Average Error: 0.9 → 0.3
Time: 32.4s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \left(-\lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right)\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2, \cos \phi_1\right)}\]
\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(\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \left(-\lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2\right)\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2, \cos \phi_1\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r41858 = lambda1;
        double r41859 = phi2;
        double r41860 = cos(r41859);
        double r41861 = lambda2;
        double r41862 = r41858 - r41861;
        double r41863 = sin(r41862);
        double r41864 = r41860 * r41863;
        double r41865 = phi1;
        double r41866 = cos(r41865);
        double r41867 = cos(r41862);
        double r41868 = r41860 * r41867;
        double r41869 = r41866 + r41868;
        double r41870 = atan2(r41864, r41869);
        double r41871 = r41858 + r41870;
        return r41871;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r41872 = lambda1;
        double r41873 = sin(r41872);
        double r41874 = lambda2;
        double r41875 = cos(r41874);
        double r41876 = -r41874;
        double r41877 = sin(r41876);
        double r41878 = cos(r41872);
        double r41879 = r41877 * r41878;
        double r41880 = fma(r41873, r41875, r41879);
        double r41881 = phi2;
        double r41882 = cos(r41881);
        double r41883 = r41880 * r41882;
        double r41884 = expm1(r41883);
        double r41885 = log1p(r41884);
        double r41886 = r41878 * r41875;
        double r41887 = sin(r41874);
        double r41888 = r41873 * r41887;
        double r41889 = r41886 + r41888;
        double r41890 = phi1;
        double r41891 = cos(r41890);
        double r41892 = fma(r41882, r41889, r41891);
        double r41893 = atan2(r41885, r41892);
        double r41894 = r41872 + r41893;
        return r41894;
}

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

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

  4. Applied sub-neg0.9

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

  5. Applied sin-sum0.8

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

  6. Simplified0.8

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

  8. Applied cos-diff0.2

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

  10. Applied log1p-expm1-u0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019325 +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)))))))