Average Error: 0.2 → 0.2
Time: 22.5s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\begin{array}{l} t_1 := \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\\ t_2 := \sin \phi_1 \cdot t_1\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(1, \cos delta, -t_2\right) + \mathsf{fma}\left(-t_1, \sin \phi_1, t_2\right)} \end{array} \]
\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 \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\\
t_2 := \sin \phi_1 \cdot t_1\\
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(1, \cos delta, -t_2\right) + \mathsf{fma}\left(-t_1, \sin \phi_1, t_2\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
          (asin
           (+
            (* (cos delta) (sin phi1))
            (* (* (sin delta) (cos phi1)) (cos theta))))))
        (t_2 (* (sin phi1) t_1)))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (+ (fma 1.0 (cos delta) (- t_2)) (fma (- t_1) (sin phi1) t_2))))))
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(asin((cos(delta) * sin(phi1)) + ((sin(delta) * cos(phi1)) * cos(theta))));
	double t_2 = sin(phi1) * t_1;
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (fma(1.0, cos(delta), -t_2) + fma(-t_1, sin(phi1), t_2)));
}

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. Applied *-un-lft-identity_binary640.2

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

  3. Applied prod-diff_binary640.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022055 
(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))))))))))