Average Error: 24.7 → 24.7
Time: 41.2s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(1 - {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right) - \left(\cos \phi_2 \cdot \left(\sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}} \cdot \cos \phi_1\right)\right) \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}\right)}^{3}}}}\right)\]
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 \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\left(1 - {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right) - \left(\cos \phi_2 \cdot \left(\sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}} \cdot \cos \phi_1\right)\right) \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}\right)}^{3}}}}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r99797 = R;
        double r99798 = 2.0;
        double r99799 = phi1;
        double r99800 = phi2;
        double r99801 = r99799 - r99800;
        double r99802 = r99801 / r99798;
        double r99803 = sin(r99802);
        double r99804 = pow(r99803, r99798);
        double r99805 = cos(r99799);
        double r99806 = cos(r99800);
        double r99807 = r99805 * r99806;
        double r99808 = lambda1;
        double r99809 = lambda2;
        double r99810 = r99808 - r99809;
        double r99811 = r99810 / r99798;
        double r99812 = sin(r99811);
        double r99813 = r99807 * r99812;
        double r99814 = r99813 * r99812;
        double r99815 = r99804 + r99814;
        double r99816 = sqrt(r99815);
        double r99817 = 1.0;
        double r99818 = r99817 - r99815;
        double r99819 = sqrt(r99818);
        double r99820 = atan2(r99816, r99819);
        double r99821 = r99798 * r99820;
        double r99822 = r99797 * r99821;
        return r99822;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r99823 = R;
        double r99824 = 2.0;
        double r99825 = phi1;
        double r99826 = phi2;
        double r99827 = r99825 - r99826;
        double r99828 = r99827 / r99824;
        double r99829 = sin(r99828);
        double r99830 = pow(r99829, r99824);
        double r99831 = cos(r99825);
        double r99832 = lambda1;
        double r99833 = lambda2;
        double r99834 = r99832 - r99833;
        double r99835 = r99834 / r99824;
        double r99836 = sin(r99835);
        double r99837 = r99831 * r99836;
        double r99838 = cos(r99826);
        double r99839 = r99837 * r99838;
        double r99840 = r99839 * r99836;
        double r99841 = r99830 + r99840;
        double r99842 = sqrt(r99841);
        double r99843 = 1.0;
        double r99844 = r99843 - r99830;
        double r99845 = 3.0;
        double r99846 = pow(r99836, r99845);
        double r99847 = cbrt(r99846);
        double r99848 = r99847 * r99831;
        double r99849 = r99838 * r99848;
        double r99850 = pow(r99847, r99845);
        double r99851 = cbrt(r99850);
        double r99852 = r99849 * r99851;
        double r99853 = r99844 - r99852;
        double r99854 = sqrt(r99853);
        double r99855 = atan2(r99842, r99854);
        double r99856 = r99824 * r99855;
        double r99857 = r99823 * r99856;
        return r99857;
}

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 24.7

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

  2. Simplified24.7

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

  4. Applied add-cbrt-cube24.7

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

  5. Simplified24.7

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

  7. Applied add-cbrt-cube24.7

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

  8. Simplified24.7

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

  10. Applied add-cbrt-cube24.7

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

  11. Simplified24.7

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

  12. Final simplification24.7

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

Reproduce

herbie shell --seed 2019194 
(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))))))))))