Average Error: 0.2 → 0.2
Time: 55.9s
Precision: 64
Internal Precision: 576
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\sqrt[3]{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \sin \phi_1 \cdot \cos delta\right)\right)} \cdot \sqrt[3]{\log \left(e^{(\left(-\sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right) + \left(\cos delta\right))_*}\right) \cdot (\left(-\sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left((\left(\cos \phi_1 \cdot \sin delta\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right) + \left(\cos delta\right))_*}} + \lambda_1\]

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. Using strategy rm

  3. Applied add-cube-cbrt0.2

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

  5. Applied cbrt-unprod0.2

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

  6. Simplified0.2

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

  8. Applied add-log-exp0.2

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

  9. Final simplification0.2

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

Runtime

Time bar (total: 55.9s)Debug logProfile

herbie shell --seed 2018258 +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))))))))))