Average Error: 23.8 → 13.2
Time: 1.0min
Precision: binary64
\[[phi1, phi2] = \mathsf{sort}([phi1, phi2]) \\]
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
↓
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\phi_2}{2}\right)\\ t_2 := \cos \left(\frac{\phi_1}{2}\right)\\ t_3 := t_1 \cdot t_2\\ t_4 := {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), -t_3\right) + \mathsf{fma}\left(-t_1, t_2, t_3\right)\right)}^{2}\\ t_5 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4 + t_5 \cdot \left(t_0 \cdot t_5\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(t_0, {\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}^{2}, t_4\right)\right)\right)}}\right) \end{array} \]
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 := \cos \phi_1 \cdot \cos \phi_2\\
t_1 := \sin \left(\frac{\phi_2}{2}\right)\\
t_2 := \cos \left(\frac{\phi_1}{2}\right)\\
t_3 := t_1 \cdot t_2\\
t_4 := {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), -t_3\right) + \mathsf{fma}\left(-t_1, t_2, t_3\right)\right)}^{2}\\
t_5 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4 + t_5 \cdot \left(t_0 \cdot t_5\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(t_0, {\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}^{2}, t_4\right)\right)\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 (* (cos phi1) (cos phi2)))
        (t_1 (sin (/ phi2 2.0)))
        (t_2 (cos (/ phi1 2.0)))
        (t_3 (* t_1 t_2))
        (t_4
         (pow
          (+
           (fma (sin (/ phi1 2.0)) (cos (/ phi2 2.0)) (- t_3))
           (fma (- t_1) t_2 t_3))
          2.0))
        (t_5 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (+ t_4 (* t_5 (* t_0 t_5))))
      (sqrt
       (expm1
        (log1p
         (-
          1.0
          (fma
           t_0
           (pow
            (-
             (* (sin (/ lambda1 2.0)) (cos (/ lambda2 2.0)))
             (* (cos (/ lambda1 2.0)) (sin (/ lambda2 2.0))))
            2.0)
           t_4))))))))))
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) * cos(phi2);
	double t_1 = sin((phi2 / 2.0));
	double t_2 = cos((phi1 / 2.0));
	double t_3 = t_1 * t_2;
	double t_4 = pow((fma(sin((phi1 / 2.0)), cos((phi2 / 2.0)), -t_3) + fma(-t_1, t_2, t_3)), 2.0);
	double t_5 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt((t_4 + (t_5 * (t_0 * t_5)))), sqrt(expm1(log1p((1.0 - fma(t_0, pow(((sin((lambda1 / 2.0)) * cos((lambda2 / 2.0))) - (cos((lambda1 / 2.0)) * sin((lambda2 / 2.0)))), 2.0), 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 23.8

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

  2. Applied egg-rr23.2

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

  3. Applied egg-rr13.6

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

  4. Applied egg-rr13.6

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), -\sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right) + \mathsf{fma}\left(-\sin \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, {\left(\mathsf{fma}\left(-\sin \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right) + \mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \left(-\sin \left(\frac{\phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)}^{2}\right)\right)\right)}}}\right) \]

  5. Applied egg-rr13.2

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), -\sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right) + \mathsf{fma}\left(-\sin \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, {\color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}^{2}, {\left(\mathsf{fma}\left(-\sin \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right) + \mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \left(-\sin \left(\frac{\phi_2}{2}\right)\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)}^{2}\right)\right)\right)}}\right) \]

  6. Final simplification13.2

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), -\sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right) + \mathsf{fma}\left(-\sin \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, {\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}^{2}, {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), -\sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right) + \mathsf{fma}\left(-\sin \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)}^{2}\right)\right)\right)}}\right) \]

Reproduce

herbie shell --seed 2022130 
(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))))))))))