Average Error: 16.6 → 3.8
Time: 13.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\log \left(e^{\cos^{-1} \left(\sqrt[3]{{\left(\sin \phi_1 \cdot \sin \phi_2\right)}^{3}} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)\right)}\right) \cdot R\]
\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
↓
\log \left(e^{\cos^{-1} \left(\sqrt[3]{{\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)}\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r26759 = phi1;
        double r26760 = sin(r26759);
        double r26761 = phi2;
        double r26762 = sin(r26761);
        double r26763 = r26760 * r26762;
        double r26764 = cos(r26759);
        double r26765 = cos(r26761);
        double r26766 = r26764 * r26765;
        double r26767 = lambda1;
        double r26768 = lambda2;
        double r26769 = r26767 - r26768;
        double r26770 = cos(r26769);
        double r26771 = r26766 * r26770;
        double r26772 = r26763 + r26771;
        double r26773 = acos(r26772);
        double r26774 = R;
        double r26775 = r26773 * r26774;
        return r26775;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r26776 = phi1;
        double r26777 = sin(r26776);
        double r26778 = phi2;
        double r26779 = sin(r26778);
        double r26780 = r26777 * r26779;
        double r26781 = 3.0;
        double r26782 = pow(r26780, r26781);
        double r26783 = cbrt(r26782);
        double r26784 = cos(r26776);
        double r26785 = cos(r26778);
        double r26786 = r26784 * r26785;
        double r26787 = lambda1;
        double r26788 = cos(r26787);
        double r26789 = lambda2;
        double r26790 = cos(r26789);
        double r26791 = r26788 * r26790;
        double r26792 = sin(r26787);
        double r26793 = -r26789;
        double r26794 = sin(r26793);
        double r26795 = r26792 * r26794;
        double r26796 = r26791 - r26795;
        double r26797 = r26786 * r26796;
        double r26798 = r26783 + r26797;
        double r26799 = acos(r26798);
        double r26800 = exp(r26799);
        double r26801 = log(r26800);
        double r26802 = R;
        double r26803 = r26801 * r26802;
        return r26803;
}

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 16.6

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

  3. Applied sub-neg16.6

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

  4. Applied cos-sum3.8

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

  5. Simplified3.8

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

  7. Applied add-cbrt-cube3.8

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

  8. Applied add-cbrt-cube3.8

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

  9. Applied cbrt-unprod3.8

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

  10. Simplified3.8

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

  12. Applied add-log-exp3.8

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

  13. Final simplification3.8

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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))