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Derivation

1. Split input into 3 regimes

2. if (- lambda1 lambda2) < -3.0921645126948295e+158

   1. Initial program 60.8

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   2. Initial simplification60.8

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R\]
   3. Using strategy rm

   4. Applied add-log-exp60.8

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)}\right)} \cdot R\]
   5. Using strategy rm

   6. Applied add-cbrt-cube60.8

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]

   7. Taylor expanded around 0 39.6

      \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R\]

   if -3.0921645126948295e+158 < (- lambda1 lambda2) < 3.179701208867931e+152

   1. Initial program 24.3

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   2. Initial simplification24.3

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R\]
   3. Using strategy rm

   4. Applied add-log-exp24.4

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)}\right)} \cdot R\]
   5. Using strategy rm

   6. Applied add-cbrt-cube24.4

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]
   7. Using strategy rm

   8. Applied add-cbrt-cube24.4

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]

   if 3.179701208867931e+152 < (- lambda1 lambda2)

   1. Initial program 60.3

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

   2. Initial simplification60.3

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot R\]
   3. Using strategy rm

   4. Applied add-log-exp60.3

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)}\right)} \cdot R\]
   5. Using strategy rm

   6. Applied add-cbrt-cube60.3

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]
   7. Using strategy rm

   8. Applied add-cbrt-cube60.3

      \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]
   9. Using strategy rm

   10. Applied cos-mult60.3

       \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)}{2}} \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]

   11. Applied associate-*l/60.3

       \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}{2}}}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]

   12. Applied cbrt-div60.3

       \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}{\sqrt[3]{2}}}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]

   13. Applied associate-*r/60.3

       \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\color{blue}{\frac{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}{\sqrt[3]{2}}}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]

   14. Applied cbrt-div60.3

       \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{\sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}}{\sqrt[3]{\sqrt[3]{2}}}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]

   15. Applied associate-*r/60.3

       \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \color{blue}{\frac{\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}}{\sqrt[3]{\sqrt[3]{2}}}} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)} \cdot R\]

   16. Applied associate-*l/60.3

       \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \color{blue}{\frac{\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)}{\sqrt[3]{\sqrt[3]{2}}}}} \cdot R\]

   17. Applied flip--60.4

       \[\leadsto \sqrt{\left(\phi_1 - \phi_2\right) \cdot \color{blue}{\frac{\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2}{\phi_1 + \phi_2}} + \frac{\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)}{\sqrt[3]{\sqrt[3]{2}}}} \cdot R\]

   18. Applied flip--60.4

       \[\leadsto \sqrt{\color{blue}{\frac{\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2}{\phi_1 + \phi_2}} \cdot \frac{\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2}{\phi_1 + \phi_2} + \frac{\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)}{\sqrt[3]{\sqrt[3]{2}}}} \cdot R\]

   19. Applied frac-times61.2

       \[\leadsto \sqrt{\color{blue}{\frac{\left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right)}{\left(\phi_1 + \phi_2\right) \cdot \left(\phi_1 + \phi_2\right)}} + \frac{\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)}{\sqrt[3]{\sqrt[3]{2}}}} \cdot R\]

   20. Applied frac-add61.3

       \[\leadsto \sqrt{\color{blue}{\frac{\left(\left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right)\right) \cdot \sqrt[3]{\sqrt[3]{2}} + \left(\left(\phi_1 + \phi_2\right) \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)\right)}{\left(\left(\phi_1 + \phi_2\right) \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \sqrt[3]{\sqrt[3]{2}}}}} \cdot R\]

   21. Applied sqrt-div61.3

       \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right)\right) \cdot \sqrt[3]{\sqrt[3]{2}} + \left(\left(\phi_1 + \phi_2\right) \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \sqrt[3]{\left(\cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2} - \frac{\phi_2 + \phi_1}{2}\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)}}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right)\right)\right)}}{\sqrt{\left(\left(\phi_1 + \phi_2\right) \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \sqrt[3]{\sqrt[3]{2}}}}} \cdot R\]

   22. Simplified55.7

       \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{\sqrt[3]{2}} \cdot \left(\left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right)\right) + \left(\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_1 + \phi_2\right) \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) + \cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2} + \frac{\phi_1 + \phi_2}{2}\right)} \cdot \left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}}}}{\sqrt{\left(\left(\phi_1 + \phi_2\right) \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \sqrt[3]{\sqrt[3]{2}}}} \cdot R\]
3. Recombined 3 regimes into one program.

4. Final simplification33.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \le -3.0921645126948295 \cdot 10^{+158}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \le 3.179701208867931 \cdot 10^{+152}:\\ \;\;\;\;R \cdot \sqrt{\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right) + \left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)}\right) \cdot \left(\log \left(e^{\cos \left(\frac{\phi_2 + \phi_1}{2}\right)}\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \frac{\sqrt{\sqrt[3]{\sqrt[3]{\cos \left(\frac{\phi_2 + \phi_1}{2}\right) + \cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2} + \frac{\phi_2 + \phi_1}{2}\right)} \cdot \left(\cos \left(\frac{\phi_2 + \phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)} \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\left(\left(\phi_2 + \phi_1\right) \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right)\right)\right) + \sqrt[3]{\sqrt[3]{2}} \cdot \left(\left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \phi_1 - \phi_2 \cdot \phi_2\right)\right)}}{\sqrt{\left(\left(\phi_2 + \phi_1\right) \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \sqrt[3]{\sqrt[3]{2}}}}\\ \end{array}\]

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed 2018286 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))