Henrywood and Agarwal, Equation (12)

Average Error: 25.6 → 11.2

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le 5.63930426562316 \cdot 10^{-278}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right) \cdot \left(1 - \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\sqrt[3]{d}}\right)\right) \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}}{\sqrt{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \sqrt{\ell}}\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r5246954 = d;
        double r5246955 = h;
        double r5246956 = r5246954 / r5246955;
        double r5246957 = 1.0;
        double r5246958 = 2.0;
        double r5246959 = r5246957 / r5246958;
        double r5246960 = pow(r5246956, r5246959);
        double r5246961 = l;
        double r5246962 = r5246954 / r5246961;
        double r5246963 = pow(r5246962, r5246959);
        double r5246964 = r5246960 * r5246963;
        double r5246965 = M;
        double r5246966 = D;
        double r5246967 = r5246965 * r5246966;
        double r5246968 = r5246958 * r5246954;
        double r5246969 = r5246967 / r5246968;
        double r5246970 = pow(r5246969, r5246958);
        double r5246971 = r5246959 * r5246970;
        double r5246972 = r5246955 / r5246961;
        double r5246973 = r5246971 * r5246972;
        double r5246974 = r5246957 - r5246973;
        double r5246975 = r5246964 * r5246974;
        return r5246975;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r5246976 = d;
        double r5246977 = 5.63930426562316e-278;
        bool r5246978 = r5246976 <= r5246977;
        double r5246979 = h;
        double r5246980 = cbrt(r5246979);
        double r5246981 = r5246976 / r5246980;
        double r5246982 = sqrt(r5246981);
        double r5246983 = 1.0;
        double r5246984 = r5246980 * r5246980;
        double r5246985 = r5246983 / r5246984;
        double r5246986 = sqrt(r5246985);
        double r5246987 = r5246982 * r5246986;
        double r5246988 = cbrt(r5246976);
        double r5246989 = l;
        double r5246990 = r5246988 / r5246989;
        double r5246991 = sqrt(r5246990);
        double r5246992 = fabs(r5246988);
        double r5246993 = r5246991 * r5246992;
        double r5246994 = M;
        double r5246995 = D;
        double r5246996 = r5246994 * r5246995;
        double r5246997 = 2.0;
        double r5246998 = r5246976 * r5246997;
        double r5246999 = r5246996 / r5246998;
        double r5247000 = cbrt(r5246989);
        double r5247001 = r5246980 / r5247000;
        double r5247002 = r5246999 * r5247001;
        double r5247003 = 0.5;
        double r5247004 = r5247002 * r5247003;
        double r5247005 = r5247002 * r5247004;
        double r5247006 = r5247005 * r5247001;
        double r5247007 = r5246983 - r5247006;
        double r5247008 = r5246993 * r5247007;
        double r5247009 = r5246987 * r5247008;
        double r5247010 = sqrt(r5246988);
        double r5247011 = r5246992 * r5247010;
        double r5247012 = r5247007 * r5247011;
        double r5247013 = r5247012 * r5246982;
        double r5247014 = sqrt(r5246984);
        double r5247015 = sqrt(r5246989);
        double r5247016 = r5247014 * r5247015;
        double r5247017 = r5247013 / r5247016;
        double r5247018 = r5246978 ? r5247009 : r5247017;
        return r5247018;
}

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 < 5.63930426562316e-278

   1. Initial program 26.1

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

   3. Applied *-un-lft-identity 26.1

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

   4. Applied add-cube-cbrt 26.3

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

   5. Applied times-frac 26.3

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

   6. Applied unpow-prod-down 22.7

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

   7. Simplified 22.7

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

   8. Simplified 22.7

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

   10. Applied add-cube-cbrt 22.8

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

   11. Applied add-cube-cbrt 22.8

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

   12. Applied times-frac 22.8

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

   13. Applied associate-*r* 20.7

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

   14. Simplified 18.1

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

   16. Applied add-cube-cbrt 18.1

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

   17. Applied *-un-lft-identity 18.1

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

   18. Applied times-frac 18.1

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

   19. Applied unpow-prod-down 12.8

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

   20. Simplified 12.8

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

   21. Simplified 12.8

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

   23. Applied associate-*l* 12.7

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

   24. Simplified 12.7

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

   if 5.63930426562316e-278 < d

   1. Initial program 25.1

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

   3. Applied *-un-lft-identity 25.1

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

   4. Applied add-cube-cbrt 25.4

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

   5. Applied times-frac 25.4

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

   6. Applied unpow-prod-down 21.7

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

   7. Simplified 21.7

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

   8. Simplified 21.7

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

   10. Applied add-cube-cbrt 21.8

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

   11. Applied add-cube-cbrt 21.8

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

   12. Applied times-frac 21.8

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

   13. Applied associate-*r* 19.1

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

   14. Simplified 16.3

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

   16. Applied add-cube-cbrt 16.4

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

   17. Applied *-un-lft-identity 16.4

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

   18. Applied times-frac 16.4

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

   19. Applied unpow-prod-down 11.3

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

   20. Simplified 11.3

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

   21. Simplified 11.3

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

   23. Applied sqrt-div 9.8

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

   24. Applied associate-*r/ 9.7

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

   25. Applied sqrt-div 9.7

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

   26. Applied associate-*l/ 9.7

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

   27. Applied frac-times 9.7

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

   28. Applied associate-*l/ 9.5

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

   29. Simplified 9.5

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

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 5.63930426562316 \cdot 10^{-278}:\\ \;\;\;\;\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right) \cdot \left(1 - \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(1 - \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\sqrt[3]{d}}\right)\right) \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}}{\sqrt{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \sqrt{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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)))))