Average Error: 0.2 → 0.2
Time: 2.6m
Precision: 64
Internal Precision: 384
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\sqrt[3]{{\left(\left(\sqrt[3]{\cos delta \cdot \left(\cos \phi_1 \cdot \cos \phi_1\right)} \cdot \sqrt[3]{\cos delta \cdot \left(\cos \phi_1 \cdot \cos \phi_1\right)}\right) \cdot \sqrt[3]{\cos delta \cdot \left(\cos \phi_1 \cdot \cos \phi_1\right)} - \left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\sin \phi_1 \cdot \cos theta\right)\right)}^{3}}}\]

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

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. Taylor expanded around inf 0.2

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

  4. Applied add-cbrt-cube0.2

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

  5. Applied simplify0.2

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

  7. Applied add-cube-cbrt0.2

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

  8. Applied simplify0.2

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

  9. Applied simplify0.2

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

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(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))))))))))