Average Error: 13.1 → 0.2
Time: 16.6s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right) \cdot \sin \lambda_2\right)}\]
\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(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right) \cdot \sin \lambda_2\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r135156 = lambda1;
        double r135157 = lambda2;
        double r135158 = r135156 - r135157;
        double r135159 = sin(r135158);
        double r135160 = phi2;
        double r135161 = cos(r135160);
        double r135162 = r135159 * r135161;
        double r135163 = phi1;
        double r135164 = cos(r135163);
        double r135165 = sin(r135160);
        double r135166 = r135164 * r135165;
        double r135167 = sin(r135163);
        double r135168 = r135167 * r135161;
        double r135169 = cos(r135158);
        double r135170 = r135168 * r135169;
        double r135171 = r135166 - r135170;
        double r135172 = atan2(r135162, r135171);
        return r135172;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r135173 = lambda1;
        double r135174 = sin(r135173);
        double r135175 = lambda2;
        double r135176 = cos(r135175);
        double r135177 = r135174 * r135176;
        double r135178 = cos(r135173);
        double r135179 = -r135175;
        double r135180 = sin(r135179);
        double r135181 = r135178 * r135180;
        double r135182 = r135177 + r135181;
        double r135183 = phi2;
        double r135184 = cos(r135183);
        double r135185 = r135182 * r135184;
        double r135186 = phi1;
        double r135187 = cos(r135186);
        double r135188 = sin(r135183);
        double r135189 = r135187 * r135188;
        double r135190 = log1p(r135189);
        double r135191 = expm1(r135190);
        double r135192 = r135178 * r135176;
        double r135193 = sin(r135186);
        double r135194 = r135193 * r135184;
        double r135195 = r135192 * r135194;
        double r135196 = r135194 * r135174;
        double r135197 = sin(r135175);
        double r135198 = r135196 * r135197;
        double r135199 = r135195 + r135198;
        double r135200 = r135191 - r135199;
        double r135201 = atan2(r135185, r135200);
        return r135201;
}

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 13.1

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

  3. Applied sub-neg13.1

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

  4. Applied sin-sum6.5

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

  5. Simplified6.5

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

  7. Applied cos-diff0.2

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

  8. Applied distribute-lft-in0.2

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

  9. Simplified0.2

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

  11. Applied associate-*r*0.2

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

  13. Applied expm1-log1p-u0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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))))))