Average Error: 13.5 → 0.2
Time: 49.3s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\left(\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) - \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}\]
\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 \left(-\lambda_2\right)\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 \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r78357 = lambda1;
        double r78358 = lambda2;
        double r78359 = r78357 - r78358;
        double r78360 = sin(r78359);
        double r78361 = phi2;
        double r78362 = cos(r78361);
        double r78363 = r78360 * r78362;
        double r78364 = phi1;
        double r78365 = cos(r78364);
        double r78366 = sin(r78361);
        double r78367 = r78365 * r78366;
        double r78368 = sin(r78364);
        double r78369 = r78368 * r78362;
        double r78370 = cos(r78359);
        double r78371 = r78369 * r78370;
        double r78372 = r78367 - r78371;
        double r78373 = atan2(r78363, r78372);
        return r78373;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r78374 = lambda1;
        double r78375 = sin(r78374);
        double r78376 = lambda2;
        double r78377 = cos(r78376);
        double r78378 = r78375 * r78377;
        double r78379 = log1p(r78378);
        double r78380 = expm1(r78379);
        double r78381 = cos(r78374);
        double r78382 = -r78376;
        double r78383 = sin(r78382);
        double r78384 = r78381 * r78383;
        double r78385 = r78380 + r78384;
        double r78386 = phi2;
        double r78387 = cos(r78386);
        double r78388 = r78385 * r78387;
        double r78389 = phi1;
        double r78390 = cos(r78389);
        double r78391 = sin(r78386);
        double r78392 = r78390 * r78391;
        double r78393 = sin(r78389);
        double r78394 = r78393 * r78387;
        double r78395 = r78381 * r78377;
        double r78396 = r78394 * r78395;
        double r78397 = r78392 - r78396;
        double r78398 = sin(r78376);
        double r78399 = r78375 * r78398;
        double r78400 = r78399 * r78394;
        double r78401 = r78397 - r78400;
        double r78402 = atan2(r78388, r78401);
        return r78402;
}

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 sub-neg13.5

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

  4. Applied sin-sum6.9

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

  5. Simplified6.9

    \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\sin \lambda_1 \cdot \cos \lambda_2} + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
  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-rgt-in0.2

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

  9. Applied associate--r+0.2

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

  10. Simplified0.2

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

  12. Applied expm1-log1p-u0.2

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

  13. Final simplification0.2

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

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))))))