Average Error: 13.0 → 0.2
Time: 25.5s
Precision: binary64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \cos \lambda_1\\ \tan^{-1}_* \frac{\left(\left(\sin \lambda_1 \cdot \cos \lambda_2 - \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, t_0\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)} \end{array} \]
\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\begin{array}{l}
t_0 := \sin \lambda_2 \cdot \cos \lambda_1\\
\tan^{-1}_* \frac{\left(\left(\sin \lambda_1 \cdot \cos \lambda_2 - \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, t_0\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)}
\end{array}

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
↓
(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda2) (cos lambda1))))
   (atan2
    (*
     (+
      (- (* (sin lambda1) (cos lambda2)) (log1p (expm1 t_0)))
      (fma (- (sin lambda2)) (cos lambda1) t_0))
     (cos phi2))
    (-
     (* (cos phi1) (sin phi2))
     (*
      (* (cos phi2) (sin phi1))
      (fma (sin lambda2) (sin lambda1) (* (cos lambda2) (cos lambda1))))))))
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}
↓
double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda2) * cos(lambda1);
	return atan2(((((sin(lambda1) * cos(lambda2)) - log1p(expm1(t_0))) + fma(-sin(lambda2), cos(lambda1), t_0)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1))))));
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 13.0

    \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

  2. Applied egg-rr6.8

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, -\cos \lambda_1 \cdot \sin \lambda_2\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

  3. Applied egg-rr0.4

    \[\leadsto \tan^{-1}_* \frac{\left(\mathsf{fma}\left(\sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, \sqrt[3]{\cos \lambda_2 \cdot \sin \lambda_1}, -\cos \lambda_1 \cdot \sin \lambda_2\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

  4. Taylor expanded in lambda2 around inf 0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)} + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

  5. Applied egg-rr0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\left(\sin \lambda_1 \cdot \cos \lambda_2 - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_2 \cdot \cos \lambda_1\right)\right)}\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

  6. Final simplification0.2

    \[\leadsto \tan^{-1}_* \frac{\left(\left(\sin \lambda_1 \cdot \cos \lambda_2 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_2 \cdot \cos \lambda_1\right)\right)\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))