Average Error: 0.8 → 0.4
Time: 41.4s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right), \cos \phi_1\right)}\right)\right)\right)}\]
\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(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\mathsf{fma}\left(\cos \phi_2, \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right), \cos \phi_1\right)}\right)\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1974786 = lambda1;
        double r1974787 = phi2;
        double r1974788 = cos(r1974787);
        double r1974789 = lambda2;
        double r1974790 = r1974786 - r1974789;
        double r1974791 = sin(r1974790);
        double r1974792 = r1974788 * r1974791;
        double r1974793 = phi1;
        double r1974794 = cos(r1974793);
        double r1974795 = cos(r1974790);
        double r1974796 = r1974788 * r1974795;
        double r1974797 = r1974794 + r1974796;
        double r1974798 = atan2(r1974792, r1974797);
        double r1974799 = r1974786 + r1974798;
        return r1974799;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r1974800 = lambda1;
        double r1974801 = sin(r1974800);
        double r1974802 = lambda2;
        double r1974803 = cos(r1974802);
        double r1974804 = r1974801 * r1974803;
        double r1974805 = cos(r1974800);
        double r1974806 = sin(r1974802);
        double r1974807 = r1974805 * r1974806;
        double r1974808 = r1974804 - r1974807;
        double r1974809 = phi2;
        double r1974810 = cos(r1974809);
        double r1974811 = r1974808 * r1974810;
        double r1974812 = r1974806 * r1974801;
        double r1974813 = fma(r1974805, r1974803, r1974812);
        double r1974814 = phi1;
        double r1974815 = cos(r1974814);
        double r1974816 = fma(r1974810, r1974813, r1974815);
        double r1974817 = exp(r1974816);
        double r1974818 = log1p(r1974817);
        double r1974819 = expm1(r1974818);
        double r1974820 = log(r1974819);
        double r1974821 = atan2(r1974811, r1974820);
        double r1974822 = r1974800 + r1974821;
        return r1974822;
}

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

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

  4. Applied cos-diff0.8

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

  6. Applied sin-diff0.2

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

  8. Applied add-log-exp0.3

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

  9. Simplified0.3

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

  11. Applied expm1-log1p-u0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019149 +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)))))))