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Derivation

1. Split input into 3 regimes

2. if (- phi2 phi1) < -1.6024288027903094e+163

   1. Initial program 60.8

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

   2. Taylor expanded around inf 60.9

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

   3. Simplified60.8

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

   4. Taylor expanded around inf 53.4

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

   5. Simplified52.7

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

   if -1.6024288027903094e+163 < (- phi2 phi1) < 6.334975968862399e+126

   1. Initial program 23.0

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

   2. Taylor expanded around inf 23.1

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

   3. Simplified23.0

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

   if 6.334975968862399e+126 < (- phi2 phi1)

   1. Initial program 54.9

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

   2. Taylor expanded around 0 14.4

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification26.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 - \phi_1 \le -1.6024288027903094 \cdot 10^{+163}:\\ \;\;\;\;\left(\left(-\phi_2\right) - \frac{\left(\lambda_2 \cdot \lambda_1 + \frac{\phi_2 \cdot \lambda_2}{\frac{\phi_1}{\lambda_1}}\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}{\frac{\phi_1}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}}\right) \cdot R\\ \mathbf{elif}\;\phi_2 - \phi_1 \le 6.334975968862399 \cdot 10^{+126}:\\ \;\;\;\;\sqrt{\left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \cdot \left(\lambda_2 \cdot \lambda_2 + \left(\lambda_1 - \lambda_2 \cdot 2\right) \cdot \lambda_1\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array}\]

Runtime

Time bar (total: 5.3m)Debug logProfile

herbie shell --seed 2018215 
(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))))))