Average Error: 0.8 → 0.3
Time: 26.1s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\log \left(e^{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)}\]
\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{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\log \left(e^{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r49841 = lambda1;
        double r49842 = phi2;
        double r49843 = cos(r49842);
        double r49844 = lambda2;
        double r49845 = r49841 - r49844;
        double r49846 = sin(r49845);
        double r49847 = r49843 * r49846;
        double r49848 = phi1;
        double r49849 = cos(r49848);
        double r49850 = cos(r49845);
        double r49851 = r49843 * r49850;
        double r49852 = r49849 + r49851;
        double r49853 = atan2(r49847, r49852);
        double r49854 = r49841 + r49853;
        return r49854;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r49855 = lambda1;
        double r49856 = phi2;
        double r49857 = cos(r49856);
        double r49858 = lambda2;
        double r49859 = cos(r49858);
        double r49860 = sin(r49855);
        double r49861 = r49859 * r49860;
        double r49862 = cos(r49855);
        double r49863 = sin(r49858);
        double r49864 = r49862 * r49863;
        double r49865 = r49861 - r49864;
        double r49866 = r49857 * r49865;
        double r49867 = expm1(r49866);
        double r49868 = log1p(r49867);
        double r49869 = r49860 * r49863;
        double r49870 = fma(r49862, r49859, r49869);
        double r49871 = phi1;
        double r49872 = cos(r49871);
        double r49873 = fma(r49870, r49857, r49872);
        double r49874 = exp(r49873);
        double r49875 = log(r49874);
        double r49876 = atan2(r49868, r49875);
        double r49877 = r49855 + r49876;
        return r49877;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.8

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

  2. Simplified0.8

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

  4. Applied sub-neg0.8

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

  5. Applied cos-sum0.8

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

  6. Simplified0.8

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

  8. Applied sin-diff0.2

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

  9. Simplified0.2

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

  11. Applied log1p-expm1-u0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right), \cos \phi_1\right)}\]
  12. Using strategy rm

  13. Applied add-log-exp0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\color{blue}{\log \left(e^{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right), \cos \phi_1\right)}\right)}}\]

  14. Simplified0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\log \color{blue}{\left(e^{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)}}\]

  15. Final simplification0.3

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\log \left(e^{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right), \cos \phi_2, \cos \phi_1\right)}\right)}\]

Reproduce

herbie shell --seed 2019208 +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)))))))