Average Error: 14.6 → 8.7
Time: 12.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r190942 = w0;
        double r190943 = 1.0;
        double r190944 = M;
        double r190945 = D;
        double r190946 = r190944 * r190945;
        double r190947 = 2.0;
        double r190948 = d;
        double r190949 = r190947 * r190948;
        double r190950 = r190946 / r190949;
        double r190951 = pow(r190950, r190947);
        double r190952 = h;
        double r190953 = l;
        double r190954 = r190952 / r190953;
        double r190955 = r190951 * r190954;
        double r190956 = r190943 - r190955;
        double r190957 = sqrt(r190956);
        double r190958 = r190942 * r190957;
        return r190958;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r190959 = w0;
        double r190960 = 1.0;
        double r190961 = M;
        double r190962 = D;
        double r190963 = r190961 * r190962;
        double r190964 = 2.0;
        double r190965 = d;
        double r190966 = r190964 * r190965;
        double r190967 = r190963 / r190966;
        double r190968 = 2.0;
        double r190969 = r190964 / r190968;
        double r190970 = pow(r190967, r190969);
        double r190971 = h;
        double r190972 = cbrt(r190971);
        double r190973 = l;
        double r190974 = cbrt(r190973);
        double r190975 = r190972 / r190974;
        double r190976 = r190970 * r190975;
        double r190977 = cbrt(r190972);
        double r190978 = r190977 * r190977;
        double r190979 = r190974 * r190974;
        double r190980 = cbrt(r190979);
        double r190981 = r190978 / r190980;
        double r190982 = r190970 * r190981;
        double r190983 = cbrt(r190974);
        double r190984 = r190977 / r190983;
        double r190985 = r190982 * r190984;
        double r190986 = r190976 * r190985;
        double r190987 = r190986 * r190975;
        double r190988 = r190960 - r190987;
        double r190989 = sqrt(r190988);
        double r190990 = r190959 * r190989;
        return r190990;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.6

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

  3. Applied add-cube-cbrt14.6

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

  4. Applied add-cube-cbrt14.6

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

  5. Applied times-frac14.6

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

  6. Applied associate-*r*11.3

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

  8. Applied times-frac11.3

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

  9. Applied sqr-pow11.3

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

  10. Applied unswap-sqr8.7

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

  12. Applied add-cube-cbrt8.7

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

  13. Applied cbrt-prod8.7

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

  14. Applied add-cube-cbrt8.7

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

  15. Applied times-frac8.7

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

  16. Applied associate-*r*8.7

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

  17. Final simplification8.7

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))