Henrywood and Agarwal, Equation (12)

Average Error: 25.4 → 9.4

Time: 5.8m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le 1.676306393372541 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{h}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} - \frac{\frac{\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\sqrt{2}} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\frac{\sqrt{2}}{\frac{M}{2} \cdot \frac{D}{d}}}}{\frac{1}{h}}\right)\\ \mathbf{elif}\;d \le 8.058937538114404 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left|\sqrt[3]{d}\right| \cdot \left(\left(\frac{\sqrt{d}}{h} - \sqrt{\ell} \cdot \left(\frac{\frac{\left|\sqrt[3]{d}\right|}{\sqrt{2}}}{\frac{\ell}{\frac{\frac{M}{2} \cdot D}{d}}} \cdot \left(\frac{\frac{M}{2} \cdot D}{d} \cdot \frac{\sqrt{\frac{\sqrt[3]{d}}{\ell}}}{\sqrt{2}}\right)\right)\right) \cdot \sqrt{\sqrt[3]{d}}\right)}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{h}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} - \frac{\frac{\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\sqrt{2}} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\frac{\sqrt{2}}{\frac{M}{2} \cdot \frac{D}{d}}}}{\frac{1}{h}}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r15378862 = d;
        double r15378863 = h;
        double r15378864 = r15378862 / r15378863;
        double r15378865 = 1.0;
        double r15378866 = 2.0;
        double r15378867 = r15378865 / r15378866;
        double r15378868 = pow(r15378864, r15378867);
        double r15378869 = l;
        double r15378870 = r15378862 / r15378869;
        double r15378871 = pow(r15378870, r15378867);
        double r15378872 = r15378868 * r15378871;
        double r15378873 = M;
        double r15378874 = D;
        double r15378875 = r15378873 * r15378874;
        double r15378876 = r15378866 * r15378862;
        double r15378877 = r15378875 / r15378876;
        double r15378878 = pow(r15378877, r15378866);
        double r15378879 = r15378867 * r15378878;
        double r15378880 = r15378863 / r15378869;
        double r15378881 = r15378879 * r15378880;
        double r15378882 = r15378865 - r15378881;
        double r15378883 = r15378872 * r15378882;
        return r15378883;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r15378884 = d;
        double r15378885 = 1.676306393372541e-308;
        bool r15378886 = r15378884 <= r15378885;
        double r15378887 = cbrt(r15378884);
        double r15378888 = r15378887 * r15378887;
        double r15378889 = sqrt(r15378888);
        double r15378890 = h;
        double r15378891 = r15378887 / r15378890;
        double r15378892 = sqrt(r15378891);
        double r15378893 = r15378889 * r15378892;
        double r15378894 = l;
        double r15378895 = cbrt(r15378894);
        double r15378896 = r15378887 / r15378895;
        double r15378897 = fabs(r15378896);
        double r15378898 = sqrt(r15378896);
        double r15378899 = r15378897 * r15378898;
        double r15378900 = 2.0;
        double r15378901 = sqrt(r15378900);
        double r15378902 = r15378889 / r15378901;
        double r15378903 = M;
        double r15378904 = r15378903 / r15378900;
        double r15378905 = D;
        double r15378906 = r15378905 / r15378884;
        double r15378907 = r15378904 * r15378906;
        double r15378908 = r15378902 * r15378907;
        double r15378909 = r15378908 / r15378894;
        double r15378910 = 1.0;
        double r15378911 = r15378895 * r15378895;
        double r15378912 = r15378910 / r15378911;
        double r15378913 = sqrt(r15378912);
        double r15378914 = r15378898 * r15378913;
        double r15378915 = r15378901 / r15378907;
        double r15378916 = r15378914 / r15378915;
        double r15378917 = r15378910 / r15378890;
        double r15378918 = r15378916 / r15378917;
        double r15378919 = r15378909 * r15378918;
        double r15378920 = r15378899 - r15378919;
        double r15378921 = r15378893 * r15378920;
        double r15378922 = 8.058937538114404e+37;
        bool r15378923 = r15378884 <= r15378922;
        double r15378924 = fabs(r15378887);
        double r15378925 = sqrt(r15378884);
        double r15378926 = r15378925 / r15378890;
        double r15378927 = sqrt(r15378894);
        double r15378928 = r15378924 / r15378901;
        double r15378929 = r15378904 * r15378905;
        double r15378930 = r15378929 / r15378884;
        double r15378931 = r15378894 / r15378930;
        double r15378932 = r15378928 / r15378931;
        double r15378933 = r15378887 / r15378894;
        double r15378934 = sqrt(r15378933);
        double r15378935 = r15378934 / r15378901;
        double r15378936 = r15378930 * r15378935;
        double r15378937 = r15378932 * r15378936;
        double r15378938 = r15378927 * r15378937;
        double r15378939 = r15378926 - r15378938;
        double r15378940 = sqrt(r15378887);
        double r15378941 = r15378939 * r15378940;
        double r15378942 = r15378924 * r15378941;
        double r15378943 = sqrt(r15378890);
        double r15378944 = r15378943 * r15378927;
        double r15378945 = r15378944 / r15378890;
        double r15378946 = r15378942 / r15378945;
        double r15378947 = r15378923 ? r15378946 : r15378921;
        double r15378948 = r15378886 ? r15378921 : r15378947;
        return r15378948;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 2 regimes

2. if d < 1.676306393372541e-308 or 8.058937538114404e+37 < d

   1. Initial program 24.7

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

   2. Simplified 24.9

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

   4. Applied *-un-lft-identity 24.9

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

   5. Applied add-cube-cbrt 25.1

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

   6. Applied times-frac 25.1

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

   7. Applied sqrt-prod 19.2

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

   8. Simplified 19.2

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

   10. Applied div-inv 19.2

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

   11. Applied add-sqr-sqrt 19.2

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

   12. Applied times-frac 19.1

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

   13. Applied *-un-lft-identity 19.1

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

   14. Applied add-cube-cbrt 19.1

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

   15. Applied times-frac 19.1

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

   16. Applied sqrt-prod 18.8

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

   17. Applied times-frac 17.9

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

   18. Applied times-frac 14.7

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

   19. Simplified 14.7

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

   21. Applied add-cube-cbrt 14.9

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

   22. Applied add-cube-cbrt 15.1

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

   23. Applied times-frac 15.1

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

   24. Applied sqrt-prod 9.9

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

   25. Simplified 9.7

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

   27. Applied add-cube-cbrt 9.7

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

   28. Applied *-un-lft-identity 9.7

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

   29. Applied cbrt-prod 9.7

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

   30. Applied times-frac 9.7

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

   31. Applied sqrt-prod 8.6

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

   32. Simplified 8.6

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

   if 1.676306393372541e-308 < d < 8.058937538114404e+37

   1. Initial program 27.5

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

   2. Simplified 28.3

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

   4. Applied *-un-lft-identity 28.3

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

   5. Applied add-cube-cbrt 28.5

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

   6. Applied times-frac 28.5

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

   7. Applied sqrt-prod 26.3

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

   8. Simplified 26.3

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

   10. Applied div-inv 26.3

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

   11. Applied add-sqr-sqrt 26.4

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

   12. Applied times-frac 26.2

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

   13. Applied *-un-lft-identity 26.2

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

   14. Applied add-cube-cbrt 26.2

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

   15. Applied times-frac 26.2

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

   16. Applied sqrt-prod 25.4

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

   17. Applied times-frac 23.3

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

   18. Applied times-frac 18.4

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

   19. Simplified 18.4

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

   21. Applied sqrt-div 17.2

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

   22. Applied associate-*r/ 17.2

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

   23. Applied associate-*r/ 19.6

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

   24. Applied sqrt-div 15.3

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

   25. Applied frac-sub 20.4

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

   26. Applied frac-times 19.1

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

   27. Simplified 16.2

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

   28. Simplified 11.5

       \[\leadsto \frac{\left(\left(\frac{\sqrt{d}}{h} - \sqrt{\ell} \cdot \left(\frac{\frac{\left|\sqrt[3]{d}\right|}{\sqrt{2}}}{\frac{\ell}{\frac{D \cdot \frac{M}{2}}{d}}} \cdot \left(\frac{D \cdot \frac{M}{2}}{d} \cdot \frac{\sqrt{\frac{\sqrt[3]{d}}{\ell}}}{\sqrt{2}}\right)\right)\right) \cdot \sqrt{\sqrt[3]{d}}\right) \cdot \left|\sqrt[3]{d}\right|}{\color{blue}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{h}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 1.676306393372541 \cdot 10^{-308}:\\ \;\;\;\;\left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{h}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} - \frac{\frac{\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\sqrt{2}} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\frac{\sqrt{2}}{\frac{M}{2} \cdot \frac{D}{d}}}}{\frac{1}{h}}\right)\\ \mathbf{elif}\;d \le 8.058937538114404 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left|\sqrt[3]{d}\right| \cdot \left(\left(\frac{\sqrt{d}}{h} - \sqrt{\ell} \cdot \left(\frac{\frac{\left|\sqrt[3]{d}\right|}{\sqrt{2}}}{\frac{\ell}{\frac{\frac{M}{2} \cdot D}{d}}} \cdot \left(\frac{\frac{M}{2} \cdot D}{d} \cdot \frac{\sqrt{\frac{\sqrt[3]{d}}{\ell}}}{\sqrt{2}}\right)\right)\right) \cdot \sqrt{\sqrt[3]{d}}\right)}{\frac{\sqrt{h} \cdot \sqrt{\ell}}{h}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}} \cdot \sqrt{\frac{\sqrt[3]{d}}{h}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} - \frac{\frac{\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\sqrt{2}} \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)}{\ell} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\frac{\sqrt{2}}{\frac{M}{2} \cdot \frac{D}{d}}}}{\frac{1}{h}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(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)))))