Average Error: 0.1 → 0.1
Time: 18.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)}\]
\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{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r129840 = lambda1;
        double r129841 = theta;
        double r129842 = sin(r129841);
        double r129843 = delta;
        double r129844 = sin(r129843);
        double r129845 = r129842 * r129844;
        double r129846 = phi1;
        double r129847 = cos(r129846);
        double r129848 = r129845 * r129847;
        double r129849 = cos(r129843);
        double r129850 = sin(r129846);
        double r129851 = r129850 * r129849;
        double r129852 = r129847 * r129844;
        double r129853 = cos(r129841);
        double r129854 = r129852 * r129853;
        double r129855 = r129851 + r129854;
        double r129856 = asin(r129855);
        double r129857 = sin(r129856);
        double r129858 = r129850 * r129857;
        double r129859 = r129849 - r129858;
        double r129860 = atan2(r129848, r129859);
        double r129861 = r129840 + r129860;
        return r129861;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r129862 = lambda1;
        double r129863 = delta;
        double r129864 = sin(r129863);
        double r129865 = phi1;
        double r129866 = cos(r129865);
        double r129867 = theta;
        double r129868 = sin(r129867);
        double r129869 = r129866 * r129868;
        double r129870 = r129864 * r129869;
        double r129871 = cos(r129863);
        double r129872 = sin(r129865);
        double r129873 = cos(r129867);
        double r129874 = r129866 * r129873;
        double r129875 = r129864 * r129874;
        double r129876 = fma(r129872, r129871, r129875);
        double r129877 = r129872 * r129876;
        double r129878 = r129871 - r129877;
        double r129879 = atan2(r129870, r129878);
        double r129880 = r129862 + r129879;
        return r129880;
}

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 0.1

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

  2. Simplified0.1

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

  3. Taylor expanded around 0 0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 +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))))))))))