Average Error: 0.2 → 0.2
Time: 14.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}\]
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r83770 = lambda1;
        double r83771 = theta;
        double r83772 = sin(r83771);
        double r83773 = delta;
        double r83774 = sin(r83773);
        double r83775 = r83772 * r83774;
        double r83776 = phi1;
        double r83777 = cos(r83776);
        double r83778 = r83775 * r83777;
        double r83779 = cos(r83773);
        double r83780 = sin(r83776);
        double r83781 = r83780 * r83779;
        double r83782 = r83777 * r83774;
        double r83783 = cos(r83771);
        double r83784 = r83782 * r83783;
        double r83785 = r83781 + r83784;
        double r83786 = asin(r83785);
        double r83787 = sin(r83786);
        double r83788 = r83780 * r83787;
        double r83789 = r83779 - r83788;
        double r83790 = atan2(r83778, r83789);
        double r83791 = r83770 + r83790;
        return r83791;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r83792 = lambda1;
        double r83793 = theta;
        double r83794 = sin(r83793);
        double r83795 = delta;
        double r83796 = sin(r83795);
        double r83797 = r83794 * r83796;
        double r83798 = phi1;
        double r83799 = cos(r83798);
        double r83800 = r83797 * r83799;
        double r83801 = cos(r83795);
        double r83802 = cos(r83793);
        double r83803 = r83799 * r83802;
        double r83804 = sin(r83798);
        double r83805 = r83804 * r83801;
        double r83806 = fma(r83796, r83803, r83805);
        double r83807 = r83806 * r83804;
        double r83808 = r83801 - r83807;
        double r83809 = atan2(r83800, r83808);
        double r83810 = r83792 + r83809;
        return r83810;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Derivation

  1. Initial program 0.2

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

  3. Applied add-log-exp0.2

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

  5. Applied add-log-exp0.2

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

  6. Taylor expanded around inf 0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))