Average Error: 39.9 → 3.7
Time: 33.6s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]
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 \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r56223 = R;
        double r56224 = lambda1;
        double r56225 = lambda2;
        double r56226 = r56224 - r56225;
        double r56227 = phi1;
        double r56228 = phi2;
        double r56229 = r56227 + r56228;
        double r56230 = 2.0;
        double r56231 = r56229 / r56230;
        double r56232 = cos(r56231);
        double r56233 = r56226 * r56232;
        double r56234 = r56233 * r56233;
        double r56235 = r56227 - r56228;
        double r56236 = r56235 * r56235;
        double r56237 = r56234 + r56236;
        double r56238 = sqrt(r56237);
        double r56239 = r56223 * r56238;
        return r56239;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r56240 = lambda1;
        double r56241 = lambda2;
        double r56242 = r56240 - r56241;
        double r56243 = phi1;
        double r56244 = phi2;
        double r56245 = r56243 + r56244;
        double r56246 = 2.0;
        double r56247 = r56245 / r56246;
        double r56248 = cos(r56247);
        double r56249 = log1p(r56248);
        double r56250 = expm1(r56249);
        double r56251 = r56242 * r56250;
        double r56252 = r56243 - r56244;
        double r56253 = hypot(r56251, r56252);
        double r56254 = R;
        double r56255 = r56253 * r56254;
        return r56255;
}

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 39.9

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

  2. Simplified3.7

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

  4. Applied expm1-log1p-u3.7

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  5. Final simplification3.7

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]

Reproduce

herbie shell --seed 2019305 +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))))))