Average Error: 13.3 → 0.2
Time: 27.7s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)}\]
\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{\mathsf{fma}\left(\sin \lambda_1, \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) + \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r480 = lambda1;
        double r481 = lambda2;
        double r482 = r480 - r481;
        double r483 = sin(r482);
        double r484 = phi2;
        double r485 = cos(r484);
        double r486 = r483 * r485;
        double r487 = phi1;
        double r488 = cos(r487);
        double r489 = sin(r484);
        double r490 = r488 * r489;
        double r491 = sin(r487);
        double r492 = r491 * r485;
        double r493 = cos(r482);
        double r494 = r492 * r493;
        double r495 = r490 - r494;
        double r496 = atan2(r486, r495);
        return r496;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r497 = lambda1;
        double r498 = sin(r497);
        double r499 = lambda2;
        double r500 = cos(r499);
        double r501 = cos(r497);
        double r502 = sin(r499);
        double r503 = r501 * r502;
        double r504 = -r503;
        double r505 = fma(r498, r500, r504);
        double r506 = phi2;
        double r507 = cos(r506);
        double r508 = r505 * r507;
        double r509 = phi1;
        double r510 = cos(r509);
        double r511 = sin(r506);
        double r512 = r510 * r511;
        double r513 = sin(r509);
        double r514 = r513 * r507;
        double r515 = r501 * r500;
        double r516 = r514 * r515;
        double r517 = r502 * r498;
        double r518 = r507 * r517;
        double r519 = r513 * r518;
        double r520 = r516 + r519;
        double r521 = r512 - r520;
        double r522 = atan2(r508, r521);
        return r522;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 13.3

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

  8. Applied fma-neg0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2020025 +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))))))