Average Error: 13.3 → 0.2
Time: 47.1s
Precision: 64
\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\]
↓
\[\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \left(\sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1} \cdot \sqrt[3]{\sin \lambda_2 \cdot \sin \lambda_1}\right)\right)\right)}\]
\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \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)\right)\right)}
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r5317525 = lambda1;
        double r5317526 = lambda2;
        double r5317527 = r5317525 - r5317526;
        double r5317528 = sin(r5317527);
        double r5317529 = phi2;
        double r5317530 = cos(r5317529);
        double r5317531 = r5317528 * r5317530;
        double r5317532 = phi1;
        double r5317533 = cos(r5317532);
        double r5317534 = sin(r5317529);
        double r5317535 = r5317533 * r5317534;
        double r5317536 = sin(r5317532);
        double r5317537 = r5317536 * r5317530;
        double r5317538 = cos(r5317527);
        double r5317539 = r5317537 * r5317538;
        double r5317540 = r5317535 - r5317539;
        double r5317541 = atan2(r5317531, r5317540);
        return r5317541;
}
↓
double f(double lambda1, double lambda2, double phi1, double phi2) {
        double r5317542 = lambda2;
        double r5317543 = cos(r5317542);
        double r5317544 = lambda1;
        double r5317545 = sin(r5317544);
        double r5317546 = r5317543 * r5317545;
        double r5317547 = cos(r5317544);
        double r5317548 = sin(r5317542);
        double r5317549 = r5317547 * r5317548;
        double r5317550 = r5317546 - r5317549;
        double r5317551 = phi2;
        double r5317552 = cos(r5317551);
        double r5317553 = r5317550 * r5317552;
        double r5317554 = sin(r5317551);
        double r5317555 = phi1;
        double r5317556 = cos(r5317555);
        double r5317557 = r5317554 * r5317556;
        double r5317558 = r5317543 * r5317547;
        double r5317559 = sin(r5317555);
        double r5317560 = r5317559 * r5317552;
        double r5317561 = r5317558 * r5317560;
        double r5317562 = r5317548 * r5317545;
        double r5317563 = cbrt(r5317562);
        double r5317564 = r5317563 * r5317563;
        double r5317565 = r5317563 * r5317564;
        double r5317566 = r5317560 * r5317565;
        double r5317567 = r5317561 + r5317566;
        double r5317568 = r5317557 - r5317567;
        double r5317569 = atan2(r5317553, r5317568);
        return r5317569;
}

Error

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 13.3

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

  3. Applied sin-diff6.8

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

  5. Applied cos-diff0.2

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

  6. Applied distribute-lft-in0.2

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

  8. Applied add-cube-cbrt0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))