Distance on a great circle

Average Error: 24.5 → 13.9

Time: 1.0min

Precision: binary64

\[[phi1, phi2]=\mathsf{sort}([phi1, phi2])\]

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(\phi_1 \cdot 0.5\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sin \left(\phi_1 \cdot 0.5\right)\\ t_3 := \sin \left(\phi_2 \cdot 0.5\right)\\ t_4 := \cos \left(\phi_2 \cdot 0.5\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t_1, t_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\mathsf{fma}\left({t_3}^{2}, {t_0}^{2}, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\right)\right) + t_2 \cdot \left(t_2 \cdot {t_4}^{2} - \left(2 \cdot t_0\right) \cdot \left(t_3 \cdot t_4\right)\right)\right)}}\right) \end{array} \]

FPCore

(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 (cos (* phi1 0.5)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (sin (* phi1 0.5)))
        (t_3 (sin (* phi2 0.5)))
        (t_4 (cos (* phi2 0.5))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (fma
        t_1
        (* t_1 (* (cos phi1) (cos phi2)))
        (pow
         (-
          (* (sin (/ phi1 2.0)) (cos (/ phi2 2.0)))
          (* (cos (/ phi1 2.0)) (sin (/ phi2 2.0))))
         2.0)))
      (sqrt
       (-
        1.0
        (+
         (fma
          (pow t_3 2.0)
          (pow t_0 2.0)
          (*
           (cos phi2)
           (* (cos phi1) (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))))
         (* t_2 (- (* t_2 (pow t_4 2.0)) (* (* 2.0 t_0) (* t_3 t_4))))))))))))

C

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 = cos(phi1 * 0.5);
	double t_1 = sin((lambda1 - lambda2) / 2.0);
	double t_2 = sin(phi1 * 0.5);
	double t_3 = sin(phi2 * 0.5);
	double t_4 = cos(phi2 * 0.5);
	return R * (2.0 * atan2(sqrt(fma(t_1, (t_1 * (cos(phi1) * cos(phi2))), pow(((sin(phi1 / 2.0) * cos(phi2 / 2.0)) - (cos(phi1 / 2.0) * sin(phi2 / 2.0))), 2.0))), sqrt(1.0 - (fma(pow(t_3, 2.0), pow(t_0, 2.0), (cos(phi2) * (cos(phi1) * pow(sin((lambda1 - lambda2) * 0.5), 2.0)))) + (t_2 * ((t_2 * pow(t_4, 2.0)) - ((2.0 * t_0) * (t_3 * t_4))))))));
}

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 24.5

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

2. Simplified 24.4

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

3. Applied div-sub_binary64 24.4

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

4. Applied sin-diff_binary64 23.8

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

5. Applied div-sub_binary64 23.8

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

6. Applied sin-diff_binary64 13.9

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

7. Taylor expanded in lambda1 around inf 13.9

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

8. Simplified 13.9

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

9. Final simplification 13.9

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

Reproduce

herbie shell --seed 2022004 
(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))))))))))