Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 18.8s

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(\cos delta\right)}^{2} \cdot {\left(\cos delta\right)}^{2} - \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right) \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)}{\left(\cos delta + \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin \phi_1 \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)\right)\right) \cdot \left({\left(\cos delta\right)}^{2} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r123721 = lambda1;
        double r123722 = theta;
        double r123723 = sin(r123722);
        double r123724 = delta;
        double r123725 = sin(r123724);
        double r123726 = r123723 * r123725;
        double r123727 = phi1;
        double r123728 = cos(r123727);
        double r123729 = r123726 * r123728;
        double r123730 = cos(r123724);
        double r123731 = sin(r123727);
        double r123732 = r123731 * r123730;
        double r123733 = r123728 * r123725;
        double r123734 = cos(r123722);
        double r123735 = r123733 * r123734;
        double r123736 = r123732 + r123735;
        double r123737 = asin(r123736);
        double r123738 = sin(r123737);
        double r123739 = r123731 * r123738;
        double r123740 = r123730 - r123739;
        double r123741 = atan2(r123729, r123740);
        double r123742 = r123721 + r123741;
        return r123742;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r123743 = lambda1;
        double r123744 = theta;
        double r123745 = sin(r123744);
        double r123746 = delta;
        double r123747 = sin(r123746);
        double r123748 = r123745 * r123747;
        double r123749 = phi1;
        double r123750 = cos(r123749);
        double r123751 = r123748 * r123750;
        double r123752 = cos(r123746);
        double r123753 = 2.0;
        double r123754 = pow(r123752, r123753);
        double r123755 = r123754 * r123754;
        double r123756 = sin(r123749);
        double r123757 = 3.0;
        double r123758 = pow(r123756, r123757);
        double r123759 = cos(r123744);
        double r123760 = r123747 * r123759;
        double r123761 = r123752 * r123760;
        double r123762 = r123750 * r123761;
        double r123763 = r123758 * r123762;
        double r123764 = pow(r123756, r123753);
        double r123765 = pow(r123750, r123753);
        double r123766 = pow(r123759, r123753);
        double r123767 = pow(r123747, r123753);
        double r123768 = r123766 * r123767;
        double r123769 = r123765 * r123768;
        double r123770 = r123764 * r123769;
        double r123771 = 4.0;
        double r123772 = pow(r123756, r123771);
        double r123773 = r123772 * r123754;
        double r123774 = r123752 * r123759;
        double r123775 = r123747 * r123774;
        double r123776 = r123750 * r123775;
        double r123777 = r123758 * r123776;
        double r123778 = r123773 + r123777;
        double r123779 = r123770 + r123778;
        double r123780 = r123763 + r123779;
        double r123781 = r123780 * r123780;
        double r123782 = r123755 - r123781;
        double r123783 = r123764 * r123752;
        double r123784 = r123759 * r123747;
        double r123785 = r123750 * r123784;
        double r123786 = r123756 * r123785;
        double r123787 = r123783 + r123786;
        double r123788 = r123752 + r123787;
        double r123789 = r123754 + r123780;
        double r123790 = r123788 * r123789;
        double r123791 = r123782 / r123790;
        double r123792 = atan2(r123751, r123791);
        double r123793 = r123743 + r123792;
        return r123793;
}

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 flip-- 0.2

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

4. Taylor expanded around -inf 0.2

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

6. Applied flip-- 0.2

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

7. 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(\cos delta\right)}^{2} \cdot {\left(\cos delta\right)}^{2} - \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right) \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)}{\left(\cos delta + \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin \phi_1 \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)\right)\right) \cdot \left({\left(\cos delta\right)}^{2} + \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\cos delta \cdot \left(\sin delta \cdot \cos theta\right)\right)\right) + \left({\left(\sin \phi_1\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin delta\right)}^{2}\right)\right) + \left({\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2} + {\left(\sin \phi_1\right)}^{3} \cdot \left(\cos \phi_1 \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right)\right)\right)\right)}}}\]

8. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019353 
(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))))))))))