Average Error: 0.2 → 0.2
Time: 23.0s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}{\sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}\right) \cdot \sin \phi_1 + \cos delta}}\]
\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}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}{\sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}\right) \cdot \sin \phi_1 + \cos delta}}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r84453 = lambda1;
        double r84454 = theta;
        double r84455 = sin(r84454);
        double r84456 = delta;
        double r84457 = sin(r84456);
        double r84458 = r84455 * r84457;
        double r84459 = phi1;
        double r84460 = cos(r84459);
        double r84461 = r84458 * r84460;
        double r84462 = cos(r84456);
        double r84463 = sin(r84459);
        double r84464 = r84463 * r84462;
        double r84465 = r84460 * r84457;
        double r84466 = cos(r84454);
        double r84467 = r84465 * r84466;
        double r84468 = r84464 + r84467;
        double r84469 = asin(r84468);
        double r84470 = sin(r84469);
        double r84471 = r84463 * r84470;
        double r84472 = r84462 - r84471;
        double r84473 = atan2(r84461, r84472);
        double r84474 = r84453 + r84473;
        return r84474;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r84475 = lambda1;
        double r84476 = theta;
        double r84477 = sin(r84476);
        double r84478 = delta;
        double r84479 = sin(r84478);
        double r84480 = r84477 * r84479;
        double r84481 = phi1;
        double r84482 = cos(r84481);
        double r84483 = r84480 * r84482;
        double r84484 = cos(r84478);
        double r84485 = 2.0;
        double r84486 = pow(r84484, r84485);
        double r84487 = pow(r84479, r84485);
        double r84488 = pow(r84482, r84485);
        double r84489 = cos(r84476);
        double r84490 = pow(r84489, r84485);
        double r84491 = sin(r84481);
        double r84492 = pow(r84491, r84485);
        double r84493 = r84490 * r84492;
        double r84494 = r84488 * r84493;
        double r84495 = r84487 * r84494;
        double r84496 = 3.0;
        double r84497 = pow(r84491, r84496);
        double r84498 = r84484 * r84489;
        double r84499 = r84497 * r84498;
        double r84500 = r84482 * r84499;
        double r84501 = r84479 * r84500;
        double r84502 = r84485 * r84501;
        double r84503 = 4.0;
        double r84504 = pow(r84491, r84503);
        double r84505 = r84504 * r84486;
        double r84506 = r84502 + r84505;
        double r84507 = r84495 + r84506;
        double r84508 = r84486 - r84507;
        double r84509 = r84482 * r84489;
        double r84510 = r84479 * r84509;
        double r84511 = r84491 * r84484;
        double r84512 = r84510 + r84511;
        double r84513 = asin(r84512);
        double r84514 = pow(r84513, r84496);
        double r84515 = cbrt(r84514);
        double r84516 = sin(r84515);
        double r84517 = r84516 * r84491;
        double r84518 = r84517 + r84484;
        double r84519 = r84508 / r84518;
        double r84520 = atan2(r84483, r84519);
        double r84521 = r84475 + r84520;
        return r84521;
}

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. Simplified0.2

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

  5. Simplified0.2

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

  6. Taylor expanded around inf 0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}}{\sin \left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \sin \phi_1 + \cos delta}}\]
  7. Using strategy rm

  8. Applied add-cbrt-cube0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}{\sin \color{blue}{\left(\sqrt[3]{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right) \cdot \sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right) \cdot \sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)}\right)} \cdot \sin \phi_1 + \cos delta}}\]

  9. Simplified0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}{\sin \left(\sqrt[3]{\color{blue}{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}}\right) \cdot \sin \phi_1 + \cos delta}}\]

  10. Final simplification0.2

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\cos delta\right)}^{2} - \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \left({\left(\cos theta\right)}^{2} \cdot {\left(\sin \phi_1\right)}^{2}\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{3} \cdot \left(\cos delta \cdot \cos theta\right)\right)\right)\right) + {\left(\sin \phi_1\right)}^{4} \cdot {\left(\cos delta\right)}^{2}\right)\right)}{\sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \sin \phi_1 \cdot \cos delta\right)\right)}^{3}}\right) \cdot \sin \phi_1 + \cos delta}}\]

Reproduce

herbie shell --seed 2020046 
(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))))))))))