8

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

16. Applied times-frac 12.6

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

17. Applied associate-*l* 11.5

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

18. Simplified 11.5

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

20. Applied div-inv 11.5

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

21. Final simplification 11.5

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

Reproduce

herbie shell --seed 2019194 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))