Average Error: 0.1 → 0.1
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\left(-\sin \phi_1\right) \cdot \sin \left(\sin^{-1} \left((\left(\cos theta\right) \cdot \left(\cos \phi_1 \cdot \sin delta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right) + \cos delta} + \lambda_1\]

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 0.1

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

  2. Applied simplify0.1

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

  4. Applied fma-udef0.1

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +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))))))))))