Average Error: 13.2 → 0.2
Time: 34.7s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sqrt[3]{\cos \lambda_2} \cdot \left(\sin \lambda_1 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos \lambda_2}\right)\right) \cdot \sqrt[3]{\cos \lambda_2}\right)\right) - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\]
\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(\sqrt[3]{\cos \lambda_2} \cdot \left(\sin \lambda_1 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos \lambda_2}\right)\right) \cdot \sqrt[3]{\cos \lambda_2}\right)\right) - \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 \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3810092 = lambda1;
        double r3810093 = lambda2;
        double r3810094 = r3810092 - r3810093;
        double r3810095 = sin(r3810094);
        double r3810096 = phi2;
        double r3810097 = cos(r3810096);
        double r3810098 = r3810095 * r3810097;
        double r3810099 = phi1;
        double r3810100 = cos(r3810099);
        double r3810101 = sin(r3810096);
        double r3810102 = r3810100 * r3810101;
        double r3810103 = sin(r3810099);
        double r3810104 = r3810103 * r3810097;
        double r3810105 = cos(r3810094);
        double r3810106 = r3810104 * r3810105;
        double r3810107 = r3810102 - r3810106;
        double r3810108 = atan2(r3810098, r3810107);
        return r3810108;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r3810109 = lambda2;
        double r3810110 = cos(r3810109);
        double r3810111 = cbrt(r3810110);
        double r3810112 = lambda1;
        double r3810113 = sin(r3810112);
        double r3810114 = expm1(r3810111);
        double r3810115 = log1p(r3810114);
        double r3810116 = r3810115 * r3810111;
        double r3810117 = r3810113 * r3810116;
        double r3810118 = r3810111 * r3810117;
        double r3810119 = cos(r3810112);
        double r3810120 = sin(r3810109);
        double r3810121 = r3810119 * r3810120;
        double r3810122 = r3810118 - r3810121;
        double r3810123 = phi2;
        double r3810124 = cos(r3810123);
        double r3810125 = r3810122 * r3810124;
        double r3810126 = phi1;
        double r3810127 = cos(r3810126);
        double r3810128 = sin(r3810123);
        double r3810129 = r3810127 * r3810128;
        double r3810130 = sin(r3810126);
        double r3810131 = r3810130 * r3810124;
        double r3810132 = r3810119 * r3810110;
        double r3810133 = r3810131 * r3810132;
        double r3810134 = r3810120 * r3810113;
        double r3810135 = r3810134 * r3810131;
        double r3810136 = r3810133 + r3810135;
        double r3810137 = r3810129 - r3810136;
        double r3810138 = atan2(r3810125, r3810137);
        return r3810138;
}

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 13.2

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

  3. Applied sin-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-rgt-in0.2

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

  8. Applied add-cube-cbrt0.2

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

  9. Applied associate-*r*0.2

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

  11. Applied log1p-expm1-u0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\left(\sin \lambda_1 \cdot \left(\sqrt[3]{\cos \lambda_2} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos \lambda_2}\right)\right)}\right)\right) \cdot \sqrt[3]{\cos \lambda_2} - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\]

  12. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\sqrt[3]{\cos \lambda_2} \cdot \left(\sin \lambda_1 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos \lambda_2}\right)\right) \cdot \sqrt[3]{\cos \lambda_2}\right)\right) - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)\right)}\]

Reproduce

herbie shell --seed 2019172 +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))))))