Average Error: 39.2 → 3.9
Time: 1.0m
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \cdot R\]
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_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3576315 = R;
        double r3576316 = lambda1;
        double r3576317 = lambda2;
        double r3576318 = r3576316 - r3576317;
        double r3576319 = phi1;
        double r3576320 = phi2;
        double r3576321 = r3576319 + r3576320;
        double r3576322 = 2.0;
        double r3576323 = r3576321 / r3576322;
        double r3576324 = cos(r3576323);
        double r3576325 = r3576318 * r3576324;
        double r3576326 = r3576325 * r3576325;
        double r3576327 = r3576319 - r3576320;
        double r3576328 = r3576327 * r3576327;
        double r3576329 = r3576326 + r3576328;
        double r3576330 = sqrt(r3576329);
        double r3576331 = r3576315 * r3576330;
        return r3576331;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3576332 = lambda1;
        double r3576333 = lambda2;
        double r3576334 = r3576332 - r3576333;
        double r3576335 = phi2;
        double r3576336 = phi1;
        double r3576337 = r3576335 + r3576336;
        double r3576338 = 2.0;
        double r3576339 = r3576337 / r3576338;
        double r3576340 = cos(r3576339);
        double r3576341 = r3576334 * r3576340;
        double r3576342 = r3576336 - r3576335;
        double r3576343 = hypot(r3576341, r3576342);
        double r3576344 = R;
        double r3576345 = r3576343 * r3576344;
        return r3576345;
}

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 39.2

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

  2. Simplified3.9

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

  3. Final simplification3.9

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

Reproduce

herbie shell --seed 2019168 +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) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))