Bearing on a great circle

Average Error: 13.2 → 0.2

Time: 38.8s

Precision: 64

Math

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

↓

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

C

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4997944 = lambda1;
        double r4997945 = lambda2;
        double r4997946 = r4997944 - r4997945;
        double r4997947 = sin(r4997946);
        double r4997948 = phi2;
        double r4997949 = cos(r4997948);
        double r4997950 = r4997947 * r4997949;
        double r4997951 = phi1;
        double r4997952 = cos(r4997951);
        double r4997953 = sin(r4997948);
        double r4997954 = r4997952 * r4997953;
        double r4997955 = sin(r4997951);
        double r4997956 = r4997955 * r4997949;
        double r4997957 = cos(r4997946);
        double r4997958 = r4997956 * r4997957;
        double r4997959 = r4997954 - r4997958;
        double r4997960 = atan2(r4997950, r4997959);
        return r4997960;
}

↓

double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r4997961 = lambda2;
        double r4997962 = cos(r4997961);
        double r4997963 = lambda1;
        double r4997964 = sin(r4997963);
        double r4997965 = r4997962 * r4997964;
        double r4997966 = cos(r4997963);
        double r4997967 = sin(r4997961);
        double r4997968 = r4997966 * r4997967;
        double r4997969 = r4997965 - r4997968;
        double r4997970 = phi2;
        double r4997971 = cos(r4997970);
        double r4997972 = r4997969 * r4997971;
        double r4997973 = sin(r4997970);
        double r4997974 = phi1;
        double r4997975 = cos(r4997974);
        double r4997976 = r4997973 * r4997975;
        double r4997977 = r4997962 * r4997966;
        double r4997978 = sin(r4997974);
        double r4997979 = r4997978 * r4997971;
        double r4997980 = r4997977 * r4997979;
        double r4997981 = cbrt(r4997967);
        double r4997982 = cbrt(r4997981);
        double r4997983 = r4997981 * r4997982;
        double r4997984 = r4997982 * r4997982;
        double r4997985 = r4997983 * r4997984;
        double r4997986 = r4997964 * r4997985;
        double r4997987 = r4997986 * r4997981;
        double r4997988 = r4997979 * r4997987;
        double r4997989 = r4997980 + r4997988;
        double r4997990 = r4997976 - r4997989;
        double r4997991 = atan2(r4997972, r4997990);
        return r4997991;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

1. Initial program 13.2

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

3. Applied sin-diff 6.8

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

5. Applied cos-diff 0.2

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

6. Applied distribute-lft-in 0.2

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

8. Applied add-cube-cbrt 0.2

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

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. Using strategy rm

11. Applied add-cube-cbrt 0.2

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

12. Applied associate-*l* 0.2

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

13. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019164 
(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))))))