Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 7.5m

Precision: 64

• unused variable phi2

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{\cos delta \cdot \left(\cos delta \cdot \cos delta\right) - \left(\left(\left(\left(2 \cdot \sin delta\right) \cdot \cos \phi_1\right) \cdot \left(\cos theta \cdot \left(\left(\cos delta \cdot \cos delta\right) \cdot {\left(\sin \phi_1\right)}^{5}\right)\right) + \left(\log \left(e^{\left(\cos theta \cdot \cos \phi_1\right) \cdot \left(\cos theta \cdot \cos \phi_1\right)}\right) \cdot \cos delta\right) \cdot \left(3 \cdot \left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin delta\right) \cdot \left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin delta\right)\right)\right)\right) + \left(\left(\left(\cos theta \cdot \cos theta\right) \cdot \cos theta\right) \cdot \left(\left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\left(\left(\sin delta \cdot \left(\sin delta \cdot \sin delta\right)\right) \cdot \cos \phi_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_1\right)\right)\right) + \left(\left(\cos delta \cdot \left(\cos delta \cdot \cos delta\right)\right) \cdot \left(\left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right)\right) + {\left(\sin \phi_1\right)}^{5} \cdot \left(\left(\cos theta \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos delta\right)\right)\right) \cdot \cos \phi_1\right)\right)\right)\right)}{\cos delta \cdot \cos delta + \left(\left(\sin \left(\sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)\right) \cdot \sin \phi_1\right) + \cos delta \cdot \left(\sin \left(\sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)\right) \cdot \sin \phi_1\right)\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r17158467 = lambda1;
        double r17158468 = theta;
        double r17158469 = sin(r17158468);
        double r17158470 = delta;
        double r17158471 = sin(r17158470);
        double r17158472 = r17158469 * r17158471;
        double r17158473 = phi1;
        double r17158474 = cos(r17158473);
        double r17158475 = r17158472 * r17158474;
        double r17158476 = cos(r17158470);
        double r17158477 = sin(r17158473);
        double r17158478 = r17158477 * r17158476;
        double r17158479 = r17158474 * r17158471;
        double r17158480 = cos(r17158468);
        double r17158481 = r17158479 * r17158480;
        double r17158482 = r17158478 + r17158481;
        double r17158483 = asin(r17158482);
        double r17158484 = sin(r17158483);
        double r17158485 = r17158477 * r17158484;
        double r17158486 = r17158476 - r17158485;
        double r17158487 = atan2(r17158475, r17158486);
        double r17158488 = r17158467 + r17158487;
        return r17158488;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r17158489 = lambda1;
        double r17158490 = phi1;
        double r17158491 = cos(r17158490);
        double r17158492 = delta;
        double r17158493 = sin(r17158492);
        double r17158494 = theta;
        double r17158495 = sin(r17158494);
        double r17158496 = r17158493 * r17158495;
        double r17158497 = r17158491 * r17158496;
        double r17158498 = cos(r17158492);
        double r17158499 = r17158498 * r17158498;
        double r17158500 = r17158498 * r17158499;
        double r17158501 = 2.0;
        double r17158502 = r17158501 * r17158493;
        double r17158503 = r17158502 * r17158491;
        double r17158504 = cos(r17158494);
        double r17158505 = sin(r17158490);
        double r17158506 = 5.0;
        double r17158507 = pow(r17158505, r17158506);
        double r17158508 = r17158499 * r17158507;
        double r17158509 = r17158504 * r17158508;
        double r17158510 = r17158503 * r17158509;
        double r17158511 = r17158504 * r17158491;
        double r17158512 = r17158511 * r17158511;
        double r17158513 = exp(r17158512);
        double r17158514 = log(r17158513);
        double r17158515 = r17158514 * r17158498;
        double r17158516 = 3.0;
        double r17158517 = r17158505 * r17158505;
        double r17158518 = r17158517 * r17158493;
        double r17158519 = r17158518 * r17158518;
        double r17158520 = r17158516 * r17158519;
        double r17158521 = r17158515 * r17158520;
        double r17158522 = r17158510 + r17158521;
        double r17158523 = r17158504 * r17158504;
        double r17158524 = r17158523 * r17158504;
        double r17158525 = r17158505 * r17158517;
        double r17158526 = r17158493 * r17158493;
        double r17158527 = r17158493 * r17158526;
        double r17158528 = r17158527 * r17158491;
        double r17158529 = r17158491 * r17158491;
        double r17158530 = r17158528 * r17158529;
        double r17158531 = r17158525 * r17158530;
        double r17158532 = r17158524 * r17158531;
        double r17158533 = r17158525 * r17158525;
        double r17158534 = r17158500 * r17158533;
        double r17158535 = r17158493 * r17158499;
        double r17158536 = r17158504 * r17158535;
        double r17158537 = r17158536 * r17158491;
        double r17158538 = r17158507 * r17158537;
        double r17158539 = r17158534 + r17158538;
        double r17158540 = r17158532 + r17158539;
        double r17158541 = r17158522 + r17158540;
        double r17158542 = r17158500 - r17158541;
        double r17158543 = r17158491 * r17158493;
        double r17158544 = r17158543 * r17158504;
        double r17158545 = r17158498 * r17158505;
        double r17158546 = r17158544 + r17158545;
        double r17158547 = asin(r17158546);
        double r17158548 = sin(r17158547);
        double r17158549 = r17158548 * r17158505;
        double r17158550 = r17158549 * r17158549;
        double r17158551 = r17158498 * r17158549;
        double r17158552 = r17158550 + r17158551;
        double r17158553 = r17158499 + r17158552;
        double r17158554 = r17158542 / r17158553;
        double r17158555 = atan2(r17158497, r17158554);
        double r17158556 = r17158489 + r17158555;
        return r17158556;
}

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 flip3-- 0.2

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

4. Taylor expanded around inf 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{{\left(\cos delta\right)}^{3} - \left({\left(\cos \phi_1\right)}^{3} \cdot \left({\left(\sin delta\right)}^{3} \cdot \left({\left(\cos theta\right)}^{3} \cdot {\left(\sin \phi_1\right)}^{3}\right)\right) + \left({\left(\sin \phi_1\right)}^{6} \cdot {\left(\cos delta\right)}^{3} + \left(\cos \phi_1 \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\sin delta \cdot \left(\cos theta \cdot {\left(\cos delta\right)}^{2}\right)\right)\right) + \left(2 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot {\left(\cos delta\right)}^{2}\right)\right)\right)\right) + 3 \cdot \left({\left(\sin delta\right)}^{2} \cdot \left({\left(\sin \phi_1\right)}^{4} \cdot \left({\left(\cos theta\right)}^{2} \cdot \left({\left(\cos \phi_1\right)}^{2} \cdot \cos delta\right)\right)\right)\right)\right)\right)\right)\right)}}{\cos delta \cdot \cos delta + \left(\left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) + \cos delta \cdot \left(\sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)\right)}}\]

5. Simplified 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 \cdot \cos delta\right) \cdot \cos delta - \left(\left(\left(\cos theta \cdot \left(\cos theta \cdot \cos theta\right)\right) \cdot \left(\left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\left(\left(\sin delta \cdot \left(\sin delta \cdot \sin delta\right)\right) \cdot \cos \phi_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_1\right)\right)\right) + \left(\left(\cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos delta\right)\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{5} + \left(\left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right)\right) \cdot \left(\left(\cos delta \cdot \cos delta\right) \cdot \cos delta\right)\right)\right) + \left(\left(\left(2 \cdot \sin delta\right) \cdot \cos \phi_1\right) \cdot \left(\cos theta \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\cos delta \cdot \cos delta\right)\right)\right) + \left(3 \cdot \left(\left(\sin delta \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\sin delta \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right)\right)\right) \cdot \left(\left(\left(\cos \phi_1 \cdot \cos theta\right) \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \cos delta\right)\right)\right)}}{\cos delta \cdot \cos delta + \left(\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 \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)\right)}}\]
6. Using strategy rm

7. Applied add-log-exp 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\left(\cos delta \cdot \cos delta\right) \cdot \cos delta - \left(\left(\left(\cos theta \cdot \left(\cos theta \cdot \cos theta\right)\right) \cdot \left(\left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\left(\left(\sin delta \cdot \left(\sin delta \cdot \sin delta\right)\right) \cdot \cos \phi_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_1\right)\right)\right) + \left(\left(\cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos delta\right)\right)\right)\right) \cdot {\left(\sin \phi_1\right)}^{5} + \left(\left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right)\right) \cdot \left(\left(\cos delta \cdot \cos delta\right) \cdot \cos delta\right)\right)\right) + \left(\left(\left(2 \cdot \sin delta\right) \cdot \cos \phi_1\right) \cdot \left(\cos theta \cdot \left({\left(\sin \phi_1\right)}^{5} \cdot \left(\cos delta \cdot \cos delta\right)\right)\right) + \left(3 \cdot \left(\left(\sin delta \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\sin delta \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right)\right)\right) \cdot \left(\color{blue}{\log \left(e^{\left(\cos \phi_1 \cdot \cos theta\right) \cdot \left(\cos \phi_1 \cdot \cos theta\right)}\right)} \cdot \cos delta\right)\right)\right)}{\cos delta \cdot \cos delta + \left(\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 \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)\right)}}\]

8. Final simplification 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{\cos delta \cdot \left(\cos delta \cdot \cos delta\right) - \left(\left(\left(\left(2 \cdot \sin delta\right) \cdot \cos \phi_1\right) \cdot \left(\cos theta \cdot \left(\left(\cos delta \cdot \cos delta\right) \cdot {\left(\sin \phi_1\right)}^{5}\right)\right) + \left(\log \left(e^{\left(\cos theta \cdot \cos \phi_1\right) \cdot \left(\cos theta \cdot \cos \phi_1\right)}\right) \cdot \cos delta\right) \cdot \left(3 \cdot \left(\left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin delta\right) \cdot \left(\left(\sin \phi_1 \cdot \sin \phi_1\right) \cdot \sin delta\right)\right)\right)\right) + \left(\left(\left(\cos theta \cdot \cos theta\right) \cdot \cos theta\right) \cdot \left(\left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\left(\left(\sin delta \cdot \left(\sin delta \cdot \sin delta\right)\right) \cdot \cos \phi_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_1\right)\right)\right) + \left(\left(\cos delta \cdot \left(\cos delta \cdot \cos delta\right)\right) \cdot \left(\left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right) \cdot \left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \sin \phi_1\right)\right)\right) + {\left(\sin \phi_1\right)}^{5} \cdot \left(\left(\cos theta \cdot \left(\sin delta \cdot \left(\cos delta \cdot \cos delta\right)\right)\right) \cdot \cos \phi_1\right)\right)\right)\right)}{\cos delta \cdot \cos delta + \left(\left(\sin \left(\sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)\right) \cdot \sin \phi_1\right) + \cos delta \cdot \left(\sin \left(\sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)\right) \cdot \sin \phi_1\right)\right)}}\]

Reproduce

herbie shell --seed 2019144 
(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))))))))))