Average Error: 16.5 → 3.6
Time: 1.4m
Precision: 64
Internal Precision: 1856
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\left(\frac{\pi}{2} - \left(\frac{\pi}{2} - \cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left((e^{\log_* (1 + \sin \phi_2 \cdot \sin \phi_1)} - 1)^*\right))_*\right)\right)\right) \cdot R\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 16.5

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]

  2. Initial simplification16.5

    \[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right)\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]
  3. Using strategy rm

  4. Applied sub-neg16.5

    \[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]

  5. Applied cos-sum3.6

    \[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]

  6. Simplified3.6

    \[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)\]
  7. Using strategy rm

  8. Applied expm1-log1p-u3.6

    \[\leadsto R \cdot \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \color{blue}{\left((e^{\log_* (1 + \sin \phi_2 \cdot \sin \phi_1)} - 1)^*\right)})_*\right)\]
  9. Using strategy rm

  10. Applied acos-asin3.7

    \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left((e^{\log_* (1 + \sin \phi_2 \cdot \sin \phi_1)} - 1)^*\right))_*\right)\right)}\]
  11. Using strategy rm

  12. Applied asin-acos3.6

    \[\leadsto R \cdot \left(\frac{\pi}{2} - \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left((e^{\log_* (1 + \sin \phi_2 \cdot \sin \phi_1)} - 1)^*\right))_*\right)\right)}\right)\]

  13. Final simplification3.6

    \[\leadsto \left(\frac{\pi}{2} - \left(\frac{\pi}{2} - \cos^{-1} \left((\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + \left((e^{\log_* (1 + \sin \phi_2 \cdot \sin \phi_1)} - 1)^*\right))_*\right)\right)\right) \cdot R\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018220 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))