Average Error: 0.2 → 0.2
Time: 50.3s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \left(\sqrt[3]{\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right) \cdot \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right) \cdot \log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right)}\right) \cdot \sin \phi_1\right)}^{3}}{\cos delta \cdot \cos delta + \left(\cos delta \cdot \left(\sin \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right) \cdot \sin \phi_1\right) + \left(\sin \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right) \cdot \sin \phi_1\right)\right)}} + \lambda_1\]
\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(\cos delta\right)}^{3} - {\left(\sin \left(\sqrt[3]{\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right) \cdot \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right) \cdot \log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right)}\right) \cdot \sin \phi_1\right)}^{3}}{\cos delta \cdot \cos delta + \left(\cos delta \cdot \left(\sin \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right) \cdot \sin \phi_1\right) + \left(\sin \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right) \cdot \sin \phi_1\right)\right)}} + \lambda_1
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r1718455 = lambda1;
        double r1718456 = theta;
        double r1718457 = sin(r1718456);
        double r1718458 = delta;
        double r1718459 = sin(r1718458);
        double r1718460 = r1718457 * r1718459;
        double r1718461 = phi1;
        double r1718462 = cos(r1718461);
        double r1718463 = r1718460 * r1718462;
        double r1718464 = cos(r1718458);
        double r1718465 = sin(r1718461);
        double r1718466 = r1718465 * r1718464;
        double r1718467 = r1718462 * r1718459;
        double r1718468 = cos(r1718456);
        double r1718469 = r1718467 * r1718468;
        double r1718470 = r1718466 + r1718469;
        double r1718471 = asin(r1718470);
        double r1718472 = sin(r1718471);
        double r1718473 = r1718465 * r1718472;
        double r1718474 = r1718464 - r1718473;
        double r1718475 = atan2(r1718463, r1718474);
        double r1718476 = r1718455 + r1718475;
        return r1718476;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r1718477 = phi1;
        double r1718478 = cos(r1718477);
        double r1718479 = delta;
        double r1718480 = sin(r1718479);
        double r1718481 = theta;
        double r1718482 = sin(r1718481);
        double r1718483 = r1718480 * r1718482;
        double r1718484 = r1718478 * r1718483;
        double r1718485 = cos(r1718479);
        double r1718486 = 3.0;
        double r1718487 = pow(r1718485, r1718486);
        double r1718488 = sin(r1718477);
        double r1718489 = r1718485 * r1718488;
        double r1718490 = cos(r1718481);
        double r1718491 = r1718478 * r1718480;
        double r1718492 = r1718490 * r1718491;
        double r1718493 = r1718489 + r1718492;
        double r1718494 = asin(r1718493);
        double r1718495 = exp(r1718494);
        double r1718496 = log(r1718495);
        double r1718497 = r1718496 * r1718496;
        double r1718498 = r1718496 * r1718497;
        double r1718499 = cbrt(r1718498);
        double r1718500 = sin(r1718499);
        double r1718501 = r1718500 * r1718488;
        double r1718502 = pow(r1718501, r1718486);
        double r1718503 = r1718487 - r1718502;
        double r1718504 = r1718485 * r1718485;
        double r1718505 = sin(r1718496);
        double r1718506 = r1718505 * r1718488;
        double r1718507 = r1718485 * r1718506;
        double r1718508 = r1718506 * r1718506;
        double r1718509 = r1718507 + r1718508;
        double r1718510 = r1718504 + r1718509;
        double r1718511 = r1718503 / r1718510;
        double r1718512 = atan2(r1718484, r1718511);
        double r1718513 = lambda1;
        double r1718514 = r1718512 + r1718513;
        return r1718514;
}

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-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. Using strategy rm

  5. Applied flip3--0.2

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

  7. Applied add-cbrt-cube0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019153 
(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))))))))))