Henrywood and Agarwal, Equation (9a)

Average Error: 14.0 → 8.4

Time: 35.5s

Precision: 64

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

↓

\[\sqrt{1 - \frac{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \left(\frac{M}{2} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)\right)}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}} \cdot w0\]

C

double f(double w0, double M, double D, double h, double l, double d) {
        double r4516933 = w0;
        double r4516934 = 1.0;
        double r4516935 = M;
        double r4516936 = D;
        double r4516937 = r4516935 * r4516936;
        double r4516938 = 2.0;
        double r4516939 = d;
        double r4516940 = r4516938 * r4516939;
        double r4516941 = r4516937 / r4516940;
        double r4516942 = pow(r4516941, r4516938);
        double r4516943 = h;
        double r4516944 = l;
        double r4516945 = r4516943 / r4516944;
        double r4516946 = r4516942 * r4516945;
        double r4516947 = r4516934 - r4516946;
        double r4516948 = sqrt(r4516947);
        double r4516949 = r4516933 * r4516948;
        return r4516949;
}

↓

double f(double w0, double M, double D, double h, double l, double d) {
        double r4516950 = 1.0;
        double r4516951 = M;
        double r4516952 = 2.0;
        double r4516953 = r4516951 / r4516952;
        double r4516954 = D;
        double r4516955 = d;
        double r4516956 = r4516954 / r4516955;
        double r4516957 = r4516953 * r4516956;
        double r4516958 = l;
        double r4516959 = cbrt(r4516958);
        double r4516960 = h;
        double r4516961 = cbrt(r4516960);
        double r4516962 = r4516959 / r4516961;
        double r4516963 = r4516957 / r4516962;
        double r4516964 = cbrt(r4516954);
        double r4516965 = cbrt(r4516955);
        double r4516966 = r4516964 / r4516965;
        double r4516967 = r4516953 * r4516966;
        double r4516968 = r4516966 * r4516967;
        double r4516969 = r4516966 * r4516968;
        double r4516970 = r4516969 / r4516962;
        double r4516971 = r4516963 * r4516970;
        double r4516972 = r4516971 / r4516962;
        double r4516973 = r4516950 - r4516972;
        double r4516974 = sqrt(r4516973);
        double r4516975 = w0;
        double r4516976 = r4516974 * r4516975;
        return r4516976;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

1. Initial program 14.0

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

2. Simplified 14.0

   \[\leadsto \color{blue}{\sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \cdot w0}\]
3. Using strategy rm

4. Applied add-cube-cbrt 14.0

   \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \cdot w0\]

5. Applied add-cube-cbrt 14.0

   \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot w0\]

6. Applied times-frac 14.0

   \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}} \cdot w0\]

7. Applied associate-*r* 10.8

   \[\leadsto \sqrt{1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}} \cdot w0\]

8. Simplified 8.3

   \[\leadsto \sqrt{1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]
9. Using strategy rm

10. Applied associate-/r* 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\color{blue}{\frac{\frac{M \cdot D}{2}}{d}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]
11. Using strategy rm

12. Applied *-un-lft-identity 8.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\frac{M \cdot D}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \color{blue}{\left(1 \cdot w0\right)}\]

13. Applied associate-*r* 8.3

    \[\leadsto \color{blue}{\left(\sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\frac{M \cdot D}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot 1\right) \cdot w0}\]

14. Simplified 8.4

    \[\leadsto \color{blue}{\sqrt{1 - \frac{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}}} \cdot w0\]
15. Using strategy rm

16. Applied add-cube-cbrt 8.4

    \[\leadsto \sqrt{1 - \frac{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{M}{2} \cdot \frac{D}{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}} \cdot w0\]

17. Applied add-cube-cbrt 8.4

    \[\leadsto \sqrt{1 - \frac{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{M}{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{D} \cdot \sqrt[3]{D}\right) \cdot \sqrt[3]{D}}}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}} \cdot w0\]

18. Applied times-frac 8.4

    \[\leadsto \sqrt{1 - \frac{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{M}{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{D} \cdot \sqrt[3]{D}}{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}} \cdot w0\]

19. Applied associate-*r* 8.4

    \[\leadsto \sqrt{1 - \frac{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\color{blue}{\left(\frac{M}{2} \cdot \frac{\sqrt[3]{D} \cdot \sqrt[3]{D}}{\sqrt[3]{d} \cdot \sqrt[3]{d}}\right) \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}} \cdot w0\]

20. Simplified 8.4

    \[\leadsto \sqrt{1 - \frac{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\color{blue}{\left(\left(\frac{M}{2} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right) \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}} \cdot w0\]

21. Final simplification 8.4

    \[\leadsto \sqrt{1 - \frac{\frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \left(\frac{\sqrt[3]{D}}{\sqrt[3]{d}} \cdot \left(\frac{M}{2} \cdot \frac{\sqrt[3]{D}}{\sqrt[3]{d}}\right)\right)}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}} \cdot w0\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))