Average Error: 24.3 → 14.1
Time: 1.2min
Precision: binary64
\[[phi1, phi2] = \mathsf{sort}([phi1, phi2]) \\]
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
↓
\[\begin{array}{l} t_0 := \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \cos \left(\phi_2 \cdot 0.5\right)\\ t_4 := \cos \left(0.5 \cdot \phi_1\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_2 \cdot \left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1\right) + \left({t_1}^{2} \cdot {t_3}^{2} + {t_4}^{2} \cdot {t_0}^{2}\right)\right) - 2 \cdot \left(t_4 \cdot \left(t_1 \cdot \left(t_3 \cdot t_0\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(t_2, t_2 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}\right) \end{array} \]
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)
↓
\begin{array}{l}
t_0 := \sin \left(\phi_2 \cdot 0.5\right)\\
t_1 := \sin \left(0.5 \cdot \phi_1\right)\\
t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\
t_3 := \cos \left(\phi_2 \cdot 0.5\right)\\
t_4 := \cos \left(0.5 \cdot \phi_1\right)\\
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_2 \cdot \left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1\right) + \left({t_1}^{2} \cdot {t_3}^{2} + {t_4}^{2} \cdot {t_0}^{2}\right)\right) - 2 \cdot \left(t_4 \cdot \left(t_1 \cdot \left(t_3 \cdot t_0\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(t_2, t_2 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}\right)
\end{array}

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt
     (+
      (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
      (*
       (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
       (sin (/ (- lambda1 lambda2) 2.0)))))
    (sqrt
     (-
      1.0
      (+
       (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
       (*
        (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
        (sin (/ (- lambda1 lambda2) 2.0))))))))))
↓
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* phi2 0.5)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (cos (* phi2 0.5)))
        (t_4 (cos (* 0.5 phi1))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (-
        (+
         (*
          (cos phi2)
          (* (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0) (cos phi1)))
         (+ (* (pow t_1 2.0) (pow t_3 2.0)) (* (pow t_4 2.0) (pow t_0 2.0))))
        (* 2.0 (* t_4 (* t_1 (* t_3 t_0))))))
      (sqrt
       (-
        1.0
        (fma
         t_2
         (* t_2 (* (cos phi2) (cos phi1)))
         (pow
          (-
           (* (sin (/ phi1 2.0)) (cos (/ phi2 2.0)))
           (* (cos (/ phi1 2.0)) (sin (/ phi2 2.0))))
          2.0)))))))))
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt(pow(sin((phi1 - phi2) / 2.0), 2.0) + (((cos(phi1) * cos(phi2)) * sin((lambda1 - lambda2) / 2.0)) * sin((lambda1 - lambda2) / 2.0))), sqrt(1.0 - (pow(sin((phi1 - phi2) / 2.0), 2.0) + (((cos(phi1) * cos(phi2)) * sin((lambda1 - lambda2) / 2.0)) * sin((lambda1 - lambda2) / 2.0))))));
}
↓
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi2 * 0.5);
	double t_1 = sin(0.5 * phi1);
	double t_2 = sin((lambda1 - lambda2) / 2.0);
	double t_3 = cos(phi2 * 0.5);
	double t_4 = cos(0.5 * phi1);
	return R * (2.0 * atan2(sqrt(((cos(phi2) * (pow(sin(0.5 * (lambda1 - lambda2)), 2.0) * cos(phi1))) + ((pow(t_1, 2.0) * pow(t_3, 2.0)) + (pow(t_4, 2.0) * pow(t_0, 2.0)))) - (2.0 * (t_4 * (t_1 * (t_3 * t_0))))), sqrt(1.0 - fma(t_2, (t_2 * (cos(phi2) * cos(phi1))), pow(((sin(phi1 / 2.0) * cos(phi2 / 2.0)) - (cos(phi1 / 2.0) * sin(phi2 / 2.0))), 2.0)))));
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Derivation

  1. Initial program 24.3

    \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

  2. Simplified24.3

    \[\leadsto \color{blue}{R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}\right)} \]

  3. Applied div-sub_binary6424.3

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}\right) \]

  4. Applied sin-diff_binary6423.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}\right) \]

  5. Applied div-sub_binary6423.8

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2}\right)}}\right) \]

  6. Applied sin-diff_binary6414.1

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2}\right)}}\right) \]

  7. Taylor expanded in lambda1 around inf 14.1

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\left(\cos \phi_2 \cdot \left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1\right) + \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\cos \left(0.5 \cdot \phi_2\right)}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)\right) - 2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}\right) \]

  8. Final simplification14.1

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_2 \cdot \left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1\right) + \left({\sin \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\cos \left(\phi_2 \cdot 0.5\right)}^{2} + {\cos \left(0.5 \cdot \phi_1\right)}^{2} \cdot {\sin \left(\phi_2 \cdot 0.5\right)}^{2}\right)\right) - 2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}\right) \]

Reproduce

herbie shell --seed 2022087 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))