Average Error: 4.2 → 4.3
Time: 1.3m
Precision: 64
Internal Precision: 128
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\frac{{\left(\cos delta\right)}^{3} - \left(\left(2 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\left(\sin delta \cdot \left(\cos theta \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \cos \phi_1\right)\right) + \left(\left({\left(\cos delta\right)}^{3} \cdot \left(\sqrt{{\left(\sin \phi_1\right)}^{6}} \cdot \log \left(e^{\sqrt{{\left(\sin \phi_1\right)}^{6}}}\right)\right) + \left(\left({\left(\cos theta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot {\left(\sin delta\right)}^{3}\right)\right) \cdot {\left(\sin \phi_1\right)}^{3} + \left(\left(\cos theta \cdot \left(\cos \phi_1 \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \sin delta\right) \cdot {\left(\sin \phi_1\right)}^{5}\right)\right) + \left(\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \cos delta\right)\right) \cdot {\left(\cos theta\right)}^{2}\right) \cdot {\left(\sin \phi_1\right)}^{4}\right)\right) + 2 \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot \cos delta\right)\right)\right)\right)\right)}{\cos delta \cdot \cos delta + \left(\cos delta \cdot \left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \sin \phi_1\right) + \left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \sin \phi_1\right)\right)}}\]

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 4.2

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

  2. Initial simplification4.2

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

  4. Applied flip3--4.3

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

  5. Taylor expanded around -inf 4.2

    \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\frac{\color{blue}{{\left(\cos delta\right)}^{3} - \left(2 \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot \cos delta\right)\right)\right)\right) + \left(2 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos theta\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \cos delta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{6} \cdot {\left(\cos delta\right)}^{3} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot \left({\left(\sin delta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right)\right) + {\left(\sin \phi_1\right)}^{5} \cdot \left(\sin delta \cdot \left(\cos theta \cdot \left(\cos \phi_1 \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)\right)}}{\cos delta \cdot \cos delta + \left(\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) + \cos delta \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right)\right)}} + \lambda_1\]
  6. Using strategy rm

  7. Applied add-log-exp4.3

    \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\frac{{\left(\cos delta\right)}^{3} - \left(2 \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot \cos delta\right)\right)\right)\right) + \left(2 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos theta\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \cos delta\right)\right)\right) + \left(\color{blue}{\log \left(e^{{\left(\sin \phi_1\right)}^{6}}\right)} \cdot {\left(\cos delta\right)}^{3} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot \left({\left(\sin delta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right)\right) + {\left(\sin \phi_1\right)}^{5} \cdot \left(\sin delta \cdot \left(\cos theta \cdot \left(\cos \phi_1 \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)\right)}{\cos delta \cdot \cos delta + \left(\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) + \cos delta \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right)\right)}} + \lambda_1\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt4.3

    \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\frac{{\left(\cos delta\right)}^{3} - \left(2 \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot \cos delta\right)\right)\right)\right) + \left(2 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos theta\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \cos delta\right)\right)\right) + \left(\log \left(e^{\color{blue}{\sqrt{{\left(\sin \phi_1\right)}^{6}} \cdot \sqrt{{\left(\sin \phi_1\right)}^{6}}}}\right) \cdot {\left(\cos delta\right)}^{3} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot \left({\left(\sin delta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right)\right) + {\left(\sin \phi_1\right)}^{5} \cdot \left(\sin delta \cdot \left(\cos theta \cdot \left(\cos \phi_1 \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)\right)}{\cos delta \cdot \cos delta + \left(\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) + \cos delta \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right)\right)}} + \lambda_1\]

  10. Applied exp-prod4.3

    \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\frac{{\left(\cos delta\right)}^{3} - \left(2 \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot \cos delta\right)\right)\right)\right) + \left(2 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos theta\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \cos delta\right)\right)\right) + \left(\log \color{blue}{\left({\left(e^{\sqrt{{\left(\sin \phi_1\right)}^{6}}}\right)}^{\left(\sqrt{{\left(\sin \phi_1\right)}^{6}}\right)}\right)} \cdot {\left(\cos delta\right)}^{3} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot \left({\left(\sin delta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right)\right) + {\left(\sin \phi_1\right)}^{5} \cdot \left(\sin delta \cdot \left(\cos theta \cdot \left(\cos \phi_1 \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)\right)}{\cos delta \cdot \cos delta + \left(\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) + \cos delta \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right)\right)}} + \lambda_1\]

  11. Applied log-pow4.3

    \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\frac{{\left(\cos delta\right)}^{3} - \left(2 \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot \cos delta\right)\right)\right)\right) + \left(2 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos theta \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos theta\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \cos delta\right)\right)\right) + \left(\color{blue}{\left(\sqrt{{\left(\sin \phi_1\right)}^{6}} \cdot \log \left(e^{\sqrt{{\left(\sin \phi_1\right)}^{6}}}\right)\right)} \cdot {\left(\cos delta\right)}^{3} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot \left({\left(\sin delta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}\right)\right) + {\left(\sin \phi_1\right)}^{5} \cdot \left(\sin delta \cdot \left(\cos theta \cdot \left(\cos \phi_1 \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)\right)}{\cos delta \cdot \cos delta + \left(\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right) + \cos delta \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)\right)\right)}} + \lambda_1\]

  12. Final simplification4.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\frac{{\left(\cos delta\right)}^{3} - \left(\left(2 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\left(\sin delta \cdot \left(\cos theta \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \cos \phi_1\right)\right) + \left(\left({\left(\cos delta\right)}^{3} \cdot \left(\sqrt{{\left(\sin \phi_1\right)}^{6}} \cdot \log \left(e^{\sqrt{{\left(\sin \phi_1\right)}^{6}}}\right)\right) + \left(\left({\left(\cos theta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot {\left(\sin delta\right)}^{3}\right)\right) \cdot {\left(\sin \phi_1\right)}^{3} + \left(\left(\cos theta \cdot \left(\cos \phi_1 \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \sin delta\right) \cdot {\left(\sin \phi_1\right)}^{5}\right)\right) + \left(\left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \cos delta\right)\right) \cdot {\left(\cos theta\right)}^{2}\right) \cdot {\left(\sin \phi_1\right)}^{4}\right)\right) + 2 \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot \cos delta\right)\right)\right)\right)\right)}{\cos delta \cdot \cos delta + \left(\cos delta \cdot \left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \sin \phi_1\right) + \left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \sin \phi_1\right)\right)}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018277 
(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))))))))))