Average Error: 39.3 → 3.8
Time: 7.4s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\]
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)}
↓
R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r73027 = R;
        double r73028 = lambda1;
        double r73029 = lambda2;
        double r73030 = r73028 - r73029;
        double r73031 = phi1;
        double r73032 = phi2;
        double r73033 = r73031 + r73032;
        double r73034 = 2.0;
        double r73035 = r73033 / r73034;
        double r73036 = cos(r73035);
        double r73037 = r73030 * r73036;
        double r73038 = r73037 * r73037;
        double r73039 = r73031 - r73032;
        double r73040 = r73039 * r73039;
        double r73041 = r73038 + r73040;
        double r73042 = sqrt(r73041);
        double r73043 = r73027 * r73042;
        return r73043;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r73044 = R;
        double r73045 = lambda1;
        double r73046 = lambda2;
        double r73047 = r73045 - r73046;
        double r73048 = phi1;
        double r73049 = phi2;
        double r73050 = r73048 + r73049;
        double r73051 = 2.0;
        double r73052 = r73050 / r73051;
        double r73053 = cos(r73052);
        double r73054 = r73047 * r73053;
        double r73055 = r73048 - r73049;
        double r73056 = hypot(r73054, r73055);
        double r73057 = r73044 * r73056;
        return r73057;
}

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 39.3

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

  2. Simplified3.8

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

  4. Applied *-commutative3.8

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

  5. Final simplification3.8

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

Reproduce

herbie shell --seed 2020046 +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))))))