Average Error: 16.8 → 3.6
Time: 1.1m
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left((\left((\left(\cos \lambda_2\right) \cdot \left(\cos \lambda_1\right) + \left((e^{\log_* (1 + \sin \lambda_1 \cdot \sin \lambda_2)} - 1)^*\right))_* \cdot \cos \phi_2\right) \cdot \left(\cos \phi_1\right) + \left(\sin \phi_1 \cdot \sin \phi_2\right))_*\right) \cdot R\]

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 16.8

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

  2. Simplified16.8

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

  4. Applied cos-diff3.6

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

  6. Applied *-un-lft-identity3.6

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

  7. Applied associate-*r*3.6

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

  8. Simplified3.6

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

  10. Applied expm1-log1p-u3.6

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

  11. Final simplification3.6

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

Reproduce

herbie shell --seed 2019088 +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))