Error

Derivation

1. Initial program 0.1

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

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

6. Applied add-cbrt-cube0.2

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

7. Applied cbrt-unprod0.2

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

8. Applied simplify0.2

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

10. Applied add-cbrt-cube0.2

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

11. Applied simplify0.2

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

13. Applied add-cube-cbrt0.3

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

14. Applied unpow-prod-down0.3

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

15. Applied cbrt-prod0.3

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

16. Applied simplify0.2

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

17. Applied simplify0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\sqrt[3]{\left(\cos delta \cdot \cos delta - \sqrt[3]{{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)\right)}^{\left(1 + \left(3 + 2\right)\right)}}\right) \cdot \left(\cos delta \cdot \cos delta - \sqrt[3]{{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)\right)}^{\left(1 + \left(3 + 2\right)\right)}}\right)} \cdot \color{blue}{\sqrt[3]{\cos delta \cdot \cos delta - \sqrt[3]{{\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos theta \cdot \sin delta\right) \cdot \cos \phi_1\right)\right)\right)}^{\left(\left(3 + 1\right) + 2\right)}}}}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]
18. Removed slow pow expressions.

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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))))))))))