Average Error: 13.0 → 0.2
Time: 32.4s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot \sqrt[3]{\sin \lambda_2}\right)\right)}\]
\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot \sqrt[3]{\sin \lambda_2}\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r120990 = lambda1;
        double r120991 = lambda2;
        double r120992 = r120990 - r120991;
        double r120993 = sin(r120992);
        double r120994 = phi2;
        double r120995 = cos(r120994);
        double r120996 = r120993 * r120995;
        double r120997 = phi1;
        double r120998 = cos(r120997);
        double r120999 = sin(r120994);
        double r121000 = r120998 * r120999;
        double r121001 = sin(r120997);
        double r121002 = r121001 * r120995;
        double r121003 = cos(r120992);
        double r121004 = r121002 * r121003;
        double r121005 = r121000 - r121004;
        double r121006 = atan2(r120996, r121005);
        return r121006;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r121007 = lambda1;
        double r121008 = sin(r121007);
        double r121009 = lambda2;
        double r121010 = cos(r121009);
        double r121011 = r121008 * r121010;
        double r121012 = cos(r121007);
        double r121013 = sin(r121009);
        double r121014 = r121012 * r121013;
        double r121015 = r121011 - r121014;
        double r121016 = phi2;
        double r121017 = cos(r121016);
        double r121018 = r121015 * r121017;
        double r121019 = phi1;
        double r121020 = cos(r121019);
        double r121021 = sin(r121016);
        double r121022 = r121020 * r121021;
        double r121023 = sin(r121019);
        double r121024 = r121023 * r121017;
        double r121025 = r121012 * r121010;
        double r121026 = r121024 * r121025;
        double r121027 = cbrt(r121013);
        double r121028 = r121027 * r121027;
        double r121029 = r121008 * r121028;
        double r121030 = r121029 * r121027;
        double r121031 = r121024 * r121030;
        double r121032 = r121026 + r121031;
        double r121033 = r121022 - r121032;
        double r121034 = atan2(r121018, r121033);
        return r121034;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.0

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

  3. Applied sin-diff6.5

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

  5. Applied cos-diff0.2

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

  6. Applied distribute-lft-in0.2

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied associate-*r*0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019294 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))