Average Error: 13.2 → 0.2
Time: 22.1s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right) \cdot \cos \lambda_2 + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_1}\right) \cdot \left(\sqrt[3]{\sin \lambda_1} \cdot \sin \lambda_2\right)\right)\right)}\]
\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{\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(\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right) \cdot \cos \lambda_2 + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\sqrt[3]{\sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_1}\right) \cdot \left(\sqrt[3]{\sin \lambda_1} \cdot \sin \lambda_2\right)\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r96431 = lambda1;
        double r96432 = lambda2;
        double r96433 = r96431 - r96432;
        double r96434 = sin(r96433);
        double r96435 = phi2;
        double r96436 = cos(r96435);
        double r96437 = r96434 * r96436;
        double r96438 = phi1;
        double r96439 = cos(r96438);
        double r96440 = sin(r96435);
        double r96441 = r96439 * r96440;
        double r96442 = sin(r96438);
        double r96443 = r96442 * r96436;
        double r96444 = cos(r96433);
        double r96445 = r96443 * r96444;
        double r96446 = r96441 - r96445;
        double r96447 = atan2(r96437, r96446);
        return r96447;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r96448 = lambda1;
        double r96449 = sin(r96448);
        double r96450 = lambda2;
        double r96451 = cos(r96450);
        double r96452 = r96449 * r96451;
        double r96453 = cos(r96448);
        double r96454 = sin(r96450);
        double r96455 = r96453 * r96454;
        double r96456 = r96452 - r96455;
        double r96457 = phi2;
        double r96458 = cos(r96457);
        double r96459 = r96456 * r96458;
        double r96460 = phi1;
        double r96461 = cos(r96460);
        double r96462 = sin(r96457);
        double r96463 = r96461 * r96462;
        double r96464 = sin(r96460);
        double r96465 = r96464 * r96458;
        double r96466 = r96465 * r96453;
        double r96467 = r96466 * r96451;
        double r96468 = cbrt(r96449);
        double r96469 = r96468 * r96468;
        double r96470 = r96468 * r96454;
        double r96471 = r96469 * r96470;
        double r96472 = r96465 * r96471;
        double r96473 = r96467 + r96472;
        double r96474 = r96463 - r96473;
        double r96475 = atan2(r96459, r96474);
        return r96475;
}

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-diff6.7

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

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

  10. Applied add-cube-cbrt0.2

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

  11. Applied associate-*l*0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2020042 
(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))))))