Average Error: 0.2 → 0.2
Time: 52.1s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right) \cdot \left(\log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right) \cdot \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right)}\right)\right) \cdot \left(\sin \left(\log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right) \cdot \sin \phi_1\right)}{\cos delta + \sin \left(\log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right) \cdot \sin \phi_1}}\]
\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)}
↓
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{\cos delta \cdot \cos delta - \left(\sin \phi_1 \cdot \sin \left(\sqrt[3]{\log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right) \cdot \left(\log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right) \cdot \log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right)}\right)\right) \cdot \left(\sin \left(\log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right) \cdot \sin \phi_1\right)}{\cos delta + \sin \left(\log \left(e^{\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\right)\right) \cdot \sin \phi_1}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r1832424 = lambda1;
        double r1832425 = theta;
        double r1832426 = sin(r1832425);
        double r1832427 = delta;
        double r1832428 = sin(r1832427);
        double r1832429 = r1832426 * r1832428;
        double r1832430 = phi1;
        double r1832431 = cos(r1832430);
        double r1832432 = r1832429 * r1832431;
        double r1832433 = cos(r1832427);
        double r1832434 = sin(r1832430);
        double r1832435 = r1832434 * r1832433;
        double r1832436 = r1832431 * r1832428;
        double r1832437 = cos(r1832425);
        double r1832438 = r1832436 * r1832437;
        double r1832439 = r1832435 + r1832438;
        double r1832440 = asin(r1832439);
        double r1832441 = sin(r1832440);
        double r1832442 = r1832434 * r1832441;
        double r1832443 = r1832433 - r1832442;
        double r1832444 = atan2(r1832432, r1832443);
        double r1832445 = r1832424 + r1832444;
        return r1832445;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r1832446 = lambda1;
        double r1832447 = phi1;
        double r1832448 = cos(r1832447);
        double r1832449 = delta;
        double r1832450 = sin(r1832449);
        double r1832451 = theta;
        double r1832452 = sin(r1832451);
        double r1832453 = r1832450 * r1832452;
        double r1832454 = r1832448 * r1832453;
        double r1832455 = cos(r1832449);
        double r1832456 = r1832455 * r1832455;
        double r1832457 = sin(r1832447);
        double r1832458 = r1832448 * r1832450;
        double r1832459 = cos(r1832451);
        double r1832460 = r1832458 * r1832459;
        double r1832461 = fma(r1832457, r1832455, r1832460);
        double r1832462 = asin(r1832461);
        double r1832463 = exp(r1832462);
        double r1832464 = log(r1832463);
        double r1832465 = r1832464 * r1832464;
        double r1832466 = r1832464 * r1832465;
        double r1832467 = cbrt(r1832466);
        double r1832468 = sin(r1832467);
        double r1832469 = r1832457 * r1832468;
        double r1832470 = sin(r1832464);
        double r1832471 = r1832470 * r1832457;
        double r1832472 = r1832469 * r1832471;
        double r1832473 = r1832456 - r1832472;
        double r1832474 = r1832455 + r1832471;
        double r1832475 = r1832473 / r1832474;
        double r1832476 = atan2(r1832454, r1832475);
        double r1832477 = r1832446 + r1832476;
        return r1832477;
}

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

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

  6. Applied flip--0.2

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

  8. Applied add-cbrt-cube0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 +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))))))))))