Average Error: 16.6 → 3.9
Time: 37.7s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[\cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \left(\cos \lambda_2 \cdot \cos \lambda_1 - \log \left(e^{\sqrt[3]{-{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}}\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\]
\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(\sin \phi_2 \cdot \sin \phi_1 + \left(\cos \lambda_2 \cdot \cos \lambda_1 - \log \left(e^{\sqrt[3]{-{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}}\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r24645 = phi1;
        double r24646 = sin(r24645);
        double r24647 = phi2;
        double r24648 = sin(r24647);
        double r24649 = r24646 * r24648;
        double r24650 = cos(r24645);
        double r24651 = cos(r24647);
        double r24652 = r24650 * r24651;
        double r24653 = lambda1;
        double r24654 = lambda2;
        double r24655 = r24653 - r24654;
        double r24656 = cos(r24655);
        double r24657 = r24652 * r24656;
        double r24658 = r24649 + r24657;
        double r24659 = acos(r24658);
        double r24660 = R;
        double r24661 = r24659 * r24660;
        return r24661;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r24662 = phi2;
        double r24663 = sin(r24662);
        double r24664 = phi1;
        double r24665 = sin(r24664);
        double r24666 = r24663 * r24665;
        double r24667 = lambda2;
        double r24668 = cos(r24667);
        double r24669 = lambda1;
        double r24670 = cos(r24669);
        double r24671 = r24668 * r24670;
        double r24672 = sin(r24669);
        double r24673 = sin(r24667);
        double r24674 = r24672 * r24673;
        double r24675 = 3.0;
        double r24676 = pow(r24674, r24675);
        double r24677 = -r24676;
        double r24678 = cbrt(r24677);
        double r24679 = exp(r24678);
        double r24680 = log(r24679);
        double r24681 = r24671 - r24680;
        double r24682 = cos(r24664);
        double r24683 = cos(r24662);
        double r24684 = r24682 * r24683;
        double r24685 = r24681 * r24684;
        double r24686 = r24666 + r24685;
        double r24687 = acos(r24686);
        double r24688 = R;
        double r24689 = r24687 * r24688;
        return r24689;
}

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-sum3.9

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

  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. Simplified3.9

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

  8. Applied add-cbrt-cube3.9

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

  9. Applied add-cbrt-cube3.9

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

  10. Applied cbrt-unprod3.9

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

  11. Simplified3.9

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

  13. Applied add-log-exp3.9

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

  14. Simplified3.9

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

  15. Final simplification3.9

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))