Average Error: 23.7 → 23.7
Time: 54.7s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[2 \cdot \left(R \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right) + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_2\right) \cdot \left(\cos \phi_1 \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\cos \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1 - \phi_2}{2}\right) - \left(\cos \phi_2 \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right) \cdot \left(\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \cos \phi_1\right)}}\right)\]
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)
↓
2 \cdot \left(R \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\phi_1 - \phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1 - \phi_2}{2}\right) + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_2\right) \cdot \left(\cos \phi_1 \cdot \sin \left(\frac{\lambda_1 - \lambda_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(\cos \phi_2 \cdot \log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right) \cdot \left(\log \left(e^{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \cos \phi_1\right)}}\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3461637 = R;
        double r3461638 = 2.0;
        double r3461639 = phi1;
        double r3461640 = phi2;
        double r3461641 = r3461639 - r3461640;
        double r3461642 = r3461641 / r3461638;
        double r3461643 = sin(r3461642);
        double r3461644 = pow(r3461643, r3461638);
        double r3461645 = cos(r3461639);
        double r3461646 = cos(r3461640);
        double r3461647 = r3461645 * r3461646;
        double r3461648 = lambda1;
        double r3461649 = lambda2;
        double r3461650 = r3461648 - r3461649;
        double r3461651 = r3461650 / r3461638;
        double r3461652 = sin(r3461651);
        double r3461653 = r3461647 * r3461652;
        double r3461654 = r3461653 * r3461652;
        double r3461655 = r3461644 + r3461654;
        double r3461656 = sqrt(r3461655);
        double r3461657 = 1.0;
        double r3461658 = r3461657 - r3461655;
        double r3461659 = sqrt(r3461658);
        double r3461660 = atan2(r3461656, r3461659);
        double r3461661 = r3461638 * r3461660;
        double r3461662 = r3461637 * r3461661;
        return r3461662;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r3461663 = 2.0;
        double r3461664 = R;
        double r3461665 = phi1;
        double r3461666 = phi2;
        double r3461667 = r3461665 - r3461666;
        double r3461668 = r3461667 / r3461663;
        double r3461669 = sin(r3461668);
        double r3461670 = r3461669 * r3461669;
        double r3461671 = lambda1;
        double r3461672 = lambda2;
        double r3461673 = r3461671 - r3461672;
        double r3461674 = r3461673 / r3461663;
        double r3461675 = sin(r3461674);
        double r3461676 = cos(r3461666);
        double r3461677 = r3461675 * r3461676;
        double r3461678 = cos(r3461665);
        double r3461679 = r3461678 * r3461675;
        double r3461680 = r3461677 * r3461679;
        double r3461681 = r3461670 + r3461680;
        double r3461682 = sqrt(r3461681);
        double r3461683 = cos(r3461668);
        double r3461684 = r3461683 * r3461683;
        double r3461685 = exp(r3461675);
        double r3461686 = log(r3461685);
        double r3461687 = r3461676 * r3461686;
        double r3461688 = r3461686 * r3461678;
        double r3461689 = r3461687 * r3461688;
        double r3461690 = r3461684 - r3461689;
        double r3461691 = sqrt(r3461690);
        double r3461692 = atan2(r3461682, r3461691);
        double r3461693 = r3461664 * r3461692;
        double r3461694 = r3461663 * r3461693;
        return r3461694;
}

Error

Bits error versus R

Bits error versus lambda1

Bits error versus lambda2

Bits error versus phi1

Bits error versus phi2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified23.7

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

  4. Applied add-log-exp23.7

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

  6. Applied add-log-exp23.7

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

  7. Final simplification23.7

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

Reproduce

herbie shell --seed 2019158 
(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))))))))))