Average Error: 3.5 → 3.5
Time: 1.0m
Precision: 64
Internal Precision: 128
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\frac{\left(\sin \left(\sin^{-1} \left((e^{\log \left(e^{\log_* (1 + (\left(\cos theta\right) \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*)}\right)} - 1)^*\right)\right) \cdot \sin \phi_1\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left((e^{\log_* (1 + (\left(\cos theta\right) \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*)} - 1)^*\right)\right)\right) - \cos delta \cdot \cos delta}{\sin \left(\sin^{-1} \left((e^{\log_* (1 + (\left(\cos theta\right) \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*)} - 1)^*\right)\right) \cdot \left(-\sin \phi_1\right) - \cos delta}}\]

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 3.5

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

  2. Initial simplification3.5

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

  4. Applied expm1-log1p-u3.5

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

  6. Applied fma-udef3.5

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

  8. Applied flip-+3.5

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

  10. Applied add-log-exp3.5

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

  11. Final simplification3.5

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018346 +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))))))))))