Average Error: 38.5 → 3.7
Time: 8.4s
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 r96963 = R;
        double r96964 = lambda1;
        double r96965 = lambda2;
        double r96966 = r96964 - r96965;
        double r96967 = phi1;
        double r96968 = phi2;
        double r96969 = r96967 + r96968;
        double r96970 = 2.0;
        double r96971 = r96969 / r96970;
        double r96972 = cos(r96971);
        double r96973 = r96966 * r96972;
        double r96974 = r96973 * r96973;
        double r96975 = r96967 - r96968;
        double r96976 = r96975 * r96975;
        double r96977 = r96974 + r96976;
        double r96978 = sqrt(r96977);
        double r96979 = r96963 * r96978;
        return r96979;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r96980 = R;
        double r96981 = lambda1;
        double r96982 = lambda2;
        double r96983 = r96981 - r96982;
        double r96984 = phi1;
        double r96985 = phi2;
        double r96986 = r96984 + r96985;
        double r96987 = 2.0;
        double r96988 = r96986 / r96987;
        double r96989 = cos(r96988);
        double r96990 = r96983 * r96989;
        double r96991 = r96984 - r96985;
        double r96992 = hypot(r96990, r96991);
        double r96993 = r96980 * r96992;
        return r96993;
}

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 38.5

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

  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 *-un-lft-identity3.7

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

  5. Final simplification3.7

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

Reproduce

herbie shell --seed 2020001 +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))))))