Average Error: 0.2 → 0.2
Time: 42.4s
Precision: 64
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \left(\sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}\]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \left(\sqrt[3]{{\left(\sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \cos delta, \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}^{3}}\right) \cdot \sin \phi_1}\]
\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)}
↓
\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \left(\sqrt[3]{{\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)}^{3}}\right) \cdot \sin \phi_1}
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r65373 = lambda1;
        double r65374 = theta;
        double r65375 = sin(r65374);
        double r65376 = delta;
        double r65377 = sin(r65376);
        double r65378 = r65375 * r65377;
        double r65379 = phi1;
        double r65380 = cos(r65379);
        double r65381 = r65378 * r65380;
        double r65382 = cos(r65376);
        double r65383 = sin(r65379);
        double r65384 = r65383 * r65382;
        double r65385 = r65380 * r65377;
        double r65386 = cos(r65374);
        double r65387 = r65385 * r65386;
        double r65388 = r65384 + r65387;
        double r65389 = asin(r65388);
        double r65390 = sin(r65389);
        double r65391 = r65383 * r65390;
        double r65392 = r65382 - r65391;
        double r65393 = atan2(r65381, r65392);
        double r65394 = r65373 + r65393;
        return r65394;
}
↓
double f(double lambda1, double phi1, double __attribute__((unused)) phi2, double delta, double theta) {
        double r65395 = lambda1;
        double r65396 = theta;
        double r65397 = sin(r65396);
        double r65398 = delta;
        double r65399 = sin(r65398);
        double r65400 = r65397 * r65399;
        double r65401 = phi1;
        double r65402 = cos(r65401);
        double r65403 = r65400 * r65402;
        double r65404 = cos(r65398);
        double r65405 = sin(r65401);
        double r65406 = r65402 * r65399;
        double r65407 = cos(r65396);
        double r65408 = r65406 * r65407;
        double r65409 = fma(r65405, r65404, r65408);
        double r65410 = asin(r65409);
        double r65411 = 3.0;
        double r65412 = pow(r65410, r65411);
        double r65413 = cbrt(r65412);
        double r65414 = sin(r65413);
        double r65415 = r65414 * r65405;
        double r65416 = r65404 - r65415;
        double r65417 = atan2(r65403, r65416);
        double r65418 = r65395 + r65417;
        return r65418;
}

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. Simplified0.2

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

  4. Applied add-cbrt-cube0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019325 +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))))))))))