Average Error: 17.0 → 3.6
Time: 1.7m
Precision: 64
Internal Precision: 1856
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\log_* (1 + (e^{\cos^{-1} \left((\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 - \log_* (1 + (e^{\sin \left(-\lambda_2\right) \cdot \sin \lambda_1} - 1)^*)\right) + \left(\sin \phi_2 \cdot \sin \phi_1\right))_*\right)} - 1)^*) \cdot R\]

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 17.0

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

  2. Initial simplification17.0

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

  4. Applied sub-neg17.0

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

  5. Applied cos-sum3.6

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

  6. Simplified3.6

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

  8. Applied log1p-expm1-u3.6

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

  10. Applied log1p-expm1-u3.6

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

  11. Final simplification3.6

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018215 +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))