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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{1}{\ell} \le -1.2250626974477426 \cdot 10^{+35}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{if}\;\frac{1}{\ell} \le 6.341168922503132 \cdot 10^{-12}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \sqrt[3]{{\left(\frac{h \cdot \frac{D \cdot M}{d + d}}{\frac{\ell}{\frac{D \cdot M}{d + d}}}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \end{array}\]

Derivation

Split input into 2 regimes
if (/ 1 l) < -1.2250626974477426e+35 or 6.341168922503132e-12 < (/ 1 l)
1. Initial program 18.8
  \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
2. Using strategy rm
3. Applied div-inv18.8
  \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]
4. Applied associate-*r*10.2
  \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
5. Using strategy rm
6. Applied times-frac10.3
  \[\leadsto w0 \cdot \sqrt{1 - \left({\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\]
if -1.2250626974477426e+35 < (/ 1 l) < 6.341168922503132e-12
1. Initial program 9.3
  \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
2. Using strategy rm
3. Applied div-inv9.3
  \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]
4. Applied associate-*r*10.0
  \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
5. Using strategy rm
6. Applied add-cbrt-cube10.9
  \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\sqrt[3]{\left(\left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right) \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right)}}}\]
7. Applied simplify8.9
  \[\leadsto w0 \cdot \sqrt{1 - \sqrt[3]{\color{blue}{{\left(\frac{h \cdot \frac{D \cdot M}{d + d}}{\frac{\ell}{\frac{D \cdot M}{d + d}}}\right)}^{3}}}}\]
Recombined 2 regimes into one program.

Error

Derivation

`if (/ 1 l) < -1.2250626974477426e+35 or 6.341168922503132e-12 < (/ 1 l)`

`if -1.2250626974477426e+35 < (/ 1 l) < 6.341168922503132e-12`

Runtime