Average Error: 0.8 → 0.3
Time: 32.8s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}\right) \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right)\right)}^{3}} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]
\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{\cos \phi_2 \cdot \left(\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}\right) \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sqrt[3]{{\left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_2 \cdot \cos \lambda_2, \cos \phi_1\right)\right)}^{3}} + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r45975 = lambda1;
        double r45976 = phi2;
        double r45977 = cos(r45976);
        double r45978 = lambda2;
        double r45979 = r45975 - r45978;
        double r45980 = sin(r45979);
        double r45981 = r45977 * r45980;
        double r45982 = phi1;
        double r45983 = cos(r45982);
        double r45984 = cos(r45979);
        double r45985 = r45977 * r45984;
        double r45986 = r45983 + r45985;
        double r45987 = atan2(r45981, r45986);
        double r45988 = r45975 + r45987;
        return r45988;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r45989 = lambda1;
        double r45990 = phi2;
        double r45991 = cos(r45990);
        double r45992 = lambda2;
        double r45993 = cos(r45992);
        double r45994 = sin(r45989);
        double r45995 = r45993 * r45994;
        double r45996 = cbrt(r45995);
        double r45997 = r45996 * r45996;
        double r45998 = r45997 * r45996;
        double r45999 = sin(r45992);
        double r46000 = cos(r45989);
        double r46001 = r45999 * r46000;
        double r46002 = r45998 - r46001;
        double r46003 = r45991 * r46002;
        double r46004 = r45991 * r45993;
        double r46005 = phi1;
        double r46006 = cos(r46005);
        double r46007 = fma(r46000, r46004, r46006);
        double r46008 = 3.0;
        double r46009 = pow(r46007, r46008);
        double r46010 = cbrt(r46009);
        double r46011 = r45994 * r45999;
        double r46012 = r45991 * r46011;
        double r46013 = r46010 + r46012;
        double r46014 = atan2(r46003, r46013);
        double r46015 = r45989 + r46014;
        return r46015;
}

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. Using strategy rm

  3. Applied sin-diff0.8

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

  4. Simplified0.8

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

  5. Simplified0.8

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

  7. Applied cos-diff0.2

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

  8. Applied distribute-lft-in0.2

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

  9. Applied associate-+r+0.2

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

  10. Simplified0.2

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

  12. Applied add-cube-cbrt0.3

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

  14. Applied add-cbrt-cube0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 +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)))))))