Average Error: 23.7 → 23.7
Time: 56.7s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[\left(R + R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right), \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}}\]
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)
↓
\left(R + R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right), \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), \sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \left(\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}}
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3301695 = R;
        double r3301696 = 2.0;
        double r3301697 = phi1;
        double r3301698 = phi2;
        double r3301699 = r3301697 - r3301698;
        double r3301700 = r3301699 / r3301696;
        double r3301701 = sin(r3301700);
        double r3301702 = pow(r3301701, r3301696);
        double r3301703 = cos(r3301697);
        double r3301704 = cos(r3301698);
        double r3301705 = r3301703 * r3301704;
        double r3301706 = lambda1;
        double r3301707 = lambda2;
        double r3301708 = r3301706 - r3301707;
        double r3301709 = r3301708 / r3301696;
        double r3301710 = sin(r3301709);
        double r3301711 = r3301705 * r3301710;
        double r3301712 = r3301711 * r3301710;
        double r3301713 = r3301702 + r3301712;
        double r3301714 = sqrt(r3301713);
        double r3301715 = 1.0;
        double r3301716 = r3301715 - r3301713;
        double r3301717 = sqrt(r3301716);
        double r3301718 = atan2(r3301714, r3301717);
        double r3301719 = r3301696 * r3301718;
        double r3301720 = r3301695 * r3301719;
        return r3301720;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3301721 = R;
        double r3301722 = r3301721 + r3301721;
        double r3301723 = lambda1;
        double r3301724 = lambda2;
        double r3301725 = r3301723 - r3301724;
        double r3301726 = 2.0;
        double r3301727 = r3301725 / r3301726;
        double r3301728 = sin(r3301727);
        double r3301729 = phi1;
        double r3301730 = cos(r3301729);
        double r3301731 = phi2;
        double r3301732 = cos(r3301731);
        double r3301733 = r3301730 * r3301732;
        double r3301734 = r3301728 * r3301733;
        double r3301735 = r3301729 - r3301731;
        double r3301736 = r3301735 / r3301726;
        double r3301737 = sin(r3301736);
        double r3301738 = r3301737 * r3301737;
        double r3301739 = fma(r3301734, r3301728, r3301738);
        double r3301740 = sqrt(r3301739);
        double r3301741 = cos(r3301736);
        double r3301742 = r3301741 * r3301741;
        double r3301743 = cbrt(r3301728);
        double r3301744 = r3301743 * r3301743;
        double r3301745 = r3301743 * r3301744;
        double r3301746 = expm1(r3301745);
        double r3301747 = log1p(r3301746);
        double r3301748 = expm1(r3301728);
        double r3301749 = log1p(r3301748);
        double r3301750 = r3301749 * r3301733;
        double r3301751 = r3301747 * r3301750;
        double r3301752 = r3301742 - r3301751;
        double r3301753 = sqrt(r3301752);
        double r3301754 = atan2(r3301740, r3301753);
        double r3301755 = r3301722 * r3301754;
        return r3301755;
}

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 23.7

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

  2. Simplified23.6

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

  4. Applied log1p-expm1-u23.7

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

  6. Applied log1p-expm1-u23.7

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

  8. Applied add-cube-cbrt23.7

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

  9. Final simplification23.7

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  (* 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))))))))))