Average Error: 0.2 → 0.2
Time: 43.1s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\frac{{\left(\cos delta\right)}^{3} - {\left(\sin \left(\sqrt[3]{\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right) \cdot \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right) \cdot \log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right)}\right) \cdot \sin \phi_1\right)}^{3}}{\cos delta \cdot \cos delta + \left(\cos delta \cdot \left(\sin \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right) \cdot \sin \phi_1\right) + \left(\sin \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right) \cdot \sin \phi_1\right) \cdot \left(\sin \left(\log \left(e^{\sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\right)\right) \cdot \sin \phi_1\right)\right)}} + \lambda_1\]
\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 r1676375 = lambda1;
        double r1676376 = theta;
        double r1676377 = sin(r1676376);
        double r1676378 = delta;
        double r1676379 = sin(r1676378);
        double r1676380 = r1676377 * r1676379;
        double r1676381 = phi1;
        double r1676382 = cos(r1676381);
        double r1676383 = r1676380 * r1676382;
        double r1676384 = cos(r1676378);
        double r1676385 = sin(r1676381);
        double r1676386 = r1676385 * r1676384;
        double r1676387 = r1676382 * r1676379;
        double r1676388 = cos(r1676376);
        double r1676389 = r1676387 * r1676388;
        double r1676390 = r1676386 + r1676389;
        double r1676391 = asin(r1676390);
        double r1676392 = sin(r1676391);
        double r1676393 = r1676385 * r1676392;
        double r1676394 = r1676384 - r1676393;
        double r1676395 = atan2(r1676383, r1676394);
        double r1676396 = r1676375 + r1676395;
        return r1676396;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r1676397 = phi1;
        double r1676398 = cos(r1676397);
        double r1676399 = delta;
        double r1676400 = sin(r1676399);
        double r1676401 = theta;
        double r1676402 = sin(r1676401);
        double r1676403 = r1676400 * r1676402;
        double r1676404 = r1676398 * r1676403;
        double r1676405 = cos(r1676399);
        double r1676406 = 3.0;
        double r1676407 = pow(r1676405, r1676406);
        double r1676408 = sin(r1676397);
        double r1676409 = r1676405 * r1676408;
        double r1676410 = cos(r1676401);
        double r1676411 = r1676398 * r1676400;
        double r1676412 = r1676410 * r1676411;
        double r1676413 = r1676409 + r1676412;
        double r1676414 = asin(r1676413);
        double r1676415 = exp(r1676414);
        double r1676416 = log(r1676415);
        double r1676417 = r1676416 * r1676416;
        double r1676418 = r1676416 * r1676417;
        double r1676419 = cbrt(r1676418);
        double r1676420 = sin(r1676419);
        double r1676421 = r1676420 * r1676408;
        double r1676422 = pow(r1676421, r1676406);
        double r1676423 = r1676407 - r1676422;
        double r1676424 = r1676405 * r1676405;
        double r1676425 = sin(r1676416);
        double r1676426 = r1676425 * r1676408;
        double r1676427 = r1676405 * r1676426;
        double r1676428 = r1676426 * r1676426;
        double r1676429 = r1676427 + r1676428;
        double r1676430 = r1676424 + r1676429;
        double r1676431 = r1676423 / r1676430;
        double r1676432 = atan2(r1676404, r1676431);
        double r1676433 = lambda1;
        double r1676434 = r1676432 + r1676433;
        return r1676434;
}

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))))))))))