Henrywood and Agarwal, Equation (12)

Average Error: 25.8 → 12.1

Time: 4.6m

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 r59150953 = d;
        double r59150954 = h;
        double r59150955 = r59150953 / r59150954;
        double r59150956 = 1.0;
        double r59150957 = 2.0;
        double r59150958 = r59150956 / r59150957;
        double r59150959 = pow(r59150955, r59150958);
        double r59150960 = l;
        double r59150961 = r59150953 / r59150960;
        double r59150962 = pow(r59150961, r59150958);
        double r59150963 = r59150959 * r59150962;
        double r59150964 = M;
        double r59150965 = D;
        double r59150966 = r59150964 * r59150965;
        double r59150967 = r59150957 * r59150953;
        double r59150968 = r59150966 / r59150967;
        double r59150969 = pow(r59150968, r59150957);
        double r59150970 = r59150958 * r59150969;
        double r59150971 = r59150954 / r59150960;
        double r59150972 = r59150970 * r59150971;
        double r59150973 = r59150956 - r59150972;
        double r59150974 = r59150963 * r59150973;
        return r59150974;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r59150975 = 1.0;
        double r59150976 = h;
        double r59150977 = M;
        double r59150978 = D;
        double r59150979 = r59150977 * r59150978;
        double r59150980 = d;
        double r59150981 = 2.0;
        double r59150982 = r59150980 * r59150981;
        double r59150983 = r59150979 / r59150982;
        double r59150984 = r59150983 / r59150981;
        double r59150985 = l;
        double r59150986 = cbrt(r59150985);
        double r59150987 = r59150984 / r59150986;
        double r59150988 = r59150976 * r59150987;
        double r59150989 = r59150986 * r59150986;
        double r59150990 = r59150983 / r59150989;
        double r59150991 = r59150988 * r59150990;
        double r59150992 = r59150975 - r59150991;
        double r59150993 = cbrt(r59150976);
        double r59150994 = r59150993 * r59150993;
        double r59150995 = r59150975 / r59150994;
        double r59150996 = sqrt(r59150995);
        double r59150997 = r59150980 / r59150993;
        double r59150998 = 0.5;
        double r59150999 = pow(r59150997, r59150998);
        double r59151000 = r59150996 * r59150999;
        double r59151001 = cbrt(r59150980);
        double r59151002 = fabs(r59151001);
        double r59151003 = r59151001 / r59150985;
        double r59151004 = pow(r59151003, r59150998);
        double r59151005 = r59151002 * r59151004;
        double r59151006 = r59151000 * r59151005;
        double r59151007 = r59150992 * r59151006;
        return r59151007;
}

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 +o rules:numerics
(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)))))