Average Error: 0.2 → 0.2
Time: 39.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)\right)}\]
\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(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)\right)\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r2661526 = lambda1;
        double r2661527 = theta;
        double r2661528 = sin(r2661527);
        double r2661529 = delta;
        double r2661530 = sin(r2661529);
        double r2661531 = r2661528 * r2661530;
        double r2661532 = phi1;
        double r2661533 = cos(r2661532);
        double r2661534 = r2661531 * r2661533;
        double r2661535 = cos(r2661529);
        double r2661536 = sin(r2661532);
        double r2661537 = r2661536 * r2661535;
        double r2661538 = r2661533 * r2661530;
        double r2661539 = cos(r2661527);
        double r2661540 = r2661538 * r2661539;
        double r2661541 = r2661537 + r2661540;
        double r2661542 = asin(r2661541);
        double r2661543 = sin(r2661542);
        double r2661544 = r2661536 * r2661543;
        double r2661545 = r2661535 - r2661544;
        double r2661546 = atan2(r2661534, r2661545);
        double r2661547 = r2661526 + r2661546;
        return r2661547;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r2661548 = lambda1;
        double r2661549 = phi1;
        double r2661550 = cos(r2661549);
        double r2661551 = theta;
        double r2661552 = sin(r2661551);
        double r2661553 = r2661550 * r2661552;
        double r2661554 = delta;
        double r2661555 = sin(r2661554);
        double r2661556 = r2661553 * r2661555;
        double r2661557 = cos(r2661554);
        double r2661558 = sin(r2661549);
        double r2661559 = cos(r2661551);
        double r2661560 = r2661550 * r2661555;
        double r2661561 = r2661559 * r2661560;
        double r2661562 = fma(r2661558, r2661557, r2661561);
        double r2661563 = r2661558 * r2661562;
        double r2661564 = r2661557 - r2661563;
        double r2661565 = expm1(r2661564);
        double r2661566 = log1p(r2661565);
        double r2661567 = atan2(r2661556, r2661566);
        double r2661568 = r2661548 + r2661567;
        return r2661568;
}

Error

Bits error versus lambda1

Bits error versus phi1

Bits error versus phi2

Bits error versus delta

Bits error versus theta

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. Simplified0.2

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

  4. Applied log1p-expm1-u0.2

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

  5. Taylor expanded around inf 0.2

    \[\leadsto \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\mathsf{log1p}\left(\color{blue}{e^{\cos delta - \left({\left(\sin \phi_1\right)}^{2} \cdot \cos delta + \sin delta \cdot \left(\sin \phi_1 \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)} - 1}\right)} + \lambda_1\]

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(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))))))))))