Average Error: 0.2 → 0.2
Time: 2.4m
Precision: 64
Internal Precision: 576
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left((e^{\log_* (1 + \sin \left(\sin^{-1} \left((\left(\cos \phi_1\right) \cdot \left(\sin delta \cdot \cos theta\right) + \left(\cos delta \cdot \sin \phi_1\right))_*\right)\right) \cdot \sin \phi_1)} - 1)^*\right)}^{3}}{(\left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right) \cdot \sin \phi_1\right) + \left((\left(\sin \phi_1 \cdot \cos delta\right) \cdot \left(\sin \left(\sin^{-1} \left((\left(\sin delta \cdot \cos \phi_1\right) \cdot \left(\cos theta\right) + \left(\sin \phi_1 \cdot \cos delta\right))_*\right)\right)\right) + \left(\cos delta \cdot \cos delta\right))_*\right))_*}}\]

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 expm1-log1p-u0.2

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

  4. Applied simplify0.2

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

  6. Applied flip3--0.2

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

  7. Applied simplify0.2

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +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))))))))))