Average Error: 0.2 → 0.2
Time: 6.5m
Precision: 64
Internal Precision: 576
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \cos \phi_1}{(e^{\log_* (1 + \left(\cos delta - \sin \phi_1 \cdot (\left(\cos \phi_1\right) \cdot \left(\cos theta \cdot \sin delta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right))} - 1)^*} + \lambda_1\]

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. Initial simplification0.2

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

  4. Applied expm1-log1p-u0.2

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

  5. Final simplification0.2

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

Runtime

Time bar (total: 6.5m)Debug logProfile

herbie shell --seed 2018214 +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))))))))))