Average Error: 13.3 → 0.2
Time: 37.3s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)}\]
\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(\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(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4281988 = lambda1;
        double r4281989 = lambda2;
        double r4281990 = r4281988 - r4281989;
        double r4281991 = sin(r4281990);
        double r4281992 = phi2;
        double r4281993 = cos(r4281992);
        double r4281994 = r4281991 * r4281993;
        double r4281995 = phi1;
        double r4281996 = cos(r4281995);
        double r4281997 = sin(r4281992);
        double r4281998 = r4281996 * r4281997;
        double r4281999 = sin(r4281995);
        double r4282000 = r4281999 * r4281993;
        double r4282001 = cos(r4281990);
        double r4282002 = r4282000 * r4282001;
        double r4282003 = r4281998 - r4282002;
        double r4282004 = atan2(r4281994, r4282003);
        return r4282004;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4282005 = lambda2;
        double r4282006 = cos(r4282005);
        double r4282007 = lambda1;
        double r4282008 = sin(r4282007);
        double r4282009 = r4282006 * r4282008;
        double r4282010 = cos(r4282007);
        double r4282011 = sin(r4282005);
        double r4282012 = r4282010 * r4282011;
        double r4282013 = r4282009 - r4282012;
        double r4282014 = phi2;
        double r4282015 = cos(r4282014);
        double r4282016 = r4282013 * r4282015;
        double r4282017 = sin(r4282014);
        double r4282018 = phi1;
        double r4282019 = cos(r4282018);
        double r4282020 = r4282017 * r4282019;
        double r4282021 = r4282006 * r4282010;
        double r4282022 = sin(r4282018);
        double r4282023 = r4282022 * r4282015;
        double r4282024 = r4282021 * r4282023;
        double r4282025 = r4282011 * r4282008;
        double r4282026 = cbrt(r4282025);
        double r4282027 = r4282026 * r4282026;
        double r4282028 = r4282026 * r4282027;
        double r4282029 = r4282023 * r4282028;
        double r4282030 = r4282024 + r4282029;
        double r4282031 = r4282020 - r4282030;
        double r4282032 = atan2(r4282016, r4282031);
        return r4282032;
}

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 13.3

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

  3. Applied sin-diff6.8

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

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +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))))))