Average Error: 17.0 → 4.0
Time: 3.7m
Precision: 64
Internal Precision: 2112
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\log \left(e^{e^{\log \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right)}}\right) \cdot R\]

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 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. Using strategy rm

  3. Applied sub-neg17.0

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

  4. Applied cos-sum3.9

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

  5. Applied simplify3.9

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

  7. Applied add-log-exp3.9

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

  8. Applied simplify3.9

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

  10. Applied acos-asin4.0

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

  12. Applied add-exp-log4.0

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

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))