Average Error: 0.2 → 0.2
Time: 20.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(-\sin \phi_1, \sin \left(\frac{\pi}{2} - \cos^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right), \cos delta\right) + 0 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)}\]
\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(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(-\sin \phi_1, \sin \left(\frac{\pi}{2} - \cos^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right), \cos delta\right) + 0 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r112440 = lambda1;
        double r112441 = theta;
        double r112442 = sin(r112441);
        double r112443 = delta;
        double r112444 = sin(r112443);
        double r112445 = r112442 * r112444;
        double r112446 = phi1;
        double r112447 = cos(r112446);
        double r112448 = r112445 * r112447;
        double r112449 = cos(r112443);
        double r112450 = sin(r112446);
        double r112451 = r112450 * r112449;
        double r112452 = r112447 * r112444;
        double r112453 = cos(r112441);
        double r112454 = r112452 * r112453;
        double r112455 = r112451 + r112454;
        double r112456 = asin(r112455);
        double r112457 = sin(r112456);
        double r112458 = r112450 * r112457;
        double r112459 = r112449 - r112458;
        double r112460 = atan2(r112448, r112459);
        double r112461 = r112440 + r112460;
        return r112461;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r112462 = lambda1;
        double r112463 = theta;
        double r112464 = sin(r112463);
        double r112465 = delta;
        double r112466 = sin(r112465);
        double r112467 = r112464 * r112466;
        double r112468 = phi1;
        double r112469 = cos(r112468);
        double r112470 = r112467 * r112469;
        double r112471 = sin(r112468);
        double r112472 = -r112471;
        double r112473 = atan2(1.0, 0.0);
        double r112474 = 2.0;
        double r112475 = r112473 / r112474;
        double r112476 = cos(r112463);
        double r112477 = r112469 * r112476;
        double r112478 = cos(r112465);
        double r112479 = r112471 * r112478;
        double r112480 = fma(r112466, r112477, r112479);
        double r112481 = acos(r112480);
        double r112482 = r112475 - r112481;
        double r112483 = sin(r112482);
        double r112484 = fma(r112472, r112483, r112478);
        double r112485 = 0.0;
        double r112486 = asin(r112480);
        double r112487 = sin(r112486);
        double r112488 = r112485 * r112487;
        double r112489 = r112484 + r112488;
        double r112490 = atan2(r112470, r112489);
        double r112491 = r112462 + r112490;
        return r112491;
}

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. Using strategy rm

  3. Applied add-log-exp0.2

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

  4. Simplified0.2

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

  6. Applied add-sqr-sqrt15.9

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

  7. Applied prod-diff15.9

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

  8. Simplified0.2

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

  9. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(-\sin \phi_1, \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right), \cos delta\right) + \color{blue}{0 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)}}\]
  10. Using strategy rm

  11. Applied asin-acos0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(-\sin \phi_1, \sin \color{blue}{\left(\frac{\pi}{2} - \cos^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)}, \cos delta\right) + 0 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)}\]

  12. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(-\sin \phi_1, \sin \left(\frac{\pi}{2} - \cos^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right), \cos delta\right) + 0 \cdot \sin \left(\sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \sin \phi_1 \cdot \cos delta\right)\right)\right)}\]

Reproduce

herbie shell --seed 2020042 +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))))))))))