Average Error: 0.9 → 0.2
Time: 16.6s
Precision: binary64
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\cos \lambda_2}\\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_0 \cdot \left(\sin \lambda_1 \cdot \left(t_0 \cdot t_0\right)\right) - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2, \cos \phi_1\right)} \end{array} \]
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\begin{array}{l}
t_0 := \sqrt[3]{\cos \lambda_2}\\
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_0 \cdot \left(\sin \lambda_1 \cdot \left(t_0 \cdot t_0\right)\right) - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2, \cos \phi_1\right)}
\end{array}

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

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 0.9

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

  2. Simplified0.9

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

  3. Applied sin-diff_binary640.8

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

  4. Applied cos-diff_binary640.2

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

  5. Simplified0.2

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

  6. Applied add-cube-cbrt_binary640.2

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

  7. Applied associate-*r*_binary640.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2022081 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))