Destination given bearing on a great circle

Average Error: 0.1 → 0.2

Time: 16.5s

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\left({\left(\cos delta\right)}^{3} - \mathsf{fma}\left({\left(\sin delta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}, {\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}, {\left(\sin \phi_1\right)}^{6} \cdot {\left(\cos delta\right)}^{3}\right)\right) - 3 \cdot \mathsf{fma}\left(\cos \phi_1 \cdot \sin delta, {\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right), {\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r123700 = lambda1;
        double r123701 = theta;
        double r123702 = sin(r123701);
        double r123703 = delta;
        double r123704 = sin(r123703);
        double r123705 = r123702 * r123704;
        double r123706 = phi1;
        double r123707 = cos(r123706);
        double r123708 = r123705 * r123707;
        double r123709 = cos(r123703);
        double r123710 = sin(r123706);
        double r123711 = r123710 * r123709;
        double r123712 = r123707 * r123704;
        double r123713 = cos(r123701);
        double r123714 = r123712 * r123713;
        double r123715 = r123711 + r123714;
        double r123716 = asin(r123715);
        double r123717 = sin(r123716);
        double r123718 = r123710 * r123717;
        double r123719 = r123709 - r123718;
        double r123720 = atan2(r123708, r123719);
        double r123721 = r123700 + r123720;
        return r123721;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r123722 = lambda1;
        double r123723 = theta;
        double r123724 = sin(r123723);
        double r123725 = delta;
        double r123726 = sin(r123725);
        double r123727 = r123724 * r123726;
        double r123728 = phi1;
        double r123729 = cos(r123728);
        double r123730 = r123727 * r123729;
        double r123731 = cos(r123725);
        double r123732 = 3.0;
        double r123733 = pow(r123731, r123732);
        double r123734 = pow(r123726, r123732);
        double r123735 = pow(r123729, r123732);
        double r123736 = r123734 * r123735;
        double r123737 = cos(r123723);
        double r123738 = pow(r123737, r123732);
        double r123739 = sin(r123728);
        double r123740 = pow(r123739, r123732);
        double r123741 = r123738 * r123740;
        double r123742 = 6.0;
        double r123743 = pow(r123739, r123742);
        double r123744 = r123743 * r123733;
        double r123745 = fma(r123736, r123741, r123744);
        double r123746 = r123733 - r123745;
        double r123747 = r123729 * r123726;
        double r123748 = 5.0;
        double r123749 = pow(r123739, r123748);
        double r123750 = 2.0;
        double r123751 = pow(r123731, r123750);
        double r123752 = r123751 * r123737;
        double r123753 = r123749 * r123752;
        double r123754 = pow(r123726, r123750);
        double r123755 = pow(r123729, r123750);
        double r123756 = 4.0;
        double r123757 = pow(r123739, r123756);
        double r123758 = pow(r123737, r123750);
        double r123759 = r123731 * r123758;
        double r123760 = r123757 * r123759;
        double r123761 = r123755 * r123760;
        double r123762 = r123754 * r123761;
        double r123763 = fma(r123747, r123753, r123762);
        double r123764 = r123732 * r123763;
        double r123765 = r123746 - r123764;
        double r123766 = r123739 * r123731;
        double r123767 = r123747 * r123737;
        double r123768 = r123766 + r123767;
        double r123769 = asin(r123768);
        double r123770 = sin(r123769);
        double r123771 = fma(r123739, r123770, r123731);
        double r123772 = r123770 * r123771;
        double r123773 = r123739 * r123772;
        double r123774 = fma(r123731, r123731, r123773);
        double r123775 = r123765 / r123774;
        double r123776 = atan2(r123730, r123775);
        double r123777 = r123722 + r123776;
        return r123777;
}

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. 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. Simplified 0.2

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

6. Applied expm1-log1p-u 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{3} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}\right)\right)}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}\]

7. Taylor expanded around inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{{\left(\cos delta\right)}^{3} - \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right) + \left({\left(\sin \phi_1\right)}^{6} \cdot {\left(\cos delta\right)}^{3} + \left(3 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right)\right)\right)\right) + 3 \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}\]

8. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\left({\left(\cos delta\right)}^{3} - \mathsf{fma}\left({\left(\sin delta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}, {\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}, {\left(\sin \phi_1\right)}^{6} \cdot {\left(\cos delta\right)}^{3}\right)\right) - 3 \cdot \mathsf{fma}\left(\cos \phi_1 \cdot \sin delta, {\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right), {\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)}}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}\]

9. Final simplification 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\left({\left(\cos delta\right)}^{3} - \mathsf{fma}\left({\left(\sin delta\right)}^{3} \cdot {\left(\cos \phi_1\right)}^{3}, {\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}, {\left(\sin \phi_1\right)}^{6} \cdot {\left(\cos delta\right)}^{3}\right)\right) - 3 \cdot \mathsf{fma}\left(\cos \phi_1 \cdot \sin delta, {\left(\sin \phi_1\right)}^{5} \cdot \left({\left(\cos delta\right)}^{2} \cdot \cos theta\right), {\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left(\cos delta \cdot {\left(\cos theta\right)}^{2}\right)\right)\right)\right)}{\mathsf{fma}\left(\cos delta, \cos delta, \sin \phi_1 \cdot \left(\sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1, \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right), \cos delta\right)\right)\right)}}\]

Reproduce

herbie shell --seed 2020036 +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))))))))))