Error

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-sum3.7

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

5. Applied simplify3.7

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

7. Applied acos-asin3.8

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

8. Applied simplify3.8

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

10. Applied flip--3.9

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

11. Applied associate-*l/3.9

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))