Average Error: 24.8 → 24.8
Time: 51.5s
Precision: 64
\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\]
↓
\[\left(\tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \frac{\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) - \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt[3]{2}}\right)\right)}} \cdot 2\right) \cdot R\]
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(\tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\sin \left(\frac{\phi_1 - \phi_2}{2}\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \frac{\sqrt[3]{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) - \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt[3]{2}}\right)\right)}} \cdot 2\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r4514639 = R;
        double r4514640 = 2.0;
        double r4514641 = phi1;
        double r4514642 = phi2;
        double r4514643 = r4514641 - r4514642;
        double r4514644 = r4514643 / r4514640;
        double r4514645 = sin(r4514644);
        double r4514646 = pow(r4514645, r4514640);
        double r4514647 = cos(r4514641);
        double r4514648 = cos(r4514642);
        double r4514649 = r4514647 * r4514648;
        double r4514650 = lambda1;
        double r4514651 = lambda2;
        double r4514652 = r4514650 - r4514651;
        double r4514653 = r4514652 / r4514640;
        double r4514654 = sin(r4514653);
        double r4514655 = r4514649 * r4514654;
        double r4514656 = r4514655 * r4514654;
        double r4514657 = r4514646 + r4514656;
        double r4514658 = sqrt(r4514657);
        double r4514659 = 1.0;
        double r4514660 = r4514659 - r4514657;
        double r4514661 = sqrt(r4514660);
        double r4514662 = atan2(r4514658, r4514661);
        double r4514663 = r4514640 * r4514662;
        double r4514664 = r4514639 * r4514663;
        return r4514664;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r4514665 = lambda1;
        double r4514666 = lambda2;
        double r4514667 = r4514665 - r4514666;
        double r4514668 = 2.0;
        double r4514669 = r4514667 / r4514668;
        double r4514670 = sin(r4514669);
        double r4514671 = phi2;
        double r4514672 = cos(r4514671);
        double r4514673 = phi1;
        double r4514674 = cos(r4514673);
        double r4514675 = r4514672 * r4514674;
        double r4514676 = r4514675 * r4514670;
        double r4514677 = r4514670 * r4514676;
        double r4514678 = r4514673 - r4514671;
        double r4514679 = r4514678 / r4514668;
        double r4514680 = sin(r4514679);
        double r4514681 = pow(r4514680, r4514668);
        double r4514682 = r4514677 + r4514681;
        double r4514683 = sqrt(r4514682);
        double r4514684 = 1.0;
        double r4514685 = r4514669 + r4514669;
        double r4514686 = cos(r4514685);
        double r4514687 = r4514686 * r4514670;
        double r4514688 = r4514670 - r4514687;
        double r4514689 = cbrt(r4514688);
        double r4514690 = 2.0;
        double r4514691 = cbrt(r4514690);
        double r4514692 = r4514689 / r4514691;
        double r4514693 = r4514675 * r4514692;
        double r4514694 = r4514670 * r4514693;
        double r4514695 = r4514681 + r4514694;
        double r4514696 = r4514684 - r4514695;
        double r4514697 = sqrt(r4514696);
        double r4514698 = atan2(r4514683, r4514697);
        double r4514699 = r4514698 * r4514668;
        double r4514700 = R;
        double r4514701 = r4514699 * r4514700;
        return r4514701;
}

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 24.8

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

  3. Applied add-cbrt-cube24.8

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

  5. Applied sin-mult24.8

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

  6. Applied associate-*l/24.8

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

  7. Applied cbrt-div24.8

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

  8. Simplified24.8

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

  9. Final simplification24.8

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

Reproduce

herbie shell --seed 2019171 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  (* 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))))))))))