Average Error: 13.5 → 0.2
Time: 43.1s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}\]
\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(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(\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\right)\right) - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r81282 = lambda1;
        double r81283 = lambda2;
        double r81284 = r81282 - r81283;
        double r81285 = sin(r81284);
        double r81286 = phi2;
        double r81287 = cos(r81286);
        double r81288 = r81285 * r81287;
        double r81289 = phi1;
        double r81290 = cos(r81289);
        double r81291 = sin(r81286);
        double r81292 = r81290 * r81291;
        double r81293 = sin(r81289);
        double r81294 = r81293 * r81287;
        double r81295 = cos(r81284);
        double r81296 = r81294 * r81295;
        double r81297 = r81292 - r81296;
        double r81298 = atan2(r81288, r81297);
        return r81298;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r81299 = lambda1;
        double r81300 = sin(r81299);
        double r81301 = lambda2;
        double r81302 = cos(r81301);
        double r81303 = r81300 * r81302;
        double r81304 = log1p(r81303);
        double r81305 = expm1(r81304);
        double r81306 = cos(r81299);
        double r81307 = sin(r81301);
        double r81308 = r81306 * r81307;
        double r81309 = r81305 - r81308;
        double r81310 = phi2;
        double r81311 = cos(r81310);
        double r81312 = r81309 * r81311;
        double r81313 = phi1;
        double r81314 = cos(r81313);
        double r81315 = sin(r81310);
        double r81316 = r81314 * r81315;
        double r81317 = sin(r81313);
        double r81318 = r81317 * r81311;
        double r81319 = r81306 * r81302;
        double r81320 = r81318 * r81319;
        double r81321 = r81316 - r81320;
        double r81322 = r81300 * r81307;
        double r81323 = r81318 * r81322;
        double r81324 = r81321 - r81323;
        double r81325 = atan2(r81312, r81324);
        return r81325;
}

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 13.5

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

  3. Applied sin-diff6.9

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  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. Applied associate--r+0.2

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

  9. Applied expm1-log1p-u0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +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))))))