Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 18.4s

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}^{3} - {\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)}^{3}}{\left(\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right) \cdot \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right) + \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r103734 = lambda1;
        double r103735 = theta;
        double r103736 = sin(r103735);
        double r103737 = delta;
        double r103738 = sin(r103737);
        double r103739 = r103736 * r103738;
        double r103740 = phi1;
        double r103741 = cos(r103740);
        double r103742 = r103739 * r103741;
        double r103743 = cos(r103737);
        double r103744 = sin(r103740);
        double r103745 = r103744 * r103743;
        double r103746 = r103741 * r103738;
        double r103747 = cos(r103735);
        double r103748 = r103746 * r103747;
        double r103749 = r103745 + r103748;
        double r103750 = asin(r103749);
        double r103751 = sin(r103750);
        double r103752 = r103744 * r103751;
        double r103753 = r103743 - r103752;
        double r103754 = atan2(r103742, r103753);
        double r103755 = r103734 + r103754;
        return r103755;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r103756 = lambda1;
        double r103757 = theta;
        double r103758 = sin(r103757);
        double r103759 = delta;
        double r103760 = sin(r103759);
        double r103761 = r103758 * r103760;
        double r103762 = phi1;
        double r103763 = cos(r103762);
        double r103764 = r103761 * r103763;
        double r103765 = cos(r103759);
        double r103766 = 2.0;
        double r103767 = pow(r103765, r103766);
        double r103768 = sin(r103762);
        double r103769 = pow(r103768, r103766);
        double r103770 = pow(r103763, r103766);
        double r103771 = r103769 * r103770;
        double r103772 = cos(r103757);
        double r103773 = pow(r103772, r103766);
        double r103774 = pow(r103760, r103766);
        double r103775 = r103773 * r103774;
        double r103776 = 4.0;
        double r103777 = pow(r103768, r103776);
        double r103778 = r103777 * r103767;
        double r103779 = fma(r103771, r103775, r103778);
        double r103780 = r103767 - r103779;
        double r103781 = 3.0;
        double r103782 = pow(r103780, r103781);
        double r103783 = pow(r103768, r103781);
        double r103784 = r103765 * r103772;
        double r103785 = r103783 * r103784;
        double r103786 = r103763 * r103785;
        double r103787 = r103760 * r103786;
        double r103788 = r103766 * r103787;
        double r103789 = pow(r103788, r103781);
        double r103790 = r103782 - r103789;
        double r103791 = r103788 + r103780;
        double r103792 = r103788 * r103791;
        double r103793 = r103780 * r103780;
        double r103794 = r103792 + r103793;
        double r103795 = r103768 * r103765;
        double r103796 = r103763 * r103760;
        double r103797 = r103796 * r103772;
        double r103798 = r103795 + r103797;
        double r103799 = asin(r103798);
        double r103800 = sin(r103799);
        double r103801 = fma(r103768, r103800, r103765);
        double r103802 = r103794 * r103801;
        double r103803 = r103790 / r103802;
        double r103804 = atan2(r103764, r103803);
        double r103805 = r103756 + r103804;
        return r103805;
}

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-exp 0.2

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

5. Applied flip-- 0.2

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

6. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\mathsf{fma}\left(\cos delta, \cos delta, -{\left(\sin \phi_1\right)}^{2} \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}}{\cos delta + \log \left(e^{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)}}\]

7. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\mathsf{fma}\left(\cos delta, \cos delta, -{\left(\sin \phi_1\right)}^{2} \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}{\color{blue}{\mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)}}}\]

8. Taylor expanded around inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}}{\mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)}}\]

9. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) - 2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)}}{\mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)}}\]
10. Using strategy rm

11. Applied flip3-- 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\frac{{\left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}^{3} - {\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)}^{3}}{\left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) + \left(\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right) \cdot \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right) + \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)}}}{\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)}}\]

12. Applied associate-/l/ 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{{\left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}^{3} - {\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)}^{3}}{\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) \cdot \left(\left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) + \left(\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right) \cdot \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right) + \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)\right)}}}\]

13. Simplified 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}^{3} - {\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)}^{3}}{\color{blue}{\left(\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right) \cdot \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right) + \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)}}}\]

14. Final simplification 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}^{3} - {\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)}^{3}}{\left(\left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right) \cdot \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right) + \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right) \cdot \left({\left(\cos delta\right)}^{2} - \mathsf{fma}\left({\left(\sin \phi_1\right)}^{2} \cdot {\left(\cos \phi_1\right)}^{2}, {\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}, {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)}}\]

Reproduce

herbie shell --seed 2020024 +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))))))))))