Average Error: 0.2 → 0.2
Time: 39.9s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
↓
\[\begin{array}{l} t_1 := \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos delta - t_1\right) + \mathsf{fma}\left(-\sin \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}\right), \sin \phi_1, t_1\right)} \end{array} \]
\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 \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)\\
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos delta - t_1\right) + \mathsf{fma}\left(-\sin \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)\right)}\right), \sin \phi_1, t_1\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 phi1)
          (sin
           (asin
            (+
             (* (cos delta) (sin phi1))
             (* (* (sin delta) (cos phi1)) (cos theta))))))))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (+
      (- (cos delta) t_1)
      (fma
       (-
        (sin
         (log
          (exp
           (asin
            (fma
             (sin phi1)
             (cos delta)
             (* (sin delta) (* (cos phi1) (cos theta)))))))))
       (sin phi1)
       t_1))))))
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(phi1) * sin(asin((cos(delta) * sin(phi1)) + ((sin(delta) * cos(phi1)) * cos(theta))));
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), ((cos(delta) - t_1) + fma(-sin(log(exp(asin(fma(sin(phi1), cos(delta), (sin(delta) * (cos(phi1) * cos(theta)))))))), sin(phi1), t_1)));
}

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. Applied add-log-exp_binary640.2

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

  5. Simplified0.2

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

  6. Applied fma-udef_binary640.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2021215 
(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))))))))))