Average Error: 24.1 → 24.1
Time: 14.1s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}\right)}}\right)\]
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)
↓
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 \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}\right)}}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r70862 = R;
        double r70863 = 2.0;
        double r70864 = phi1;
        double r70865 = phi2;
        double r70866 = r70864 - r70865;
        double r70867 = r70866 / r70863;
        double r70868 = sin(r70867);
        double r70869 = pow(r70868, r70863);
        double r70870 = cos(r70864);
        double r70871 = cos(r70865);
        double r70872 = r70870 * r70871;
        double r70873 = lambda1;
        double r70874 = lambda2;
        double r70875 = r70873 - r70874;
        double r70876 = r70875 / r70863;
        double r70877 = sin(r70876);
        double r70878 = r70872 * r70877;
        double r70879 = r70878 * r70877;
        double r70880 = r70869 + r70879;
        double r70881 = sqrt(r70880);
        double r70882 = 1.0;
        double r70883 = r70882 - r70880;
        double r70884 = sqrt(r70883);
        double r70885 = atan2(r70881, r70884);
        double r70886 = r70863 * r70885;
        double r70887 = r70862 * r70886;
        return r70887;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r70888 = R;
        double r70889 = 2.0;
        double r70890 = phi1;
        double r70891 = phi2;
        double r70892 = r70890 - r70891;
        double r70893 = r70892 / r70889;
        double r70894 = sin(r70893);
        double r70895 = pow(r70894, r70889);
        double r70896 = cos(r70890);
        double r70897 = cos(r70891);
        double r70898 = r70896 * r70897;
        double r70899 = lambda1;
        double r70900 = lambda2;
        double r70901 = r70899 - r70900;
        double r70902 = r70901 / r70889;
        double r70903 = sin(r70902);
        double r70904 = r70898 * r70903;
        double r70905 = r70904 * r70903;
        double r70906 = r70895 + r70905;
        double r70907 = sqrt(r70906);
        double r70908 = 1.0;
        double r70909 = exp(r70903);
        double r70910 = log(r70909);
        double r70911 = r70898 * r70910;
        double r70912 = 3.0;
        double r70913 = pow(r70903, r70912);
        double r70914 = cbrt(r70913);
        double r70915 = r70911 * r70914;
        double r70916 = r70895 + r70915;
        double r70917 = r70908 - r70916;
        double r70918 = sqrt(r70917);
        double r70919 = atan2(r70907, r70918);
        double r70920 = r70889 * r70919;
        double r70921 = r70888 * r70920;
        return r70921;
}

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 24.1

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

  3. Applied add-cbrt-cube24.1

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

  4. Simplified24.1

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

  6. Applied add-log-exp24.1

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

  7. Final simplification24.1

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

Reproduce

herbie shell --seed 2019362 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))) (sqrt (- 1 (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))))))))