Average Error: 13.5 → 0.2
Time: 26.9s
Precision: binary64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right) + \left(e^{\mathsf{log1p}\left(\sin \lambda_1 \cdot \left(\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)} - 1\right)\right)} \]
\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)}
↓
\tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right) + \left(e^{\mathsf{log1p}\left(\sin \lambda_1 \cdot \left(\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_2\right)\right)\right)} - 1\right)\right)}

(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
 (atan2
  (*
   (log1p
    (expm1
     (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (sin (- lambda2))))))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (+
    (* (cos lambda2) (* (cos lambda1) (* (cos phi2) (sin phi1))))
    (-
     (exp
      (log1p (* (sin lambda1) (* (sin phi1) (* (cos phi2) (sin lambda2))))))
     1.0)))))
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) {
	return atan2((log1p(expm1(fma(sin(lambda1), cos(lambda2), (cos(lambda1) * sin(-lambda2))))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(lambda2) * (cos(lambda1) * (cos(phi2) * sin(phi1)))) + (exp(log1p(sin(lambda1) * (sin(phi1) * (cos(phi2) * sin(lambda2))))) - 1.0))));
}

Error

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 13.5

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

  3. Applied sin-diff_binary647.0

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

  4. Applied flip--_binary649.8

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

  5. Simplified8.9

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

  6. Simplified8.9

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

  8. Applied cos-diff_binary642.1

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

  9. Applied distribute-rgt-in_binary642.1

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

  10. Simplified2.1

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

  11. Simplified2.1

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

  13. Applied log1p-expm1-u_binary642.1

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

  14. Simplified0.2

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

  16. Applied expm1-log1p-u_binary640.2

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

  17. Applied expm1-udef_binary640.2

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

  18. Simplified0.2

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

  19. Final simplification0.2

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

Reproduce

herbie shell --seed 2021211 
(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))))))