Average Error: 16.8 → 3.9
Time: 44.5s
Precision: 64
Internal precision: 2176
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
⬇
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left({\left(\cos \lambda_2 \cdot \cos \lambda_1\right)}^3 - {\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^3\right)}{{\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^2 + \left({\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}\right) \cdot R\]

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 16.8

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

  3. Applied sub-neg 16.8

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

  4. Applied cos-sum 3.8

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

  5. Applied simplify 3.8

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

  7. Applied flip3-- 3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\frac{{\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} - {\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^{3}}{{\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^2 + \left({\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}}\right) \cdot R\]

  8. Applied associate-*r/ 3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\frac{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left({\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} - {\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^{3}\right)}{{\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^2 + \left({\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}}\right) \cdot R\]

  9. Applied simplify 3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left({\left(\cos \lambda_2 \cdot \cos \lambda_1\right)}^3 - {\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^3\right)}}{{\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^2 + \left({\left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}^2 + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}\right) \cdot R\]
  10. Removed slow pow expressions

Runtime

Time bar (total: 44.5s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))