Average Error: 36.8 → 3.9
Time: 20.0s
Precision: 64
Internal Precision: 128
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \log_* (1 + (e^{(e^{\log_* (1 + \cos \left(\frac{\phi_2 + \phi_1}{2}\right))} - 1)^*} - 1)^*)\right)^2 + \left(\phi_1 - \phi_2\right)^2}^* \cdot R\]

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 36.8

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

  2. 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. Using strategy rm

  4. Applied expm1-log1p-u3.8

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

  6. Applied log1p-expm1-u3.9

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

  7. Final simplification3.9

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

Reproduce

herbie shell --seed 2019030 +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))))))