Average Error: 24.6 → 24.6
Time: 23.2s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[2 \cdot \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}} \cdot R\right)\]
R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\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(\sin \left(\frac{\phi_1 - \phi_2}{2}\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)
↓
2 \cdot \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}} \cdot R\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r108971 = R;
        double r108972 = 2.0;
        double r108973 = phi1;
        double r108974 = phi2;
        double r108975 = r108973 - r108974;
        double r108976 = r108975 / r108972;
        double r108977 = sin(r108976);
        double r108978 = pow(r108977, r108972);
        double r108979 = cos(r108973);
        double r108980 = cos(r108974);
        double r108981 = r108979 * r108980;
        double r108982 = lambda1;
        double r108983 = lambda2;
        double r108984 = r108982 - r108983;
        double r108985 = r108984 / r108972;
        double r108986 = sin(r108985);
        double r108987 = r108981 * r108986;
        double r108988 = r108987 * r108986;
        double r108989 = r108978 + r108988;
        double r108990 = sqrt(r108989);
        double r108991 = 1.0;
        double r108992 = r108991 - r108989;
        double r108993 = sqrt(r108992);
        double r108994 = atan2(r108990, r108993);
        double r108995 = r108972 * r108994;
        double r108996 = r108971 * r108995;
        return r108996;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r108997 = 2.0;
        double r108998 = phi1;
        double r108999 = cos(r108998);
        double r109000 = phi2;
        double r109001 = cos(r109000);
        double r109002 = r108999 * r109001;
        double r109003 = lambda1;
        double r109004 = lambda2;
        double r109005 = r109003 - r109004;
        double r109006 = r109005 / r108997;
        double r109007 = sin(r109006);
        double r109008 = r109007 * r109007;
        double r109009 = r108998 - r109000;
        double r109010 = r109009 / r108997;
        double r109011 = sin(r109010);
        double r109012 = pow(r109011, r108997);
        double r109013 = fma(r109002, r109008, r109012);
        double r109014 = sqrt(r109013);
        double r109015 = 1.0;
        double r109016 = exp(r109007);
        double r109017 = log(r109016);
        double r109018 = log1p(r109017);
        double r109019 = expm1(r109018);
        double r109020 = 3.0;
        double r109021 = pow(r109007, r109020);
        double r109022 = cbrt(r109021);
        double r109023 = r109019 * r109022;
        double r109024 = fma(r109002, r109023, r109012);
        double r109025 = r109015 - r109024;
        double r109026 = sqrt(r109025);
        double r109027 = atan2(r109014, r109026);
        double r109028 = R;
        double r109029 = r109027 * r109028;
        double r109030 = r108997 * r109029;
        return r109030;
}

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 24.6

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

  2. Simplified24.6

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

  4. Applied add-cbrt-cube24.6

    \[\leadsto 2 \cdot \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\sqrt[3]{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}} \cdot R\right)\]

  5. Simplified24.6

    \[\leadsto 2 \cdot \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sqrt[3]{\color{blue}{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}}, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}} \cdot R\right)\]
  6. Using strategy rm

  7. Applied expm1-log1p-u24.6

    \[\leadsto 2 \cdot \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)} \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}} \cdot R\right)\]
  8. Using strategy rm

  9. Applied add-log-exp24.6

    \[\leadsto 2 \cdot \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}\right)\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}} \cdot R\right)\]

  10. Final simplification24.6

    \[\leadsto 2 \cdot \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)\right) \cdot \sqrt[3]{{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}^{3}}, {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}\right)}} \cdot R\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))) (sqrt (- 1 (+ (pow (sin (/ (- phi1 phi2) 2)) 2) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2))) (sin (/ (- lambda1 lambda2) 2))))))))))