Average Error: 39.0 → 3.8
Time: 7.5s
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 r63803 = R;
        double r63804 = lambda1;
        double r63805 = lambda2;
        double r63806 = r63804 - r63805;
        double r63807 = phi1;
        double r63808 = phi2;
        double r63809 = r63807 + r63808;
        double r63810 = 2.0;
        double r63811 = r63809 / r63810;
        double r63812 = cos(r63811);
        double r63813 = r63806 * r63812;
        double r63814 = r63813 * r63813;
        double r63815 = r63807 - r63808;
        double r63816 = r63815 * r63815;
        double r63817 = r63814 + r63816;
        double r63818 = sqrt(r63817);
        double r63819 = r63803 * r63818;
        return r63819;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r63820 = lambda1;
        double r63821 = lambda2;
        double r63822 = r63820 - r63821;
        double r63823 = phi1;
        double r63824 = phi2;
        double r63825 = r63823 + r63824;
        double r63826 = 2.0;
        double r63827 = r63825 / r63826;
        double r63828 = cos(r63827);
        double r63829 = r63822 * r63828;
        double r63830 = r63823 - r63824;
        double r63831 = hypot(r63829, r63830);
        double r63832 = R;
        double r63833 = r63831 * r63832;
        return r63833;
}

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 39.0

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

  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 *-un-lft-identity3.8

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

  5. Final simplification3.8

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

Reproduce

herbie shell --seed 2019354 +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))))))