Average Error: 39.8 → 3.8
Time: 7.3s
Precision: 64
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\]
↓
\[\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\]
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{log1p}\left(\mathsf{expm1}\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 r58818 = R;
        double r58819 = lambda1;
        double r58820 = lambda2;
        double r58821 = r58819 - r58820;
        double r58822 = phi1;
        double r58823 = phi2;
        double r58824 = r58822 + r58823;
        double r58825 = 2.0;
        double r58826 = r58824 / r58825;
        double r58827 = cos(r58826);
        double r58828 = r58821 * r58827;
        double r58829 = r58828 * r58828;
        double r58830 = r58822 - r58823;
        double r58831 = r58830 * r58830;
        double r58832 = r58829 + r58831;
        double r58833 = sqrt(r58832);
        double r58834 = r58818 * r58833;
        return r58834;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r58835 = lambda1;
        double r58836 = lambda2;
        double r58837 = r58835 - r58836;
        double r58838 = phi1;
        double r58839 = phi2;
        double r58840 = r58838 + r58839;
        double r58841 = 2.0;
        double r58842 = r58840 / r58841;
        double r58843 = cos(r58842);
        double r58844 = expm1(r58843);
        double r58845 = log1p(r58844);
        double r58846 = r58837 * r58845;
        double r58847 = r58838 - r58839;
        double r58848 = hypot(r58846, r58847);
        double r58849 = R;
        double r58850 = r58848 * r58849;
        return r58850;
}

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 39.8

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

  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 add-cbrt-cube3.8

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

  5. Simplified3.8

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

  7. Applied log1p-expm1-u3.8

    \[\leadsto \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{{\left(\cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right)}^{3}}\right)\right)}, \phi_1 - \phi_2\right) \cdot R\]

  8. Simplified3.8

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

  9. Final simplification3.8

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

Reproduce

herbie shell --seed 2020083 +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))))))