Average Error: 16.8 → 3.9
Time: 34.8s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\log \left({\left(e^{\sin \phi_1}\right)}^{\left(\sin \phi_2\right)}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot R\]
\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R
↓
\cos^{-1} \left(\log \left({\left(e^{\sin \phi_1}\right)}^{\left(\sin \phi_2\right)}\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r21783 = phi1;
        double r21784 = sin(r21783);
        double r21785 = phi2;
        double r21786 = sin(r21785);
        double r21787 = r21784 * r21786;
        double r21788 = cos(r21783);
        double r21789 = cos(r21785);
        double r21790 = r21788 * r21789;
        double r21791 = lambda1;
        double r21792 = lambda2;
        double r21793 = r21791 - r21792;
        double r21794 = cos(r21793);
        double r21795 = r21790 * r21794;
        double r21796 = r21787 + r21795;
        double r21797 = acos(r21796);
        double r21798 = R;
        double r21799 = r21797 * r21798;
        return r21799;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r21800 = phi1;
        double r21801 = sin(r21800);
        double r21802 = exp(r21801);
        double r21803 = phi2;
        double r21804 = sin(r21803);
        double r21805 = pow(r21802, r21804);
        double r21806 = log(r21805);
        double r21807 = cos(r21800);
        double r21808 = cos(r21803);
        double r21809 = r21807 * r21808;
        double r21810 = lambda1;
        double r21811 = cos(r21810);
        double r21812 = lambda2;
        double r21813 = cos(r21812);
        double r21814 = r21811 * r21813;
        double r21815 = sin(r21810);
        double r21816 = -r21812;
        double r21817 = sin(r21816);
        double r21818 = r21815 * r21817;
        double r21819 = r21814 - r21818;
        double r21820 = r21809 * r21819;
        double r21821 = r21806 + r21820;
        double r21822 = acos(r21821);
        double r21823 = R;
        double r21824 = r21822 * r21823;
        return r21824;
}

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 16.8

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

  3. Applied sub-neg16.8

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right) \cdot R\]

  4. Applied cos-sum3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}\right) \cdot R\]

  5. Simplified3.9

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot R\]
  6. Using strategy rm

  7. Applied add-cbrt-cube3.9

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

  8. Applied add-cbrt-cube3.9

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

  9. Applied cbrt-unprod3.9

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

  10. Simplified3.9

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

  12. Applied add-log-exp3.9

    \[\leadsto \cos^{-1} \left(\color{blue}{\log \left(e^{\sqrt[3]{{\left(\sin \phi_1 \cdot \sin \phi_2\right)}^{3}}}\right)} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right) \cdot R\]

  13. Simplified3.9

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

  14. Final simplification3.9

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

Reproduce

herbie shell --seed 2019294 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))