Average Error: 0.2 → 0.2
Time: 26.1s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\begin{array}{l} t_1 := \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - {\sin \phi_1}^{2} \cdot {\left(\cos delta \cdot \sin \phi_1 + t_1\right)}^{2}}{\mathsf{fma}\left(\sin \phi_1, \sin \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, t_1\right)\right), \cos delta\right)}} \end{array} \]
\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 := \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\\
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos delta \cdot \cos delta - {\sin \phi_1}^{2} \cdot {\left(\cos delta \cdot \sin \phi_1 + t_1\right)}^{2}}{\mathsf{fma}\left(\sin \phi_1, \sin \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, t_1\right)\right), \cos delta\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 (* (sin delta) (* (cos phi1) (cos theta)))))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (/
      (-
       (* (cos delta) (cos delta))
       (* (pow (sin phi1) 2.0) (pow (+ (* (cos delta) (sin phi1)) t_1) 2.0)))
      (fma
       (sin phi1)
       (sin (asin (fma (sin phi1) (cos delta) t_1)))
       (cos delta)))))))
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 = sin(delta) * (cos(phi1) * cos(theta));
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (((cos(delta) * cos(delta)) - (pow(sin(phi1), 2.0) * pow(((cos(delta) * sin(phi1)) + t_1), 2.0))) / fma(sin(phi1), sin(asin(fma(sin(phi1), cos(delta), t_1))), cos(delta))));
}

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

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

  2. Simplified0.2

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

  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. Simplified0.2

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

  5. Simplified0.2

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

  6. Taylor expanded in phi1 around inf 0.2

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

  7. Final simplification0.2

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

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))))))))))