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

↓

\[\begin{array}{l} \mathbf{if}\;d \le -6.084062671205626 \cdot 10^{-79}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\frac{1}{2}}\right)\right)\\ \mathbf{elif}\;d \le 6.739049195027558 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left(\left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right) \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \frac{h}{\ell}\right)\right)\right) + \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\frac{1}{2}}\right)\right)\\ \end{array}\]

double f(double d, double h, double l, double M, double D) {
        double r40650906 = d;
        double r40650907 = h;
        double r40650908 = r40650906 / r40650907;
        double r40650909 = 1.0;
        double r40650910 = 2.0;
        double r40650911 = r40650909 / r40650910;
        double r40650912 = pow(r40650908, r40650911);
        double r40650913 = l;
        double r40650914 = r40650906 / r40650913;
        double r40650915 = pow(r40650914, r40650911);
        double r40650916 = r40650912 * r40650915;
        double r40650917 = M;
        double r40650918 = D;
        double r40650919 = r40650917 * r40650918;
        double r40650920 = r40650910 * r40650906;
        double r40650921 = r40650919 / r40650920;
        double r40650922 = pow(r40650921, r40650910);
        double r40650923 = r40650911 * r40650922;
        double r40650924 = r40650907 / r40650913;
        double r40650925 = r40650923 * r40650924;
        double r40650926 = r40650909 - r40650925;
        double r40650927 = r40650916 * r40650926;
        return r40650927;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r40650928 = d;
        double r40650929 = -6.084062671205626e-79;
        bool r40650930 = r40650928 <= r40650929;
        double r40650931 = 1.0;
        double r40650932 = l;
        double r40650933 = r40650931 / r40650932;
        double r40650934 = h;
        double r40650935 = D;
        double r40650936 = r40650935 / r40650928;
        double r40650937 = M;
        double r40650938 = 2.0;
        double r40650939 = r40650937 / r40650938;
        double r40650940 = r40650936 * r40650939;
        double r40650941 = pow(r40650940, r40650938);
        double r40650942 = 0.5;
        double r40650943 = r40650941 * r40650942;
        double r40650944 = r40650934 * r40650943;
        double r40650945 = r40650933 * r40650944;
        double r40650946 = r40650931 - r40650945;
        double r40650947 = cbrt(r40650928);
        double r40650948 = cbrt(r40650934);
        double r40650949 = r40650947 / r40650948;
        double r40650950 = pow(r40650949, r40650942);
        double r40650951 = fabs(r40650949);
        double r40650952 = r40650950 * r40650951;
        double r40650953 = fabs(r40650947);
        double r40650954 = r40650947 / r40650932;
        double r40650955 = pow(r40650954, r40650942);
        double r40650956 = r40650953 * r40650955;
        double r40650957 = r40650952 * r40650956;
        double r40650958 = r40650946 * r40650957;
        double r40650959 = 6.739049195027558e+112;
        bool r40650960 = r40650928 <= r40650959;
        double r40650961 = sqrt(r40650949);
        double r40650962 = sqrt(r40650954);
        double r40650963 = r40650953 * r40650962;
        double r40650964 = -0.5;
        double r40650965 = r40650938 / r40650936;
        double r40650966 = r40650937 / r40650965;
        double r40650967 = r40650966 * r40650966;
        double r40650968 = r40650964 * r40650967;
        double r40650969 = r40650934 / r40650932;
        double r40650970 = r40650951 * r40650969;
        double r40650971 = r40650968 * r40650970;
        double r40650972 = r40650963 * r40650971;
        double r40650973 = r40650961 * r40650972;
        double r40650974 = r40650973 + r40650957;
        double r40650975 = r40650960 ? r40650974 : r40650958;
        double r40650976 = r40650930 ? r40650958 : r40650975;
        return r40650976;
}

\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 -6.084062671205626 \cdot 10^{-79}:\\
\;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\frac{1}{2}}\right)\right)\\

\mathbf{elif}\;d \le 6.739049195027558 \cdot 10^{+112}:\\
\;\;\;\;\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left(\left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right) \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \frac{h}{\ell}\right)\right)\right) + \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\frac{1}{2}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\frac{1}{2}}\right)\right)\\

\end{array}

Derivation

Split input into 2 regimes
if d < -6.084062671205626e-79 or 6.739049195027558e+112 < d
1. Initial program 23.0
  \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
2. Using strategy rm
3. Applied add-cube-cbrt23.3
  \[\leadsto \left({\left(\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
4. Applied add-cube-cbrt23.4
  \[\leadsto \left({\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
5. Applied times-frac23.4
  \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
6. Applied unpow-prod-down15.2
  \[\leadsto \left(\color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
7. Simplified13.9
  \[\leadsto \left(\left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
8. Using strategy rm
9. Applied *-un-lft-identity13.9
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\color{blue}{1 \cdot \ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
10. Applied add-cube-cbrt14.1
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{1 \cdot \ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
11. Applied times-frac14.1
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1} \cdot \frac{\sqrt[3]{d}}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
12. Applied unpow-prod-down11.2
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
13. Simplified11.2
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left|\sqrt[3]{d}\right|} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
14. Using strategy rm
15. Applied div-inv11.2
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}\right)\]
16. Applied associate-*r*8.2
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h\right) \cdot \frac{1}{\ell}}\right)\]
17. Using strategy rm
18. Applied times-frac8.0
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\left(\frac{1}{2} \cdot {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2}\right) \cdot h\right) \cdot \frac{1}{\ell}\right)\]
if -6.084062671205626e-79 < d < 6.739049195027558e+112
1. Initial program 28.5
  \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
2. Using strategy rm
3. Applied add-cube-cbrt28.8
  \[\leadsto \left({\left(\frac{d}{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
4. Applied add-cube-cbrt28.8
  \[\leadsto \left({\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
5. Applied times-frac28.8
  \[\leadsto \left({\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
6. Applied unpow-prod-down26.6
  \[\leadsto \left(\color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
7. Simplified26.6
  \[\leadsto \left(\left(\color{blue}{\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
8. Using strategy rm
9. Applied *-un-lft-identity26.6
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{\color{blue}{1 \cdot \ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
10. Applied add-cube-cbrt26.8
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{1 \cdot \ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
11. Applied times-frac26.8
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\color{blue}{\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1} \cdot \frac{\sqrt[3]{d}}{\ell}\right)}}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
12. Applied unpow-prod-down23.0
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
13. Simplified23.0
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left|\sqrt[3]{d}\right|} \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
14. Using strategy rm
15. Applied div-inv23.0
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}\right)\]
16. Applied associate-*r*21.5
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(1 - \color{blue}{\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h\right) \cdot \frac{1}{\ell}}\right)\]
17. Using strategy rm
18. Applied sub-neg21.5
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \color{blue}{\left(1 + \left(-\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h\right) \cdot \frac{1}{\ell}\right)\right)}\]
19. Applied distribute-lft-in21.5
  \[\leadsto \color{blue}{\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot 1 + \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot \left(-\left(\left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot h\right) \cdot \frac{1}{\ell}\right)}\]
20. Simplified22.2
  \[\leadsto \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right) \cdot 1 + \color{blue}{\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left(\left(\left(\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right) \cdot \frac{-1}{2}\right) \cdot \left(\frac{h}{\ell} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification15.4
\[\leadsto \begin{array}{l} \mathbf{if}\;d \le -6.084062671205626 \cdot 10^{-79}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\frac{1}{2}}\right)\right)\\ \mathbf{elif}\;d \le 6.739049195027558 \cdot 10^{+112}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left(\left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right) \cdot \left(\left(\frac{-1}{2} \cdot \left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \frac{h}{\ell}\right)\right)\right) + \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{\ell} \cdot \left(h \cdot \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{2} \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot {\left(\frac{\sqrt[3]{d}}{\ell}\right)}^{\frac{1}{2}}\right)\right)\\ \end{array}\]

Error

Derivation

`if d < -6.084062671205626e-79 or 6.739049195027558e+112 < d`

`if -6.084062671205626e-79 < d < 6.739049195027558e+112`

Reproduce