Error

Derivation

1. Initial program 37.0

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]

2. Simplified3.8

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

3. Taylor expanded around -inf 3.8

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

5. Applied distribute-lft-in3.8

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

6. Applied cos-sum0.1

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

8. Applied fma-neg0.1

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

9. Taylor expanded around inf 0.1

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

10. Simplified0.1

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

11. Final simplification0.1

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

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))