Distance on a great circle

Average Error: 24.5 → 24.6

Time: 45.1s

Precision: 64

Math

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

↓

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

C

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r60880 = R;
        double r60881 = 2.0;
        double r60882 = phi1;
        double r60883 = phi2;
        double r60884 = r60882 - r60883;
        double r60885 = r60884 / r60881;
        double r60886 = sin(r60885);
        double r60887 = pow(r60886, r60881);
        double r60888 = cos(r60882);
        double r60889 = cos(r60883);
        double r60890 = r60888 * r60889;
        double r60891 = lambda1;
        double r60892 = lambda2;
        double r60893 = r60891 - r60892;
        double r60894 = r60893 / r60881;
        double r60895 = sin(r60894);
        double r60896 = r60890 * r60895;
        double r60897 = r60896 * r60895;
        double r60898 = r60887 + r60897;
        double r60899 = sqrt(r60898);
        double r60900 = 1.0;
        double r60901 = r60900 - r60898;
        double r60902 = sqrt(r60901);
        double r60903 = atan2(r60899, r60902);
        double r60904 = r60881 * r60903;
        double r60905 = r60880 * r60904;
        return r60905;
}

↓

double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r60906 = R;
        double r60907 = 2.0;
        double r60908 = phi1;
        double r60909 = phi2;
        double r60910 = r60908 - r60909;
        double r60911 = r60910 / r60907;
        double r60912 = sin(r60911);
        double r60913 = pow(r60912, r60907);
        double r60914 = cos(r60908);
        double r60915 = cos(r60909);
        double r60916 = r60914 * r60915;
        double r60917 = lambda1;
        double r60918 = lambda2;
        double r60919 = r60917 - r60918;
        double r60920 = r60919 / r60907;
        double r60921 = sin(r60920);
        double r60922 = r60916 * r60921;
        double r60923 = r60922 * r60921;
        double r60924 = r60913 + r60923;
        double r60925 = sqrt(r60924);
        double r60926 = 1.0;
        double r60927 = cbrt(r60921);
        double r60928 = 2.0;
        double r60929 = pow(r60927, r60928);
        double r60930 = r60929 * r60927;
        double r60931 = r60916 * r60930;
        double r60932 = r60931 * r60921;
        double r60933 = r60913 + r60932;
        double r60934 = r60926 - r60933;
        double r60935 = sqrt(r60934);
        double r60936 = atan2(r60925, r60935);
        double r60937 = r60907 * r60936;
        double r60938 = r60906 * r60937;
        return r60938;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 24.5

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

3. Applied add-cbrt-cube 24.6

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

4. Simplified 24.6

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

6. Applied add-cube-cbrt 24.6

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

7. Applied unpow-prod-down 24.6

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

8. Applied cbrt-prod 24.6

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

9. Simplified 24.6

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

10. Simplified 24.6

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

11. Final simplification 24.6

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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))))))))))