Average Error: 0.9 → 0.3
Time: 27.9s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\left(-\sin \lambda_2 \cdot \cos \lambda_1\right) + \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\log \left(e^{\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)}\]
\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{\left(\left(-\sin \lambda_2 \cdot \cos \lambda_1\right) + \cos \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_2}{\log \left(e^{\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r36691 = lambda1;
        double r36692 = phi2;
        double r36693 = cos(r36692);
        double r36694 = lambda2;
        double r36695 = r36691 - r36694;
        double r36696 = sin(r36695);
        double r36697 = r36693 * r36696;
        double r36698 = phi1;
        double r36699 = cos(r36698);
        double r36700 = cos(r36695);
        double r36701 = r36693 * r36700;
        double r36702 = r36699 + r36701;
        double r36703 = atan2(r36697, r36702);
        double r36704 = r36691 + r36703;
        return r36704;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r36705 = lambda1;
        double r36706 = lambda2;
        double r36707 = sin(r36706);
        double r36708 = cos(r36705);
        double r36709 = r36707 * r36708;
        double r36710 = -r36709;
        double r36711 = cos(r36706);
        double r36712 = sin(r36705);
        double r36713 = r36711 * r36712;
        double r36714 = r36710 + r36713;
        double r36715 = phi2;
        double r36716 = cos(r36715);
        double r36717 = r36714 * r36716;
        double r36718 = r36708 * r36711;
        double r36719 = fma(r36707, r36712, r36718);
        double r36720 = phi1;
        double r36721 = cos(r36720);
        double r36722 = fma(r36719, r36716, r36721);
        double r36723 = exp(r36722);
        double r36724 = log(r36723);
        double r36725 = atan2(r36717, r36724);
        double r36726 = r36705 + r36725;
        return r36726;
}

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. Simplified0.9

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

  4. Applied cos-diff0.9

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

  5. Simplified0.9

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

  6. Simplified0.9

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

  8. Applied sub-neg0.9

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

  9. Applied sin-sum0.2

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

  10. Simplified0.2

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

  11. Simplified0.2

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

  13. Applied add-log-exp0.3

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

  14. Simplified0.3

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

  16. Applied pow10.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))