Henrywood and Agarwal, Equation (12)

Average Error: 25.8 → 12.1

Time: 4.5m

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r48122421 = d;
        double r48122422 = h;
        double r48122423 = r48122421 / r48122422;
        double r48122424 = 1.0;
        double r48122425 = 2.0;
        double r48122426 = r48122424 / r48122425;
        double r48122427 = pow(r48122423, r48122426);
        double r48122428 = l;
        double r48122429 = r48122421 / r48122428;
        double r48122430 = pow(r48122429, r48122426);
        double r48122431 = r48122427 * r48122430;
        double r48122432 = M;
        double r48122433 = D;
        double r48122434 = r48122432 * r48122433;
        double r48122435 = r48122425 * r48122421;
        double r48122436 = r48122434 / r48122435;
        double r48122437 = pow(r48122436, r48122425);
        double r48122438 = r48122426 * r48122437;
        double r48122439 = r48122422 / r48122428;
        double r48122440 = r48122438 * r48122439;
        double r48122441 = r48122424 - r48122440;
        double r48122442 = r48122431 * r48122441;
        return r48122442;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r48122443 = 1.0;
        double r48122444 = h;
        double r48122445 = M;
        double r48122446 = D;
        double r48122447 = r48122445 * r48122446;
        double r48122448 = d;
        double r48122449 = 2.0;
        double r48122450 = r48122448 * r48122449;
        double r48122451 = r48122447 / r48122450;
        double r48122452 = r48122451 / r48122449;
        double r48122453 = l;
        double r48122454 = cbrt(r48122453);
        double r48122455 = r48122452 / r48122454;
        double r48122456 = r48122444 * r48122455;
        double r48122457 = r48122454 * r48122454;
        double r48122458 = r48122451 / r48122457;
        double r48122459 = r48122456 * r48122458;
        double r48122460 = r48122443 - r48122459;
        double r48122461 = cbrt(r48122444);
        double r48122462 = r48122461 * r48122461;
        double r48122463 = r48122443 / r48122462;
        double r48122464 = sqrt(r48122463);
        double r48122465 = r48122448 / r48122461;
        double r48122466 = 0.5;
        double r48122467 = pow(r48122465, r48122466);
        double r48122468 = r48122464 * r48122467;
        double r48122469 = cbrt(r48122448);
        double r48122470 = fabs(r48122469);
        double r48122471 = r48122469 / r48122453;
        double r48122472 = pow(r48122471, r48122466);
        double r48122473 = r48122470 * r48122472;
        double r48122474 = r48122468 * r48122473;
        double r48122475 = r48122460 * r48122474;
        return r48122475;
}

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 25.8

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

3. Applied add-cube-cbrt 26.1

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

4. Applied *-un-lft-identity 26.1

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

5. Applied times-frac 26.1

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

6. Applied unpow-prod-down 21.2

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

7. Simplified 21.2

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

9. Applied *-un-lft-identity 21.2

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

10. Applied add-cube-cbrt 21.3

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

11. Applied times-frac 21.3

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

12. Applied unpow-prod-down 17.9

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

13. Simplified 17.9

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

15. Applied pow1 17.9

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

16. Applied pow1 17.9

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

17. Applied pow-prod-down 17.9

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

18. Simplified 15.7

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

20. Applied add-cube-cbrt 15.8

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

21. Applied *-un-lft-identity 15.8

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

22. Applied times-frac 15.8

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

23. Applied times-frac 13.8

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

24. Applied associate-*l* 12.1

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

25. Final simplification 12.1

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

Reproduce

herbie shell --seed 2019112 
(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)))))