Average Error: 38.9 → 3.5
Time: 13.4s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]
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 \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r70818 = R;
        double r70819 = lambda1;
        double r70820 = lambda2;
        double r70821 = r70819 - r70820;
        double r70822 = phi1;
        double r70823 = phi2;
        double r70824 = r70822 + r70823;
        double r70825 = 2.0;
        double r70826 = r70824 / r70825;
        double r70827 = cos(r70826);
        double r70828 = r70821 * r70827;
        double r70829 = r70828 * r70828;
        double r70830 = r70822 - r70823;
        double r70831 = r70830 * r70830;
        double r70832 = r70829 + r70831;
        double r70833 = sqrt(r70832);
        double r70834 = r70818 * r70833;
        return r70834;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r70835 = lambda1;
        double r70836 = lambda2;
        double r70837 = r70835 - r70836;
        double r70838 = phi1;
        double r70839 = phi2;
        double r70840 = r70838 + r70839;
        double r70841 = 2.0;
        double r70842 = r70840 / r70841;
        double r70843 = cos(r70842);
        double r70844 = log1p(r70843);
        double r70845 = expm1(r70844);
        double r70846 = r70837 * r70845;
        double r70847 = r70838 - r70839;
        double r70848 = hypot(r70846, r70847);
        double r70849 = R;
        double r70850 = r70848 * r70849;
        return r70850;
}

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. Simplified3.5

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

  4. Applied expm1-log1p-u3.5

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  5. Final simplification3.5

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]

Reproduce

herbie shell --seed 2020047 +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))))))