Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 16.8s

Precision: binary64

Math

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

↓

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :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))))))))))

↓

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (/
    (-
     (* (- 1.0 (pow (sin phi1) 4.0)) (pow (cos delta) 2.0))
     (*
      (sin delta)
      (+
       (*
        (sin delta)
        (*
         (pow (cos phi1) 2.0)
         (* (pow (sin phi1) 2.0) (pow (cos theta) 2.0))))
       (*
        2.0
        (* (cos phi1) (* (pow (sin phi1) 3.0) (* (cos delta) (cos theta))))))))
    (+
     (cos delta)
     (*
      (sin phi1)
      (sin
       (asin
        (+
         (* (sin phi1) (cos delta))
         (* (cos theta) (* (sin delta) (cos phi1))))))))))))

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) (1.0 - ((double) pow(((double) sin(phi1)), 4.0)))) * ((double) pow(((double) cos(delta)), 2.0)))) - ((double) (((double) sin(delta)) * ((double) (((double) (((double) sin(delta)) * ((double) (((double) pow(((double) cos(phi1)), 2.0)) * ((double) (((double) pow(((double) sin(phi1)), 2.0)) * ((double) pow(((double) cos(theta)), 2.0)))))))) + ((double) (2.0 * ((double) (((double) cos(phi1)) * ((double) (((double) pow(((double) sin(phi1)), 3.0)) * ((double) (((double) cos(delta)) * ((double) cos(theta)))))))))))))))) / ((double) (((double) cos(delta)) + ((double) (((double) sin(phi1)) * ((double) sin(((double) asin(((double) (((double) (((double) sin(phi1)) * ((double) cos(delta)))) + ((double) (((double) cos(theta)) * ((double) (((double) sin(delta)) * ((double) cos(phi1)))))))))))))))))))));
}

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--_binary64 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. Simplified 0.2

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

5. Simplified 0.2

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

6. Taylor expanded around inf 0.2

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

7. Simplified 0.2

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

9. Applied associate--r+_binary64 0.2

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

10. Simplified 0.2

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

11. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020210 
(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))))))))))