Henrywood and Agarwal, Equation (12)

Average Error: 26.4 → 11.5

Time: 1.1m

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r256505 = d;
        double r256506 = h;
        double r256507 = r256505 / r256506;
        double r256508 = 1.0;
        double r256509 = 2.0;
        double r256510 = r256508 / r256509;
        double r256511 = pow(r256507, r256510);
        double r256512 = l;
        double r256513 = r256505 / r256512;
        double r256514 = pow(r256513, r256510);
        double r256515 = r256511 * r256514;
        double r256516 = M;
        double r256517 = D;
        double r256518 = r256516 * r256517;
        double r256519 = r256509 * r256505;
        double r256520 = r256518 / r256519;
        double r256521 = pow(r256520, r256509);
        double r256522 = r256510 * r256521;
        double r256523 = r256506 / r256512;
        double r256524 = r256522 * r256523;
        double r256525 = r256508 - r256524;
        double r256526 = r256515 * r256525;
        return r256526;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r256527 = d;
        double r256528 = cbrt(r256527);
        double r256529 = r256528 * r256528;
        double r256530 = h;
        double r256531 = cbrt(r256530);
        double r256532 = r256531 * r256531;
        double r256533 = r256529 / r256532;
        double r256534 = 1.0;
        double r256535 = 2.0;
        double r256536 = r256534 / r256535;
        double r256537 = pow(r256533, r256536);
        double r256538 = r256528 / r256531;
        double r256539 = pow(r256538, r256536);
        double r256540 = r256537 * r256539;
        double r256541 = l;
        double r256542 = cbrt(r256541);
        double r256543 = r256542 * r256542;
        double r256544 = r256529 / r256543;
        double r256545 = pow(r256544, r256536);
        double r256546 = r256528 / r256542;
        double r256547 = pow(r256546, r256536);
        double r256548 = r256545 * r256547;
        double r256549 = M;
        double r256550 = D;
        double r256551 = r256527 / r256550;
        double r256552 = r256535 * r256551;
        double r256553 = r256549 / r256552;
        double r256554 = 2.0;
        double r256555 = r256535 / r256554;
        double r256556 = pow(r256553, r256555);
        double r256557 = r256535 * r256527;
        double r256558 = r256557 / r256550;
        double r256559 = r256549 / r256558;
        double r256560 = pow(r256559, r256555);
        double r256561 = 1.0;
        double r256562 = r256561 / r256541;
        double r256563 = r256560 * r256562;
        double r256564 = r256530 * r256563;
        double r256565 = r256556 * r256564;
        double r256566 = r256536 * r256565;
        double r256567 = r256534 - r256566;
        double r256568 = r256548 * r256567;
        double r256569 = r256540 * r256568;
        return r256569;
}

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)))))