Average Error: 24.1 → 24.1
Time: 53.0s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right) + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \cos \phi_2\right) \cdot \left(\cos \phi_1 \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)}}\right)\]
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 \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{\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right) + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \left(\cos \phi_2 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \cos \phi_2\right) \cdot \left(\cos \phi_1 \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)}}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2869752 = R;
        double r2869753 = 2.0;
        double r2869754 = phi1;
        double r2869755 = phi2;
        double r2869756 = r2869754 - r2869755;
        double r2869757 = r2869756 / r2869753;
        double r2869758 = sin(r2869757);
        double r2869759 = pow(r2869758, r2869753);
        double r2869760 = cos(r2869754);
        double r2869761 = cos(r2869755);
        double r2869762 = r2869760 * r2869761;
        double r2869763 = lambda1;
        double r2869764 = lambda2;
        double r2869765 = r2869763 - r2869764;
        double r2869766 = r2869765 / r2869753;
        double r2869767 = sin(r2869766);
        double r2869768 = r2869762 * r2869767;
        double r2869769 = r2869768 * r2869767;
        double r2869770 = r2869759 + r2869769;
        double r2869771 = sqrt(r2869770);
        double r2869772 = 1.0;
        double r2869773 = r2869772 - r2869770;
        double r2869774 = sqrt(r2869773);
        double r2869775 = atan2(r2869771, r2869774);
        double r2869776 = r2869753 * r2869775;
        double r2869777 = r2869752 * r2869776;
        return r2869777;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r2869778 = R;
        double r2869779 = 2.0;
        double r2869780 = phi1;
        double r2869781 = phi2;
        double r2869782 = r2869780 - r2869781;
        double r2869783 = r2869782 / r2869779;
        double r2869784 = sin(r2869783);
        double r2869785 = r2869784 * r2869784;
        double r2869786 = lambda1;
        double r2869787 = lambda2;
        double r2869788 = r2869786 - r2869787;
        double r2869789 = r2869788 / r2869779;
        double r2869790 = sin(r2869789);
        double r2869791 = cos(r2869780);
        double r2869792 = r2869790 * r2869791;
        double r2869793 = cos(r2869781);
        double r2869794 = r2869793 * r2869790;
        double r2869795 = r2869792 * r2869794;
        double r2869796 = r2869785 + r2869795;
        double r2869797 = sqrt(r2869796);
        double r2869798 = cos(r2869783);
        double r2869799 = r2869798 * r2869798;
        double r2869800 = r2869790 * r2869790;
        double r2869801 = r2869790 * r2869800;
        double r2869802 = cbrt(r2869801);
        double r2869803 = r2869802 * r2869793;
        double r2869804 = exp(r2869790);
        double r2869805 = log(r2869804);
        double r2869806 = r2869791 * r2869805;
        double r2869807 = r2869803 * r2869806;
        double r2869808 = r2869799 - r2869807;
        double r2869809 = sqrt(r2869808);
        double r2869810 = atan2(r2869797, r2869809);
        double r2869811 = r2869779 * r2869810;
        double r2869812 = r2869778 * r2869811;
        return r2869812;
}

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. Simplified24.1

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

  4. Applied add-log-exp24.1

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

  6. Applied add-cbrt-cube24.1

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

  7. Final simplification24.1

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

Reproduce

herbie shell --seed 2019149 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  (* 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))))))))))