Average Error: 0.1 → 0.2
Time: 24.2s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\begin{array}{l} t_1 := {\cos delta}^{2}\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{t_1 - \mathsf{fma}\left({\sin delta}^{2}, {\sin \phi_1}^{2} \cdot \left({\cos theta}^{2} \cdot {\cos \phi_1}^{2}\right), \mathsf{fma}\left(2, \sin delta \cdot \left(\log \left(e^{{\sin \phi_1}^{3}}\right) \cdot \left(\cos delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right), t_1 \cdot {\sin \phi_1}^{4}\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)}} \end{array} \]
\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)}
↓
\begin{array}{l}
t_1 := {\cos delta}^{2}\\
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{t_1 - \mathsf{fma}\left({\sin delta}^{2}, {\sin \phi_1}^{2} \cdot \left({\cos theta}^{2} \cdot {\cos \phi_1}^{2}\right), \mathsf{fma}\left(2, \sin delta \cdot \left(\log \left(e^{{\sin \phi_1}^{3}}\right) \cdot \left(\cos delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right), t_1 \cdot {\sin \phi_1}^{4}\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\cos delta, \sin \phi_1, \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)\right)}}
\end{array}

(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
 (let* ((t_1 (pow (cos delta) 2.0)))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (/
      (-
       t_1
       (fma
        (pow (sin delta) 2.0)
        (* (pow (sin phi1) 2.0) (* (pow (cos theta) 2.0) (pow (cos phi1) 2.0)))
        (fma
         2.0
         (*
          (sin delta)
          (*
           (log (exp (pow (sin phi1) 3.0)))
           (* (cos delta) (* (cos phi1) (cos theta)))))
         (* t_1 (pow (sin phi1) 4.0)))))
      (+
       (cos delta)
       (*
        (sin phi1)
        (sin
         (asin
          (fma
           (cos delta)
           (sin phi1)
           (* (cos theta) (* (sin delta) (cos 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) {
	double t_1 = pow(cos(delta), 2.0);
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), ((t_1 - fma(pow(sin(delta), 2.0), (pow(sin(phi1), 2.0) * (pow(cos(theta), 2.0) * pow(cos(phi1), 2.0))), fma(2.0, (sin(delta) * (log(exp(pow(sin(phi1), 3.0))) * (cos(delta) * (cos(phi1) * cos(theta))))), (t_1 * pow(sin(phi1), 4.0))))) / (cos(delta) + (sin(phi1) * sin(asin(fma(cos(delta), sin(phi1), (cos(theta) * (sin(delta) * 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 0.1

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

  2. Simplified0.1

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

  3. Applied flip--_binary640.2

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

  4. Taylor expanded in delta around inf 0.2

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

  5. Simplified0.1

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

  6. Applied add-log-exp_binary640.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2022088 
(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))))))))))