Error

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 add-log-exp0.2

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

4. Applied add-log-exp0.2

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

5. Applied diff-log0.2

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

6. Applied simplify0.2

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

8. Applied add-sqr-sqrt0.2

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

9. Applied *-un-lft-identity0.2

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

10. Applied times-frac0.2

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

11. Applied log-prod0.2

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

12. Applied simplify0.2

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

13. Applied simplify0.2

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

15. Applied add-sqr-sqrt0.2

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

16. Applied unpow-prod-down0.2

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

17. Applied sqrt-prod0.2

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

18. Applied log-prod0.2

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

20. Applied add-cube-cbrt0.3

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

21. Applied simplify0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\sqrt[3]{(2 \cdot \left(-\log \left(\sqrt{{\left(\sqrt{e^{\sin \phi_1}}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right) + \left(\cos delta - \log \left(\sqrt{{\left(e^{\sin \phi_1}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right))_*} \cdot \sqrt[3]{(2 \cdot \left(-\log \left(\sqrt{{\left(\sqrt{e^{\sin \phi_1}}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right) + \left(\cos delta - \log \left(\sqrt{{\left(e^{\sin \phi_1}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right))_*}\right)} \cdot \sqrt[3]{\left(-\left(\log \left(\sqrt{{\left(\sqrt{e^{\sin \phi_1}}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right)}}\right) + \log \left(\sqrt{{\left(\sqrt{e^{\sin \phi_1}}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right)}}\right)\right)\right) + \left(\cos delta - \log \left(\sqrt{{\left(e^{\sin \phi_1}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right)}}\]

22. Applied simplify0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\sqrt[3]{(2 \cdot \left(-\log \left(\sqrt{{\left(\sqrt{e^{\sin \phi_1}}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right) + \left(\cos delta - \log \left(\sqrt{{\left(e^{\sin \phi_1}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right))_*} \cdot \sqrt[3]{(2 \cdot \left(-\log \left(\sqrt{{\left(\sqrt{e^{\sin \phi_1}}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right) + \left(\cos delta - \log \left(\sqrt{{\left(e^{\sin \phi_1}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right))_*}\right) \cdot \color{blue}{\sqrt[3]{(2 \cdot \left(-\log \left(\sqrt{{\left(\sqrt{e^{\sin \phi_1}}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right) + \left(\cos delta - \log \left(\sqrt{{\left(e^{\sin \phi_1}\right)}^{\left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right)\right)}}\right)\right))_*}}}\]

Runtime

Time bar (total: 51.4s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +o rules:numerics
(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))))))))))