Average Error: 38.8 → 3.6
Time: 26.8s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \cdot R\]
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)}
↓
\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r68262 = R;
        double r68263 = lambda1;
        double r68264 = lambda2;
        double r68265 = r68263 - r68264;
        double r68266 = phi1;
        double r68267 = phi2;
        double r68268 = r68266 + r68267;
        double r68269 = 2.0;
        double r68270 = r68268 / r68269;
        double r68271 = cos(r68270);
        double r68272 = r68265 * r68271;
        double r68273 = r68272 * r68272;
        double r68274 = r68266 - r68267;
        double r68275 = r68274 * r68274;
        double r68276 = r68273 + r68275;
        double r68277 = sqrt(r68276);
        double r68278 = r68262 * r68277;
        return r68278;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r68279 = lambda1;
        double r68280 = lambda2;
        double r68281 = r68279 - r68280;
        double r68282 = phi1;
        double r68283 = phi2;
        double r68284 = r68282 + r68283;
        double r68285 = 2.0;
        double r68286 = r68284 / r68285;
        double r68287 = cos(r68286);
        double r68288 = r68281 * r68287;
        double r68289 = r68282 - r68283;
        double r68290 = hypot(r68288, r68289);
        double r68291 = R;
        double r68292 = r68290 * r68291;
        return r68292;
}

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 38.8

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

  2. Simplified3.6

    \[\leadsto \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \cdot R}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity3.6

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

  5. Final simplification3.6

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))