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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.7292671685933904 \cdot 10^{+277}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{if}\;\frac{h}{\ell} \le -2.020589533065359 \cdot 10^{-239}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \sqrt[3]{{\left(\frac{h \cdot \frac{D \cdot M}{d + d}}{\frac{\ell}{\frac{D \cdot M}{d + d}}}\right)}^{3}}}\\ \end{array}\]

Derivation

Split input into 3 regimes
if (/ h l) < -1.7292671685933904e+277
1. Initial program 55.4
  \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
2. Using strategy rm
3. Applied div-inv55.4
  \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]
4. Applied associate-*r*23.9
  \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
if -1.7292671685933904e+277 < (/ h l) < -2.020589533065359e-239
1. Initial program 12.6
  \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
2. Using strategy rm
3. Applied times-frac12.6
  \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}}^{2} \cdot \frac{h}{\ell}}\]
if -2.020589533065359e-239 < (/ h l)
1. Initial program 7.9
  \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
2. Using strategy rm
3. Applied div-inv7.9
  \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]
4. Applied associate-*r*4.9
  \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
5. Using strategy rm
6. Applied add-cbrt-cube4.9
  \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\sqrt[3]{\left(\left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right) \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}\right)}}}\]
7. Applied simplify3.5
  \[\leadsto w0 \cdot \sqrt{1 - \sqrt[3]{\color{blue}{{\left(\frac{h \cdot \frac{D \cdot M}{d + d}}{\frac{\ell}{\frac{D \cdot M}{d + d}}}\right)}^{3}}}}\]
Recombined 3 regimes into one program.

Error

Derivation

`if (/ h l) < -1.7292671685933904e+277`

`if -1.7292671685933904e+277 < (/ h l) < -2.020589533065359e-239`

`if -2.020589533065359e-239 < (/ h l)`

Runtime