Henrywood and Agarwal, Equation (12)

Average Error: 26.5 → 15.5

Time: 28.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r315966 = d;
        double r315967 = h;
        double r315968 = r315966 / r315967;
        double r315969 = 1.0;
        double r315970 = 2.0;
        double r315971 = r315969 / r315970;
        double r315972 = pow(r315968, r315971);
        double r315973 = l;
        double r315974 = r315966 / r315973;
        double r315975 = pow(r315974, r315971);
        double r315976 = r315972 * r315975;
        double r315977 = M;
        double r315978 = D;
        double r315979 = r315977 * r315978;
        double r315980 = r315970 * r315966;
        double r315981 = r315979 / r315980;
        double r315982 = pow(r315981, r315970);
        double r315983 = r315971 * r315982;
        double r315984 = r315967 / r315973;
        double r315985 = r315983 * r315984;
        double r315986 = r315969 - r315985;
        double r315987 = r315976 * r315986;
        return r315987;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r315988 = d;
        double r315989 = cbrt(r315988);
        double r315990 = r315989 * r315989;
        double r315991 = 1.0;
        double r315992 = 2.0;
        double r315993 = r315991 / r315992;
        double r315994 = pow(r315990, r315993);
        double r315995 = cbrt(r315989);
        double r315996 = r315995 * r315995;
        double r315997 = pow(r315996, r315993);
        double r315998 = h;
        double r315999 = r315995 / r315998;
        double r316000 = pow(r315999, r315993);
        double r316001 = r315997 * r316000;
        double r316002 = r315994 * r316001;
        double r316003 = 1.0;
        double r316004 = l;
        double r316005 = cbrt(r316004);
        double r316006 = r316005 * r316005;
        double r316007 = r316003 / r316006;
        double r316008 = pow(r316007, r315993);
        double r316009 = r315988 / r316005;
        double r316010 = pow(r316009, r315993);
        double r316011 = r316008 * r316010;
        double r316012 = r316002 * r316011;
        double r316013 = M;
        double r316014 = D;
        double r316015 = r316013 * r316014;
        double r316016 = r315992 * r315988;
        double r316017 = r316015 / r316016;
        double r316018 = pow(r316017, r315992);
        double r316019 = r315993 * r316018;
        double r316020 = cbrt(r315998);
        double r316021 = r316020 * r316020;
        double r316022 = r316019 * r316021;
        double r316023 = r316020 / r316004;
        double r316024 = r316022 * r316023;
        double r316025 = r315991 - r316024;
        double r316026 = r316012 * r316025;
        return r316026;
}

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 26.5

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

3. Applied add-cube-cbrt 26.8

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

4. Applied *-un-lft-identity 26.8

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

5. Applied times-frac 26.8

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

6. Applied unpow-prod-down 22.8

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

8. Applied *-un-lft-identity 22.8

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

9. Applied add-cube-cbrt 22.9

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

10. Applied times-frac 22.9

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

11. Applied unpow-prod-down 18.6

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

12. Simplified 18.6

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

14. Applied *-un-lft-identity 18.6

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

15. Applied add-cube-cbrt 18.6

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

16. Applied times-frac 18.6

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

17. Applied associate-*r* 16.5

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

18. Simplified 16.5

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

20. Applied *-un-lft-identity 16.5

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

21. Applied add-cube-cbrt 16.6

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

22. Applied times-frac 16.6

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

23. Applied unpow-prod-down 15.5

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

24. Simplified 15.5

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

25. Final simplification 15.5

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

Reproduce

herbie shell --seed 2020046 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))