Destination given bearing on a great circle

Average Error: 0.1 → 0.2

Time: 23.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

• 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{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\mathsf{fma}\left(\frac{{\left(\cos delta\right)}^{2}}{\mathsf{fma}\left(\sin \phi_1, \sin \left(\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), \cos delta\right)}, \frac{{\left(\cos delta\right)}^{2}}{\mathsf{fma}\left(\sin \phi_1, \sin \left(\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), \cos delta\right)}, -\left(\frac{{\left(\sin \phi_1\right)}^{2}}{\mathsf{fma}\left(\sin \phi_1, \sin \left(\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), \cos delta\right)} \cdot \frac{{\left(\sin \phi_1\right)}^{2}}{\mathsf{fma}\left(\sin \phi_1, \sin \left(\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), \cos delta\right)}\right) \cdot \left({\left(\sin \left(\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)}^{3} \cdot \sin \left(\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)}{\mathsf{fma}\left(\sin \left(\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 \sin \left(\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), \frac{{\left(\sin \phi_1\right)}^{2}}{\mathsf{fma}\left(\sin \phi_1, \sin \left(\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), \cos delta\right)}, \frac{{\left(\cos delta\right)}^{2}}{\mathsf{fma}\left(\sin \phi_1, \sin \left(\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), \cos delta\right)}\right)}}\]

C

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r133348 = lambda1;
        double r133349 = theta;
        double r133350 = sin(r133349);
        double r133351 = delta;
        double r133352 = sin(r133351);
        double r133353 = r133350 * r133352;
        double r133354 = phi1;
        double r133355 = cos(r133354);
        double r133356 = r133353 * r133355;
        double r133357 = cos(r133351);
        double r133358 = sin(r133354);
        double r133359 = r133358 * r133357;
        double r133360 = r133355 * r133352;
        double r133361 = cos(r133349);
        double r133362 = r133360 * r133361;
        double r133363 = r133359 + r133362;
        double r133364 = asin(r133363);
        double r133365 = sin(r133364);
        double r133366 = r133358 * r133365;
        double r133367 = r133357 - r133366;
        double r133368 = atan2(r133356, r133367);
        double r133369 = r133348 + r133368;
        return r133369;
}

↓

double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r133370 = lambda1;
        double r133371 = theta;
        double r133372 = sin(r133371);
        double r133373 = delta;
        double r133374 = sin(r133373);
        double r133375 = r133372 * r133374;
        double r133376 = phi1;
        double r133377 = cos(r133376);
        double r133378 = r133375 * r133377;
        double r133379 = cos(r133373);
        double r133380 = 2.0;
        double r133381 = pow(r133379, r133380);
        double r133382 = sin(r133376);
        double r133383 = r133377 * r133374;
        double r133384 = cos(r133371);
        double r133385 = r133383 * r133384;
        double r133386 = fma(r133382, r133379, r133385);
        double r133387 = asin(r133386);
        double r133388 = sin(r133387);
        double r133389 = fma(r133382, r133388, r133379);
        double r133390 = r133381 / r133389;
        double r133391 = pow(r133382, r133380);
        double r133392 = r133391 / r133389;
        double r133393 = r133392 * r133392;
        double r133394 = 3.0;
        double r133395 = pow(r133388, r133394);
        double r133396 = r133395 * r133388;
        double r133397 = r133393 * r133396;
        double r133398 = -r133397;
        double r133399 = fma(r133390, r133390, r133398);
        double r133400 = r133388 * r133388;
        double r133401 = fma(r133400, r133392, r133390);
        double r133402 = r133399 / r133401;
        double r133403 = atan2(r133378, r133402);
        double r133404 = r133370 + r133403;
        return r133404;
}

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 0.1

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

2. Simplified 0.1

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

4. Applied flip-- 0.2

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

5. Simplified 0.2

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

6. Simplified 0.2

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

8. Applied div-sub 0.2

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

9. Simplified 0.2

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

11. Applied flip-- 0.2

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

12. Simplified 0.2

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

13. Simplified 0.2

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

14. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))))))))))