Average Error: 12.9 → 0.2
Time: 42.6s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}\]
\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 \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r121609 = lambda1;
        double r121610 = lambda2;
        double r121611 = r121609 - r121610;
        double r121612 = sin(r121611);
        double r121613 = phi2;
        double r121614 = cos(r121613);
        double r121615 = r121612 * r121614;
        double r121616 = phi1;
        double r121617 = cos(r121616);
        double r121618 = sin(r121613);
        double r121619 = r121617 * r121618;
        double r121620 = sin(r121616);
        double r121621 = r121620 * r121614;
        double r121622 = cos(r121611);
        double r121623 = r121621 * r121622;
        double r121624 = r121619 - r121623;
        double r121625 = atan2(r121615, r121624);
        return r121625;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r121626 = lambda1;
        double r121627 = sin(r121626);
        double r121628 = lambda2;
        double r121629 = cos(r121628);
        double r121630 = r121627 * r121629;
        double r121631 = cos(r121626);
        double r121632 = -r121628;
        double r121633 = sin(r121632);
        double r121634 = r121631 * r121633;
        double r121635 = r121630 + r121634;
        double r121636 = phi2;
        double r121637 = cos(r121636);
        double r121638 = r121635 * r121637;
        double r121639 = phi1;
        double r121640 = cos(r121639);
        double r121641 = sin(r121636);
        double r121642 = r121640 * r121641;
        double r121643 = log1p(r121642);
        double r121644 = expm1(r121643);
        double r121645 = sin(r121639);
        double r121646 = r121645 * r121637;
        double r121647 = sin(r121628);
        double r121648 = r121627 * r121647;
        double r121649 = fma(r121631, r121629, r121648);
        double r121650 = r121646 * r121649;
        double r121651 = expm1(r121650);
        double r121652 = log1p(r121651);
        double r121653 = r121644 - r121652;
        double r121654 = atan2(r121638, r121653);
        return r121654;
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 12.9

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

  3. Applied sub-neg12.9

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

  4. Applied sin-sum6.4

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

  5. Simplified6.4

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

  9. Applied log1p-expm1-u0.2

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

  10. Simplified0.2

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

  12. Applied expm1-log1p-u0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 +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))))))