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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \le 2.827185544827511 \cdot 10^{-189}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot (\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{-1}{2}\right) \cdot \left(\frac{h}{\ell}\right) + 1)_*}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array}\]

Derivation

Split input into 2 regimes
if l < 2.827185544827511e-189
1. Initial program 26.9
  \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
if 2.827185544827511e-189 < l
1. Initial program 25.4
  \[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
2. Initial simplification25.7
  \[\leadsto (\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{-1}{2}\right) \cdot \left(\frac{h}{\ell}\right) + 1)_* \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \sqrt{\frac{d}{h}}\right)\]
3. Using strategy rm
4. Applied sqrt-div19.0
  \[\leadsto (\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{-1}{2}\right) \cdot \left(\frac{h}{\ell}\right) + 1)_* \cdot \left(\sqrt{\frac{d}{\ell}} \cdot \color{blue}{\frac{\sqrt{d}}{\sqrt{h}}}\right)\]
5. Applied sqrt-div15.1
  \[\leadsto (\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{-1}{2}\right) \cdot \left(\frac{h}{\ell}\right) + 1)_* \cdot \left(\color{blue}{\frac{\sqrt{d}}{\sqrt{\ell}}} \cdot \frac{\sqrt{d}}{\sqrt{h}}\right)\]
6. Applied frac-times15.1
  \[\leadsto (\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{-1}{2}\right) \cdot \left(\frac{h}{\ell}\right) + 1)_* \cdot \color{blue}{\frac{\sqrt{d} \cdot \sqrt{d}}{\sqrt{\ell} \cdot \sqrt{h}}}\]
7. Applied associate-*r/15.2
  \[\leadsto \color{blue}{\frac{(\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{-1}{2}\right) \cdot \left(\frac{h}{\ell}\right) + 1)_* \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\sqrt{\ell} \cdot \sqrt{h}}}\]
Recombined 2 regimes into one program.
Final simplification21.6
\[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le 2.827185544827511 \cdot 10^{-189}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \left({\left(\frac{D \cdot M}{d \cdot 2}\right)}^{2} \cdot \frac{1}{2}\right) \cdot \frac{h}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot (\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{-1}{2}\right) \cdot \left(\frac{h}{\ell}\right) + 1)_*}{\sqrt{h} \cdot \sqrt{\ell}}\\ \end{array}\]

Error

Derivation

`if l < 2.827185544827511e-189`

`if 2.827185544827511e-189 < l`

Runtime