Average Error: 37.4 → 4.0
Time: 28.7s
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 r2969903 = R;
        double r2969904 = lambda1;
        double r2969905 = lambda2;
        double r2969906 = r2969904 - r2969905;
        double r2969907 = phi1;
        double r2969908 = phi2;
        double r2969909 = r2969907 + r2969908;
        double r2969910 = 2.0;
        double r2969911 = r2969909 / r2969910;
        double r2969912 = cos(r2969911);
        double r2969913 = r2969906 * r2969912;
        double r2969914 = r2969913 * r2969913;
        double r2969915 = r2969907 - r2969908;
        double r2969916 = r2969915 * r2969915;
        double r2969917 = r2969914 + r2969916;
        double r2969918 = sqrt(r2969917);
        double r2969919 = r2969903 * r2969918;
        return r2969919;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2969920 = lambda1;
        double r2969921 = lambda2;
        double r2969922 = r2969920 - r2969921;
        double r2969923 = phi2;
        double r2969924 = phi1;
        double r2969925 = r2969923 + r2969924;
        double r2969926 = 2.0;
        double r2969927 = r2969925 / r2969926;
        double r2969928 = cos(r2969927);
        double r2969929 = r2969922 * r2969928;
        double r2969930 = r2969924 - r2969923;
        double r2969931 = hypot(r2969929, r2969930);
        double r2969932 = R;
        double r2969933 = r2969931 * r2969932;
        return r2969933;
}

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 37.4

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

  2. Simplified4.0

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

  4. Applied *-un-lft-identity4.0

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

  5. Applied associate-*r*4.0

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

  6. Simplified4.0

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

  7. Final simplification4.0

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

Reproduce

herbie shell --seed 2019163 +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))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2)))) (* (- phi1 phi2) (- phi1 phi2))))))