Average Error: 0.2 → 0.2
Time: 14.3s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta + \left(-\sin \phi_1\right) \cdot \sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}\right)}\]
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta + \left(-\sin \phi_1\right) \cdot \sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r67659 = lambda1;
        double r67660 = theta;
        double r67661 = sin(r67660);
        double r67662 = delta;
        double r67663 = sin(r67662);
        double r67664 = r67661 * r67663;
        double r67665 = phi1;
        double r67666 = cos(r67665);
        double r67667 = r67664 * r67666;
        double r67668 = cos(r67662);
        double r67669 = sin(r67665);
        double r67670 = r67669 * r67668;
        double r67671 = r67666 * r67663;
        double r67672 = cos(r67660);
        double r67673 = r67671 * r67672;
        double r67674 = r67670 + r67673;
        double r67675 = asin(r67674);
        double r67676 = sin(r67675);
        double r67677 = r67669 * r67676;
        double r67678 = r67668 - r67677;
        double r67679 = atan2(r67667, r67678);
        double r67680 = r67659 + r67679;
        return r67680;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r67681 = lambda1;
        double r67682 = theta;
        double r67683 = sin(r67682);
        double r67684 = delta;
        double r67685 = sin(r67684);
        double r67686 = r67683 * r67685;
        double r67687 = phi1;
        double r67688 = cos(r67687);
        double r67689 = r67686 * r67688;
        double r67690 = cos(r67684);
        double r67691 = sin(r67687);
        double r67692 = -r67691;
        double r67693 = cos(r67682);
        double r67694 = r67688 * r67693;
        double r67695 = r67685 * r67694;
        double r67696 = r67691 * r67690;
        double r67697 = r67695 + r67696;
        double r67698 = asin(r67697);
        double r67699 = 3.0;
        double r67700 = pow(r67698, r67699);
        double r67701 = cbrt(r67700);
        double r67702 = sin(r67701);
        double r67703 = r67692 * r67702;
        double r67704 = r67690 + r67703;
        double r67705 = atan2(r67689, r67704);
        double r67706 = r67681 + r67705;
        return r67706;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

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

  3. Applied add-cbrt-cube0.2

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

  4. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sqrt[3]{\color{blue}{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}}\right)}\]
  5. Using strategy rm

  6. Applied expm1-log1p-u0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}\right)\right)\right)}}\]
  7. Using strategy rm

  8. Applied sub-neg0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta + \left(-\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}\right)\right)\right)\right)}}\]

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))