Average Error: 38.9 → 4.0
Time: 19.3s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \cdot R\]
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_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r69854 = R;
        double r69855 = lambda1;
        double r69856 = lambda2;
        double r69857 = r69855 - r69856;
        double r69858 = phi1;
        double r69859 = phi2;
        double r69860 = r69858 + r69859;
        double r69861 = 2.0;
        double r69862 = r69860 / r69861;
        double r69863 = cos(r69862);
        double r69864 = r69857 * r69863;
        double r69865 = r69864 * r69864;
        double r69866 = r69858 - r69859;
        double r69867 = r69866 * r69866;
        double r69868 = r69865 + r69867;
        double r69869 = sqrt(r69868);
        double r69870 = r69854 * r69869;
        return r69870;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r69871 = lambda1;
        double r69872 = lambda2;
        double r69873 = r69871 - r69872;
        double r69874 = phi1;
        double r69875 = phi2;
        double r69876 = r69874 + r69875;
        double r69877 = 2.0;
        double r69878 = r69876 / r69877;
        double r69879 = cos(r69878);
        double r69880 = r69873 * r69879;
        double r69881 = r69874 - r69875;
        double r69882 = hypot(r69880, r69881);
        double r69883 = R;
        double r69884 = r69882 * r69883;
        return r69884;
}

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 38.9

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

  2. Simplified4.0

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

  4. Applied pow14.0

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

  5. Applied pow14.0

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

  6. Applied pow-prod-down4.0

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

  7. Final simplification4.0

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

Reproduce

herbie shell --seed 2019212 +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))))))