Average Error: 0.2 → 0.2
Time: 15.5s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}\]
\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{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r117733 = lambda1;
        double r117734 = theta;
        double r117735 = sin(r117734);
        double r117736 = delta;
        double r117737 = sin(r117736);
        double r117738 = r117735 * r117737;
        double r117739 = phi1;
        double r117740 = cos(r117739);
        double r117741 = r117738 * r117740;
        double r117742 = cos(r117736);
        double r117743 = sin(r117739);
        double r117744 = r117743 * r117742;
        double r117745 = r117740 * r117737;
        double r117746 = cos(r117734);
        double r117747 = r117745 * r117746;
        double r117748 = r117744 + r117747;
        double r117749 = asin(r117748);
        double r117750 = sin(r117749);
        double r117751 = r117743 * r117750;
        double r117752 = r117742 - r117751;
        double r117753 = atan2(r117741, r117752);
        double r117754 = r117733 + r117753;
        return r117754;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r117755 = lambda1;
        double r117756 = theta;
        double r117757 = sin(r117756);
        double r117758 = delta;
        double r117759 = sin(r117758);
        double r117760 = r117757 * r117759;
        double r117761 = phi1;
        double r117762 = cos(r117761);
        double r117763 = r117760 * r117762;
        double r117764 = cos(r117758);
        double r117765 = 3.0;
        double r117766 = pow(r117764, r117765);
        double r117767 = sin(r117761);
        double r117768 = pow(r117767, r117765);
        double r117769 = cos(r117756);
        double r117770 = r117762 * r117769;
        double r117771 = r117759 * r117770;
        double r117772 = r117767 * r117764;
        double r117773 = r117771 + r117772;
        double r117774 = pow(r117773, r117765);
        double r117775 = r117768 * r117774;
        double r117776 = r117766 - r117775;
        double r117777 = r117762 * r117759;
        double r117778 = r117777 * r117769;
        double r117779 = r117772 + r117778;
        double r117780 = asin(r117779);
        double r117781 = sin(r117780);
        double r117782 = fma(r117767, r117781, r117764);
        double r117783 = r117781 * r117782;
        double r117784 = r117767 * r117783;
        double r117785 = fma(r117764, r117764, r117784);
        double r117786 = r117776 / r117785;
        double r117787 = atan2(r117763, r117786);
        double r117788 = r117755 + r117787;
        return r117788;
}

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 flip3--0.2

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

  4. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}}\]

  5. Taylor expanded around inf 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \color{blue}{{\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)}^{3}}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}\]

  6. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \phi_1\right)}^{3} \cdot {\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)}^{3}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}\]

Reproduce

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