Average Error: 12.7 → 0.2
Time: 40.8s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \left(\sin \lambda_1 \cdot \left(\sqrt[3]{\sin \lambda_2} \cdot \sqrt[3]{\sin \lambda_2}\right)\right) \cdot \sqrt[3]{\sin \lambda_2}\right)}\]
\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(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \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)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2405990 = lambda1;
        double r2405991 = lambda2;
        double r2405992 = r2405990 - r2405991;
        double r2405993 = sin(r2405992);
        double r2405994 = phi2;
        double r2405995 = cos(r2405994);
        double r2405996 = r2405993 * r2405995;
        double r2405997 = phi1;
        double r2405998 = cos(r2405997);
        double r2405999 = sin(r2405994);
        double r2406000 = r2405998 * r2405999;
        double r2406001 = sin(r2405997);
        double r2406002 = r2406001 * r2405995;
        double r2406003 = cos(r2405992);
        double r2406004 = r2406002 * r2406003;
        double r2406005 = r2406000 - r2406004;
        double r2406006 = atan2(r2405996, r2406005);
        return r2406006;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r2406007 = lambda2;
        double r2406008 = cos(r2406007);
        double r2406009 = lambda1;
        double r2406010 = sin(r2406009);
        double r2406011 = r2406008 * r2406010;
        double r2406012 = cos(r2406009);
        double r2406013 = sin(r2406007);
        double r2406014 = r2406012 * r2406013;
        double r2406015 = r2406011 - r2406014;
        double r2406016 = phi2;
        double r2406017 = cos(r2406016);
        double r2406018 = r2406015 * r2406017;
        double r2406019 = sin(r2406016);
        double r2406020 = phi1;
        double r2406021 = cos(r2406020);
        double r2406022 = r2406019 * r2406021;
        double r2406023 = sin(r2406020);
        double r2406024 = r2406023 * r2406017;
        double r2406025 = r2406008 * r2406012;
        double r2406026 = cbrt(r2406013);
        double r2406027 = r2406026 * r2406026;
        double r2406028 = r2406010 * r2406027;
        double r2406029 = r2406028 * r2406026;
        double r2406030 = r2406025 + r2406029;
        double r2406031 = r2406024 * r2406030;
        double r2406032 = r2406022 - r2406031;
        double r2406033 = atan2(r2406018, r2406032);
        return r2406033;
}

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 12.7

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

  3. Applied sin-diff6.6

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

  7. Applied add-cube-cbrt0.2

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

  8. Applied associate-*r*0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))