Average Error: 24.8 → 24.8
Time: 44.3s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[\left(\tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \frac{\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) - \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt[3]{2}}\right)\right)}} \cdot 2\right) \cdot R\]
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)
↓
\left(\tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \frac{\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) - \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt[3]{2}}\right)\right)}} \cdot 2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2334874 = R;
        double r2334875 = 2.0;
        double r2334876 = phi1;
        double r2334877 = phi2;
        double r2334878 = r2334876 - r2334877;
        double r2334879 = r2334878 / r2334875;
        double r2334880 = sin(r2334879);
        double r2334881 = pow(r2334880, r2334875);
        double r2334882 = cos(r2334876);
        double r2334883 = cos(r2334877);
        double r2334884 = r2334882 * r2334883;
        double r2334885 = lambda1;
        double r2334886 = lambda2;
        double r2334887 = r2334885 - r2334886;
        double r2334888 = r2334887 / r2334875;
        double r2334889 = sin(r2334888);
        double r2334890 = r2334884 * r2334889;
        double r2334891 = r2334890 * r2334889;
        double r2334892 = r2334881 + r2334891;
        double r2334893 = sqrt(r2334892);
        double r2334894 = 1.0;
        double r2334895 = r2334894 - r2334892;
        double r2334896 = sqrt(r2334895);
        double r2334897 = atan2(r2334893, r2334896);
        double r2334898 = r2334875 * r2334897;
        double r2334899 = r2334874 * r2334898;
        return r2334899;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2334900 = lambda1;
        double r2334901 = lambda2;
        double r2334902 = r2334900 - r2334901;
        double r2334903 = 2.0;
        double r2334904 = r2334902 / r2334903;
        double r2334905 = sin(r2334904);
        double r2334906 = phi2;
        double r2334907 = cos(r2334906);
        double r2334908 = phi1;
        double r2334909 = cos(r2334908);
        double r2334910 = r2334907 * r2334909;
        double r2334911 = r2334910 * r2334905;
        double r2334912 = r2334905 * r2334911;
        double r2334913 = r2334908 - r2334906;
        double r2334914 = r2334913 / r2334903;
        double r2334915 = sin(r2334914);
        double r2334916 = pow(r2334915, r2334903);
        double r2334917 = r2334912 + r2334916;
        double r2334918 = sqrt(r2334917);
        double r2334919 = 1.0;
        double r2334920 = r2334904 + r2334904;
        double r2334921 = cos(r2334920);
        double r2334922 = r2334921 * r2334905;
        double r2334923 = r2334905 - r2334922;
        double r2334924 = cbrt(r2334923);
        double r2334925 = 2.0;
        double r2334926 = cbrt(r2334925);
        double r2334927 = r2334924 / r2334926;
        double r2334928 = r2334910 * r2334927;
        double r2334929 = r2334905 * r2334928;
        double r2334930 = r2334916 + r2334929;
        double r2334931 = r2334919 - r2334930;
        double r2334932 = sqrt(r2334931);
        double r2334933 = atan2(r2334918, r2334932);
        double r2334934 = r2334933 * r2334903;
        double r2334935 = R;
        double r2334936 = r2334934 * r2334935;
        return r2334936;
}

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 24.8

    \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
  2. Using strategy rm

  3. Applied add-cbrt-cube24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
  4. Using strategy rm

  5. Applied sin-mult24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sqrt[3]{\color{blue}{\frac{\cos \left(\frac{\lambda_1 - \lambda_2}{2} - \frac{\lambda_1 - \lambda_2}{2}\right) - \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

  6. Applied associate-*l/24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sqrt[3]{\color{blue}{\frac{\left(\cos \left(\frac{\lambda_1 - \lambda_2}{2} - \frac{\lambda_1 - \lambda_2}{2}\right) - \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}{2}}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

  7. Applied cbrt-div24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\frac{\sqrt[3]{\left(\cos \left(\frac{\lambda_1 - \lambda_2}{2} - \frac{\lambda_1 - \lambda_2}{2}\right) - \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt[3]{2}}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

  8. Simplified24.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) - \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt[3]{2}}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]

  9. Final simplification24.8

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \frac{\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) - \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt[3]{2}}\right)\right)}} \cdot 2\right) \cdot R\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))