Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 14.2s

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\frac{\left(\left(-{\left({\left(\sin \phi_1\right)}^{4}\right)}^{3}\right) + 1\right) \cdot {\left(\cos delta\right)}^{6}}{{\left(\cos delta\right)}^{4} \cdot \left(\left({\left(\sin \phi_1\right)}^{8} + 1\right) + {\left(\sin \phi_1\right)}^{4}\right)} - \sin delta \cdot \left(\left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right) \cdot 2 + \sin delta \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return ((double) (lambda1 + ((double) atan2(((double) (((double) (((double) sin(theta)) * ((double) sin(delta)))) * ((double) cos(phi1)))), ((double) (((double) cos(delta)) - ((double) (((double) sin(phi1)) * ((double) sin(((double) asin(((double) (((double) (((double) sin(phi1)) * ((double) cos(delta)))) + ((double) (((double) (((double) cos(phi1)) * ((double) sin(delta)))) * ((double) cos(theta))))))))))))))))));
}

↓

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return ((double) (lambda1 + ((double) atan2(((double) (((double) (((double) sin(theta)) * ((double) sin(delta)))) * ((double) cos(phi1)))), ((double) (((double) (((double) (((double) (((double) (((double) -(((double) pow(((double) pow(((double) sin(phi1)), 4.0)), 3.0)))) + 1.0)) * ((double) pow(((double) cos(delta)), 6.0)))) / ((double) (((double) pow(((double) cos(delta)), 4.0)) * ((double) (((double) (((double) pow(((double) sin(phi1)), 8.0)) + 1.0)) + ((double) pow(((double) sin(phi1)), 4.0)))))))) - ((double) (((double) sin(delta)) * ((double) (((double) (((double) (((double) cos(phi1)) * ((double) (((double) pow(((double) sin(phi1)), 3.0)) * ((double) (((double) cos(delta)) * ((double) cos(theta)))))))) * 2.0)) + ((double) (((double) sin(delta)) * ((double) (((double) pow(((double) cos(phi1)), 2.0)) * ((double) (((double) pow(((double) cos(theta)), 2.0)) * ((double) pow(((double) sin(phi1)), 2.0)))))))))))))) / ((double) (((double) cos(delta)) + ((double) (((double) sin(phi1)) * ((double) sin(((double) asin(((double) (((double) (((double) sin(phi1)) * ((double) cos(delta)))) + ((double) (((double) (((double) cos(phi1)) * ((double) sin(delta)))) * ((double) cos(theta))))))))))))))))))));
}

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

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

5. Simplified 0.2

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

7. Applied flip3-- 0.2

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

8. Simplified 0.2

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

9. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\frac{\left(\left(-{\left({\left(\sin \phi_1\right)}^{4}\right)}^{3}\right) + 1\right) \cdot {\left(\cos delta\right)}^{6}}{\color{blue}{{\left(\cos delta\right)}^{4} \cdot \left(\left({\left(\sin \phi_1\right)}^{8} + 1\right) + {\left(\sin \phi_1\right)}^{4}\right)}} - \sin delta \cdot \left(\left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right) \cdot 2 + \sin delta \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

10. Final simplification 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\frac{\left(\left(-{\left({\left(\sin \phi_1\right)}^{4}\right)}^{3}\right) + 1\right) \cdot {\left(\cos delta\right)}^{6}}{{\left(\cos delta\right)}^{4} \cdot \left(\left({\left(\sin \phi_1\right)}^{8} + 1\right) + {\left(\sin \phi_1\right)}^{4}\right)} - \sin delta \cdot \left(\left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right) \cdot 2 + \sin delta \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

Reproduce

herbie shell --seed 2020121 
(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))))))))))