Average Error: 0.2 → 0.1
Time: 48.5s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \sin delta \cdot \left({\left({\sin \phi_1}^{2}\right)}^{0.3333333333333333} \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sqrt[3]{\sin \phi_1}\right)\right)\right)} \]
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \sin delta \cdot \left({\left({\sin \phi_1}^{2}\right)}^{0.3333333333333333} \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sqrt[3]{\sin \phi_1}\right)\right)\right)}

(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)))
   (-
    (* (* (cos phi1) (cos phi1)) (cos delta))
    (*
     (sin delta)
     (*
      (pow (pow (sin phi1) 2.0) 0.3333333333333333)
      (* (cos phi1) (* (cos theta) (cbrt (sin phi1))))))))))
double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta)))))));
}
↓
double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (((cos(phi1) * cos(phi1)) * cos(delta)) - (sin(delta) * (pow(pow(sin(phi1), 2.0), 0.3333333333333333) * (cos(phi1) * (cos(theta) * cbrt(sin(phi1))))))));
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

    Proof

  3. Taylor expanded in phi1 around inf 0.2

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

  4. Applied associate--r+_binary640.2

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

  5. Simplified0.1

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

    Proof

  6. Applied add-cube-cbrt_binary640.1

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

  7. Applied associate-*l*_binary640.1

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

  8. Simplified0.1

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

    Proof

  9. Taylor expanded in phi1 around inf 0.1

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \sin delta \cdot \left(\color{blue}{{\left({\sin \phi_1}^{2}\right)}^{0.3333333333333333}} \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sqrt[3]{\sin \phi_1}\right)\right)\right)} \]

  10. Final simplification0.1

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \sin delta \cdot \left({\left({\sin \phi_1}^{2}\right)}^{0.3333333333333333} \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sqrt[3]{\sin \phi_1}\right)\right)\right)} \]

Reproduce

herbie shell --seed 2022004 
(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))))))))))