Average Error: 13.1 → 0.4
Time: 13.9s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sqrt[3]{{\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{3}} + \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right) \cdot \sin \lambda_2\right)}\]
\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sqrt[3]{{\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)}^{3}} + \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right) \cdot \sin \lambda_2\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r125759 = lambda1;
        double r125760 = lambda2;
        double r125761 = r125759 - r125760;
        double r125762 = sin(r125761);
        double r125763 = phi2;
        double r125764 = cos(r125763);
        double r125765 = r125762 * r125764;
        double r125766 = phi1;
        double r125767 = cos(r125766);
        double r125768 = sin(r125763);
        double r125769 = r125767 * r125768;
        double r125770 = sin(r125766);
        double r125771 = r125770 * r125764;
        double r125772 = cos(r125761);
        double r125773 = r125771 * r125772;
        double r125774 = r125769 - r125773;
        double r125775 = atan2(r125765, r125774);
        return r125775;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r125776 = lambda1;
        double r125777 = sin(r125776);
        double r125778 = lambda2;
        double r125779 = cos(r125778);
        double r125780 = r125777 * r125779;
        double r125781 = cos(r125776);
        double r125782 = -r125778;
        double r125783 = sin(r125782);
        double r125784 = r125781 * r125783;
        double r125785 = r125780 + r125784;
        double r125786 = phi2;
        double r125787 = cos(r125786);
        double r125788 = r125785 * r125787;
        double r125789 = phi1;
        double r125790 = cos(r125789);
        double r125791 = sin(r125786);
        double r125792 = r125790 * r125791;
        double r125793 = sin(r125789);
        double r125794 = r125793 * r125787;
        double r125795 = r125781 * r125779;
        double r125796 = r125794 * r125795;
        double r125797 = 3.0;
        double r125798 = pow(r125796, r125797);
        double r125799 = cbrt(r125798);
        double r125800 = r125794 * r125777;
        double r125801 = sin(r125778);
        double r125802 = r125800 * r125801;
        double r125803 = r125799 + r125802;
        double r125804 = r125792 - r125803;
        double r125805 = atan2(r125788, r125804);
        return r125805;
}

Error

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 13.1

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

  3. Applied sub-neg13.1

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

  4. Applied sin-sum6.5

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

  5. Simplified6.5

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

  7. Applied cos-diff0.2

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

  8. Applied distribute-lft-in0.2

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

  10. Applied associate-*r*0.2

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

  12. Applied add-cbrt-cube0.2

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

  13. Applied add-cbrt-cube0.2

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

  14. Applied cbrt-unprod0.2

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

  15. Applied add-cbrt-cube0.2

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

  16. Applied add-cbrt-cube0.4

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

  17. Applied cbrt-unprod0.4

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

  18. Applied cbrt-unprod0.4

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

  19. Simplified0.4

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

  20. Final simplification0.4

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

Reproduce

herbie shell --seed 2020046 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))