Average Error: 0.9 → 0.3
Time: 10.6s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right)\right)}^{3}} + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}\]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right)\right)}^{3}} + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r59728 = lambda1;
        double r59729 = phi2;
        double r59730 = cos(r59729);
        double r59731 = lambda2;
        double r59732 = r59728 - r59731;
        double r59733 = sin(r59732);
        double r59734 = r59730 * r59733;
        double r59735 = phi1;
        double r59736 = cos(r59735);
        double r59737 = cos(r59732);
        double r59738 = r59730 * r59737;
        double r59739 = r59736 + r59738;
        double r59740 = atan2(r59734, r59739);
        double r59741 = r59728 + r59740;
        return r59741;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r59742 = lambda1;
        double r59743 = phi2;
        double r59744 = cos(r59743);
        double r59745 = sin(r59742);
        double r59746 = lambda2;
        double r59747 = cos(r59746);
        double r59748 = r59745 * r59747;
        double r59749 = cos(r59742);
        double r59750 = -r59746;
        double r59751 = sin(r59750);
        double r59752 = r59749 * r59751;
        double r59753 = r59748 + r59752;
        double r59754 = r59744 * r59753;
        double r59755 = r59744 * r59747;
        double r59756 = phi1;
        double r59757 = cos(r59756);
        double r59758 = fma(r59749, r59755, r59757);
        double r59759 = 3.0;
        double r59760 = pow(r59758, r59759);
        double r59761 = cbrt(r59760);
        double r59762 = sin(r59746);
        double r59763 = r59745 * r59762;
        double r59764 = r59763 * r59744;
        double r59765 = r59761 + r59764;
        double r59766 = atan2(r59754, r59765);
        double r59767 = r59742 + r59766;
        return r59767;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.9

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

  3. Applied sub-neg0.9

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

  4. Applied sin-sum0.8

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

  5. Simplified0.8

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

  7. Applied cos-diff0.2

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

  8. Applied distribute-rgt-in0.2

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

  9. Applied associate-+r+0.2

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

  10. Simplified0.2

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

  12. Applied add-log-exp0.3

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

  14. Applied add-cbrt-cube0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020003 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))