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Derivation

1. Initial program 0.2

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
2. Using strategy rm

3. Applied flip--0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}}\]
4. Using strategy rm

5. Applied add-cbrt-cube0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

6. Applied add-cbrt-cube0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1}} \cdot \sqrt[3]{\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

7. Applied cbrt-unprod0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

8. Applied add-cbrt-cube0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\right) \cdot \sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

9. Applied add-cbrt-cube0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - \left(\color{blue}{\sqrt[3]{\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1}} \cdot \sqrt[3]{\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right) \cdot \sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

10. Applied cbrt-unprod0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - \color{blue}{\sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}} \cdot \sqrt[3]{\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

11. Applied cbrt-unprod0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - \color{blue}{\sqrt[3]{\left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right) \cdot \left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin \phi_1\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

12. Applied simplify0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - \sqrt[3]{\color{blue}{\left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right)}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]
13. Using strategy rm

14. Applied add-sqr-sqrt0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - \color{blue}{\sqrt{\sqrt[3]{\left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right)}} \cdot \sqrt{\sqrt[3]{\left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right)}}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

15. Applied difference-of-squares0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\left(\cos delta + \sqrt{\sqrt[3]{\left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right)}}\right) \cdot \left(\cos delta - \sqrt{\sqrt[3]{\left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right) \cdot \left(\left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{3}\right) \cdot \left(\sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right) \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)\right)\right)}}\right)}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018178 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))