Average Error: 16.9 → 4.1
Time: 43.1s
Precision: 64
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\]
↓
\[R \cdot \cos^{-1} \left(\left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}\right) + \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_2 \cdot \sin \phi_1\right)\]
\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
↓
R \cdot \cos^{-1} \left(\left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}\right) + \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_2 \cdot \sin \phi_1\right)
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r351540 = phi1;
        double r351541 = sin(r351540);
        double r351542 = phi2;
        double r351543 = sin(r351542);
        double r351544 = r351541 * r351543;
        double r351545 = cos(r351540);
        double r351546 = cos(r351542);
        double r351547 = r351545 * r351546;
        double r351548 = lambda1;
        double r351549 = lambda2;
        double r351550 = r351548 - r351549;
        double r351551 = cos(r351550);
        double r351552 = r351547 * r351551;
        double r351553 = r351544 + r351552;
        double r351554 = acos(r351553);
        double r351555 = R;
        double r351556 = r351554 * r351555;
        return r351556;
}
↓
double f(double R, double lambda1, double lambda2, double phi1, double phi2) {
        double r351557 = R;
        double r351558 = lambda2;
        double r351559 = sin(r351558);
        double r351560 = lambda1;
        double r351561 = sin(r351560);
        double r351562 = r351559 * r351561;
        double r351563 = cbrt(r351562);
        double r351564 = r351563 * r351563;
        double r351565 = r351563 * r351564;
        double r351566 = cos(r351558);
        double r351567 = cos(r351560);
        double r351568 = r351566 * r351567;
        double r351569 = r351565 + r351568;
        double r351570 = phi1;
        double r351571 = cos(r351570);
        double r351572 = phi2;
        double r351573 = cos(r351572);
        double r351574 = r351571 * r351573;
        double r351575 = r351569 * r351574;
        double r351576 = sin(r351572);
        double r351577 = sin(r351570);
        double r351578 = r351576 * r351577;
        double r351579 = r351575 + r351578;
        double r351580 = acos(r351579);
        double r351581 = r351557 * r351580;
        return r351581;
}

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 16.9

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

  3. Applied cos-diff4.0

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

  5. Applied add-cube-cbrt4.1

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

  6. Final simplification4.1

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

Reproduce

herbie shell --seed 2019154 +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))