Average Error: 0.2 → 0.2
Time: 1.0m
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{\left(\frac{1}{2} \cdot \cos \left(2 \cdot delta\right) - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)\right)\right) + \frac{1}{2}}{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right) + \cos delta}} + \lambda_1\]
\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)}
↓
\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{\left(\frac{1}{2} \cdot \cos \left(2 \cdot delta\right) - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)\right)\right) + \frac{1}{2}}{\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right) + \cos delta}} + \lambda_1
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r5187565 = lambda1;
        double r5187566 = theta;
        double r5187567 = sin(r5187566);
        double r5187568 = delta;
        double r5187569 = sin(r5187568);
        double r5187570 = r5187567 * r5187569;
        double r5187571 = phi1;
        double r5187572 = cos(r5187571);
        double r5187573 = r5187570 * r5187572;
        double r5187574 = cos(r5187568);
        double r5187575 = sin(r5187571);
        double r5187576 = r5187575 * r5187574;
        double r5187577 = r5187572 * r5187569;
        double r5187578 = cos(r5187566);
        double r5187579 = r5187577 * r5187578;
        double r5187580 = r5187576 + r5187579;
        double r5187581 = asin(r5187580);
        double r5187582 = sin(r5187581);
        double r5187583 = r5187575 * r5187582;
        double r5187584 = r5187574 - r5187583;
        double r5187585 = atan2(r5187573, r5187584);
        double r5187586 = r5187565 + r5187585;
        return r5187586;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r5187587 = phi1;
        double r5187588 = cos(r5187587);
        double r5187589 = delta;
        double r5187590 = sin(r5187589);
        double r5187591 = theta;
        double r5187592 = sin(r5187591);
        double r5187593 = r5187590 * r5187592;
        double r5187594 = r5187588 * r5187593;
        double r5187595 = 0.5;
        double r5187596 = 2.0;
        double r5187597 = r5187596 * r5187589;
        double r5187598 = cos(r5187597);
        double r5187599 = r5187595 * r5187598;
        double r5187600 = sin(r5187587);
        double r5187601 = cos(r5187589);
        double r5187602 = r5187601 * r5187600;
        double r5187603 = cos(r5187591);
        double r5187604 = r5187588 * r5187590;
        double r5187605 = r5187603 * r5187604;
        double r5187606 = r5187602 + r5187605;
        double r5187607 = asin(r5187606);
        double r5187608 = sin(r5187607);
        double r5187609 = r5187600 * r5187608;
        double r5187610 = r5187609 * r5187609;
        double r5187611 = r5187599 - r5187610;
        double r5187612 = r5187611 + r5187595;
        double r5187613 = r5187609 + r5187601;
        double r5187614 = r5187612 / r5187613;
        double r5187615 = atan2(r5187594, r5187614);
        double r5187616 = lambda1;
        double r5187617 = r5187615 + r5187616;
        return r5187617;
}

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 flip--0.2

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

  5. Applied sqr-cos0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot delta\right)\right)} - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

  6. Applied associate--l+0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\frac{1}{2} + \left(\frac{1}{2} \cdot \cos \left(2 \cdot delta\right) - \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}}{\cos delta + \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\]

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))